Survey of Major Correlational Models/Questions simple correlation questions obtaining and comparing bivariate correlations statistical control questions what “would be” the bivariate correlation is all participants had the same score on some “control variable”? multiple correlation questions obtaining and comparing models with multiple predictors

Simple Correlation questions (old friends) r y,x1 simple correlation of y and x1 r y,x1 vs. r y,x2 comparing “correlated” correlations within a population/group (uses Hotelling’s t-test or M,R&R Z-test) r y,x1 vs. r y,x1 comparing the same bivariate correlation in 2 populations/grps (uses Fisher’s Z-test) Examples… Is there a relationship between # therapy sessions and symptomatic improvement? Is # therapy sessions a better predictor of symptomatic improvement than initial level of depression? Is # therapy sessions a better predictor of symptomatic improvement for adults than for adolescents? r imp,#ses r imp,#ses vs. r imp,init r imp,#ses for adults vs r imp,#ses for adolescents.

Does amount of practice predict performance better for novices that for experiences individuals? Does amount of practice predict level of performance? Does amount of practice predict performance better then prior experience? Same r, dif populations r perf,pract for novices vs. r perf,pract for experienced Use Fisher’s Z-test Simple r -- r perf,pract Compare r’s in the same pop r perf,pract vs. r perf,exp Hotelling’s t-test & Steiger’s Z-test Which type is each of the following? Use the notation & tell test used for each type of comparison

Statistical Control questions partial correlation questions -- is the relationship BOTH variables have with some 3rd variable(s) “producing” the bivariate relationship between them? r y,x1.x2 partial correlation of y & x1 controlling both for x2 (control var listed to right of “.” ) r y,x1.x2,x3 multiple partial correlation of y & x1 controlling both for x2 and x3 semi-partial (part) questions -- is the relationship JUST ONE of the variables have with some 3rd variable(s) “producing” the bivariate relationship between them? r y,(x1.x2) semi-partial correlation of y & x1, controlling the latter for x2 “( )” around variable being controlled and control variable r y,(x1.x2,x3) multiple semi-partial correlation of y & x1, controlling latterfor x2 & x3 “( )” around variable being controlled and control variables

Examples … Are # sessions and symptomatic improvement correlated after controlling symptomatic improvement for initial depression. Are # sessions and symptomatic improvement correlated after controlling both for initial depression? Are # sessions and symptomatic improvement correlated after controlling both for initial depression and age? Are # sessions and symptomatic improvement correlated after controlling symptomatic improvement for initial depression and age. Partial correlation r #ses,imp.init Multiple partial r #ses,imp.init,age Semi-partial r #ses (imp.init) Multiple semi-partial r #ses,imp (init,age)

Does practice predict performance after controlling both for prior experience? Does practice predict performance after controlling performance for prior experience and age? Does practice predict performance after controlling performance for age? Does practice predict performance after controlling both for prior experience and age? partial r perf,prac.exp multiple semi-partial r prac(perf.exp,age) semi-partial r prac(perf.age) multiple partial r perf,prac.exp,age Which type is each of the following?

Multiple Regression questions R y.x1,x2,x3,x4 ² multiple correlation with y as the criterion and x1, x2, x3 and x4 as predictors predictors to right of “.” R y.x1,x2,x3,x4 ² vs. R y.x1,x2, ² comparing nested models (uses R 2 change F-test) R y.x1,x2 ² vs. R y.,x3,x4 ² comparing non-nested models (uses Hotelling’s t-test or M,F&R Z-test) R y.x1,x2,x3,x4 ² vs. R y.x1,x2,x3,x4 ² comparing the same multiple regression model in two different populations (uses Fisher’s Z-test & H’s t or M,R&R Z-test) R y.x1,x2,x3,x4 ² vs. R z.x1,x2,x3,x4 ² comparing the same multiple regression model with two different criterion, in the same population ( Hotelling’s t-test & Steiger’s Z-test )

Examples… Symptomatic improvement is predicted from a combination of # sessions, initial depression and age. Symptomatic improvement is predicted from a combination of # sessions, initial depression and age and prediction is improved by adding # of prior therapists. Symptomatic improvement is predicted better from a combination of # sessions, initial depression and age than from # sessions & # of prior therapists. Symptomatic improvement is predicted from a combination of # sessions, initial depression and age better for adults than for adolescents. A combination of # sessions, initial depression and age predicts symptomatic improvement better than it predicts treatment satisfaction. R imp.#ses,init,age 2 vs. R imp.#ses,init,age,#ther 2 R imp.#ses,init,age 2 R imp.#ses,init,age 2 vs. R imp.#ses,#ther 2 R imp.#ses,init,age 2 for adults vs. for adolescents R imp.#ses,init,age 2 vs. R tsat.#ses,init,age 2

Do practice, prior skill and motivation predict performance? Do practice, prior skill and motivation predict performance on a speeded task as well as they they predict performance on an accuracy task? Do practice, prior skill and motivation predict performance as well as do prior skill and motivation? Do practice, prior skill and motivation predict performance as well as do practice, motivation and age? Do practice, prior skill and motivation predict performance as well for amateurs as for professionals? single model R perf.prac,skill 2 single model for 2 criterion H & MMR R speed.prac,skill 2 vs. R acc.prac,skill 2 nested model comparisons R 2 -  F-test R perf.prac,skill,mot 2 vs. R perf.prior,mot 2 non-nested models H & MMR R perf.prac,skill,mot 2 vs. R perf.prac,mot,age 2 R 2 for 2 populations F’s Z-test - H&MM R perf.prac,skill,mot 2 Which type is each of the following? Use the notation & tell the test used for each model comparison

Two important things to remember: 1.Simple (bivariate) correlation and simple (bivariate) regression ask exactly the same question – is there a bivariate relationship between these variables? Even the H0:s and significant tests are equivalent… Correlation H0: r=0 Regression H0: b=0 2. Whether a variable contributes to a particular model can be tested two ways … a.The test of that variables regression (b) weight in the multiple regression model H0: b 3 =0 in y’ = b 1 x 1 + b 2 x 2 + b 3 x 3 + a b.The multiple semi-partial between the criterion and that predictor, after controlling that predictor for all of the other predictors H0: r y(x3.x1, x2) = 0