Abstract
Background
Outcome-dependent two-phase sampling designs permit cost-effective measurement of expensive covariates in large prospective cohorts. Weighted Cox regression is commonly used for these designs, but applications requiring smooth hazard estimation or direct absolute risk prediction motivate a complementary parametric approach.
Methods
We extend the casebase framework, which represents hazard regression as logistic regression on sampled person-moments, by combining the casebase sampling offset with inverse-probability weights for phase-two inclusion. We formulate the estimator as a participant-level M-estimator, establish consistency and asymptotic normality under standard conditions, and develop a computationally efficient dfbeta-based jackknife/sandwich variance estimator. Performance was evaluated against weighted Cox regression in simulations, and implementation was illustrated using a frequency-matched biomarker substudy within the Ontario Health Study.
Results
Weighted casebase and weighted Cox regression had broadly similar coefficient performance across the scenarios studied. Both methods showed small bias and approximately nominal coverage in the binary-covariate and targeted sensitivity scenarios, whereas continuous or highly skewed covariates combined with fixed phase-two sample sizes and large weights produced finite-sample instability for both approaches. In the illustrative application, the two methods showed similar discrimination and weighted Brier score patterns, while the casebase model additionally produced smooth absolute-risk curves.
Conclusions
Weighted casebase regression provides a practical GLM-based approach to smooth parametric or flexible-parametric hazard estimation and direct absolute risk prediction under outcome-dependent two-phase sampling. It is complementary to weighted Cox regression rather than a replacement for it.
Keywords: Two-phase studies, Case-cohort design, Weighted casebase analysis, Dfbeta-based sandwich variance
Background
Large prospective cohort studies are central to modern epidemiology, providing longitudinal follow-up and enabling estimation of disease risk and identification of environmental, clinical, or genomic determinants of health. The Ontario Health Study (OHS) [1], with over 225,000 participants, exemplifies such efforts. While such cohorts collect detailed baseline and outcome data, resource constraints often limit the measurement of expensive exposures such as genomic, proteomic, metabolomic, or other biomarker measurements on all participants. Outcome-dependent two-phase sampling designs provide an efficient strategy in these settings by measuring costly covariates in a subset of participants while retaining information from the full cohort. Case-cohort sampling, in which a random subcohort and all incident cases are selected, represents one commonly used example [2].
Standard analyses of case-cohort and related two-phase designs are typically conducted using weighted Cox proportional hazards models, where inverse-probability weights correct for unequal inclusion probabilities [3]. This approach is well established and remains the natural default when the primary target is a relative hazard parameter and the baseline hazard can remain unspecified. However, in many applications involving risk prediction, calibration, or decision support, researchers require estimates of hazard, survival, or absolute risk over clinically relevant time horizons. In these settings, the baseline hazard is not merely a nuisance quantity; it is part of the target of inference. Weighted Cox regression can also produce absolute-risk estimates after estimating the baseline cumulative hazard, but these estimates are typically step functions tied to observed failure times. The weighted casebase approach is therefore most attractive when investigators want a directly parameterized smooth hazard or risk curve, for example for calibrated 5- or 10-year risk prediction, interpolation between observed event times, or decisions based on absolute-risk thresholds.
The casebase framework offers a reformulation of hazard regression as logistic regression applied to sampled person-moments [4, 5]. In full cohort settings, this approach provides a convenient way to fit parametric hazard models using standard generalized linear model (GLM) software, with an offset term accounting for the sampling of person-moments. The resulting model can accommodate parametric or flexible functions of time, allowing smooth estimation of hazards and absolute risks. However, extension to outcome-dependent two-phase designs requires two additional considerations: inverse-probability weighting for the individual-level sampling design and appropriate variance estimation when individuals contribute repeated sampled person-moments.
Recent methodological work has emphasized unified modeling strategies for complex survival data structures, including clustering, cure fractions, and competing risks [6]. The problem considered here is complementary. We focus on outcome-dependent two-phase sampling designs, such as those used for expensive covariates, where the main inferential challenge is to combine individual-level sampling weights with the person-time sampling structure of the casebase representation.
In this paper, we extend the casebase framework to outcome-dependent two-phase sampling designs, including case-cohort and related stratified or frequency-matched sampling schemes when the relevant inclusion probabilities are known or consistently estimable. The proposed estimator combines two sampling adjustments that operate at different levels: the casebase offset corrects for sampling of person-moments from the observed at-risk person-time used to generate the base series, whereas inverse-probability weights correct for unequal sampling of individuals into the phase-two sample. This separation is useful in applications where investigators require smooth estimates of hazard, survival, or absolute risk functions using standard GLM software. The goal is not to replace weighted Cox regression, which remains the natural default for estimating relative hazard parameters without specifying the baseline hazard, but to provide a complementary approach when the baseline hazard or absolute risk function is itself part of the estimand.
Our main contributions are as follows. First, we formulate a weighted casebase estimating equation for outcome-dependent two-phase samples with known or consistently estimable inclusion probabilities. Second, we express the estimator as an individual-level M-estimator that allows repeated sampled person-moments within participants. Third, we develop a computationally efficient variance estimator based on aggregation of dfbeta diagnostics from weighted logistic regression, corresponding to an individual-level sandwich covariance estimator. Fourth, we evaluate finite-sample performance relative to weighted Cox regression through simulation studies that include correctly specified univariate settings as well as targeted sensitivity scenarios involving multiple covariates, baseline hazard misspecification, independent random censoring, and estimated inclusion probabilities. Finally, we illustrate implementation using data from a biomarker substudy within the Ontario Health Study.
The OHS analysis is intended as an illustrative implementation rather than as a definitive matched-design epidemiologic analysis. The original biomarker substudy used frequency matching, and a fully design-consistent analysis would require matching-stratum-specific inclusion probabilities, conditional or stratified methods, or other analyses explicitly incorporating the matching structure. We use this example to demonstrate the weighted casebase workflow, including estimated inclusion probabilities, repeated sampled person-moments, and smooth absolute risk prediction.
The remainder of the paper is organized as follows. The Methods section develops the weighted casebase estimator, its large-sample justification, and variance estimation. The Simulation studies section presents simulation studies evaluating finite-sample performance under both primary and sensitivity scenarios. The Illustrative implementation using an Ontario Health Study biomarker substudy section provides an illustrative implementation using OHS biomarker data. The Discussion section provides a discussion of the implications and limitations, followed by the conclusions in the Conclusions section.
Methods
This section develops a weighted casebase estimator for outcome-dependent two-phase sampling designs. We first review the standard casebase representation of hazard regression as logistic regression on sampled person-moments. We then extend this representation to two-phase samples by combining the casebase offset, which accounts for sampling of person-moments, with inverse-probability weights, which account for unequal sampling of individuals. We finally describe the large-sample justification and a computationally efficient individual-level variance estimator based on aggregated dfbeta diagnostics.
The basic casebase framework
We begin with the full cohort setting. Let i=1,… ,N index independent cohort members, let 
denote observed follow-up time, and let 
denote the event indicator. The total observed at-risk person-time is
 |
A person-moment is a sampled at-risk time point 
paired with the covariate history 
for individual i at that time [4].
The casebase dataset consists of two components:
the case series, containing all
observed failure times, andthe base series, containing b sampled person-moments drawn from the observed at-risk person-time in the cohort.
The base series is generated using the standard two-step sampling scheme [4]. For each of the b base-series records, first sample an individual with replacement with probability proportional to follow-up time, 
, and then draw a time point uniformly from 
for that individual. This yields a sample of person-moments representing the covariate distribution over observed at-risk person-time. Individuals may contribute multiple base-series person-moments. Individuals who experience events contribute their failure time to the case series and may also contribute base-series person-moments from their observed pre-event follow-up time.
For a sampled casebase record, let Y=1 indicate membership in the case series and Y=0 indicate membership in the base series. Under uniform sampling of person-moments over observed at-risk person-time, the odds of case-series membership at covariate value 
and time t satisfy
 |
1 |
where 
denotes the hazard function. This follows by comparing the first-order case-series and base-series contributions for the same infinitesimal element of observed at-risk person-time. The case-series contribution is 
, because all failures are included, whereas the expected base-series contribution is 
under uniform sampling over observed at-risk person-time.
Assume a log-linear hazard model
 |
Taking logarithms of (1) gives
 |
2 |
Thus, parametric hazard regression can be implemented as logistic regression with linear predictor 
and offset 
.
Common parametric hazard specifications arise from particular choices of 
. For example, with a single covariate x,
 |
3 |
 |
4 |
 |
5 |
Flexible functions of time, including spline bases, can be incorporated in the same way.
The number of base person-moments b is typically chosen proportional to the number of cases c, with larger ratios improving numerical precision at the cost of increased computation [4]. Theoretical justification for the casebase representation in full cohort settings is provided by Hanley and Miettinen [4] and Saarela [5, 7].
Weighted casebase estimator for two-phase designs
We now consider an outcome-dependent two-phase sampling design. Let 
indicate whether individual i is included in the phase-two sample, where the expensive covariates required for modeling are observed. Let 
denote the phase-one information used by the sampling design, such as event status or sampling stratum, and define
 |
We assume design ignorability conditional on this phase-one information. Specifically, for any individual-level contribution 
that would enter the target estimating equation,
 |
Thus, after conditioning on 
, phase-two inclusion does not depend further on the expensive covariates or on other unobserved features of the estimating contribution. This is the usual inverse-probability-weighting condition for two-phase designs and implies
 |
The corresponding inverse-probability weight is
 |
When 
is unknown, a consistent estimator 
may be substituted, giving 
.
For a simple outcome-dependent two-phase design in which all 
cases are included and 
controls are sampled from the N-c non-cases, the first-order inclusion probabilities are
 |
More generally, in stratified or frequency-matched two-phase designs, 
should be the first-order inclusion probability defined within the relevant sampling strata or matching classes used by the phase-two design. The estimator described below applies when these inclusion probabilities are known or consistently estimable. Designs requiring conditional matching likelihoods or pair-specific sampling probabilities require design-specific extensions and are outside the present development.
The casebase sampling mechanism is applied after the phase-two sample has been selected. Let
 |
denote the observed phase-two at-risk person-time from which the base series is sampled. For each of the b base-series records, sample an individual with replacement with probability 
and then sample a time uniformly from 
. Since the base-series sampling rate is 
per unit of observed phase-two at-risk person-time, the casebase offset is
 |
The offset uses the unweighted phase-two at-risk person-time 
because 
is the denominator of the actual base-series sampling distribution. The inverse-probability weights enter the estimating equations below because they correct the individual-level phase-two sampling design.
For each selected individual i, let 
denote the set of case-series and base-series records contributed by that individual, with 
when 
. For record 
, let 
for a case-series record and 
for a base-series record. Let 
denote the corresponding time and let 
denote the covariate vector at that time. Define
 |
The weighted casebase pseudo-log-likelihood is
 |
6 |
For designs in which all cases are sampled with certainty, 
for cases. Case individuals may also contribute base-series person-moments from their observed follow-up time; those base-series records inherit the same individual-level weight as the case-series record.
Let
 |
The corresponding weighted score equation is
 |
7 |
The weighted casebase estimator 
is the solution to (7), or equivalently the maximizer of (6). In practice, estimation is performed using standard weighted logistic regression with offset 
and observation weights inherited from the source individual.
When inclusion probabilities are estimated, the same estimating equation is solved after replacing 
with 
. Point estimation proceeds by substitution. The additional variance component from estimating 
is not included by this substitution alone and is discussed in the Dfbeta-based jackknife/sandwich variance estimation section and Appendix A.
Large-sample justification
The weighted casebase estimator can be viewed as an individual-level M-estimator. Although the fitted logistic regression is applied to case-series and base-series records, inference is organized at the participant level because each participant may contribute more than one sampled person-moment. For individual i, define the aggregated score contribution
 |
where 
indicates phase-two inclusion, 
is the inverse inclusion probability weight, and 
contains all case-series and base-series records contributed by participant i. The estimator solves
 |
This representation separates the two sampling mechanisms. The casebase offset accounts for sampling person-moments from the observed phase-two at-risk time, whereas the inverse-probability weight accounts for unequal selection of individuals into the phase-two sample. Under the usual two-phase sampling condition,
 |
weighted phase-two contributions recover the corresponding full-cohort estimating-equation contributions.
The large-sample argument uses standard regularity conditions for inverse-probability weighted M-estimation: independent cohort members; correctly specified and positive inclusion probabilities; a base-series size satisfying 
for 
; bounded second moments for the aggregated individual-level score contributions; smoothness of the hazard model in 
; and nonsingularity of the population sensitivity matrix. Under correct model specification, 
is defined as the unique solution of the population estimating equation,
 |
Under these conditions, the fixed-weight estimator is consistent and asymptotically normal:
 |
8 |
where
 |
When the individual-level contributions are asymptotically independent, 
.
If inclusion probabilities are estimated, 
is replaced by 
. Consistency follows by standard plug-in M-estimation arguments when 
is uniformly consistent. However, the default dfbeta-based variance estimator described below treats the supplied weights as fixed. To propagate uncertainty from estimating the inclusion probabilities, one may use a stacked estimating-equation sandwich estimator or a subject-level bootstrap that re-estimates the inclusion-probability model and repeats the casebase sampling step. Appendix A explains how this formulation maps onto standard inverse-probability weighted M-estimation theory and summarizes the stacked estimating-equation variance calculation for estimated inclusion probabilities.
Dfbeta-based jackknife/sandwich variance estimation
For analyses that condition on the supplied weights, the covariance in (8) can be estimated using the aggregated individual-level score contributions. Define
 |
where 
when the design probabilities are known. Let
 |
denote the observed sensitivity matrix for the total score, and write
 |
The corresponding empirical sandwich estimator is
 |
Because the model is fit as a weighted logistic regression, a first-order computational route is available through dfbeta diagnostics.
Let 
denote the estimator that would be obtained after deleting all records contributed by individual i. A first-order Taylor expansion of the score equation around 
gives
 |
or equivalently,
 |
Thus, the first-order delete-one-individual jackknife contribution is obtained by aggregating the observation-level deletion approximations over all case-series and base-series records contributed by that individual.
In weighted logistic regression, the unstandardized dfbeta diagnostic for an observation is a first-order approximation to the coefficient change induced by deleting that observation, with the fitted offset and supplied weights held fixed. Let 
denote this observation-level dfbeta contribution for record 
, using the same coefficient ordering as 
. The individual-level dfbeta contribution is
 |
The sign convention for dfbeta affects 
but not the outer-product estimator below.
The proposed dfbeta-based jackknife/sandwich variance estimator is then
 |
9 |
Thus, (9) is an individual-level sandwich covariance estimator computed through first-order delete-one-individual jackknife contributions. It is a first-order jackknife rather than a refitted jackknife: the dfbeta diagnostics approximate deletion of individual records, and these approximations are aggregated to approximate deletion of all records from the same source individual. This aggregation accounts for repeated sampled person-moments within individuals rather than treating person-moments as independent observations.
The estimator in (9) is conditional on the supplied weights. If weights are known design quantities, this corresponds to the fixed-weight large-sample variance in (8). If weights are estimated, (9) estimates the variance conditional on the estimated weights and does not by itself propagate variability from estimating 
. In that setting, the full variance can be estimated using the stacked estimating-equation sandwich summarized in Appendix A, or approximated by a subject-level bootstrap that re-estimates the weights and regenerates the base series within each bootstrap replicate.
Practical implementation
The estimator can be implemented using any weighted logistic regression routine that permits an offset and record-level weights inherited from source individuals. A practical implementation proceeds as follows. After the phase-two sample is identified, compute 
, generate the base series from observed phase-two at-risk person-time using the sampling mechanism in the Weighted casebase estimator for two-phase designs section, construct the case-series and base-series records, and assign each record the supplied inverse-probability weight of its source individual, 
or 
. Then fit the weighted logistic regression with offset 
, extract unstandardized dfbeta diagnostics, aggregate those diagnostics by source individual, and compute the cross-product in (9). This requires only one model fit and avoids repeatedly refitting the model after deleting each individual. The casebaseweights R package implements this workflow for fitting weighted casebase models and computing the individual-level dfbeta aggregation variance estimator.
When inclusion probabilities are estimated, the dfbeta-based estimator remains conditional on the estimated weights. To propagate uncertainty from weight estimation, the stacked estimating-equation sandwich is the natural default when the inclusion probabilities are obtained from a standard parametric model fitted to the phase-one data, because it avoids repeated resampling and refitting. A subject-level bootstrap is useful when the weight-construction procedure is complex or multistep but can be reproduced in each replicate, or as a sensitivity analysis. In each bootstrap replicate, individuals are resampled at the subject level, respecting phase-two sampling strata when applicable; the inclusion probabilities are re-estimated; 
and 
are recomputed; the base series is regenerated; and the weighted logistic regression is refit. Thus, K bootstrap replicates require K complete analysis workflows, whereas the stacked sandwich uses the original inclusion-probability and weighted casebase model fits. If weights are supplied externally and their estimation procedure cannot be reproduced, inference remains conditional on the supplied weights.
Simulation studies
We conducted simulation studies to evaluate the finite-sample performance of the weighted casebase estimator compared to weighted Cox regression under outcome-dependent two-phase sampling. The primary objectives were to examine bias, variance estimation, confidence interval coverage, and convergence across different covariate types, cohort sizes, and sampling-weight magnitudes.
The primary simulations reproduce the original correctly specified univariate setting. To address additional operating characteristics relevant to applied two-phase studies, we added targeted sensitivity simulations involving multiple covariates, baseline hazard misspecification, independent random censoring, and estimated inclusion probabilities. These additional scenarios were designed to address specific limitations of the primary simulations without creating a full factorial expansion that would obscure interpretation.
Primary design and data-generating mechanism
We simulated full cohorts of sizes 
. Each individual had a single covariate generated from one of three distributions: Bernoulli(0.3), Normal(0, 1), and log-normal(0, 1) (Table 1 in Appendix A). These represent common discrete, symmetric continuous, and skewed continuous covariate settings.
Latent event times were generated from a Weibull proportional hazards model following Bender et al. (2005) [8]:
 |
10 |
with hazard
 |
11 |
The covariate effect was fixed at β = 1.5. Parameters 
were chosen so that approximately 200 events occurred per cohort. Follow-up was administratively truncated at the time of the 200th event to maintain a fixed number of cases per simulated cohort.
Two-phase sampling
From each full cohort, we constructed an outcome-dependent two-phase sample by including all 
cases and randomly sampling 
controls from the non-cases. This design yields sampling weights equal to 1 for cases and
 |
for sampled controls. Control weights ranged from 24 for 
to 499 for 
, allowing us to examine performance under increasing weight magnitude.
Estimators compared
For each simulated dataset, we fitted both Cox and casebase models under three analysis strategies:
Full cohort (oracle) analysis using the entire cohort without two-phase sampling.
Naive two-phase analysis using the sampled data but ignoring inverse-probability weights.
Weighted two-phase analysis using inverse-probability weights to account for outcome-dependent sampling.
The Cox model specified a proportional hazards model with unspecified baseline hazard. The casebase model specified a Weibull proportional hazards model,
 |
For casebase models in the primary simulations, we considered base-to-case ratios of 100:1, 200:1, and 500:1. For weighted Cox models, robust sandwich standard errors were used. For weighted casebase models, standard errors were computed using the proposed dfbeta-based jackknife/sandwich estimator.
For each primary scenario, 1,000 replicate datasets were generated. We evaluated mean coefficient estimates, bias, empirical standard error, mean estimated standard error, and 95% confidence interval coverage (Table 2 in Appendix A). Coverage between 93.6% and 96.4% was considered compatible with the nominal 95% level for 1,000 simulations, corresponding approximately to a two-Monte-Carlo-standard-error interval around 95% coverage.
Aggregation of simulation results.
For each estimator, summary statistics were computed using simulation replicates with finite coefficient and standard error estimates for that estimator. This estimator-specific aggregation prevents isolated numerical failures in one model from propagating to other estimators in the same scenario. In the settings considered here, this affected only a small number of weighted Cox replicates in the most extreme log-normal scenarios.
Primary simulation results
For binary covariates (Tables 4–7 in Appendix A), weighted casebase regression and weighted Cox regression performed similarly across all cohort sizes. Bias was small for both weighted estimators, estimated standard errors were close to empirical standard errors, and coverage remained close to nominal even when the control weight increased to 499. In contrast, naive two-phase analyses that ignored inverse-probability weights were substantially biased and had poor coverage, illustrating the need to account for the outcome-dependent sampling design.
For normally distributed covariates (Tables 8–11 in Appendix A), weighting reduced the large selection bias seen in the naive analyses, but finite-sample performance was less stable than in the binary-covariate setting. As the cohort size increased while the number of sampled controls remained fixed, the inverse-probability weights became larger without increasing the amount of phase-two covariate information. Both weighted Cox and weighted casebase estimators showed upward bias and under-coverage in these settings, indicating that the instability reflects the weighted two-phase design rather than the casebase formulation specifically.
For log-normal covariates (Tables 12–15 in Appendix A), performance was again affected by the combination of skewed covariates, fixed phase-two sample size, and large inverse-probability weights. Weighted casebase and weighted Cox regression showed broadly similar qualitative behaviour, although the most extreme scenarios exhibited greater numerical instability, particularly for weighted Cox regression. These results support the interpretation that the main finite-sample limitation is weight variability under sparse phase-two sampling, not the casebase representation itself.
Figure 1 summarizes the bias and relative standard-error behaviour for the 500:1 casebase models across covariate distributions and cohort sizes. The figure reinforces the table-based findings: binary covariate settings remain stable, whereas continuous and especially skewed covariate settings become more sensitive as inverse-probability weights increase.
Additional sensitivity simulations
The primary simulations assess the proposed estimator in a correctly specified univariate setting across binary, normal, and log-normal covariate distributions. We therefore added four targeted sensitivity scenarios to evaluate performance under more realistic features of applied two-phase studies. These scenarios were not intended to form a full factorial expansion over all covariate distributions. Instead, each sensitivity analysis was anchored to a stable binary-covariate primary design unless the feature being studied required an additional continuous variable. Thus, the multivariable scenario used one binary and one continuous covariate, whereas the baseline misspecification, random censoring, and estimated-inclusion-probability scenarios used a binary target covariate. These scenarios are summarized in Table 3 in Appendix A. To limit computational burden and improve interpretability, these additional simulations used cohort sizes 
, included 200 cases and 200 sampled controls unless otherwise noted, and used a fixed casebase base-to-case ratio of 200:1. Each scenario was replicated 1,000 times.
Multivariable covariate scenario
To assess performance with more than one covariate, we simulated one binary and one continuous covariate. Specifically,
 |
and
 |
so that the two covariates were mildly correlated. Event times were generated from a Weibull proportional hazards model,
 |
with 
and 
. These moderate effects were chosen to avoid near-deterministic event selection while still producing meaningful covariate associations. Weighted Cox and weighted casebase models were fitted using both covariates. Bias, empirical standard error, mean estimated standard error, and coverage were evaluated separately for 
and 
.
Baseline hazard misspecification scenario
To evaluate robustness to misspecification of the baseline hazard, we generated event times from a Gompertz proportional hazards model but fitted the primary casebase analysis using a Weibull time specification. To isolate the effect of baseline hazard misspecification from the instability observed in the continuous-covariate primary settings, this sensitivity analysis used a binary covariate,
 |
The data-generating hazard was
 |
with β = 1.5 and 
. Event times were generated using the inverse cumulative hazard,
 |
The value of 
was calibrated to produce approximately 200 observed events per cohort after administrative truncation.
We compared three estimators: weighted Cox regression, weighted casebase regression with a misspecified Weibull time term łog (t), and weighted casebase regression with a flexible natural cubic spline in łog (t) with three degrees of freedom. This comparison separates the effect of using the casebase framework from the effect of imposing an overly restrictive baseline hazard model. In addition to coefficient bias and coverage, we evaluated absolute risk calibration at a fixed prediction horizon 
, chosen as the median event time among cases in the corresponding full cohort simulation and evaluated at the reference covariate value x=1.
Random censoring sensitivity scenario
The primary simulations used administrative truncation to obtain a fixed number of events. To examine robustness to additional random censoring beyond administrative truncation, we added independent exponential censoring to the binary-covariate primary design. Specifically,
 |
and event times were generated using the same Weibull proportional hazards model and parameter values as in the corresponding binary primary scenario.
Independent censoring times were generated as
 |
where 
was calibrated to achieve a censoring proportion of approximately 10–15% before administrative truncation. The censoring distribution did not depend on 
, event time, or any auxiliary variable.
The observed follow-up time was defined as
 |
with event indicator
 |
where τ was chosen as the time of the 200th observed event after applying the random censoring mechanism. This preserves the fixed-case structure of the primary two-phase design.
Under this construction, censoring is independent and non-informative. Weighted Cox and weighted casebase models were then fitted using the same two-phase sampling scheme as in the primary simulations, with all 200 observed cases included and 200 controls sampled from the observed non-cases.
Estimated inclusion probability scenario
The primary simulations used known design-based inclusion probabilities. To examine the impact of estimated weights, we simulated a setting in which control inclusion probabilities depended on an observed phase-one auxiliary variable. To keep this sensitivity analysis anchored to the binary primary setting, the target covariate was binary. Specifically,
 |
and
 |
We set 
and calibrated 
so that 
. Event times were generated from a Weibull proportional hazards model with hazard
 |
using the same target covariate effect as in the corresponding binary primary scenario, namely β =1.5.
All cases were included with probability 1. Among non-cases, the phase-two sampling indicator 
was generated from
 |
where 
and 
was calibrated within each scenario so that the expected number of sampled controls was approximately 200. We compared two weighted analyses:
Known-weight analysis, using the true inclusion probabilities
.Estimated-weight analysis, using
obtained from a logistic regression model for phase-two control selection as a function of 
among non-cases.
Cases were assigned weight 1 in both analyses. For sampled controls, weights were 
in the known-weight analysis and 
in the estimated-weight analysis.
This scenario evaluates whether substituting estimated inclusion probabilities materially changes bias, empirical variability, standard error estimation, or coverage when the target regression covariate follows the binary primary design.
Sensitivity simulation results
Multivariable covariate sensitivity analysis.
Across both sample sizes, weighted Cox and weighted casebase regression produced similar results for both coefficients (Table 16 in Appendix A). At 
, bias ranged from 0.011 to 0.014 and coverage ranged from 0.948 to 0.952. At 
, bias remained small, ranging from 0.007 to 0.016, with coverage between 0.942 and 0.954.
Baseline hazard misspecification.
Under Gompertz data generation with a binary target covariate, all three estimators showed similar coefficient performance (Table 17 in Appendix A) and very small absolute risk bias (Table 18 in Appendix A). At 
, bias was approximately 0.010–0.011 and coverage was 0.958–0.961. At 
, bias was approximately 0.014 and coverage was 0.948–0.952. Absolute risk calibration bias at the reference covariate value remained very small.
Random censoring.
Under light independent exponential censoring, both methods remained close to their corresponding binary primary-simulation behavior (Table 19 in Appendix A). The achieved mean censoring proportion was 0.120 at 
and 0.117 at 
. At 
, bias was approximately 0.020 and coverage ranged from 0.938 to 0.944. At 
, bias was approximately 0.008 and coverage ranged from 0.955 to 0.957. No convergence failures occurred.
Estimated inclusion probabilities.
In the estimated-inclusion-probability scenario, known-weight and estimated-weight analyses gave nearly identical results (Table 20 in Appendix A). At 
, bias ranged from 0.002 to 0.004 and coverage from 0.950 to 0.954 across weighted Cox and weighted casebase fits. At 
, bias was approximately 0.009 and coverage ranged from 0.949 to 0.954. The average number of sampled controls remained close to 200 at both cohort sizes, and the estimated-weight analyses closely tracked the corresponding known-weight analyses.
Simulation summary
Weighted casebase and weighted Cox regression had similar finite-sample performance across the scenarios studied. Both performed well for binary covariates and in the targeted multivariable, baseline-misspecification, censoring, and estimated-weight sensitivity analyses, whereas continuous or highly skewed covariates combined with fixed phase-two sample sizes and large weights produced instability for both methods. These findings support weighted casebase regression as a complementary approach when smooth hazard or absolute-risk estimation is required, rather than as a replacement for weighted Cox regression when the baseline hazard may remain unspecified.
Illustrative implementation using an Ontario Health Study biomarker substudy
We illustrate the computational implementation of the weighted casebase methodology using data from a biomarker substudy within the Ontario Health Study (OHS). The OHS is a prospective cohort of over 225,000 Ontario residents designed to investigate environmental, lifestyle, and genetic factors associated with chronic disease development [1].
This analysis is intended as an illustrative implementation of the weighted casebase workflow rather than as a definitive epidemiologic analysis of cell-free DNA (cfDNA) methylation and breast cancer risk. The original biomarker substudy used an outcome-dependent frequency-matched sampling design. A fully design-consistent analysis of that study would require stratum-specific inclusion probabilities, conditional or stratified methods, or other analyses explicitly incorporating the matching structure. Because those features are not fully available in the present analysis, we focus on demonstrating how the proposed weighted casebase estimator can be implemented with estimated inverse-probability weights, repeated sampled person-moments, and smooth absolute risk prediction. The resulting estimates should therefore be interpreted as a methodological illustration rather than as definitive evidence about breast cancer risk prediction.
Study design and data
The biomarker substudy selected 80 women who developed breast cancer after enrollment and 70 women who remained cancer-free from OHS cohort participants with available plasma samples. The original study employed frequency matching on age in 5-year categories, sex, date of plasma collection in 6-month windows, ethnicity, smoking status, and alcohol consumption. Because all participants in this substudy were women, sex was fixed in the analytic sample.
For each participant, cfDNA was extracted from stored plasma samples and methylation levels were measured across 1,760,313 binned genomic regions profiled using cfMeDIP-Seq. Following quality control and preprocessing, the analysis focused on 1,000 CpG sites with the largest variance across samples. These measurements were summarized using principal component-based dimension reduction, with candidate models considering up to five methylation principal components.
Controls were selected through frequency matching, yielding an outcome-dependent two-phase sample. Inclusion probabilities were not directly observed from the original sampling scheme. We therefore used the estimated inclusion probabilities reported by Cheng et al. (2023) [9], which were derived using age-specific cumulative breast cancer incidence rates from the Canadian Cancer Registry and all-cause mortality rates in Ontario reported by Statistics Canada. These estimated inclusion probabilities were used to construct inverse-probability weights. The same weights were incorporated consistently for parameter estimation, risk prediction, and performance evaluation throughout this illustrative analysis.
Because the original frequency-matching strata were not explicitly incorporated into the weighted casebase likelihood, the analysis should be viewed as a simplified weighted two-phase analysis. Available matching variables were included as adjustment covariates where possible, but this adjustment is not equivalent to a fully matched or stratified design-consistent analysis.
Analysis methods
The dataset was partitioned into a training set and a test set. The training set consisted of samples from batches 1 through 7, while the test set consisted of samples from batch 8. This split was chosen because batch 8 was analyzed later. The training set contained 124 samples, and the test set contained 26 samples. The test set was held out during model tuning and was used only after final tuning parameters were selected.
Given the high dimensionality of the methylation data relative to the sample size, we used supervised principal components for dimension reduction. In the training set, univariate weighted Cox score statistics were computed for individual methylation regions. Regions with score statistics exceeding a threshold θ were retained, and principal components were then computed using the retained regions. Candidate values were 
for the number of principal components and 
for the score statistic threshold.
Tuning was performed using repeated 10-fold cross-validation within the training set. Each fold contained a balanced mix of cases and controls, with fold sizes ranging from 11 to 13 individuals. For each candidate pair 
, the supervised PCA procedure was performed within the training data for each cross-validation split. Weighted Cox and weighted casebase models were then fit using the selected principal components, and prediction performance was evaluated in the held-out fold.
For the weighted Cox model, the linear predictor took the form
 |
where j denotes the number of principal components selected for the candidate model. For the weighted casebase model, the corresponding linear predictor was
 |
This Weibull casebase specification permits direct estimation of smooth hazard, survival, and absolute risk functions over time.
For the weighted casebase analysis, we used a base-to-case ratio of 100:1. Standard errors were computed using the proposed dfbeta-based jackknife/sandwich estimator, which accounts for repeated sampled person-moments within individuals conditional on the fitted sampling weights. Because the inclusion probabilities in this application were estimated externally, the reported standard errors should be interpreted as conditional on these estimated weights. They do not propagate uncertainty from the external incidence and mortality rates used to estimate the inclusion probabilities.
The weighted Cox model was fit using the survival package in R with robust variance estimation. Both weighted Cox and weighted casebase analyses incorporated inverse-probability weights to account for the outcome-dependent two-phase sampling design. Available matching variables, including age at enrollment, ethnicity indicators, smoking status, and alcohol consumption, were included as adjustment covariates where appropriate. These covariate adjustments partially address baseline differences induced by the matching design, but they do not constitute a full matched-design analysis.
Model performance during cross-validation was evaluated using a weighted concordance metric [10],
 |
where I(· ) is an indicator function, 
is the event indicator, 
and 
are sampling weights, 
and 
are follow-up times, 
and 
are covariate vectors, and 
denotes the fitted coefficient vector. The final Cox and casebase models were selected using the highest mean weighted concordance from repeated cross-validation and were then applied once to the independent test set.
Prediction performance was also evaluated using a weighted Brier score,
 |
where 
is the censoring time, 
is the event indicator, 
is the Kaplan-Meier estimate of the censoring survival function, and 
is the predicted probability of breast cancer diagnosis by time t for individual i. Predicted absolute risks were plotted against weighted Kaplan-Meier estimates to provide a descriptive assessment of calibration.
Results
Cross-validation within the training set selected two principal components 
and a score statistic threshold of θ =1.89, yielding 24 methylation regions for the final analysis. These tuning parameters were then used to fit both the weighted Cox and weighted casebase models.
Table 21 in Appendix A presents weighted concordance values for the training and test sets. Both methods achieved identical discrimination in the training set, with concordance equal to 0.70. In the test set, the weighted casebase model had a slightly higher point estimate of concordance than the weighted Cox model, 0.68 versus 0.65. However, the test set contained only 26 individuals, so this difference should not be interpreted as evidence that the casebase model has superior predictive performance. The more relevant conclusion is that the weighted casebase workflow produced predictions with discrimination comparable to the weighted Cox workflow in this small illustrative example.
The weighted Brier scores shown in Figure 2 provide a time-varying descriptive assessment of prediction error. Across follow-up time, the weighted Cox and weighted casebase models produced very similar Brier score curves in both the training and test sets. This suggests comparable overall predictive performance in this dataset, while reinforcing that the primary distinction between the methods is not discrimination but the form of the fitted risk function.
Figure 3 compares predicted absolute risks with weighted Kaplan-Meier estimates. For this display, individuals were stratified by the median fitted risk score, using the Cox model risk score for consistency across methods. The casebase model produced smooth estimated risk curves over time, whereas the Cox model produced step functions tied to observed event times through its estimated baseline cumulative hazard. Both methods showed reasonable agreement for the lower-risk groups. In the higher-risk groups, both methods tended to overestimate risk, particularly in the test set.
These calibration patterns should be interpreted cautiously. The test set was small, the supervised PCA procedure may have introduced overfitting despite internal cross-validation, and the matching structure of the original study was only partially addressed through covariate adjustment and inverse-probability weights. In addition, the instability of the weighted Kaplan-Meier estimate at later follow-up times reflects the limited number of individuals remaining at risk, especially in subgroup-specific calibration plots.
Overall, this application demonstrates the practical implementation of the weighted casebase framework in a realistic biomarker substudy with estimated inverse-probability weights. The analysis illustrates how weighted casebase regression can be embedded in a high-dimensional prediction workflow, how repeated person-moments can be handled through individual-level variance aggregation, and how smooth absolute risk curves can be obtained from a GLM-based hazard model. The application should not be interpreted as establishing improved prediction for cfDNA methylation markers or as a complete matched-design analysis of the original OHS substudy.
The weighted casebase analysis was implemented using the casebaseweights R package, available at https://github.com/karinakwan/casebaseweights/. The package provides functions for fitting weighted casebase models and computing the individual-level dfbeta aggregation variance estimator described in this paper.
Discussion
We developed a weighted extension of the casebase framework for outcome-dependent two-phase survival designs. The proposed approach combines two distinct sampling adjustments: the casebase offset accounts for sampling of person-moments from observed person-time [4, 5], whereas inverse-probability weights correct for unequal sampling of individuals into the phase-two sample [3]. This yields a GLM-based implementation of parametric or flexible-parametric hazard regression under outcome-dependent sampling.
The methodological contribution is not that weighted casebase regression replaces weighted Cox regression. Rather, the contribution is to extend the casebase representation to settings in which individuals are selected through outcome-dependent two-phase sampling and may then contribute repeated sampled person-moments. This creates a two-level sampling structure: individuals are sampled into the phase-two dataset, and person-moments are sampled from the observed follow-up time of those individuals. By formulating the estimator as an individual-level M-estimator, the repeated person-moments can be aggregated at the source-individual level for inference.
The proposed dfbeta-based jackknife/sandwich variance estimator follows from this individual-level influence representation. By aggregating observation-level dfbeta diagnostics within individuals, the estimator accounts for dependence among repeated sampled person-moments from the same participant and provides a computationally efficient fixed-weight sandwich covariance estimator. This is useful when the casebase dataset contains many sampled person-moments or when models are embedded within cross-validation or high-dimensional workflows. When inclusion probabilities are estimated rather than known design quantities, the default dfbeta-based estimator should be interpreted as conditional on the supplied weights. Uncertainty from estimating inclusion probabilities can be incorporated using a stacked estimating-equation sandwich estimator or by using a subject-level bootstrap in which the inclusion probability model is re-estimated within each bootstrap replicate.
The simulation studies showed that weighted casebase regression had finite-sample behaviour broadly comparable to weighted Cox regression in the settings examined. In the primary correctly specified simulations, both methods performed well for binary covariates, with negligible bias and approximately nominal coverage even when sampling weights were large. For continuous covariates, increasing weight magnitude led to greater finite-sample instability for both weighted casebase and weighted Cox regression. This pattern is consistent with the known sensitivity of inverse-probability weighted estimators to extreme weights [11], and indicates that the observed instability reflects the weighting structure rather than a feature unique to the casebase formulation. In these scenarios, the number of sampled controls was fixed while the full cohort size increased, so larger cohorts increased the control weights without increasing the amount of phase-two covariate information. In our simulations, this was most evident for weighted Cox regression under log-normal covariates, where occasional numerical instability and heavy-tailed robust variance estimates were observed.
The additional sensitivity simulations were included to assess settings closer to those encountered in applied two-phase studies. These simulations considered multiple covariates, baseline hazard misspecification, independent random censoring, and estimated inclusion probabilities. In these sensitivity analyses, weighted casebase and weighted Cox regression again showed similar coefficient performance, with small bias, close agreement between empirical and estimated standard errors, and near-nominal coverage. The misspecification analysis also showed that flexible time bases can be used within the casebase framework to reduce reliance on a strict Weibull baseline specification. These additional scenarios are important because the casebase approach requires specification of the time component of the hazard model, whereas Cox regression leaves the baseline hazard unspecified.
The main practical distinction between weighted Cox and weighted casebase regression therefore concerns the role of the baseline hazard. Weighted Cox regression remains the natural default when the scientific goal is estimation of relative hazard ratios and the baseline hazard can be treated as a nuisance function. Weighted casebase regression is complementary: it is useful when smooth estimation of the hazard, survival function, or absolute risk function is itself part of the inferential target. This feature is particularly relevant in risk prediction, calibration, or decision-support settings, where absolute risks over clinically meaningful time horizons are often more directly interpretable than relative hazards alone.
The Ontario Health Study biomarker substudy illustrates this practical distinction. The weighted Cox and weighted casebase analyses produced similar discrimination and Brier score patterns in this small example, while the casebase model yielded smooth absolute risk curves through its parametric time specification. However, this application should be interpreted as a methodological illustration rather than as a definitive epidemiologic analysis of cfDNA methylation and breast cancer risk. The original substudy used frequency matching, and a fully design-consistent analysis would require stratum-specific inclusion probabilities, conditional or stratified methods, or another approach that explicitly incorporates the matching structure. In addition, the test set was small, the supervised PCA workflow may have introduced overfitting despite internal cross-validation, and uncertainty in the externally estimated inclusion probabilities was not fully propagated in the main analysis.
A natural extension is to stratified two-phase designs. If base person-moments are sampled separately within design stratum h, records in that stratum use the offset 
and the corresponding stratum-specific first-order inclusion weight; participant-level influence contributions are then aggregated as above, with the variance calculation respecting dependence induced by the sampling design. Fully matched designs are less direct: matched-set membership and set-specific selection probabilities must enter 
, and conditional or stratified nuisance components may be required. Because a purely conditional analysis removes the baseline component needed for absolute-risk prediction, absolute risks would additionally require calibration of the baseline hazard from the full cohort or an external source. Formal development and evaluation of this matched-design extension are beyond the present scope.
Several limitations warrant emphasis. First, as with other inverse-probability weighted methods, very large sampling weights may lead to instability, particularly for continuous and highly skewed covariates, where we observed occasional numerical failures and heavy-tailed variance estimates in weighted Cox regression [11]. In practice, this may be mitigated by increasing the number of sampled controls, improving the sampling design, using stabilized or calibrated weights, or conducting sensitivity analyses for influential weights. Second, weighted casebase regression requires specification of the time component of the hazard model. Flexible time bases can reduce sensitivity to strict parametric assumptions, but misspecification can affect absolute risk estimation. Third, when inclusion probabilities are estimated, variance estimation should account for the weight-estimation step if uncertainty in those probabilities is material. Fourth, all simulations were low dimensional: the primary scenarios used one covariate, and the multivariable sensitivity analysis used two. They therefore do not establish performance when the number of candidate predictors is large relative to the phase-two sample size, under regularization, or with complex nonlinear and interaction structures. Fifth, the current development focuses on right-censored survival outcomes under two-phase sampling; extensions to time-dependent covariates, competing risks, recurrent events, and more complex clustered survival settings remain useful directions for future work. Recent work on unified survival models for clustered data, cure fractions, and competing risks highlights the broader need for methods that remain computationally accessible while accommodating complex survival-data structures [6].
Conclusions
Weighted casebase regression provides a theoretically grounded and computationally practical approach to parametric or flexible-parametric hazard modeling under outcome-dependent two-phase sampling. Its finite-sample performance was broadly comparable to weighted Cox regression in the scenarios studied, while offering direct smooth estimation of hazards and absolute risks within a familiar GLM framework. The method should be viewed as a complementary tool rather than a competitor to weighted Cox regression: Cox regression is preferable when the baseline hazard can remain unspecified, whereas weighted casebase regression is useful when the baseline hazard or absolute risk function is part of the scientific target.
Acknowledgements
The authors thank Sahir Bhatnagar for consultations regarding the casebase methodology.
Abbreviations
- cfDNA
cell-free DNA
- cfMeDIP-Seq
cell-free methylated DNA immunoprecipitation sequencing
- GLM
generalized linear model
- IPW
inverse-probability weighting
- OHS
Ontario Health Study
- PCA
principal component analysis
Appendix
Connection with standard inverse-probability weighted M-estimation
This appendix explains how the weighted casebase estimator in the Large-sample justification section fits within standard inverse-probability weighted M-estimation theory. The key point is that the rows of the casebase dataset are not treated as independent observations for inference. Instead, all case-series and base-series records contributed by the same participant are aggregated into a single individual-level score contribution.
For known inclusion probabilities, the estimating equation can be written as
where
 |
Here 
is the phase-two selection indicator, 
, and 
contains all records contributed by participant i. Repeated sampled person-moments from the same participant are therefore handled within 
, rather than by treating the corresponding rows as independent observations.
The two sampling adjustments enter at different levels. The casebase offset accounts for sampling person-moments from observed phase-two at-risk time. Conditional on the realized phase-two sample, the base series is sampled at rate 
per unit of observed phase-two person-time, which gives the offset
 |
The inverse-probability weight accounts for unequal selection of individuals into the phase-two sample. Under the usual two-phase sampling condition,
 |
for any individual-level contribution 
entering the target estimating equation. Thus, after aggregation at the participant level, the weighted phase-two estimating equation has the same population target as the corresponding full-cohort estimating equation.
At the correctly specified parameter value 
, the casebase construction implies the odds representation
 |
Equivalently, the expected base-series contribution, after accounting for the sampling rate 
, balances the expected case-series contribution. Combining this casebase identity with the inverse-probability-weighting identity gives
 |
Therefore 
is the population solution of the individual-level estimating equation.
Under the standard regularity conditions for inverse-probability weighted M-estimation summarized in the Large-sample justification section, the usual linearization gives
 |
where
 |
If
 |
then
 |
as stated in the Large-sample justification section. The covariance matrix 
is defined at the level of the aggregated individual-level contributions, so it includes the dependence induced by repeated sampled person-moments from the same participant and any dependence induced by the sampling design.
When inclusion probabilities are estimated, write 
and define 
as the preceding score contribution with 
replaced by 
. Let 
solve
 |
where 
is the individual-level estimating-equation contribution for the inclusion-probability model. Define
 |
The joint estimator solves the stacked estimating equation
 |
Applying the same M-estimation argument to this stacked system gives
 |
where
 |
The covariance for 
is the 
-block of this sandwich covariance matrix.
The dfbeta-based estimator in the Dfbeta-based jackknife/sandwich variance estimation section estimates the fixed-weight covariance, conditional on the supplied weights. If inclusion probabilities are estimated and uncertainty from that step is to be propagated, one may use the stacked sandwich estimator above or a subject-level bootstrap that re-estimates the inclusion-probability model, regenerates the base series, and refits the weighted casebase model in each bootstrap replicate.
Table 1.
Parameters used in the primary simulation studies. All scenarios used a single covariate, covariate effect β = 1.5, 200 cases, 200 sampled controls in the two-phase analyses, casebase base-to-case ratios of 100, 200, and 500, and 1,000 simulation replicates per setting
| Scenario | Covariate distribution | N | Weibull scale (λ ) | Weibull shape (ν ) | β | Cases | Controls |
|---|---|---|---|---|---|---|---|
| Binary | Bernoulli(0.3) | 5,000 |
|
6.74 | 1.5 | 200 | 200 |
| Binary | Bernoulli(0.3) | 10,000 |
|
5.01 | 1.5 | 200 | 200 |
| Binary | Bernoulli(0.3) | 50,000 |
|
4.20 | 1.5 | 200 | 200 |
| Binary | Bernoulli(0.3) | 100,000 |
|
4.18 | 1.5 | 200 | 200 |
| Normal | Normal(0, 1) | 5,000 |
|
7.00 | 1.5 | 200 | 200 |
| Normal | Normal(0, 1) | 10,000 |
|
7.00 | 1.5 | 200 | 200 |
| Normal | Normal(0, 1) | 50,000 |
|
7.00 | 1.5 | 200 | 200 |
| Normal | Normal(0, 1) | 100,000 |
|
7.00 | 1.5 | 200 | 200 |
| Log-normal | Log-normal(0, 1) | 5,000 |
|
5.00 | 1.5 | 200 | 200 |
| Log-normal | Log-normal(0, 1) | 10,000 |
|
5.00 | 1.5 | 200 | 200 |
| Log-normal | Log-normal(0, 1) | 50,000 |
|
5.00 | 1.5 | 200 | 200 |
| Log-normal | Log-normal(0, 1) | 100,000 |
|
5.00 | 1.5 | 200 | 200 |
Table 2.
Performance measures used in the primary simulation study. For estimator e, let 
denote the set of simulation replicates with finite coefficient and standard error estimates, and let 
| Performance measure | Estimate |
|---|---|
| Mean coef |
|
| Bias |
|
| EmpSE |
|
| Mean estimated SE |
|
| Coverage |
|
| Coverage SE |
|
Mean coef: mean coefficient estimate. EmpSE: empirical standard error across successful replicates for that estimator. Mean estimated SE: root mean squared reported standard error across successful replicates for that estimator. Coverage SE: Monte Carlo standard error of the estimated coverage
Table 3.
Additional sensitivity simulation scenarios. The primary simulations considered univariate covariates, known sampling weights, administrative truncation, and correctly specified Weibull casebase models. The additional scenarios were designed to evaluate performance under targeted departures from that setting
| Scenario | Purpose | Data-generating mechanism | Main comparison |
|---|---|---|---|
| Multivariable covariates | Evaluate performance with more than one covariate | Weibull proportional hazards model with one binary covariate and one continuous covariate, mildly correlated | Weighted Cox vs weighted casebase for both regression coefficients |
| Baseline misspecification | Evaluate sensitivity to incorrect baseline hazard specification | Gompertz proportional hazards model with a binary target covariate; casebase model fitted with Weibull and spline time terms | Weighted Cox vs Weibull casebase vs spline casebase |
| Random censoring | Evaluate robustness to additional independent censoring | Binary-covariate Weibull proportional hazards model with independent exponential censoring calibrated to approximately 10–15% censoring before administrative truncation | Weighted Cox vs weighted casebase |
| Estimated inclusion probabilities | Evaluate the effect of estimating phase-two control inclusion probabilities | Binary target covariate with auxiliary phase-one variable Z; control sampling probability depends on Z | Known weights vs estimated weights for weighted Cox and weighted casebase |
Table 4.
Results for a binary covariate, N = 5,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Binary covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.513 | 0.013 | 0.149 | 0.146 | 0.959 (0.006) |
| Ratio 100 full | 1.514 | 0.014 | 0.152 | 0.149 | 0.952 (0.007) |
| Ratio 200 full | 1.514 | 0.014 | 0.150 | 0.148 | 0.956 (0.006) |
| Ratio 500 full | 1.513 | 0.013 | 0.149 | 0.146 | 0.955 (0.007) |
| Cox naive | 1.054 | -0.446 | 0.149 | 0.151 | 0.170 (0.012) |
| Ratio 100 naive | 1.032 | -0.468 | 0.153 | 0.152 | 0.138 (0.011) |
| Ratio 200 naive | 1.034 | -0.466 | 0.151 | 0.150 | 0.141 (0.011) |
| Ratio 500 naive | 1.037 | -0.463 | 0.150 | 0.149 | 0.145 (0.011) |
| Cox weighted | 1.508 | 0.008 | 0.210 | 0.215 | 0.947 (0.007) |
| Ratio 100 weighted | 1.511 | 0.011 | 0.214 | 0.219 | 0.946 (0.007) |
| Ratio 200 weighted | 1.508 | 0.008 | 0.212 | 0.217 | 0.944 (0.007) |
| Ratio 500 weighted | 1.509 | 0.009 | 0.211 | 0.216 | 0.948 (0.007) |
Table 5.
Results for a binary covariate, N = 10,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Binary covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.508 | 0.008 | 0.149 | 0.145 | 0.956 (0.006) |
| Ratio 100 full | 1.508 | 0.008 | 0.152 | 0.147 | 0.966 (0.006) |
| Ratio 200 full | 1.508 | 0.008 | 0.150 | 0.147 | 0.956 (0.006) |
| Ratio 500 full | 1.508 | 0.008 | 0.150 | 0.146 | 0.961 (0.006) |
| Cox naive | 1.046 | -0.454 | 0.150 | 0.138 | 0.136 (0.011) |
| Ratio 100 naive | 1.024 | -0.476 | 0.152 | 0.138 | 0.111 (0.010) |
| Ratio 200 naive | 1.026 | -0.474 | 0.151 | 0.136 | 0.100 (0.009) |
| Ratio 500 naive | 1.029 | -0.471 | 0.150 | 0.135 | 0.107 (0.010) |
| Cox weighted | 1.515 | 0.015 | 0.213 | 0.199 | 0.968 (0.006) |
| Ratio 100 weighted | 1.517 | 0.017 | 0.216 | 0.201 | 0.963 (0.006) |
| Ratio 200 weighted | 1.516 | 0.016 | 0.214 | 0.201 | 0.961 (0.006) |
| Ratio 500 weighted | 1.516 | 0.016 | 0.213 | 0.200 | 0.967 (0.006) |
Table 6.
Results for a binary covariate, N = 50,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Binary covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.495 | -0.005 | 0.149 | 0.147 | 0.953 (0.007) |
| Ratio 100 full | 1.496 | -0.004 | 0.151 | 0.148 | 0.950 (0.007) |
| Ratio 200 full | 1.494 | -0.006 | 0.150 | 0.148 | 0.954 (0.007) |
| Ratio 500 full | 1.494 | -0.006 | 0.150 | 0.147 | 0.954 (0.007) |
| Cox naive | 1.029 | -0.471 | 0.150 | 0.148 | 0.112 (0.010) |
| Ratio 100 naive | 1.006 | -0.494 | 0.152 | 0.147 | 0.098 (0.009) |
| Ratio 200 naive | 1.010 | -0.490 | 0.151 | 0.145 | 0.096 (0.009) |
| Ratio 500 naive | 1.012 | -0.488 | 0.150 | 0.145 | 0.095 (0.009) |
| Cox weighted | 1.504 | 0.004 | 0.215 | 0.213 | 0.958 (0.006) |
| Ratio 100 weighted | 1.506 | 0.006 | 0.217 | 0.215 | 0.958 (0.006) |
| Ratio 200 weighted | 1.505 | 0.005 | 0.216 | 0.215 | 0.959 (0.006) |
| Ratio 500 weighted | 1.504 | 0.004 | 0.215 | 0.214 | 0.960 (0.006) |
Table 7.
Results for a binary covariate, N = 100,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Binary covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.499 | -0.001 | 0.149 | 0.145 | 0.962 (0.006) |
| Ratio 100 full | 1.500 | 0.000 | 0.151 | 0.147 | 0.963 (0.006) |
| Ratio 200 full | 1.500 | 0.000 | 0.150 | 0.145 | 0.963 (0.006) |
| Ratio 500 full | 1.499 | -0.001 | 0.150 | 0.145 | 0.961 (0.006) |
| Cox naive | 1.033 | -0.467 | 0.150 | 0.148 | 0.127 (0.011) |
| Ratio 100 naive | 1.012 | -0.488 | 0.152 | 0.147 | 0.102 (0.010) |
| Ratio 200 naive | 1.015 | -0.485 | 0.151 | 0.146 | 0.095 (0.009) |
| Ratio 500 naive | 1.015 | -0.485 | 0.150 | 0.146 | 0.092 (0.009) |
| Cox weighted | 1.505 | 0.005 | 0.215 | 0.213 | 0.954 (0.007) |
| Ratio 100 weighted | 1.507 | 0.007 | 0.217 | 0.217 | 0.955 (0.007) |
| Ratio 200 weighted | 1.506 | 0.006 | 0.216 | 0.215 | 0.952 (0.007) |
| Ratio 500 weighted | 1.505 | 0.005 | 0.216 | 0.213 | 0.956 (0.006) |
Table 8.
Normal covariate results, N = 5,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Normal covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.502 | 0.002 | 0.073 | 0.074 | 0.949 (0.007) |
| Ratio 100 full | 1.504 | 0.004 | 0.084 | 0.082 | 0.958 (0.006) |
| Ratio 200 full | 1.503 | 0.003 | 0.080 | 0.079 | 0.958 (0.006) |
| Ratio 500 full | 1.504 | 0.004 | 0.076 | 0.077 | 0.952 (0.007) |
| Cox naive | 0.841 | -0.659 | 0.066 | 0.061 | 0.000 (0.000) |
| Ratio 100 naive | 0.821 | -0.679 | 0.069 | 0.063 | 0.000 (0.000) |
| Ratio 200 naive | 0.819 | -0.681 | 0.067 | 0.060 | 0.000 (0.000) |
| Ratio 500 naive | 0.819 | -0.681 | 0.066 | 0.059 | 0.000 (0.000) |
| Cox weighted | 1.539 | 0.039 | 0.140 | 0.158 | 0.860 (0.011) |
| Ratio 100 weighted | 1.538 | 0.038 | 0.142 | 0.162 | 0.870 (0.011) |
| Ratio 200 weighted | 1.541 | 0.041 | 0.141 | 0.159 | 0.860 (0.011) |
| Ratio 500 weighted | 1.540 | 0.040 | 0.140 | 0.159 | 0.855 (0.011) |
Table 9.
Normal covariate results, N = 10,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Normal covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.504 | 0.004 | 0.072 | 0.068 | 0.967 (0.006) |
| Ratio 100 full | 1.505 | 0.005 | 0.084 | 0.080 | 0.965 (0.006) |
| Ratio 200 full | 1.506 | 0.006 | 0.079 | 0.077 | 0.958 (0.006) |
| Ratio 500 full | 1.504 | 0.004 | 0.076 | 0.073 | 0.962 (0.006) |
| Cox naive | 0.785 | -0.715 | 0.063 | 0.057 | 0.000 (0.000) |
| Ratio 100 naive | 0.763 | -0.737 | 0.066 | 0.058 | 0.000 (0.000) |
| Ratio 200 naive | 0.762 | -0.738 | 0.064 | 0.056 | 0.000 (0.000) |
| Ratio 500 naive | 0.761 | -0.739 | 0.063 | 0.055 | 0.000 (0.000) |
| Cox weighted | 1.564 | 0.064 | 0.167 | 0.205 | 0.826 (0.012) |
| Ratio 100 weighted | 1.566 | 0.066 | 0.169 | 0.206 | 0.820 (0.012) |
| Ratio 200 weighted | 1.563 | 0.063 | 0.168 | 0.206 | 0.818 (0.012) |
| Ratio 500 weighted | 1.564 | 0.064 | 0.167 | 0.206 | 0.822 (0.012) |
Table 10.
Normal covariate results, N = 50,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Normal covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.500 | 0.000 | 0.071 | 0.071 | 0.948 (0.007) |
| Ratio 100 full | 1.505 | 0.005 | 0.085 | 0.087 | 0.942 (0.007) |
| Ratio 200 full | 1.504 | 0.004 | 0.080 | 0.079 | 0.951 (0.007) |
| Ratio 500 full | 1.502 | 0.002 | 0.076 | 0.077 | 0.949 (0.007) |
| Cox naive | 0.727 | -0.773 | 0.060 | 0.058 | 0.000 (0.000) |
| Ratio 100 naive | 0.708 | -0.792 | 0.063 | 0.059 | 0.000 (0.000) |
| Ratio 200 naive | 0.706 | -0.794 | 0.061 | 0.057 | 0.000 (0.000) |
| Ratio 500 naive | 0.704 | -0.796 | 0.060 | 0.055 | 0.000 (0.000) |
| Cox weighted | 1.616 | 0.116 | 0.214 | 0.283 | 0.804 (0.013) |
| Ratio 100 weighted | 1.620 | 0.120 | 0.215 | 0.288 | 0.809 (0.012) |
| Ratio 200 weighted | 1.619 | 0.119 | 0.214 | 0.285 | 0.802 (0.013) |
| Ratio 500 weighted | 1.617 | 0.117 | 0.214 | 0.283 | 0.799 (0.013) |
Table 11.
Normal covariate results, N = 100,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Normal covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.498 | -0.002 | 0.072 | 0.070 | 0.965 (0.006) |
| Ratio 100 full | 1.501 | 0.001 | 0.085 | 0.082 | 0.966 (0.006) |
| Ratio 200 full | 1.498 | -0.002 | 0.080 | 0.078 | 0.962 (0.006) |
| Ratio 500 full | 1.498 | -0.002 | 0.077 | 0.075 | 0.961 (0.006) |
| Cox naive | 0.731 | -0.769 | 0.061 | 0.059 | 0.000 (0.000) |
| Ratio 100 naive | 0.711 | -0.789 | 0.064 | 0.060 | 0.000 (0.000) |
| Ratio 200 naive | 0.709 | -0.791 | 0.062 | 0.057 | 0.000 (0.000) |
| Ratio 500 naive | 0.707 | -0.793 | 0.061 | 0.056 | 0.000 (0.000) |
| Cox weighted | 1.633 | 0.133 | 0.234 | 0.326 | 0.827 (0.012) |
| Ratio 100 weighted | 1.638 | 0.138 | 0.235 | 0.332 | 0.821 (0.012) |
| Ratio 200 weighted | 1.635 | 0.135 | 0.235 | 0.328 | 0.812 (0.012) |
| Ratio 500 weighted | 1.634 | 0.134 | 0.234 | 0.327 | 0.816 (0.012) |
Table 12.
Log-normal covariate results, N = 5,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Log-normal covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.507 | 0.007 | 0.066 | 0.069 | 0.936 (0.008) |
| Ratio 100 full | 1.519 | 0.019 | 0.086 | 0.088 | 0.955 (0.007) |
| Ratio 200 full | 1.514 | 0.014 | 0.079 | 0.081 | 0.949 (0.007) |
| Ratio 500 full | 1.512 | 0.012 | 0.073 | 0.079 | 0.927 (0.008) |
| Cox naive | 1.159 | -0.341 | 0.068 | 0.072 | 0.014 (0.004) |
| Ratio 100 naive | 1.183 | -0.317 | 0.073 | 0.078 | 0.029 (0.005) |
| Ratio 200 naive | 1.195 | -0.305 | 0.071 | 0.075 | 0.032 (0.006) |
| Ratio 500 naive | 1.209 | -0.291 | 0.070 | 0.073 | 0.037 (0.006) |
| Cox weighted | 1.526 | 0.026 | 0.291 | 0.091 | 0.962 (0.006) |
| Ratio 100 weighted | 1.522 | 0.022 | 0.085 | 0.095 | 0.932 (0.008) |
| Ratio 200 weighted | 1.519 | 0.019 | 0.083 | 0.093 | 0.926 (0.008) |
| Ratio 500 weighted | 1.519 | 0.019 | 0.082 | 0.092 | 0.927 (0.008) |
Table 13.
Log-normal covariate results, N = 10,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Log-normal covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.504 | 0.004 | 0.071 | 0.069 | 0.958 (0.006) |
| Ratio 100 full | 1.514 | 0.014 | 0.107 | 0.107 | 0.950 (0.007) |
| Ratio 200 full | 1.514 | 0.014 | 0.096 | 0.097 | 0.950 (0.007) |
| Ratio 500 full | 1.513 | 0.013 | 0.086 | 0.083 | 0.959 (0.006) |
| Cox naive | 1.180 | -0.320 | 0.073 | 0.073 | 0.023 (0.005) |
| Ratio 100 naive | 1.208 | -0.292 | 0.078 | 0.075 | 0.058 (0.007) |
| Ratio 200 naive | 1.222 | -0.278 | 0.076 | 0.074 | 0.062 (0.008) |
| Ratio 500 naive | 1.235 | -0.265 | 0.074 | 0.072 | 0.072 (0.008) |
| Cox weighted | 1.532 | 0.032 | 0.324 | 0.098 | 0.959 (0.006) |
| Ratio 100 weighted | 1.520 | 0.020 | 0.106 | 0.110 | 0.939 (0.008) |
| Ratio 200 weighted | 1.519 | 0.019 | 0.103 | 0.108 | 0.936 (0.008) |
| Ratio 500 weighted | 1.517 | 0.017 | 0.100 | 0.104 | 0.933 (0.008) |
Table 14.
Log-normal covariate results, N = 50,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Log-normal covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.427 | -0.073 | 0.075 | 0.071 | 0.813 (0.012) |
| Ratio 100 full | 1.549 | 0.049 | 0.188 | 0.198 | 0.961 (0.006) |
| Ratio 200 full | 1.540 | 0.040 | 0.159 | 0.159 | 0.962 (0.006) |
| Ratio 500 full | 1.524 | 0.024 | 0.131 | 0.129 | 0.953 (0.007) |
| Cox naive | 1.205 | -0.295 | 0.079 | 0.075 | 0.052 (0.007) |
| Ratio 100 naive | 1.319 | -0.181 | 0.088 | 0.089 | 0.424 (0.016) |
| Ratio 200 naive | 1.331 | -0.169 | 0.085 | 0.089 | 0.468 (0.016) |
| Ratio 500 naive | 1.340 | -0.160 | 0.082 | 0.084 | 0.484 (0.016) |
| Cox weighted | 1.421 | -0.079 | 0.153 | 0.154 | 0.637 (0.015) |
| Ratio 100 weighted | 1.480 | -0.020 | 0.136 | 0.177 | 0.788 (0.013) |
| Ratio 200 weighted | 1.481 | -0.019 | 0.131 | 0.170 | 0.780 (0.013) |
| Ratio 500 weighted | 1.480 | -0.020 | 0.126 | 0.163 | 0.780 (0.013) |
Table 15.
Log-normal covariate results, N = 100,000. Top: full cohort analysis. Middle: naive two-phase analysis. Bottom: weighted two-phase analysis. 
. Log-normal covariate with β = 1.5. Ratio 100, 200, 500: casebase models with ratio parameter set to 100, 200, and 500, respectively. Coef: coefficient. EmpSE: empirical standard error. SE: standard error. Mean estimated SE: root mean squared reported standard error across successful replicates. For weighted casebase models, this standard error was computed using the proposed dfbeta-based jackknife/sandwich estimator; for weighted Cox models, robust sandwich standard errors were used
| Mean coef | Bias | Mean estimated SE | EmpSE | Coverage (SE) | |
|---|---|---|---|---|---|
| Cox full | 1.423 | -0.077 | 0.077 | 0.069 | 0.837 (0.012) |
| Ratio 100 full | 1.581 | 0.081 | 0.245 | 0.245 | 0.975 (0.005) |
| Ratio 200 full | 1.563 | 0.063 | 0.200 | 0.201 | 0.962 (0.006) |
| Ratio 500 full | 1.538 | 0.038 | 0.158 | 0.163 | 0.965 (0.006) |
| Cox naive | 1.210 | -0.290 | 0.080 | 0.072 | 0.055 (0.007) |
| Ratio 100 naive | 1.323 | -0.177 | 0.088 | 0.088 | 0.454 (0.016) |
| Ratio 200 naive | 1.334 | -0.166 | 0.085 | 0.083 | 0.480 (0.016) |
| Ratio 500 naive | 1.345 | -0.155 | 0.083 | 0.079 | 0.508 (0.016) |
| Cox weighted | 1.364 | -0.136 | 0.293 | 0.189 | 0.626 (0.015) |
| Ratio 100 weighted | 1.442 | -0.058 | 0.133 | 0.209 | 0.622 (0.015) |
| Ratio 200 weighted | 1.446 | -0.054 | 0.128 | 0.198 | 0.657 (0.015) |
| Ratio 500 weighted | 1.451 | -0.049 | 0.125 | 0.190 | 0.662 (0.015) |
Table 16.
Multivariable binary-plus-continuous sensitivity results
| N | Method/parameter | True | Mean coef | Bias | Mean estimated SE | EmpSE | Coverage |
|---|---|---|---|---|---|---|---|
| 10,000 | Weighted casebase, 
|
0.693 | 0.705 | 0.011 | 0.222 | 0.222 | 0.950 |
| 10,000 | Weighted casebase, 
|
0.405 | 0.419 | 0.014 | 0.117 | 0.120 | 0.952 |
| 10,000 | Weighted Cox, 
|
0.693 | 0.704 | 0.011 | 0.220 | 0.221 | 0.948 |
| 10,000 | Weighted Cox, 
|
0.405 | 0.418 | 0.013 | 0.116 | 0.119 | 0.951 |
| 50,000 | Weighted casebase, 
|
0.693 | 0.709 | 0.016 | 0.223 | 0.229 | 0.942 |
| 50,000 | Weighted casebase, 
|
0.405 | 0.413 | 0.007 | 0.117 | 0.119 | 0.952 |
| 50,000 | Weighted Cox, 
|
0.693 | 0.708 | 0.015 | 0.222 | 0.227 | 0.943 |
| 50,000 | Weighted Cox, 
|
0.405 | 0.413 | 0.007 | 0.116 | 0.118 | 0.954 |
Table 17.
Binary baseline-hazard misspecification sensitivity results for coefficient estimation. Event times were generated from a Gompertz proportional hazards model and fitted using weighted Cox regression, weighted casebase regression with a Weibull time term, and weighted casebase regression with a spline time term
| N | Method | True | Mean coef | Bias | Mean estimated SE | EmpSE | Coverage |
|---|---|---|---|---|---|---|---|
| 10,000 | Weighted casebase spline | 1.500 | 1.511 | 0.011 | 0.213 | 0.210 | 0.960 |
| 10,000 | Weighted casebase Weibull | 1.500 | 1.510 | 0.010 | 0.213 | 0.209 | 0.961 |
| 10,000 | Weighted Cox | 1.500 | 1.510 | 0.010 | 0.213 | 0.209 | 0.958 |
| 50,000 | Weighted casebase spline | 1.500 | 1.514 | 0.014 | 0.215 | 0.218 | 0.951 |
| 50,000 | Weighted casebase Weibull | 1.500 | 1.514 | 0.014 | 0.215 | 0.219 | 0.948 |
| 50,000 | Weighted Cox | 1.500 | 1.514 | 0.014 | 0.215 | 0.218 | 0.952 |
Table 18.
Binary baseline-hazard misspecification sensitivity results for absolute risk calibration at the reference covariate value x=1
| N | Method | Mean 
|
Mean true risk | Mean predicted risk | Mean risk bias |
|---|---|---|---|---|---|
| 10,000 | Weighted casebase spline | 0.294 | 0.02174 | 0.02168 | -0.00006 |
| 10,000 | Weighted casebase Weibull | 0.294 | 0.02174 | 0.02179 | 0.00005 |
| 10,000 | Weighted Cox | 0.294 | 0.02174 | 0.02179 | 0.00006 |
| 50,000 | Weighted casebase spline | 0.297 | 0.00439 | 0.00444 | 0.00004 |
| 50,000 | Weighted casebase Weibull | 0.297 | 0.00439 | 0.00447 | 0.00007 |
| 50,000 | Weighted Cox | 0.297 | 0.00439 | 0.00446 | 0.00006 |
Table 19.
Binary independent random censoring sensitivity results. Independent exponential censoring was calibrated to achieve approximately 10–15% censoring before administrative truncation
| N | Method | Mean censoring | Mean coef | Bias | Mean estimated SE | EmpSE | Coverage |
|---|---|---|---|---|---|---|---|
| 10,000 | Weighted casebase | 0.120 | 1.520 | 0.020 | 0.220 | 0.228 | 0.938 |
| 10,000 | Weighted Cox | 0.120 | 1.520 | 0.020 | 0.219 | 0.227 | 0.944 |
| 50,000 | Weighted casebase | 0.117 | 1.508 | 0.008 | 0.221 | 0.219 | 0.957 |
| 50,000 | Weighted Cox | 0.117 | 1.508 | 0.008 | 0.220 | 0.217 | 0.955 |
Table 20.
Binary estimated-inclusion-probability sensitivity results. Among non-cases, phase-two sampling depended on an auxiliary phase-one variable Z; analyses used either the known inclusion probabilities or probabilities estimated from a logistic sampling model. Mean weight refers to the mean inverse-inclusion weight among sampled controls
| N | Method | Mean n controls | Mean weight | True | Mean coef | Bias | Mean estimated SE | EmpSE | Coverage |
|---|---|---|---|---|---|---|---|---|---|
| 10,000 | Weighted casebase, estimated weights | 199.5 | 49.248 | 1.500 | 1.504 | 0.004 | 0.234 | 0.226 | 0.951 |
| 10,000 | Weighted casebase, known weights | 199.5 | 48.940 | 1.500 | 1.502 | 0.002 | 0.234 | 0.229 | 0.954 |
| 10,000 | Weighted Cox, estimated weights | 199.5 | 49.248 | 1.500 | 1.504 | 0.004 | 0.233 | 0.224 | 0.953 |
| 10,000 | Weighted Cox, known weights | 199.5 | 48.940 | 1.500 | 1.503 | 0.003 | 0.232 | 0.227 | 0.950 |
| 50,000 | Weighted casebase, estimated weights | 200.6 | 249.258 | 1.500 | 1.508 | 0.008 | 0.238 | 0.235 | 0.954 |
| 50,000 | Weighted casebase, known weights | 200.6 | 249.058 | 1.500 | 1.509 | 0.009 | 0.237 | 0.239 | 0.951 |
| 50,000 | Weighted Cox, estimated weights | 200.6 | 249.258 | 1.500 | 1.509 | 0.009 | 0.236 | 0.234 | 0.951 |
| 50,000 | Weighted Cox, known weights | 200.6 | 249.058 | 1.500 | 1.509 | 0.009 | 0.236 | 0.239 | 0.949 |
Table 21.
Weighted concordances from weighted Cox and casebase models
| Training1 | Test | |
|---|---|---|
| Cox | 0.70 | 0.65 |
| Casebase | 0.70 | 0.68 |
1 Mean concordance in the held-out “test” sets across repeated cross-validation.
Fig. 1.
Bias and relative difference in standard error for weighted casebase models (ratio 500) under two-phase sampling across cohort sizes and covariate distributions Bias: coefficient bias as defined in Table 2; Rel diff in SE: relative difference in standard error 
. Positive values indicate overestimation of the empirical standard error. Bias (A) and rel diff in SE (B) of casebase ratio 500 weighted models fitted on case-cohort samples across cohort sizes N={5,000, 10,000, 50,000,100,000} and binary, log-normal, and normal covariate distributions
Fig. 2.
Weighted time-dependent Brier score for Cox and casebase models in the (A) training and (B) test sets CB denotes the casebase results
Fig. 3.
Predicted absolute risk curves from weighted Cox and weighted casebase models compared with weighted Kaplan–Meier estimates in (A) training and (B) test sets CB = casebase. KM = Kaplan-Meier
Authors’ contributions
DS and CMTG conceived and developed the methodology. KK and DS conducted the analyses, prepared the figures, and developed the R package. NC curated the OHS dataset and supported the analyses. DS, CMTG, and KK drafted the manuscript. All authors contributed to critical revisions of the manuscript and read and approved the final manuscript.
Funding
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC; RGPIN-2019-05912 to DS and RGPIN-2019-04482 to CMTG) and the Canadian Statistical Sciences Institute (CANSSI; Collaborative Research Team Project #19 grant to DS and CMTG). The funding bodies had no role in the design of the study; the collection, analysis, and interpretation of data; or the writing of the manuscript.
Data availability
The simulation code and analysis scripts used in this study are available at https://github.com/dsoave/weighted-casebase-repro. The weighted casebase implementation is available through the casebaseweights R package at https://github.com/karinakwan/casebaseweights. The Ontario Health Study data used in the illustrative application cannot be made publicly available due to privacy restrictions but may be accessed through application to the Ontario Health Study.
Declarations
Ethics approval and consent to participate
Patient plasma samples were obtained from the Ontario Health Study (OHS) under protocols approved by the University of Toronto Health Sciences Research Ethics Board (protocol #34088). All participants gave written informed consent before participation. All samples and participant data were deidentified and assigned unique research IDs. Original OHS participant and Cancer Care Ontario IDs are not known to anyone outside the research group.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Footnotes
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
- 1.Kirsh VA, Skead K, McDonald K, et al. Cohort profile: the Ontario Health Study (OHS). Int J Epidemiol. 2022. 10.1093/ije/dyac156. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 2.Prentice RL. A case-cohort design for epidemiologic cohort studies and disease prevention trials. Biometrika. 1986;73(1):1–11. 10.1093/biomet/73.1.1. [DOI] [Google Scholar]
- 3.Kalbfleisch JD, Lawless JF. Likelihood analysis of multi-state models for disease incidence and mortality. Stat Med. 1988;7(1–2):149–60. 10.1002/sim.4780070116. [DOI] [PubMed] [Google Scholar]
- 4.Hanley JA, Miettinen OS. Fitting smooth-in-time prognostic risk functions via logistic regression. Int J Biostat. 2009. 10.2202/1557-4679.1125. [DOI] [Google Scholar]
- 5.Saarela O. A case-base sampling method for estimating recurrent event intensities. Lifetime Data Anal. 2016;22(4):589–605. 10.1007/s10985-015-9352-x. [DOI] [PubMed] [Google Scholar]
- 6.Wang Y, Li J, Zhou Q, Wang W. A unified framework for complex survival data: accounting for clustering, cure fractions, and competing risks. BMC Med Res Methodol. 2026. 10.1186/s12874-026-02842-z. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 7.Saarela O, Liu ZA. A flexible parametric approach for estimating continuous-time inverse probability of treatment and censoring weights. Stat Med. 2016;35(23):4238–51. 10.1002/sim.6979. [DOI] [PubMed] [Google Scholar]
- 8.Bender R, Augustin T, Blettner M. Generating survival times to simulate Cox proportional hazards models. Stat Med. 2005;24(11):1713–23. 10.1002/sim.2059. [DOI] [PubMed] [Google Scholar]
- 9.Cheng N, Skead K, Singhawansa A, et al. Pre-diagnosis plasma cell-free DNA methylome profiling up to seven years prior to clinical detection reveals early signatures of breast cancer. medRxiv 2023. Pages: 2023.01.30.23285027. 10.1101/2023.01.30.23285027. [DOI]
- 10.Soave D, Lawless JF. Regularized regression for two phase failure time studies. Comput Stat Data Anal. 2023;182:107703. 10.1016/j.csda.2023.107703. [DOI] [Google Scholar]
- 11.Robins JM, Hernán MA, Brumback B. Marginal Structural Models and Causal Inference in Epidemiology. Epidemiology. 2000;11(5):550–60. 10.1097/00001648-200009000-00011. [DOI] [PubMed] [Google Scholar]
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The simulation code and analysis scripts used in this study are available at https://github.com/dsoave/weighted-casebase-repro. The weighted casebase implementation is available through the casebaseweights R package at https://github.com/karinakwan/casebaseweights. The Ontario Health Study data used in the illustrative application cannot be made publicly available due to privacy restrictions but may be accessed through application to the Ontario Health Study.