Response to reviewer - Inmaculada Aban
The topic is very relevant in planning a clinical trial especially if it will help
minimize the cost. However, this reviewer has serious issues about the derivation
of the methodology. For model (1) on page 6, it is my understanding that this is a
conventional linear model with binary treatment predictor variable (TRT), a
continuous covariate (X) and their interaction. This implies that the distribution of
Y conditional on TRT = t and X = x is normal with mean and variance E(Y|TRT =
t;X = x) = 0 + 1 t + 2 x + 3 t x and V (Y|TRT = t;X = x) = 2: This model setting
and assumption avoid the issue of assuming the joint distribution of X and Y
because the interest is on the conditional distribution of Y given the variables in
the model. Hence it is not clear to this reviewer how equations (2) and (3) on page
7 were obtained. If one assumes X a, why did X disappeared in equation (2) since
it is still a covariate value even if it was restricted to this range? In equation (3),
what does E(X|X>a) mean, i.e., with respect to which distribution, and how did
this follow from equation (1)? The authors need to provide details of the
derivation of their results. Since the subsequent results depend on these model
setting and assumptions, these issues need to be clarified. If the interest is on the
marginal distribution of Y, should there be an assumption on the joint distribution
of Y and X where the marginal distribution of Y can be derived? This joint
distribution will describe the correlation between X and Y which should be nonzero in this case because X is the baseline severity and should in fluence the
value of Y.
We would like to thank you for spending efforts on reading our manuscript. We have
provided more details to your question in the manuscript.
Yes, we have assumed that both X and Y are random (please see the paragraph below
equation (5)). Equation (2) is not obtained from (1). It is a model resulted from fitting Y
only using treatment. Equation (3) is obtained from the fact that we fit model (2) to
subjects with X>=a under the assumption of (1). That is (1) is assumed to be the global
true model. In this case, we have  0  1 Trt  E [Y | Trt , X  a] . From (1), the right hand
side E[Y|Trt, X>=a] = 0 + 1 Trt + 2 E[X|X>=a] + 3 E[X|X>=a]Trt. Compare the
coefficients of Trt, we obtain (3).
We have now added this to our text. Thanks for your comments.