Causal Inference in Case-Control Studies
The goal of ciccr is to implement methods for carrying out causal
inference in case-control and case-population studies (Jun and Lee,
2023).
You can install the released version of ciccr from
CRAN with:
install.packages("ciccr")
Alternatively, you can install the development version from
GitHub with:
# install.packages("devtools") # uncomment this line if devtools is not installed yetdevtools::install_github("sokbae/ciccr")
We first call the ciccr package.
library(ciccr)
To illustrate the usefulness of the package, we use the dataset ACS_CC
that is included the package. This dataset is an extract from American
Community Survey (ACS) 2018, restricted to white males residing in
California with at least a bachelor’s degree. The ACS is an ongoing
annual survey by the US Census Bureau that provides key information
about US population. We use the following variables:
y = ACS_CC$topincomet = ACS_CC$baplusx = ACS_CC$age
The binary outcome y is defined to be one if a respondent’s annual
total pre-tax wage and salary income is top-coded. In the sample
extract, the top-coded income bracket has median income \$565,000 and
the next highest income that is not top-coded is \$327,000.
The binary treatment t is defined to be one if a respondent has a
master’s degree, a professional degree, or a doctoral degree.
The covariate x is age in years and is restricted to be between 25
and 70.
The original ACS survey is not from case-control sampling but we
construct a case-control sample by the following procedure:
We now construct cubic b-spline terms with three inner knots using the
age variable.
x = splines::bs(x, df = 6)
Using the retrospective sieve logistic regression model, we estimate the
average of the log odds ratio conditional on the case sample by
results_case = avg_RR_logit(y, t, x, 'case')results_case$est#> y#> 0.7286012results_case$se#> y#> 0.1013445
Here, option 'case' refers to conditioning on the event that income is
top-coded.
Similarly, we estimate the average of the log odds ratio conditional on
the control sample by
results_control = avg_RR_logit(y, t, x, 'control')results_control$est#> y#> 0.5469094results_control$se#> y#> 0.1518441
Here, option 'control' refers to conditioning on the event that income
is not top-coded.
We carry out causal inference by
results = cicc_RR(y, t, x, 'cc', 0.95)
Here, ‘cc’ refers to case-control sampling and 0.95 refers to the level
of the uniform confidence band (0.95 is the default choice).
est = results$estprint(est)#> y y#> 0.5469094 0.7286012se = results$seprint(se)#> y y#> 0.1518441 0.1013445ci = results$ciprint(ci)#> y y#> 0.8445183 1.0262101
The S3 object results contains estimates est, standard errors se,
and one-sided confidence bands ci at p = 0 and p = 1. The point
estimates and confidence interval estimates of the cicc_RR command are
based on the scale of log relative risk. It is more conventional to look
at the results in terms of relative risk. To do so, we plot the results
in the following way:
cicc_plot(results)
To interpret the results, we assume both marginal treatment response
(MTR) and marginal treatment selection (MTS). In this setting, MTR means
that everyone will not earn less by obtaining a degree higher than
bachelor’s degree; MTS indicates that those who selected into higher
education have higher potential to earn top incomes. Based on the MTR
and MTS assumptions, we can conclude that the treatment effect lies in
between 1 and the upper end point of the one-sided confidence interval
with high probability. Thus, the estimates in the graph above suggest
that the effect of obtaining a degree higher than bachelor’s degree is
anywhere between 1 and the upper end points of the uniform confidence
band. This roughly implies that the chance of earning top incomes may
increase up to by a factor as large as the upper end points of the
uniform confidence band, but allowing for possibility of no positive
effect at all. The results are shown over the range of the unknown true
case probability. See Jun and Lee,
2020 for more detailed explanations
regarding how to interpret the estimation results.
We can compare these results with estimates obtained from logistic
regression.
logit = stats::glm(y~t+x, family=stats::binomial("logit"))est_logit = stats::coef(logit)ci_logit = stats::confint(logit, level = 0.9)#> Waiting for profiling to be done...# point estimateexp(est_logit)#> (Intercept) t x1 x2 x3 x4#> 0.05461156 2.06117153 4.42179639 12.99601849 19.03962976 26.83565737#> x5 x6#> 6.42381406 26.14359394# confidence intervalexp(ci_logit)#> 5 % 95 %#> (Intercept) 0.01960819 0.1304108#> t 1.75166056 2.4271287#> x1 1.05679997 21.6604223#> x2 5.50583091 33.8909622#> x3 6.79458010 61.3258710#> x4 10.22943808 78.7353953#> x5 2.00536450 22.8509008#> x6 8.66983039 87.6311482
Here, the relevant coefficient is 2.06 (t) and its two-sided 90%
confidence interval is [1.75, 2.43]. If we assume strong ignorability,
the treatment effect is about 2 and its two-sided confidence interval is
between [1.75, 2.43]. However, it is unlikely that the
higher-degrees-than-BA treatment satisfies the strong ignorability
condition.
See the vignette for other examples that include inference on
attributable risk and how to work with case-population samples.
Jun, S.J. and Lee, S. (2023). Causal Inference under Outcome-Based
Sampling with Monotonicity Assumptions.
https://arxiv.org/abs/2004.08318, accepted for publication in Journal
of Business & Economic Statistics.
Manski, C.F. (1997). Monotone Treatment Response. Econometrica, 65(6),
1311-1334.
Manski, C.F. and Pepper, J.V. (2000). Monotone Instrumental Variables:
With an Application to the Returns to Schooling. Econometrica, 68(4),
997-1010.