Including Injured Workers Without Compensated Time Loss in Cox Regression Models: Analyzing Time Loss Using All Available Data

Abstract

Introduction Cox proportional hazards regression is commonly used to analyze time loss duration, but statistical packages conventionally exclude cases with no recorded follow-up time. For this and other substantive reasons, many researchers limit time loss analyses to the subset of workers who received time loss compensation. This can exclude both injured workers who missed no work days and those missing up to a week of work. For some research questions, excluding cases where injury is reported but no time loss is recorded may result in significant ascertainment bias. We present a novel technique based on standard survival analysis methods to allow for the inclusion of all cases when appropriate. Methods A simple technique to allow standard statistical software to include both medical-only and time loss claims in Cox regression is illustrated by example and compared with a two-part model using a time-varying step function to allow regression effects to change over time. Results We showed that a pooled analysis is obtained by simply adding a small constant to the time loss duration variable. This technique produced appropriate estimates while accounting for censoring when a suitable method was used for tied event times. Using a formal statistical framework, the combined model was justified as a special case of the more standard two-part model approach. Conclusions When it is desirable to have a single pooled outcome estimate for injured workers with both medical-only and time loss claims, all claims can be combined into one statistical model. This may have particular utility for research questions where the risk factor or intervention of interest would be expected to affect time loss duration beginning upstream of claim filing or statutory compensation waiting periods. This novel alternative modeling strategy expands the tool kit available for analyzing time loss data.

Appendix: Statistical Models

In this appendix, we briefly review the statistical details of the models under discussion. The goal of this detail is to provide formal justification for our novel approach, and to explicitly show the relationship between the two-part model approach and the combined Cox regression approach. Finally, we show how a two-part model can be represented as a special case of the combined model when certain interactions are included in the combined model.

In the discussion that follows we use T = t to denote that an event occurs at time “t”. We do not represent the outcome as censored times in the presentation of the underlying regression models but rather discuss censoring as an issue that would be addressed in the estimation of these models. Finally, for ease of presentation we focus on discrete event times such that the probability that T = t is meaningful. The two-part model consists of a first-part logistic regression model and a second-part Cox proportional hazards regression model and can be represented as follows:

$$ {\text{Part 1}}\left( {{\text{odds ratio}}} \right){: \log }\left[ {{\text{P}}\left( {{\text{T}} = 0|{\text{X}}} \right)/\left\{ { 1- {\text{P}}\left( {{\text{T}} = 0|{\text{X}}} \right)} \right\}} \right] = \beta _{0} + \beta _{1}* {\text{X}} $$

(1)

$$ {\text{Part 2}}\left( {{\text{hazard ratio}}} \right){: \text{log hazard}}\left( {{\text{t}}|{\text{X}}} \right) = { \log }\left[ {\lambda _{0} \left( {\text{t}} \right)} \right] + \beta _{1}*{\text{X}} $$

(2)

$$ { \log }\left[ {{\text{P}}\left( {{\text{T}} = {\text{t}}|{\text{X}}} \right)/{\text{P}}\left( {{\text{T}} \ge {\text{t}}|{\text{X}}} \right)} \right] = { \log }\left[ {\lambda 0\left( {\text{t}} \right)} \right] + \beta _{ 1}*{\text{X}} = \beta _{0} \left( {\text{t}} \right) + \beta _{ 1} *{\text{X}} $$

The first part can be alternatively represented using a relative risk regression model rather than a logistic regression model:

$$ {\text{Part 1}}\left( {{\text{relative risk}}} \right){: \log }\left[ {{\text{P}}\left( {{\text{T}} = 0|{\text{X}}} \right)} \right] = \beta _{0} + \beta _{ 1} *{\text{X}} $$

(3)

Furthermore, the relative risk regression is equivalent to a hazard regression model when focusing on the first event time:

$$ {\text{Part 1}}\left( {{\text{hazard ratio}}} \right){: \log }\left[ {{\text{P}}\left( {{\text{T}} = 0|{\text{X}}} \right)} \right] = { \log }\left[ {{\text{P}}\left( {{\text{T}} = 0|{\text{X}}} \right)/{\text{P}}\left( {{\text{T}} \ge 0|{\text{X}}} \right)} \right] $$

(4)

The equivalence is due to the fact that P(T ≥ 0) = 1 by definition (of time loss or any event time). Now we can re-write and subsequently combine these two models, labeling each coefficient to show from which part of the model it originates and how it is represented in the combined model:

$$ {\text{Part 1}}\left( {{\text{hazard ratio}}} \right){: \log }\left[ {{\text{P}}\left( {{\text{T}} = 0|{\text{X}}} \right)/{\text{P}}\left( {{\text{T}} \ge 0|{\text{X}}} \right)} \right] = \beta _{0} \left[ 1\right] + \beta _{ 1} \left[ 1\right]*{\text{X}} $$

(5)

$$ {\text{Part 2}}\left( {{\text{hazard ratio}}} \right){: \log }\left[ {{\text{P}}\left( {{\text{T}} = {\text{t}}|{\text{X}}} \right)/{\text{P}}\left( {{\text{T}} \ge {\text{t}}|{\text{X}}} \right)} \right] = \beta _{0} \left[ 2\right]\left( {\text{t}} \right) + \beta _{ 1} \left[ 2\right]*{\text{X}} $$

(6)

$$ {\text{Combined}}{: \log }\left[ {{\text{P}}\left( {{\text{T}} = {\text{t}}|{\text{X}}} \right)/{\text{P}}\left( {{\text{T}} \ge {\text{t}}|{\text{X}}} \right)} \right] = \beta _{0} \left( {\text{t}} \right) + \beta _{ 1} \left[ 1\right]*{\text{X}} + \gamma *\left( {{\text{t}} > 0} \right)*{\text{X}} $$

(7)

where γ = β₁[2] − β₁[1].

Thus the covariate effect for “early” is β₁[1] and the covariate effect for “late” is β₁[1] + γ. This shows that fitting a combined model with appropriate interaction terms is equivalent to a two-part model using relative risk regression for the first part. In addition, a test of γ = 0 allows one to test whether the part[1] and part[2] model coefficients are equal and could therefore be used to justify the fitting of a combined model when they are found not to be significantly different. Finally, if appropriate the effect modification by time (i.e., γ) can be removed and a single overall pooled estimate obtained, β₁ = β₁[1] = β₁[2].

The antilog of the coefficient β₁ in the part 1 model (Eq. 5) can be interpreted as the relative risk (or “hazard”) of having no time loss (versus some time loss) comparing those in the intervention group with those in the reference group (whereas the antilog of the coefficient for the part 1 logistic model, Eq. 1, can be interpreted as the odds of having no time loss for the intervention group relative to the odds for the reference group, which approximates relative risk in limited circumstances). The antilog of the coefficient β₁ in the part 2 model (Eq. 6) can be interpreted as the relative risk (or “hazard”) of ending time loss comparing those in the intervention group with those in the reference group, among those having some compensated time loss. The antilog of the coefficient β₁ in the combined model (Eq. 7) can be interpreted as the relative risk (or “hazard”) of ending time loss among all injured workers, comparing those in the intervention group with those in the reference group, assuming that γ = 0 (no difference in the relative risk of ending time loss among those who have no compensated time loss compared with those who have compensated time loss).