that could be observed for each type of plot. Graphical analysis of residuals 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 heteroskedasticity (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 plots of residuals versus predictors non-constant variation across the data: scatter plots of residuals versus predictors; for data collected over time, also plots of residuals against time drift in the errors (data collected over time): run charts 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 Numerical methods also play an important role in model validation. For example, the 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 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 experiments. 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 several ways. Out-of-sample evaluation Cross-validation is the process of assessing how the results of a statistical analysis will generalize to an independent data set. If