Question: Could the authors clarify the distinction between their approach and that of Shepherd et al?
Evidence: As mentioned in and will be described in Section 4.4, when the parameters of model () are identified, the causal estimands can be identified.
Lemma 1 For any x ∈ Γ = {0, 1, . . . , K} and y ∈ {0, 1}, let
Proof For all x ∈ Γ, we have
Hence
We have proposed a method under the principal stratification framework to estimate causal effects of a treatment on a binary long-term endpoint conditional on a post-treatment binary marker in randomized controlled clinical trials. We also extend our method to address censored outcome data. In our motivating study, we demonstrate the causal effect of the new regimen in the longterm survival for patients who would achieve pCR. Other principal stratum causal effects can be estimated in a similar fashion. Our approach can play an important role in a sensitivity analysis. Identification of causal effects is achieved through two assumptions. First, a subject who responds under the control would respond if given the treatment. This monotonicity assumption could prove valuable and can be justified in many scenarios that the additional therapy would help to improve the response. When the auxiliary variable X is discrete, we can identify and estimate Pr{S(1) = 1|S(0) = 0, X} under the monotonicity assumption. Second, a parametric model is used to describe the counterfactual response under the treatment for a control non-respondent. Both the future long-term outcome and a baseline covariate are predictors in this parametric model. does not consider when the auxiliary X is discrete, the parameters of model () can be identified when the level of the discrete covariate is at least of the same dimension of model parameters. Instead they perform sensitivity analyses by varying the values of those model parameters in order to estimate the causal estimands. It is recognized that no diagnostic tool is available to verify the validity of this counterfactual model.
In the motivating dataset, we discretize a continuous baseline variable into several levels. In practice, the linearity assumption may not hold. We would consider a two-pronged approach: 1) to estimate G L (x) and G R (x, y) by nonparametric estimates such as spline or kernel density estimates for a univariate continuous X; 2) to use a more flexible model for the counterfactual response such as a logistic regression with natural cubic spline with fixed and even-spaced knots along the domain of X. For each given x, we can still use the same probabilistic argument to link those estimates and the model parameters. The objective function would be a weighted sum of the squared difference of those probabilistic estimates.
Then p 00x is the proportion of non-respondents among all subjects with X = x; p 11x is the proportion of respondents among all subjects with X = x.
The identifiability of model parameter β depends on the availability of ax = Pr{S(1) = 1|S(0) = 0, X = x} and b_xy = Pr{Y (0) = y|S(0) = 0, X = x}, for x ∈ Γ; y = 0, 1.****Identification of causal effects is achieved through two assumptions. First, a subject who responds under the control would respond if given the treatment. This monotonicity assumption could prove valuable (Bartolucci and Grilli, 2011) and can be justified in many scenarios that the additional therapy would help to improve the response. When the auxiliary variable X is discrete, we can identify and estimate Pr{S(1) = 1|S(0) = 0, X} under the monotonicity assumption. Second, a parametric model is used to describe the counterfactual response under the treatment for a control non-respondent (Shepherd et al., 2006). Both the future long-term outcome and a baseline covariate are predictors in this parametric model. Shepherd et al. (2006) does not consider when the auxiliary X is discrete, the parameters of model (1) can be identified when the level of the discrete covariate is at least of the same dimension of model parameters. Instead they perform sensitivity analyses by varying the values of those model parameters in order to estimate the causal estimands. It is recognized that no diagnostic tool is available to verify the validity of this counterfactual model.
Answer: We adopted the same logistic regression model and assumptions as in Shepherd et al. (2006). In general, regression coefficients are not identifiable without any additional assumptions. In that paper and subsequent works, Shepherd and his co-authors focused on sensitivity analyses by varying model parameters in an imposed model on counterfactual outcomes. Noticing the probabilistic equation (7), we figured out how to identify the regression coefficients when the linearity assumption holds in the logistic regression. The identification was discussed in the last paragraph of page 6 in Section 4.2 and required assumptions were discussed in Section 7. We will revise the paper by providing a sub-section on the identifiability in Section 4.