Research Design - - Topic 9 Fundamentals of Bivariate Regression and Correlation 2010 R. C. Gardner, Ph.D. Bivariate Regression and Correlation.

Transcription

1 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 An eaple showing the distinction etween and Intepetations of coelation Thee liited tuths Factos that influence the agnitude of pecial cases of the Peason coelation Tests of significance Coelations with siple aggegates Bivaiate Regession and Coelation Bivaiate egession efes to an equation that elates a dependent vaiale to an independent vaiale, o a citeion to a pedicto. The fundaental equation in aw scoe fo is: a with a and deteined such that Σ- ² a iniu. and a The foula in standad scoe fo is: whee is as defined on the net slide ² Bivaiate coelation efes to covaiation etween two vaiales, and. The ost coon easue is the Peason poduct-oent coelation coefficient defined as: n n n n using iased and uniased u estiates of the standad deviations espectivel. O altenatives: ² ² u u N N ² ² N ² ² 3 Given -, we can copute: ouce Regession Residual Total ² ² df ² TOTAL REGREION REIDUAL And with soe algea, we can constuct the following sua tale n n us of quaes TOTAL TOTAL TOTAL F TOTAL n n

2 Conside the saple data set: Mean u Coputing Regession Coefficients and Coelation a a 0.5 ² ² n u u Regession of on a In each case: a the intecept --the value of the dependent vaiale when the independent vaiale is 0. the slope of the egession line the ise ove the un The two egession plots in aw scoe fo Regession of on a Regession of on Note. The slope of the egession line in standad scoe fo is the sae in each case and is equal to the coelation coefficient.8. The intecept is 0..8 The two egession plots in standad scoe fo Regession of on 8

3 3 Diffeent Intepetations of Coelation. Coelation is a easue of the linea elation etween and : whee: a and a 0. Coelation is a easue of the slope of the egession line in standad scoe fo: slope Best fit line iniu 0 d d 0 0 n n Note: 3. Coelation is a easue of the accuac of pedicting given : Given whee and ae independent Defining ± Thee Liited Tuths* Given N and N can equal, onl if and -, onl if - Thus, fo this to e tue, the standadized distiutions of and ust e: a Identical etical not necessail noal. The Peason poduct-oent coelation vaies fo - to. Tue, onl unde ve specific cicustances. Poof: * Adapted fo Gadne 000.

4 . Given a lage enough saple size, the coelation will alwas e significant. Tue, onl ecause of atifacts. Poof: Given ρ T T it is possile that ρ T EM T ER EM T ER EM i.e., the easues of and consist of tue scoes T & T, ando eo E R and E R and easueent eo E M & E M. Given: ecause the coelations 0, ρ 0., ρ and ρ ae not 0 T EM EM EM Thus, even with two vaiales that ae tul independent, the coelation etween easues of those vaiales a not e 0, and given a lage enough saple size it a e significant Coelation does not ean causation. This is not a liitation of the statistic, ut athe the natue of the undeling design. Conside an epeient on the effects of the aount of alcohol consued in the aftenoon and nue of hous slept that night. This stud could e un in contolled conditions with caeful attention to detail, etc. The coelation etween the two could e consideed an inde of the linea effects of alcohol on hous slept and an indication of causalit if the aount consued was andol deteined and adinisteed the epeiente. The coelation etween the two would sipl e an inde of the covaiation etween the two if the aount consued was not deteined andol. The egession equation would descie the natue of the linea elationship. Factos That Affect the Peason Poduct Moent Coelation Coefficient. Non-linea elationships.. Heteoscedasticit. will appoach 0, even though thee is a non-linea elationship etween the vaiales. An assuption undeling is hooscedasticit viz., that the vaiation aound the egession line is elativel constant 3. Restiction in Range. elativel low Tuncated Range Entie Range elativel high. Asetical Distiutions. If selection on one vaiale has esticted its ange, this could influence the agnitude of. To coect fo this, calculate co whee stand. dev. uncutailed s stand. dev. cutailed If the distiutions aen t setical, It can influence the value of. s s concentation of cases Hooscedastic Heteoscedastic 5

5 5. The Pesence of Outlies. A few ve etee scoes can influence the coelation.. Reliailit. The coelation etween two easued vaiales undeestiates the coelation etween the tue vaiales in the population as the eliailit of the easues deceases. The coection fo such attenuation is:. A Confounding Vaiale. Miing saples can soeties eliinate eaningful elationships. Assue IQ, gades. Two classes a e siila on IQ, ut diffeences in level of teache gading ight oscue elation etween gades and IQ if the saples ae eged. T T whee and ae the eliailities of the easued vaiales.: 8 pecial Cases of the Peason Poduct Moent Coelation peaan Rank Ode ρ s Coelation etween two vaiales, anked fo to N. N N N N N Given: d d d N d [ N ][ N ] N N ρ s Point Biseial p 0 i i Coelation etween a dichotoous and continuous vaiale. i i n n N n n n n p n N q n N N pq [ N ][ N ] p 0 5

6 Phi Coefficient Φ Effect tength and Powe Coelation etween two dichotoous vaiales. 0 a a a c 0 d c c d a d a c N a c d a N [ N ][ N ] ad c a c d d a c φ Cohen s 88 definitions: all ρ.0. an elationships pusued in soft ehavioal sciences ae of this agnitude p.. Mediu ρ.30. this degee of elationship would e peceptile to the naked ee of a easonal sensitive oseve p. 80. Lage ρ.50. aound the uppe end of the ange of noneliailit s one encountes in those fields of ehavioal science which use the etensivel p.80. and φ fo tales N Powe can e calculated using Cohen 88 o G*Powe3. Note that G*Powe3 has thee outines that can e used fo this pupose, one with t, one with F, and one with Eact tests. Testing the significance of a single ivaiate coelation coefficient. Ho: ρ 0. F df ; df N o its df N.. N t Testing the significance of a single ultiple coelation coefficient 3. Ho: ρ 0. R p F R N df p; df N p. Ho : ρ 0. Fo l ag e N. N 3

7 Testing the diffeence etween two coelation coefficients. Ho : ρ ρ fo independent saples. Copaing two coelations fo the sae saple with a coon vaiale 5. Ho : ρ ρ3 fo coelated coelatio ns.. Test poposed Dunn and Clak. Fishe s loge N loge. Test poposed Meng, Rosenthal & Ruin. N 3 h 3 and n 3 n 3 5 whee each: loge and 3 f 3 f 3 h 3 Copaing two coelations fo the sae saple with diffeent vaiales Coss Lagged Panel Analsis. Ho : ρ ρ5 fo coelated coelatio ns. Testing the ignificance of an aveage coelation. Ho : ρ av 0 Atts Tie Tie 5 Atts AV n 3 n 3 K nk 3 n 3 n 3 L n 3 k k Behav Behav 5 N C 5 5 whee: then: log e whee: AV k n 3 n 3 L n 3 C

8 Testing the significance of a patial coelation 8. Ho : ρ t N 3.3. Ho : ρ.3 0 N 3 F R.3.3 df N 3. Testing the significance of a seipatial pat coelation df, N 3 Note. These two statistics ield identical esults, ecept that F t² oth at N-3 df. Copaing two ivaiate egession coefficients 0. Ho : in the population whee: and: D t D C and df n n. n n n n C n s ns Coelations Involving Aggegates Raw Data 3 T T Fo these data: T T 3. T T Coelation Mati T j j jk j k Aggegated tandad coes: L

9 3 Covaiance Mati T cov j cov j k j jk Aggegated Raw coes: L Coelations Involving Diffeence coes Coelation of Initial coe with the Diffeence [ ] N M M whee : M M 3 Coelation of one vaiale A with a Diffeence 3 Coelation etween two Diffeence coes A A A[ ] N A B A [ B A B A][ ] N B A M A M A M whee : M [ L M L B L L A AB ][ M B A M ] 35 whee : M ; L B A 3

10 Refeences Cohen, J. 88. tatistical Powe Analsis fo the Behavioal ciences econd Edition. Hillsdale, NJ: Lawence Elau. Dunn, O.J. & Clak, V.A.. Coelation coefficients easued on the sae individuals. Jounal of the Aeican tatistical Association,, 3-3. Gadne, R. C Coelation, causation, otivation and second language acquisition. Canadian Pscholog,, 0-. Gadne, R.C. & Edle,. 8. Aggegating scoes: To standadize o not to standadize? Educational and Pschological Measueent,,83-8. Gadne, R.C. & Neufeld, R.W.J. 8. Use of the siple change scoe in coelational analses. Educational and Pschological Measueent,, 8-8. Meng, -L, Rosenthal, R. & Ruin, D.B.. Copaing coelated coelation coefficients. Pschological Bulletin,, -5. Taachnick, B. G. & Fidell, L Using Multivaiate tatistics Fifth Edition. Needha Heights, MA: Alln & Bacon. 3 0