Psychology 282 Lectue #2 Outline Review of Peason coelation coefficient: z z ( n 1) Measue of linea elationship. Magnitude Stength Sign Diection Bounded by +1.0 and -1.0. Independent of scales of measuement. Conside intepetation of magnitude when elationship is not close to pefectly linea. Thee exists a tendency in pactice to ove-estimate stength of elationship implied by intemediate values of. May be elated to significance testing (late). Applet: http://noppa5.pc.helsinki.fi/koe/co/co7.html Be consevative in intepeting stength of association. Conside othe infomation (late), and obtain and inspect scatteplots.
2 Types of coelation coefficients Peason: Usually conceived of as applicable to situations whee and ae inteval o atio scales (quantitative vaiables). But actually moe geneal: Applicable to othe kinds of vaiables. Conside dichotomous (binay) vaiables and anks. Phi coefficient: Suppose and ae both binay; e.g., test items. Data in fom of 2 2 fequency table: 0 1 0 A B 1 C D Peason coelation fo binay and can be deived as: z z ( n 1) BC AD ( A + B)( C + D)( A + C)( B + D)
3 Called phi coefficient, designated φ. A special case of Peason coelation. Eithe fomula gives same esult. No need fo special softwae. Relationship between φ and χ 2 fo 2 2 table: φ 2 χ n Measues of association between two binay vaiables. Point-biseial coelation is binay, is quantitative (e.g., is a test item, is total test scoe). Let have values 0 and 1. p is popotion of sample scoing 1. q is popotion of sample scoing 0. sd p q 1 is mean of fo individuals scoing 1 on. 0 is mean of fo individuals scoing 0 on. Peason coelation between and can be deived as:
4 z z ( n 1) ( 1 0 ) sd p q Called point-biseial coelation; designated pb. Special case of Peason coelation; eithe fomula gives same esult; no need fo special softwae. Now conside two othe types of coelation coefficients involving binay vaiables, but not special cases of Peason coelation. Tetachoic coelation and ae binay (e.g., two test items). Define p, q, p, q. Phi coefficient gives obseved coelation. Tetachoic is an estimate of an unobseved coelation based on following concepts: Fo, assume thee exists a continuous undelying vaiable that follows a nomal distibution. A latent vaiable, call it L. is a binay measue of L.
5 A cutoff o theshold point on L divides the aea unde the cuve into two pats, p and q. -3-2 -1 0 1 2 3 Latent Same assumption fo, defining L. -3-2 -1 0 1 2 3 Latent Question: What is coelation between L and L?
This coelation is unobsevable, but can be estimated. Estimate is called tetachoic coelation. No simple fomula exists; no closed-fom algebaic expession. Special softwae equied fo computation. Not a special case of Peason coelation. Tetachoic is an estimated o infeed coelation. Fo same data, tetachoic > phi, based on assumption of impoved measuement of L. Can extend this appoach to case whee and each have multiple odeed categoies; e.g., Liket scales. Assume undelying continuous nomally-distibuted vaiables L and L. Nomal distibutions have multiple theshold points that epesent cutoffs fo categoies on and scales. Can estimate coelation between L and L as polychoic coelation. 6
7 Biseial coelation is binay, is quantitative; e.g., is a test item, is total test scoe. Obseved coelation between and is given by point-biseial. Biseial coelation is an estimate of an obseved coelation. Fo, assume thee exists a continuous undelying vaiable that follows a nomal distibution. A latent vaiable, call it L. is a binay measue of L. A cutoff o theshold point on L divides the aea unde the cuve into two pats, p and q. -3-2 -1 0 1 2 3 Latent Question: What is coelation between L and?
8 This coelation is unobseved. Cannot be computed exactly, but can be estimated as biseial coelation. bis ( ) 0 1 ( h) sd p q whee h is odinate (height) of nomal cuve at value of z dividing aea into p and q. (Fom Table C). Relationship between biseial and point-biseial: bis pb p h q Fo same data, bis > pb. Biseial is not a special case of Peason coelation. It is an estimate of an unobsevable coelation. Requies special fomula.
9 Rank-Ode Coelation (Speaman) Given scoes on and, suppose we ae inteested only in ank ode of obsevations on these vaiables. Conside coelation between ank-odes. Convet and to anks, designated R and R. Obtain Peason coelation between anks. Simplified fomula: Let d be a new vaiable epesenting the diffeence in anks fo each peson: d R R Then the following expession fo Speaman s ankode coelation can be deived fom fomula fo Peason coelation: S 2 6 d 1 2 n( n 1) This is anothe special case of Peason coelation. Fomulas fo and S poduce same esult.
10 Summay: Have defined following coelation coefficients: Peason Phi Point-biseial Tetachoic Polychoic Biseial Rank-Ode Questions: What type of vaiables ae each of these designed fo? Which ones ae special cases of Peason coelation, and which ae not? Which ae obseved coelations, and which ae estimated o infeed coelations?