Survey of Major Correlational Models/Questions simple correlation questions obtaining and comparing bivariate correlat

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

2. Simple Correlation questions (old friends) ry,x1 simple correlation of y and x1 ry,x1 vs. ry,x2 comparing “correlated” correlations within a population/group (uses Hotelling’s t-test or Steiger’sZ-test) ry,x1 vs. ry,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? rimp,#ses rimp,#ses vs. rimp,init rimp,#ses for adults vs rimp,#ses for adolescents.

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

4. Multiple Regression questions Ry.x1,x2,x3,x4² multiple correlation with y as the criterion and x1, x2, x3 and x4 as predictors predictors to right of “.” Ry.x1,x2,x3,x4² vs. Ry.x1,x2,² comparing nested models (uses R2 change F-test) Ry.x1,x2² vs. Ry.,x3,x4² comparing non-nested models (uses Hotelling’s t-test or Steiger’sZ-test) Ry.x1,x2,x3,x4² vs. Ry.x1,x2,x3,x4² comparing the same multiple regression model in two different populations (uses Fisher’s Z-test & H’s t or Steiger’sZ-test) Ry.x1,x2,x3,x4² vs. Rz.x1,x2,x3,x4² comparing the same multiple regression model with two different criterion, in the same population (Hotelling’s t-test & Steiger’s Z-test)

5. 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. Rimp.#ses,init,age2 Rimp.#ses,init,age2 vs. Rimp.#ses,init,age,#ther2 Rimp.#ses,init,age2 vs. Rimp.#ses,#ther2 Rimp.#ses,init,age2 for adults vs. for adolescents Rimp.#ses,init,age2 vs. Rtsat.#ses,init,age2

6. Which type is each of the following? Use the notation & tell the test used for each model comparison single model Rperf.prac,skill2 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 for 2 criterion H & S Rspeed.prac,skill2vs.Racc.prac,skill2 nested model comparisons R2- F-test Rperf.prac,skill,mot2 vs. Rperf.prior,mot2 non-nested models H & S Rperf.prac,skill,mot2 vs. Rperf.prac,mot,age2 R2 for 2 populations F’s Z-test Rperf.prac,skill,mot2

7. Statistical control questions are always about the “causal relationships” that “produce” or “modify” bivariate correlations - called “3rd variable problems”. Here’s a well-known example ... • When plotted by week or by month -- there is a +r between ice cream sales & amount of violent crime. Huh? • Does eating ice cream make you violent ? • Does being violent make you crave ice cream ? • Is there some “3rd variable” variable that is “producing” the bivariate correlation? What might it be ??? Violent crimes Ice cream sales Here’s the same scatterplot, but with the size of each data point representing the average temperature of that period. We can see that there is no relationship between violent crimes and ice cream sales after controlling both for temperature. Temperature might be that “3rd variable”. Violent crimes Ice cream sales

8. Another example ... • We found a -r between # therapy sessions and amount of symptomatic improvement! Huh?!? Let’s think through this... • The sample is heterogeneous with respect to initial level of depression • Initial level of depression is likely to be related to # sessions they attend • Initial level of depression is likely to be related to symptomatic improvement • So, is the relationship each of these variables has with initial level of depression “producing” the bivariate correlation we found? symptomatic improvement # sessions dotsize indicates initial depression… Those who are more depressed come to more sessions and show less improvement. Those who are less depressed come to fewer sessions and show more improvement. symptomatic improvement # sessions

9. Statistical Control questions partial correlation questions-- is the relationship BOTH variables have with some 3rd variable(s) “producing” the bivariate relationship between them? ry,x1.x2partial correlation of y & x1 controlling both for x2 (control var listed to right of “.” ) ry,x1.x2,x3multiple 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? ry,(x1.x2)semi-partial correlation of y & x1, controlling the latter for x2“( )” around variable being controlled and control variable ry,(x1.x2,x3)multiple semi-partial correlation of y & x1, controlling latter for x2 & x3“( )” around variable being controlled and control variables

10. 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. Semi-partial r#ses (imp.init) Partial correlation r#ses,imp.init Multiple partial r#ses,imp.init,age Multiple semi-partial r#ses,imp (init,age)

11. Which type is each of the following? partial rperf,prac.exp 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? multiple semi-partial rprac(perf.exp,age) semi-partial rprac(perf.age) multiple partial rperf,prac.exp,age

12. Two important things to remember: • 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 … • The test of that variables regression (b) weight in the multiple regression model H0: b3=0 in y’ = b1x1 + b2x2 + b3x3 + a • The multiple semi-partial between the criterion and that predictor, after controlling that predictor for all of the other predictors H0: ry(x3.x1, x2)= 0