Various regression techniques are used to characterize dose-response relationships via a mathematical function. Dose-response relationships are probabilistic and will therefore take a value between 0 and 1. A dose-response analysis begins with a **best-fit** test of dose-response data. These data are usually provided in the literature as a comparison of the median dose concentration and number of organisms that experienced a given effect (infection, illness, death; known as the **endpoint**) at that dose. The statistical technique **maximum likelihood estimation (MLE)** is used to fit the data to theoretical distributions, typically either **Beta-Poisson** or **Exponential** due to their biologic plausibility. This process calculates the probability of obtaining the observed data given a theoretical distribution by minimizing **deviance (Y)** of each of these model fits:

Y=-2(lnM_{1} - lnM_{2})

Where L_{1}, L_{2} are maximized likelihood estimates for the full (L_{2}) and restricted (L_{1}) models. Optimized deviance follows a Χ^{2} distribution with *k – m* degrees of freedom, where *k* is the number of doses and *m* is the number of dose-response parameters of a given model. This allows the analyst to reject the model if Y > Χ^{2}_{k-m,α}. If both models are significant, the model with the lowest deviance when compared to the full (for example the empirical model with a separate parameter for each dose group) model is chosen. **Bootstrapping** is performed to characterize the uncertainty of parameter estimates (r, α, N_{50}, etc.) of the distribution, most commonly by generating confidence intervals. The estimates from this approach approximate the uncertainty associated with the “true” distribution by repeatedly sampling the data and re-computing a statistic.

In some cases, it is necessary to **pool** data from different studies to compare strains or increase confidence in a dose-response model. The ability to pool data is assessed via a **hypothesis test** (null: no difference in dose-response parameter(s)), where the deviance of the pooled dataset (Y_{T}) is added to each individual optimized deviance (Y_{1}, Y_{2}..) and Δ is compared to a Χ^{2} distribution with df= (number of parameters in each dataset) ‒ (total number of parameters):

Δ=Y_{T} - (Y_{1} + Y_{2} +...)

If these approaches are not sufficient to describe the model fit, more complex approaches must be applied.

Download R software: http://cran.mtu.edu/