Example 58.2 Freeman-Tukey and t Tests with Bootstrap Resampling

The data for this example are the same as for Example 58.1, except that a continuous variable T, which indicates the time of death of the animal, has been added.

data a;
   input S1 S2 T Dose @@;
   datalines;
0 1 104 1   1 0  80 1   0 1 104 1
0 1 104 1   0 1 100 1   1 0 104 1
1 0  85 2   1 0  60 2   0 1  89 2
1 0  96 2   0 1  96 2   1 0  99 2
1 0  60 3   1 0  50 3   1 0  80 3
0 1  98 3   0 1  99 3   1 0  50 3
;

proc multtest data=a bootstrap nsample=10000 seed=37081 outsamp=res;
   test ft(S1 S2 / lowertailed) mean(T / lowertailed);
   class Dose;
   contrast 'Linear Trend' 0 1 2;
run;

proc print data=res(obs=36);
run;

The BOOTSTRAP option in the PROC MULTTEST statement requests bootstrap resampling, and NSAMPLE=10000 requests 10,000 bootstrap samples. The SEED=37081 option provides a starting value for the random number generator. The OUTSAMP=res option creates an output SAS data set res containing the 10,000 bootstrap samples.

The TEST statement specifies the Freeman-Tukey test for S1 and S2 and specifies the t test for T. Both tests are lower-tailed. The grouping variable in the CLASS statement is Dose, and the coefficients across the levels of Dose are 0, 1, and 2, as specified in the CONTRAST statement. The PROC PRINT statement displays the first 36 observations of the res data set containing the bootstrap samples.

The results from this analysis are listed in Output 58.2.1 through Output 58.2.5.

The "Model Information" table in Output 58.2.1 corresponds to the specifications in the invocation of PROC MULTTEST.

The "Contrast Coefficients" table in Output 58.2.2 shows the coefficients from the CONTRAST statement after centering, and they model a linear trend.

The summary statistics are displayed in Output 58.2.3. The values for the discrete variables S1 and S2 are the same as those from Example 58.1. The mean, standard deviation, and sample size for the continuous variable T at each level of Dose are displayed in the "Continuous Variable Tabulations" table.

The p-values, displayed in Output 58.2.4, are from the Freeman-Tukey test for S1 and S2, and are from the t test for T.

The Raw column in Output 58.2.4 contains the results from the tests on the original data, while the Bootstrap column contains the bootstrap resampled adjustment to raw_p. Note that the adjusted p-values are larger than the raw p-values for all three variables. The adjusted p-values more accurately reflect the correlation of the raw p-values, the small size of the data, and the multiple testing.

Output 58.2.5 displays the first 36 observations of the SAS data set resulting from the OUTSAMP=RES option in the PROC MULTTEST statement. The entire data set has 180,000 observations, which is 10,000 times the number of observations in the data set.

The _sample_ variable is the sample indicator and _class_ indicates the resampling group—that is, the level of the CLASS variable Dose assigned to the new observation. The number of the observation in the original data set is represented by _obs_. Also listed are the values of the original test variables, S1 and S2, and the mean-centered values of T.