# Validation

## Test Suite

The following fictitious example of a prospective cohort study will be used to validate the correct estimation of the BSW package in R.

Exposed Non-Exposed
Cases 200 50
Non-Cases 50 200
library(testthat)
library(BSW)
df <- data.frame(y = rep(c(0, 1), each = 250),
x = rep(c(0, 1, 0, 1), times = c(200, 50, 50, 200))
)
RR <- (200 * 250) / (50 * 250)
SE <- sqrt((1/200 + 1/50) - (1/250 + 1/250))
fit <- bsw(y ~ x, df)
out <- summary(fit)
Call:
bsw(formula = y ~ x, data = df)

Convergence: TRUE
Coefficients:
Estimate Std. Error   z value     Pr(>|z|)  RR      2.5%    97.5%
(Intercept) -1.609438  0.1264911 -12.72372 4.365487e-37 0.2 0.1560848 0.256271
x            1.386294  0.1303840  10.63239 2.106386e-26 4.0 3.0979677 5.164676

Iterations: 37 

The relative risk for exposed individuals compared to non-exposed individuals can be calculated from

$$RR = \displaystyle\frac{200 * 250}{50*250} = 4$$.
test_that(desc = "Estimated relative risk is equal to 4",
code = {
expect_equal(object = unname(exp(coef(fit))),
expected = RR)
}
)
Test passed 🥇

The standard error of the natural logarithm of the relative risk can be calculated from

$$SE(ln(RR)) = \displaystyle\sqrt{\Big(\frac{1}{200} + \frac{1}{50}\Big) - \Big(\frac{1}{250}+\frac{1}{250}\Big)} = 0.130384$$.
test_that(desc = "Estimated standard error is equal to 0.1303840",
code = {
expect_equal(object = unname(out$std.err), expected = SE) } ) Test passed 🌈 The z-value can be calculated from $$z = \displaystyle\frac{1.386294}{0.130384} = 10.63239$$. test_that(desc = "Estimated z-value is equal to 10.63239", code = { expect_equal(object = unname(out$z.value),
expected = log(RR) / SE)
}
)
Test passed 😀

The 95% confidence interval limits can be calculated from

$$exp(1.386294 \pm 1.959964 * 0.1303840) = [3.097968; 5.164676]$$.
test_that(desc = "Estimated 95% confidence interval limits are equal to 3.097968 and 5.164676",
code = {
expect_equal(object = unname(exp(confint(fit)[2,])),
expected = exp(log(RR) + SE * qnorm(c(0.025, 0.975))))
}
)
Test passed 😀