A compound representation of the multiple treatment propensity score with applications to marginal structural modeling

Demonstration of confounding if p(t|x)=0

The following example shows how estimates of treatment effect for different treatment types can be biased if the positivity constraint is violated. Assume two time periods and two treatments, t ₁ and t ₂. Untreated is denoted by t ₀. Assume that the probability that a subject appears in interval 2 is twice the probability that a subject appears in interval 1. The probabilities of a subject receiving no treatment or the treatments and the probabilities of the adverse outcome, p _j (Y=1), for the two time intervals, j=1, 2, and for the combined interval, j=C, are given in Table 6. Treatment 1 is given in interval 1 but not in interval 2, whereas treatment 2 is given in both intervals. The disease becomes more contagious and more virulent in interval 2 as compared with interval 1. This table also presents the odds ratios of the adverse outcomes, comparing treated vs. non-treated patients, for each of the time periods and for the combined time period. One sees that, in time interval 1, treatments 1 and 2 are of equal effectiveness, with odds ratios of 0.11. The odds ratio for treatment 2 in interval 2 is 0.08 – note that the probability of the adverse outcome increases for both untreated and those given treatment 2 in interval 2 as compared with interval 1. The combined odds ratio is also shown in the table. From the combined odds ratios, treatment 1 is more effective than treatment 2, whereas they are of equal effectiveness in interval 1, and treatment 1 is not used against the more virulent strain in interval 2. The combined odds ratio for treatment 1 is confounded by comparing treatment effectiveness of the treated population in interval 1 with untreated patients from interval 2, a period of time during which treatment 1 was not available.

An alternate derivation of SW ¹

Robins [11] defines marginal structural models (MSM) to predict outcomes conditioned on baseline covariates of interest from observational data applicable to both failure time studies and outcomes measured at the end of follow up. The MSM incorporates time varying treatments subject to conditions [11], Estimators are derived using influence functions, and The models are applicable to of outcomes measured after a pre-set duration or time-to-event observations. time-to-event and non-linear least squares, semiparametric regression, models to predict outcomes from observational data general treatment and response time dependent processes, where the response process includes the outcome process of interest and the process of other recorded variables. Examples include treatment processes such that the treatment is given at multiple discrete time points and outcome process that are measured at a fixed time or are failure time process. He shows, using influence functions, that weighting observation by the inverse of a subject’s probability of having had his observed treatment history allows for the estimation of causal effects from non-randomized observations. A simpler proof using importance sampling that is applicable to the case of one of several possible treatments, including the null treatment, given at a single time point is presented.

Assume an independent set of N samples {(y _j , t _j , x _j )∣1≤j≤N}, where y _j is the outcome, and t j = A ( x j ) ∈ T . The samples can be expressed as {(y _j , a _j , t _j , x _j )∣1≤j≤N} where a _j ∈ {0, 1} and t j ∈ T 1 . If a _j =0, t _j is interpreted as the potential treatment. Assume that there is a discrete set L and a mapping ϕ: X → L such that p(t _j ∣x _j )=p(t _j ∣ϕ(x _j )) – that is, t j ∼ ψ ϕ ( x j ) . Let Y ⁱ denote the outcome random variable if all patients have exposure A=i, for i=0, 1.

The observations can be used to estimate causal effects if the following assumptions hold:

1. Y i ∐ A | X , T .

2. p(A|X)p(T|X)>0.

3. The outcome of one individual is independent of the treatment assignment of any other.

From assumption A.1, within strata of (X, T) treated and untreated patients are exchangeable [14], and from assumption A.3., outcomes of different patients are independent. Thus, the expected value of the causal variables can be computed from observations, according to

and

Importance sampling [31] states that if {z _j } is a sample drawn from probability density function f, and g is a probability density function such that g(x) = 0 if f(x) = 0, then { g ( z j ) f ( z j ) z j } is a sample drawn from g. This result is used to transform samples drawn from f(a=1, t _j , x) and f(a=0, x) to samples drawn from f(x) so that A.1 holds for the weighted samples.

Let S W j 1 ⊙ ( a j , t j , x j ) denote the sample (a _j , t _j , x _j ) counted with multiplicity S W j 1 . Let a _o ∈ {0, 1} and t _o ∈ {t ₁, …, t _M }. If a _o =1, define S _o = {(a _j , t _j , x _j )| a _j =a _o , t _j =t _o }, and if a _o =0, define S _o = {(a _j , t _j , x _j )| a _j =a _o }. Let N _o = |S _o |. Let I _o be the indicator function for S _o defined by I _o (j) = 1 if (a _j , t _j , x _j ) ∈ S _o , and I _o (j) = 0, otherwise.

Define

Additional simulation results

Table 7 lists the log odds, probabilities, and odds ratios of Y=1 for all combinations of the input variables. The outcome probabilities were selected and the coefficients were obtained by solving a system of equations.