Applied Statistics and Probability for Engineers

2.086. Minitab implements Fisher’s LSD method by computing confidence intervalson all

pairs of treatment means using Equation 13-12. The lower and upper 95% confidence limits

are shown at the bottom of the table. Notice that the only pair of means for which the

confidence interval includes zero is for  10 and  15. This implies that  10 and  15 are not

significantly different, the same result found in Example 13-2.

Table 13-5 also provides a “family error rate,” equal to 0.192 in this example. When all

possible pairs of means are tested, the probability of at least one type I error can be much

greater than for a single test. We can interpret the family error rate as follows. The probability

is 10.1920.808 that there are no type I errors in the six comparisons. The family error

rate in Table 13-5 is based on the distribution of the range of the sample means. See

Montgomery (2001) for details. Alternatively, Minitab permits you to specify a family error

rate and will then calculate an individual error rate for each comparison.

13-2.4 More About Multiple Comparisons (CD Only)

13-2.5 Residual Analysis and Model Checking

The analysis of variance assumes that the observations are normally and independently dis-

tributed with the same variance for each treatment or factor level. These assumptions should

be checked by examining the residuals. A residualis the difference between an observation yij

and its estimated (or fitted) value from the statistical model being studied, denoted as. For

the completely randomized design and each residual is , that is, the dif-

ference between an observation and the corresponding observed treatment mean. The residuals

for the paper tensile strength experiment are shown in Table 13-6. Using to calculate each

residual essentially removes the effect of hardwood concentration from the data; consequently,

the residuals contain information about unexplained variability.

The normality assumption can be checked by constructing a normal probability plotof

the residuals. To check the assumption of equal variances at each factor level, plot the residu-

als against the factor levels and compare the spread in the residuals. It is also useful to plot the

residuals against (sometimes called the fitted value); the variability in the residuals should

not depend in any way on the value of. Most statistics software packages will construct

these plots on request. When a pattern appears in these plots, it usually suggests the need for

a transformation, that is, analyzing the data in a different metric. For example, if the variabil-

ity in the residuals increases with , a transformation such as log yor should be consid-

ered. In some problems, the dependency of residual scatter on the observed mean is very

important information. It may be desirable to select the factor level that results in maximum

response; however, this level may also cause more variation in response from run to run.

The independence assumption can be checked by plotting the residuals against the time

or run order in which the experiment was performed. A pattern in this plot, such as sequences

of positive and negative residuals, may indicate that the observations are not independent.

yi.

yi. 1 y

yi

yi.

yi.

yˆijyi. eijyijyi.

yˆij

13-2 THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT 481

Table 13-6 Residuals for the Tensile Strength Experiment

Hardwood

Concentration (%) Residuals

5 3.00 2.00 5.00 1.00 1.00 0.00

10 3.67 1.33 2.67 2.33 3.33 0.67

15 3.00 1.00 2.00 0.00 1.00 1.00

20 2.17 3.83 0.83 1.83 3.17 1.17

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