Applied Statistics and Probability for Engineers

13-2.4 More About Multiple Comparisons

As noted in the previous section, there are many ways to investigate the treatment means

following rejection of the null hypothesis with an ANOVA. The Fisher LSD method is easy

and very widely used. It is consider to be a very “liberal” procedure in that although each

test is at significance level , the type I error for the entire set of comparisons (called the

experimentwise error rate) is much greater than . In this section we briefly describe three

other approaches.

Graphical Comparison of Means

It is easy to compare treatment means graphically, following the analysis of variance. Suppose

that the factor has alevels and that are the observed averages for these factor

levels. Each treatment average has standard deviation , where is the standard devia-

tion of an individual observation. If all treatment means are equal, the observed means

would behave as if they were a set of observations drawn at random from a normal distribu-

tion with mean and standard deviation.

Visualize this normal distribution capable of being slid along an axis below which the

treatment means are plotted. If all treatment means are equal, there should

be some position for this distribution that makes it obvious that the values were drawn

from the same distribution. If this is not the case, the values that do not appear to have

been drawn from this distribution are associated with treatments that produce different

mean responses.

The only flaw in this logic is that is unknown. However, we can use from

the analysis of variance to estimate . This implies that a t-distribution should be used

instead of the normal in making the plot, but since the tlooks so much like the normal,

sketching a normal curve that is approximately units wide will usually work

very well.

Figure S13-1 shows this arrangement for the hardwood concentration experiment in

Example 13-1. The standard deviation of this normal distribution is

If we visualize sliding this distribution along the horizontal axis, we note that there is no lo-

cation for the distribution that would suggest that all four observations (the plotted means) are

typical, randomly selected values from that distribution. This, of course, should be expected,

because the analysis of variance has indicated that the means differ, and the display in

Fig. S13-1 is just a graphical representation of the analysis of variance results. The figure does

1 MSEn 1 6.51 6 1.04

61 MSEn

1 MSE

yi.

yi.

y 1 ., y 2 .,p, ya.

 1 n

yi.

 1 n

y 1 ., y 2 .,p, ya.

0 5 10 15 20 25 30

σ/√n = 1.04

1234

∧

Figure S13-1 Tensile strength averages from the hardwood concentration

experiment in relation to a normal distribution with standard deviation

1 MSEn 1 6.51 6 1.04.

13-1

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