Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Survey 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”?Simple Correlation questions (old friends)ry,x1 simple correlation of y and x1ry,x1 vs. ry,x2 comparing “correlated” correlations within a population/group (uses Hotelling’s t-test or Steiger’s Z-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,#sesrimp,#ses vs. rimp,initrimp,#ses for adults vsrimp,#ses for adolescents.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?Same r, dif populationsrperf,pract for novices vs. rperf,pract for experiencedUse Fisher’s Z-testSimple r -- rperf,pract Compare r’s in the same poprperf,pract vs. rperf,expHotelling’s t-test & Steiger’s Z-test Which type is each of the following? Use the notation & tell test used for each type of comparisonMultiple Regression questionsRy.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’s Z-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’s Z-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 )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 vs. Rimp.#ses,init,age,#ther2 Rimp.#ses,init,age2Rimp.#ses,init,age2 vs. Rimp.#ses,#ther2 Rimp.#ses,init,age2 for adultsvs. for adolescentsRimp.#ses,init,age2 vs.Rtsat.#ses,init,age2Do 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 Rperf.prac,skill2single model for 2 criterion H & SRspeed.prac,skill2 vs. Racc.prac,skill2nested model comparisons R2- F-testRperf.prac,skill,mot2 vs. Rperf.prior,mot2non-nested models H & SRperf.prac,skill,mot2 vs. Rperf.prac,mot,age2R2 for 2 populations F’s Z-test Rperf.prac,skill,mot2Which type is each of the following? Use the notation & tell the test used for each model comparisonStatistical control questions are always about the “causal relationships” that “produce” or “modify” bivariate correlations - called “3rd variable problems”.Here’s a well-known example ...Ice cream salesViolent crimesWhen 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 crimesIce cream salesHere’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”.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? # sessionssymptomatic improvementsymptomatic improvement# sessionsdotsize 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.Another example ...Statistical Control questionspartial correlation questions -- is the relationship BOTH variables have with some 3rd variable(s) “producing” the bivariate relationship between them?ry,x1.x2 partial correlation of y & x1 controlling both for x2 (control var listed to right of “.” )ry,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?ry,(x1.x2)