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.