Events per variable for risk differences and relative risks using pseudo-observations

Abstract

A method based on pseudo-observations has been proposed for direct regression modeling of functionals of interest with right-censored data, including the survival function, the restricted mean and the cumulative incidence function in competing risks. The models, once the pseudo-observations have been computed, can be fitted using standard generalized estimating equation software. Regression models can however yield problematic results if the number of covariates is large in relation to the number of events observed. Guidelines of events per variable are often used in practice. These rules of thumb for the number of events per variable have primarily been established based on simulation studies for the logistic regression model and Cox regression model. In this paper we conduct a simulation study to examine the small sample behavior of the pseudo-observation method to estimate risk differences and relative risks for right-censored data. We investigate how coverage probabilities and relative bias of the pseudo-observation estimator interact with sample size, number of variables and average number of events per variable.

Appendix

1.1 The risk difference model

Let $\tau >0$ be a given time point, and let us introduce two functions $f$ and $g$ by

$$\begin{aligned} f(y,z)=\frac{y}{y+z},\quad g(y,z)=e^{-(y+z)\tau },\quad y,z>0. \end{aligned}$$

Let $T, C$ and $X$ be given as in Sect. 3 and let $q_x=P(X=x)$. The model in (2) can equivalently be formulated as $T\mid X=x\sim \exp (\lambda _x)$, where $\lambda _x=-\log (1-p-dx)/\tau$ and $\tau =1$. Under this model, the probability of observing an event before time $\tau$ is

$$\begin{aligned} P(T\le C,T&\le \tau )\nonumber \\&= \sum \limits _{x=0,1}\big (1+f(\lambda _c,\lambda _x)g(\lambda _c,\lambda _x)-f(\lambda _c,\lambda _x)-g(\lambda _c,\lambda _x)\big )q_x, \end{aligned}$$

(5)

and the probability of observing an event after time $\tau$ is

$$\begin{aligned} P(T\le C,T>\tau )=\sum \limits _{x=0,1}\big (g(\lambda _c,\lambda _x)-f(\lambda _c,\lambda _x)g(\lambda _c,\lambda _x)\big )q_x. \end{aligned}$$

(6)

If we denote the probability of observing an event before time $\tau$ by $p_e(\lambda _c)$, where $\lambda _c$ is the censoring rate parameter, then

$$\begin{aligned} \lim _{\lambda _c\rightarrow \infty }p_e(\lambda _c)=0 \end{aligned}$$

(7)

since the function inside the summation tends to $0$. Furthermore we have that

$$\begin{aligned} \lim _{\lambda _c\downarrow 0}p_e(\lambda _c)=\sum _{x=0,1}\left( 1-g(0,\lambda _x)\right) q_x&= \sum _{x=0,1}\left( 1-(1-p-dx)\right) q_x\nonumber \\&= p+dq , \end{aligned}$$

(8)

where $q=q_1$ is the mean of $X$.

Let $X_1$ be a binary covariate with mean $q$ and assume that $X_2,\ldots ,X_k$ are i.i.d. normally distributed with mean $0$ and variance $\sigma ^2>0$. The model (3) can now be formulated as $T\mid X_1=x_1,\tilde{X}=\tilde{x}\sim \exp (\lambda _{x_1,\tilde{x}})$, where $\tilde{X}=(X_2,\ldots ,X_k)$ and

$$\begin{aligned} \lambda _{x_1,\tilde{x}}=-\log (1-p-dx_1-x_2-\cdots -x_k)/\tau . \end{aligned}$$

Under this model, the probability, $P(T\le C,T\le \tau )$, of observing an event before $\tau$ is

$$\begin{aligned} \mathrm{E}\left[ 1+f(\lambda _c,\lambda _{X_1,\tilde{X}})g(\lambda _c,\lambda _{X_1,\tilde{X}})-f(\lambda _c,\lambda _{X_1,\tilde{X}})-g(\lambda _c,\lambda _{X_1,\tilde{X}})\mid A\right] , \end{aligned}$$

where $A$ is the event that $p+dX_1+X_2+\cdots +X_k\in (0,1)$. The probability of observing an event after $\tau$ is thus

$$\begin{aligned} P(T\le C,T>\tau )=\mathrm{E}\left[ g(\lambda _c,\lambda _{X_1,\tilde{X}})-f(\lambda _c,\lambda _{X_1,\tilde{X}})g(\lambda _c,\lambda _{X_1,\tilde{X}})\mid A\right] . \end{aligned}$$

These probabilities are simulated in our study since explicit formulas for them do not exist.

To obtain a risk difference model such that $P(A)$ is close to one, we choose the variance $\sigma ^2$ of $X_2,\ldots ,X_k$ such that

$$\begin{aligned} \sigma = \frac{\min (p,1-p-d)}{3\sqrt{k-1}}, \end{aligned}$$

(9)

since then

$$\begin{aligned} P(A)=P(p+dX_1+X_2+\cdots +X_k\in (0,1))\ge 0.9973. \end{aligned}$$