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I was given a database with prostate cancer patients measured for PSA (continuous) at two time points, prior and after treatment, using three drugs (categorical), including information about age (continuous) and tumor size (continuous).

The task is to propose a linear model for analyzing the data using PSA as an endpoint, address its limitations and propose at least three (3) alternative methods for dealing with them.

I thought ANCOVA could be a simple method to use for modeling the three independent variables, but I'm not sure on how to describe alterantive methods for addressing its limitations like non-parallel slopes, non-linear outcome etc.

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You just named two:

  1. Slopes that aren’t parallel

  2. Nonlinear trends

Two remedies should address these, perhaps by using an interaction term to remedy the former and a spline for the latter.

These can be combined to allow for different nonlinear behavior in each group. Instead of ANCOVA-with-interaction multiplying just the factor levels by the covariate to get different slopes, multiply the factor levels by splines of the covariate to allow one level to exhibit rise-fall-rise while another exhibits fall-rise-fall (for instance).

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  • $\begingroup$ Thanks for the quick response! Is there any way to formally describe the combination of the two methods you described in your last paragraph? $\endgroup$
    – civy
    Mar 9 at 11:32
  • $\begingroup$ @civy Multiply the factor indicator variables and the splines, same as you would multiply the factor indicator variables by the covariate to allow for different slopes. $\endgroup$
    – Dave
    Mar 9 at 11:33
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When ANCOVA is taken to mean a parametric model such as the linear regression model, a very impactful assumption of ANCOVA is that $Y$ has been nearly perfectly transformed so as to achieve Gaussian residuals and equal variability in $Y$ across levels of $X$. An efficient alternative is a semiparametric ordinal regression model such as the proportional odds or proportional hazards models. These are $Y$-transformation invariant and are discussed in BBR and RMS. I don't have experience with PSA as an outcome but many lab measurements (HbA1c and serum creatinine being two examples) do not work well either untransformed or log transformed.

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