# Binary Endpoint Case - Borrowing from an External Control Arm

In this example, we illustrate how to use Bayesian dynamic borrowing (BDB) with the inclusion of inverse probability weighting to balance baseline covariate distributions between external and internal datasets. This particular example considers a hypothetical trial with a binary endpoint, and our objective is to use BDB with IPWs to construct a posterior distribution for the control response rate $$\theta_C$$. We will use simulated internal and external datasets from the package where each dataset has a binary response variable (1: positive response, 0: otherwise) and four baseline covariates which we will balance.

The external control dataset has a sample size of 150 participants, and the distributions of the four covariates are as follows: - Covariate 1: normal with a mean and standard deviation of approximately 65 and 10, respectively - Covariate 2: binary (0 vs. 1) with approximately 30% of participants with level 1 - Covariate 3: binary (0 vs. 1) with approximately 40% of participants with level 1 - Covariate 4: binary (0 vs. 1) with approximately 50% of participants with level 1

The internal dataset has 160 participants with 80 participants in each of the control arm and the active treatment arms. The covariate distributions of each arm are as follows: - Covariate 1: normal with a mean and standard deviation of approximately 62 and 8, respectively - Covariate 2: binary (0 vs. 1) with approximately 40% of participants with level 1 - Covariate 3: binary (0 vs. 1) with approximately 40% of participants with level 1 - Covariate 4: binary (0 vs. 1) with approximately 60% of participants with level 1

library(beastt)
library(distributional)
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
library(ggplot2)
summary(int_binary_df)
#>      subjid            cov1            cov2          cov3
#>  Min.   :  1.00   Min.   :40.00   Min.   :0.0   Min.   :0.0000
#>  1st Qu.: 40.75   1st Qu.:59.00   1st Qu.:0.0   1st Qu.:0.0000
#>  Median : 80.50   Median :64.00   Median :0.0   Median :0.0000
#>  Mean   : 80.50   Mean   :62.98   Mean   :0.4   Mean   :0.3937
#>  3rd Qu.:120.25   3rd Qu.:68.00   3rd Qu.:1.0   3rd Qu.:1.0000
#>  Max.   :160.00   Max.   :80.00   Max.   :1.0   Max.   :1.0000
#>       cov4             trt            y
#>  Min.   :0.0000   Min.   :0.0   Min.   :0.000
#>  1st Qu.:0.0000   1st Qu.:0.0   1st Qu.:0.000
#>  Median :1.0000   Median :0.5   Median :1.000
#>  Mean   :0.5938   Mean   :0.5   Mean   :0.575
#>  3rd Qu.:1.0000   3rd Qu.:1.0   3rd Qu.:1.000
#>  Max.   :1.0000   Max.   :1.0   Max.   :1.000
summary(ex_binary_df)
#>      subjid            cov1            cov2             cov3
#>  Min.   :  1.00   Min.   :40.00   Min.   :0.0000   Min.   :0.0000
#>  1st Qu.: 38.25   1st Qu.:57.25   1st Qu.:0.0000   1st Qu.:0.0000
#>  Median : 75.50   Median :66.00   Median :0.0000   Median :0.0000
#>  Mean   : 75.50   Mean   :65.21   Mean   :0.2733   Mean   :0.3667
#>  3rd Qu.:112.75   3rd Qu.:71.00   3rd Qu.:1.0000   3rd Qu.:1.0000
#>  Max.   :150.00   Max.   :93.00   Max.   :1.0000   Max.   :1.0000
#>       cov4              y
#>  Min.   :0.0000   Min.   :0.0000
#>  1st Qu.:0.0000   1st Qu.:0.0000
#>  Median :1.0000   Median :0.0000
#>  Mean   :0.5267   Mean   :0.4867
#>  3rd Qu.:1.0000   3rd Qu.:1.0000
#>  Max.   :1.0000   Max.   :1.0000

With the covariate data from both the external and internal datasets, we can calculate the propensity scores and ATT inverse probability weights (IPWs) for the internal and external controls using the calc_prop_scr function. This creates a propensity score object which we can use for calculating a inverse probability weighted power prior in the next step.

Note: when reading external and internal datasets into calc_prop_scr, be sure to include only the treatment arms across which you want to balance the covariate distributions. In this example, we want to balance the covariate distributions of the external control arm to be similar to those of the internal control arm, so we will exclude the internal active treatment arm data from this function.

ps_obj <- calc_prop_scr(internal_df = filter(int_binary_df, trt == 0),
external_df = ex_binary_df,
id_col = subjid,
model = ~ cov1 + cov2 + cov3 + cov4)
ps_obj
#>
#> ── Model ───────────────────────────────────────────────────────────────────────
#> • cov1 + cov2 + cov3 + cov4
#>
#> ── Propensity Scores and Weights ───────────────────────────────────────────────
#> # A tibble: 150 × 4
#>    subjid Internal Propensity Score Inverse Probability Weight
#>     <int> <lgl>                 <dbl>                        <dbl>
#>  1      1 FALSE                 0.342                        0.520
#>  2      2 FALSE                 0.312                        0.453
#>  3      3 FALSE                 0.221                        0.284
#>  4      4 FALSE                 0.461                        0.854
#>  5      5 FALSE                 0.531                        1.13
#>  6      6 FALSE                 0.444                        0.798
#>  7      7 FALSE                 0.424                        0.735
#>  8      8 FALSE                 0.254                        0.340
#>  9      9 FALSE                 0.334                        0.501
#> 10     10 FALSE                 0.242                        0.319
#> # ℹ 140 more rows
#>
#> ── Absolute Standardized Mean Difference ───────────────────────────────────────
#> # A tibble: 4 × 3
#>   <chr>          <dbl>    <dbl>
#> 1 cov1          0.323  0.0365
#> 2 cov2          0.192  0.000289
#> 3 cov3          0.0173 0.0132
#> 4 cov4          0.226  0.00715

In order to check the suitability of the external data, we can create a variety of diagnostic plots. The first plot we might want is a histogram of the overlapping propensity score distributions from both datasets. To get this, we use the prop_scr_hist function. This function takes in the propensity score object made in the previous step, and we can optionally supply the variable we want to look at (either the propensity score or the IPW). By default, it will plot the propensity scores. Additionally, we can look at the densities rather than histograms by using the prop_scr_dens function. When looking at the IPWs with either the histogram or the density functions, it is important to note that only the IPWs for external control participants will be shown because the ATT IPWs of all internal control participants are equal to 1.

prop_scr_hist(ps_obj)

prop_scr_dens(ps_obj, variable = "ipw")

The final plot we might want to look at is a love plot to visualize the absolute standard mean differences (both unadjusted and adjusted by the IPWs) of the covariates between the internal and external data. To do this, we use the prop_scr_love function. Like the previous function, the only required parameter for this function is the propensity score object, but we can also provide a location along the x-axis for a vertical reference line.

prop_scr_love(ps_obj, reference_line = 0.1)

Now that we are happy with our propensity score object, we can use it to calculate a beta inverse probability weighted power prior for $$\theta_C$$. To calculate the power prior, we need to supply:

• weighted object, the propensity score we created above

• response variable, in this case $$y$$

• initial prior, in the form of a beta distributional object

Once we have a power prior, we might want to plot it. To do that, we use the plot_dist function.

pwr_prior <- calc_power_prior_beta(ps_obj,
response = y,
prior = dist_beta(0.5, 0.5))
plot_dist(pwr_prior)

Now that we have a beta power prior, we can calculate the posterior distribution for $$\theta_C$$ using the calc_post_beta function. By defining our prior to be a beta distribution, the resulting posterior distribution will also be beta.

Note: if reading internal data directly into calc_post_beta instead of a propensity score object, be sure to include only the treatment arm of interest (e.g., the internal control arm if creating a posterior distribution for $$\theta_C$$).

post <- calc_post_beta(ps_obj,
response = y,
prior = pwr_prior)
plot_dist(post)

If we want to robustify our power prior for $$\theta_C$$, we can add a vague component to the power prior distribution we previously created to construct a robust mixture prior which we can then pass to calc_post_beta. In general, we can define our prior to be a mixture distribution with an arbitrary number of beta components, in which case the resulting posterior distribution will also be a mixture of beta components.

mix_prior <- dist_mixture(pwr_prior,
dist_beta(0.5, 0.5),
weights = c(0.5, 0.5))
post_mixed <- calc_post_beta(ps_obj,
response = y,
prior = mix_prior)
plot_dist("Control Posterior" = post,
"Mixed Posterior" = post_mixed)

Lastly, we can also create a posterior distribution for the response rate of the active treatment arm by reading the internal data directly into the calc_post_beta function. In this case, we assume a vague beta prior.

As noted earlier, be sure to read in only the data for the internal active treatment arm while excluding the internal control data.

post_treated <- calc_post_beta(internal_data = filter(int_binary_df, trt == 1),
response = y,
prior = dist_beta(0.5, 0.5))
plot_dist("Control Posterior" = post,
"Mixed Posterior" = post_mixed,
"Treatment Posterior" = post_treated)