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Formulae relevant to ANOVA assumptions. Levene’s test. d = X-M A1. skew. N ∑(x-M) 3 N-2 (N-1)s 3. F-independent variances.  2 /  2. Epsilon: . Geiser-Greenhouse/Box. a = number of levels of within IV s jj = mean of entries from diagonal s = mean of all entries in matrix

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Formulae relevant to ANOVA assumptions


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slide1

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

slide2

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)

slide3

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

slide4

Effect Sizes

µ1 - µ2

Cohen’s d =

SStreatment

SStotal

2 =

SStreatment - (k-1)MSerror

SStotal + MSerror

2 =

slide5

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

slide6

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

slide7

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

slide8

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)

slide9

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

slide10

Prediction Error

sy x

∑(y-y)2

N-2 df

.

SSy(1-r2 )

df

^

SSresidual

=

=

slide11

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 )

slide12

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

slide13

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)

~

slide14

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)

slide15

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

slide16

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