Two-phase outcome-dependent studies for failure times and testing for effects of expensive covariates

Abstract

Two- or multi-phase study designs are often used in settings involving failure times. In most studies, whether or not certain covariates are measured on an individual depends on their failure time and status. For example, when failures are rare, case–cohort or case–control designs are used to increase the number of failures relative to a random sample of the same size. Another scenario is where certain covariates are expensive to measure, so they are obtained only for selected individuals in a cohort. This paper considers such situations and focuses on cases where we wish to test hypotheses of no association between failure time and expensive covariates. Efficient score tests based on maximum likelihood are developed and shown to have a simple form for a wide class of models and sampling designs. Some numerical comparisons of study designs are presented.

Appendix: Variance estimate for score statistic (7)

The individual components of the score functions given by (4) and (5) can be written as

$$\begin{aligned} U_{\theta i}= & {} R_i \phi ^{\prime }_{\theta }(y_i | x_i, z_i) + (1 - R_i) E \{ \phi ^{\prime }_{\theta }(y_i | X_i, z_i) | y_i, z_i \} \\ U_{\alpha i}= & {} R_i \phi ^{\prime }_{\alpha }(x_i | z_i) + (1 - R_i) E \{ \phi ^{\prime }_{\alpha }(X_i | z_i) | y_i, z_i \} \end{aligned}$$

where $i = 1, \ldots , N$. The information matrix for $(\theta ^{\prime }, \alpha ^{\prime })^{\prime }$ then has component matrices as follows, where we define $A_{\theta }(y | x, z) = \partial ^2 \log f(y | x, z) / \partial \theta \partial \theta ^{\prime }$ and $A_{\alpha }(x | z) = \partial ^2 \log g(x | z) / \partial \alpha \partial \alpha ^{\prime }$:

$$\begin{aligned} I_{\theta \theta i} = -\frac{\partial U_{\theta i}}{\partial \theta ^{\prime }}= & {} R_i A_{\theta }(y_i | x_i, z_i) + (1 - R_i) E \{ A_{\theta }(y_i | X_i, z_i) | y_i, z_i \} \\&- (1 - R_i) E \{ \phi ^{\prime }_{\theta }(y_i | X_i, z_i) \phi ^{\prime }_{\theta }(y_i | X_i, z_i)^{\prime } | y_i, z_i \} \\&+ (1 - R_i) E \{ \phi ^{\prime }_{\theta }(y_i | X_i, z_i) |y_i, z_i \} E\{ \phi ^{\prime }_{\theta }(y_i | X_i, z_i)^{\prime } | y_i, z_i \} \\ I_{\alpha \alpha i} = -\frac{\partial U_{\alpha i}}{\partial \alpha ^{\prime }}= & {} R_i A_{\alpha }(x_i | z_i) + (1 - R_i) E \{A_{\alpha }(X_i | z_i) | y_i, z_i \} \\&- (1 - R_i) E \{ \phi ^{\prime }_{\alpha }(X_i | z_i) \phi ^{\prime }_{\alpha }(X_i | z_i)^{\prime } | y_i, z_i \} \\&+ (1 - R_i) E \{ \phi ^{\prime }_{\alpha }(X_i | z_i) |y_i, z_i \} E\{ \phi ^{\prime }_{\alpha }(X_i | z_i)^{\prime } | y_i, z_i \} \\ I_{\theta \alpha i} = -\frac{\partial U_{\theta i}}{\partial \alpha ^{\prime }}= & {} (1 - R_i) E \{ \phi ^{\prime }_{\theta }(y_i | X_i, z_i) \} E \{ \phi ^{\prime }_{\alpha }(X_i | z_i) | y_i, z_i \}^{\prime } \\&- (1 - R_i) E \{ \phi ^{\prime }_{\theta }(y_i | X_i, z_i) \phi ^{\prime }_{\alpha }(X_i | z_i)^{\prime } | y_i, z_i \} \; . \end{aligned}$$

Define $A_{\mu \mu i} = -\partial ^2 \log f_0(y_i; \mu _i, \sigma ) / \partial \mu _i^2$, $A_{\sigma \sigma i} = -\partial ^2 \log f_0(y_i; \mu _i, \sigma ) / \partial \sigma \partial \sigma ^{\prime }$, $A_{\mu \sigma i} = -\partial ^2 \log f_0(y_i; \mu _i, \sigma ) / \partial \mu _i \partial \sigma ^{\prime }$, $A_{\alpha \alpha i} = - \partial ^2 \log g(x_i | z_i) / \partial \alpha \partial \alpha ^{\prime }$, where $\mu _i = \beta ^{\prime } x_i + \gamma ^{\prime } z_i$. We note that under $H_0$: $\beta = 0$ we have (i) $g(x|y,z) = g(x|z)$, (ii) $f(y | x, z; \theta ) = f(y | z; \gamma , \sigma )$, and (iii) all of $\phi ^{\prime }_{\mu }(y | x, z)$, $\phi ^{\prime }_{\sigma }(y | x, z)$, $A_{\mu \mu i}$, $A_{\sigma \sigma i}$ and $A_{\mu \sigma i}$ depend only on y and z. The components of $I_{\theta \theta i}$, $I_{\alpha \alpha i}$ and $I_{\theta \alpha i}$ then simplify as follows:

$$\begin{aligned} I_{\beta \beta i}= & {} R_i A_{\mu \mu i} x_i x^{\prime }_i + (1 - R_i) \{ A_{\mu \mu i} E(X_i X^{\prime }_i | z_i) - \phi ^{\prime }_{\mu }(y_i | z_i)^2 \mathrm{var}(X_i | z_i) \} \\ I_{\beta \gamma i}= & {} R_i A_{\mu \mu i} x_i z^{\prime }_i + (1 - R_i) A_{\mu \mu i} E(X_i | z_i) z^{\prime }_i \\ I_{\beta \sigma i}= & {} R_i x_i A_{\mu \sigma i} + (1 - R_i) E(X_i | z_i) A_{\mu \sigma i} \\ I_{\beta \alpha i}= & {} - (1 - R_i) \phi ^{\prime }_{\mu }(y_i | z_i) E \{X_i \phi ^{\prime }_{\alpha }(X_i | z_i)^{\prime } | z_i \} \\ I_{\gamma \gamma i}= & {} A_{\mu \mu i} z_i z^{\prime }_i \;,~~ I_{\gamma \sigma i} = z_i A_{\mu \sigma i} \;, ~~ I_{\gamma \alpha i} = 0 \;, ~~ I_{\sigma \alpha i} = 0 \;, ~~ I_{\alpha \alpha i} = R_i A_{\alpha \alpha i} \; . \end{aligned}$$

The variance estimate (8) is then obtained by noting that under $H_0$, the asymptotic variance of $N^{1/2} U_{\beta }(\tilde{\theta }, \tilde{\alpha })$ is the limit of

$$\begin{aligned} N^{-1} E \left\{ I_{\beta \beta } - \left( I_{\beta \eta } I_{\beta \alpha } \right) \left( \begin{array}{cc} I_{\eta \eta } &{} 0 \\ 0 &{} I_{\alpha \alpha } \\ \end{array} \right) ^{-1} \left( \begin{array}{c} I^{\prime }_{\beta \eta } \\ I^{\prime }_{\beta \alpha } \\ \end{array} \right) \right\} \; , \end{aligned}$$

where $I_{\beta \beta } = \sum _{i=1}^N I_{\beta \beta i}$, $I_{\beta \gamma } = \sum _{i=1}^N I_{\beta \gamma i}$ etc. and replacing $\theta$, $\alpha$ in the entries of $I_{\beta \beta }$, $I_{\beta \eta }$, $I_{\beta \alpha }$, $I_{\eta \eta }$ and $I_{\alpha \alpha }$ given above with $\tilde{\theta }$, $\tilde{\alpha }$.