Research Design - - Topic 16 Bivariate Correlation Continued 2009 R.C. Gardner, Ph.D.
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1 Reseach Design - - Topic 6 Bivaiate Coelation Continued 009 R.C. Gadne, Ph.D. Factos that Influence magnitude of pecial Cases of the Peason Coelation Effect tength and Powe Tests of ignificance Involving Coelation and Regession (Available on Copack in Depatmental Liba Coelation with imple Aggegates
2 Factos That Affect the Peason Poduct Moment Coelation Coefficient. on-linea elationships. will appoach 0, even though thee is a non-linea elationship between the vaiables.. Heteoscedasticit. An assumption undeling is homoscedasticit viz., that the vaiation aound the egession line is elativel constant Homoscedastic Heteoscedastic
3 3. Restiction in Range. If selection on one vaiable has esticted its ange, this could influence the magnitude of. To coect fo this, calculate Entie Range elativel low Tuncated Range elativel high co s whee stand. dev. uncutailed s stand. dev. cutailed s 4. Asmmetical Distibutions. If the distibutions aen t smmetical, It can influence the value of. concentation of cases 3
4 5. The Pesence of Outlies. A few ve eteme scoes can influence the coelation. 6. A Confounding Vaiable. Miing samples can sometimes eliminate meaningful elationships. Assume X IQ, Y gades. Two classes ma be simila on IQ, but diffeences in level of teache gading might obscue elation between gades and IQ if the samples ae meged. 4
5 7. Reliabilit. The coelation between two measued vaiables undeestimates the coelation between the tue vaiables in the population as the eliabilit of the measues deceases. The coection fo such attenuation is: T T whee and ae the eliabilities of the measued vaiables.: 5
6 6 pecial Cases of the Peason Poduct Moment Coelation peaman Rank Ode ρ s Coelation between two vaiables, anked fom to. ( ( ( 6 Given: d d d s d ρ ( 6 ] ( ][ ( [
7 Phi Coefficient Φ 0 Coelation between two dichotomous vaiables. 0 b a a b d c c d b d a c a b c d a c a a b [ ad bc ( ][ ( ] ( a b( c d( b d( a c φ and φ fo tables 7
8 Point Biseial pb Coelation between a dichotomous and continuous vaiable. 0 i i i i n n n n p n q n ( pq [ ( ][ ( ] pb n n 8
9 Effect tength and Powe Cohen s (988 definitions: mall ρ.0. man elationships pusued in soft behavioal sciences ae of this magnitude ( p. 79. Medium ρ.30. this degee of elationship would be peceptible to the naked ee of a easonabl sensitive obseve (p. 80. Lage ρ.50. aound the uppe end of the ange of (noneliabilit s one encountes in those fields of behavioal science which use them etensivel (p.80. Powe can be calculated using Cohen (988 o G*Powe (available on the intenet. 9
10 Testing the significance of a single bivaiate coelation coefficient. Ho: ρ 0. F ( df ; df. o its equivalent: df.. Ho : ρ 0. ( Fo l ag e. 0
11 Testing the significance of a single multiple coelation coefficient 3. Ho: ρ 0. F ( R R p ( df p; df p
12 Testing the diffeence between two coelation coefficients 4. Ho : ρ ρ fo independent samples. Fishe s log e ( ( log e ( ( and n 3 n 3
13 Compaing two coelations fom the same sample (with a common vaiable 5. Ho : ρ ρ3 fo coelated coelations. Test poposed b Dunn and Clak (969.. ( ( 3 ( ( 3 3 ( 3 3. Test poposed b Meng, Rosenthal & Rubin (99. ( 3 ( h 3 whee each: log e ( ( and f ( ( 3 3 h f ( ( 3 3 3
14 Compaing two coelations fom the same sample (with diffeent vaiables (Coss Lagged Panel Analsis 6. Ho : ρ ρ45 fo coelated coelations. Atts Time Time 5 Atts Behav 4 Behav ( ( ( ( ( C whee: C (
15 Testing the ignificance of an aveage coelation 7. Ho : ρ av 0 AV ( n 3 ( n 3 K ( nk 3 ( n 3 ( n 3 L ( n 3 k k whee: log e then: AV k (( n 3 ( n 3 L ( n 3 5
16 Testing the significance of a patial coelation 8. Ho : ρ. 3 0 t (.3.3 ( df 3. 3 Testing the significance of a semipatial (pat coelation 9. Ho : ρ (.3 0 F ( 3 R (.3.3 df, 3 ote. These two statistics ield identical esults, ecept that F t² (both at -3 df. 6
17 Compaing two bivaiate egession coefficients 0. Ho : b b in the b b t Db population whee: Db C n ( ( n n n 4 ( and: C n s ns and df n n 4. 7
18 Coelations Involving Aggegates Raw Data X X X 3 Y T X Fo these data: IF T T X X X IF T X X X X TX
19 Coelation Mati Y X X X 3 Y X X X Aggegated tandad coes: T j m m m j j k jk L
20 Covaiance Mati Y X X X 3 Y X X X Aggegated Raw coes: T X m j m cov m j k j cov jk ( L
21 Coelations Involving Diffeence coes ( Coelation of Initial coe with the Diffeence ( ( [( ( ] M M M whee : M
22 ( Coelation of one vaiable (A with a Diffeence ( A( ( A A[( A ( ] M M A A M whee : M
23 3 (3 Coelation between two Diffeence coes A B A B A B A B ] ( ][( ( [( ( ( ] ][ [ ( ( AB A B A B M M L L L L M A B L M whee ; :
24 Refeences Dunn, O.J. & Clak, V.A. (969. Coelation coefficients measued on the same individuals. Jounal of the Ameican tatistical Association, 64, Gadne, R.C. & Edle,. (984. Aggegating scoes: To standadize o not to standadize? Educational and Pschological Measuement, 44,83-8. Gadne, R.C. & eufeld, R.W.J. (987. Use of the simple change scoe in coelational analses. Educational and Pschological Measuement, 47, Meng, X-L, Rosenthal, R. & Rubin, D.B. (99. Compaing coelated coelation coefficients. Pschological Bulletin,,
Research Design - - Topic 9 Fundamentals of Bivariate Regression and Correlation 2010 R. C. Gardner, Ph.D. Bivariate Regression and Correlation.
Reseach Design - - Topic Fundaentals of Bivaiate Regession and Coelation 00 R. C. Gadne, Ph.D. Bivaiate egession - - defining foulae Bivaiate coelation - - defining foulae Test of significance fo egession
Research Design - - Topic 17 Multiple Regression & Multiple Correlation: Two Predictors 2009 R.C. Gardner, Ph.D.
Reseach Design - - Topic 7 Multiple Regession & Multiple Coelation: Two Pedictos 009 R.C. Gadne, Ph.D. Geneal Rationale and Basic Aithmetic fo two pedictos Patial and semipatial coelation Regession coefficients
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