Checking for outliers in regression

According to Hoaglin and Welsch (1978) leverage values above 2(p+1)/n where p predictors are in the regression on n observations (items) are influential values. If the sample size is < 30 a stiffer criterion such as 3(p+1)/n is suggested.

Leverage is also related to the i-th observation's Mahalanobis distance, MD(i), such that for sample size, N

Leverage for observation i = MD(i)/(N-1) + 1/N

so

Critical MD(i) = (2(p+1)/N - 1/N)(N-1)

(See Tabachnick and Fidell)

Other outlier detection methods using boxplots are in the Exploratory Data Analysis Graduate talk located here or by using z-scores using tests such as Grubb's test - further details and an on-line calculator are located here.

Hair, Anderson, Tatham and Black (1998) suggest Cook's distances greater than 1 are influential. Hair et al mention that some people also use 4/(N-k-1) for k predictors and N points as a threshold for Cook’s distance which gives a lower threshold than 1 (e.g. with 1 predictor and 27 observations this gives 4/(27-1-1) = 0.16). A third threshold of 4/N is also mentioned (Bollen and Jackman (1990)) which would give a threshold of 4/27 = 0.14 in the above example.

References

Bollen, K. A. and Jackman, R. W. (1990) Regression diagnostics: An expository treatment of outliers and influential cases, in Fox, John; and Long, J. Scott (eds.); Modern Methods of Data Analysis (pp. 257-91). Newbury Park, CA: Sage

Hair, J., Anderson, R., Tatham, R. and Black W. (1998). Multivariate Data Analysis (fifth edition). Englewood Cliffs, NJ: Prentice-Hall.

Hoaglin, D. C. and Welsch, R. E. (1978). The hat matrix in regression and ANOVA. The American Statistician 32, 17-22.

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