Launch the Multivariate platform by selecting Analyze > Multivariate Methods > Multivariate.

Several estimation methods for the correlations options are available to provide flexibility and to accommodate personal preferences. REML and Pairwise are the methods used most frequently. You can also estimate missing values by using the estimated covariance matrix, and then using the Impute Missing Data command. See Impute Missing Data.

The Default option uses either the REML, Row-wise, or Pairwise methods:

REML (restricted maximum likelihood) estimates are less biased than the ML (maximum likelihood) estimation method. The REML method maximizes marginal likelihoods based upon error contrasts. The REML method is often used for estimating variances and covariances.The REML method in the Multivariate platform is the same as the REML estimation of mixed models for repeated measures data with an unstructured covariance matrix. See the documentation for SAS PROC MIXED about REML estimation of mixed models. REML uses all of your data, even if missing cells are present, and is most useful for smaller datasets. Because of the bias-correction factor, this method is slow if your dataset is large and there are many missing data values. If there are no missing cells in the data, then the REML estimate is equivalent to the sample covariance matrix.

The maximum likelihood estimation method (ML) is useful for large data tables with missing cells. The ML estimates are similar to the REML estimates, but the ML estimates are generated faster. Observations with missing values are not excluded. For small data tables, REML is preferred over ML because REML’s variance and covariance estimates are less biased.

Robust estimation is useful for data tables that might have outliers. For statistical details, see Robust.

Rowwise estimation does not use observations containing missing cells. This method is useful in the following situations:

Pairwise estimation performs correlations for all rows for each pair of columns with nonmissing values.

The default multivariate report shows the standard correlation matrix and the scatterplot matrix. The platform menu lists additional correlation options and other techniques for looking at multiple variables. See Multivariate Platform Options.

In most of the analysis options, a missing value in an observation does not cause the entire observation to be deleted. However, the Pairwise Correlations option excludes rows that are missing for either of the variables under consideration. The Simple Statistics > Univariate option calculates its statistics column-by-column, without regard to missing values in other columns.

The Nonparametric Correlations menu offers three nonparametric measures:

A scatterplot matrix helps you visualize the correlations between each pair of response variables. The scatterplot matrix is shown by default, and can be hidden or shown by selecting Scatterplot Matrix from the red triangle menu for Multivariate.

Re-sizing any cell resizes all the cells.

The red triangle menu for the Scatterplot Matrix lets you tailor the matrix with color and density ellipses and by setting the α-level.

The Outlier Analysis menu contains options that show or hide plots that measure distance in the multivariate sense using one of these methods:

The Mahalanobis Outlier Distance plot shows the Mahalanobis distance of each point from the multivariate mean (centroid). The standard Mahalanobis distance depends on estimates of the mean, standard deviation, and correlation for the data. The distance is plotted for each observation number. Extreme multivariate outliers can be identified by highlighting the points with the largest distance values. See Computations and Statistical Details, for more information.

The Jackknife Distances plot shows distances that are calculated using a jackknife technique. The distance for each observation is calculated with estimates of the mean, standard deviation, and correlation matrix that do not include the observation itself. The jack-knifed distances are useful when there is an outlier. In this case, the Mahalanobis distance is distorted and tends to disguise the outlier or make other points look more outlying than they are.

The T2 plot shows distances that are the square of the Mahalanobis distance. This plot is preferred for multivariate control charts. The plot includes the value of the calculated T2 statistic, as well as its upper control limit. Values that fall outside this limit might be outliers.

You can save any of the distances to the data table by selecting the Save option from the red triangle menu for the plot.

Item reliability indicates how consistently a set of instruments measures an overall response. Cronbach’s α (Cronbach 1951) is one measure of reliability. Two primary applications for Cronbach’s α are industrial instrument reliability and questionnaire analysis.

To impute missing data, select Impute Missing Data from the red triangle menu for Multivariate. A new data table is created that duplicates your data table and replaces all missing values with estimated values.

This example uses the Danger.jmp data in the Sample Data folder. This table lists 30 items having some level of inherent danger. Three groups of people (students, nonstudents, and experts) ranked the items according to perceived level of danger. Note that Nuclear power is rated as very dangerous (1) by both students and nonstudents, but is ranked low (20) by experts. On the other hand, motorcycles are ranked either fifth or sixth by all three judging groups.

This method essentially ignores any outlying values by substantially down-weighting them. A sequence of iteratively reweighted fits of the data is done using the weight:

The Pearson product-moment correlation coefficient measures the strength of the linear relationship between two variables. For response variables X and Y, it is denoted as r and computed as

For the Spearman, Kendall, or Hoeffding correlations, the data are first ranked. Computations are then performed on the ranks of the data values. Average ranks are used in case of ties.

Spearman’s ρ correlation coefficient is computed on the ranks of the data using the formula for the Pearson’s correlation previously described.

Kendall’s τb coefficients are based on the number of concordant and discordant pairs. A pair of rows for two variables is concordant if they agree in which variable is greater. Otherwise they are discordant, or tied.

The formula for Hoeffding’s D (1948) is

The inverse correlation matrix provides useful multivariate information. The diagonal elements of the inverse correlation matrix, sometimes called the variance inflation factors (VIF), are a function of how closely the variable is a linear function of the other variables. Specifically, if the correlation matrix is denoted R and the inverse correlation matrix is denoted R-1, the diagonal element is denoted and is computed as

The Outlier Distance plot shows the Mahalanobis distance of each point from the multivariate mean (centroid). The Mahalanobis distance takes into account the correlation structure of the data as well as the individual scales. For each value, the distance is denoted di and is computed as

Cronbach’s α is defined as