Authors: Lucy D’Agostino McGowan,
Malcolm Barrett
License:
MIT
Install the CRAN version
install.packages("tipr")Or install the development version from GitHub:
# install.packages(devtools)
devtools::install_github("r-causal/tipr")library(tipr)After fitting your model, you can determine the unmeasured confounder
needed to tip your analysis. This unmeasured confounder is determined by
two quantities, the relationship between the exposure and the unmeasured
confounder (if the unmeasured confounder is continuous, this is
indicated with exposure_confounder_effect, if binary, with
exposed_confounder_prev and unexposed_confounder_prev), and the
relationship between the unmeasured confounder and outcome
confounder_outcome_effect. Using this 📦, we can fix one of these and
solve for the other. Alternatively, we can fix both and solve for n,
that is, how many unmeasured confounders of this magnitude would tip the
analysis.
This package comes with a few example data sets. For this example, we
will use exdata_rr. This data set was simulated such that there are
two confounders, one that was “measured” (and thus usable in the main
analysis, this is called measured_confounder) and one that is
“unmeasured” (we have access to it because this is simulated data, but
ordinarily we would not, this variable is called
.unmeasured_confounder).
Using the example data exdata_rr, we can estimate the exposure-outcome
relationship using the measured confounder as follows:
mod <- glm(outcome ~ exposure + measured_confounder, data = exdata_rr,
family = poisson)
mod |>
broom::tidy(exponentiate = TRUE, conf.int = TRUE)## # A tibble: 3 × 7
## term estimate std.error statistic p.value conf.low conf.high
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 0.0366 0.151 -21.9 2.56e-106 0.0269 0.0486
## 2 exposure 1.49 0.166 2.43 1.52e- 2 1.09 2.08
## 3 measured_confounder 2.43 0.0754 11.7 7.51e- 32 2.09 2.81
We see the above example, the exposure-outcome relationship is 1.5 (95%
CI: 1.1, 2.1). Note, in practice when estimating the effect of an
exposure on a binary outcome using a GLM with the Poisson distribution
and log link function, it is important to use a sandwich estimator to
appropriately estimate the variability (this can be done in R using the
sandwich package), which in this case gives a very similar result (95%
CI: 1.1, 2.0).
We are interested in a continuous unmeasured confounder, so we will use
the tip_with_continuous() function.
Let’s assume the unmeasured confounder is normally distributed with a
mean of 0.5 in the exposed group and 0 in the unexposed (and unit
variance in both), resulting in a mean difference of 0.5
(exposure_confounder_effect = 0.5), let’s solve for the relationship
between the unmeasured confounder and outcome needed to tip the analysis
(in this case, we are solving for confounder_outcome_effect).
tip(effect_observed = 1.5, exposure_confounder_effect = 0.5)## ℹ The observed effect (1.5) WOULD be tipped by 1 unmeasured confounder with the
## following specifications:
## • estimated difference in scaled means between the unmeasured confounder in the
## exposed population and unexposed population: 0.5
## • estimated relationship between the unmeasured confounder and the outcome:
## 2.25
## # A tibble: 1 × 5
## effect_adjusted effect_observed exposure_confounder_e…¹ confounder_outcome_e…²
## <dbl> <dbl> <dbl> <dbl>
## 1 1 1.5 0.5 2.25
## # ℹ abbreviated names: ¹exposure_confounder_effect, ²confounder_outcome_effect
## # ℹ 1 more variable: n_unmeasured_confounders <dbl>
A hypothetical unobserved continuous confounder a scaled mean difference
between exposure groups of 0.5 would need a relationship of at least
2.25 with the outcome to tip this analysis at the point estimate.
tip(effect_observed = 1.09, exposure_confounder_effect = 0.5)## ℹ The observed effect (1.09) WOULD be tipped by 1 unmeasured confounder with
## the following specifications:
## • estimated difference in scaled means between the unmeasured confounder in the
## exposed population and unexposed population: 0.5
## • estimated relationship between the unmeasured confounder and the outcome:
## 1.19
## # A tibble: 1 × 5
## effect_adjusted effect_observed exposure_confounder_e…¹ confounder_outcome_e…²
## <dbl> <dbl> <dbl> <dbl>
## 1 1 1.09 0.5 1.19
## # ℹ abbreviated names: ¹exposure_confounder_effect, ²confounder_outcome_effect
## # ℹ 1 more variable: n_unmeasured_confounders <dbl>
A hypothetical unobserved continuous confounder a scaled mean difference
between exposure groups of 0.5 would need a relationship of at least
1.19 with the outcome to tip this analysis at the 5% level, rendering it
inconclusive.
Because this is simulated data, we can see what the true unmeasured confounder looked like. First we will calculate the difference in scaled means.
exdata_rr |>
dplyr::group_by(exposure) |>
dplyr::summarise(m = mean(.unmeasured_confounder / sd(.unmeasured_confounder))) |>
tidyr::pivot_wider(names_from = exposure,
values_from = m,
names_prefix = "u_") |>
dplyr::summarise(estimate = u_1 - u_0)## # A tibble: 1 × 1
## estimate
## <dbl>
## 1 0.494
Now we can refit the above model with this unmeasured confounder included. According to our tipping point result, as long as the risk ratio of the unmeasured confounder and outcome in the model is greater than 2.25, the result that we observed will be “tipped” (the point estimate will cross the null).
mod_true <- glm(
outcome ~ exposure + measured_confounder + .unmeasured_confounder,
data = exdata_rr,
family = poisson)
mod_true |>
broom::tidy(exponentiate = TRUE, conf.int = TRUE)## # A tibble: 4 × 7
## term estimate std.error statistic p.value conf.low conf.high
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 0.0245 0.163 -22.7 1.49e-114 0.0176 0.0334
## 2 exposure 0.921 0.172 -0.477 6.34e- 1 0.660 1.30
## 3 measured_confounder 2.44 0.0746 11.9 6.95e- 33 2.11 2.82
## 4 .unmeasured_confoun… 2.42 0.0742 11.9 1.35e- 32 2.09 2.80
Notice here the .unmeasured_confounder effect is 2.42 (which is
greater than the 2.25 we calculated that would be needed to render our
result null) and, as expected, the point estimate for the exposure has
crossed the null (and now is less than 1).
Now we are interested in the binary unmeasured confounder, so we will
use the tip_with_binary() function.
Let’s assume the unmeasured confounder is prevalent in 25% of the
exposed population (exposed_confounder_prev = 0.25) and in 10% of the
unexposed population (unexposed_confounder_prev = 0.10) – let’s solve
for the relationship between the unmeasured confounder and the outcome
needed to tip the analysis (confounder_outcome_effect).
tip_with_binary(effect_observed = 1.09,
exposed_confounder_prev = 0.25,
unexposed_confounder_prev = 0.10)## ℹ The observed effect (1.09) WOULD be tipped by 1 unmeasured confounder with
## the following specifications:
## • estimated prevalence of the unmeasured confounder in the exposed population:
## 0.25
## • estimated prevalence of the unmeasured confounder in the unexposed
## population: 0.1
## • estimated relationship between the unmeasured confounder and the outcome:
## 1.64
## # A tibble: 1 × 6
## effect_adjusted effect_observed exposed_confounder_prev unexposed_confounder…¹
## <dbl> <dbl> <dbl> <dbl>
## 1 1 1.09 0.25 0.1
## # ℹ abbreviated name: ¹unexposed_confounder_prev
## # ℹ 2 more variables: confounder_outcome_effect <dbl>,
## # n_unmeasured_confounders <dbl>
A hypothetical unobserved binary confounder that is prevalent in 10% of the unexposed population and 25% of the exposed population would need to have a relationship with the outcome of 1.64 to tip this analysis at the 5% level, rendering it inconclusive.
Suppose we are concerned that there are many small, independent, continuous, unmeasured confounders present.
tip(effect_observed = 1.09,
exposure_confounder_effect = 0.25,
confounder_outcome_effect = 1.05)## ℹ The observed effect (1.09) WOULD be tipped by 7 unmeasured confounders with
## the following specifications:
## • estimated difference in scaled means between the unmeasured confounder in the
## exposed population and unexposed population: 0.25
## • estimated relationship between the unmeasured confounder and the outcome:
## 1.05
## # A tibble: 1 × 5
## effect_adjusted effect_observed exposure_confounder_e…¹ confounder_outcome_e…²
## <dbl> <dbl> <dbl> <dbl>
## 1 1 1.09 0.25 1.05
## # ℹ abbreviated names: ¹exposure_confounder_effect, ²confounder_outcome_effect
## # ℹ 1 more variable: n_unmeasured_confounders <dbl>
It would take about 7 independent standardized Normal unmeasured
confounders with a mean difference between exposure groups of 0.25 and a
relationship with the outcome of 1.05 tip the observed analysis at the
5% level, rendering it inconclusive.
These functions were created to easily integrate with models tidied
using the broom package. This is not necessary to use these
functions, but a nice feature if you choose to do so. Here is an example
of a logistic regression fit with glm and tidied with the tidy
function broom that can be directly fed into the tip() function.
if (requireNamespace("broom", quietly = TRUE) && requireNamespace("dplyr", quietly = TRUE)) {
glm(outcome ~ exposure + measured_confounder, data = exdata_rr,
family = poisson) |>
broom::tidy(conf.int = TRUE, exponentiate = TRUE) |>
dplyr::filter(term == "exposure") |>
dplyr::pull(conf.low) |>
tip(confounder_outcome_effect = 2.5)
}## ℹ The observed effect (1.09) WOULD be tipped by 1 unmeasured confounder with
## the following specifications:
## • estimated difference in scaled means between the unmeasured confounder in the
## exposed population and unexposed population: 0.09
## • estimated relationship between the unmeasured confounder and the outcome: 2.5
## # A tibble: 1 × 5
## effect_adjusted effect_observed exposure_confounder_e…¹ confounder_outcome_e…²
## <dbl> <dbl> <dbl> <dbl>
## 1 1 1.09 0.0907 2.5
## # ℹ abbreviated names: ¹exposure_confounder_effect, ²confounder_outcome_effect
## # ℹ 1 more variable: n_unmeasured_confounders <dbl>
Please note that the tipr project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.