Using instrumental variables to estimate treatment effects has the potential to bypass bias due to confounding (endogeneity). This method is far from a panacea as it is not possible to empirically validate an instrumental variable: it is not possible to prove or disprove that a putative instrument does indeed satisfy the definition of an instrumental variable, with the exception of randomization as designed into clinical and other trials. Despite this weakness, the method has been steadily gaining in popularity as a way to resolve confounding in observational studies.
A property and criticism of instrumental variable estimators is that they provide an estimate of the treatment effect on a subset of the population, sometimes referred to as the compliers1 or population on the margin.2 It is plausible that the treatment effect is different in the remainder of the population, for example, heterogeneity of the treatment effect. The concept of the complier is clear when the instrumental variable is dichotomous and causal.3 In this case, it is possible to define a latent categorization known as principal strata4–7 consisting of three categories: compliers, takers, and nontakers (also known as “always takers” and “never takers”8,9).
The compliers are individuals whose treatment is determined by the instrument. The takers are those who would receive treatment regardless of the value of the instrument; the nontakers are those who would not. The property that the instrumental variable estimator provides an estimate of the treatment effect on the compliers opens a criticism: the treatment effect in the takers or nontakers may be different. The effect of treatment in the nontakers is relevant because there may be scenarios in which they would take the treatment. For instance, those who are not affected by the instrument of one study (e.g., provider preference) may be affected by the instrument of another study (e.g., differential geographic distance to treatments being compared, gene in another location). Due to the latency of the principal strata, it is not possible to definitively categorize each subject in a sample. For this reason, instrumental variable estimators have been criticized as applying to an unknowable strata. However, it is possible to characterize the compliers.1,10–12 Parametric models combined with Bayesian methods have been proposed for modeling the probability of being a complier in terms of baseline characteristics.8
In this brief report, we show nonparametric methods for presenting and comparing baseline characteristics among the compliers, takers, and nontakers for the setting of a binary fixed-in-time instrument that has a causal effect on a binary fixed-in-time exposure.
METHODS
Let R represent the binary instrument (e.g., randomization). Let ;)
be the potential treatment indicators, and the observed treatment indicator. The potential treatment indicators stratify the population into four principal strata4: compliers , takers , nontakers , and defiers . We assume that the defier strata has probability zero, known as the Monotonicity assumption, which facilitates partial identification of the taker and nontaker strata using observations of R and X as shown in Table 1.
TABLE 1.: Identification of the Principal Strata According to the Observations of the Instrument, R, and exposure, X
Let pC, pT, and pN be the probabilities of sampling subjects from the complier, taker, and nontaker strata. From Table 1, it follows that and , and the equation, implies (or equivalently ). Estimators of the frequencies of the three strata pN, pT, and pC are found in Table 2.
TABLE 2.: Estimators for Baseline Characteristics of the Compliers, Takers, and Nontakers
Table 2 also presents estimators of the means in the compliers, takers, and nontakers, where is the sample mean, and is the sample mean in subjects for whom , . The formula for the compliers follows from the identity, . Estimators for the variance in the three strata are presented next in the table, where is the sample variance and is the sample variance in subjects for whom , . The formula for the variance in the compliers is based on the law of total variance.
Formulas for the standard error of the mean are presented in the row labeled SEM, where is the SD in subjects for whom , . The Appendix A provides a derivation of the variance of the mean in the compliers, as well as expressions for the covariances of the three means.
The final row of Table 2 provides estimators of the cumulative distribution function for each strata, where F is the empirical distribution function (EDF) of the baseline characteristics, i.e., , F10 is the EDF among subjects for whom , , and F01 is the EDF among subjects for whom , . The formula for the distribution in the compliers follows from the identity F=pCFC + pTFT + pNFN. Quantiles such as the median can be calculated for the takers and nontakers as usual. An estimator of the q-quantile in the compliers is .
A test of equality of means (or proportions) between the three strata, can be based on the vector which has an expected value of (0,0) and covariance matrix equal to
Under the null hypothesis and the assumption that pC > 0 the statistic −1 has a chi-square distribution with 2 degrees of freedom in large samples.
R code for these statistics can be found at https://github.com/toddamackenzie/Baseline-Characteristics-for-Instrumental-Variable-Analyses. or at https://links.lww.com/EDE/B402.
Example
Bekelis et al3 used the method of instrumental variables to estimate the effect of teaching versus nonteaching hospitals on outcomes in 186,483 patients from New York undergoing spine and cranial operations. They used a continuously valued geographic instrument; the proportion in each of the 62 counties of New York of surgical patients who received their surgery at a teaching hospital. Some limitations of using geographic and preference instruments are given in Ertefaie et al5 and Hernán and Robins.7 We found that teaching hospitals increased mortality, discharge to a facility, and length of stay.
We conjecture that this instrument is causal3 as the local (i.e., county) preference for teaching hospitals affects a resident’s choice of where they access surgery. Therefore, the methods we advocate in this paper for causal instruments are applicable. To illustrate the method for characterizing the compliers, we dichotomized the instrument at its median value of 92.6%. The complier average causal effect of teaching versus nonteaching hospitals on mortality was 4.9 (95% confidence interval = 2.0, 7.8) more deaths per 1000 surgeries.
The proportion of patients who were compliers, takers, and nontakers for this instrument was 26.0%, 71.3%, and 2.6%, respectively: Only 26% of patients are influenced by living in a county with a teaching hospital preference above or below the median value of 92.6%. Table 3 reports mean age, and the proportion of females, and race, for the compliers, takers and nontakers. It also reports the difference between the compliers and takers, as well as the difference between the compliers and nontakers. There was a higher proportion of females among the compliers than among the takers and nontakers. In addition the compliers included considerably fewer whites. The median age in the compliers was 56, 55 in the takers, and 53 in the nontakers.
TABLE 3.: Characteristics and Comparisons of the Compliers, Takers, and Nontakers Using Means and Proportions (Standard Error in Brackets)
Discussion
We recommend that the statistics proposed in this report should accompany the reporting of instrumental variable analyses. An alternative is to model the strata an individual is in according to their baseline characteristics.8 The advantage of our approach is that it is nonparametric and matches the convention of reporting descriptive statistics for baseline characteristics, such as the first table of most clinical and epidemiological studies.
If there is treatment effect heterogeneity, it is possible that different instrumental variable analyses will result in substantively different estimates depending on the instrument. For instance, two popular choices for instrumental variables are formed from distances traveled and clinic (or physician) preference. The subpopulation of individuals whose treatment choice is affected by the difference in distance may be quite different from the subpopulation of individuals whose treatment choice is affected by the preference of their clinic.
Mendelian Randomization studies employ a genotype or genotypic score as an instrument for the effect of a phenotype on an outcome. In this design, the subpopulation of compliers is those for whom the genotypic instrument determines the phenotype as opposed to being unaffected by the instrument. The methods used in our paper should be reported in Mendelian randomization studies to determine this subpopulation.
ACKNOWLEDGMENTS
All statements in this report, including its findings and conclusions, are solely those of the authors and do not necessarily represent the views of the Patient-Centered Outcomes Research Institute (PCORI), its Board of Governors or Methodology Committee.
REFERENCES
1. Baiocchi M, Cheng J, Small DS. Instrumental variable methods for causal inference. Stat Med. 2014;33:22972340.
2. Bartolucci F, Farcomeni A. Causal inference in paired two-arm experimental studies under noncompliance with application to prognosis of myocardial infarction. Stat Med. 2013;32:43484366.
3. Bekelis K, Missios S, Coy S, MacKenzie TA. Association of hospital teaching status with neurosurgical outcomes: an instrumental variable analysis. World Neurosurg. 2018;110:e689e698.
4. Card D. The causal effect of education on earnings. In: Handbook of Labor Economics. 1999:18011863.
5. Ertefaie A, Small DS, Rosenbaum PR. Quantitative evaluation of the trade-off of strengthened instruments and sample size in observational studies. J Am Stat Assoc. 2018:113.
6. Frangakis CE, Rubin DB. Principal stratification in causal inference. Biometrics. 2002;58:2129.
7. Hernán MA, Robins JM. Instruments for causal inference: an epidemiologist’s dream? Epidemiology. 2006;17:360372.
8. Hirano K, Imbens GW, Rubin DB, Zhou X-H. Assessing the effect of an influenza vaccine in an encouragement design. Biostatistics. 2000;1:6988.
9. Imbens GW, Rubin DB. Estimating outcome distributions for compliers in instrumental variables models. Rev Econ Stud. 1997;64:555574.
10. Swanson SA, Hernán MA. The challenging interpretation of instrumental variable estimates under monotonicity. Int J Epidemiol. 2018;47(4):12891297.
11. Angrist JD, Pischke J-S, Pischke J-S. Mostly Harmless Econometrics: An Empiricists Companion. 2013.Cram101 Publishing;
12. Swanson SA, Hernán MA. Commentary: how to report instrumental variable analyses (suggestions welcome). Epidemiology. 2013;24:370374.
APPENDIX A
Derivations of and the Covariance of the Means
The estimator can be written as follows:
where
Therefore, the variance of the estimator of the mean in the compliers is as follows:
Conditional on R and X, the covariance of estimators and is follows:
The covariance of and is as follows:
Using a similar derivation, the covariance of and is as follows:
