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This paper (open access) Babies born at the threshold of viability: changes in survival and workload over 20 years (table 2) presents data for survival of extremely premature babies over the last 20 years in 4-year time periods. They state that the survival of 23-week gestation babies has not improved over this time (p=0.08). Does this test of trend have sufficient power to detect a trend if one existed? How could I determine that?

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You have misinterpreted the results of the statistical test just as the authors of that paper somewhat misinterpreted it in the conclusion section of their paper. It is concerning that we continue to quote $P$-values without effect estimates and confidence limits (which also lessen the need to talk about power). The problem is well summarized in the classic paper Absence of Evidence is not Evidence of Absence.

The trend test the authors used in that paper inappropriately made time into a categorical variable.

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I don't know of any easy formulas to compute power in survival analyses. My first impulse would be to run a simulation. Assume a certain effect size, simulate data based on this effect size, run your analysis, see whether the effect is detected at your prespecified alpha. Do this many times and count whether the effect was detected in 80% or in 20% of your sample.

Note that you must specify an effect size in advance here. So-called post hoc power calculations ("is the study sufficiently powerful to detect the effect we did observe as statistically significant?") are meaningless, because there is a monotone relationship between the observed effect size and the p value. If the effect was statistically sigificant, then the study was powerful enough to detect the observed effect as significant. If not, then not. This is nothing more than a tautology dressed up in formal-sounding statistics jargon.

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