# Estimating synergy of treatments in time series data

I am studying the growth (in volume) of tumors in mice when they are given 1) saline solution (control), 2) treatment A, 3) treatment B, and 4) treatment A and B. The treatments are designed to inhibit the growth of the tumors.

I am hoping to estimate whether there is a "synergistic" relationship between A and B. By synergy, I mean that the combination treatment of A and B leads to what we would call, above some uncertainty threshold, a greater than additive effect. My data will be in time series format, as the measurements will be made longitudinally over many days, with replicates in each group.

Here's a quick literature review. Many studies cite and use Chou's method, but that will not work for us because it requires a separate dose-response curve for each drug, which we will not have.

I also found one study that uses a regression-based model (Tan et al), which employs the model

$y_i = \alpha + y_{i0}1_{12}\beta_0 + x^1_i\beta_1 + x_i^2\beta_2 + + x_i^3\beta_3 + \epsilon_i$

where $y_i$ is the log tumor size, the $x_i$'s refer to the quantity of the treatments used, and the $\beta$'s refer to the parameters. They then do expectation-maximization to get MLE's of the parameters. In this setting, $\beta_3$ is the crucial interaction parameter which we want to estimate as either significantly less than zero or not.

However, the authors do not mention access to their implementation of the method and it seems difficult to re-implement. Moreover, our treatments will be binary rather than two-leveled as in their data set.

I also found a study that uses an ANOVA (Soundararajan et al) and presumably post-hoc tests, but I didn't exactly understand their methods.

I am not looking to reinvent the wheel and I would prefer to use existing R packages and/or software. If the latter, I'd prefer open-source, to boost reproducibility. Any suggestions?

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