Marginal Structural Models for Comparing Alternative Treatment Strategies in Ophthalmology using Observational Data

Figures & data

FIGURE 1. Causal diagrams depicting relationships among key variables, illustrating the issue of time-dependent confounding. (a) Actual relationships in a population, where the intermediate variable L is a time-dependent confounder. (b) Relationships in the pseudo-population created by weighting (condition 2 is removed). A_0, initial treatment; A₁, subsequent treatment; L, intermediate variable (disease activity); Y, outcome (visual acuity); U, common cause (possibly unmeasured) of L and Y. Conditions: (1) Covariate L is an independent predictor of outcome Y because of 1A (effect of L on Y) and/or 1B (unmeasured common cause U of L and Y). (2) Covariate L influences subsequent treatment A₁. (3) Covariate L is influenced by earlier treatment A₀.

TABLE 1. Hypothetical data illustrating an observational study of the effect of IOP-lowering medication on visual acuity.

Initial use of IOP lowering medication (initial treatment status, A0)	Original population, n	Pseudo- population, n	IOP > 21 mmHg after A0 (intermediate variable, L)	Original population, n	Pseudo-population, n	Later IOP medication (subsequent treatment, A1)	Original population, n (%)	Pseudo-population, n (%)	Diminished visual acuity (%) (outcome, Y)
No	1000	4000	No	250	1000	No	200 (10.00)	500 (6.25)	35
						Yes	50 (2.50)	500 (6.25)	25
			Yes	750	3000	No	375 (18.75)	1500 (18.75)	55
						Yes	375 (18.75)	1500 (18.75)	45
Yes	1000	4000	No	750	3000	No	375 (18.75)	1500 (18.75)	35
						Yes	375 (18.75)	1500 (18.75)	25
			Yes	250	1000	No	50 (2.50)	500 (6.25)	55
						Yes	200 (10.00)	500 (6.25)	45

IOP, intraocular pressure.

The labels for the variables (A₀, A₁, L, and Y) refer to the corresponding labels in Figure 1. The pseudo-population represents what would have been seen had each person been able to participate simultaneously in each of the four arms of a clinical trial with treatment sequences no IOP medication-no IOP medication, no-yes, yes-no, and yes-yes; it essentially represents what might have been if IOP did not influence later treatment and so the effects of earlier and later treatment were un-confounded (note that initial treatment and IOP are associated with later treatment in the original population, column 5, but not in the pseudopopulation, column 6). In the weighted analysis, we simply regress the outcome on the treatment sequence, which is equivalent to the average in the weighted population of the outcome. Under the sequence no medication each time (“no-no”), this average is with diminished acuity. Similarly, the averages for the “no-yes,” “yes-no,” and “yes-yes” sequences are 40%, 40%, and 30%, respectively.