Can I use the early response to a treatment to predict the full effect? - Dealing with regression towards the mean Suppose I have a 6 week weight-loss program, that I know is only effective in a fraction of the population. I weight the participants at baseline, after 1 week and after the full 6 weeks.
$$
T_0 = Baseline
$$
$$
T_1 = 1\ Week
$$
$$
T_2 = 6\ Weeks
$$
I want to investigate if the effect at after 1 week can indicate if it is worthwhile to continue the program.
A naive approach would be to see if 
$$
WLoss_{T1} = \frac{W_{T0} - W_{T1}}{W_{T0}}
$$ 
is correlated with 
$$
WLoss_{T2} = \frac{W_{T1} - W_{T2}}{W_{T1}}$$ 
and maybe decide some minimal meaningful weight loss (eg $WLoss_{T2} > 2 \%$) and 
use a receiver operating characteristic (ROC) analysis to see how well $WLoss_{T1}$ can predict this weight loss.
This would, however, underestimate the correlation between $WLoss_{T1}$ and $WLoss_{T2}$ due to the mathematical coupling between $WLoss_{T1}$ and $WLoss_{T2}$ (both use $W_{T1}$). A persons weight loss after 1 week could simply be due to measurement error and random variation in body weight (random error), and the next measurement would likely be closer to baseline, due to the same random variation in body weight measurement, giving a negative correlation between $WLoss_{T1}$ and $WLoss_{T2}$.
The problem can also be viewed as regression towards the mean. 
What are appropriate ways of avoiding the bias caused by this mathematical coupling?
 A: Followig comments below rolando2 post, I was about to suggest you to make the regression:
$$Weight_{i,t} = c_i + \alpha . 1(T_1) + \gamma . 1(T_2) + \epsilon_{i,t}$$
Then you would get a matrix of variance-covariace of the coefficients.
Maybe it would be enough to look at $cov(\alpha,\beta)$ (and to check somehow that it is significantly positive). But, if I make no mistake, you would have in fact the same problem as described in the question.
Otherwise, if you have a lot of observations, you might divide you sample into cells c (with lots of people; for instance by age-gender). Make on each cell the following regression:
$$log(Weight_{i \in c,t}) = c_i + \alpha_c . 1(T_1) + \gamma_c . 1(T_2) + \epsilon_{i,t}$$
In each cell c, there is no reason why the mean effect of $T_1$ and the mean effect of $T_2$ would be subject to individual measurement error.
Then you regress: $\gamma_c = \delta . \gamma_c + \zeta_{i,t}$.
You are interested in $\delta$.
Please note that significancy of $\delta$ will be underestimated...because the dependent variable is itself estimated (and the explanatory variable). I would say this does not require to be dealt with if you have enough people in each cell so that the first steps estimations are precise enough. If you do not have many people, there are formula to de-bias the variance when the dependent variable is estimated - but this might be more difficult here as both the dependent variable and the explanatory variable are estimated.
A: You seem to be talking about an intervention working. If it is only worth persisting with the intervention, if the intervention works, then how do we tell whether it works? This leads to the counter-factual of "What would have happened, had the person not received the intvention?" Otherwise, you might just attribute random fluctuation and regression to the mean to the intervention
To tell what would have happened, if a person had not taken an intervention, you would need a pretty reliable prediction model for what happens without intervention including the uncertainty around that. E.g. such a model might give you the (correlated) prediction for your two time points and you can check whether the change is larger than what you expect for most patients without intervention.
Difficulities include that you would need a pretty large dataset representative of the relevant target population to build a good model (alternatively, you could replace a model with elicited expert opinions about trajectories without intervention, but that is also not very easy). Additionally, it may be that for an individual it might often be quite hard to tell the difference between chance variation and a large causal effect (depends, also depends a lot on the time scale and the size of the intervention effect).
A: I have the feeling you are over-complicating things:  differences...correlation...ROC.  You have longitudinal data collected at 3 time points; you could take advantage of this to create a single model, a random-effects model (mixed model; longitudinal model).  I think it would take into account everything you have mentioned and would do so in, as John Willett likes to say, one swell foop.
