R/binary-sim.R
calc_cond_binary.RdIn order to properly generate binary response data for the internal trial as part of a simulation study that investigates inverse probability weighting, we need to translate the desired marginal drift and treatment effect to the corresponding conditional drift and treatment effect that can then be added into a binary outcome model (e.g., logistic regression model) used to simulate response data.
calc_cond_binary(population, glm, marg_drift, marg_trt_eff)A very large data frame (e.g., number of rows \(\ge\) 100,000) where the columns correspond to the covariates defined in the logistic regression model object. This data frame should be constructed to represent the population of the internal trial according to the assumed covariate distributions (possibly imbalanced from the external data).
Logistic regression model object fit using the external data
Vector of marginal drift values
Vector of marginal treatment effect values
tibble of all combinations of the marginal drift and treatment effect. For each row the conditional drift and treatment effect has been calculated as well as the true control response rate and true treatment effect.
In simulation studies that investigate the properties of inverse probability weighted Bayesian dynamic borrowing, scenarios should be considered in which the underlying response rates for the internal and external control populations differ by varying amounts due to unmeasured confounding (i.e., drift, where positive values indicate a higher response rate for the internal population). While values of drift and treatment effect (i.e., risk difference) can be defined on the marginal scale for simulation studies, we must first convert these values to the conditional scale and then include these terms, along with covariates, in a logistic regression outcome model when generating response data for the internal arms. Doing so allows us to assume a relationship between the covariates and the response variable while properly accounting for drift and treatment effect.
To identify the conditional drift and treatment effect that correspond to
specified values of marginal drift and treatment effect, we first bootstrap
covariate vectors from the external data (e.g., \(N \ge 100,000\)) to
construct a "population" that represents both the internal trial
(possibly incorporating intentional covariate imbalance) and the external
trial after standardizing it to match the covariate distributions
of the internal trial (allowing us to control for measured confounding
from potential imbalance in the covariate distributions). Measured
confounding can be incorporated into the data generation by bootstrapping
a very large data frame (population) in which the distribution of at
least one covariate is intentionally varied from that of the external data;
additional unmeasured drift can be incorporated through the
translation of specified marginal values (marg_drift) to conditional
values.
Let \(\Delta\) and \(\delta\) denote the marginal and conditional drift, respectively. For a specified value of \(\Delta\), we can identify the corresponding \(\delta\) as the value that, when added as an additional term in the logistic regression model (i.e., change in the intercept) for each individual in the population, increases/decreases the population-averaged conditional probabilities of response by an amount approximately equal to \(\Delta\). That is, the optimal \(\delta\) minimizes
$$\left| \left( \frac{1}{N} \sum_{i=1}^N \frac{\exp \left( \boldsymbol{x}_i^\prime \boldsymbol{\beta}_{EC} + \delta \right)}{1 + \exp\left(\boldsymbol{x}_i^\prime \boldsymbol{\beta}_{EC} + \delta \right)} - \frac{1}{N} \sum_{i=1}^N \frac{\exp \left( \boldsymbol{x}_i^\prime \boldsymbol{\beta}_{EC} \right)}{1 + \exp \left(\boldsymbol{x}_i^\prime \boldsymbol{\beta}_{EC} \right)} \right) - \Delta \right|,$$
where \(\boldsymbol{\beta}_{EC}\) is the vector of regression coefficients
from the logistic regression model (glm) fit to the external control data
(assumed here to be the "true" covariate effects when generating response
data) and \(\boldsymbol{x}_i\) is a vector of covariates from the
bootstrapped population of size \(N\). In the formula above, the first
and second terms correspond to the population-averaged conditional
probabilities (i.e., the marginal response rates) of the internal control
population with drift and the external control population (with covariate
distributions standardized to match the internal trial), respectively.
If we now denote the marginal and conditional treatment effect by \(\Gamma\) and \(\gamma\), respectively, we can use a similar process to identify the optimal \(\gamma\) that approximately corresponds to the specified value of \(\Gamma\), which is done by minimizing the following:
$$\left| \left( \frac{1}{N} \sum_{i=1}^N \frac{\exp \left( \boldsymbol{x}_i^\prime \boldsymbol{\beta}_{EC} + \delta + \gamma \right)}{1 + \exp\left(\boldsymbol{x}_i^\prime \boldsymbol{\beta}_{EC} + \delta + \gamma \right)} - \frac{1}{N} \sum_{i=1}^N \frac{\exp \left( \boldsymbol{x}_i^\prime \boldsymbol{\beta}_{EC} + \delta \right)}{1 + \exp \left(\boldsymbol{x}_i^\prime \boldsymbol{\beta}_{EC} + \delta \right)} \right) - \Gamma \right|,$$
where the first term is the population-averaged conditional probabilities (i.e., the marginal response rate) of the internal treated population.
library(dplyr)
#>
#> Attaching package: ‘dplyr’
#> The following objects are masked from ‘package:stats’:
#>
#> filter, lag
#> The following objects are masked from ‘package:base’:
#>
#> intersect, setdiff, setequal, union
# Model "true" regression coefficients using the external data
logit_mod <- glm(y ~ cov1 + cov2 + cov3 + cov4, data = ex_binary_df, family = binomial)
# Bootstrap internal control "population" with imbalance w.r.t. covariate 2
pop_int_ctrl <- bootstrap_cov(ex_binary_df, n = 100000, imbal_var = cov2,
imbal_prop = 0.25, ref_val = 0) |>
select(-subjid, -y) # keep only covariate columns
# Convert the marginal drift and treatment effects to conditional
calc_cond_binary(population = pop_int_ctrl, glm = logit_mod,
marg_drift = c(-.1, 0, .1), marg_trt_eff = c(0, .15))
#> # A tibble: 6 × 6
#> marg_drift marg_trt_eff conditional_drift true_control_RR conditional_trt_eff
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 -0.1 0 -0.413 0.446 0
#> 2 -0.1 0.15 -0.413 0.446 0.623
#> 3 0 0 0 0.546 0
#> 4 0 0.15 0 0.546 0.660
#> 5 0.1 0 0.428 0.646 0
#> 6 0.1 0.15 0.428 0.646 0.776
#> # ℹ 1 more variable: true_trt_RR <dbl>