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Example 3
Cross-over Clinical Trial
The data below are taken from a three-treatment, three-period cross-over clinical trial. The three drugs are A=New Drug, B=Aspirin, C=Placebo. The primary end-point was analgesic efficacy, here dichotomized as 0 for relief and 1 for no-relief, during periods P1, P2, and P3 respectively. See Snapinn and Small (1986) for details.
| Patient |
Drug Sequence |
Response |
| |
|
P1 |
P2 |
P3 |
| 1 |
ABC |
0 |
1 |
1 |
| 7 |
ABC |
0 |
1 |
1 |
| 2 |
BCA |
0 |
1 |
1 |
| 8 |
BCA |
0 |
0 |
0 |
| 3 |
CAB |
1 |
0 |
0 |
| 9 |
CAB |
1 |
0 |
1 |
| 4 |
CBA |
1 |
0 |
1 |
| 10 |
CBA |
1 |
0 |
0 |
| 5 |
ACB |
0 |
0 |
0 |
| 11 |
ACB |
0 |
1 |
0 |
| 6 |
BAC |
1 |
0 |
0 |
| 12 |
BAC |
0 |
0 |
1 |
A logistic regression model of the form:
RESPONSE = DRUG + PERIOD
in which DRUG and PERIOD are each treated as 2 degree of freedom factor variables (i.e., DRUG consists of the two indicator variables A versus C and B versus C, while PERIOD consists of the two indicator variables P1 versus P3 and P2 versus P3) is appropriate for modelling the response. However the analysis must take into account the fact that each patient provides three binary responses. In other words, this is a repeated measures data set. The usual large-sample techniques for handling repeated measures data are unreliable here since the data set is rather small. We solve the problem by treating each patient as a separate stratum or matched-set, and using the stratified logistic regression option of LogXact. The question of interest is whether the three treatments are different. We answer this question by performing a 2 degree of freedom test on DRUG in the above stratified logistic regression model. Both exact and asymptotic tests were performed and the results are tabulated below:
| Type of Test |
Chi Squared Value |
P-value |
| Likelihood Ratio |
8.74 |
.013 |
| Bivariate Wald |
5.09 |
.079 |
| Unconditional Scores |
7.8 |
.020 |
| Exact |
7.06 |
.029 |
Notice that there are variations among the three asymptotic tests (Likelihood ratio, Wald, and Scores), even though all three tests are supposed to be equivalent asymptotically. This suggests that the asymptotics are performing poorly. What conclusion are we to draw about the treatment effect, when the Wald test is not statistically significant but the Likelihood Ratio and Scores tests are? The exact test reports a p-value of 0.029 and resolves the dilemma.
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