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Stratify despite no interaction?

Q: I have a statistically significant association between y and x, adjusting for z. If I add an x*z interaction term the p value is well above 0.10, but if I choose to stratify one stratum has a statistically significant association and one does not. How can this be?

A: First I though this could be figured out by plotting the data, suspecting the ranges didn't overlap. The scatterplots were similar. Then I looked at the regression coefficients and confidence intervals. The coefficients were almost identical (and very small in clinical terms). One group had a slightly wider CI.

Figure A: The slopes are similar, though not identical

Figure B: Add the CI and one crosses horizontal, so the slope is not statistically significant

Figure C: Overlay the two slopes and you clearly cannot say they are different, so an interaction term will be non-significant


Conclusion: This could be put in the "Beware of p-values" category. There is probably an inverse but weak association between X and Y. At an intuitive level there is no difference between the two groups, so stratifying the data is not justified.