Formulae relevant to ANOVA assumptions Levenes test d

Formulae relevant to ANOVA assumptions Levene’s test d = X-MA 1 skew N ∑(x-M)3 N-2 (N-1)s 3 F-independent variances 2 / 2

Epsilon: Geiser-Greenhouse/Box a 2(sjj -s)2 (a-1)(∑sjk 2 -2 a∑ sj 2 +a 2 s 2) a = number of levels of within IV sjj = mean of entries from diagonal s = mean of all entries in matrix ∑ sj 2 = sum of the squared means of each row ∑sjk 2 = sum of squared entries from matrix Huyhn-Feldt n(a-1)( ) - 2 (a-1)[(n-1) - (a-1) ] Lower limit: 1/(a-1)

Power Cohen’s d = µ 1 - µ 2 Repd Measures 2 levels µ 1 - µ 2 √ 2(1 - ) 2 Independent Groups = d√n/2 2( /d )2 = n 50% power: 8/d 2 = d√n ( /d)2 = n 50% power: 4/d 2 Noncentral F parameter = ’√n ’ = ∑(µj - µ)2/k e

Effect Sizes Cohen’s d = µ 1 - µ 2 2 SStreatment = SStotal 2 = SStreatment - (k-1)MSerror SStotal + MSerror

Agreement k= ∑ƒO - ∑ƒE N - ∑ƒE where ƒO are observed frequencies on diagonal ƒE are expected frequencies on diagonal intraclass correlation For each pair of judges find m=(x 1 + x 2)/2 d= (x 1 -x 2) a= F(N-1, N) = a/b [2∑(m-M)2] /(N-1) intraclass correlation = (a-b)/(a+b) b= ∑d 2/(2 N) Where N number of things judged M is the mean of m

Post-tests A priori contrast L 2[/n∑a 2] MSerror where L = ∑a. T Range [M-M] / √MSerror/n Satterthwaite-Welch Correction SS s/a + SS s/axb df s/a + df s/axb = MSerror (SSs/a + SS s/axb)2 SSs/a 2 SS s/axb 2 = df error df s/axb

Nested Factors Compute 1 -way ANOVA on Job 1 -way ANOVA on Organization Use SSJob - SSOrg = SSnest. Job Use SSOrg Use this error term Same rule with df Source Total Org Job error SS ∑x 2 - (∑x)2/N ∑O 2/jn - (∑x)2/N ∑J 2/n - ∑O 2/jn ∑x 2 - ∑J 2/n df N-1 o(j-1) N-oj MS F

One random effect IV Source SS Row r-1 error df nr-r Two random effects IVs Source SS Row r-1 error(Rx. C) Column error(Rx. C) Rx. Cr-1(c-1) error ( e 2 ) df r-1(c-1) c-1 r-1(c-1) n(c)(r)-(c)(r)

Pearson r and simple regression r = COVxy sxsy F= r 2 = = r 2 (N-2) 1 -r 2 N∑xy - ∑x∑y √[N∑x 2 - (∑x)2][N∑y 2 - (∑y)2] = r 2 (1 - r 2 ) /(N-2) = SSreg/k SSresid/N-k-1 SSy - SSresidual SSy b = N∑xy - ∑x∑y [N∑x 2 - (∑x)2] = COVxy s x 2 = 2 s y r sx 2 t-test for slope b sb

Prediction Error sy. x ^2 ∑(y-y) = N-2 SSresidual df = SSy(1 -r 2 ) df

Comparing Correlations independent correlations Zr 1 - Zr 2 =z 1 1 + N 1 -3 N 2 -3 dependent correlations t= (N-1)(1 + r 23) (r 12 - r 13) 2 N-1 R + (r 12 + r 13)2(1 - r )3 23 N-3 4 where R = ( 1 - r 212 - r 213 - r 223 ) + (2 r 13 r 23 )

Range Restriction Sx ~ rt(xy) St(x) rxy = ___________ S 2 x 1 + r 2 t(xy) S 2 t(x) - r 2 t(xy) Where rt(xy) is correlation when x is truncated Sx is the unrestricted standard deviation of x St(x) is the truncated standard deviation of x

Miscellaneous Regression Statistics leverage > 2(k +1)/N is high variance inflation factor = d N-1 1/(1 - R 2 j) short-cut for 50% 4/r 2 = N-1 Wherry’s correction for Shrinkage: ~ R 2 = 1 - (1 -R 2) N-1 N-k-1 k/(N-1)

Partial and Semi-Partial Correlations Partial Correlation ry 1. 2 = rx 1 y - rx 2 y rx 1 x 2 (1 -r 2 x 2 y)(1 -r 2 x 1 x 2) Semi-partial Correlation ry(1. 2) = rx 1 y - rx 2 y rx 1 x 2 (1 -r 2 x 1 x 2)

Multiple R F= R 2/k (1 -R 2)/(N-k-1) SSreg/k SSresid/N-k-1 Change in R 2 r 2 yx 1 + r 2 yx 2 - 2 ryx 1 ryx 2 rx 1 x 2 (R 2 c - R 2 r)/(kc-kr) (1 -R 2 c)/(N- kc -1) Simple slope √ varb 1 + (2 M)covb 1 b 3 + M 2 varb 3 1 - r 2 x 1 x 2 = t(N-k-1)

B-K Modification of Sobel Baron & Kenny’s modification of Sobel’s test of the indirect path (ab)/√ s 2 as 2 b + b 2 sa 2 + a 2 sb 2 where a = unstandardized regression weight of IV-->Mediator b = unstandardized regression weight of Mediator-->DV s 2 a = squared standard error of a s 2 b = squared standard error of b