Formulae relevant to ANOVA assumptions

1. Formulae relevant to ANOVA assumptions Levene’s test d = X-MA1 skew N ∑(x-M)3 N-2 (N-1)s3 F-independent variances 2 /2

2. Epsilon:  Geiser-Greenhouse/Box a = number of levels of within IV sjj = mean of entries from diagonal s = mean of all entries in matrix ∑ sj2 = sum of the squared means of each row ∑sjk2= sum of squared entries from matrix a2(sjj -s)2 (a-1)(∑sjk2 -2a∑ sj2 +a2 s 2) Huyhn-Feldt n(a-1)() - 2 (a-1)[(n-1) - (a-1)] Lower limit: 1/(a-1)

3. Power µ1 - µ2  Repd Measures 2 levels Cohen’s d = µ1 - µ2 √2(1-) 2 Independent Groups  = d√n/2 2(/d )2 = n 50% power: 8/d2 • = d√n (/d)2 = n 50% power: 4/d2 Noncentral F parameter ∑(µj - µ)2/k ’ = e  = ’√n

4. Effect Sizes µ1 - µ2  Cohen’s d = SStreatment SStotal 2 = SStreatment - (k-1)MSerror SStotal + MSerror 2 =

5. Agreement ∑ƒO - ∑ƒE k = N - ∑ƒE where ƒO are observed frequencies on diagonal ƒE are expected frequencies on diagonal intraclass correlation For each pair of judges find m=(x1 + x2)/2 d= (x1-x2) F(N-1, N) = a/b intraclass correlation = (a-b)/(a+b) a= [2∑(m-M)2] /(N-1) b= ∑d2/(2N) Where N number of things judged M is the mean of m

6. Post-tests A priori contrast where L = ∑aT L2[/n∑a2] MSerror Range [M-M] / √MSerror/n Satterthwaite-Welch Correction (SSs/a + SS s/axb)2 SSs/a2 SS s/axb2 df s/a df s/axb SS s/a + SS s/axb df s/a + df s/axb = df error = MSerror

7. Nested Factors Compute 1-way ANOVA on Job 1-way ANOVA on Organization Use SSJob - SSOrg = SSnestJob Use this error term Use SSOrg Same rule with df Source SS df MS F Total ∑x2 - (∑x)2/N N-1 Org ∑O2/jn - (∑x)2/N o-1 Job ∑J2/n - ∑O2/jn o(j-1) error ∑x2 - ∑J2/n N-oj

8. One random effect IV Source SS df Row r-1 error nr-r Two random effects IVs Source SS df Row r-1 error(RxC) r-1(c-1) Column c-1 error(RxC) r-1(c-1) RxC r-1(c-1) error (e2) n(c)(r)-(c)(r)

9. Pearson r and simple regression N∑xy - ∑x∑y √[N∑x2 - (∑x)2][N∑y2 - (∑y)2] r = COVxy = sxsy r2 (N-2) r2 F = 1-r2 = (1- r2 )/(N-2) SSreg/k SSresid/N-k-1 = r2 = SSy - SSresidual SSy t-test for slope b sb sy2 b = N∑xy - ∑x∑y = COVxy = r [N∑x2 - (∑x)2] sx2 sx2

10. Prediction Error sy x ∑(y-y)2 N-2 df . SSy(1-r2 ) df ^ SSresidual = =

11. Comparing Correlations independent correlations Zr1 - Zr2 1 1 N1-3 N2-3 = z + (N-1)(1 + r23) (r12 - r13) N-1 + (r12 + r13)2 N-3 4 dependent correlations t = 2 R (1 - r23)3 where R = ( 1 - r212 - r213 - r223 ) + (2r12r13r23 )

12. Range Restriction rxy = _____________________ S2x 1 + r2t(xy) S2t(x) - r2t(xy) Sx rt(xy) St(x) ~ 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

13. Miscellaneous Regression Statistics leverage > 2(k +1)/N is high variance inflation factor 1/(1 - R2j)  = d N-1 short-cut for 50% 4/r2 = N-1 Wherry’s correction for Shrinkage: R2 = 1- (1-R2) N-1 N-k-1 k/(N-1) ~

14. Partial and Semi-Partial Correlations Partial Correlation ry1.2 = rx1y - rx2y rx1x2 (1-r2x2y)(1-r2x1x2) Semi-partial Correlation ry(1.2) = rx1y - rx2y rx1x2 (1-r2x1x2)

15. Multiple R F= R2/k (1-R2)/(N-k-1) SSreg/k SSresid/N-k-1 Change in R2 r2yx1 + r2yx2 - 2ryx1ryx2rx1x2 1 - r2x1x2 (R2c - R2r)/(kc-kr) (1-R2c)/(N- kc -1) Simple slope = t(N-k-1) √ varb1 + (2M)covb1b3 + M2varb3

16. B-K Modification of Sobel Baron & Kenny’s modification of Sobel’s test of the indirect path (ab)/√ s2as2b + b2sa2 + a2sb2 where a = unstandardized regression weight of IV-->Mediator b = unstandardized regression weight of Mediator-->DV s2a = squared standard error of a s2b = squared standard error of b