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In statistics, the variance inflation factor (VIF) quantifies the severity of multicollinearity in an ordinary least squares regression analysis. It provides an index that measures how much the variance (the square of the estimate's standard deviation) of an estimated regression coefficient is increased because of collinearity. ==Definition== Consider the following linear model with ''k'' independent variables: : ''Y'' = ''β''0 + ''β''1 ''X''1 + ''β''2 ''X'' 2 + ... + ''β''''k'' ''X''''k'' + ''ε''. The standard error of the estimate of ''β''''j'' is the square root of the ''j''+1, ''j''+1 element of ''s2''(''X''′''X'')−1, where ''s'' is the root mean squared error (RMSE) (note that RMSE2 is an unbiased estimator of the true variance of the error term, ); ''X'' is the regression design matrix — a matrix such that ''X''''i'', ''j''+1 is the value of the ''j''''th'' independent variable for the ''i''''th'' case or observation, and such that ''X''''i'', 1 equals 1 for all ''i''. It turns out that the square of this standard error, the estimated variance of the estimate of ''β''''j'', can be equivalently expressed as : where ''R''''j''2 is the multiple ''R''2 for the regression of ''X''''j'' on the other covariates (a regression that does not involve the response variable ''Y''). This identity separates the influences of several distinct factors on the variance of the coefficient estimate: * ''s''2: greater scatter in the data around the regression surface leads to proportionately more variance in the coefficient estimates * ''n'': greater sample size results in proportionately less variance in the coefficient estimates * : greater variability in a particular covariate leads to proportionately less variance in the corresponding coefficient estimate The remaining term, 1 / (1 − ''R''''j''2) is the VIF. It reflects all other factors that influence the uncertainty in the coefficient estimates. The VIF equals 1 when the vector ''X''''j'' is orthogonal to each column of the design matrix for the regression of ''X''''j'' on the other covariates. By contrast, the VIF is greater than 1 when the vector ''X''''j'' is not orthogonal to all columns of the design matrix for the regression of ''X''''j'' on the other covariates. Finally, note that the VIF is invariant to the scaling of the variables (that is, we could scale each variable ''X''''j'' by a constant ''c''''j'' without changing the VIF). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「variance inflation factor」の詳細全文を読む スポンサード リンク
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