Model suggestion

Question

Could anyone give me hints as to a model framework that can be used in the following setting:

The outcome A is dichotomous. I want to investigate the effect of a continuous variable B and a continous variable C on A in a longitudinal setting. So far so good.

The problem is that C depends on B, in that values of B above a threshold (Unfortunately we do not know the threshold and the threshold itself might not be constant over time) will change C. The change in C will furthermore reduce B in the next measurement.

How can I decipher the isolated effect og B and C on A?

Would a mixed-logistic-regression model be OK?

ie;t=time

A~B*t+C*t+B*C*t

Any other ideas?

//M

Answer 1

Not easy at all. This starts to sound like the sort of thing that Jamie Robins and colleagues have done a lot of work on. To quote the start of the abstract of one of their papers:

"In observational studies with exposures or treatments that vary over time, standard approaches for adjustment of confounding are biased when there exist time-dependent confounders that are also affected by previous treatment."

(Robins JM, Hernán MA, Brumback B. Marginal Structural Models and Causal Inference in Epidemiology. Epidemiology 2000;11:550-560. http://www.jstor.org/stable/3703997)

In their example, low CD4 count means you're more likely to get anti-HIV drugs, which (hopefully) increase your CD4 count. Sounds very much like your setting.

There's been a lot of work in this area recently (i.e. more recently than that paper), and it's not easy to get to grips with from the journal papers. Hernán & Robins are writing a book, which should help a lot -- it's not finished yet, but there's a draft of the first 10 chapters available.

Answer 2

You have clearly stated a part of your model:

C depends on B, in that values of B above a threshold will change C. The change in C will furthermore reduce B in the next measurement.

By "next measurement" I understand you mean next in time. Let's index time as $t = 0, 1, 2, \ldots$. Then the dependence of C on B sounds like a contemporaneous one. If we adopt a simple linear model (which can readily be expanded to incorporate covariates and variable but predetermined thresholds) and let $u$ be a constant threshold,

$$C(t) = \beta_1 I_{B(t) \gt u} + \epsilon$$

with random deviations $\epsilon$ (which I won't bother to index; you know the drill). Here, $I$ is the indicator function.

Also,

$$B(t+1) = B(t) - \beta_2 (C(t) - C(t-1)) + \delta$$

and again $\delta$ represents random (independent) deviations. I'm stuck here because you haven't specified more precisely just how B changes in response to a change in C; I have merely provided one possible interpretation.

Finally,

$$A(t) = \beta_3 B(t) + \beta_4 C(t) + \beta_5 + \gamma$$

with independent random deviations $\gamma$.

The presence of that indicator function in the first formula is problematic: it makes this a nonlinear problem. However, this seems to be an essential feature of the situation; I would be loth to ignore it in the name of simplicity or ease of calculation (although both are important considerations). The lags $B(t+1) - B(t)$ and $C(t) - C(t-1)$ also point towards autoregressive models, another complication. Because of these issues, the most tractable approach might be with Bayesian techniques: parameterize the distributions of $\epsilon$, $\delta$, and $\gamma$, provide priors for those parameters and the $\beta$s, and let the machinery (e.g., WinBUGS or RBUGS) roll.

Answer 3

Your problem is one of multi-collinear regressors (since B and C are correlated). I would suggest that you look at the answers to the question: Dealing with correlated regressors.

The following paper may also be relevant in your context: Using principal components for estimating logistic regression with high-dimensional multicollinear data