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StatXact Example 1
Meta Analysis in StatXact
Introduction
In meta analysis one is usually interested in asking the following types of questions. Is it appropriate to aggregate the data from several studies, and estimate a single summary measure of treatment effect? If so, what is the estimate of the summary measure. Is the estimate statistically significant? The StatXact package makes it extremely convenient to answer these questions for studies involving dichotomous outcomes, under the assumption of a fixed-effects model. Suppose the objective is to aggregate results from K independent studies with binary end points, each comparing a treatment group and a control group. For study k, let tk be the probability of a response in the treatment group, and tk be the probability of a response in the control group. Then the odds ratio for study k is defined as:
StatXact enables you to make the following types of inferences about these odds ratios:
- Exact test of homogeneity of odds ratios across K studies (Zelen, Biometrika, 58, 1971).
- Asymptotic test of homogeneity of odds ratios across K studies (Breslow and Day, IARC Scientific Publications, 1980).
- Exact test and exact confidence interval for the common odds ratio across K studies (Menta, Patel, Gray, JASA, 1985).
- Asymptotic test and asymptotic confidence interval for the common odds ratio across K studies (Mantel and Haenszel, JNCI, 1959).
The first two statistical procedures determine whether it is appropriate to pool the results of the K studies under the fixed effects model. In situations where such pooling is appropriate, the last two statistical procedures determine if tlie treatment group is significantly different from the control group, and estimate the magnitude of the difference on the odds ratio scale.
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Example:
Does taking an anti-arrhythmic drug instead of a placebo affect the chance of death?
Adams et al. (Draft Manuscript, Technology Assessment Group, Harvard School of Public Health, 1990) conducted a meta analysis of placebo-controlled randomized clinical trials of anti-arrhythmic drugs. The results from 4 such trials are shown below. A patient is said to have responded if death occurred within one year.These data can be entered directly into StatXact through its convenient Table editor.
|
Drug |
Placebo |
| Study Number |
No Response |
Response |
No Response |
Response |
| 1 |
82 |
3 |
37 |
0 |
| 2 |
140 |
3 |
45 |
0 |
| 3 |
42 |
0 |
43 |
0 |
| 4 |
1 |
0 |
11 |
0 |
First we test whether the odds ratios for the above 4 studies are the same. The asymptotic test fails to provide any answer, because all the placebo responses are zero. However the exact test produces a p-value of 1, implying that there is no evidence whatsoever to reject the null hypothesis of a common odds ratio. Hence it is appropriate to pool the four studies.
The next step is to calculate the p-value for the null hypothesis that the common odds ratio is 1 (there is no difference between the drugs and the placebo). The p-values and the 95% confidence interval for the common odds ratio are tabulated below:
| Method |
P-value |
95% Interval |
| Mantel-Haenszel |
undefined |
undefined |
| Exact |
0.293 |
(0.44,+Inf) |
| Mid-p |
0.195 |
(0.58,+Inf) |
Again the fact that all the placebo responses are zero causes the asymptotic (Mantel-Haenszel) method to fail. However the exact method produces a p-value of 0.293, supporting the null hypothesis that there is no difference between the drugs and the placebo in terms of response, and rules out, with 95% confidence, the possibility that the odds of response for the treatment group are any lower than 44% of the odds of response for the control group. The mid-p method is less conservative, producing a p-value of 0.195 (leading to the same conclusion as the exact) and a lower bound of 58% on the odds ratio.
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