(1984) is seriously affected by the underlying p( x i). For example, the local mean deviance plot suggested by Landwehr et al. The argument based on difference in likelihoods to support such claims as made by Pregibon (1981) and alluded to by Beckman and Cook (1983) relies on asymptotic conditions that are not obtained in strictly binary data.Дiagnostics must consider the effects of differing residual distributions. In this setting claims that вЂњstandardized residualsвЂќ exist that are asymptotically normal or chi squared are not appropriate. That is, each residual has a unique distribution. Thus any reasonable definition of a residual from the true model, for example, Y - p( x), has a two-point distribution that depends on x through p( x). There Y has a Bernoulli distribution with probability of success p( x) = Pr( Y = 1 | x). This is not true in logistic regression with binary data. Thus the distribution of this residual from the true model does not depend on the explanatory variables x. In linear regression with normal errors, Y - E ( Y | x) is normally distributed with mean 0 and variance Пѓ 2. Let y be the response variable and let x be a pГ—1 vector of explanatory variable. The second difference examined deals with the distribution of residuals. This suggests that the motives for outlier detection should be carefully considered. The simple conclusion is that to estimate the probability of success p in an area in which p is small, one must observe events of small probability. The first examines the effect of events of small probability on estimation in logistic regression. Unfortunately the similarities that allow such techniques to adapt to logistic regression seem, in addition, to hide many of the differences. Several innovative attempts in this direction have been made by authors such as Tsiatis (1980), Pregibon (1979, 1981), Landwehr, Pregibon, and Shoemaker (1984), and Cook and Weisberg (1982). Thus the wealth of work done in linear regression provides guides and suggestions that may, with care and ingenuity, be applied to logistic regression. Certainly logistic regression requires procedures to detect global and local model weaknesses. Logistic regression has been both blessed and hindered by this development. Detection of outliers and other diagnostics based on residuals have gained widespread use in linear regression.

Residual definition statistics