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Concept
Statistical assumption
Résumé
Statistics, like all mathematical disciplines, does not infer valid conclusions from nothing. Inferring interesting conclusions about real statistical populations almost always requires some background assumptions. Those assumptions must be made carefully, because incorrect assumptions can generate wildly inaccurate conclusions.
Here are some examples of statistical assumptions:
Independence of observations from each other (this assumption is an especially common error).
Independence of observational error from potential confounding effects.
Exact or approximate normality of observations (or errors).
Linearity of graded responses to quantitative stimuli, e.g., in linear regression.
There are two approaches to statistical inference: model-based inference and design-based inference. Both approaches rely on some statistical model to represent the data-generating process. In the model-based approach, the model is taken to be initially unknown, and one of the goals is to select an appropriate model for inference. In the design-based approach, the model is taken to be known, and one of the goals is to ensure that the sample data are selected randomly enough for inference.
Statistical assumptions can be put into two classes, depending upon which approach to inference is used.
Model-based assumptions. These include the following three types:
Distributional assumptions. Where a statistical model involves terms relating to random errors, assumptions may be made about the probability distribution of these errors. In some cases, the distributional assumption relates to the observations themselves.
Structural assumptions. Statistical relationships between variables are often modelled by equating one variable to a function of another (or several others), plus a random error. Models often involve making a structural assumption about the form of the functional relationship, e.g. as in linear regression. This can be generalised to models involving relationships between underlying unobserved latent variables.
Cross-variation assumptions.
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