![]() ![]() That both takes care of the censoring problem and spaces cases more evenly. You examine the survival function of the $s_j$ (the complement of the CDF) with a Kaplan-Meier plot that incorporates the censoring and compare that against the survival function of the standard distribution $W$. Second is the nature of the plots typically used to evaluate the agreement with the theoretical exponential form, which tend to overemphasize the larger values at the upper right while squishing together most of the cases.īoth of these problems are solved by working directly with the (potentially censored) standardized residuals $s_j$ instead. The answer from shows problems with using Cox-Snell residuals to evaluate a model. This 1:1 relationship between the residual types means that evaluation of a model via Cox-Snell residuals $r_j$ is equivalent to evaluation via standardized residuals $s_j$, as Klein and Moeschberger note on page 415. For example, if you want a Cox-Snell residual for a lognormal model, $r_j = -\ln(1-\Phi(s_j))$ where $\Phi$ is the standard normal CDF. ![]() The survival function $S(T)$ is the complement of the cumulative distribution function (CDF) of the survival times, so the Cox-Snell residual can be written $r_j = -\ln(1-\widehat, $$Īs those $s_j$ should be distributed according to $W$. These notes provide a concise introduction to such parametric modeling. With a log link, $W$ is standard minimum extreme value for Weibull and exponential models ( $\sigma = 1$ for exponential), standard logistic for log-logistic, and standard normal for lognormal. Where $\sigma$ is a scale factor, $W$ is the standard probability distribution corresponding to the parametric survival model, and $f(T)$ is a link function, typically $\ln(T)$. There is a simple relationship between Cox-Snell residuals and standardized residuals for models that can be expressed in a location-scale form: The Survival() function in the R rms package, for example, provides an easy way to get survival probability estimates from a psm parametric model fit for specified times and linear-predictor values, which then can be transformed into Cox-Snell residuals. That simple multiplicative form for $\hat H$, however, doesn't hold without proportional hazards.Īs $S(T|X)=\exp(-H(T|X))$ for continuous-time survival models, once you have a parametric equation for $\hat S(T|X)$ you can calculate a Cox-Snell residual directly from that equation: $r_j =\hat H(T_j|X_j) = -\ln(\hat S(T_j|X_j))$. That's a convenient form for proportional hazard models, for which $\hat H(T_j|X_j)= \hat H_0(T) \exp(X_j' \hat\beta)$, the product of an estimated baseline hazard $H_0(T)$ and the hazard ratio for the case calculated from the vector $\hat \beta$ of regression coefficients. The Cox-Snell residual for case $j$ in a survival model is $r_j=\hat H(T_j|X_j)$, where $\hat H(T|X)$ is the estimated cumulative hazard function, $T_j$ is the event/censoring time for the case and $X_j$ its vector of covariate values. Any improvement on this answer should be welcomed. I haven't studied the survival analysis thoroughly so I am not sure how to improve this model. This makes the mean to be unity though it makes the qqplot seem worse. I guess the residuals are more spread out than the exponential distribution with parameter 1 and some of them are smaller than they supposed to be.Īccording to the reference book, we need to modify rC for censored observations because they will be too small. This graph seems not bad but the mean and the variance isn't close to 1 and deletion of the 6th row which looks like outlier lower the variance. Based on the book's formulas, I wrote this code : #rC : Cox-Snell residuals
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