Deriving relative risk from logistic regression

Binary or continuous exposure variable

Let us first define adjusted relative risks of binary exposure \(E\) on binary outcome \(D\) conditional on \(\mathbf{X}\).

\[\frac{P(D = 1 \mid E = 1, \mathbf{X} )}{P(D = 1 \mid E = 0, \mathbf{X})}\]

Generally speaking, when exposure variable of \(E\) is continuous or ordinal, we can define adjusted relative risks as ratio between probability of observing \(D = 1\) when \(E = z+1\) over \(E = z\) conditional on \(\mathbf{X}\). Unlike adjusted odds ratio, these ratio depend on baseline value of exposure \(z\) under logistic regression.

\[\frac{P(D = 1 \mid E = z+1, \mathbf{X} )}{P(D = 1 \mid E = z, \mathbf{X})}\]

Nominal exposure variable

On the other hand, when exposure variable is nominal, it is impossible to compare the probabilities in one unit change. Therefore, when a type of exposure variable is factor, we allow users to specify two values of exposure variable including baseline (\(z_{0}\)) and comparative level (\(z_{1}\)) and derive the relative risks given those two exposure levels.

\[\frac{P(D = 1 \mid E = z_{1}, \mathbf{X} )}{P(D = 1 \mid E = z_{0}, \mathbf{X})}\] The above is more generalized version. By setting \(z_{1} = 1\) and \(z_{0} = 0\) we can go back to binary case.

Delta method

Let \(\boldsymbol{\beta}\) be a vector of coefficients used in logistic regression and among them \(\beta_{1}\) is a coefficient associated with an exposure variable of interest taking a value of \(z_{0}\) as baseline level and \(z_{1}\) as comparative level. Then we can represent the adjusted relative risk as a function of \(\boldsymbol{\beta}\) conditional on \(\mathbf{X}\):

\[g(\boldsymbol{\beta}) = \frac{1 + \exp(-\beta_{0} - \beta_{1} z_{0} - \boldsymbol{\beta}^{T}_{2:p} \mathbf{X}) }{ 1 + \exp (-\beta_{0} - \beta_{1} z_{1} - \boldsymbol{\beta}^{T}_{2:p} \mathbf{X}) }\]

Then by Delta method, \[var[g(\boldsymbol{\beta})] = \left\{\frac{\partial g(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} \right\}^{T} var(\boldsymbol{\beta}) \left\{\frac{\partial g(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} \right\}\] Note that \(\frac{\partial g(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}\) is \(p \times 1\) and \(var(\boldsymbol{\beta})\) is \(p \times p\), so \(var[g(\boldsymbol{\beta})]\) is a scalar value.

A \(p \times p\) matrix of \(var{(\boldsymbol{\beta})}\) is obtained by summary(fit)$cov.unscaled when fit is a glm object.

Let \(e_{0} = \exp(-\beta_{0} - \beta_{1} z_{0} - \boldsymbol{\beta}^{T}_{2:p} \mathbf{X})\) and \(e_{1} = \exp (-\beta_{0} - \beta_{1} z_{1} - \boldsymbol{\beta}^{T}_{2:p} \mathbf{X})\).

\[\frac{\partial g(\boldsymbol{\beta})}{\partial \beta_{0}} = \frac{- e_{1} + e_{0}}{(1 + e_{1})^2 } = \frac{e_{0}(1 - \exp(-\beta_{1}(z_{1} - z_{0}) ) ) }{(1 + e_{1})^2}\] \[\frac{\partial g(\boldsymbol{\beta})}{\partial \beta_{1}} = \frac{-z_{1} e_{1}( 1 + e_{0}) + z_{0} e_{0}(1 + e_{1}) }{(1 + e_{1})^2 }\] For any \(j = 2,3,\ldots, p\) where \(X_{j}\) is a covariate of which effect is associated with \(\beta_{j}\): \[\frac{\partial g(\boldsymbol{\beta})}{\partial \beta_{j}} = \frac{x_{j}(e_{0} - e_{1} ) }{ (1+e_{1})^2} = \frac{1 - \exp(-(z_{1} - z_{0})\beta_{1}) }{(1 + e_{1})^2}\]

By combining information of estimated \(var{(\boldsymbol{\beta})}\) and \(\frac{\partial g(\boldsymbol{\hat{\beta}})}{\partial \boldsymbol{\hat{\beta}}}\), we can derive the estimated variance of \(g(\boldsymbol{\beta})\).

Bootstrap

In both of logisticRR and nominalRR, we add a logical input of boot: by setting boot = TRUE those functions print out a vector of n.boot number of (adjusted) relative risks.

Adjusted relative risks depending on confounders

We have a total of six combination of confounder variables. By the assumption made in logistic regression, adjusted odds ratio is consistent against of these confounders but adjusted relative risk is not.

levels(dat$W)
#> [1] "0" "1" "2"
levels(dat$Z)
#> [1] "female" "male"
adjusted <- cbind(rep(levels(dat$W), 2), rep(levels(dat$Z), each = 3))
adjusted <- as.data.frame(adjusted)
names(adjusted) <- c("W", "Z")

Adjusted relative risk tends to be higher for male and for higher level of W.

## compare with odds ratio 
results <- list()
for(i in 1:nrow(adjusted)){
  results[[i]] <- logisticRR(Y ~ X + W + Z, data = dat, fixcov = adjusted[i,], boot = FALSE)
}


## adjusted relative risk
# female
print(c(results[[1]]$RR, results[[2]]$RR, results[[3]]$RR))
#>        1        1        1 
#> 1.076041 1.206734 1.494543
# male
print(c(results[[4]]$RR, results[[5]]$RR, results[[6]]$RR))
#>        1        1        1 
#> 1.650889 2.248270 2.811120


## adjusted odds ratio
## all the same : by the assumption of logistic regression
print(exp(coefficients(results[[1]]$fit)[2]))
#>        X 
#> 3.441811

# betas <- coefficients(fit)
# exposed <- exp(-predict(fit, expose.cov, type = "link"))
# unexposed <- exp(-predict(fit, unexpose.cov, type = "link"))
# RR <- (1 + unexposed) / (1 + exposed)

Let us change the prevalence of exposure variable (\(X\)).

dat2 <- dat 
dat2$Y <- ifelse(dat$Y == 1, rbinom(n, 1, 0.2), rbinom(n, 1, 0.01))
## compare with odds ratio 
results2 <- list()
for(i in 1:nrow(adjusted)){
  results2[[i]] <- logisticRR(Y ~ X + W + Z, data = dat2, fixcov = adjusted[i,], boot = TRUE, n.boot = 1000)
}


## adjusted relative risk
# female
print(c(results2[[1]]$RR, results2[[2]]$RR, results2[[3]]$RR))
#>        1        1        1 
#> 1.173466 1.177487 1.186902
# male
print(c(results2[[4]]$RR, results2[[5]]$RR, results2[[6]]$RR))
#>        1        1        1 
#> 1.184617 1.187547 1.194245


## adjusted odds ratio
## all the same : by the assumption of logistic regression
print(exp(coefficients(results2[[1]]$fit)[2]))
#>        X 
#> 1.209375

# betas <- coefficients(fit)
# exposed <- exp(-predict(fit, expose.cov, type = "link"))
# unexposed <- exp(-predict(fit, unexpose.cov, type = "link"))
# RR <- (1 + unexposed) / (1 + exposed)

Plotting the distribution of adjusted relative risks for each combination of confounders. Each black dot denotes point estimate of adjusted relative risk and red horizontal line denotes adjusted odds ratio which is independent of the levels of confounders.