Checking Model Assumptions
5.6 Checking the Equal Variance Assumption
5.6.2 Data Transformations to Equalize Variances
Finding a transformation of the data to equalize the variances of the error variables involves finding some functionh(yi t)of the data so that the model
h(Yi t)=μ∗+τi∗+i t∗
holds and∗i t ∼N(0,σ2)and the∗i t’s are mutually independent for allt=1, . . . ,riandi=1, . . . , v.
An appropriate transformation can generally be found if there is a clear relationship between the error varianceσ2i =Var(i t)and the mean responseE[Yi t] =μ+τi, fori =1, . . . , v. If the variance and the mean increase together, as suggested by the megaphone-shaped residual plot in Fig.5.6, or if one increases as the other decreases, then the relationship betweenσ2i andμ+τiis often of the form
σ2i =k(μ+τi)q, (5.6.2)
wherekandqare constants. In this case, the functionh(yi t)should be chosen to be
h(yi t)=
⎧⎨
⎩
(yi t)1−(q/2)ifq =2,
ln(yi t) ifq =2 and allyi t’s are nonzero, ln(yi t+1) ifq =2 and someyi t’s are zero.
(5.6.3)
Here “ln” denotes the natural logarithm, which is the logarithm to the basee. Usually, the value ofq is not known, but a reasonable approximation can be obtained empirically as follows. Substituting the least squares estimates for the parameters into Eq. (5.6.2) and taking logs of both sides gives
ln(si2)=ln(k)+q(ln(yi.)) .
Therefore, the slope of the line obtained by plotting ln(si2)against ln(yi.)gives an estimate forq. This will be illustrated in Example5.6.2.
The value ofq is sometimes suggested by theoretical considerations. For example, if the normal distribution assumed in the model is actually an approximation to the Poisson distribution, then the variance would be equal to the mean, andq =1. The square-root transformation h(yi t)= (yi t)1/2 would then be appropriate. The binomial distribution provides another commonly occurring case for which an appropriate transformation can be obtained theoretically. If eachYi thas a binomial distribution with meanmpand variancemp(1−p), then a variance-stabilizing transformation is
h(yi t)=sin−1
yi t/m=arcsin yi t/m
.
When a transformation is found that equalizes the variances, then it is necessary to check or recheck the other model assumptions, since a transformation that cures one problem could cause others. If there are no problems with the other model assumptions, then analysis can proceed using the techniques of the previous two chapters, but using the transformed datah(yi t).
Example 5.6.2 Choosing a transformation: battery experiment
In Sect.2.5.2, the response variable considered for the battery experiment was “battery life per unit cost,” and a plot of the residuals versus the fitted values looks similar to Fig.5.3and shows fairly constant error variances.
5.6 Checking the Equal Variance Assumption 113
Table 5.3 Life data for the battery experiment
Battery Lifetime (minutes) yi. si2
1 602 529 534 585 562.50 1333.71
2 863 743 773 840 804.75 3151.70
3 232 255 200 215 225.50 557.43
4 235 282 238 228 245.75 601.72
Fig. 5.7 Residual plot for the battery life data
3 2 1 0 1 2 3
y^
it
zit
225 425 625 825
Type 1 Type 2 Type 3 Type 4
Suppose, however, that the response variable of interest had been “battery life” regardless of cost.
The corresponding data are given in Table5.3. The battery types are 1=alkaline, name brand
2=alkaline, store brand 3=heavy duty, name brand 4=heavy duty, store brand
Figure5.7shows a plot of the standardized residuals versus the fitted values. Variability seems to be increasing modestly with mean response, suggesting that a transformation can be found to stabilize the error variance. The ratio of extreme variance estimates issmax2 /smin2 =s22/s32=3151.70/557.43≈ 5.65. Hence, based on the rule of thumb, a variance stabilizing transformation should be used. Using the treatment sample means and variances from Table5.3, we have
i yi. ln(yi.) si2 ln(si2)
1 562.50 6.3324 1333.71 7.1957
2 804.75 6.6905 3151.70 8.0557
3 225.50 5.4183 557.43 6.3233
4 245.75 5.5043 601.72 6.3998
Figure5.8shows a plot of ln(si2)against ln(yi.). This plot is nearly linear, so the slope will provide an estimate of q in (5.6.2). A line can be drawn by eye or by the regression methods of Chap.8.
Both methods give a slope approximately equal toq =1.25. From Eq. (5.6.3) a variance-stabilizing transformation is
h(yi t)=(yi t)0.375.
114 5 Checking Model Assumptions
Fig. 5.8 Plot of ln(s2i) versus ln(yi.)for the battery life data
ln(yi.) ln(si
2)
5.3 5.7 6.1 6.5 6.9
6.2 6.7 7.2 7.7 8.2
Table 5.4 Transformed life data√yi tfor the battery experiment
Brand xi t=h(yi t)= √yi t xi. si2
1 24.536 23.000 23.108 24.187 23.708 0.592
2 29.377 27.258 27.803 28.983 28.355 0.982
3 15.232 15.969 14.142 14.663 15.001 0.614
4 15.330 16.793 15.427 15.100 15.662 0.587
Since (yi t)0.375 is close to (yi t)0.5, and since the square root of the data values is perhaps more meaningful than(yi t)0.375, we will try taking the square root transformation. The square roots of the data are shown in Table5.4.
The transformation has stabilized the variances considerably, as evidenced by s2max/smin2 = 0.982/0.587 ≈ 1.67. Checks of the other model assumptions for the transformed data also reveal no severe problems. The analysis can now proceed using the transformed data. The stated significance level and confidence levels will now be approximate, since the model has been changed based on the data. For the transformed data,msE = 0.6936. Using Tukey’s method of multiple comparisons to compare the lives of the four battery types (regardless of cost) at an overall confidence level of 99%, the minimum significant difference obtained from Eq. (4.4.28) is
msd=q4,12,0.01
msE/4=5.50
0.6936/4=2.29.
Comparing msdwith the differences in the sample means xi. of the transformed data in Table5.4, we can conclude that at an overall 99% level of confidence, all pairwise differences are significantly different from zero except for the comparison of battery types 3 and 4. Furthermore, it is reasonable to conclude that type 2 (alkaline, store brand) is best, followed by type 1 (alkaline, name brand).
However, any more detailed interpretation of the results is muddled by use of the transformation, since the comparisons use mean values of√
life. A more natural transformation, which also provided approximately equal error variances, was used in Sect.2.5.2. There, the response variable was taken to be “life per unit cost,” and confidence intervals were able to be calculated in meaningful units.
5.6 Checking the Equal Variance Assumption 115