# Survival analysis using meta-analysis of linear regression for censored and non-censored data

I am trying to determine whether the concentration of some compound in the blood can help predict the life expectancy of cancer patients. Compound concentration were assessed as the patients entered the study and one knows how long the each patient survived. The data is right-censored as some patients did not die by the end of the study. My model is the following: \begin{align} y &= u(1+zw)\\ y &= y_{0}x^{b} \epsilon \end{align}

where $y$ is the real (potentially unknown) survival time, $u$ is the observed survival time (until death or censoring), $x$ is the compound concentration, $z$ is a indicator function telling whether the patient died before the end of the trial, $w$ is an unknown quantity describing how much longer the patient will live after the trial ends and $e$ is some log-normal distributed multiplicative noise. Combining the equations above and taking the log yields $$u(1+zw) = y_{0}x^{b} \epsilon \Rightarrow \log(u) = log(y_0)+b\log(x)+\log(\epsilon)+\log(1+zw)$$ Separating out the censored from the non censored data: \begin{align} \log(u) &= \log(y_0)+b\log(x)+\log(\epsilon) \quad\text{for non censored data}\\ \log(u) &= \log(y_0)+b\log(x)+\log(\epsilon) +\log(1+w) \quad\text{for censored data} \end{align} If we assume that, under the null hypothesis $w$ is independent from $x$, is it reasonable to do a linear regression separately for the censored and non-censored data and meta-analyze the two results as in both cases the effect size $b$ is the same (although you expect different intercepts and variances)?

I suspect this is not a standard procedure in survival analysis, and I would be glad to know if/where it is flawed. My dataset is pretty small, so having a simple parametric model is an advantage (provided its assumptions are reasonable)

Many thanks.

• What is wrong with the standard procedures for survival analysis (such as a cox survival model)? You seem to have the required data for it. Moreover, there are parametric survival models available (the Weibull distribution for example). – IWS Aug 29 '17 at 12:35
• @ IWS, Nothing wrong with the standard procedures. On my data, cox proportional hazard give a similat p-value. I was just curious to know if my alternative is also justifiable. – citronrose Aug 29 '17 at 13:21