Example Using Standard Least Squares

You have data that contains the responses of 30 subjects after treatment by three drugs (Snedecor and Cochran 1967). You want to use the Standard Least Squares personality of the Fit Model platform to test whether there is a difference in the mean response among the three drugs.

1. Open the Drug.jmp sample data table.

2. Select Analyze > Fit Model.

See the chapter Introduction to the Fit Model Platform, for details about the Fit Model launch window.

3. Select y and click Y.

When you select the column y to be the Y response, the Fitting Personality becomes Standard Least Squares and the Emphasis is Effect Leverage. You can change these options in other situations.

4. Select Drug and x. Click Add.

Figure 3.2 Completed Fit Model Launch Window

5. Click Run.

Figure 3.3 Standard Least Squares Report Window

After accounting for the effect of the x variable, since the p-value for Drug is large (0.1384), you cannot conclude that there a difference in the mean response among the three drugs.

The Standard Least Squares Report and Options

You can show or hide each report and plot in the Standard Least Squares report window using the options in the red triangle menus. Each response and each effect that you specify in the Fit Model launch window has an associated red triangle menu.

`Table 3.1` Description of Response Options
Regression Reports	Show or hide reports and report options. See Regression Reports.
Estimates	Provides options for further analysis on parameter estimates. See Estimates.
Effect Screening	Provides a report and plots for investigating significant effects. See Effect Screening.
Factor Profiling	Provides profilers to examine the response surface. See Factor Profiling.
Row Diagnostics	Provides plots and reports for examining residuals. See Row Diagnostics.
Save Columns	Saves model results as columns in the data table. See Save Columns.
Model Dialog	Shows the completed launch window for the current analysis.
Script	Contains options that are available to all platforms. See `Using JMP`.

For a description of the red triangle options available for an effect, see Effect Options.

Regression Reports

Regression reports provide summary information about model fit, effect significance, and model parameters.

`Table 3.2` Description of Regression Reports and Options
Summary of Fit	Shows or hides a summary of model fit. See Summary of Fit.
Analysis of Variance	Shows or hides the calculations for comparing the fitted model to a simple mean model. See Analysis of Variance.
Parameter Estimates	Shows or hides the estimates of the parameters and a t-test for the hypothesis that each parameter is zero. See Parameter Estimates.
Effect Tests	Shows or hides tests for all of the effects in the model. See Effect Tests.
Effect Details	Shows or hides the Effect Details report when `Effect Screening` or `Minimal` `Report` is selected in Fit Model launch window. See Effect Options. `Note:` If you select the `Effect Leverage Emphasis` option, each effect has its own report at the top of the report window.
Lack of Fit	Shows or hides a test assessing if the model has the appropriate effects. See Lack of Fit.
Show All Confidence Intervals	Shows or hides confidence intervals for the following: • Parameter estimates in the Parameter Estimates report • Least squares means in the Least Squares Means Table
AICc	Shows or hides AICc and BIC values in the Summary of Fit report.

Summary of Fit

The Summary of Fit report provides a summary of model fit.

`Table 3.3` Description of the Summary of Fit Report
RSquare	Estimates the proportion of variation in the response that can be attributed to the model rather than to random error. Using quantities from the corresponding Analysis of Variance table, R2 is calculated as follows: An R2 closer to 1 indicates a better fit. An R2 closer to 0 indicates that the fit predicts the response no better than the overall response mean.
Rsquare Adj	Adjusts R2 to make it more comparable between models with different numbers of parameters. The computation uses the degrees of freedom, and is the ratio of mean squares instead of sums of squares. The mean square for Error appears in the Analysis of Variance report. The mean square for C. Total is computed as the C. Total sum of squares divided by its respective degrees of freedom.
Root Mean Square Error	Estimates the standard deviation of the random error. It is the square root of the Mean Square for Error in the Analysis of Variance report. `Note:` Root Mean Square Error (RMSE) is commonly denoted as s.
Mean of Response	Overall mean of the response values.
Observations (or Sum Wgts)	Records the number of observations used in the model. If there are no missing values and no excluded rows, this is the same as the number of rows in the data table. If there is a column assigned to the role of weight, this is the sum of the weight column values.
AICc	`Note:` Appears only if you have the `AICc` option selected. Shows or hides the corrected Akaike Information Criterion value, which is a measure of model fit that is helpful when comparing different models.
BIC	`Note:` Appears only if you have the `AICc` option selected. Shows or hides the Bayesian Information Criterion value, which is a measure of model fit that is helpful when comparing different models.

Analysis of Variance

The Analysis of Variance report provides the calculations for comparing the fitted model to a simple mean model.

`Table 3.4` Description of the Analysis of Variance Report
Source	Lists the three sources of variation: Model, Error, and C. Total (Corrected Total).
DF	Records an associated degrees of freedom (DF) for each source of variation. • The `C.` `Total` DF is always one less than the number of observations. • The total DF are partitioned into the `Model` and `Error` terms: – The `Model` degrees of freedom is the number of parameters (except for the intercept) used to fit the model. – The `Error` DF is the difference between the `C.` `Total` DF and the `Model` DF.
Sum of Squares	Records an associated sum of squares (SS) for each source of variation. The SS column accounts for the variability measured in the response. • The total (`C.` `Total`) SS is the sum of squared distances of each response from the sample mean. • The `Error` SS is the sum of squared differences between the fitted values and the actual values. This corresponds to the unexplained variability after fitting the model. • The `Model` SS is the difference between the `C.` `Total` SS and the `Error` SS.
Mean Square	Lists the mean squares, which is the sum of squares divided by its associated degrees of freedom. `Note:` The square root of the Mean Square for `Error` is the same as the RMSE in the Summary of Fit report.
F Ratio	Lists the model mean square divided by the error mean square.
Prob > F	Lists the p-value for the test. Values of 0.05 or less are often considered evidence that there is at least one significant effect in the model.

Parameter Estimates

The Parameter Estimates report shows the estimates of the parameters and a t-test for the hypothesis that each parameter is zero.

`Table 3.5` Description of the Parameter Estimates Report
Term	Names the estimated parameter. The first parameter is always the intercept. Continuous variables show as the name of the data table column. Nominal or ordinal effects appear with the names of the levels in brackets. See the appendix Statistical Details for additional information.
Estimate	Lists the parameter estimates for each term. They are the coefficients of the model used to predict the response.
Std Error	Lists the estimate of the standard error of the coefficient.
t Ratio	Tests whether the true parameter is zero. It is the ratio of the estimate to its standard error and has a Student’s t-distribution under the hypothesis, given the usual assumptions about the model.
Prob > \|t\|	Lists the p-value for the test. This is the two-tailed test against the alternatives in each direction.
Lower 95%	`Note:` Only appears if you have the `Regression Reports > Show All` `Confidence Intervals` option selected or if you right-click in the report and select `Columns > Lower 95%`. Shows the lower 95% confidence interval for the parameter estimate.
Upper 95%	`Note:` Only appears if you have the `Regression Reports > Show All` `Confidence Intervals` option selected or if you right-click in the report and select `Columns > Upper 95%`. Shows the upper 95% confidence interval for the parameter estimate.
Std Beta	`Note:` Only appears if you right-click in the report and select `Columns >` `Std Beta`. Shows parameter estimates that would have resulted from the regression if all of the variables had been standardized to a mean of 0 and a variance of 1.
VIF	`Note:` Only appears if you right-click in the report and select `Columns >` `VIF`. Shows the variance inflation factors. High VIFs indicate a collinearity problem. The VIF is defined as follows: where Ri 2 is the coefficient of multiple determination for the regression of xi as a function of the other explanatory variables.
Design Std Error	`Note:` Only appears if you right-click in the report and select `Columns >` `Design Std Error`. Shows standard error without being scaled by sigma (RMSE). Design Std Error is equal to the following:

Note: For reports that do not appear by default, you can set them to always appear. Select File > Preferences > Platforms > Fit Least Squares.

Effect Tests

The effect tests are joint tests in which all parameters for an individual effect are zero. If an effect has only one parameter, as with simple regressors, then the tests are no different from the t-tests in the Parameter Estimates report.

Note: Parameterization and handling of singularities are different from the SAS GLM procedure. For details about parameterization and handling of singularities, see the Statistical Details appendix.

`Table 3.6` Description of the Effect Tests Report
Source	Lists the names of the effects in the model.
Nparm	Shows the number of parameters associated with the effect. Continuous effects have one parameter. Nominal and Ordinal effects have one less parameter than the number of levels. Crossed effects multiply the number of parameters for each term.
DF	Shows the degrees of freedom for the effect test. Ordinarily `Nparm` and `DF` are the same. They are different if there are linear combinations found among the regressors so that an effect cannot be tested to its fullest extent. Sometimes the DF is zero, indicating that no part of the effect is testable. Whenever DF is less than Nparm, the note `Lost DFs` appears to the right of the line in the report.
Sum of Squares	Lists the SS for the hypothesis that the listed effect is zero.
F Ratio	Lists the F statistic for testing that the effect is zero. It is the ratio of the mean square for the effect divided by the mean square for error. The mean square for the effect is the sum of squares for the effect divided by its degrees of freedom.
Prob > F	Lists the p-value for the Effect test.
`Mean Square`	Only appears if you right-click in the report and select `Columns > Mean` `Square`. Shows the mean square for the effect, which is the SS divided by the DF.

Lack of Fit

The Lack of Fit report shows or hides a test assessing if the model has the appropriate effects. In the following situations, the Lack of Fit report does not appear:

• There are no exactly replicated points with respect to the X data, and therefore there are no degrees of freedom for pure error.

• The model is saturated, meaning that the model itself has a degree of freedom for each different x value. Therefore, there are no degrees of freedom for lack of fit.

When observations are exact replicates of each other in terms of the X variables, you can estimate the error variance whether you have the right form of the model. The error that you can measure for these exact replicates is called pure error. This is the portion of the sample error that cannot be explained or predicted by the form that the model uses for the X variables. However, a lack of fit test is not very useful if it has only a few degrees of freedom (not many replicated x values).

The difference between the residual error from the model and the pure error is called the lack of fit error. The lack of fit error can be significantly greater than pure error if you have the wrong functional form of a regressor, or if you do not have enough interaction effects in an analysis of variance model. In that case, you should consider adding interaction terms, if appropriate, or try to better capture the functional form of a regressor.

Table 3.7 shows information about the calculations in the Lack of Fit report.

`Table 3.7` Description of the Lack of Fit Report
Source	Lists the three sources of variation: Lack of Fit, Pure Error, and Total Error.
DF	Records an associated DF for each source of error: • The DF for Total Error are also found on the Error line of the Analysis of Variance table. It is the difference between the C. Total DF and the Model DF found in that table. The Error DF is partitioned into degrees of freedom for lack of fit and for pure error. • The Pure Error `DF` is pooled from each group where there are multiple rows with the same values for each effect. For example, in the sample data, `Big Class.jmp`, there is one instance where two subjects have the same values of `age` and `weight` (Chris and Alfred are both 14 and have a weight of 99). This gives 1(2 - 1) = 1 DF for Pure Error. In general, if there are g groups having multiple rows with identical values for each effect, the pooled DF, denoted DFp, can be calculated as follows: where ni is the number of replicates in the ith group. • The Lack of Fit DF is the difference between the Total Error and Pure Error DFs.
Sum of Squares	Records an associated sum of squares (SS) for each source of error: • The Total Error SS is the sum of squares found on the `Error` line of the corresponding Analysis of Variance table. • The Pure Error SS is pooled from each group where there are multiple rows with the same values for each effect. This estimates the portion of the true random error that is not explained by model effects. In general, if there are g groups having multiple rows with like values for each effect, the pooled SS, denoted SSp, is calculated as follows: where SSi is the sum of squares for the ith group corrected for its mean. • The Lack of Fit SS is the difference between the Total Error and Pure Error sum of squares. If the lack of fit SS is large, it is possible that the model is not appropriate for the data. The F-ratio described below tests whether the variation due to lack of fit is small, relative to the Pure Error.
Mean Square	Shows the mean square for the source, which is the SS divided by the DF.
F Ratio	Shows the ratio of the Mean Square for the Lack of Fit to the Mean Square for `Pure Error`. It tests the hypothesis that the lack of fit error is zero.
Prob > F	Lists the p-value for the Lack of Fit test. A small p-value indicates a significant lack of fit.
Max RSq	Lists the maximum R2 that can be achieved by a model using only the variables in the model. Because Pure Error is invariant to the form of the model and is the minimum possible variance, Max RSq is calculated as follows:

Estimates

Estimates options provide additional reports and tests for the model parameters.

`Table 3.8` Description of Estimates Options
Show Prediction Expression	Places the prediction expression in the report. See Show Prediction Expression.
Sorted Estimates	Produces a different version of the Parameter Estimates report that is more useful in screening situations. See Sorted Estimates.
Expanded Estimates	Use this option when there are categorical (nominal) terms in the model and you want a full set of effect coefficients. See Expanded Estimates.
Indicator Parameterization Estimates	Displays the estimates using the Indicator Variable parameterization. See Indicator Parameterization Estimates.
Sequential Tests	Show the reduction in residual sum of squares as each effect is entered into the fit. See Sequential Tests.
Custom Test	Enables you to test a custom hypothesis. See Custom Test.
Joint Factor Tests	For each main effect in the model, JMP produces a joint test on all of the parameters involving that main effect. See Joint Factor Tests.
Inverse Prediction	Finds the value of x for a given y. See Inverse Prediction.
Cox Mixtures	`Note:` Appears only for mixture models. Produces parameter estimates from which inference can be made about factor effects and the response surface shape, relative to a reference point in the design space. See Cox Mixtures.
Parameter Power	Calculates statistical power and other details relating to a given hypothesis test. The results appear in the Parameter Estimates report. See Parameter Power.
Correlation of Estimates	Produces a correlation matrix for all coefficients in a model. See Correlation of Estimates.

Show Prediction Expression

The Show Prediction Expression shows the equation used to predict the response.

Example of a Prediction Expression

1. Open the Drug.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select y and click Y.

4. Select Drug and x. Click Add.

5. Click Run.

6. From the red triangle menu next to Response, select Estimates > Show Prediction Expression.

Figure 3.4 Prediction Expression

Sorted Estimates

The Sorted Estimates option produces a different version of the Parameter Estimates report that is more useful in screening situations. This version of the report is especially useful if the design is saturated, since typical reports are less informative.

Example of a Sorted Estimates Report

1. Open the Reactor.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. Make sure that the Degree box has a 2 in it.

5. Select F, Ct, A, T, and Cn and click Macros > Factorial to Degree.

6. Click Run.

Figure 3.5 Sorted Parameter Estimates

The Sorted Parameter Estimates report also appears automatically if the Emphasis is set to Effect Screening and all of the effects have only one parameter.

Note the following differences between this report and the Parameter Estimates report:

• This report does not show the intercept.

• The effects are sorted by the absolute value of the t-ratio, showing the most significant effects at the top.

• A bar chart shows the t-ratio, with lines showing the 0.05 significance level.

• If JMP cannot obtain standard errors for the estimates, relative standard errors appear.

• If there are no degrees of freedom for residual error, JMP constructs t-ratios and p-values using Lenth’s Pseudo-Standard Error (PSE). These quantities are labeled with Pseudo in their name. A note explains the change and shows the PSE. To calculate p-values, JMP uses a degrees of freedom for error of m/3, where m is the number of parameter estimates excluding the intercept.

Expanded Estimates

Use the Expanded Estimates option when there are nominal terms in the model and you want to see the full set of coefficients.

Example of Expanded Estimates

To recreate the reports in Figure 3.6, follow the steps in Example of a Prediction Expression, except instead of selecting Show Prediction Expression, select Expanded Estimates.

Figure 3.6 Comparison of Parameter Estimates and Expanded Estimates

Notice that the coefficient for Drug(f) appears in the Expanded Estimates report.

Statistical Details for Expanded Estimates

When you have nominal terms in your model, the platform constructs a set of dummy columns to represent the levels in the classification. Full details are shown in the appendix Statistical Details in the Statistical Details appendix. For n levels, there are n - 1 dummy columns. Each dummy variable is a zero-or-one indicator for a particular level, except for the last level, which is coded -1 for all dummy variables. The following table shows an example of the A1 and A2 dummy columns when column A has levels A1, A2, and A3.

A	A1 dummy	A2 dummy
A1	1	0
A2	0	1
A3	-1	-1

These columns do not appear in the report, but they help conceptualize the fitting process. The parameter estimates are the coefficients fit to these columns. In this case, there are two, labeled A[A1] and A[A2]. This coding causes the parameter estimates to be interpreted as how much the response for each level differs from the average across all levels. Suppose, however, that you want the coefficient for the last level, A[A3]. The coefficient for the last level is the negative of the sum across the other levels, because the sum across all levels is constrained to be zero. Although many other codings are possible, this coding has proven to be practical and interpretable.

However, you probably do not want to hand calculate the estimate for the last level. The Expanded Estimates option in the Estimates menu calculates these missing estimates and shows them in a text report. You can verify that the mean (or sum) of the estimates across a classification is zero.

Keep in mind that the Expanded Estimates option with high-degree interactions of two-level factors produces a lengthy report. For example, a five-way interaction of two-level factors produces only one parameter but has 25 = 32 expanded coefficients, which are all the same except for sign changes.

Indicator Parameterization Estimates

This option displays the estimates using the Indicator Variable parameterization. To re-create the report in Figure 3.7, follow the steps in Show Prediction Expression with one exception: instead of selecting Estimates > Show Prediction Expression, select Estimates > Indicator Parameterization Estimates.

Figure 3.7 Indicator Parameterization Estimates

This parameterization is inspired by the PROC GLM parameterization. Some models match, but others, such as no-intercept models, models with missing cells, and mixture models, will most likely show differences.

Sequential Tests

Sequential Tests show the reduction in the residual sum of squares as each effect is entered into the fit. The sequential tests are also called Type I sums of squares (Type I SS). One benefit of the Type I SS is that they are independent and sum to the regression SS. One disadvantage is that they depend on the order of terms in the model; each effect is adjusted only for the preceding effects in the model.

The following models are considered appropriate for the Type I hypotheses:

• balanced analysis of variance models specified in proper sequence (that is, interactions do not precede main effects in the effects list, and so on)

• purely nested models specified in the proper sequence

• polynomial regression models specified in the proper sequence.

Custom Test

If you want to test a custom hypothesis, select Custom Test from the Estimates menu. To jointly test several linear functions, click on Add Column. This displays the window shown to the left in Figure 3.8 for constructing the test in terms of all the parameters. After filling in the test, click Done. The dialog then changes to a report of the results, as shown on the right in Figure 3.8.

The space beneath the Custom Test title bar is an editable area for entering a test name. Use the Custom Test dialog as follows:

Parameter

lists the names of the model parameters. To the right of the list of parameters are columns of zeros corresponding to these parameters. Click a cell here, and enter a new value corresponding to the test that you want.

Add Column

adds a column of zeros so that you can test jointly several linear functions of the parameters. Use the Add Column button to add as many columns to the test as you want.

The last line in the Parameter list is labeled =. Enter a constant into this box to test the linear constraint against. For example, to test the hypothesis β0=1, enter a 1 in the = box. In Figure 3.8, the constant is equal to zero.

When you finish specifying the test, click Done to see the test performed. The results are appended to the bottom of the dialog.

When the custom test is done, the report lists the test name, the function value of the parameters tested, the standard error, and other statistics for each test column in the dialog. A joint F-test for all columns shows at the bottom.

Caution: The test is always done with respect to residual error. If you have random effects in your model, this test might not be appropriate if you use EMS instead of REML.

Note: If you have a test within a classification effect, consider using the contrast dialog (which tests hypotheses about the least squares means) instead of a custom test.

Figure 3.8 The Custom Test Dialog and Test Results for Age Variable

Joint Factor Tests

This option appears when interaction effects are present. For each main effect in the model, JMP produces a joint test on all the parameters involving that main effect.

Example of a Joint Factor Tests Report

1. Open the Big Class.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select weight and click Y.

4. Select age, sex, and height and click Macros > Factorial to degree.

5. Click Run.

6. From the red triangle next to Response weight, select Estimates > Joint Factor Tests.

Figure 3.9 Joint Factor Tests for Big Class model

Note that age has 15 degrees of freedom; it is testing the five parameters for age, the five parameters for age*sex, and the five parameters for height*age. All parameters tested to be zero.

Inverse Prediction

To find the value of x for a given y requires inverse prediction, sometimes called calibration. The Inverse Prediction option on the Estimates menu lets you ask for the specific value of one independent (X) variable given a specific value of a dependent variable and other x values (Figure 3.11). The inverse prediction computation includes confidence limits (fiducial limits) on the prediction.

Example of Inverse Prediction

1. Open the Fitness.jmp sample data table.

2. Select Analyze > Fit Y by X.

3. Select Oxy and click Y, Response.

4. Select Runtime and click X, Factor.

When there is only a single X, as in this example, the Fit Y by X platform can give you a visual approximation of the inverse prediction values.

5. Click OK.

6. From the red triangle menu, select Fit Line.

Use the crosshair tool to approximate inverse prediction.

7. Select Tools > Crosshairs.

8. Click on the Oxy axis at about 50 and then drag the cursor so that the crosshairs intersect with the prediction line.

Figure 3.10 Bivariate Fit for Fitness.jmp

Figure 3.10 shows which value of Runtime gives an Oxy value of 50, intersecting the Runtime axis at about 9.779, which is an approximate inverse prediction. However, to see the exact prediction of Runtime, use the Fit Model dialog as follows:

1. From the Fitness.jmp sample data table, select Analyze > Fit Model.

2. Select Oxy and click Y.

3. Add Runtime as the single model effect.

4. Click Run.

5. From the red triangle menu next to Response Oxy, select Estimates > Inverse Prediction.

Figure 3.11 Inverse Prediction Given by the Fit Model Platform

6. Type the values for Oxy as shown in Figure 3.11.

7. Click OK.

The Inverse Prediction report (Figure 3.12) gives the exact predictions for each Oxy value specified with upper and lower 95% confidence limits. The exact prediction for Runtime when Oxy is 50 is 9.7935. This value is close to the approximate prediction of 9.779 found in Figure 3.10.

Figure 3.12 Inverse Prediction Given by the Fit Model Platform

Note: The fiducial confidence limits are formed by Fieller’s method. Sometimes this method results in a degenerate (outside) interval, or an infinite interval, for one or both sides of an interval. When this happens for both sides, Wald intervals are used. If it happens for only one side, the Fieller method is still used and a missing value is returned. See the Statistical Details appendix.

The inverse prediction option also predicts a single x value when there are multiple effects in the model. To predict a single x, you supply one or more y values of interest. You also set the x value that you predict is missing. By default, the other x values are set to the regressor’s means but can be changed to any desirable value.

Example Predicting a Single X Value with Multiple Model Effects

1. From the Fitness.jmp sample data table, select Analyze > Fit Model.

2. Select Oxy and click Y.

3. Add Runtime, RunPulse, and RstPulse as effects.

4. Click Run.

5. From the red triangle menu next to Response Oxy, select Estimates > Inverse Prediction.

Figure 3.13 Inverse Prediction Dialog for a Multiple Regression Model

6. Type the values for Oxy, as shown in Figure 3.13.

7. Delete the value for Runtime, because you want to predict that value.

8. Click OK.

Figure 3.14 Inverse Prediction for a Multiple Regression Model

Cox Mixtures

Note: This option is available only for mixture models.

In mixture designs, the model parameters cannot easily be used to judge the effects of the mixture components. The Cox Mixture model (a reparameterized and constrained version of the Scheffé model) produces parameter estimates. You can then derive factor effects and the response surface shape relative to a reference point in the design space. See Cornell (1990) for a complete discussion.

Example of Cox Mixtures

1. Open the Five Factor Mixture.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select Y1 and click Y.

4. Select X1, X2, and X3.

5. Click on Macros > Mixture Response Surface.

6. Click Run.

7. From the red triangle menu next to Response Y1, select Estimates > Cox Mixtures.

Figure 3.15 Cox Mixtures Window

8. Type the reference mixture points shown in Figure 3.15. Note that if the components of the reference point do not equal one, then the values are scaled so that they do equal one.

9. Click OK.

Figure 3.16 Cox Mixtures

The report shows the parameter estimates along with standard errors, hypothesis tests, and the reference mixture.

Parameter Power

Suppose that you want to know how likely it is that your experiment will detect some difference at a given α-level. The probability of getting a significant test result is called the power. The power is a function of the unknown parameter values tested, the sample size, and the unknown residual error variance.

Alternatively, suppose that you already did an experiment and the effect was not significant. You think that the effect might have been significant if you had more data. How much more data do you need?

JMP offers the following calculations of statistical power and other details related to a given hypothesis test:

• LSV, the least significant value, is the value of some parameter or function of parameters that would produce a certain p-value alpha.

• LSN, the least significant number, is the number of observations that would produce a specified p-value alpha if the data has the same structure and estimates as the current sample.

• Power values are also available. See The Power for details.

The LSV, LSN, and power values are important measuring sticks that should be available for all test statistics, especially when the test statistics are not significant. If a result is not significant, you should at least know how far from significant the result is in the space of the estimate (rather than in the probability). YOu should also know how much additional data is needed to confirm significance for a given value of the parameters.

Sometimes a novice confuses the role of the null hypotheses, thinking that failure to reject the null hypothesis is equivalent to proving it. For this reason, it is recommended that the test be presented in these other aspects (power and LSN) that show the test’s sensitivity. If an analysis shows no significant difference, it is useful to know the smallest difference that the test is likely to detect (LSV).

The power details provided by JMP are for both prospective and retrospective power analyses. In the planning stages of a study, a prospective analysis helps determine how large your sample size must be to obtain a desired power in tests of hypothesis. During data analysis, however, a retrospective analysis helps determine the power of hypothesis tests that have already been conducted.

Technical details for power, LSN, and LSV are covered in the section Power Calculations in the Power Calculations appendix.

Calculating retrospective power at the actual sample size and estimated effect size is somewhat non-informative, even controversial [Hoenig and Heisey, 2001]. The calculation does not provide additional information about the significance test, but rather shows the test in a different perspective. However, we believe that many studies fail to detect a meaningful effect size due to insufficient sample size. There should be an option to help guide for the next study, for specified effect sizes and sample sizes.

For more information, see John M. Hoenig and Dinnis M. Heisey, (2001) “The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis.”, American Statistician (v55 No 1, 19-24).

Power Options

Power options are available only for continuous-response models. Power and other test details are available in the following contexts:

• If you want the 0.05 level details for all parameter estimates, select Estimates > Parameter Power from the report’s red triangle menu. This option produces the LSV, LSN, and adjusted power for an alpha of 0.05 for each parameter in the linear model.

• If you want the details for an F-test for a certain effect, select Power Analysis from the effect’s red triangle menu.

• If you want the details for a Contrast, create the contrast and then select Power Analysis from the effect’s red triangle menu.

• If you want the details for a custom test, create the test that you want with the Custom Test option from the platform popup menu and then select Power Analysis option in the popup menu next to the Custom Test.

In all cases except the first, you enter information in the Power Details window for the calculations that you want.

Example of Power Analysis

The Power Details window (Figure 3.17) shows the contexts and options for test detailing. After you fill in the values, the results are added to the end of the report.

1. Open the Big Class.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select weight and click Y.

4. Add age, sex, and height as the effects.

5. Click Run.

6. From the red triangle next to age, select Power Analysis.

Figure 3.17 Power Details Window

7. Replace the Delta values with 3,6, and 1 as shown in Figure 3.17.

8. Replace the Number values with 20, 60, and 10 as shown in Figure 3.17.

9. Select Solve for Power and Solve for Least Significant Number.

10. Click Done.

For details about the power results shown in Figure 3.17, see Text Reports for Power Analysis. For details about the Power Details options, see Power Details Window.

Power Details Window

In the Alpha, Sigma, Delta, and Number columns, you can enter a single value, two values, or the start, stop, and increment for a sequence of values. Power calculations are done on all possible combinations of the values that you specify.

`Table 3.9` Description of the Power Details Window
Alpha (α)	Represents the significance level between 0 and 1, usually 0.05, 0.01, or 0.10. Initially, `Alpha` automatically has a value of 0.05. Click in one of the three boxes to enter one, two, or a sequence of values.
Sigma (σ)	Represents the standard error of the residual error in the model. Initially, RMSE (the square root of the mean square error) is in the first box. Click in one of the three boxes to enter one, two, or a sequence of values.
Delta (δ)	Represents the raw effect size. See Effect Size, for details. Initially, the square root of the sum of squares for the hypothesis divided by n is in the first box. Click in one of the three boxes to enter or edit one, two, or a sequence of values.
Number (n)	Represents the sample size. Initially, the actual sample size is in the first box. Click in one of the three boxes to enter or edit one, two, or a sequence of values.
Solve for Power	Solves for the power (the probability of a significant result) as a function of α, σ, δ, and n.
Solve for Least Significant Number	Solves for the number of observations expected to achieve significance alpha given α, σ, and δ.
Solve for Least Significant Value	Solves for the value of the parameter or linear test that produces a p-value of alpha. This is a function of α, σ, and n.This option is available only for one-degree-of-freedom tests and is used for individual parameters.
Adjusted Power and Confidence Interval	To look at power retrospectively, you use estimates of the standard error and the test parameters. Adjusted power is the power calculated from a more unbiased estimate of the noncentrality parameter. The confidence interval for the adjusted power is based on the confidence interval for the noncentrality estimate. Adjusted power and confidence limits are computed only for the original δ where the random variation is.

Effect Size

The power is the probability that an F achieves its α-critical value given a noncentrality parameter related to the hypothesis. The noncentrality parameter is zero when the null hypothesis is true (that is, when the effect size is zero). The noncentrality parameter λ can be factored into the three components that you specify in the JMP Power window as

λ = (nδ2)/σ2 .

Power increases with λ. This means that the power increases with sample size n and raw effect size δ, and decreases with error variance σ2. Some books (Cohen 1977) use standardized rather than raw Effect Size, Δ = δ/σ, which factors the noncentrality into two components: λ = nΔ2.

Delta (δ) is initially set to the value implied by the estimate given by the square root of SSH/n, where SSH is the sum of squares for the hypothesis. If you use this estimate for delta, you might want to correct for bias by asking for the Adjusted Power.

In the special case for a balanced one-way layout with k levels:

Because JMP uses parameters of the following form:

with

the delta for a two-level balanced layout appears as follows:

Text Reports for Power Analysis

The Power Analysis option calculates power as a function of every combination of α, σ, δ, and n values that you specify in the Power Details window.

• For every combination of α, σ, and δ, the least significant number is calculated.

• For every combination of α, σ, and n, the least significant value is calculated.

Perform the example shown in Figure 3.17 to produce the results in Figure 3.18.

Figure 3.18 The Power Analysis Tables

If you check Adjusted Power and Confidence Interval in the Power Details window, the Power report includes the AdjPower, LowerCL, and UpperCL columns.

Plot of Power by Sample Size

To see a plot of power by sample size, select the Power Plot option from the red triangle menu at the bottom of the Power report. See Figure 3.18. JMP plots the Power by N columns from the Power table. The plot in Figure 3.19 is a result of plotting the Power table in Figure 3.18.

Figure 3.19 Plot of Power by Sample Size

The Least Significant Value (LSV)

After a single-degree-of-freedom hypothesis test is performed, you often want to know how sensitive the test was. In other words, you want to know how small would a significant effect be at some p-value alpha.

The LSV provides a significant measuring stick on the scale of the parameter rather than on a probability scale. It shows the sensitivity of the design and data. LSV also encourages proper statistical intuition concerning the null hypothesis by highlighting how small a value would be detected as significant by the data.

• The LSV is the value that the parameter must be greater than or equal to in absolute value, in order to give the p-value of the significance test a value less than or equal to alpha.

• The LSV is the radius of the confidence interval for the parameter. A 1–alpha confidence interval is derived by taking the parameter estimate plus or minus the LSV.

• The absolute value of the parameter or function of the parameters tested is equal to the LSV, if and only if the p-value for its significance test is exactly alpha.

Compare the absolute value of the parameter estimate to the LSV. If the absolute parameter estimate is bigger, it is significantly different from zero. If the LSV is bigger, the parameter is not significantly different from zero.

The Least Significant Number (LSN)

The LSN or least significant number is the number of observations needed to decrease the variance of the estimates enough to achieve a significant result with the given values of alpha, sigma, and delta (the significance level, the standard deviation of the error, and the effect size, respectively).If you need more data points (a larger sample size) to achieve significance, the LSN indicates how many more data points are necessary.

Note: LSN is not a recommendation of how large a sample to take because the probability of significance (power) is only about 0.5 at the LSN.

The LSN has these characteristics:

• If the LSN is less than the actual sample size n, then the effect is significant. This means that you have more data than you need to detect the significance at the given alpha level.

• If the LSN is greater than n, the effect is not significant. In this case, if you believe that more data will show the same standard errors and structural results as the current sample, the LSN suggests how much data you would need to achieve significance.

• If the LSN is equal to n, then the p-value is equal to the significance level alpha. The test is on the border of significance.

• Power (described next) calculated when n = LSN is always greater than or equal to 0.5.

The Power

The power is the probability of getting a significant result. It is a function of the sample size n, the effect size δ, the standard deviation of the error σ, and the significance level α. The power tells you how likely your experiment will detect a difference at a given α-level.

Power has the following characteristics:

• If the true value of the parameter is the hypothesized value, the power should be alpha, the size of the test. You do not want to reject the hypothesis when it is true.

• If the true value of the parameters is not the hypothesized value, you want the power to be as great as possible.

• The power increases with the sample size, as the error variance decreases, and as the true parameter gets farther from the hypothesized value.

The Adjusted Power and Confidence Intervals

You typically substitute sample estimates in power calculations, because power is a function of unknown population quantities (Wright and O’Brien 1988). If you regard these sample estimates as random, you can adjust them to have a more proper expectation.

You can also construct a confidence interval for this adjusted power. However, the confidence interval is often very wide. The adjusted power and confidence intervals can be computed only for the original δ, because that is where the random variation is. For details about adjusted power see Computations for Adjusted Power in the Computations for Adjusted Power appendix

Correlation of Estimates

The Correlation of Estimates command on the Estimates menu creates a correlation matrix for all parameter estimates.

Example of Correlation of Estimates

1. Open the Tiretread.jmp data table.

2. Select Analyze > Fit Model.

3. Select ABRASION and click Y.

4. Select SILICA and SILANE and click Add.

5. Select SILICA and SILANE and click Cross.

6. Click Run.

7. From the Response red triangle menu, select Estimates > Correlation of Estimates.

Figure 3.20 Correlation of Estimates Report

Effect Screening

The Effect Screening options on the Response red triangle menu examine the sizes of the effects.

Scaled Estimates	Gives coefficients corresponding to factors that are scaled to have a mean of zero and a range of two. See Scaled Estimates and the Coding Of Continuous Terms.
Normal Plot	Helps you identify effects that deviate from the normal lines. See Plot Options.
Bayes Plot	Computes posterior probabilities using a Bayesian approach. See Plot Options.
Pareto Plot	Shows the absolute values of the orthogonalized estimates showing their composition relative to the sum of the absolute values.See Plot Options.

Scaled Estimates and the Coding Of Continuous Terms

The parameter estimates are highly dependent on the scale of the factor. When you convert a factor from grams to kilograms, the parameter estimates change by a multiple of a thousand. When you apply the same change to a squared (quadratic) term, the scale changes by a multiple of a million.

To learn more about the effect size, examine the estimates in a more scale-invariant fashion. This means converting from an arbitrary scale to a meaningful one. Then the sizes of the estimates relate to the size of the effect on the response. There are many approaches to doing this. In JMP, the Effect Screening > Scaled Estimates command on the report’s red triangle menu gives coefficients corresponding to factors that are scaled to have a mean of zero and a range of two. If the factor is symmetrically distributed in the data, then the scaled factor has a range from -1 to 1. This factor corresponds to the scaling used in the design of experiments (DOE) tradition. Therefore, for a simple regressor, the scaled estimate is half of the predicted response change as the regression factor travels its whole range.

Scaled estimates are important in assessing effect sizes for experimental data that contains uncoded values. If you use coded values (–1 to 1), then the scaled estimates are no different from the regular estimates. Scaled estimates also take care of the issues for polynomial (crossed continuous) models, even if they are not centered by the Center Polynomials command on the launch window’s red triangle menu.

You also do not need scaled estimates if your factors have the Coding column property. In that case, they are converted to uncoded form when the model is estimated and the results are already in an interpretable form for effect sizes.

Example of Scaled Estimates

1. Open the Drug.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select y and click Y.

4. Add Drug and x as the effects.

5. Click Run.

6. From the red triangle menu next to Response y, select Effect Screening > Scaled Estimates.

As noted in the report, the estimates are parameter-centered by the mean and scaled by range/2.

Figure 3.21 Scaled Estimates

Plot Options

The Normal, Bayes, and Pareto Plot options correct for scaling and for correlations among the estimates. These three Effect Screening options point out the following:

1. The process of fitting can be thought of as converting one set of realizations of random values (the response values) into another set of realizations of random values (the parameter estimates). If the design is balanced with an equal number of levels per factor, these estimates are independent and identically distributed, just as the responses are.

2. If you are fitting a screening design that has many effects and only a few runs, you expect that only a few effects are active. That is, a few effects have a sizable impact and the rest of them are inactive (they are estimating zeros). This is called the assumption of effect sparsity.

3. Given points 1 and 2 above, you can think of screening as a way to determine which effects are inactive with random values around zero and which ones are outliers (not part of the distribution of inactive effects).

Therefore, you treat the estimates themselves as a set of data to help you judge which effects are active and which are inactive. If there are few runs, with little or no degrees of freedom for error, then there are no classical significance tests, and this approach is especially needed.

The final Effect Screening options provide more reports:

• The Effect Screening report contains the Lenth PSE (pseudo-standard error) value.

• The Parameter Estimate Population report highlights significant factors.

You can also look at the model parameters from different angles using scaled estimates and three plots. See Parameter Estimate Population Report for details.

Normal Plot

The Normal Plot option displays a Parameter Estimate Population report (see Parameter Estimate Population Report) and shows a normal plot of these parameter estimates (Daniel 1959). The estimates are on the vertical axis, and the normal quantiles on the horizontal axis.

Example of a Normal Plot

1. Open the Reactor.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. Make sure that the Degree box has a 2 in it.

5. Select F, Ct, A, T, and Cn and click Macros > Factorial to Degree.

6. Click Run.

7. From the red triangle menu next to Response Y, select Effect Screening > Normal Plot.

Figure 3.22 Normal Plot

If all effects are due to random noise, they tend to follow a straight line with slope σ, the standard error. In the plot, the line with slope equal to the Lenth’s PSE estimate is blue.

The Normal plot helps you identify effects that deviate from the normal lines. Estimates that deviate substantially are labeled.

Half-Normal Plot

Below the Normal Plot report title, select Half Normal Plot from the list to plot the absolute values of the estimates against the normal quantiles for the absolute value normal distribution (shown in Figure 3.23).

Figure 3.23 Half Normal Plot

Bayes Plot

Another approach to resolving which effects are important (sometimes referred to as active contrasts) is computing posterior probabilities using a Bayesian approach. This method, due to Box and Meyer (1986), assumes that the estimates are a mixture from two distributions. Some portion of the effects is assumed to come from pure random noise with a small variance. The remaining terms are assumed to come from a contaminating distribution that has a variance K times larger than the error variance.

An effect’s prior probability is the chance you give that effect of being nonzero (or being in the contaminating distribution). These priors are usually set to equal values for each effect. 0.2 is a commonly recommended prior probability value. The K contamination coefficient is often set at 10. This value indicates that the contaminating distribution has a variance that is 10 times the error variance.

The Bayes plot is done with respect to normalized estimates (JMP lists as Orthog t-Ratio), which have been transformed to be uncorrelated and have equal variance.

Example of a Bayes Plot

1. Open the Reactor.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. Make sure that the Degree box has a 2 in it.

5. Select F, Ct, A, T, and Cn and click Macros > Factorial to Degree.

6. Click Run.

7. From the red triangle menu next to Response Y, select Effect Screening > Bayes Plot.

Figure 3.24 Bayes Plot Specifications

8. Click Go to start the calculation of the posterior probabilities.

Figure 3.25 Bayes Plot Report

The Std Err Est value is set to 0 for a saturated model with no estimate of error. If there is an estimate of standard error (the root mean square error), this value is set to 1 because the estimates have already been transformed and scaled to unit variance. If you specify a different value, it should be done as a scale factor of the RMSE estimate.

The resulting posterior probabilities are listed and plotted with bars as shown in Figure 3.25. An overall posterior probability is also shown for the outcome that the sample is uncontaminated.

Bayes Plot for Factor Activity

JMP includes a script that helps you determine which factors are active in the design.

1. Open the Reactor.jmp sample data table.

1. Open the BayesPlotforFactors.jsl file in the Sample Scripts folder.

2. Select Edit > Run Script.

3. Select Y and click Y, Response.

4. Select F, Ct, A, T, and Cn and click X, Factor.

5. Click OK.

Figure 3.26 Bayes Plot for Factor Activity

In this case, we specified that the highest order interaction to consider is two. Therefore, all possible models that include (up to) second-order interactions are constructed. Based on the value assigned to Prior Probability (see the Controls section of the plot), a posterior probability is computed for each of the possible models. The probability for a factor is the sum of the probabilities for each of the models where it was involved.

In this example, we see that Ct, T, and Cn are active and that A and F are not. These results match those of the Bayes Plot shown in Figure 3.25.

Note: If the ridge parameter were zero (not allowed), all the models would be fit by least squares. As the ridge parameter gets large, the parameter estimates for any model shrink toward zero. Details on the ridge parameter (and why it cannot be zero) are explained in Box and Meyer (1993).

Pareto Plot

The Pareto Plot selection gives plots of the absolute values of the orthogonalized estimates showing their composition relative to the sum of the absolute values. The estimates are orthogonalized to be uncorrelated and standardized to have equal variances by default. If your data set has estimates that are correlated and/or have unequal variances, then your data is transformed, by default, to have equal variances and to be uncorrelated. However, you have the option of undoing the transformations. (See Parameter Estimate Population Report.) In this case, the Pareto Plot represents your selection of equal variances or unequal variances and uncorrelated or correlated estimates.

Example of a Pareto Plot

1. Open the Reactor.jmp sample data table.

For this data set, the estimates have equal variances and are not correlated.

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. Make sure that the Degree box has a 2 in it.

5. Select F, Ct, A, T, and Cn and click Macros > Factorial to Degree.

6. Click Run.

7. From the red triangle menu next to Response Y, select Effect Screening > Pareto Plot.

Figure 3.27 Pareto Plot for Reactor.jmp

Parameter Estimate Population Report

Most inferences about effect size first assume that the estimates are uncorrelated and have equal variances. This is true for fractional factorials and many classical experimental designs. However, these assumptions are not true for some designs.

The last three Effect Screening options in the report’s red triangle menu show a plot and the Parameter Estimate Population report. The Parameter Population report finds the correlation of the estimates and tells you whether the estimates are uncorrelated and have equal variances.

• If the estimates are correlated, a normalizing transformation can be applied make them uncorrelated and have equal variances.

• If an estimate of error variance is unavailable, the relative standard error for estimates is calculated by setting the error variance to one.

• If the estimates are uncorrelated and have equal variances, then the following notes appear under the Effect Screening report title:

– The parameter estimates have equal variances.

– The parameter estimates are not correlated.

If the estimates are correlated and/or have unequal variances, then each of these two notes can appear as list items. The selected list item shows that JMP has transformed the estimates.

To undo both transformations, click these list items.

The transformation to uncorrelate the estimates is the same as that used to calculate sequential sums of squares. The estimates measure the additional contribution of the variable after all previous variables have been entered into the model.

Example of Data with Equal Variances and Uncorrelated Estimates

1. Open the Reactor.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. Make sure that the Degree box has a 2 in it.

5. Select Ct, A, T, and Cn and click Macros > Factorial to Degree.

6. Click Run.

7. From the red triangle menu next to Response Y, select Effect Screening > Normal Plot.

8. Open the Parameter Estimate Population report.

Figure 3.28 Parameter Estimate Population Report for Reactor.jmp

Estimate	Lists the parameter estimates for the fitted linear model. These estimates can be compared in size only if the X variables are scaled the same.
t Ratio	The t-test associated with this parameter.
Orthog Coded	`Note:` This column appears only for unequal variances and correlated estimates. Contains the orthogonalized estimates. These estimates are shown in the Pareto plot because this plot partitions the sum of the effects, which requires orthogonality. The orthogonalized values are computed by premultiplying the column vector of the `Original` estimates by the Cholesky root of X´X. These estimates depend on the model order unless the original design is orthogonal. If the design was orthogonal and balanced, then these estimates are identical to the original estimates. Otherwise, each effect’s contribution is measured after it is made orthogonal to the effects before it.
Orthog t-Ratio	Lists the parameter estimates after a transformation that makes them independent and identically distributed. These values are used in the Normal plot (discussed next), which requires uncorrelated estimates of equal variance. The p-values associated with Orthog t-ratio estimates (given in the `Prob>\|t\|` column) are equivalent to Type I sequential tests. This means that if the parameters of the model are correlated, the estimates and their p-values depend on the order of terms in the model. The `Orthog t-ratio` estimates are computed by dividing the orthogonalized estimates by their standard errors. The `Orthog t-ratio` values let JMP treat the estimates as if they were from a random sample for use in Normal plots or Bayes plots.
Prob>\|t\|	Significance level or p-value associated with the values in the `Orthog t-Ratio` column.

Saturated Models

Screening experiments often involve fully saturated models, where there are not enough degrees of freedom to estimate error. Because of this, neither standard errors for the estimates, nor t-ratios, nor p-values can be calculated in the traditional way.

For these cases, JMP uses the relative standard error, corresponding to a residual standard error of 1. In cases where all the variables are identically coded (say, [–1,1] for low and high levels), these relative standard errors are identical.

JMP also displays a Pseudo-t-ratio, calculated as follows:

using Lenth’s PSE (pseudo standard-error) and degrees of freedom for error (DFE) equal to one-third the number of parameters. The value for Lenth’s PSE is shown at the bottom of the report.

Example of a Saturated Model

1. Open the Reactor.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. Select the following five columns: F, Ct, A, T, and Cn.

5. Click on the Macros button and select Full Factorial.

6. Click Run.

The parameter estimates are presented in sorted order, with smallest p-values listed first. The sorted parameter estimates are presented in Figure 3.29.

Figure 3.29 Saturated Report

In cases where the relative standard errors are different (perhaps due to unequal scaling), a similar report appears. However, there is a different value for Lenth’s PSE for each estimate.

Description of Lenth’s Method

An estimate of standard error is calculated using the method of Lenth (1989) and shows in the Effect Screening report (shown above). This estimate, called the pseudo standard error, is formed by taking 1.5 times the median absolute value of the estimates after removing all the estimates greater than 3.75 times the median absolute estimate in the complete set of estimates.

Factor Profiling

Assuming that the prediction equation is estimated well, you still must explore the equation itself to answer a number of questions:

• What type of curvature does the response surface have?

• What are the predicted values at the corners of the factor space?

• Would a transformation on the response produce a better fit?

The tools described in this section explore the prediction equation to answer these questions assuming that the equation is correct enough to work with.

Profiler	Shows prediction traces for each X variable. See The Profiler.
Interaction Plots	Shows a matrix of interaction plots when there are interaction effects in the model. See Interaction Plots.
Contour Profiler	Provides an interactive contour profiler, which is useful for optimizing response surfaces graphically. See Contour Profiler.
Mixture Profiler	`Note:` This option appears only if you specify the `Macros` `>` `Mixture` `Response Surface` option for an effect. Shows response contours of mixture experiment models on a ternary plot. Select this option when three or more factors in the experiment are components in a mixture. See Mixture Profiler.
Cube Plots	Shows a set of predicted values for the extremes of the factor ranges. The values are laid out on the vertices of cubes. See Cube Plots.
Box Cox Y Transformation	Finds a power transformation of the response that fits the best. See Box Cox Y Transformations.
Surface Profiler	Shows a three-dimensional surface plot of the response surface. See Surface Profiler.

The Profiler

Note: For details about using the Profiler, see the Visualizing, Optimizing, and Simulating Response Surfaces.

The Profiler (or Prediction Profiler) shows prediction traces for each X variable (Figure 3.30). The vertical dotted line for each X variable shows its current value or current setting. Use the Profiler to change one variable at a time and examine the effect on the predicted response.

Figure 3.30 Illustration of Prediction Traces

If the variable is nominal, the x-axis identifies categories.

For each X variable, the value above the factor name is its current value. You change the current value by clicking in the graph or by dragging the dotted line where you want the new current value to be.

• The horizontal dotted line shows the current predicted value of each Y variable for the current values of the X variables.

• The black lines within the plots show how the predicted value changes when you change the current value of an individual X variable. The 95% confidence interval for the predicted values is shown by a dotted curve surrounding the prediction trace (for continuous variables) or an error bar (for categorical variables).

Contour Profiler

Note: For details about using the Contour Profiler, see the Visualizing, Optimizing, and Simulating Response Surfaces.

Use the interactive Contour Profiler for optimizing response surfaces graphically. The Contour Profiler shows contours for the fitted model for two factors at a time. The report also includes a surface plot.

Example Using the Contour Profiler

1. Open the Tiretread.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select ABRASION, MODULUS, ELONG, and HARDNESS and click Y.

4. Select SILICA, SILANE, and SULFUR and click Macros > Response Surface.

5. Click Run.

6. From the red triangle menu next to Least Squares Fit, select Profilers > Contour Profiler.

Figure 3.31 Contour Profiler

Mixture Profiler

Note: This option appears only if you specify the Macros > Mixture Response Surface option for an effect. For details about using the Mixture Profiler, see the Visualizing, Optimizing, and Simulating Response Surfaces.

The Mixture Profiler shows response contours of mixture experiment models on a ternary plot. Use the Mixture Profiler when three or more factors in your experiment are components in a mixture. The Mixture Profiler enables you to visualize and optimize the response surfaces of your experiment.

Example Using the Mixture Profiler

1. Open the Plasticizer.jmp sample data table.

2. Select Analyze > Fit Model.

The response (Y) and predictor variables (p1, p2, and p3) are automatically populated. The predictor variables use the Macros > Mixture Response Surface option.

3. Click Run.

4. From the red triangle menu next to Response Y, select Factor Profiling > Mixture Profiler.

Figure 3.32 Mixture Profiler

You modify plot axes for the factors by selecting different radio buttons at the top left of the plot. The Lo and Hi Limit columns at the upper right of the plot let you enter constraints for both the factors and the response.

Surface Profiler

Note: For details about using the Surface Profiler, see the Visualizing, Optimizing, and Simulating Response Surfaces. For details about Surface Plots, see the Plotting Surfaces.

The Surface Profiler shows a three-dimensional surface plot of the response surface.

Example Using the Surface Profiler

1. Open the Odor.jmp sample data table.

2. From the red triangle next to Model, select Run Script.

3. Click Run.

4. From the red triangle menu next to Response odor, select Factor Profiling > Surface Profiler.

Figure 3.33 Surface Plot

Interaction Plots

The Interaction Plots option shows a matrix of interaction plots when there are interaction effects in the model.

Example of Interaction Plots

1. Open the Reactor.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. Make sure that the Degree box has a 2 in it.

5. Select Ct, A, T, and Cn and click Macros > Factorial to Degree.

6. Click Run.

7. From the red triangle menu next to Response Y, select Factor Profiling > Interaction Plots.

Figure 3.34 Interaction Plots

In an Interaction Plot, evidence of interaction shows as nonparallel lines. For example, in the T*Cn plot in the bottom row of plots, the effect of Cn is very small at the low values of temperature. However, the Cn effect diverges widely for the high values of temperature.

Cube Plots

The Cube Plots option displays a set of predicted values for the extremes of the factor ranges. These values appear on the vertices of cubes (Figure 3.35). If a factor is nominal, then the vertices are the first and last level.

Example of Cube Plots

1. Open the Reactor.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. Make sure that the Degree box has a 2 in it.

5. Select Ct, A, T, and Cn and click Macros > Factorial to Degree.

6. Click Run.

7. From the red triangle menu next to Response Y, select Factor Profiling > Cube Plots.

Figure 3.35 Cube Plots

To change the layout so that the factors are mapped to different cube coordinates, click one of the factor names in the first cube and drag it to the desired axis. For example, in Figure 3.35, if you click T and drag it over Ct, then T and Ct (and their corresponding coordinates) exchange places. When there is more than one response, the multiple responses are shown stacked at each vertex.

Box Cox Y Transformations

Sometimes a transformation on the response fits the model better than the original response. A commonly used transformation raises the response to some power. Box and Cox (1964) formalized and described this family of power transformations. The formula for the transformation is constructed so that it provides a continuous definition and the error sums of squares are comparable.

where

is the geometric mean

The plot shown here illustrates the effect of this family of power transformations on Y.

Figure 3.36 Power Transformations on Y

The Box-Cox Y Transformation option fits transformations from λ = –2 to 2 in increments of 0.2. This transformation also plots the sum of squares error (SSE) across the λ power.

Example of Box-Cox Y Transformation

1. Open the Reactor.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select Y and click Y.

4. Make sure that the Degree box has a 2 in it.

5. Select F, Ct, A, T, and Cn and click Macros > Factorial to Degree.

6. Click Run.

7. From the red triangle menu next to Response Y, select Factor Profiling > Box Cox Y Transformation.

Figure 3.37 Box Cox Y Transformation

Figure 3.37 shows the best fit when λ is between 1.0 and 1.5. The best transformation is found on the plot by finding the lowest point on the curve.

Save Best Transformation	Creates a new column in the data table and saves the formula for the best transformation.
Save Specific Transformation	Prompts you for a lambda value and then does the same thing as the `Save` `Best Transformation`.
Table of Estimates	Creates a new data table containing parameter estimates and SSE for all values of λ from -2 to 2.

Row Diagnostics

Plot Regression	Produces a regression plot of the response and the continuous X variable. If there is exactly one continuous term in a model, and no more than one categorical term, then JMP plots the regression line (or lines).
Plot Actual by Predicted	Displays the observed values by the predicted values of Y. This is the leverage plot for the whole model. See Leverage Plots.
Plot Effect Leverage	Produces a leverage plot for each effect in the model showing the point-by-point composition of the test for that effect. See Leverage Plots.
Plot Residual By Predicted	Displays the residual values by the predicted values of Y. You typically want to see the residual values scattered randomly about zero.
Plot Residual By Row	Displays the residual value by the row number of its observation.
Press	Displays a Press statistic, which computes the residual sum of squares where the residual for each row is computed after dropping that row from the computations. The Press statistic is the total prediction error sum of squares and is given by where n is the number of variables in the model, yi is the observed response value of the ith observation, and is the predicted response value of the omitted observation. The Press RMSE is defined as . The Press statistic is useful when comparing multiple models. Models with lower Press statistics are favored.
Durbin-Watson Test	Displays the Durbin-Watson statistic to test whether the errors have first-order autocorrelation. The autocorrelation of the residuals is also shown. The Durbin-Watson table has a popup option that computes and displays the exact probability associated with the statistic. This Durbin-Watson table is appropriate only for time series data when you suspect that the errors are correlated across time. `Note:` The computation of the Durbin-Watson exact probability can be time-intensive if there are many observations. The space and time needed for the computation increase with the square and the cube of the number of observations, respectively.

Leverage Plots

To graphically view the significance of the model or focus attention on whether an effect is significant, you want to display the data by focusing on the hypothesis for that effect. You might say that you want more of an X-ray picture showing the inside of the data rather than a surface view from the outside. The leverage plot gives this view of your data; it offers maximum insight into how the fit carries the data.

The effect in a model is tested for significance by comparing the sum of squared residuals to the sum of squared residuals of the model with that effect removed. Residual errors that are much smaller when the effect is included in the model confirm that the effect is a significant contribution to the fit.

The graphical display of an effect’s significance test is called a leverage plot. See Sall (1990). This type of plot shows for each point what the residual would be both with and without that effect in the model. Leverage plots are found in the Row Diagnostics submenu of the Fit Model report.

A leverage plot is constructed as illustrated in Figure 3.38. The distance from a point to the line of fit shows the actual residual. The distance from the point to the horizontal line of the mean shows what the residual error would be without the effect in the model. In other words, the mean line in this leverage plot represents the model where the hypothesized value of the parameter (effect) is constrained to zero.

Historically, leverage plots are referred to as a partial-regression residual leverage plot by Belsley, Kuh, and Welsch (1980) or an added variable plot by Cook and Weisberg (1982).

The term leverage is used because a point exerts more influence on the fit if it is farther away from the middle of the plot in the horizontal direction. At the extremes, the differences of the residuals before and after being constrained by the hypothesis are greater and contribute a larger part of the sums of squares for that effect’s hypothesis test.

The fitting platform produces a leverage plot for each effect in the model. In addition, there is one special leverage plot titled Whole Model that shows the actual values of the response plotted against the predicted values. This Whole Model leverage plot dramatizes the test that all the parameters (except intercepts) in the model are zero. The same test is reported in the Analysis of Variance report.

Figure 3.38 Illustration of a General Leverage Plot

The leverage plot for the linear effect in a simple regression is the same as the traditional plot of actual response values and the regressor.

Example of a Leverage Plot for a Linear Effect

1. Open the Big Class.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select weight and click Y.

4. Select height and click Add.

5. Click Run.

Figure 3.39 Whole Model and Effect Leverage Plots

The plot on the left is the Whole Model test for all effects, and the plot on the right is the leverage plot for the effect height.

The points on a leverage plot for simple regression are actual data coordinates, and the horizontal line for the constrained model is the sample mean of the response. But when the leverage plot is for one of multiple effects, the points are no longer actual data values. The horizontal line then represents a partially constrained model instead of a model fully constrained to one mean value. However, the intuitive interpretation of the plot is the same whether for simple or multiple regression. The idea is to judge if the line of fit on the effect’s leverage plot carries the points significantly better than does the horizontal line.

Figure 3.38 is a general diagram of the plots in Figure 3.39. Recall that the distance from a point to the line of fit is the actual residual and that the distance from the point to the mean is the residual error if the regressor is removed from the model.

Confidence Curves

The leverage plots are shown with confidence curves. These indicate whether the test is significant at the 5% level by showing a confidence region for the line of fit. If the confidence region between the curves contains the horizontal line, then the effect is not significant. If the curves cross the line, the effect is significant. Compare the examples shown in Figure 3.40.

Figure 3.40 Comparison of Significance Shown in Leverage Plots

Interpretation of X Scales

If the modeling type of the regressor is continuous, then the x-axis is scaled like the regressor and the slope of the line of fit in the leverage plot is the parameter estimate for the regressor. (See the left illustration in Figure 3.41.)

If the effect is nominal or ordinal, or if a complex effect like an interaction is present instead of a simple regressor, then the x-axis cannot represent the values of the effect directly. In this case the x-axis is scaled like the y-axis, and the line of fit is a diagonal with a slope of 1. The whole model leverage plot is a version of this. The x-axis turns out to be the predicted response of the whole model, as illustrated by the right-hand plot in Figure 3.41.

Figure 3.41 Leverage Plots for Simple Regression and Complex Effects

The influential points in all leverage plots are the ones far out on the x-axis. If two effects in a model are closely related, then these effects as a whole do not have much leverage. This problem is called collinearity. By scaling regressor axes by their original values, collinearity is shown by shrinkage of the points in the x direction.

See the appendix Statistical Details for the details of leverage plot construction.

Save Columns

Each Save Columns option generates one or more new columns in the current data table, where <colname> is the name of the response variable.

`Table 3.10` Description of the Save Columns Options
Prediction Formula	Creates a new column called `Pred Formula colname` that contains the predicted values computed by the specified model. It differs from the `Predicted colname` column, because it contains the prediction formula. This is useful for predicting values in new rows or for obtaining a picture of the fitted model. Use the `Column Info` option and click the `Edit Formula` button to see the prediction formula. The prediction formula can require considerable space if the model is large. If you do not need the formula with the column of predicted values, use the `Save Columns > Predicted Values` option. For information about formulas, see the `Using JMP` book. `Note:` When using this option, an attempt is first made to find a Response Limits property containing desirability functions. The desirability functions are determined from the profiler, if that option has been used. Otherwise, the desirability functions are determined from the response columns. If you reset the desirabilities later, it affects only subsequent saves. (It does not affect columns that has already been saved.)
Predicted Values	Creates a new column called `Predicted colname` that contains the predicted values computed by the specified model.
Residuals	Creates a new column called `Residual colname` that contains the residuals, which are the observed response values minus predicted values.
Mean Confidence Interval	Creates two new columns called `Lower 95% Mean colname` and `Upper` `95%` `Mean` `colname`. The new columns contain the lower and upper 95% confidence limits for the line of fit. `Note:` If you hold down the SHIFT key and select the option, you are prompted to enter an α-level for the computations.
Indiv Confidence Interval	Creates two new columns called `Lower` `95%` `Indiv colname` and `Upper` `95%` `Indiv colname`. The new columns contain lower and upper 95% confidence limits for individual response values. `Note:` If you hold down the SHIFT key and select the option, you are prompted to enter an α-level for the computations.
Studentized Residuals	Creates a new column called `Studentized Resid colname`. The new column values are the residuals divided by their standard error.
Hats	Creates a new column called `h colname`. The new column values are the diagonal values of the matrix , sometimes called hat values.
Std Error of Predicted	Creates a new column called `StdErr Pred colname` that contains the standard errors of the predicted values.
Std Error of Residual	Creates a new column called `StdErr Resid colname` that contains the standard errors of the residual values.
Std Error of Individual	Creates a new column called `StdErr Indiv colname` that contains the standard errors of the individual predicted values.
Effect Leverage Pairs	Creates a set of new columns that contain the values for each leverage plot. The new columns consist of an X and Y column for each effect in the model. The columns are named as follows. If the response column name is `R` and the effects are `X1` and `X2,` then the new column names are: `X Leverage of X1 for R` `Y Leverage of X1 for R` `X Leverage of X2 for R` `Y Leverage of X2 for R`
Cook’s D Influence	Saves the Cook’s D influence statistic. Influential observations are those that, according to various criteria, appear to have a large influence on the parameter estimates.
StdErr Pred Formula	Creates a new column called `PredSE colname` that contains the standard error of the predicted values. It is the same as the `Std Error of Predicted` option, but it saves the formula with the column. Also, it can produce very large formulas.
Mean Confidence Limit Formula	Creates a new column in the data table containing a formula for the mean confidence intervals. `Note:` If you hold down the SHIFT key and select the option, you are prompted to enter an α-level for the computations.
Indiv Confidence Limit Formula	creates a new column in the data table containing a formula for the individual confidence intervals. `Note:` If you hold down the SHIFT key and select the option, you are prompted to enter an α-level for the computations.
Save Coding Table	Produces a new data table showing the intercept, all continuous terms, and coded values for nominal terms.

Note: If you are using the Graph option to invoke the Profiler, then you should first save the columns Prediction Formula and StdErr Pred Formula to the data table. Then, place both of these formulas into the Y,Prediction Formula role in the Profiler launch window. The resulting window asks whether you want to use the PredSE colname to make confidence intervals for the Pred Formula colname, instead of making a separate profiler plot for the PredSE colname.

Effect Options

Use the red triangle options next to an effect to request reports, plots, and tests for the effect. You can close results or dismiss the results by deselecting the item in the menu.

`Table 3.11` Description of Effect Options
LSMeans Table	Shows the statistics that are compared when effects are tested. See LSMeans Table.
LSMeans Plot	Plots least squares means for nominal and ordinal main effects and for interactions. See LSMeans Plot.
LSMeans Contrast	`Note:` This option is enabled only for pure classification effects. Specify contrasts for pure classification effects to compare levels. See LSMeans Contrast.
LSMeans Student’s t	Computes individual pairwise comparisons of least squares means in the model using Student’s t-tests. This test is sized for individual comparisons. If you make many pairwise tests, there is no protection across the inferences. Thus, the alpha-size (Type I) error rate across the hypothesis tests is higher than that for individual tests. See the `Basic Analysis and Graphing` book.
LSMeans Tukey HSD	Shows a test that is sized for all differences among the least squares means. This is the Tukey or Tukey-Kramer HSD (Honestly Significant Difference) test. (Tukey 1953, Kramer 1956). This test is an exact alpha-level test if the sample sizes are the same and conservative if the sample sizes are different (Hayter 1984). See the `Basic Analysis and Graphing` book.
LSMeans Dunnett	Performs multiple comparisons of each level against a control level. This test is called Dunnett’s test. In the Fit Model platform, Hsu’s factor analytical approximation is used for the calculation of p-values and confidence intervals. The test results are also plotted.
Test Slices	`Note:` This option is available only for interaction effects. For each level of each classification column in the interaction, it makes comparisons among all the levels of the other classification columns in the interaction. For example, if an interaction is ABC, then there is a slice called A=1, which tests all the B*C levels when A=1. There is another slice called A=2, and so on, for all the levels of B, and C. This is a way to detect the importance of levels inside an interaction.
Power Analysis	Shows the Power Details report, which enables you to analyze the power for the effect test. For details, see Parameter Power.

LSMeans Table

Least squares means are predicted values from the specified model across the levels of a categorical effect where the other model factors are controlled by being set to neutral values. The neutral values are the sample means (possibly weighted) for regressors with interval values, and the average coefficient over the levels for unrelated nominal effects.

Least squares means are the values that let you see which levels produce higher or lower responses, holding the other variables in the model constant. Least squares means are also called adjusted means or population marginal means. Least squares means can differ from simple means when there are other effects in the model.

Least squares means are the statistics that are compared when effects are tested. They might not reflect typical real-world values of the response if the values of the factors do not reflect prevalent combinations of values in the real world. Least squares means are useful as comparisons in experimental situations.

Example of a Least Squares Means Table

1. Open the Big Class.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select height and click Y.

4. Select age and click Add.

5. Click Run.

Figure 3.42 Least Squares Mean Table

A Least Squares Means table with standard errors is produced for all categorical effects in the model. For main effects, the Least Squares Means table also includes the sample mean. It is common for the least squares means to be closer together than the sample means. For further details about least squares means, see the appendix Statistical Details.

`Table 3.12` Description of the Least Squares Means Table
Level	Lists the names of each categorical level.
Least Sq Mean	Lists the least squares mean for each level of the categorical variable.
Estimability	Displays a warning if a least squares mean is not estimable. The column only appears when a message has to be displayed.
Std Error	Lists the standard error of the least square mean for each level of the categorical variable.
Lower 95%	`Note:` Only appears if you have the `Regression Reports > Show All` `Confidence Intervals` option selected or if you right-click in the report and select `Columns > Lower 95%`. Lower 95% confidence interval for the mean.
Upper 95%	`Note:` Only appears if you have the `Regression Reports > Show All` `Confidence Intervals` option selected or if you right-click in the report and select `Columns > Upper 95%`. Upper 95% confidence interval for the mean.
Mean	Lists the response sample mean for each level of the categorical variable. This is different from the least squares mean if the values of other effects in the model do not balance out across this effect.

LSMeans Plot

The LSMeans Plot option plots least squares means (LSMeans) plots for nominal and ordinal main effects and interactions.

Example of an LS Means Plot

1. Open the Popcorn.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select yield and click Y.

4. Select popcorn, oil amt, and batch and click Add.

5. Select popcorn in the Construct Model Effects section and batch in the Select Columns section.

6. Click Cross to obtain the popcorn*batch interaction.

7. Click Run.

8. Hold down the CTRL key. From the red triangle menu for one of the effects, select LSMeans Plot.

Tip: Holding down the CTRL key broadcasts the command to all of the effects.

Figure 3.43 LSMeans Plots for Main Effects and Interactions

Figure 3.43 shows the effect plots for main effects and the two-way interaction.

To transpose the factors of the LSMeans plot for a two-factor interaction, hold down the SHIFT key and select the LSMeans Plot option. (In this example, you must first deselect the LSMeans Plot option and then select it again, holding down the SHIFT key.) Figure 3.44 shows both the default and the transposed factors plots.

Figure 3.44 LSMeans Plot Comparison with Transposed Factors

LSMeans Contrast

A contrast is a set of linear combinations of parameters that you want to jointly test to be zero. JMP builds contrasts in terms of the least squares means of the effect. By convention, each column of the contrast is normalized to have sum zero and an absolute sum equal to two. If a contrast involves a covariate, you can specify the value of the covariate at which to test the contrast.

Example of the LSMeans Contrast Option

To illustrate using the LSMeans Contrast option:

1. Open the Big Class.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select height and click Y.

4. Select age and click Add.

5. Click Run.

6. From the red triangle menu next to age, select LSMeans Contrast.

Figure 3.45 LSMeans Contrast Specification Window

This Contrast Specification window shows the name of the effect and the names of the levels in the effect. Beside the levels is an area enclosed in a rectangle that has a column of numbers next to boxes of + and - signs.

To construct a contrast, click the + and - boxes beside the levels that you want to compare. If possible, the window normalizes each time to make the sum for a column zero and the absolute sum equal to two each time you click; it adds to the plus or minus score proportionately.

For example, to form a contrast that compares the first two age levels with the second two levels:

1. Click + for the ages 12 and 13

2. Click - for ages 14 and 15

If you wanted to do more comparisons, you would click the New Column button for a new column to define the new contrast.

Figure 3.46 LSMeans Contrast Specification for Age

3. Click Done.

The contrast is estimated, and the Contrast report appears.

Figure 3.47 LSMeans Contrast Report

The Contrast report shows the following:

• Contrast as a function of the least squares means

• Estimates and standard errors of the contrast for the least squares means

• t-tests for each column of the contrast

• F-test for all columns of the contrast tested jointly

• Parameter Function report that shows the contrast expressed in terms of the parameters. In this example, the parameters are for the ordinal variable, age.

LSMeans Student’s t and LSMeans Tukey’s HSD

The LSMeans Student’s t and LSMeans Tukey’s HSD options give multiple comparison tests for model effects. For details about these tests, see the Basic Analysis and Graphing book.

The red triangle options that appear in each report window show or hide optional reports.

`Table 3.13` Description of the LSMeans Student’s t and LSMeans Tukey’s HSD Options
Crosstab Report	A two-way table that highlights significant differences in red.
Connecting Letters Report	Illustrates significant differences with letters (similar to traditional SAS GLM output). Levels not connected by the same letter are significantly different.
Ordered Differences Report	Ranks the differences from lowest to highest. It also plots the differences on a histogram that has overlaid confidence interval lines.
Detailed Comparisons	Compares each level of the effect with all other levels in a pairwise fashion. Each section shows the difference between the levels, standard error and confidence intervals, t-ratios, p-values, and degrees of freedom. A plot illustrating the comparison appears on the right of each report.
Equivalence Test	Uses the TOST (Two One-Sided Tests) method to test for practical equivalence. For details, see the `Basic Analysis and Graphing` book.

Test Slices

The Test Slices option, which is enabled for interaction effects, is a quick way to do many contrasts at the same time. For each level of each classification column in the interaction, it makes comparisons among all the levels of the other classification columns in the interaction. For example, if an interaction is A*B*C, then there is a slice called A=1, which tests all the B*C levels when A=1. There is another slice called A=2, and so on, for all the levels of B, and C. This is a way to detect the importance of levels inside an interaction.

Restricted Maximum Likelihood (REML) Method

Random effects are those where the effect levels are chosen randomly from a larger population of levels. These random effects represent a sample from the larger population. In contrast, the levels of fixed effects are of direct interest rather than representative. If you have both random and fixed (nonrandom) effects in a model, it is called a mixed model.

Note: It is very important to declare random effects. Otherwise, the test statistics produced from the fitted model are calculated with the wrong assumptions.

Typical random effects can be any of the following:

• Subjects in a repeated measures experiment, where the subject is measured at several times.

• Plots in a split plot experiment, where an experimental unit is subdivided and multiple measurements are taken, usually with another treatment applied to the subunits.

• Measurement studies, where multiple measurements are taken in order to study measurement variation.

• Random coefficients models, where the random effect is built with a continuous term crossed with categories.

The Fit Model platform in JMP fits mixed models using these modern methods, now generally regarded as best practice:

• REML estimation method (Restricted Maximum Likelihood)

• Kenward-Roger tests

For historical interest only, the platform also offers the Method of Moments (EMS), but this is no longer a recommended method, except in special cases where it is equivalent to REML.

If you have a model where all of the effects are random, you can also fit it in the Variability Chart platform.

The REML method for fitting mixed models is now the mainstream, state-of-the-art method, supplanting older methods.

In the days before availability of powerful computers, researchers needed to restrict their interest to situations in which there were computational short cuts to obtain estimates of variance components and tests on fixed effects in a mixed model. Most books today introduce mixed models using these short cuts that work on balanced data. See McCulloch, Searle, and Neuhaus (2008), Poduri (1997), and Searle, Casella, and McCulloch(1992). The Method of Moments provided a way to calculate what the expected value of Mean Squares (EMS) were in terms of the variance components, and then back-solve to obtain the variance components. It was also possible using these techniques to obtain expressions for test statistics that had the right expected value under the null hypotheses that were synthesized from mean squares.

If your model satisfies certain conditions (that is, it has random effects that contain the terms of the fixed effects that they provide random structure for) then you can use the EMS choice to produce these traditional analyses. However, since the newer REML method produces identical results to these models, but is considerably more general, the EMS method is never recommended.

The REML approach was pioneered by Patterson and Thompson in 1974. See also Wolfinger, Tobias, and Sall (1994) and Searle, Casella, and McCulloch(1992). The reason to prefer REML is that it works without depending on balanced data, or shortcut approximations, and it gets all the tests right, even contrasts that work across interactions. Most packages that use the traditional EMS method are either not able to test some of these contrasts, or compute incorrect variances for them.

Introduction to Random Effects

Levels in random effects are randomly selected from a larger population of levels. For the purpose of testing hypotheses, the distribution of the effect on the response over the levels is assumed to be normal, with mean zero and some variance (called a variance component).

In one sense, every model has at least one random effect, which is the effect that makes up the residual error. The units making up individual observations are assumed to be randomly selected from a much larger population, and the effect sizes are assumed to have a mean of zero and some variance, σ2.

The most common model that has random effects other than residual error is the repeated measures or split plot model. Types of Effects in a Split Plot Model, lists the types of effects in a split plot model. In these models the experiment has two layers. Some effects are applied on the whole plots or subjects of the experiment. Then these plots are divided or the subjects are measured at different times and other effects are applied within those subunits. The effects describing the whole plots or subjects are one random effect, and the subplots or repeated measures are another random effect. Usually the subunit effect is omitted from the model and absorbed as residual error.

`Table 3.14` Types of Effects in a Split Plot Model
Split Plot Model	Type of Effect	Repeated Measures Model
whole plot treatment	fixed effect	across subjects treatment
whole plot ID	random effect	subject ID
subplot treatment	fixed effect	within subject treatment
subplot ID	random effect	repeated measures ID

Each of these cases can be treated as a layered model, and there are several traditional ways to fit them in a fair way. The situation is treated as two different experiments:

1. The whole plot experiment has whole plot or subjects as the experimental unit to form its error term.

2. Subplot treatment has individual measurements for the experimental units to form its error term (left as residual error).

The older, traditional way to test whole plots is to do any one of the following:

• Take means across the measurements and fit these means to the whole plot effects.

• Form an F-ratio by dividing the whole plot mean squares by the whole plot ID mean squares.

• Organize the data so that the split or repeated measures form different columns and do a MANOVA model, and use the univariate statistics.

These approaches work if the structure is simple and the data are complete and balanced. However, there is a more general model that works for any structure of random effects. This more generalized model is called the mixed model, because it has both fixed and random effects.

Another common situation that involves multiple random effects is in measurement systems where there are multiple measurements with different parts, different operators, different gauges, and different repetitions. In this situation, all the effects are regarded as random.

Generalizability

Random effects are randomly selected from a larger population, where the distribution of their effect on the response is assumed to be a realization of a normal distribution with a mean of zero and a variance that can be estimated.

Often, the exact effect sizes are not of direct interest. It is the fact that they represent the larger population that is of interest. What you learn about the mean and variance of the effect tells you something about the general population from which the effect levels were drawn. That is different from fixed effects, where you only know about the levels you actually encounter in the data.

Unrestricted Parameterization for Variance Components in JMP

Note: Read this section only if you are concerned about matching the results of certain textbooks.

There are two different statistical traditions for parameterizing the variance components: the unrestricted and the restricted approaches. JMP and SAS use the unrestricted approach. In this approach, while the estimated effects always sum to zero, the true effects are not assumed to sum to zero over a particular random selection made of the random levels. This is the same assumption as for residual error. The estimates make the residual errors have mean zero, and the true mean is zero. But a random draw of data using the true parameters is some random event that might not have a mean of exactly zero.

You need to know about this assumption because many statistics textbooks use the restricted approach. Both approaches have been widely taught for 50 years. A good source that explains both sides is Cobb (1998, section 13.3).

Negative Variances

Note: Read this section only when you are concerned about negative variance components.

Though variances are always positive, it is possible to have a situation where the unbiased estimate of the variance is negative. This happens in experiments when an effect is very weak, and by chance the resulting data causes the estimate to be negative. This usually happens when there are few levels of a random effect that correspond to a variance component.

JMP can produce negative estimates for both REML and EMS. For REML, there are two check boxes in the model launch window: Unbounded Variance Components and Estimate Only Variance Components. Unchecking the box beside Unbounded Variance Components constrains the estimate to be nonnegative. We recommend that you do not uncheck this if you are interested in fixed effects. Constraining the variance estimates leads to bias in the tests for the fixed effects. If, however, you are interested only in variance components, and you do not want to see negative variance components, then checking the box beside Estimate Only Variance Components is appropriate.

If you remain uncomfortable about negative estimates of variances, please consider that the random effects model is statistically equivalent to the model where the variance components are really covariances across errors within a whole plot. It is not hard to think of situations in which the covariance estimate can be negative, either by random happenstance, or by a real process in which deviations in some observations in one direction would lead to deviations in the other direction in other observations. When random effects are modeled this way, the covariance structure is called compound symmetry.

So, consider negative variance estimates as useful information. If the negative value is small, it can be considered happenstance in the case of a small true variance. If the negative value is larger (the variance ratio can get as big as 0.5), it is a troubleshooting sign that the rows are not as independent as you had assumed, and some process worth investigating is happening within blocks.

Scripting Note

The JSL option for Unbounded Variance Components is No Bounds( Boolean ). Setting this option to true (1) is equivalent to checking the Unbounded Variance Components option.

Random Effects BLUP Parameters

Random effects have a dual character. In one perspective, they appear like residual error, often the error associated with a whole-plot experimental unit. In another respect, they are like fixed effects with a parameter to associate with each effect category level. As parameters, you have extra information about them—they are derived from a normal distribution with mean zero and the variance estimated by the variance component. The effect of this extra information is that the estimates of the parameters are shrunk toward zero. The parameter estimates associated with random effects are called BLUPs (Best Linear Unbiased Predictors). Some researchers consider these BLUPs as parameters of interest, and others consider them by-products of the method that are not interesting in themselves. In JMP, these estimates are available, but in an initially closed report.

BLUP parameter estimates are used to estimate random-effect least squares means, which are therefore also shrunken toward the grand mean, at least compared to what they would be if the effect were treated as a fixed effect. The degree of shrinkage depends on the variance of the effect and the number of observations per level in the effect. With large variance estimates, there is little shrinkage. If the variance component is small, then more shrinkage takes place. If the variance component is zero, the effect levels are shrunk to exactly zero. It is even possible to obtain highly negative variance components where the shrinkage is reversed. You can consider fixed effects as a special case of random effects where the variance component is very large.

If the number of observations per level is large, the estimate shrinks less. If there are very few observations per level, the estimates shrink more. If there are infinite observations, there is no shrinkage and the answers are the same as fixed effects.

The REML method balances the information about each individual level with the information about the variances across levels.

For example, suppose that you have batting averages for different baseball players. The variance component for the batting performance across player describes how much variation is usual between players in their batting averages. If the player only plays a few times and if the batting average is unusually small or large, then you tend not to trust that estimate, because it is based on only a few at-bats; the estimate has a high standard error. But if you mixed it with the grand mean, that is, shrunk the estimate toward the grand mean, you would trust the estimate more. For players that have a long batting record, you would shrink much less than those with a short record.

You can run this example and see the results for yourself. The example batting average data are in the Baseball.jmp sample data file. To compare the Method of Moments (EMS) and REML, run the model twice. Assign Batting as Y and Player as an effect. Select Player in the Construct Model Effects box, and select Random Effect from the Attributes pop-up menu.

Run the model and select REML (Recommended) from the Method popup menu. The results show the best linear unbiased prediction (BLUP) for each level of random effects.

Run the model again with EMS (Traditional) as Method. The least square mean for fixed affects and each level of random effects and is the same.

Comparison of Estimates between Method of Moments and REML, summarizes the estimates between Method of Moments and REML across a set of baseball players in this simulated example. Note that Suarez, with only three at-bats, is shrunk more than the others with more at-bats.

`Table 3.15` Comparison of Estimates between Method of Moments and REML
	Method of Moments	REML	N
Variance Component	0.01765	0.019648
Anderson Jones Mitchell Rodriguez Smith Suarez	0.29500000 0.20227273 0.32333333 0.55000000 0.35681818 0.55000000	0.29640407 0.20389793 0.32426295 0.54713393 0.35702094 0.54436227	6 11 6 6 11 3
Least Squares Means	same as ordinary means	shrunken from means

REML and Traditional Methods Agree on the Standard Cases

It turns out that in balanced designs, the REML F-test values for fixed effects is the same as with the Method of Moments (Expected Means Squares) approach. The degrees of freedom could differ in some cases. There are a number of methods of obtaining the degrees of freedom for REML F-tests; the one that JMP uses is the smallest degrees of freedom associated with a containing effect.

F-Tests in Mixed Models

Note: This section details the tests produced with REML.

The REML method obtains the variance components and parameter estimates, but there are a few additional steps to obtain tests on fixed effects in the model. The objective is to construct the F statistic and associated degrees of freedom to obtain a p-value for the significance test.

Historically, in simple models using the Method of Moments (EMS), standard tests were derived by construction of quadratic forms that had the right expectation under the null hypothesis. Where a mean square had to be synthesized from a linear combination of mean squares to have the right expectation, Satterthwaite's method could be used to obtain the degrees of freedom to get the p-value. Sometimes these were fractional degrees of freedom, just as you might find in a modern (Aspin-Welch) Student's t-test.

With modern computing power and recent methods, we have much improved techniques to obtain the tests. First, Kackar, and Harville (1984) found a way to estimate a bias-correction term for small samples. This was refined by Kenward and Roger (1997) to correct further and obtain the degrees of freedom that gave the closest match of an F-distribution to the distribution of the test statistic. These are not easy calculations. Consequently, they can take some time to perform for larger models.

If you have a simple balanced model, the results from REML-Kenward-Roger agree with the results from the traditional approach, provided that the estimates are not bounded at zero.

• These results do not depend on analyzing the syntactic structure of the model. There are no rules about finding containing effects. The method does not care if your whole plot fixed effects are purely nested in whole plot random effects. It gets the right answer regardless.

• These results do not depend on having categorical factors. It handles continuous (random coefficient) models just as easily.

• These methods produce different (and better) results than older versions of JMP (that is, earlier than JMP 6) that implemented older, less precise, technology to do these tests.

• These methods do not depend on having positive variance components. Negative variance components are not only supported, but need to be allowed in order for the tests to be unbiased.

Our goal in implementing these methods was not just to handle general cases, but to handle cases without the user needing to know very much about the details. Just declare which effects are random, and everything else is automatic. It is particularly important that engineers learn to declare random effects, because they have a history of performing inadvertent split-plot experiments where the structure is not identified.

Specifying Random Effects

Models with Random Effects use the same Fit Model dialog as other models. To specify a random effect, highlight it in the Construct Model Effects list and select Attributes > Random Effect. This appends &Random to the effect name in the model effect list.

Split Plot

The most common type of layered design is a balanced split plot, often in the form of repeated measures across time. One experimental unit for some of the effects is subdivided (sometimes by time period) and other effects are applied to these subunits.

Example of a Split Plot

Consider the data in the Animals.jmp sample data table (the data are fictional). The study collected information about differences in the seasonal hunting habits of foxes and coyotes. Each season for one year, three foxes and three coyotes were marked and observed periodically. The average number of miles that they wandered from their dens during different seasons of the year was recorded (rounded to the nearest mile). The model is defined by the following aspects:

• The continuous response variable called miles

• The species effect with values fox or coyote

• The season effect with values fall, winter, spring, and summer

• An animal identification code called subject, with nominal values 1, 2, and 3 for both foxes and coyotes

There are two layers to the model:

1. The top layer is the between-subject layer, in which the effect of being a fox or coyote (species effect) is tested with respect to the variation from subject to subject. The bottom layer is the within-subject layer, in which the repeated-measures factor for the four seasons (season effect) is tested with respect to the variation from season to season within a subject. The within-subject variability is reflected in the residual error.

2. The season effect can use the residual error for the denominator of its F-statistics. However, the between-subject variability is not measured by residual error and must be captured with the subject within species (subject[species]) effect in the model. The F-statistic for the between-subject effect species uses this nested effect instead of residual error for its F-ratio denominator.

To create the split plot for this data, follow these steps:

1. Open the Animals.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select miles and click Y.

4. Select species and subject and click Add.

5. In the Select Columns list, select species.

6. In the Construct Model Effects list, select subject.

7. Click Nest.

This adds the subject within species (subject[species]) effect to the model.

8. Select the nested effect subject[species].

9. Select Attributes > Random Effect.

This nested effect is now identified as an error term for the species effect and shows as subject[species]&Random.

10. In the Select Columns list, select season and click Add.

Figure 3.48 Fit Model Dialog

When you assign any effect as random from the Attributes popup menu, the Method options (REML and EMS) appear at the top right of the dialog, with REML selected as the default.

A new option, Convergence Settings, is available from the red triangle menu next to Model Specification. For details, see Model Specification Options.

11. Click Run.

Figure 3.49 Partial Report of REML Analysis

REML Results

A nice feature of REML is that the report does not need qualification (Figure 3.49). The estimates are all properly shrunk and the standard errors are properly scaled (SAS Institute Inc. 1996). The variance components are shown as a ratio to the error variance, and as a portion of the total variance.

There is no special table of synthetic test creation, because all the adjustments are automatically taken care of by the model itself. There is no table of expected means squares, because the method does not need this.

If you have random effects in the model, the analysis of variance report is not shown. This is because the variance does not partition in the usual way, nor do the degrees of freedom attribute in the usual way, for REML-based estimates. You can obtain the residual variance estimate from the REML report rather than from the analysis of variance report.

To obtain the standard deviations (square-root of the variance components), right-click on the REML Variance Component Estimates report, and select Columns > Sqrt Variance Component.

The Variance Component Estimates table shows 95% confidence intervals for the variance components using the Satterthwaite (1946) approximation. If the Unbounded Variance Components options is used, the confidence intervals are +/-Wald style. You can right-click on the Variance Components Estimates table to toggle on the Norm KHC (Kackar-Harville correction). This value is an approximation of the magnitude of the increase in the mean squared errors of the estimators for the mixed model. See Kackar and Harville (1984) for a discussion of approximating standard errors in mixed models.

REML Save Columns Options

When you use the REML method, additional options appear in the Save Columns menu. All of these options allow the random effects predicted values to participate in the formulas, rather than using their expected value (zero).

`Table 3.16` Description of REML Save Columns Options
Conditional Pred Formula	Saves the prediction formula to a new column in the data table.
Conditional Pred Values	Saves the predicted values to a new column in the data table.
Conditional Residuals	Saves the model residuals to a new column in the data table.
Conditional Mean CI	Saves the mean confidence interval.
Conditional Indiv CI	Saves the confidence interval for an individual.

REML Profiler Option

When you use REML method and select Factor Profiling > Profiler, a new option, Conditional Predictions, appears on the red triangle menu next to Prediction Profiler. For details, see Profiler Options.

Method of Moments Results

Note: This section is only of use in matching historical results.

We no longer recommend the Method of Moments, but we understand the need to support it for teaching use, in order to match the results of many textbooks still in use.

You have the option of choosing the EMS (Traditional) approach from the Method popup menu on the Fit Model dialog. This is also called the Method of Moments method.

Results from the steps for the Method of Moments are as follows:

• For each effect, the coefficients of the expected mean squares for that effect are calculated. This is a linear combination of the variance components and fixed effect values that describes the expected value of the mean square for that effect. All effects also have a unit coefficient on the residual variance.

• The coefficients of expected mean squares for all the random effects, including the residual error, are gathered into a matrix, and this is used to solve for variance components for each random effect.

• For each effect to be tested, a denominator for that effect is synthesized using the terms of the linear combination of mean squares in the numerator that do not contain the effect to be tested or other fixed effects. Thus, the expectation is equal for those terms common to the numerator and denominator. The remaining terms in the numerator then constitute the effect test.

• Degrees of freedom for the synthesized denominator are constructed using Satterthwaite’s method.

• The effect tests use the synthetic denominator.

JMP handles random effects like the SAS GLM procedure with a Random statement and the Test option. Figure 3.50, shows example results.

Caution: Standard errors for least squares means and denominators for contrast F-tests also use the synthesized denominators. Contrasts using synthetic denominators might not be appropriate, especially in crossed effects compared at common levels. The leverage plots and custom tests are done with respect to the residual, so they might not be appropriate.

Caution: Crossed and nested relationships must be declared explicitly. For example, if knowing a subject ID also identifies the group that contains the subject (that is, if each subject is in only one group), then subject must be declared as nested within group. In that situation, the nesting must be explicitly declared to define the design structure.

Note: JMP cannot fit a layered design if the effect for a layer’s error term cannot be specified under current effect syntax. An example of this is a design with a Latin Square on whole plots for which the error term would be Row*Column–Treatment. Fitting such special cases with JMP requires constructing your own F-tests using sequential sums of squares from several model runs.

For the Animals example above, the EMS reports are as follows.

Figure 3.50 Report of Method of Moments Analysis for Animals Data

The random submatrix from the EMS table is inverted and multiplied into the mean squares to obtain variance component estimates. These estimates are usually (but not necessarily) positive. The variance component estimate for the residual is the Mean Square Error.

Note that the CV of the variance components is initially hidden in the Variance Components Estimates report. To reveal it, right-click (or hold down CTRL and click on the Macintosh) and select Columns > CV from the menu that appears.

Singularity Details

When there are linear dependencies between model effects, the Singularity Details report appears.

Figure 3.51 Singularity Details Report

Examples with Statistical Details

The examples in this section are based on the following example:

1. Open the Drug.jmp sample data table.

2. Select Analyze > Fit Model.

3. Select y and click Y.

4. Select Drug and click Add.

5. Click Run.

One-Way Analysis of Variance with Contrasts

In a one-way analysis of variance, a different mean is fit to each of the different sample (response) groups, as identified by a nominal variable. To specify the model for JMP, select a continuous Y and a nominal X variable, such as Drug. In this example, Drug has values a, d, and f. The standard least squares fitting method translates this specification into a linear model as follows: The nominal variables define a sequence of dummy variables, which have only values 1, 0, and –1. The linear model is written as follows:

where:

– yi is the observed response in the ith trial

– x1i is the level of the first predictor variable in the ith trial

– x2i is the level of the second predictor variable in the ith trial

– β0, β1, and β2 are parameters for the intercept, the first predictor variable, and the second predictor variable, respectively

– ει are the independent and normally distributed error terms in the ith trial

As shown here, the first dummy variable denotes that Drug=a contributes a value 1 and Drug=f contributes a value –1 to the dummy variable:

The second dummy variable is given the following values:

The last level does not need a dummy variable because in this model, its level is found by subtracting all the other parameters. Therefore, the coefficients sum to zero across all the levels.

The estimates of the means for the three levels in terms of this parameterization are as follows:

Solving for βi yields the following:

(the average over levels)

Therefore, if regressor variables are coded as indicators for each level minus the indicator for the last level, then the parameter for a level is interpreted as the difference between that level’s response and the average response across all levels. See the appendix Statistical Details for additional information about the interpretation of the parameters for nominal factors.

Figure 3.52 shows the Parameter Estimates and the Effect Tests reports from the one-way analysis of the drug data. Figure 3.1, at the beginning of the chapter, shows the Least Squares Means report and LS Means Plot for the Drug effect.

Figure 3.52 Parameter Estimates and Effect Tests for Drug.jmp

The Drug effect can be studied in more detail by using a contrast of the least squares means, as follows:

1. From the red triangle menu next to Drug, select LSMeans Contrast.

2. Click the + boxes for drugs a and d, and the - box for drug f to define the contrast that compares the average of drugs a and d to f (shown in Figure 3.53).

3. Click Done.

Figure 3.53 Contrast Example for the Drug Experiment

The Contrast report shows that the Drug effect looks more significant using this one-degree-of-freedom comparison test. The LSMean for drug f is clearly significantly different from the average of the LSMeans of the other two drugs.

Analysis of Covariance

An analysis of variance model with an added regressor term is called an analysis of covariance. Suppose that the data are the same as above, but with one additional term, x3i, in the formula as a new regressor. Both x1i and x2i continue to be dummy variables that index over the three levels of the nominal effect. The model is written as follows:

There is an intercept plus two effects: one is a nominal main effect using two parameters, and the other is an interval covariate regressor using one parameter.

Re-run the Snedecor and Cochran Drug.jmp example, but add the x to the model effects as a covariate. Compared with the main effects model (Drug effect only), the R2 increases from 22.8% to 67.6%, and the standard error of the residual reduces from 6.07 to 4.0. As shown in Figure 3.54, the F-test significance probability for the whole model decreases from 0.03 to less than 0.0001.

Figure 3.54 ANCOVA Drug Results

Sometimes you can investigate the functional contribution of a covariate. For example, some transformation of the covariate might fit better. If you happen to have data where there are exact duplicate observations for the regressor effects, it is possible to partition the total error into two components. One component estimates error from the data where all the x values are the same. The other estimates error that can contain effects for unspecified functional forms of covariates, or interactions of nominal effects. This is the basis for a lack of fit test. If the lack of fit error is significant, then the fit model platform warns that there is some effect in your data not explained by your model. Note that there is no significant lack of fit error in this example, as seen by the large probability value of 0.7507.

The covariate, x, has a substitution effect with respect to Drug. It accounts for much of the variation in the response previously accounted for by the Drug variable. Thus, even though the model is fit with much less error, the Drug effect is no longer significant. The effect previously observed in the main effects model now appears explainable to some extent in terms of the values of the covariate.

The least squares means are now different from the ordinary mean because they are adjusted for the effect of x, the covariate, on the response, y. Now the least squares means are the predicted values that you expect for each of the three values of Drug, given that the covariate, x, is held at some constant value. The constant value is chosen for convenience to be the mean of the covariate, which is 10.7333.

So, the prediction equation gives the least squares means as follows:

fit equation:

-2.696 - 1.185*Drug[a - f] - 1.0761*Drug[d - f] + 0.98718*x

for a:

-2.696 - 1.185*(1) -1.0761*(0) + 0.98718*(10.7333) = 6.71

for d:

-2.696 - 1.185*(0) -1.0761*(1) + 0.98718*(10.7333) = 6.82

for f:

-2.696 - 1.185*(-1) -1.0761*(-1) + 0.98718*(10.7333) = 10.16

Figure 3.55 shows a leverage plot for each effect. Because the covariate is significant, the leverage values for Drug are dispersed somewhat from their least squares means.

Figure 3.55 Comparison of Leverage Plots for Drug Test Data

Analysis of Covariance with Separate Slopes

This example is a continuation of the Drug.jmp example presented in the previous section. The example uses data from Snedecor and Cochran (1967, p. 422). A one-way analysis of variance for a variable called Drug, shows a difference in the mean response among the levels a, d, and f, with a significance probability of 0.03.

The lack of fit test for the model with main effect Drug and covariate x is not significant. However, for the sake of illustration, this example includes the main effects and the Drug*x effect. This model tests whether the regression on the covariate has separate slopes for different Drug levels.

This specification adds two columns to the linear model (call them x4i and x5i) that allow the slopes for the covariate to be different for each Drug level. The new variables are formed by multiplying the dummy variables for Drug by the covariate values, giving the following formula:

Coding of Analysis of Covariance with Separate Slopes, shows the coding of this Analysis of Covariance with Separate Slopes. (The mean of X is 10.7333.)

`Table 3.17` Coding of Analysis of Covariance with Separate Slopes
Regressor	Effect	Values
x1	`Drug[a]`	+1 if a, 0 if d, –1 if f
x2	`Drug[d]`	0 if a, +1 if d, –1 if f
x3	`x`	the values of x
x4	`Drug[a]*(x - 10.733)`	+x – 10.7333 if a, 0 if d, –(x – 10.7333) if f
x5	`Drug[d]*(x - 10.733)`	0 if a, +x – 10.7333 if d, –(x – 10.7333) if f

A portion of the report is shown in Figure 3.56. The Regression Plot shows fitted lines with different slopes. The Effect Tests report gives a p-value for the interaction of 0.56. This is not significant statistically, indicating the model does not need to have different slopes.

Figure 3.56 Plot with Interaction

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Table of Contents for Example Using Standard Least Squares

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Table of Contents for
Example Using Standard Least Squares