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Hazard Ratio Estimators after Terminating Observation within Matched Pairs in Sibling and Propensity Score Matched Designs

  • Tomohiro Shinozaki ORCID logo EMAIL logo and Mohammad Ali Mansournia


Similar to unmatched cohort studies, matched cohort studies may suffer from the censoring of events prior to the end of follow-up. Moreover, in some matched-pair cohort studies, observation time is prematurely terminated immediately after the follow-up of his/her matched member is completed by an event or censoring. Although the follow-up termination within matched pairs may or may not change the hazard ratio estimators, when and how the change occurs has not been clarified. We study the change in the estimates of the hazard ratio conditional on matched pairs and/or covariates by considering two types of matched-pair designs in cohort studies—sibling pair matching and propensity score matching—in which termination can be naturally considered. If all possible confounders are shared within the matched pairs, after termination, a wide range of hazard ratio estimators coincides with that obtained from a stratified Cox model. If unshared confounders should be adjusted for in the analysis, however, such coincidence is not observed. Simulation studies on sibling designs with unshared confounders suggested that the pair-stratified covariate-adjusted Cox model for the hazard ratio conditional on matched pairs and covariates is generally preferred, for which termination does not deteriorate the estimation. Conversely, the comparison between stratifying or not stratifying on pair is a more subtle issue in propensity score matching which targets a marginal or covariate-conditional hazard ratio. Based on simulation studies considering Cox models after matching based on estimated propensity scores, we discourage pair-stratified analysis and termination, particularly after data collection.

Funding statement: This work was supported by the Japan Society for the Promotion of Science (Funder Id: 10.13039/501100001691, Grant-in-Aid for Young Scientists B 16K16015).


A Random-effect models without covariates fitted to terminated data (Section 3.2)

Maximum likelihood estimates of random-effect models are obtained for joint likelihood of data and random effects marginalized over random effects. For random-effect Cox models without covariates, the marginal (with respect to α = (α1, …, αn)) partial likelihood is


where R(Yi*) is a risk set (of pairs) at time Yi* and f(α;θ) is a user-specified parametric distribution of random effects α. Estimation of β is only relevant to the components outside the integral, which is equal to the partial likelihood of the stratified Cox model without covariates (3) [8].

For random-effect Poisson models without covariates, marginal likelihood is


where g(α, β, θ) = ∑iαi(di0* + di1*) – Yi*exp(αi)(1 + eβ) + log f(α;θ). From generalized linear mixed models theory, random effects αi should be subject to ∂g(α,β,θ)/∂αi = (di0* + di1*) – Yi*exp(αi)(1 + eβ) + Qi = 0 for all i at any β (where Qi = ∂log f(α,θ)/∂αi is a penalty term): it implies exp(αi) = {Yi*(1 + β)}–1(di0* + di1* + Qi). Rearranging the marginal likelihood yields


The estimation of β is only relevant to the parts outside the integral, which is equal to the partial likelihood of the stratified Cox model without covariates (3).

B Conditional models with shared covariates fitted to terminated data (Section 3.3)

For Cox models conditional on shared covariates Xi, partial likelihood is


where R(Yi*) is a risk set (of pairs) at time Yi*. Note that di1*di0* = 0. The contribution for the partial likelihood from pair i can be decomposed into eβ/(1 +eβ)di1*, 1/(1 +eβ)di0*, and exp(γTXi)/kR(Yi)exp(γTXi)di1+di0; from the first two factors of the likelihood including β, score equation of β is ∑i{di1*di0*eβ} = 0, which is independent of γ. The partial likelihood including β is the same as that from the model (3), so are its first and second derivatives.

For Poisson models conditional on shared covariates Xi, the score equations are


The first two equations jointly imply ∑i{di0*Yi*exp(α + γTXi)} = 0 and ∑i {di1*Yi*exp(α + γTXi)eβ} = 0; hence a maximum likelihood estimate of β is the solution of ∑i {di1*di0*eβ} = 0, which is independent of α and γ.

β is estimated independent of γ in both equations; therefore, robust variance of β is estimated as {∑i mi(β)/∂β}–1i mi(β)2{∑imi(β)/∂β}–1, where mi(β) = di1*di0*eβ and β is substituted by its estimate. After some algebra, this robust variance estimator becomes 1/i= 1ndi0* + 1/i= 1ndi1*, which is also the β-component of the inverse Fisher information in both Cox and Poisson models considered here (the latter calculation is somewhat complex and deferred to Appendix C). This is true whether Xi are adjusted for or not.

C Variance estimates based on Fisher information in a Poisson model conditional on shared covariates with terminated data (Section 3.3)

For simplicity, assume shared covariates Xi as a scalar Xi (though the following result holds if Xi is a vector). As shown in the Appendix B, the score equations for the Poisson model conditional on shared covariates Xi are


where (Sα, Sβ, Sγ)T is the score function of (α, β, γ)T, i. e. the first derivative of the log-likelihood with respect to (α, β, γ)T. To derive the variance of the maximum likelihood estimator of β, we reorder the score as (Sβ, Sα, Sγ)T. Further differentiating it with (β, α, γ) yields the negative of the Fisher information matrix ∂(Sβ, Sα, Sγ)T/∂(β, α, γ) as


We partition matrix (13) as


Following basic matrix rules, the (1, 1) part of the inverse of matrix (13) is obtained as (A – BD–1C)–1. Denoting K =[(1+eβ)2idi0iXi2Yi(di0+di1)iXi(di0+di1)2]1,


Substituting maximum likelihood estimator exp(βˆ)=idi1/idi0, (A – BD–1C)–1 is equal to idi1(idi1)2idi1+idi01=1/idi0+1/idi1, as desired.

D Fixed-effect models fitted to terminated data (Section 4.2)

Fixed-effect Cox models’ partial likelihood is


Equating the first derivative of the log-partial likelihood with respect to αi (i = 1, …, n) to 0, fitting the fixed-effect Cox model imposes the following conditions for all i = 1, …, n: kR(Yi),kieαkexp(β+γTXk1)+exp(γTXk0)=0. Thus, the partial likelihood reduces to


which is the partial likelihood (2) of the stratified Cox model (1) by applying Result 1:


Fixed-effect Poisson models’ score equations are further augmented with

jin pairidijYiexp(αi+βZij+γTXij)=0fori=1,,n.

Substituting Yi*exp(αi) = (di0* + di1*)/{exp(β + γTXi1) + exp(γTXi0)} from the above conditions, ∑ijZij{dij*Yi*exp(αi + βZij + γTXij)} and ∑ijXij{dij*Yi*exp(αi + βZij + γTXij)} reduce to the first derivative of log-partial likelihood (2) with respect to β and γ, respectively. To be specific, score equations of the fixed-effect Poisson model are

Sαi=jin pairidijYiexp(αi+βZij+γTXij)=0fori=1,,n,Sβ=ijZijdijYiexp(αi+βZij+γTXij)=0,Sγ=ijXijdijYiexp(αi+βZij+γTXij)=0.

The first equation of Sαi=0 imply Yi*exp(αi) = (di0* + di1*)/{exp(β + γTXi1) + exp(γTXi0)}.

Deleting Yi*exp(αi) from the second equation, Sβ = ∑i [di1* – (di0* + di1*)exp(β + γTXi1)/{exp(β + γTXi1) + exp(γTXi0)}] = ∑i [{di1*exp(γTXi0) + di0* exp(β + γTXi1)}/{exp(β + γTXi1) + exp(γTXi0)}]. This is the same as the first derivative of the log-partial likelihood of the stratified Cox model (1)


with respect to β.

Similarly, deleting Yi*exp(αi) from the third equation, Sγ = ∑i [Xi1{di1*exp(β + γTXi1) + di1*exp(γTXi0) – di1*exp(β + γTXi1) – di0*exp(β + γTXi1)}/{exp(β + γTXi1) + exp(γTXi0)}] + ∑i [Xi0{di0*exp(β + γTXi1) + di0*exp(γTXi0) – di1*exp(γTXi0) – di0*exp(γTXi0)}/{exp(β + γTXi1) + exp(γTXi0)}] = ∑i [{Xi1di1*exp(γTXi0) – Xi1di0*exp(β + γTXi1) + Xi0di0*exp(β + γTXi1) – Xi0di1*exp(γTXi0)}/{exp(β + γTXi1) + exp(γTXi0)}]. As before, the first derivative of the log-partial likelihood of the stratified Cox model (1) with respect to γ provides the same function.


We are grateful to Dr Takahiro Tabuchi (Osaka International Cancer Institute, Japan) for discussing a statistical analysis plan of the Longitudinal Survey of Middle-aged and Elderly Persons.


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Received: 2017-12-16
Revised: 2018-12-20
Accepted: 2018-12-21
Published Online: 2019-01-15

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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