P.Mean: More discussion on instrumental variables (created 2010-05-03).

I attended the May meeting of the KUMC Statistics Journal Club. The topic of discussion was a paper outlining the properties and applications of instrumental variables.

I had written about this topic a couple of years ago at my old website

and although I was a bit uncertain when I first addressed the topic, I think that my comments weren't too bad. I said in part:

As I understand it, instrumental variables are used to control for measurement error in your independent variables. Measurement error causes bias in most regression models. In general, but not always, it tends to flatten out or dilute the impact of an independent variable. If you want to get an unbiased estimate, you have to use an alternative approach. Some of these methods require you to specify the specific amount of measurement error that is present in your independent variable. Other approaches such as Deming regression modify the traditional fitting method of least squares. A third approach is to find and use an instrumental variable.

I can't provide a formal mathematical definition of an instrumental variable, and you probably wouldn't want to see such a definition. In very simple (overly simplistic?) terms, an instrumental variable is an alternative variable which does not suffer from measurement error and which only affects the outcome variable through its relationship with the independent variable. Such a condition is extremely difficult to verify empirically. Most of the time, an instrumental variable is identified by a subject matter expert based on their general understanding of the area. So a statistician like me is incapable of telling you what instrumental variable to use.

I offered a couple of resources:

  1. Wikipedia. Instrumental variable. Excerpt: "In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible. Statistically, IV methods allow consistent estimation when the explanatory variables (covariates) are correlated with the error terms. Such correlation may occur when the dependent variable causes at least one of the of covariates ("reverse" causation), when there are relevant explanatory variables which are omitted from the model, or when the covariates are subject to measurement error. In this situation, ordinary linear regression generally produces biased and inconsistent estimates. However, if an instrument is available, consistent estimates may still be obtained. An instrument is a variable that does not itself belong in the explanatory equation and is correlated with the endogenous explanatory variables, conditional on the other covariates." [Accessed May 3, 2010]. Available at: http://en.wikipedia.org/wiki/Instrumental_variable.
  2. David A. Kenny. SEM: Instrumental Variables. Excerpt: "Denote Y as the endogenous variable, U as its disturbance, I as an instrumental variable, and Z as the set of variables that cause Y but not needing an instrumental variable. The defining feature of an instrumental variable is that I is assumed not to directly cause Y: The path from I to Y is zero. The zero path is given by theory, not by statistical analysis. That is, one should not regress Y on X, I, and Z, and select I by seeing which variables have coefficients that are not significantly different from zero" [Accessed May 3, 2010]. Available at: http://davidakenny.net/cm/iv.htm.

The papers discussed at journal club were:

  1. Edwin P Martens, Wiebe R Pestman, Anthonius de Boer, Svetlana V Belitser, Olaf H Klungel. Instrumental variables: application and limitations. Epidemiology. 2006;17(3):260-267. To correct for confounding, the method of instrumental variables (IV) has been proposed. Its use in medical literature is still rather limited because of unfamiliarity or inapplicability. By introducing the method in a nontechnical way, we show that IV in a linear model is quite easy to understand and easy to apply once an appropriate instrumental variable has been identified. We also point out some limitations of the IV estimator when the instrumental variable is only weakly correlated with the exposure. The IV estimator will be imprecise (large standard error), biased when sample size is small, and biased in large samples when one of the assumptions is only slightly violated. For these reasons, it is advised to use an IV that is strongly correlated with exposure. However, we further show that under the assumptions required for the validity of the method, this correlation between IV and exposure is limited. Its maximum is low when confounding is strong, such as in case of confounding by indication. Finally, we show that in a study in which strong confounding is to be expected and an IV has been used that is moderately or strongly related to exposure, it is likely that the assumptions of IV are violated, resulting in a biased effect estimate. We conclude that instrumental variables can be useful in case of moderate confounding but are less useful when strong confounding exists, because strong instruments cannot be found and assumptions will be easily violated. [Accessed May 3, 2010]. Available at: http://igitur-archive.library.uu.nl/bio/2008-0425-200852/UUindex.html.
  2. N Zohoori, D A Savitz. Econometric approaches to epidemiologic data: relating endogeneity and unobserved heterogeneity to confounding. Ann Epidemiol. 1997;7(4):251-257. Abstract: "The concepts of endogeneity and unobserved heterogeneity are well-known among econometricians. However, these issues are rarely addressed in epidemiologic studies. This paper explores these two concepts, their relationship to each other, and the implications for analysis in epidemiologic studies. An endogenous variable is defined as a predictor variable which is partly determined by factors within the model itself, while unobserved heterogeneity is conceptualized as a vector of missing variables acting through the error term. Under certain assumptions, the simultaneous existence of an endogenous variable and unobserved heterogeneity is shown to act in a manner analogous to confounding. Specifically, this occurs due to an association between the error term in the equation and the endogenous predictor variable. The accepted econometric solution to this problem is to replace the endogenous variable with an 'instrumental variable' which is not correlated with the error term and thus not susceptible to confounding. The validity of these concepts and of the proposed solution are discussed." [Accessed May 3, 2010]. Available at: http://www.ncbi.nlm.nih.gov/pubmed/9177107.

These articles talked about unmeasured confounders rather than measurement error. Unmeasured confounders create problems identical the problems caused by measurement error, though I was unclear on this link prior to attending journal club. If the instrumental variable meets three key assumptions, then an adjustment for the instrumental variable (effectively equivalent to two stage least squares), will remove the effect of confounding.

There were several points of contention. First, one person claimed that you could use simulation to create an instrumental variable. I and several others disagreed. Another person claimed that you should use instrumental variables for randomized studies in addition to observational studies. A third person was sharply critical of any use of instrumental variables, because the assumptions were too tenuous and the situations where you could use instrumental variables effectively are situations where other approaches to controlling for confounding would make more sense.

The person who led the discussion pointed out two valuable conclusions from the Martens et al article. First, instrumental variables that have a weak relationship to exposure tend to inflate variance substantially. Second, if there is strong confounding, then it is difficult to find an instrumental variable that works well. Confounding by indication is an example where there is typically very strong confounding effects.

I enjoyed the discussions and the controversies and felt a lot more comfortable with when and how to use instrumental variables and limitations of this approach.

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