# Combining several survival experiments

In four independent experiments, two strains (A and B) were compared in their respective survival curves. Two experiments show a clear-cut result (A survives at 100%, almost half B dies). Two further experiments show no significance (log-rank / Mantel-Cox test and Breslow/ Gehan-Wilcoxon), but even in these insignificant experiments strain A always had 100% survival and all individuals that die belong to strain B.

Unfortunately, I have no experience with survival tests. How could one combine the results of the four experiments other than just pool the data? Pooling the data seems to me like a wrong choice, because there clearly is an effect of the experiment.

To make the matter more complex, there is a parameter D (dosage) that varied slightly between the four experiments. It looks (without testing) that there is an effect of dosage on survival. How should one test it?

(the current solution proposed by the experimentalists is to repeat the experiments with more individuals)

EDIT: What I did until now was to combine the p-values from the experiments using Stouffer's method (Fisher's method gives an even lower p-value). The combined p-value in Stouffer's test is $< 10^{-4}$.

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If by pooling the data you mean simply collapsing the data set together, that is total wrong and is just a case of Simpson’s paradox. Unfortunately often what people mean by pooling refers to pooling estimates or effect sizes and that can be justified – so there is possibility for confusion.

The difficult judgement for something like this, is deciding whether a defensible likelihood can be defined that adequately reflects the commonalities and differences in this set of experiments or whether that is hopeless but still it is possible that all the experiments are assessing whether something has any effect or not (and possibly in the same direction).

The former sets up the logical basis for combining the p_values - if there is no effect (NULL is true) then the p_values are an independent sample of Uniform(0,1) and in principal a NULL distribution for any function of them is defined and a cut off value will give the desired combined type one error rate. Now if there was an effect (in any or all of the experiments in the same or differing direction) the resulting power of various choices of combination function will vary. For instance, if one suspects a common small effect in the same direction in all of the studies, a sum type function will have more power. If one suspects only a few actually had an effect, a minimum type function would be preferred.
For the later, obtaining a defensible likelihood, let me first simplify the outcome to simply present versus absent - proportions. Usual assumptions would be for two parameters, Pc and Pt in each experiment, so Pc1,Pc2,Pc3,Pc4 and Pt1,Pt2,Pt3,Pt4. Now it is always worthwhile to consider whether any of these should be common and assess this graphically and decide - no and do no pooling at all.

Now what might be considered common here? Surely not any of the Pc1,Pc2,Pc3,Pc4 but perhaps Pt1/Pc1,Pt2/Pc2,Pt3/Pc3,Pt4/Pc4 all equal R as common relative effects can be defensible (the parameters now are just 5 - R,Pc1,Pc2,Pc3,Pc4). More adventurously, you might instead think of these ratios differing haphazardly but coming from a common distribution. With that, the common thing is defined as the parameters in that distribution (aka a random effects distribution or mixing distribution). To get a sense of how to actual do something like this, either frequentist or Bayesian, you may wish to read O’Rourke and Altman. Statistics in Medicine 2005

Now you have survival data, so your less adventurous definition is still relative effects but the arbitrary control group parameters are now hazard functions which are more challenging than proportions.

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Simpson's paradox will not kick in here; the group sizes are equal, the effects is steadily in one direction. However I would not pool the data, hence this question. I'm afraid that my level of statistical education is not sufficient to understand the rest of your answer, though. What are Pc and Pt? – January Nov 16 '12 at 13:59
@January Good catch, likely true that Simpson's paradox will not kick in here (the effect will not reverse) but there still will be a bias (remember with binary data the variance depends on the group proportion). Pc is proportion in control group, Pt is proportion in treatment group. Also if you are not familiar with paramteric likelihood - my answer will seem (unavoidably) strange and opaque. Maybe I'll ask a question some time on how to make that less so - the O'Rourke and Altman paper seems to be a failed attempt to do just that. – phaneron Nov 16 '12 at 16:26

I don't see what's wrong with pooling the data. You are confusing statistical significance with effect size, I think. If you have 4 experiments that all have a similar effect size, then a test of the 4 combined will be more highly significant than any of the individual tests.

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The effect size is different between the experiments (10-20% in the the two that are not significant, 50-70% if those that are significant). – January Nov 15 '12 at 12:35
OK, that's different. Then you want to try to figure out why the effect sizes are so different. – Peter Flom Nov 15 '12 at 14:08