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Suppose I have these linear regression models:

  • Y given A (p-value=p1 << p2)
  • Y given B (p-value=p2)
  • Y given A, B, C (With A having a big p-value that shows A is meaningless in this regression, but B and C having small p-values)

How do I predict Y given A and B, without having the original data, but having those mentioned models' coefficients?

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If possible, get data on C and use the third model.

If there are correlations among the predictors, then omitting any of them from a linear regression model can bias the regression coefficients of the included predictors. That's called omitted-variable bias. Omitting any predictor associated with outcome can bias coefficients of included predictors in things like logistic and survival models. So the first 2 models might not be reliable.

One explanation of what you found could be that B and C are the "most important" predictors but that A just happens to be correlated to one or both of them, probably to C. The third model uses all the predictors so it is more likely to represent the data the best.

If you can't get data on C, then you are on shakier ground. You can't use the third model. The best you can do is to try to combine the information that the first 2 models provide. Rating the models simply by their "p-values" is unreliable unless you know something about the underlying data and analyses.

You could in principle weight the predictions of the first 2 models by their Akaike Information Criterion (AIC) values. The AIC formula for a linear regression is on this page; it's related to the mean-square error (MSE), the number of observations, and the number of parameter values that were estimated from the model. With adequate information, you could use the p-values to reconstruct the mean-square error and the AIC values for the models.

If the first and second models were all developed on the same data set and each was a single-predictor (plus intercept) model as you imply, then the relationship between AIC and p-values is direct. The number of observations and number of fitted parameters values is the same, so p1 << p2 means MSE1 << MSE2 and the second model won't add much to what you get from the first.

This fall-back approach without information on C, however, will be very risky if the data sample on which you wish to make predictions is dissimilar to the data sets upon which the models were based.

Finally, avoid saying things like "A having a big p-value ... shows A is meaningless." It means your sample doesn't provide statistically reliable evidence that A matters in the third model, but that doesn't necessarily mean that A is "meaningless." If your main interest is in prediction, it's generally best to include as many predictors potentially related to outcome as is reasonable (provided you don't overfit), even if some have large p-values. In your situation without information on C, it looks like A might nevertheless provide the best predictions you can get even if it is "meaningless" in a fundamental causal sense.

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  • $\begingroup$ How do I use the third model when I only have A and B, but not C? (The purpose is only the prediction of Y.) $\endgroup$
    – HappyFace
    Jun 23, 2021 at 13:30
  • $\begingroup$ @HappyFace I've updated the answer with a suggestion. Still best to get data on C and use the thread model, if possible. $\endgroup$
    – EdM
    Jun 23, 2021 at 18:19

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