# Model suggestion

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

• > in that values of B above a threshold will change C Do you know that threeshold in advance or does it need to be estimated as well ? Also, when you write B*t do you actually mean $B_t$ ? – user603 Oct 14 '10 at 14:20
• Unfortunately we do not know the threshold and the threshold itself might not be constant over time. – Misha Oct 14 '10 at 19:09

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.

• That is quite analogous to what I´m looking into. I´ll look into their work. Thx for the input. – Misha Oct 14 '10 at 19:05

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.

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

• How did you deduce that B and C are correlated? They might be, but this does not seem to be necessarily the case. In particular, if B rarely exceeds that threshold, B and C might be approximately independent. – whuber Oct 14 '10 at 14:23
• @whuber There is no issue at all if B rarely exceeds the threshold. We may as well assume that they are independent and use standard logistic models. I was implicitly assuming that there is some correlation between B and C given that there are threshold effects, the fact that changes in C impact B and that B and C are continuous variables. – user28 Oct 14 '10 at 14:30
• @Srikant You're right. The concern lies with intermediate situations. In those cases the correlation is probably not great enough to warrant special measures to deal with correlated regressors. Moreover, although we speak of "correlation" the relationship likely is not linear, so a lot of care is needed in dealing with it. – whuber Oct 14 '10 at 14:48