Dose Response Mathematical & Statistical Approaches Summary

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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(lnM1 - lnM2)

Where L1, L2 are maximized likelihood estimates for the full (L2) and restricted (L1) models. Optimized deviance follows a Χ2 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 > Χ2k-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, N50, 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 (YT) is added to each individual optimized deviance (Y1, Y2..) and Δ is compared to a Χ2 distribution with df= (number of parameters in each dataset) ‒ (total number of parameters):

Δ=YT - (Y1 + Y2 +...)

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

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