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Assuming that we perform a Cox regression as presented in the example here. I am wondering what we should expect in the output of the Cox regression if, for example, we perform the fit again but this we replace the values of the predictor variable 'ph.karno: Karnofsky performance score (bad=0-good=100) rated by physician' with new values given from a new team of physicians who, for example, have shown that their score is statistically significant in detecting lung cancer by its own.

          beta     HR    (95% CI for HR) wald.test   p.value
age       0.019     1         (1-1)       4.1         0.042
sex       -0.53   0.59     (0.42-0.82)    10          0.0015
ph.karno -0.016   0.98     (0.97-1)       7.9         0.005
ph.ecog    0.48   1.6       (1.3-2)       18          2.7e-05
wt.loss  0.0013    1       (0.99-1)       0.05        0.83

I am wondering what we should expect to change in the coefficients of this and the other covariates (if we can expect something to change apriori) by performing the fit again. E.g. do we expect that this known change in predicting lung cancer on its own will have a specific impact on age, sex etc.?

In a nutshell, I'm interested to know if there is a mathematical connection or empirical proof on the effect of replacing the values of a predictor variable with new values that we already know that are statistically significant in detecting the effect by themselves.

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  • $\begingroup$ I'm a little unclear about just what you are asking. These seem to be the NCCTG Lung Cancer data as provided for example in the R survival package. The ph.karno is not about "predicting lung cancer"; it's a measure of functional impairment. And these patients already had advanced lung cancer so it's not clear what point there would be in analyzing a "score ... statistically significant in detecting lung cancer by its own" instead. Do you mean replacing Karnofsky score with some other measure that helps predict lung cancer survival? $\endgroup$ – EdM Jul 18 at 0:33
  • $\begingroup$ ph.krano is just an example of what I want to convey. My question is what do we expect if we replace the values of any variable with new values of the same variable that we now know that they are statistically significant in identifying cancer. $\endgroup$ – laza Jul 18 at 0:36
  • $\begingroup$ Part of my confusion is that Cox regression has to do with survival after diagnosis with cancer, not with "identifying cancer." One might for example use a logistic regression model to estimate the probability that a certain set of characteristics means that someone has cancer, but that wouldn't be a Cox proportional hazards model. $\endgroup$ – EdM Jul 18 at 1:40
  • $\begingroup$ Exactly, so in theory if we improve the accuracy of the values of one covariate, i.e. we use a better instrument or a team of more experienced doctors etc., then we should expect some effect in the parmeters of the other covariates $\endgroup$ – laza Jul 18 at 12:18
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Let's say that we are dealing with Cox multiple regression models having several predictor variables related to outcome. There are two issues with respect to adding or removing or replacing predictors, one that applies to all multiple regressions and one that is more specific to Cox and logistic regressions. These comes down to different types of omitted-variable bias.

In all linear regression models, omitting from a model a predictor that both is related to outcome and is correlated with the included predictors will lead to bias in the estimates of the coefficients for the included predictors. That type of bias is discussed on the Wikipedia page linked above. It can go so far as to flip the signs of the regression coefficients, as nicely illustrated in this answer. So the effect of replacing one predictor with another even in a standard linear regression model depends on the relationships of those two predictors to outcome and to the predictors that are included in both models.

Things are even more complicated in event-based analyses like logistic regression and Cox regressions. In such regressions, omitting from a model a predictor that is associated with outcome will lead to bias in the coefficients of all the included predictors even if the omitted predictor is not correlated with any of the included predictors. Omitting such a predictor from a logistic or Cox regression will bias the estimated coefficient values for all the included predictors toward lower magnitudes. That's nicely covered on this page, which has an analytical explanation for the related case of a probit regression. This phenomenon has been appreciated for decades for Cox models.

So in your type of example, if you remove a predictor that is not associated with outcome from a Cox model and replace it with a new predictor that is both associated with outcome and uncorrelated with the other predictors, then the model with that new predictor will tend to have higher-magnitude coefficients (hazard ratios farther away from 1) for all of the other included predictors.

It's not possible in general, though, to say what will happen to other coefficient estimates if you replace a predictor that is related to outcome with another that is also related to outcome (as in your suggestion to replace ph.karno with another predictor). In practice, most predictors in Cox models tend to have correlations among each other, so the net effect will depend on those correlations and on the associations of all the predictors with outcome.

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