Selection of Rational Samples

The manner in which we sample the process deserves our careful attention. The sampling method should maximize differences between samples and minimize differences within samples. This means that separate control charts may have to be kept for different operators, machines, or vendors.

Lots from which samples are chosen should be homogeneous. As mentioned in Chapter 6, if our objective is to determine shifts in process parameters, samples should be made up of items produced at nearly the same time. This gives us a time reference and will be helpful if we need to determine special causes. Alternatively, if we are interested in the nonconformance of items produced since the previous sample was selected, samples should be chosen from items produced since that time.

Sample Size

Sample sizes are normally between 4 and 10, and it is quite common in industry to have sample sizes of 4 or 5. The larger the sample size, the better the chance of detecting small shifts. Other factors, such as cost of inspection or cost of shipping a nonconforming item to the customer, also influence the choice of sample size.

Frequency of Sampling

The sampling frequency depends on the cost of obtaining information compared to the cost of not detecting a nonconforming item. As processes are brought into control, the frequency of sampling is likely to diminish.

Choice of Measuring Instruments

The accuracy of the measuring instrument directly influences the quality of the data collected. Measuring instruments should be calibrated and tested for dependability under controlled conditions. Low-quality data lead to erroneous conclusions. The characteristic being controlled and the desired degree of measurement precision both have an impact on the choice of a measuring instrument. In measuring dimensions such as length, height, or thickness, something as simple as a set of calipers or a micrometer may be acceptable. On the other hand, measuring the thickness of silicon wafers may require complex optical sensory equipment.

Design of Data Recording Forms

Recording forms should be designed in accordance with the control chart to be used. Common features for data recording forms include the sample number, the date and time when the sample was selected, and the raw values of the observations. A column for comments about the process is also useful.

Development of the Charts

1. Step 1: Using a pre-selected sampling scheme and sample size, record on the appropriate forms the measurements of the quality characteristic selected.
2. Step 2: For each sample, calculate the sample mean and range using the following formulas:
   (7-1)
   (7-2)

   where X_i represents the ith observation, n is the sample size, X_max is the largest observation, and X_min is the smallest observation.

3. Step 3: Obtain and draw the centerline and the trial control limits for each chart. For the- chart, the centerline is given by
   (7-3)

   where g represents the number of samples. For the R-chart, the centerline is found from

   (7-4)

   Conceptually, the 3σ control limits for the -chart are

   (7-5)

   Rather than compute σ from the raw data, we can use the relation between the process standard deviation σ (or the standard deviation of the individual items) and the mean of the ranges . Multiplying factors used to calculate the centerline and control limits are given in Appendix A-7. When sampling from a population that is normally distributed, the distribution of the statistic W = R/σ (known as the relative range) is dependent on the sample size n. The mean of W is represented by d₂ and is tabulated in Appendix A-7. Thus, an estimate of the process standard deviation is

   (7-6)

   The control limits for an -chart are therefore estimated as

   (7-7)

   where and is tabulated in Appendix A-7. Equation (7-7) is the working equation for determining the -chart control limits, given .

   The control limits for the R-chart are conceptually given by

   (7-8)

   Since R = σW, we have σ_R = σσ_w. In Appendix A-7, σ_w is tabulated as d₃. Using eq. (7-6), we get

   

   The control limits for the R-chart are estimated as

   (7-9)

   where

   

   Equation (7-9) is the working equation for calculating the control limits for the R-chart. Values of D₄ and D₃ are tabulated in Appendix A-7.

4. Step 4: Plot the values of the range on the control chart for range, with the centerline and the control limits drawn. Determine whether the points are in statistical control. If not, investigate the special causes associated with the out-of-control points (see the rules for this in Chapter 6) and take appropriate remedial action to eliminate special causes.

   Typically, only some of the rules are used simultaneously. The most commonly used criterion for determining an out-of-control situation is the presence of a point outside the control limits.

   An R-chart is usually analyzed before an -chart to determine out-of-control situations. An R-chart reflects process variability, which should be brought into control first. As shown by eq. (7-7), the control limits for an -chart involve the process variability and hence . Therefore, if an R-chart shows an out-of-control situation, the limits on the -chart may not be meaningful.

   Let's consider Figure 7-3. On the R-chart, sample 12 plots above the upper control limit and so is out of control. The -chart, however, does not show the process to be out of control. Suppose that the special cause is identified as a problem with a new vendor who supplies raw materials and components. The task is to eliminate the cause, perhaps by choosing a new vendor or requiring evidence of statistical process control at the vendor's plant.

   

5. Step 5: Delete the out-of-control point(s) for which remedial actions have been taken to remove special causes (in this case, sample 12) and use the remaining samples (here they are samples 1–11 and 13–15) to determine the revised centerline and control limits for the - and R-charts.

   These limits are known as the revised control limits. The cycle of obtaining information, determining the trial limits, finding out-of-control points, identifying and correcting special causes, and determining revised control limits then continues. The revised control limits will serve as trial control limits for the immediate future until the limits are revised again. This ongoing process is a critical component of continuous improvement.

   A point of interest regarding the revision of R-charts concerns observations that plot below the lower control limit, when the lower control limit is greater than zero. Such points that fall below LCL_R are, statistically speaking, out of control; however, they are also desirable because they indicate unusually small variability within the sample which is, after all, one of our main objectives. It is most likely that such small variability is due to special causes.

   If the user is convinced that the small variability does indeed represent the operating state of the process during that time, an effort should be made to identify the causes. If such conditions can be created consistently, process variability will be reduced. The process should be set to match those favorable conditions, and the observations should be retained for calculating the revised centerline and the revised control limits for the R-chart.

6. Step 6: Implement the control charts.

The - and R-charts should be implemented for future observations using the revised centerline and control limits. The charts should be displayed in a conspicuous place where they will be visible to operators, supervisors, and managers. Statistical process control will be effective only if everyone is committed to it—from the operator to the chief executive officer.

Variable Sample Size

So far, our sample size has been assumed to be constant. A change in the sample size has an impact on the control limits for the - and R-charts. It can be seen from eqs. (7-7) and (7-9) that an increase in the sample size n reduces the width of the control limits. For an -chart, the width of the control limits from the centerline is inversely proportional to the square root of the sample size. Appendix A-7 shows the pattern in which the values of the control chart factors A₂, D₄, and D₃ decrease with an increase in sample size.

Standardized Control Charts

When the sample size varies, the control limits on an - and an R-chart will change, as discussed previously. With fluctuating control limits, the rules for identifying out-of-control conditions we discussed in Chapter 6 become difficult to apply—that is, except for Rule 1 (which assumes a process to be out of control when an observation plots outside the control limits). One way to overcome this drawback is to use a standardized control chart. When we standardize a statistic, we subtract its mean from its value and divide this value by its standard deviation. The standardized values then represent the deviation from the mean in units of standard deviation. They are dimensionless and have a mean of zero. The control limits on a standardized chart are at ±3 and are therefore constant. It's easier to interpret shifts in the process from a standardized chart than from a chart with fluctuating control limits.

Let the sample size for sample i be denoted by n_i, and let and s_i denote its average and standard deviation, respectively. The mean of the sample averages is found as

(7-10)

An estimate of the process standard deviation, , is the square root of the weighted average of the sample variances, where the weights are 1 less the corresponding sample sizes. So,

(7-11)

Now, for sample i, the standardized value for the mean, Z_i, is obtained from

(7-12)

where and are given by eqs. (7-10) and (7-11), respectively. A plot of the Z_i-values on a control chart, with the centerline at 0, the upper control limit at 3, and the lower control limit at −3, represents a standardized control chart for the mean.

To standardize the range chart, the range R_i for sample i is first divided by the estimate of the process standard deviation, , given by eq. (7-11), to obtain

(7-13)

The values of r_i are then standardized by subtracting its mean d₂ and dividing by its standard deviation d₃ (Nelson 1989). The factors d₂ and d₃ are tabulated for various sample sizes in Appendix A-7. So, the standardized value for the range, k_i, is given by

(7-14)

These values of k_i are plotted on a control chart with a centerline at 0 and upper and lower control limits at 3 and −3, respectively.

Control Limits for a Given Target or Standard

Management sometimes wants to specify values for the process mean and standard deviation. These values may represent goals or desirable standard or target values. Control charts based on these target values help determine whether the existing process is capable of meeting the desirable standards. Furthermore, they also help management set realistic goals for the existing process.

Let and σ₀ represent the target values of the process mean and standard deviation, respectively. The centerline and control limits based on these standard values for the -chart are given by

(7-15)

Let . Values for A are tabulated in Appendix A-7. Equation (7-15) may be rewritten as

(7-16)

For the R-chart, the centerline is found as follows, Since , we have

(7-17)

where d₂ is tabulated in Appendix A-7. The control limits are

(7-18)

where D₂ = d₂ + 3d₃ (Appendix A-7) and σ_R = d₃σ.

Similarly,

(7-19)

where D₁ = d₂ − 3d₃ (Appendix A-7).

We must be cautious when we interpret control charts based on target or standard values. Sample observations can fall outside the control limits even though no special causes are present in the process. This is because these desirable standards may not be consistent with the process conditions. Thus, we could waste time and resources looking for special causes that do not exist.

On an -chart, plotted points can fall outside the control limits because a target process mean is specified as too high or too low compared to the existing process mean. Usually, it is easier to meet a desirable target value for the process mean than it is for the process variability. For example, adjusting the mean diameter or length of a part can often be accomplished by simply changing controllable process parameters. However, correcting for R-chart points that plot above the upper control limit is generally much more difficult.

An R-chart based on target values can also indicate excessive process variability without special causes present in the system. Therefore, meeting the target value σ₀ may involve drastic changes in the process. Such an R-chart may be implying that the existing process is not capable of meeting the desired standard. This information enables management to set realistic goals.

Interpretation and Inferences from the Charts

The difficult part of analysis is determining and interpreting the special causes and selecting remedial actions. Effective use of control charts requires operators who are familiar with not only the statistical foundations of control charts but also the process itself. They must thoroughly understand how the different controllable parameters influence the dependent variable of interest. The quality assurance manager or analyst should work closely with the product design engineer and the process designer or analyst to come up with optimal policies.

In Chapter 6 we discussed five rules for determining out-of-control conditions. The presence of a point falling outside the 3σ limits is the most widely used of those rules. Determinations can also be made by interpreting typical plot patterns. Once the special cause is determined, this information plus a knowledge of the plot can lead to appropriate remedial actions.

Often, when the R-chart is brought to control, many special causes for the--chart are eliminated as well. The -chart monitors the centering of the process because is a measure of the center. Thus, a jump on the -chart means that the process average has jumped and an increasing trend indicates the process center is gradually increasing. Process centering usually takes place through adjustments in machine settings or such controllable parameters as proper tool, proper depth of cut, or proper feed. On the other hand, reducing process variability to allow an R-chart to exhibit control is a difficult task that is accomplished through quality improvement.

Once a process is in statistical control, its capability can be estimated by calculating the process standard deviation. This measure can then be used to determine how the process performs with respect to some stated specification limits. The proportion of nonconforming items can be estimated. Depending on the characteristic being considered, some of the output may be reworked, while some may become scrap. Given the unit cost of rework and scrap, an estimate of the total cost of rework and scrap can be obtained. Process capability measures are discussed in more detail in Chapter 9. From an R-chart that exhibits control, the process standard deviation can be estimated as

where is the centerline and d₂ is a factor tabulated in Appendix A-7. If the distribution of the quality characteristic can be assumed to be normal, then given some specification limits, the standard normal table can be used to determine the proportion of output that is nonconforming.

Control Chart Patterns and Corrective Actions

A nonrandom identifiable pattern in the plot of a control chart might provide sufficient reason to look for special causes in the system. Common causes of variation are inherent to a system; a system operating under only common causes is said to be in a state of statistical control. Special causes, however, could be due to periodic and persistent disturbances that affect the process intermittently. The objective is to identify the special causes and take appropriate remedial action.

Western Electric Company engineers have identified 15 typical patterns in control charts. Your ability to recognize these patterns will enable you to determine when action needs to be taken and what action to take (AT&T 19841984). We discuss 9 of these patterns here.

Natural Patterns

A natural pattern is one in which no identifiable arrangement of the plotted points exists. No points fall outside the control limits, the majority of the points are near the centerline, and few points are close to the control limits. Natural patterns are indicative of a process that is in control; that is, they demonstrate the presence of a stable system of common causes. A natural pattern is shown in Figure 7-8.

Sudden Shifts in the Level

Many causes can bring about a sudden change (or jump) in pattern level on an - or R-chart. Figure 7-9 shows a sudden shift on an -chart. Such jumps occur because of changes—intentional or otherwise—in such process settings as temperature, pressure, or depth of cut. A sudden change in the average service level, for example, could be a change in customer waiting time at a bank because the number of tellers changed. New operators, new equipment, new measuring instruments, new vendors, and new methods of processing are other reasons for sudden shifts on - and R-charts.

Gradual Shifts in the Level

Gradual shifts in level occur when a process parameter changes gradually over a period of time. Afterward, the process stabilizes. An -chart might exhibit such a shift because the incoming quality of raw materials or components changed over time, the maintenance program changed, or the style of supervision changed. An R-chart might exhibit such a shift because of a new operator, a decrease in worker skill due to fatigue or monotony, or a gradual improvement in the incoming quality of raw materials because a vendor has implemented a statistical process control system. Figure 7-10 shows an -chart exhibiting a gradual shift in the level.

Trending Pattern

Trends differ from gradual shifts in level in that trends do not stabilize or settle down. Trends represent changes that steadily increase or decrease. An -chart may exhibit a trend because of tool wear, die wear, gradual deterioration of equipment, buildup of debris in jigs and fixtures, or gradual change in temperature. An R-chart may exhibit a trend because of a gradual improvement in operator skill resulting from on-the-job training or a decrease in operator skill due to fatigue. Figure 7-11 shows a trending pattern on an -chart.

Cyclic Patterns

Cyclic patterns are characterized by a repetitive periodic behavior in the system. Cycles of low and high points will appear on the control chart. An -chart may exhibit cyclic behavior because of a rotation of operators, periodic changes in temperature and humidity (such as a cold-morning startup), periodicity in the mechanical or chemical properties of the material, or seasonal variation of incoming components. An R-chart may exhibit cyclic patterns because of operator fatigue and subsequent energization following breaks, a difference between shifts, or periodic maintenance of equipment. Figure 7-12 shows a cyclic pattern for an -chart. If samples are taken too infrequently, only the high or the low points will be represented, and the graph will not exhibit a cyclic pattern. If control chart users suspect cyclic behavior, they should take samples frequently to investigate the possibility of a cyclic pattern.

Wild Patterns

Wild patterns are divided into two categories: freaks and bunches (or groups). Control chart points exhibiting either of these two properties are, statistically speaking, significantly different from the other points. Special causes are generally associated with these points.

Freaks are caused by external disturbances that influence one or more samples. Figure 7-13 shows a control chart exhibiting a freak pattern. Freaks are plotted points too small or too large with respect to the control limits. Such points usually fall outside the control limits and are easily distinguishable from the other points on the chart. It is often not difficult to identify special causes for freaks. You should make sure, however, that there is no measurement or recording error associated with the freak point. Some special causes of freaks include sudden, very short-lived power failures; the use of a new tool for a brief test period; and the failure of a component.

Bunches, or groups, are clusters of several observations that are decidedly different from other points on the plot. Figure 7-14 shows a control chart pattern exhibiting bunching behavior. Possible special causes of such behavior include the use of a new vendor for a short period time, use of a different machine for a brief time period, and new operator used for a short period.

Mixture Patterns (or the Effect of Two or More Populations)

A mixture pattern is caused by the presence of two or more populations in the sample and is characterized by points that fall near the control limits, with an absence of points near the centerline. A mixture pattern can occur when one set of values is too high and another set too low because of differences in the incoming quality of material from two vendors. A remedial action would be to have a separate control chart for each vendor. Figure 7-15 shows a mixture pattern. On an -chart, a mixture pattern can also result from overcontrol. If an operator chooses to adjust the machine or process every time a point plots near a control limit, the result will be a pattern of large swings. Mixture patterns can also occur on both - and R-charts because of two or more machines being represented on the same control chart. Other examples include two or more operators being represented on the same chart, differences in two or more pieces of testing or measuring equipment, and differences in production methods of two or more lines.

Stratification Patterns

A stratification pattern is another possible result when two or more population distributions of the same quality characteristic are present. In this case, the output is combined, or mixed (say, from two shifts), and samples are selected from the mixed output. In this pattern, the majority of the points are very close to the centerline, with very few points near the control limits, Thus, the plot can be misinterpreted as indicating unusually good control. A stratification pattern is shown in Figure 7-16. Such a plot could have resulted from plotting data for samples composed of the combined output of two shifts, each different in its performance. It is possible for the sample average (which is really the average of parts chosen from both shifts) to fluctuate very little, resulting in a stratification pattern in the plot. Remedial measures in such situations involve having separate control charts for each shift. The method of choosing rational samples should be carefully analyzed so that component distributions are not mixed when samples are selected.

Interaction Patterns

An interaction pattern occurs when the level of one variable affects the behavior of other variables associated with the quality characteristic of interest. Furthermore, the combined effect of two or more variables on the output quality characteristic may be different from the individual effect of each variable. An interaction pattern can be detected by changing the scheme for rational sampling. Suppose that in a chemical process the temperature and pressure are two important controllable variables that affect the output quality characteristic of interest. A low pressure and a high temperature may produce a very desirable effect on the output characteristic, whereas a low pressure by itself may not have that effect. An effective sampling method would involve controlling the temperature at several high values and then determining the effect of pressure on the output characteristic for each temperature value. Samples composed of random combinations of temperature and pressure may fail to identify the interactive effect of those variables on the output characteristic. The control chart in Figure 7-17 shows interaction between variables. In the first plot, the temperature was maintained at level A; in the second plot, it was held at level B. Note that the average level and variability of the output characteristic change for the two temperature levels. Also, if the R-chart shows the sample ranges to be small, information regarding the interaction could be used to establish desirable process parameter settings.

No Given Standards

The centerline of a standard deviation chart is

(7-25)

where g is the number of samples and s_i is the standard deviation of the ith sample. The upper control limit is

In accordance with eq. (7-41), an estimate of the population standard deviation σ is

(7-26)

Substituting this estimate of in the preceding expression yields

where and is tabulated in Appendix A-7. Similarly,

where and is also tabulated in Appendix A-7. Thus, the 3σ control limits are

(7-27)

The centerline of the chart for the mean is given by

(7-28)

The control limits on the -chart are

Using eq. (7-26) to obtain , we find the control limits to be

(7-29)

where and is tabulated in Appendix A-7.

The process of constructing trial control limits, determining special causes associated with out-of-control points, taking remedial actions, and finding the revised control limits is similar to that explained in the section on - and R-charts. The s-chart is constructed first. Only if it is in control should the -chart be developed, because the standard deviation of is dependent on. If the s-chart is not in control, any estimate of the standard deviation of will be unreliable, which will in turn create unreliable control limits for .

Given Standard

If a target standard deviation is specified as σ₀, the centerline of the s-chart is found by using eq. (7-22) as

(7-30)

The upper control limit for the s-chart is found by using eq. (7-23) as

where and is tabulated in Appendix A-7. Similarly, the lower control limit for the s-chart is

where and is tabulated in Appendix A-7. Thus, the control limits for the s-chart are

(7-31)

If a target value for the mean is specified as , the centerline is given by

(7-32)

Equations for the control limits will be the same as those given by eq. (7-16) in the section on - and R-charts:

(7-33)

where and is tabulated in Appendix A-7.

No Given Standards

An estimate of the process standard deviation is given by

where is the average of the moving ranges of successive observations. Note that if we have a total of g individual observations, there will be g − 1 moving ranges. The centerline and control limits of the MR-chart are

(7-34)

For n = 2, D₄ = 3.267, and D₃ = 0, the control limits become

The centerline of the X-chart is

(7-35)

The control limits of the X-chart are

(7-36)

where (for n = 2) Appendix A-7 gives d₂ = 1.128.

Given Standard

The preceding derivation is based on the assumption that no standard values are given for either the mean or the process standard deviation. If standard values are specified as and σ₀, respectively, the centerline and control limits of the X-chart are

(7-37)

Assuming n = 2, the MR-chart for standard values has the following centerline and control limits:

(7-38)

One advantage of an X-chart is the ease with which it can be understood. It can also be used to judge the capability of a process by plotting the upper and lower specification limits on the chart itself. However, it has several disadvantages compared to an -chart. An X-chart is not as sensitive to changes in the process parameters. It typically requires more samples to detect parametric changes of the same magnitude. The main disadvantage of an X-chart, though, is that the control limits can become distorted if the individual items don't fit a normal distribution.

- and R-Charts for Short Production Runs

Where different parts may be produced in the short run, one approach is to use the deviation from the nominal value as the modified observations. The nominal value may vary from part to part. So, the deviation of the observed value O_i from the nominal value N is given by

(7-39)

The procedure for the construction of the - and R-charts is the same as before using the modified observations, X_i. Different parts are plotted on the same control chart so as to have the minimum information (usually, at least 20 samples) required to construct the charts, even though for each part there are not enough samples to justify construction of a control chart.

Several assumptions are made in this approach. First, it is assumed that the process standard deviation is approximately the same for all the parts. Second, what happens when a nominal value is not specified (which is especially true for characteristics that have one-sided specifications, such as breaking strength)? In such a situation, the process average based on historical data may have to be used.

Z-MR Chart

When individuals' data are obtained on the quality characteristic, an approach is to construct a standardized control chart for individuals (Z-chart) and a moving-range (MR) chart. The standardized value is given by

(7-40)

The moving range is calculated from the standardized values using a length of size 2. Depending on how each group (part or product) is defined, the process standard deviation for group i is estimated by

(7-41)

where represents the average moving range for group i and d₂ is a control chart factor found from Appendix A-7. Minitab provides several options for selecting computation of the process mean and process standard deviation. For each group (part or product), the mean of the observations in that group could be used as an estimate of the process mean for that group. Alternatively, historical values of estimates may be specified as an option.

In estimating the process standard deviation for each group (part or product), Minitab provides options for defining groups as follows: by runs; by parts, where all observations on the same part are combined in one group; constant (combine all observations for all parts in one group); and relative to size (transform the original data by taking the natural logarithm and then combine all into one group).

The relative-to-size option assumes that variability increases with the magnitude of the quality characteristic. The natural logarithm transformation stabilizes the variance. A common estimate () of the process standard deviation is obtained from the transformed data. The constant option that pools all data assumes that the variability associated with all groups is the same, implying that product or part type or characteristic size has no influence. This option must be used only if there is enough information to justify the assumption. It produces a single estimate () of the common process standard deviation. The option of pooling by parts assumes that all runs of a particular part have the same variability. It produces an estimate () of the process standard deviation for each part group. Finally, the option of pooling by runs assumes that part variability may change from run to run. It produces an estimate of the process standard deviation for each run, independently.

Cumulative Sum Control Chart for the Process Mean

In Shewhart control charts such as the - and R-charts, a plotted point represents information corresponding to that observation only. It does not use information from previous observations. On the other hand, a cumulative sum chart, usually called a cusum chart, uses information from all of the prior samples by displaying the cumulative sum of the deviation of the sample values (e.g., the sample mean) from a specified target value.

The cumulative sum at sample number m is given by

(7-42)

where is the sample mean for sample i and μ₀ is the target mean of the process.

Cusum charts are more effective than Shewhart control charts in detecting relatively small shifts in the process mean (of magnitude to about ). A cusum chart uses information from previous samples, so the effect of a small shift is more pronounced. For situations in which the sample size n is 1 (say, when each part is measured automatically by a machine), the cusum chart is better suited than a Shewhart control chart to determining shifts in the process mean. Because of the magnified effect of small changes, process shifts are easily found by locating the point where the slope of plotted cusum pattern changes.

There are some disadvantages to using cusum charts, however. First, because the cusum chart is designed to detect small changes in the process mean, it can be slow to detect large changes in the process parameters. Because a decision criterion is designed to do well under a specific situation does not mean that it will perform equally well under different situations. Details on modifying the decision process for a cusum chart to detect large shifts can be found in Hawkins , Lucas (1976, 1982), and Woodall and Adams 1993. Second, the cusum chart is not an effective tool in analyzing the historical performance of a process to see whether it is in control or to bring it in control. Thus, these charts are typically used for well-established processes that have a history of being stable.

Recall that for Shewhart control charts the individual points are assumed to be uncorrelated. Cumulative values are, however, related. That is, S_i−1 and S_i are related because . It is therefore possible for a cusum chart to exhibit runs or other patterns as a result of this relationship. The rules for describing out-of-control conditions based on the plot patterns of Shewhart charts may therefore not be applicable to cusum charts. Finally, training workers to use and maintain cusum charts may be more costly than for Shewhart charts.

Cumulative sum charts can model the proportion of nonconforming items, the number of nonconformities, the individual values, the sample range, the sample standard deviation, or the sample mean. In this section we focus on their ability to detect shifts in the process mean.

Suppose that the target value of a process mean when the process is in control is denoted by μ₀. If the process mean shifts upward to a higher value μ₁, an upward drift will be observed in the value of the cusum S_m given by eq. (7-42) because the old lower value μ₀ is still used in the equation even though the X-values are now higher. Similarly, if the process mean shifts to a lower value μ₂, a downward trend will be observed in S_m. The task is to determine whether the trend in S_m is significant so that we can conclude that a change has taken place in the process mean.

In the situation where individual observations (n = 1) are collected from a process to monitor the process mean, eq. (7-42) becomes

(7-43)

where S₀ = 0.

Tabular Method

Let us first consider the case of individual observations (X_i) being drawn from a process with mean μ₀ and standard deviation σ. When the process is in control, we assume that . In the tabular cusum method, deviations above μ₀ are accumulated with a statistic S⁺, and deviations below μ₀ are accumulated with a statistic S⁻. These two statistics, S⁺ and S⁻, are labeled one-sided upper and lower cusums, respectively, and are given by

(7-44)

(7-45)

where S₀⁺ = S₀⁻ = 0.

The parameter K in eqs. (7-44) and (7-45) is called the allowable slack in the process and is usually chosen as halfway between the target value μ₀ and the shifted value μ₁ that we are interested in detecting. Expressing the shift (δ) in standard deviation units, we have , leading to

(7-46)

Thus, examining eqs. (7-44) and (7-45), we find that S_m⁺ and S_m⁻ accumulate deviations from the target value μ₀ that are greater than K. Both are reset to zero upon becoming negative. In practice, K = kσδ, where k is in units of standard deviation. In eq. (7-46), k = 0.5.

A second parameter in the decision-making process using cusums is the decision interval H, to determine out-of-control conditions. As before, we set H = hσ, where h is in standard deviation units. When the value of S_m⁺ or S_m⁻ plots beyond H, the process will be considered to be out of control. When k = 0.5, a reasonable value of h is 5 (in standard deviation units), which ensures a small average run length for shifts of the magnitude of one standard deviation that we wish to detect (Hawkins 1993). It can be shown that for a small value of β, the probability of a type II error, the decision interval is given by

(7-47)

Thus, if sample averages are used to construct cusums in the above procedures, σ² will be replaced by σ²/n in eq. (7-47), assuming samples of size n.

To determine when the shift in the process mean was most likely to have occurred, we will monitor two counters, N⁺ and N⁻. The counter N⁺ notes the number of consecutive periods that S_m⁺ is above 0, whereas N⁻ tracks the number of consecutive periods that S_m⁻ is above zero. When an out-of-control condition is detected, one can count backward from this point to the time period when the cusum was above zero to find the first period in which the process probably shifted. An estimate of the new process mean may be obtained from

(7-48)

or from

(7-49)

V-Mask Method

In the V-mask approach, a template known as a V-mask, proposed by Barnard (1959)1959, is used to determine a change in the process mean through the plotting of cumulative sums. Figure 7-23 shows a V-mask, which has two parameters, the lead distance d and the angle θ of each decision line with respect to the horizontal. The V-mask is positioned such that point P coincides with the last plotted value of the cumulative sum and line OP is parallel to the horizontal axis. If the values plotted previously are within the two arms of the V-mask—that is, between the upper decision line and the lower decision line—the process is judged to be in control. If any value of the cusum lies outside the arms of the V-mask, the process is considered to be out of control.

In Figure 7-23, notice that a strong upward shift in the process mean is visible for sample 5. This shift makes sense given the fact that the cusum value for sample 1 is below the lower decision line, indicating an out-of-control situation. Similarly, the presence of a plotted value above the upper decision line indicates a downward drift in the process mean.

Determination of V-Mask Parameters

The two parameters of a V-mask, d and θ, are determined based on the levels of risk that the decision maker is willing to tolerate. These risks are the type I and type II errors described in Chapter 6. The probability of a type I error, α, is the risk of concluding that a process is out of control when it is really in control. The probability of a type II error, β, is the risk of failing to detect a change in the process parameter and concluding that the process is in control when it is really out of control. Let denote the amount of shift in the process mean that we want to be able to detect and denote the standard deviation of . Next, consider the equation

(7-50)

where δ represents the degree of shift in the process mean, relative to the standard deviation of the mean, that we wish to detect. Then, the lead distance for the V-mask is given by

(7-51)

If the probability of a type II error, β, is selected to be small, then eq. (7-51) reduces to

(7-52)

The angle of decision line with respect to the horizontal is obtained from

(7-53)

where k is a scale factor representing the ratio of a vertical-scale unit to a horizontal-scale unit on the plot. The value of k should be between and , with a preferred value of .

One measure of a control chart's performance is the average run length (ARL). (We discussed ARL in Chapter 6.) This value represents the average number of points that must be plotted before an out-of-control condition is indicated. For a Shewhart control chart, if p represents the probability that a single point will fall outside the control limits, the average run length is given by

(7-54)

For 3σ limits on a Shewhart -chart, the value of p is about 0.0026 when the process is in control. Hence, the ARL for an -chart exhibiting control is

The implication of this is that, on average, if the process is in control, every 385th sample statistic will indicate an out-of-control state. The ARL is usually larger for a cusum chart than for a Shewhart chart. For example, for a cusum chart with comparable risks, the ARL is around 500. Thus, if the process is in control, on average, every 500th sample statistic will indicate an out-of-control situation, so there will be fewer false alarms.

Sample, i	Deviation of Sample Mean from Target,	Cumulative Sum, Si	Sample, i	Deviation of Sample Mean from Target,	Cumulative Sum, Si
1	−1.0	−1.0	9	−0.1	−1.4
2	−0.5	−1.5	10	−0.2	−1.6
3	0.1	−1.4	11	0.4	−1.2
4	0.3	−1.1	12	1.3	0.1
5	1.0	−0.1	13	−0.3	−0.2
6	−0.6	−0.7	14	0.3	0.1
7	0.5	−0.2	15	0.1	0.2
8	−1.1	−1.3

Designing a Cumulative Sum Chart for a Specified ARL

The average run length can be used as a design criterion for control charts. If a process is in control, the ARL should be long, whereas if the process is out of control, the ARL should be short. Recall that δ is the degree of shift in the process mean, relative to the standard deviation of the sample mean, that we are interested in detecting; that is, . Let L(δ) denote the desired ARL when a shift in the process mean is on the order of δ. An ARL curve is a plot of δ versus its corresponding average run length, L(δ). For a process in control, when δ = 0, a large value of L(0) is desirable, For a specified value of δ, we may have a desirable value of L(δ). Thus, two points on the ARL curve, [0, L(0)] and [δ, L(δ)], are specified. The goal is to find the cusum chart parameters d and θ that will satisfy these desirable goals.

Bowker and Lieberman (1987)1982 provide a table (see Table 7-9) for selecting the V-mask parameters d and θ when the objective is to minimize L(δ) for a given δ. It is assumed that the decision maker has a specified value of L(0) in mind. Table 7-9 gives values for and d, and the minimum value of L(δ) for a specified δ. We use this table in Example 7-9.

δ = Deviation from Target Value		L(0) = Expected Run Length When Process Is in Control
(standard deviations)		50	100	200	300	400	500
0.25		0.125			0.195		0.248
	d	47.6			46.2		37.4
	L(0.25)	28.3			74.0		94.0
0.50		0.25	0.28	0.29	0.28	0.28	0.27
	d	17.5	18.2	21.4	24.7	27.3	29.6
	L(0.5)	15.8	19.0	24.0	26.7	29.0	30.0
0.75		0.375	0.375	0.375	0.375	0.375	0.375
	d	9.2	11.3	13.8	15.0	16.2	16.8
	L(0.75)	8.9	11.0	13.4	14.5	15.7	16.5
1.0		0.50	0.50	0.50	0.50	0.50	0.50
	d	5.7	6.9	8.2	9.0	9.6	10.0
	L(1.0)	6.1	7.4	8.7	9.4	10.0	10.5
1.5		0.75	0.75	0.75	0.75	0.75	0.75
	d	2.7	3.3	3.9	4.3	4.5	4.7
	L(1.5)	3.4	4.0	4.6	5.0	5.2	5.4
2.0		1.0	1.0	1.0	1.0	1.0	1.0
	d	1.5	1.9	2.2	2.4	2.5	2.7
	L(2.0)	2.26	2.63	2.96	3.15	3.3	3.4
Source: A. H. Bowker and G. J. Lieberman, Engineering Statistics, 2nd ed.,1987. Reprinted by permission of Pearson Education, Inc. Upper Saddle River, NJ.

Cumulative Sum for Monitoring Process Variability

Cusum charts may also be used to monitor process variability as discussed by Hawkins (1981)1981. Assuming that X_i ∼ N(μ₀, σ), the standardized value Y_i is obtained first as Y_i = (X_i − μ₀)/σ. A new standardized quantity (Hawkins 1993) is constructed as follows:

(7-55)

where it is suggested that the v_i are sensitive to both variance and mean changes. For an in-control process, v_i is distributed approximately N(0, 1). Two one-sided standardized cusums are constructed as follows to detect scale changes:

(7-56)

(7-57)

where . The values of h and k are selected using guidelines similar to those discussed in the section on cusum for the process mean. When the process standard deviation increases, the values of in eq. (7-56) will increase. When exceeds h, we will detect an out-of-control condition. Similarly, if the process standard deviation decreases, values of will increase.

Moving-Average Control Chart

As mentioned previously, standard Shewhart control charts are quite insensitive to small shifts, and cumulative sum charts are one way to alleviate this problem. A control chart using the moving-average method is another. Such charts are effective for detecting shifts of small magnitude in the process mean. Moving-average control charts can also be used in situations for which the sample size is 1, such as when product characteristics are measured automatically or when the time to produce a unit is long. It should be noted that, by their very nature, moving-average values are correlated.

Suppose that samples of size n are collected from the process. Let the first t sample means be denoted by . (One sample is taken for each time step.) The moving average of width w (i.e., w samples) at time step t is given by

(7-58)

At any time step t, the moving average is updated by dropping the oldest mean and adding the newest mean. The variance of each sample mean is

where σ² is the population variance of the individual values. The variance of M_t is

(7-59)

The centerline and control limits for the moving-average chart are given by

(7-60)

From eq. (7-60), we can see that as w increases, the width of the control limits decreases. So, to detect shifts of smaller magnitudes, larger values of w should be chosen.

For the startup period (when t < w), the moving average is given by

(7-61)

The control limits for this startup period are

(7-62)

Since these control limits change at each sample point during this startup period, an alternative procedure would be to use the ordinary -chart for t < w and use the moving-average chart for t ≥ w.

Exponentially Weighted Moving-Average or Geometric Moving-Average Control Chart

The preceding discussion showed that a moving-average chart can be used as an alternative to an ordinary -chart to detect small changes in process parameters. The moving-average method is basically a weighted-average scheme. For sample t, the sample means are each weighted by 1/w [see eq. (7-58)], while the sample means for time steps less than t − w + 1 are weighted by zero. Along similar lines, a chart can be constructed based on varying weights for the prior observations. More weight can be assigned to the most recent observation, with the weights decreasing for less recent observations. A geometric moving-average control chart, also known as an exponentially weighted moving-average (EWMA) chart, is based on this premise. One of the advantages of a geometric moving-average chart over a moving-average chart is that the former is more effective in detecting small changes in process parameters. The geometric moving average at time step t is given by

(7-63)

where r is a weighting constant (0 < r ≤ 1) and G₀ is . By using eq. (7-63) repeatedly, we get

(7-64)

Equation (7-64) shows that the weight associated with the ith mean from is r (1 − r)ⁱ. The weights decrease geometrically as the sample mean becomes less recent. The sum of all the weights is 1. Consider, for example, the case for which r = 0.3. This implies that, in calculating G_t, the most recent sample mean has a weight of 0.3, the next most recent observation has a weight of (0.3)(1 − 0.3) = 0.21, the next observation has a weight of 0.3(1 − 0.3)² = 0.147, and so on. Here, G₀ has a weight of (1 − 0.3)^t. Since these weights appear to decrease exponentially, eq. (7-64) describes what is known as the exponentially weighted moving-average model.

If the sample means are assumed to be independent of each other and if the population standard deviation is σ, the variance of G_t is given by

(7-65)

For large values of t, the standard deviation of G_t is

The upper and lower control limits are

(7-66)

For small values of t, the control limits are found using eq. (7-65) to be

(7-67)

A geometric moving-average control chart is based on a concept similar to that of a moving-average chart. By choosing an adequate set of weights, however, where recent sample means are more heavily weighted, the ability to detect small changes in process parameters is increased. If the weighting factor r is selected as

(7-68)

where w is the moving-average span, the moving-average method and the geometric moving-average method are equivalent. There are guidelines for choosing the value of r. If our goal is to detect small shifts in the process parameters as soon as possible, we use a small value of r (say, 0.1). If we use r = 1, the geometric moving-average chart reduces to the standard Shewhart chart for the mean.

Modified Control Chart

In our discussions of all the control charts, we have assumed that the process spread (6σ) is close to and hopefully less than the difference between the specification limits. That is, we hope that 6σ<(USL − LSL). Now we assume that the natural process spread of 6σ is significantly less than the difference between the specification limits: that is, the process capability ratio = (USL − LSL)/6σ >> 1. Figure 7-27 depicts this situation.

So far, the specification limits have not been placed on -charts. One reason for this is that the specification limits correspond to the conformance of individual items. If the distribution of individual items is plotted, it makes sense to show the specification limits on the X-chart, but a control chart for the mean deals with averages, not individual values. Therefore, plotting the specification limits on an -chart is not appropriate. For a modified control chart, however, the specification limits are shown.

Our objective here is to determine bounds on the process mean such that the proportion of nonconforming items does not exceed a desirable value δ. The focus is not on detecting the statistical state of control, because a process can drift out of control and still produce parts that conform to specifications. In fact, we assume that the process variability is in a state of statistical control. An estimate of the process standard deviation σ is obtained from either the mean () of the R-chart or the mean () of the s-chart. Furthermore, we assume that the distribution of the individual values is normal and that a change in the process mean can be accomplished without much difficulty. Our aim in constructing a modified control chart is to determine whether the process mean μ is within certain acceptable limits such that the proportion of nonconforming items does not exceed the chosen value δ.

Let's consider Figure 7-28a, which shows a distribution of individual values at two different means: one the lowest allowable mean (μ_L) and the other the highest allowable mean (μ_U). Suppose that the process standard deviation is σ. If the distribution of individual values is normal, let z_δ denote the standard normal value corresponding to a tail area of δ. By the definition of a standard normal value, z_δ represents the number of standard deviations that LSL is from μ_L and that USL is from μ_U. So the distance between LSL and μ_L is z_δσ, which is also the same as the distance between USL and μ_U. The bounds within which the process mean should be contained so that the fraction nonconforming does not exceed δ are μ_L ≤ μ ≤ μ_U. From Figure 7-28a,

(7-69)

Suppose that a type I error probability of α is chosen. The control limits are placed such that the probability of a type I error is α, as shown in Figure 7-28b. The control limits are placed at each end to show that the sampling distribution of the sample mean can vary over the entire range. Figure 7-28b shows the distribution of the sample means. Given the sampling distribution of , with standard deviation , the upper and lower control limits as shown in Figure 7-28b are

(7-70)

By substituting for μ_L and μ_U, the following equations are obtained:

(7-71)

If the process standard deviation σ is to be estimated from an R-chart, then is substituted for σ in eq. (7-71). Alternatively, if σ is to be estimated from an s-chart, is used in place of σ in eq. (7-71).

Acceptance Control Chart

In the preceding discussion we outlined a procedure for obtaining the modified control limits given the sample size n, the proportion nonconforming δ, and the acceptable level of probability of type I error, α. In this section we discuss a procedure to calculate the control chart limits when the sample size is known and when we have a specified level of proportion nonconforming (γ) that we desire to detect with a probability of (1 − β). Such a control chart is known as an acceptance control chart. The same assumptions are made here as for modified control charts. That is, we assume that the inherent process spread (6σ) is much less than the difference between the specification limits, the process variability is in control, and the distribution of the individual values is normal.

Figure 7-29a shows the distribution of the individual values and the borderline locations of the process mean so that the proportion nonconforming does not exceed the desirable level of γ. From Figure 7-29a we have

(7-72)

Figure 7-29b shows the distribution of the sample mean and the bounds within which the process mean must lie for the probability of detecting a nonconformance proportion of γ to be 1 − β. From Figure 7-29b we have

(7-73)

Substituting from eq. (7-72), we get

(7-74)

If an R-chart is used to control the process variability, σ is estimated by . If an s-chart is used, σ is estimated by . These estimates are then used in eq. (7-74).

The principles of modified control charts and acceptance control charts can be combined to determine an acceptable sample size n for chosen levels of δ, α, γ, and β. By equating the expressions for the UCL in eqs. (7-71) and (7-74), we have

which yields

(7-75)

Some examples of variables control chart applications in the service sector are shown in Table 7-13. Note that the size of the subgroup in the data collected will determine, in many instances, the use of the X-MR chart (for individuals' data), and R (when the subgroup size is small, usually less than 10), and and s (when the subgroup size is large, usually equal to or greater than 10).

Quality Characteristic	Control Chart
Response time for correspondence in a consulting firm or financial institution	and R(n < 10); and s(n ≥ 10)
Waiting time in a restaurant	and R(n < 10); and s(n ≥ 10)
Processing time of claims in an insurance company	and R(n < 10); and s(n ≥ 10)
Waiting time to have cable installed	and R(n < 10); I and MR (n = 1)
Turnaround time in a laboratory test in a hospital	I and MR (n = 1)
Use of electrical power or gas on a monthly basis	Moving average; EWMA
Admission time into an intensive care unit	I and MR (n = 1)
Blood pressure or cholesterol ratings of a patient over time	I and MR (n = 1)

Risk-Adjusted Cumulative Sum (RACUSUM) Chart

Using the regular cusum chart, as discussed earlier, one could monitor the cumulative sum of the number of deaths for each successive patient given by

(7-76)

where W_n denotes the weight applied to the nth observation, in this case the outcome associated with the nth patient. Here, W_n = 1 if the nth patient dies and W_n = 0 if the nth patient survives. The control chart may be designed to signal if C_n > h, where h is a bound selected on the basis of a chosen false-alarm rate or, equivalently, an in-control ARL. The drawback of this traditional cumulative sum chart is that it does not take into account the variation in the pre-operative risk of mortality from patient to patient.

The risk-adjusted cumulative sum chart incorporates patient characteristics based on some aggregate risk score, such as the Parsonnet score or the APACHE score. Using such a score, the predicted risk of mortality is found from a logistic regression model and is given by

where p_n denotes the pre-operative predictive mortality for patient n whose aggregate risk score is given by RSC_n and a and b are estimated parameters of the logistic regression model. The above expression may be re-expressed to estimate the predicted risk of mortality as

(7-77)

The risk-adjusted chart repeatedly tests the null and the alternative hypothesis given by

The odds ratio is the ratio of the probability of mortality to the probability of survival. Two cumulative sums may be computed, one to detect increases in the mortality rate (R_a > R₀) through an upper limit h⁺ and the other to detect decreases in the mortality rate (R_a < R₀) through a lower limit h⁻.

Let us denote and as the upper and lower cumulative sum values. The risk-adjusted weight function (W_n) for patient n is utilized in computing the corresponding risk-adjusted cumulative sum values given as follows (Steiner et al. 2000):

(7-78)

where

(7-79)

The values of the control limits, h⁺ and h⁻, are selected based on chosen values of R₀, R_a, and the associated risks of a false alarm and the probability of a type II error. The chart signals if or . In order to compare surgical performance based on pre-operative prediction, R₀ may be chosen to be 1. To detect a deterioration in performance, R_a may be selected, for example, as 2, that is, a doubling of the odds ratio. On the other hand, to detect an improvement in the performance, R_a may be selected to be less than 1, for example, 0.5. By selecting the cumulative sum functions given by eqs. (7-78) and (7-79), the absorbing barrier is at the zero line so that the chart resets itself any time this barrier is reached.

Risk-Adjusted Sequential Probability Ratio Test (RASPRT)

The resetting risk-adjusted sequential probability ratio test is quite similar to the RACUSUM chart. The null hypothesis is that the risk of mortality is accurately predicted by the chosen risk adjustment prediction equation. In this case, we assume that p_n, given by eq. (7-77) based on an aggregate risk score, is accurate. The alternative hypothesis is that the probability of mortality is better predicted by a different probability.

The RASPRT statistic is given by

(7-80)

where the risk-adjusted weight for patient n is given by eq. (7-79). Thresholds for decision making are given by bounds e and f. The upper bound e is chosen based on a selected level of a type I error, α (false-alarm rate), where one may incorrectly conclude, for example, a doubling of the mortality rate if R_n > e. Both bounds are influenced by the chosen levels α and β, the probability of a type II error, and are given by

(7-81)

Risk-Adjusted Exponentially Weighted Moving-Average (RAEWMA) Chart

The regular EWMA chart introduced previously in the chapter weights observations based on how recent they are. More weight is assigned to the most recent observation with the weights decreasing sequentially in a geometric manner for observations as they go back in history.

In the risk-adjusted EWMA chart, the computation of the charting statistic is similar to that discussed previously. This is given by

where λ is the weighting constant and Y_n represents the observed outcome. If the patient survives, Y_n = 0, while it is 1 if the patient dies.

However, the control limits are based on risk adjustment associated with the varying degree of severity of illness of the patients. As in the risk-adjusted cusum chart, the predicted risk of mortality may be found from a re-expressed logistic regression model, of the type given by eq. (7-77), based on the aggregate risk score of the patient. The centerline for patient n, using risk adjustment, is given by

(7-82)

where λ is the selected weighting constant (0 < λ ≤ 1) and p_n is found from eq. (7-77). Equation (7-82) may be re-expressed as

(7-83)

where, for , the starting estimate of the predicted mortality risk, one may use the value of p₁.

The control limits of the risk-adjusted EWMA chart may be found by calculating the estimated variance of :

Since Var(p_k) = p_k(1 − p_k), k = 1, 2,…, n, we have

(7-84)

Assuming a normal approximation, the risk-adjusted control limits, using eqs. (7-83) and (7-84), are given by

(7-85)

where z_α/2 is the standard normal variate for a chosen type I error rate of α.

The risk-adjusted EWMA chart is able to detect small and gradual changes in the mortality rate. Further, an appropriate choice of the smoothing factor λ based on the anticipated change helps the RAEWMA chart to signal quickly in the event of a change.

Variable Life-Adjusted Display (VLAD) Chart

Another control chart for monitoring mortality is the VLAD chart that displays the cumulative difference between observed and predicted deaths plotted against the patient sequence number. The predicted number of deaths incorporates the severity of illness of the patient. As discussed previously, a composite measure such as the Parsonnet score or the APACHE score may be used to predict the risk of mortality for a patient using a logistic regression model. Equation (7-77) shows a reduced form of the model.

The VLAD chart is a form of the cumulative sum control chart. For each operation performed on a patient by the surgeon, the chart statistic accrues a value equal to the predicted risk, which represents the expected number of deaths that incorporates the patients' severity of illness, minus the observed outcome, which represents the actual number of deaths. The statistic represents the net lives saved when adjusted for the patient's pre-operative risk on a cumulative basis. Hence, for a surgeon performing at the predicted level of risk, the statistic will approach the value of zero. For those performing at a level better than that predicted, as influenced by the patients' risk level, a positive score will result for the VLAD statistic. On the contrary, for a surgeon not performing at the predicted level, the VLAD statistic will show a downward trend and eventually yield a negative value.

Suppose that the predicted risk of mortality for patient n is given by p_n, as given by eq. (7-77). The statistic for patient n is obtained as

(7-86)

where o_n = 1 if patient n dies and o_n = 0 if the patient survives. For a sequence of operations, the cumulative sum of the VLAD statistic after operation t is given by

(7-87)

Controlling Several Related Quality Characteristics

Suppose we have two quality characteristics that must both be in control for the process to be in control. If control charts for the averages of these two characteristics are kept independently, the result is a rectangular control region on a two-dimensional plot. The boundaries of this region are basically the upper and lower control limits of the two quality characteristics and are calculated using eq. (7-5). If the bivariate observation of sample means () plots within the control limits, the process would seem to be in control.

Such rectangular boundaries, however, can often be incorrect. An actual control region for two characteristics is elliptic in nature (see Figure 7-32). The equation of a statistic that incorporates two characteristics is an ellipse, as we will see in eq. (7-89). If the two characteristics are independent of each other, the major and minor axes of the ellipse are parallel to the respective plot axes (see ellipse A in Figure 7-32). If the pair of sample means () falls within the boundary of the ellipse, the process is said to be in control. If two characteristics are negatively correlated, the shape of the control ellipse will be similar to that of ellipse B. If the two variables are positively correlated, the control ellipse will be similar to that of ellipse C.

Figure 7-32 shows that if the variables are positively correlated and we use the rectangular region erroneously as the control region, we draw various incorrect conclusions. For instance, if () falls in region E or region F, the process is in control even though the point falls outside the rectangular region. A point in region G, on the other hand, is within the rectangular region, but the process is nonetheless out of control.

The degree of correlation between the variables influences the magnitude of the errors encountered in making inferences. If a separate -chart is constructed for each characteristic based on a type I error probability of α and a rectangular control region is used, then for independent variables the probability of a type I error for the joint control procedure is

(7-88)

where p represents the number of jointly controlled variables. The probability of all p sample means plotting within the rectangular region is (1 − α)^p.

Moderate or large values of p have a major impact on the errors associated with inference making. Suppose that individual control chart limits are constructed using a type I error probability of 0.0026. If we have four independent characteristics (i.e., p = 4), the overall type I error probability (α′) for the joint control procedure is

If the variables are not independent, the magnitude of the type I error will be difficult to obtain. In practice, a control ellipse should be chosen so that the probability of the sample means being plotted within the elliptical region when the process is in control is 1 − α, where α is the desired overall probability of a type I error.

Hotelling's T² Control Chart and Its Variations

Suppose that we have two quality characteristics, X₁ and X₂, distributed jointly according to a bivariate normal distribution. Assume that the target mean values of the characteristics are represented by and , respectively. Let the sample means be and , with sample variances and , and the covariance between the two variables be represented by s₁₂ for a sample of size n. Under these conditions, the statistic

(7-89)

is distributed according to Hotelling's T²-distribution with 2 and (n − 1) degrees of freedom (Hotelling 1947). The 2 in this case comes from the two characteristics being considered, and the (n − 1) is the degrees of freedom associated with the sample variance. If the calculated value of T² given by eq. (7-89) exceeds , the point on the T²- distribution such that the proportion to the right is α, then at least one of the characteristics is out of control.

This procedure can be shown graphically. Equation (7-89) represents the control ellipses shown in Figure 7-32. If the variables are independent, the covariance between them is zero (i.e., s₁₂ = 0), the control ellipse is A, and the joint control region is represented by the area within the control ellipse A. If a plot of the bivariate means () falls within this control region, we can assume a state of statistical control. If the two variables are positively correlated, then s₁₂ > 0, and the control ellipse is similar to ellipse C. If the variables are negatively correlated, then s₁₂ < 0, and the control ellipse will be similar to ellipse B.

Hotelling's control ellipse procedure has several disadvantages. First, the time sequence of the plotted points () is lost. This implies that we cannot check for runs in the plotted pattern as with control charts. Second, the construction of the control ellipse becomes quite difficult for more than two characteristics. To overcome these disadvantages, the values of T² given by eq. (7-89) are plotted on a control chart on a sample-by-sample basis to preserve the time order in which the data values are obtained. Such a control chart has an upper control limit of , where p represents the number of characteristics. Patterns of nonrandom runs can be investigated in such plots.

Values of Hotelling's T² percentile points can be obtained from the percentile points of the F-distribution given in Appendix A-6 by using the relation

(7-90)

where represents the point on the F-distribution such that the area to the right is α, with p degrees of freedom in the numerator and (n − p) degrees of freedom in the denominator.

If more than two characteristics are being considered, the value of T² given by eq. (7-89) for a sample can be generalized as

(7-91)

where represents the vector of sample means of p characteristics for a sample of size n, represents the vector of target values for each characteristic, and ∑ denotes the variance–covariance matrix of the p quality characteristics.

Phase 1 and Phase 2 Charts

In multivariate control charts, the process of determining control limits from an in-control process and, thereby, using those control limits to detect a change in the process parameter, for example, the process mean, is usually conducted in two phases. In phase 1, assuming that special causes do not exist upon taking remedial actions, if necessary, based on the observations from the in-control process, estimates of the process mean μ₀ and process variance–covariance matrix Σ are obtained.

Suppose that, for an in-control process, we have m samples, each of size n, with the number of characteristics being p. The vector of sample means is given by

where represents the sample mean of the ith characteristic for the jth sample and is found from

(7-92)

where represents the value of the kth observation of the ith characteristic in the jth sample. The sample variances for the ith characteristic in the jth sample are given by

(7-93)

The covariance between characteristics i and h in the jth sample is calculated from

(7-94)

The vector μ₀ of the process means of each characteristic for m samples is estimated as

(7-95)

The elements of the variance–covariance matrix Σ in eq. (7-91) are estimated from the following average for m samples:

(7-96)

and

(7-97)

Finally, the matrix Σ is estimated using S as follows (only the upper diagonal part is shown because the matrix is symmetric):

(7-98)

Phase 1 Control Limits

The upper control limit of the T²-chart given by eq. (7-90) can be modified to take the following form (Alt 1982):

(7-99)

where m represents the number of samples, each of size n, used to estimate and S. The value of T² for each of the m samples is calculated using the estimated statistic

(7-100)

and is then compared to the UCL given by eq. (7-99). If the value of T² for the jth sample (i.e., ) is above the UCL, it is treated as an out-of-control point, and investigative action is begun.

Phase 2 Control Limits

After out-of-control points, if any, are deleted assuming adequate remedial actions are taken, the procedure is repeated until all retained observations are in control.

Now, phase 2 of the procedure is used for monitoring future observations from the process. Let us denote the number of samples retained at the end of phase 1 by m, each containing n observations. The upper control limit for the T² control chart in phase 2 is given by

(7-101)

Usage and Interpretations

A Hotelling's control chart is constructed using the upper control limit and the plotted values of T² for each sample given by eq. (7-100), where the vector and the matrix S are found using the preceding procedure. A sample value of T² above the upper control limit indicates an out-of-control situation. How do we determine which quality characteristic caused the out-of-control state?

Even with only two characteristics (p = 2), the situation can be complex. If the two quality characteristics are highly positively correlated, we expect the averages for each characteristic in the sample to maintain the same relationship relative to the process average . For example, in the jth sample, if , we could expect . Similarly, if , we would expect , which would confirm that the sample averages for each characteristic move in the same direction relative to their means.

If the two characteristics are highly positively correlated and , we would not expect that . However, should this occur, this sample may show up as an out-of-control point in Hotelling's T² procedure, thereby indicating that the bivariate process is out of control. This same inference can be made using individual 3σ control limit charts constructed for each characteristic if exceeds or exceeds . However, individual quality characteristic means can plot within the control limits on separate control charts even though the T² plots above the UCL on the joint control chart. Using joint control charts for characteristics that need to be considered simultaneously is thus advantageous. However, note that an individual chart for a quality characteristic can sometimes indicate an out-of-control condition when the joint control chart does not.

In general, larger sample sizes are needed to detect process changes with positively correlated characteristics than with negatively correlated characteristics. Furthermore, for highly positively correlated characteristics, larger sample sizes are needed to detect large positive shifts in the process means than to detect small positive shifts.

Generally speaking, if an out-of-control condition is detected by a Hotelling's control chart, individual control intervals are calculated for each characteristic for that sample. If the probability of a type I error for a joint control procedure is α, then for sample j, the individual control interval for the ith quality characteristic is

(7-102)

where and are given by eqs. (7-95) and (7-96), respectively. If falls outside this interval, the corresponding characteristic should be investigated for a lack of control. If special causes are detected, the sample that contains information relating to all the characteristics should be deleted when the upper control limit is recomputed.

As described previously, even though the T² control chart is useful in detecting shifts in the process mean vector, it does not identify which specific variables(s) are responsible. One approach, in this context, is the T² decomposition method. The concept is to determine the individual contributions of each of the p variables, or combinations thereof, to the overall T²-statistic. These individual contributions or partial T²-statistics are found as follows:

(7-103)

where denotes the T²-statistic when the ith variable is left out. Large values of D_i will indicate a significant impact of variable i for the particular observation under investigation.

Individual Observations with Unknown Process Parameters

The situation considered previously dealt with subgroups of data, where the sample size (n) for each subgroup exceeds 1. In this section we consider individual observations and assume that the process parameters, mean vector or the elements of the variance–covariance matrix, are unknown. As before, we use the two-phase approach, where in phase 1 we use the preliminary data to retain observations in control.

The value of T², when individual observations are obtained, is given by

(7-104)

In eq. (7-104), the process mean vector is estimated from the observations by , while the process variance–covariance matrix is estimated, using the data, by S. The upper control limit in this situation, in phase 1, is given by

(7-105)

where denotes the upper αth quantile of the beta distribution with parameters p/2 and (m − p − 1)/2.

If an observation vector has a value of T², given by eq. (7-104), that exceeds the value of UCL, given by eq. (7-105), it is deleted from the preliminary data set. Revised estimates of the process mean vector and variance–covariance matrix elements are found using the remaining observations and the process is repeated until no further observations are deleted. We now proceed to phase 2 to monitor future observations. The estimates and S obtained at the end of phase 1 are used to calculate T², using eq. (7-104), for new observations. Assuming that the number of observations retained at the end of phase 1 is given by m, the upper control limit for phase 2 is obtained as

(7-106)

Hence, values of T² for new observations will be compared with the UCL given by eq. (7-106) to determine out-of-control conditions.

Generalized Variance Chart

The multivariate control charts discussed previously dealt with monitoring the process mean vector. Here, we introduce a procedure to develop a multivariate dispersion chart to monitor process variability based on the sample variance–covariance matrix S. A measure of the sample generalized variance is given by |S|, the determinant of the sample variance–covariance matrix.

Denoting the mean and variance of |S| by E(|S|) and V(|S|), respectively, and using the property that most of the probability distribution of |S| is contained in the interval , expressions for the parameters of the control chart for |S| may be obtained. It is known that

(7-107)

(7-108)

where Σ represents the process variance–covariance matrix, and

(7-109)

(7-110)

Since Σ is usually unknown, it is estimated based on sample information. From eq. (7-107), an unbiased estimator of |Σ| is |S|/b₁. Using eqs. (7-107) and (7-108), the centerline and control limits for the |S| chart are given by

(7-111)

When a target value for Σ, say Σ₀, is specified, |Σ| is replaced by |Σ₀| in eq. (7-111). Alternatively, the sample estimate of |Σ| given by |S|/b₁ will be used to compute the centerline and control limits in eq. (7-111). In the event that the LCL from eq. (7-111) is computed to be less than zero, it is converted to zero.

For a given sample j, |S_j|, the determinant of the variance–covariance matrix for sample j, is computed and plotted on the generalized variance chart. If the plotted value of |S_j| is outside the control limits, we flag the process and look for special causes.

Even though the generalized sample variance chart is useful, as it aggregates the variability of several variables into one index, it has to be used with caution. This is because many different S_j matrices may give the same value of |S_j|, while the variance structure could be quite different. Hence, a univariate range (R) chart or standard deviation (s) chart may help us understand the variables that contribute to make the combined impact on the generalized variance to be significant.