2.2.1. CPH Modeling and Case Matching
CPH modeling is a survival analysis in which a multivariate regression model (Equation 1) evaluates the association between covariates (e.g., baseline prognostic factors, exposure measures) and the time until a specific event occurs. The comparison of response between treatment groups is given as a HR, and this ratio is assumed to be constant over time (Equation 2 ). The model allows for estimation of the relationship between exposure and response. Multiple covariates can be evaluated in the model for statistical significance, and it is imperative that they are included to correct for the effects of confounding factors that might otherwise bias the E-R analysis. The structure of the model assumes that the effects of these covariates are time-independent, and also depend upon the value of the covariate and a constant coefficient. This approach has been used to adjust for confounding factors in E-R analyses for T-DM1.15, 20
\(h\left(t\right)=h_{0}\left(t\right)\ \times exp(b_{1}X_{1}+\ b_{2}X_{2}+\ldots\ b_{p}X_{p})\)(Equation 1 )
where h(t) describes the hazard of an event at time t , determined by a set of covariates (X1, X2, …, Xp );h0(t) describes the baseline hazard at timet ; and the coefficients (b1, b2, …, bp ) describe the relationship between the covariates and the hazard.
\(HR=\frac{{h\left(t\right)}_{y}}{{h\left(t\right)}_{z}}=\ \frac{\exp\ \left(b_{1}X_{1y}+\ b_{2}X_{2y}+\ldots\ b_{p}X_{\text{py}}\right)\ }{\exp\ \left(b_{1}X_{1z}+\ b_{2}X_{2z}+\ldots\ b_{p}X_{\text{pz}}\right)\ }\ \)(Equation 2 )
where HR is the ratio of the expected hazards of two groups, y and z, and is time-independent. Components of this equation are the same as described in Equation 1.
Case-matching analysis has been widely used in observational studies to adjust for confounding factors. The method was more recently applied to E-R analysis for the first time by Yang et al. and has since been used for E-R analyses of multiple oncology biologics such as T-DM1 and nivolumab.2, 12, 15 Case-matching analysis adjusts for confounding factors by balancing the distribution of baseline risk factors between the control and treatment groups prior to calculating the HR. Only patients in the control arm that are similar or matched in baseline risk to patients in the treatment arm are included in the analysis. The matching process can be optimized with a variety of methods including propensity score matching, Mahalanobis distance matching, and coarsened exact matching.21-23 In the more commonly used propensity score the score is typically estimated using a logistic regression model, and patients with similar scores are matched. After case-matching the endpoint can be directly compared between the matched groups by a method of choice (e.g., Kaplan-Meier survival analysis, CPH).
CPH modeling and case-matching address confounding factors in an E-R analysis by accounting for covariates in different exposure subgroups. For both approaches to successfully account for confounding in monoclonal antibodies in oncology appropriate covariates that account for imbalances in prognostic factors must be selected. The number of covariates is limited by the increasing potential for over-parameterization of results. In a comparison of response in the Q1 exposure subgroup and the control arm, Li et al . used CPH modeling and case-matching.15 Case-matching analysis demonstrated a greater reduction of the HR. While the case-matching analysis had additional covariates included that could contribute to the reduction of HR the reduction can also be attributed to the limitation of CPH modeling where the structure of the hazards model equation imposes assumptions about the effect of covariates on the E-R relationship.
Case-matching analysis is an appealing alternative to CPH modeling, as it requires no assumptions regarding the relationship between covariates and the E-R relationship. In addition, there is no specific method to select covariates used for matching, and covariates are not screened for significance in the case-matching analysis. The retention of both statistically significant and insignificant covariates may allow for an increased capacity for correction of confounding factors compared to methods that screen for covariates. Covariates that are clinically significant may be statistically insignificant in an analysis due to factors such as small sample size, variability, and correlation with other risk factors. Case-matching analysis is limited by the difficulty in matching case to control when using a small study sample or a large number of covariates. If data from an adequate sample size are available, and there is no need for validation of covariates by statistical significance case-matching appears to correct for confounding factors more effectively than CPH modeling.