The data for this example are the same as for Example 61.1, 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 61.2.1 through Output 61.2.5.

The “Model Information” table in Output 61.2.1 corresponds to the specifications in the invocation of PROC MULTTEST.

Output 61.2.1: FT and t tests with Bootstrap Resampling

The Multtest Procedure

Model Information | |
---|---|

Test for discrete variables | Freeman-Tukey |

Test for continuous variables | Mean t-test |

Degrees of Freedom Method | Pooled |

Tails for discrete tests | Lower-tailed |

Tails for continuous tests | Lower-tailed |

Strata weights | None |

P-value adjustment | Bootstrap |

Center continuous variables | Yes |

Number of resamples | 10000 |

Seed | 37081 |

The “Contrast Coefficients” table in Output 61.2.2 shows the coefficients from the CONTRAST statement after centering, and they model a linear trend.

Output 61.2.2: Contrast Coefficients

Contrast Coefficients | ||||
---|---|---|---|---|

Contrast | Dose | |||

1 | 2 | 3 | ||

Linear Trend | Centered | -1 | 0 | 1 |

The summary statistics are displayed in Output 61.2.3. The values for the discrete variables `S1`

and `S2`

are the same as those from Example 61.1. 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.

Output 61.2.3: Summary Statistics

Discrete Variable Tabulations | ||||
---|---|---|---|---|

Variable | Dose | Count | NumObs | Percent |

S1 | 1 | 2 | 6 | 33.33 |

S1 | 2 | 4 | 6 | 66.67 |

S1 | 3 | 4 | 6 | 66.67 |

S2 | 1 | 4 | 6 | 66.67 |

S2 | 2 | 2 | 6 | 33.33 |

S2 | 3 | 2 | 6 | 33.33 |

Continuous Variable Tabulations | ||||
---|---|---|---|---|

Variable | Dose | NumObs | Mean | Standard Deviation |

T | 1 | 6 | 99.3333 | 9.6056 |

T | 2 | 6 | 87.5000 | 14.4326 |

T | 3 | 6 | 72.8333 | 22.7017 |

The p-values, displayed in Output 61.2.4, are from the Freeman-Tukey test for `S1`

and `S2`

, and are from the t test for `T`

.

Output 61.2.4: p-Values

p-Values | |||
---|---|---|---|

Variable | Contrast | Raw | Bootstrap |

S1 | Linear Trend | 0.8547 | 1.0000 |

S2 | Linear Trend | 0.1453 | 0.4605 |

T | Linear Trend | 0.0070 | 0.0281 |

The Raw column in Output 61.2.4 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 61.2.5 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.

Output 61.2.5: Resampling Data Set

Obs | _sample_ | _class_ | _obs_ | S1 | S2 | T |
---|---|---|---|---|---|---|

1 | 1 | 1 | 17 | 0 | 1 | 26.1667 |

2 | 1 | 1 | 8 | 1 | 0 | -27.5000 |

3 | 1 | 1 | 5 | 0 | 1 | 0.6667 |

4 | 1 | 1 | 9 | 0 | 1 | 1.5000 |

5 | 1 | 1 | 7 | 1 | 0 | -2.5000 |

6 | 1 | 1 | 3 | 0 | 1 | 4.6667 |

7 | 1 | 2 | 12 | 1 | 0 | 11.5000 |

8 | 1 | 2 | 12 | 1 | 0 | 11.5000 |

9 | 1 | 2 | 14 | 1 | 0 | -22.8333 |

10 | 1 | 2 | 17 | 0 | 1 | 26.1667 |

11 | 1 | 2 | 1 | 0 | 1 | 4.6667 |

12 | 1 | 2 | 15 | 1 | 0 | 7.1667 |

13 | 1 | 3 | 4 | 0 | 1 | 4.6667 |

14 | 1 | 3 | 17 | 0 | 1 | 26.1667 |

15 | 1 | 3 | 14 | 1 | 0 | -22.8333 |

16 | 1 | 3 | 15 | 1 | 0 | 7.1667 |

17 | 1 | 3 | 15 | 1 | 0 | 7.1667 |

18 | 1 | 3 | 6 | 1 | 0 | 4.6667 |

19 | 2 | 1 | 6 | 1 | 0 | 4.6667 |

20 | 2 | 1 | 17 | 0 | 1 | 26.1667 |

21 | 2 | 1 | 8 | 1 | 0 | -27.5000 |

22 | 2 | 1 | 13 | 1 | 0 | -12.8333 |

23 | 2 | 1 | 9 | 0 | 1 | 1.5000 |

24 | 2 | 1 | 8 | 1 | 0 | -27.5000 |

25 | 2 | 2 | 9 | 0 | 1 | 1.5000 |

26 | 2 | 2 | 18 | 1 | 0 | -22.8333 |

27 | 2 | 2 | 15 | 1 | 0 | 7.1667 |

28 | 2 | 2 | 14 | 1 | 0 | -22.8333 |

29 | 2 | 2 | 9 | 0 | 1 | 1.5000 |

30 | 2 | 2 | 17 | 0 | 1 | 26.1667 |

31 | 2 | 3 | 16 | 0 | 1 | 25.1667 |

32 | 2 | 3 | 11 | 0 | 1 | 8.5000 |

33 | 2 | 3 | 14 | 1 | 0 | -22.8333 |

34 | 2 | 3 | 18 | 1 | 0 | -22.8333 |

35 | 2 | 3 | 18 | 1 | 0 | -22.8333 |

36 | 2 | 3 | 10 | 1 | 0 | 8.5000 |

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`

.