# Statistical Models - Propensity Model.

A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but stochastically related. In mathematical terms, a statistical model is frequently thought of as a pair (Y,P) where Y is the set of possible observations and P the set of possible probability distributions on Y. It is assumed that there is a distinct element of P which generates the observed data. Statistical inference enables us to make statements about which element(s) of this set are likely to be the true one.

Most statistical tests can be described in the form of a statistical model. For example, the Student's t-test for comparing the means of two groups can be formulated as seeing if an estimated parameter in the model is different from 0. Another similarity between tests and models is that there are assumptions involved. Error is assumed to be normally distributed in most models.

• We can gather up all the background info that we have on our subjects before treatments are assigned, that might plausibly affect which treatment they get
• We build a model to predict the probability that they will receive the treatment instead of the control. Propensity Score = Pr (Treated | background info)
• Groups of subjects with similar propensity scores can then be expected to have similar values of all of the background information, in the aggregate.

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