Triple Bias Example
We are interested in quantifying the effect of exposure X on outcome Y. The causal system can be represented in the following directed acyclic graph (DAG):
The variables are defined:
- X: true, unmeasured exposure
- Y: outcome
- C: measured confounder(s)
- U: unmeasured confounder
- X*: misclassified, measured exposure
- S: study selection
It can be seen from this DAG that the data suffers from three sources of bias. There is uncontrolled confounding from (unobserved) variable U. The true exposure, X, is unobserved, and the misclassified exposure X* is dependent on both the exposure and outcome. Lastly, there is collider stratification at variable S since exposure and outcome both affect selection. The study naturally only assesses those who were selected into the study (i.e. those with S=1), which represents a fraction of all people in the source population from which we are trying to draw inference.
A simulated dataframe corresponding to this DAG,
df_uc_emc_sel can be loaded from the multibias
package.
head(df_uc_em_sel)
#> Xstar Y C1 C2 C3
#> 1 1 0 0 0 1
#> 2 0 0 0 0 1
#> 3 0 0 0 0 0
#> 4 0 0 1 0 1
#> 5 1 0 0 0 1
#> 6 1 0 0 1 1In this data, the true, unbiased exposure-outcome odds ratio (ORYX) equals ~2. However, when we run a logistic regression of the outcome on the exposure and confounders, we do not observe an odds ratio of 2 due to the multiple bias sources.
biased_model <- glm(Y ~ Xstar + C1 + C2 + C3, ,
family = binomial(link = "logit"),
data = df_uc_em_sel)
biased_or <- round(exp(coef(biased_model)[2]), 2)
print(paste0("Biased Odds Ratio: ", biased_or))
#> [1] "Biased Odds Ratio: 1.5"The function adjust_uc_emc_sel() can be used here to
“reconstruct” the unbiased data and return the exposure-outcome odds
ratio that would be observed in the unbiased setting.
Models for the missing variables (U, X, S) are used to facilitate this data reconstruction. For the above DAG, the corresponding bias models are:
- logit(P(U=1)) = α0 + α1X + α2Y
- logit(P(X=1)) = δ0 + δ1X* + δ2Y + δ2+jCj
- logit(P(S=1)) = β0 + β1X* + β2Y + β2+jCj
where j indicates the number of measured confounders.
To perform the bias adjustment, it is necessary to obtain values of
these bias parameters. Potential sources of these bias parameters
include internal validation data, estimates in the literature, and
expert opinion. For purposes of demonstrating the methodology, we will
obtain the exact values of these bias parameters. This is possible
because for purposes of validation we have access to the data of missing
values that would otherwise be absent in real-world practice. This
source data is available in multibias as
df_uc_emc_sel_source.
u_model <- glm(U ~ X + Y,
family = binomial(link = "logit"),
data = df_uc_em_sel_source)
x_model <- glm(X ~ Xstar + Y + C1 + C2 + C3,
family = binomial(link = "logit"),
data = df_uc_em_sel_source)
s_model <- glm(S ~ Xstar + Y + C1 + C2 + C3,
family = binomial(link = "logit"),
data = df_uc_em_sel_source)In this example we’ll perform probabilistic bias analysis,
representing each bias parameter as a single draw from a probability
distribution. For this reason, we will run the analysis over 1,000
iterations with bootstrap samples to obtain a valid confidence interval.
To improve performance we will run the for loop in parallel using the
foreach() function in the doParallel package.
Below we create a cluster, make a seed for consistent results, and
specify the desired number of bootstrap repitions.
library(doParallel)
no_cores <- detectCores() - 1
registerDoParallel(cores = no_cores)
cl <- makeCluster(no_cores)
set.seed(1234)
nreps <- 1000
est <- vector(length = nreps)Next we run the parallel for loop in which we apply the
adjust_uc_em_sel() function to bootstrap samples of the
df_uc_em_sel data. We specify the following arguments: the
data, the exposure variable, the outcome variable, the confounder(s),
the U model coefficients, the X model coefficients,
and the S model coefficients. Since knowledge of the source
data was known, the correct bias parameters can be applied. Using the
results from the fitted bias models above, we’ll use Normal distribution
draws for each bias parameter where the mean correponds to the estimated
coefficient from the bias model and the standard deviation comes from
the estimated standard deviation (i.e., standard error) of the
coefficient in the bias model. Each loop iteration will now have
slightly different values for the bias parameters, corresponding to our
uncertainty in their estimates.
or <- foreach(i = 1:nreps, .combine = c,
.packages = c("dplyr", "multibias")) %dopar% {
df_sample <- df_uc_em_sel[sample(seq_len(nrow(df_uc_em_sel)),
nrow(df_uc_em_sel),
replace = TRUE), ]
est[i] <- adjust_uc_em_sel(
data_observed = data_observed(
data = df_sample,
exposure = "Xstar",
outcome = "Y",
confounders = c("C1", "C2", "C3")
),
u_model_coefs = c(
rnorm(1, mean = u_model$coef[1], sd = summary(u_model)$coef[1, 2]),
rnorm(1, mean = u_model$coef[2], sd = summary(u_model)$coef[2, 2]),
rnorm(1, mean = u_model$coef[3], sd = summary(u_model)$coef[3, 2])
),
x_model_coefs = c(
rnorm(1, mean = x_model$coef[1], sd = summary(x_model)$coef[1, 2]),
rnorm(1, mean = x_model$coef[2], sd = summary(x_model)$coef[2, 2]),
rnorm(1, mean = x_model$coef[3], sd = summary(x_model)$coef[3, 2]),
rnorm(1, mean = x_model$coef[4], sd = summary(x_model)$coef[4, 2]),
rnorm(1, mean = x_model$coef[5], sd = summary(x_model)$coef[5, 2]),
rnorm(1, mean = x_model$coef[6], sd = summary(x_model)$coef[6, 2])
),
s_model_coefs = c(
rnorm(1, mean = s_model$coef[1], sd = summary(s_model)$coef[1, 2]),
rnorm(1, mean = s_model$coef[2], sd = summary(s_model)$coef[2, 2]),
rnorm(1, mean = s_model$coef[3], sd = summary(s_model)$coef[3, 2]),
rnorm(1, mean = s_model$coef[4], sd = summary(s_model)$coef[4, 2]),
rnorm(1, mean = s_model$coef[5], sd = summary(s_model)$coef[5, 2]),
rnorm(1, mean = s_model$coef[6], sd = summary(s_model)$coef[6, 2])
)
)$estimate
}Finally, we obtain the ORYX estimate and 95% confidence interval from the distribution of 1,000 odds ratio estimates. As expected, ORYX ~ 2, indicating that we were able to obtain an unbiased odds ratio from the biased data.