Editing Statistical assumption (section)

==Classes of assumptions==
There are two approaches to [[statistical inference]]: ''model-based inference'' and ''design-based inference''.<ref>Koch G. G., Gillings D. B. (2006), "Inference, design-based vs. model-based", ''[[Encyclopedia of Statistical Sciences]]'' (editor&mdash;Kotz S.), [[Wiley-Interscience]].</ref><ref>Cox, 2006, ch.9</ref><ref>de Gruijter et al., 2006, §2.2</ref>  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 [[model selection|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.<ref>McPherson, 1990, §3.4.1</ref> 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 variable]]s.
**Cross-variation assumptions. These assumptions involve the [[joint probability distribution]]s of either the observations themselves or the random errors in a model. Simple models may include the assumption that observations or errors are [[statistically independent]].
*Design-based assumptions.  These relate to the way observations have been gathered, and often involve an assumption of [[randomization]] during [[sampling (statistics)|sampling]].<ref>McPherson, 1990, §3.3</ref><ref>de Gruijter et al., 2006, §2.2.1</ref>

The model-based approach is the most commonly used in statistical inference; the design-based approach is used mainly with [[survey sampling]]. With the model-based approach, all the assumptions are effectively encoded in the model.