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==Analysis of residuals==
{{Main article|residual analysis}}
The [[errors and residuals in statistics|residuals]] from a fitted model are the differences between the responses observed at each combination of values of the [[explanatory variable]]s and the corresponding prediction of the response computed using the regression function. Mathematically, the definition of the residual for the ''i''<sup>th</sup> observation in the [[data set]] is written
:<math>
e_i = y_i - f(x_i;\hat{\beta}),
</math>
[[File:Plot of noisy data + Gaussian fit + plot ressduals.svg|alt=plot of a fit and residuals to illustrate how plotting residuals allows us to evaluate how good a fit is|thumb|An illustrative plot of a fit to data (green curve in top panel, data in red) plus a plot of residuals: red points in bottom plot. Dashed curve in bottom panel is a straight line fit to the residuals. If the functional form is correct then there should be little or no trend to the residuals - as seen here.]]
with ''y<sub>i</sub>'' denoting the ''i''<sup>th</sup> response in the data set and ''x<sub>i</sub>'' the vector of explanatory variables, each set at the corresponding values found in the ''i''<sup>th</sup> observation in the data set.

If the model fit to the data were correct, the residuals would approximate the random errors that make the relationship between the explanatory variables and the response variable a statistical relationship. Therefore, if the residuals appear to behave randomly, it suggests that the model fits the data well. On the other hand, if non-random structure is evident in the residuals, it is a clear sign that the model fits the data poorly. The next section details the types of plots to use to test different aspects of a model and gives the correct interpretations of different results that could be observed for each type of plot.

===Graphical analysis of residuals===
{{See also|Statistical graphics}}
A basic, though not quantitatively precise, way to check for problems that render a model inadequate is to conduct a visual examination of the residuals (the mispredictions of the data used in quantifying the model) to look for obvious deviations from randomness. If a visual examination suggests, for example, the possible presence of [[heteroscedasticity]] (a relationship between the variance of the model errors and the size of an independent variable's observations), then statistical tests can be performed to confirm or reject this hunch; if it is confirmed, different modeling procedures are called for.

Different types of plots of the residuals from a fitted model provide information on the adequacy of different aspects of the model.
#sufficiency of the functional part of the model: [[scatter plot]]s of residuals versus predictors
#non-constant variation across the data: [[scatter plot]]s of residuals versus predictors; for data collected over time, also plots of residuals against time
#drift in the errors (data collected over time): [[run chart]]s of the response and errors versus time
#independence of errors: [[lag plot]]
#normality of errors: [[histogram]] and [[normal probability plot]]
Graphical methods have an advantage over numerical methods for model validation because they readily illustrate a broad range of complex aspects of the relationship between the model and the data.

===Quantitative analysis of residuals===
{{Main|Regression diagnostic}}
Numerical methods also play an important role in model validation. For example, the [[goodness of fit|lack-of-fit test]] for assessing the correctness of the functional part of the model can aid in interpreting a borderline residual plot. One common situation when numerical validation methods take precedence over graphical methods is when the number of [[statistical parameter|parameters]] being estimated is relatively close to the size of the data set. In this situation residual plots are often difficult to interpret due to constraints on the residuals imposed by the estimation of the unknown parameters. One area in which this typically happens is in optimization applications using [[designed experiment]]s. [[Logistic regression]] with [[binary data]] is another area in which graphical residual analysis can be difficult.

[[Serial correlation]] of the residuals can indicate model misspecification, and can be checked for with the [[Durbin–Watson statistic]]. The problem of [[heteroskedasticity]] can be checked for in any of [[Heteroskedasticity#Detection|several ways]].
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significant terms missing/misspecified in the functional part of the model
necessity of all terms in the functional part of the model
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