Background stratified Poisson regression analysis of cohort data

Abstract

Background stratified Poisson regression is an approach that has been used in the analysis of data derived from a variety of epidemiologically important studies of radiation-exposed populations, including uranium miners, nuclear industry workers, and atomic bomb survivors. We describe a novel approach to fit Poisson regression models that adjust for a set of covariates through background stratification while directly estimating the radiation-disease association of primary interest. The approach makes use of an expression for the Poisson likelihood that treats the coefficients for stratum-specific indicator variables as ‘nuisance’ variables and avoids the need to explicitly estimate the coefficients for these stratum-specific parameters. Log-linear models, as well as other general relative rate models, are accommodated. This approach is illustrated using data from the Life Span Study of Japanese atomic bomb survivors and data from a study of underground uranium miners. The point estimate and confidence interval obtained from this ‘conditional’ regression approach are identical to the values obtained using unconditional Poisson regression with model terms for each background stratum. Moreover, it is shown that the proposed approach allows estimation of background stratified Poisson regression models of non-standard form, such as models that parameterize latency effects, as well as regression models in which the number of strata is large, thereby overcoming the limitations of previously available statistical software for fitting background stratified Poisson regression models.

Appendices

Appendix 1

A standard log-linear unconditional Poisson regression model of the form $\lambda (\alpha ,\beta ) = \exp (\alpha_{1} S_{1} + \alpha_{2} S_{2} + \alpha_{3} S_{3} + \alpha_{4} S_{4} + \beta_{1} Z)$ may be fitted to the data in Table 1 via the SAS statistical package as follows:

The variables P and c denote counts of person-time and events, respectively, in the grouped data structure. The ‘parms’ statement defines the parameters to be estimated, and the ‘profile’ statement requests associated 95% likelihood-based confidence intervals. The term ‘lambda’ specifies that the rate of disease conforms to an exponential function of the model covariates. The ‘LL’ statement specifies the expression for the unconditional Poisson likelihood, and the statement ‘max LL’ defines the function to be maximized.

A log-linear Poisson regression model may be fitted to the data structure in Table 2, with background stratification on covariates A and B, via the SAS statistical package as follows:

The analytical data structure has one record per stratum. The variables _ncovals and _totcases denote the total number of exposure values, and total number of cases, in each stratum. The arrays _cases, _pt, and _z index the values for the counts of events, person-time, and levels of the exposure variable(s) of interest in each stratum of the analytical data structure. The length of the arrays will depend upon the analytical data structure. The variables caseprod and sum, which are the numerator and denominator, respectively, of the expression for the conditional likelihood, are initialized at each new record in the analysis. The term ‘phi’ defines the relative rate function of the regression model. In the example above, the rate ratio function conforms to a standard log-linear model. The ‘parms’ statement defines the parameter(s) to be estimated, and the ‘profile’ statement requests associated 95% profile likelihood confidence bounds. The ‘LL’ statement specifies the expression for the log likelihood in this model, and the statement ‘max LL’ defines the function to be maximized.

The SAS procedure PROC NLP is part of the SAS/OR statistical package. Some SAS users may have access to the SAS/STAT package but not the SAS/OR package. Therefore, below, we also provide sample code for fitting background stratified Poisson regression models via the SAS PROC NLMIXED procedure which is part of the SAS/STAT package. SAS PROC NLMIXED does not directly output profile likelihood confidence intervals for estimated parameters but does report Wald-type confidence intervals.

This approach accommodates a variety of functional forms for the relative rate function, ϕ. For example, a linear excess relative rate model of the form ϕ = (1 + βz) would be fitted by replacing the statement ‘ ’ with the statement ‘ ’.

Appendix 2

With the model as in (2), the unconditional log-likelihood contribution from stratum s is $L_{s} (\alpha_{s} ,\beta ) = c_{s} \alpha_{s} + \sum\nolimits_{{z \in R_{s} }} {c_{sz} \ln (P_{sz} \varphi (z,\beta ))} - \exp (\alpha_{s} )\sum\nolimits_{{z \in R_{s} }} {P_{sz} \varphi (z,\beta )} .$ The observed information at $\hat{\alpha },I(\hat{\alpha },\beta )$ is given by

$$ - dU\left( {\hat{\alpha }_{s} } \right)/d\alpha_{s} = c_{s} $$

$$ - dU\left( {\hat{\alpha }_{s} } \right)/d\alpha_{l} = 0 $$

$$ - dU\left( {\hat{\alpha }_{s} } \right)/d\beta = c_{s} \sum\limits_{{z \in R_{s} }} {P_{sz} \varphi^{\prime } (z;\beta )} /\sum\limits_{{z \in R_{s} }} {P_{sz} \varphi (z;\beta )} $$

$$ - dU(\beta )/d\beta = \sum\limits_{s} {\left[ {\sum\limits_{{i \in D_{s} }} {\left[ {\frac{{\varphi^{\prime \prime } (Z_{i} ;\beta )}}{{\varphi (Z_{i} ;\beta )}} - \left( {\frac{{\varphi^{\prime } (Z_{i} ;\beta )}}{{\varphi (Z_{i} ;\beta )}}} \right)^{2} } \right] - c_{s} \frac{{\sum\nolimits_{{z \in R_{s} }} {P_{sz} \varphi^{\prime \prime } (z;\beta )} }}{{\sum\nolimits_{{z \in R_{s} }} {P_{sz} \varphi (z;\beta )} }}} } \right]} . $$

The variance estimate for $\hat{\beta }$ is the corner of the inverse of the observed information (evaluated at $\hat{\alpha },$ $\hat{\beta }$) which may be obtained using the well-known matrix formula

$$ (I^{ - 1} )_{\beta ,\beta } = \left[ {I_{\beta ,\beta } - I_{\beta ,\alpha } I_{\alpha ,\alpha }^{{^{ - 1} }} I_{\alpha ,\beta } } \right]^{ - 1} . $$

Since I _α,α is a diagonal matrix, it is easy to compute that

$$ \text{var} \left( {\hat{\beta }} \right)^{ - 1} = \sum\limits_{s} {\left[ {\sum\limits_{{i \in D_{s} }} {\left[ {\frac{{\varphi^{\prime \prime } (Z_{i} ;\beta )}}{{\varphi (Z_{i} ;\beta )}} - \left( {\frac{{\varphi^{\prime } (Z_{i} ;\beta )}}{{\varphi (Z_{i} ;\beta )}}} \right)^{2} } \right] - \,c_{s} \left[ {\frac{{\sum\nolimits_{{z \in R_{s} }} {P_{sz} \varphi^{\prime \prime } (z;\beta )} }}{{\sum\nolimits_{{z \in R_{s} }} {P_{sz} \varphi (z;\beta )} }}\, - \left( {\frac{{\sum\nolimits_{{z \in R_{s} }} {P_{sz} \varphi^{\prime } (z;\beta )} }}{{\sum\nolimits_{{z \in R_{s} }} {P_{sz} \varphi (z;\beta )} }}} \right)^{2} } \right]^{{}} } } \right]} $$

This expression is the same as the second derivative of the ‘conditional’ Poisson log-likelihood; consequently, estimated standard errors and associated Wald-type confidence intervals will be the same. For simplicity, we derive the expression for a single parameter β. The expressions apply to column vector β where the derivatives are as in standard vector calculus and squared terms are replaced by outer products, i.e., replace a ² by aa ^t where a ^t is the transpose of a.