Has the distribution of a covariate to be normal?

[Bernd Gruber]

Dear Colleagues,

I recently try to submit a paper which deals with survival analysis using CMR-methods. In this paper I analysed the effect of a covariate on survival and compared it with models without the covariate. One concern of one of the reviewers was that the distribution of the covariate was not normal and therefore the analysis is not valid. The distribution of the covariate is more or less U-spaped, hence many animals do have values of 0 and 1 and the distribution of animals with values between zero and one have a normal distribution.
My question is: Is it necessary that the distribution of the covariate is normal?
If so, is it valid to transform the covariate (but with an U-shaped distribution it is hardly possible to achieve a normal distribution by transformation, as far as I know).
Any comment would be very much appreciated, thanks in advance,

Bernd

p.s. From talking with statisticians I know that the distribution of the residuals from in a linear model should have a normal distribution. But how do I calculate the residuals from the model with the individual covariate. Do I calculate the survival of each indidivual and look at the distribution of these values?

[bmitchel]

Bernd -

The distribution of a covariate does NOT have to be normal for mark-recapture models (or any other statistical model, for that matter). The normality assumption refers strictly to the distribution of the residuals, as the statisticians have told you.

MARK does have various options for outputting and plotting residuals (Output... Specific Model Output... Residuals...). These options work fine for models without covariates, but MARK does not calculate residuals correctly when there are covariates.

I am assuming that your model comparison (i.e. of models with and without the covariate) used some variant of AIC. If this is the case, then the AIC-best model is still better or equal to the other models in terms of goodness of fit (and looking at the distribution of residuals is one way to assess goodness of fit). However, it could still be the case that none of your models fit particularly well (AIC may be picking the "best" of a lousy set of options).

Keep in mind, though, that if you can show that one of your models fits the data, AIC will not then pick a model that does not fit. So if some of your models have no covariates, you can perform GOF tests using MARK (if they are available for survival analysis). If the model with the covariate has a lower AIC then a model that you have shown passes GOF testing, that model should also fit the data.

Brian

[Bernd Gruber]

Brian,

thanks for your repsonse, I will quickly try to summarize your thoughts to make sure that I have understood them correctly.

1. check if models without covariates by GOFs (I already did that resulting in a low c-hat) and residuals
2. if GOF is good and if AIC for model with covariate is lower, then I can be confident that model selection of model with covariate and fit is okay.

Thanks again,

Bernd