Example. Polyamine in Children  Source: Neter e.g. p 129, 
Table 3.8 p130

Data on age (X) and plasma level of a polyamine (Y) for a portion 
of the 25 healthy children in a study are presented below. These 
data are plotted in Figure 3.16a as a scatter plot. Note the 
distinct curvilinear regression relationship, as well as the 
greater variability for younger children than for older ones.

Based on the prototype regression pattern in Figure 3.15b, we 
shall first try the logarithmic transformation Y' = log10Y. The 
transformed Y values are shown in column 3 of the Table. Figure 
3.16b contains the scatter plot with this transformation. Note 
that the transforrnation not only has led to a reasonably linear 
regression relation, but the variability at the different levels 
of X also has become reasonably constant.

To further examine the reasonableness of the transformation Y' = 
log1OY, we fitted the simple linear regression model (2.1) to the 
transformed Y data and obtained:

   Y'= 1.135-.1023X

A plot of the residuals against X is shown in Figure 3.16c, and a 
normal probability plot of the residuals is shown in Figure 3.16d. 
The coefficient of correlation between the ordered residuals and 
their expected values under normality is 0.981. For a = .05, Table 
B.6 indicates that the critical value is 0.959 so that the 
observed coefficient supports the assumption of normality of the 
error terms. All of this evidence supports the appropriateness of 
regression model (2.1) for the transfonned Y data.

If it is desired to express the estimated regression function in 
the original units of Y, we simply take the antilog of Y'hat 
and obtain:

   Yhat = antilog10(l.l35 Ð 0.1023X) 

Comments 

1
At times it may be desirable to introduce a constant into a 
transformation of Y, such as when Y may be negative. For 
instance, the logarithmic transformation to shift the origin in 
Y and make all Y observations positive would be 
  Y' = loglO(Y + k), 
where k is an appropriately chosen constant.

2
When unequal error variances are present but the regression 
relation is linear, a transformation on Y may not be sufficient. 
While such a transformation rnay stabilize the error variance, 
it will also change the linear relationship to a curvilinear 
one. A transformation on X may therefore also be required. This 
case can also be handled by using weighted least squares, a 
procedure explained in Chapter 10. 


Age Level  Log10[level]

 0  13.44   1.1284
 0  12.84   1.1086
 0  11.91   1.0759
 0  20.09   1.3030
 0  15.60   1.1931
 1  10.11   1.0048
 1  11.38   1.0561
 1  10.28   1.0120
 1   8.96    .9523
 1   8.59    .9340
 2   9.83    .9926
 2   9.00    .9542
 2   8.65    .9370
 2   7.85    .8949
 2   8.88    .9484
 3   7.94    .8998
 3   6.01    .7789
 3   5.14    .7110
 3   6.90    .8388
 3   6.77    .8306
 4   4.86    .6866
 4   5.10    .7076
 4   5.67    .7536
 4   5.75    .7597
 4   6.23    .7945