Point and interval estimates of partial population attributable risks in cohort studies: examples and software

Abstract

The concept of the population attributable risk (PAR) percent has found widespread application in public health research. This quantity describes the proportion of a disease which could be prevented if a specific exposure were to be eliminated from a target population. We present methods for obtaining point and interval estimates of partial PARs, where the impact on disease burden for some presumably modifiable determinants is estimated in, and applied to, a cohort study. When the disease is multifactorial, the partial PAR must, in general, be used to quantify the proportion of disease which can be prevented if a specific exposure or group of exposures is eliminated from a target population, while the distribution of other modifiable and non-modifiable risk factors is unchanged. The methods are illustrated in a study of risk factors for bladder cancer incidence (Michaud DS et al., New England J Med 340 (1999) 1390). A user-friendly SAS macro implementing the methods described in this paper is available via the worldwide web.

Appendices

Appendix 1: Derivation of the ${Var\left( {\widehat{PAR}_p } \right)}$

$$ Var\left({\widehat{PAR}_p }\right) = Var\left(\frac{\sum\nolimits_{t=1}^{T} \hat{p}_{.t} \widehat{RR}_{2t}}{\sum\nolimits_{s=1}^{S}\sum\nolimits_{t=1}^{T} \hat{p}_{st} \widehat{RR}_{1s} \widehat{RR}_{2t}}\right)=Var(f(\varvec{\hat{p}},\varvec{\widehat{RR}_1}, \varvec{\widehat{RR}_2})) \approx \left. \left[\frac{\partial f\left(\varvec{p,RR}_1, \varvec{RR}_2\right)}{\partial \varvec{p}}\right]^{\prime} \right|_{\varvec{\hat{p}},\varvec{\widehat{RR}}} Var\left(\varvec{\hat{p}}\right)\left. \left[\frac{\partial f\left( \varvec{p,RR}_1,\varvec{RR}_2\right)}{\partial \varvec{p}}\right] \right|_{\varvec{\hat{p}},\varvec{\widehat{RR}}} + \left . \left[ \frac{\partial f\left(\varvec{{p}}\varvec{, R}\varvec{{R_1}}\varvec{,R}\varvec{{R_2}}\right)}{\partial \left(\varvec{RR}_{1}^{\prime},\varvec{RR}_{2}^{\prime}\right)^{\prime}} \right] \right|_{\varvec{\hat{p}},\varvec{\widehat{RR}}} Var\left[\left( \varvec{\widehat{RR}}^{\prime}_1\varvec{\widehat{RR}}^{\prime}_2\right)^{\prime} \right] \left. \left[\frac{\partial f\left(\varvec{{p}},\varvec{R}\varvec{{R}}_{1},\varvec{R}\varvec{{R}}_{2}\right)} {\partial \left(\varvec{RR}_{1}^{\prime} \varvec{,RR}_{2}^{\prime}\right)^{\prime}} \right]\right|_{\varvec{\hat{p}},\varvec{\widehat{RR}}} $$

(6)

where

$$ \frac{\partial f\left(\varvec{p,RR}_1, \varvec{RR}_2\right)}{\partial p_{st} }=\frac{b\,RR_{2t} -a\,RR_{2t} RR_{1s} }{b^2}, \quad \frac{\partial f\left(\varvec{p,RR}_1, \varvec{RR}_2\right)}{\partial RR_{1s} }=-\frac{a\sum\nolimits_{t=1}^T {p_{st} RR_{2t} } } {b^2}, $$

$$ \frac{\partial f\left(\varvec{p,RR}_1, \varvec{RR}_2\right)}{\partial RR_{2t} }=\frac{bp_{.t} -a\sum\nolimits_{s=1}^S {p_{st} RR_{1s} } }{b^2}, \quad a=\sum\limits_{t=1}^T {p_{.t} RR_{2t} } , \quad b=\sum\limits_{s=1}^S {\sum\limits_{t=1}^T {p_{st} RR_{1s} RR_{2t} } } , $$

where Σ _s p _st = p _.t , and ${{\varvec{RR}_1} =\left( {RR_{1,1} ,RR_{1,2} ,\ldots,RR_{1S}}\right)^\prime }$ and ${{\varvec{RR}_2}=\left( {RR_{2,1} ,RR_{2,2} ,\ldots,RR_{2T} } \right)^\prime }$ are the vectors of the relative risks corresponding to the modifiable and unmodifiable risk factors respectively.

Under the proportional hazards model, ${RR_{1s} =e^{\varvec {\beta}_1^\prime \varvec {e_s}}}$, where e _s is the vector of values of the binary indicators corresponding to the sth combination of modifiable exposure variables, of which there are S combinations, and ${RR_{2t} =e^{\varvec {\beta}_2^\prime \varvec {c_t}}}$ where c _t is the vector of values of the tth combination of unmodifiable background risk, of which there are T combinations. Then, ${Var\left[ {\left( {\widehat{RR}}^{\prime}_1{\widehat{RR}}^{\prime}_2\right)^\prime } \right]=D\Sigma {D}^\prime}$, where ${\Sigma =Var\left[ {\left( {\varvec{\hat {{\beta_1 ^\prime}}, \hat {{\beta_2^\prime}}}} \right)^\prime } \right]}$, and D = [(D _uv ), $u = 1,\ldots, S + T, v = 1,\ldots, {p_1} + {p_2}$] where

$$ D_{uv} =\left\{ {\begin{array}{l} \frac{\partial RR_{1,u} }{\partial \beta _{1,v} }\;\quad if\;u\leqslant S\;and\;v\leqslant p_1 \\ \frac{\partial RR_{2,u-S} }{\partial \beta _{2,v-p_1 } }\;\quad if\;u > S\;and\;v> p_1 \\ 0\;\quad \quad if\;u\leqslant S\;and\;v> p_1 \\ 0\;\quad \quad if\;u > S\;and\;v\leqslant p_1 \\ \end{array}} \right. $$

Under the proportional hazards model, ${\frac{\partial RR_{1,u} }{\partial \beta_{1,v} }=e_{uv} e^{\varvec {\beta}_1^\prime \varvec {e_u}}}$, where e _uv is the vth element of the vector e _u , and ${\frac{\partial RR_{2,u-S} }{\partial \beta_{2,v-p}}=c_{u-S,v-p_1 } e^{\varvec {\beta}_2^\prime \varvec {c}_{u-S}}}$, where $c_{{u-S},{v-p_{1}}}$ is the v − p ₁ th element of the vector ${{\varvec c}_{u-S} }$.

The variance of the ${\widehat{PAR}_p }$ is estimated by replacing, in Eq. 6, ${\left( {p,RR} \right)}$ with ${\left( {\hat {p},\widehat {RR}} \right)}$, Σ with the estimated variance-covariance matrix of ${\left( {\varvec{\hat {{\beta_1 ^\prime}} \hat {{\beta_2^\prime}}}} \right)}$ obtained from the pooled logistic regression model or Poisson regression model used to fit (Eq. 5). In a cohort study, the multinomial distribution is used to estimate the variance-covariance matrix of ${\hat {p}}$ , where p = ($p_{1,1}, p_{1,2}, \ldots, p_{ST}$), and ${Cov(\hat {p}_{st} ,\hat {p}_{uv} )=\left\{ {\begin{array}{l} \hat {p}_{st} (1-\hat {p}_{st} )/n\quad \quad if\;s=u\;\& \;t=v\quad \\ -\hat {p}_{st} \hat {p}_{uv} /n\quad \quad \quad if\;s\ne u\;or\;u\ne v \\ \end{array}} \right.}$, and n is the total number of units of person-time of follow-up observed.

In the spirit of transformation suggested by Leung and Kupper [31], to improve the asymptotic behavior of the 95% confidence intervals of ${\widehat{PAR}_p }$ and to ensure that the confidence intervals remain within the range of –100% to 100%, it is useful to calculate the confidence intervals using the Fisher’s Z transformation, that is

$$ \widehat{Var}\left[ {Fisherz \left( {\widehat{PAR}_p } \right)} \right]\approx \frac{1}{\left[ \left( 1 + \widehat{PAR}_p \right) \left( {1-\widehat{PAR}_p } \right)\right]^2}\widehat{Var}\left( {\widehat{PAR}_p } \right) $$

Then the 95% confidence interval for the ${\widehat{PAR}_p }$ is estimated as

$$ \frac{e^{2\left[\widehat{PAR}_p \pm 1.96\sqrt {\widehat {Var}[Fisherz(\widehat{PAR}_p )]}\right]}-1}{e^{2\left[\widehat{PAR}_p \pm 1.96\sqrt {\widehat {Var}[Fisherz(\widehat{PAR}_p)]}\right] }+1} $$

, where Fisherz $({\widehat{PAR}_p })= \log \left[\sqrt{\frac{1+{\widehat{PAR}_p }}{1-{\widehat{PAR}_p }}}\right].$

In a cohort study, it can be shown ${Cov \left( {\varvec{\hat {{p}}, \hat {{\beta}}}} \right)\approx 0}$ by a double expectation argument: The estimators ${\varvec {\hat { \beta}}}$ and ${\varvec {\hat {p}}}$ are the solutions of the following estimating equations,

$$ {\varvec U}_\beta \left( {\varvec{ \beta, p}} \right)=\sum\limits_{i=1}^n {\frac{\partial g({\varvec e}_i , {\varvec c}_i ;{\varvec \beta})}{\partial({\varvec\beta^{\prime}},{\varvec p^{\prime}})^{\prime}}} \left[ {Y_i -\hbox{E} ( Y_i \vert g({{{\varvec e}_i , {\varvec c}_i ; {\varvec \beta}}}))} \right ]={\varvec 0} $$

$$ {\varvec U}_p \left( {\varvec{ \beta, p}} \right) = \left( \begin{array}{c} {\varvec 0}_{(S+T-p_1-p_2)\times 1}\\ \sum \limits_{i=1}^n \left[ {\varvec I}({\varvec e}_i ={\varvec E}_s \;\& \; {\varvec c}_i ={\varvec C}_t )-\hbox{E}({\varvec I}({\varvec e}_i ={\varvec E}_s \;\& \; {\varvec c}_i ={\varvec C}_t ))\right] \end{array}\right)={\varvec 0}, $$

where Y _i is 1 if the unit of person-time is a case and 0 otherwise, ${g({\varvec e}_i , {\varvec c}_i ;{\varvec \beta})}$ will typically be the expit or exponential function, depending on whether pooled logistic regression or Poisson regression is used to estimate ${{\varvec \beta }}$, E(·) is the expectation operator, and ${{\varvec I}(\cdot )}$ is an S + T vector of indicator functions which take values 1 when the condition inside the parentheses is true and 0 otherwise. Because they are unbiased score functions, ${\hbox{E}\left[ {{\varvec U}_{\beta i} \left( {\varvec{{\hat {\beta}},{\hat{p}}} }\right)} \right]={\varvec 0}}$ and ${\hbox{E}\left[ {{\varvec U}_{pi} \left( {\varvec{{\hat {\beta}},{\hat{p}}} }\right)} \right]={\varvec 0}}$, i = $1,\ldots,n$. This implies that

$$ Cov\left[ {{\varvec U}_{\beta i} \left( {\varvec{{\hat {\beta}},{\hat{p}}} }\right),{\varvec U}_{pi} \left( {\varvec{{\hat {\beta}},{\hat{p}}} }\right)} \right]=\hbox{E}_{Y_i ,{\varvec {c}}_i ,{\varvec {e}}_i } \left[ {{\varvec U}_{\beta i} \left( {\varvec{{\hat {\beta}},{\hat{p}}} }\right){\varvec U}_{pi}^{\prime}\left( {\varvec{{\hat {\beta}},{\hat{p}}} }\right)} \right]=\hbox{E}_{c,e} \hbox{E}_{Y\vert {\varvec {c}},{\varvec {e}}} \left[ {{\varvec U}_{\beta i} \left( {\varvec{{\hat {\beta}},{\hat{p}}} }\right){\varvec U}_{pi}^{\prime} \left( {\varvec{{\hat {\beta}},{\hat{p}}} }\right)} \right] $$

$$ =\hbox{E}_{{\varvec {c}}, {\varvec {e}}} \hbox{E}_{Y\vert {\varvec {c}}, {\varvec {e}}} \left[ {\frac{\partial g({ {\varvec e}_i , {\varvec c}_i ;{\varvec \beta} })}{\partial( {\varvec \beta^{\prime} },{\varvec p^{\prime}})^{\prime}}[Y_i -\hbox{E}(Y_i \vert g({ {\varvec e}_i , {\varvec c}_i ;{\varvec \beta} }))] \left( \begin{array}{c} {\varvec 0}_{(S+T-p_1-p_2)\times 1}\\ \sum \limits_{i=1}^n \left[ {\varvec I}({\varvec e}_i ={\varvec E}_s \;\& \; {\varvec c}_i ={\varvec C}_t )-\hbox{E}({\varvec I}({\varvec e}_i ={\varvec E}_s \;\& \; {\varvec c}_i ={\varvec C}_t ))\right] \end{array}\right)^{\prime}} \right] $$

$$ =\hbox{E}_{\varvec {c},\varvec {e}} \left[ {\frac{\partial g({ {\varvec e}_i , {\varvec c}_i ; {\varvec \beta} })}{\partial( {\varvec \beta^{\prime} },{\varvec p^{\prime}})^{\prime}}\left( \begin{array}{c} {\varvec 0}_{(S+T-p_1-p_2)\times 1}\\ \sum \limits_{i=1}^n \left[ {\varvec I}({\varvec e}_i ={\varvec E}_s \;\& \; {\varvec c}_i ={\varvec C}_t )-\hbox{E}({\varvec I}({\varvec e}_i ={\varvec E}_s \;\& \; {\varvec c}_i ={\varvec C}_t ))\right] \end{array}\right)^{\prime}} \right]\hbox{E}_{Y\vert {\varvec {c}},{\varvec {e}}} \left[ {Y_i -\hbox{E}(Y_i \vert g({{\varvec e}_i , {\varvec c}_i ; {\varvec \beta}}))} \right]={\varvec 0}. $$

Appendix 2. Sample SAS code for calculating the ${\widehat{PAR_P}}$

Program:
title ‘make variance-covariance matrix of beta coefficients’;
proc logistic descending data=all covout outest=betas;
model bladder=
volrnk0 volrnk1 volrnk2 volrnk3	/* lowest 4 quintiles of fluid intake */
region1 region2 region3 region4	/* geographic regions */
agegrp2 - agegrp8	/* 5-year age groups */
smkc	/* current smoking */
packyr2-packyr6	/* categories of pack-years */
period1 period2 period3 period4	/* calendar time periods */
calor2-calor5	/* highest 4 quintiles of caloric intake */
fruv1-fruv3;	/* lowest 3 categories of fruit-and-
	vegetable intake */
title ‘make dataset of joint prevalences of modifiable and unmodifiable risk
factors’;
proc sort data=all; by
volrnk0 volrnk1 volrnk2 volrnk3
region1 region2 region3 region4
agegrp2 - agegrp8
smkc
packyr2-packyr6
period1 period2 period3 period4
calor2-calor5
fruv1-fruv3;
run;
proc means noprint data=all; var bladder;
output out=phats n=fq;
by
volrnk0 volrnk1 volrnk2 volrnk3
region1 region2 region3 region4
agegrp2 - agegrp8
smkc
packyr2-packyr6
period1 period2 period3 period4
calor2-calor5
fruv1-fruv3;
run;
%par(bdata=betas, pdata=phats, n_or_p=n, n_or_pname=fq,
fixedvar=agegrp2 agegrp3 agegrp4 agegrp5 agegrp6 agegrp7 agegrp8 period1
period2 period3 period4
region2 region3 region4 region5 calor2 calor3 calor4 calor5
fruv862 fruv863 fruv861
modvar=smkc packyr2 packyr3 packyr4 packyr5 packyr6
volrnk0 volrnk1 volrnk2 volrnk3);
Output:
option for the variance-covariance matrix of the prevalences is FIXED .
Partial PAR (95% CI) for
modifiable vbls : VOLRNK0 VOLRNK1 VOLRNK2 VOLRNK3 SMKC PACKYR2
PACKYR3 PACKYR4 PACKYR5 PACKYR6
fixed vbls : AGEGRP2 AGEGRP3 AGEGRP4 AGEGRP5 AGEGRP6 AGEGRP7 AGEGRP8
PERIOD1 PERIOD2 PERIOD3 PERIOD4 REGION2 REGION3 REGION4 REGION5 CALOR2
CALOR3 CALOR4 CALOR5 FRUV862 FRUV863 FRUV861
0.692 (0.366, 0.869)