= 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 [[FAQ/mahal|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 [[StatsCourse2009|here]] or by using z-scores using tests such as Grubb's test - further details and an on-line calculator are located [[http://www.graphpad.com/quickcalcs/Grubbs1.cfm|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 usually 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.

[[FAQ|Return to Statistics FAQ page]]

[[CbuStatistics|Return to Statistics main page]]

[[http://www.mrc-cbu.cam.ac.uk/|Return to CBU main page]]

These pages are maintained by [[mailto:ian.nimmo-smith@mrc-cbu.cam.ac.uk|Ian Nimmo-Smith]] and [[mailto:peter.watson@mrc-cbu.cam.ac.uk|Peter Watson]]