Introduction to Path Analysis• Ways to “think about” path analysis• Path coefficients• A bit about direct and indirect effects• What path analysis can and can’t do for you…• Measured vs. manifested  the “when” of variables• About non-recursive cause in path models• Some ways to improve a path analysis model• Mediation analyses• Model Identification & TestingOne way to “think about” path analysis is as a way of “sorting out” the colinearity patterns amongst the predictors – asking yourself what may be the “structure” -- temporal &/or causal relationships -- among these predictors that produces the pattern of colinearity.“Structure” of a MR model – with hypotheses about which predictors will contribute A proposed structure for the colinearity among the predictors and how they relate to the criterion – with hypotheses about which paths will contribute 1122334455Crit Critearlier More recentdistal cause proximal causeWhere do the path coefficients come from?One way is to run a series of multiple regressions…for each analysis: a variable with arrows pointing at it will be the criterion variable and each of the variables having arrows pointing to it will be the predictors12345Crit1. Crit = 3 Pred = 52. Crit = 1 Preds = 3 & 53. Crit = 4 Pred = 54. Crit = Crit Preds = 1, 2, 3 & 4The path coefficients are the β weights from the respective regression analyses (remember that β = r for bivariate models)What path analysis can and can’t accomplish…Cans -- for a given structural model you can…• evaluate the contribution of any path or combination of paths tothe overall fit of that structural model• help identify sources of suppressor effects (indirect paths)Can’ts• non-recursive (bi-directional) models• help decide among alternative structural models • provide tests of causality (unless experimental data)So… You have to convince yourself and your audience of the “reasonableness” of your structural model (the placing of the predictors), and then you can test hypotheses about which arrows amongst the variables have unique contributions.Alternative ways to “think about” path analysis…• to capture the “causal paths” among the predictors and to the criterion• to capture the “temporal paths” among the predictors and to the criterion• to distinguish “direct” and “indirect” paths of relationship• to investigate “mediation effects”… to distinguish “direct” and “indirect” paths of relationship…12345Crit2 has a direct effect on Crit• a “contributor” in both the regression and the path models12345Crit5 does not have a direct effect on Crit – but does have multiple indirect effects• not “contributing” in the regression model could mistakenly lead us to conclude “5 doesn’t matter in understanding Crit”…to distinguish “direct” and “indirect” paths of relationship…, cont.12345Crit3 also has an indirect effect on Crit• there’s more to the 3  Crit relationship than was captured in the regression model12345Crit3 has a direct effect on Crit… to investigate “mediation effects”…Mediation effects and analyses highlight the difference between bivariate and multivariate relationships between a variable and a criterion (collinearity & suppressor effects).For example…For Teaching Quality & Exam Performance  r = .30, p = .01• for binary regression β = r, so we have the path model…TQ EPβ=.3It occurs to one of the researchers that there just might be something else besides Teaching Quality related to (influencing, even) Exam Performance. • The researcher decides that Study Time (ST) might be such a variable. • Thinking temporally/causally, the researcher considers that Study Time “comes in between” Teaching and Testing. • So the researcher builds a mediation model, getting the weights from a multiple regression with TQ and ST as predictors of EP… to investigate “mediation effects”…The resulting model looks like …TQEPβ=.0STβ=.4β=.3We might describe model as, “The apparent effect of Teaching Quality on ExamPerformance (r=.30) is mediated by Study Time.”We might describe the combination of the bivariate analysis and the multiple regression from which the path coefficients were obtained as, “While Teaching Quality has a bivariate relationship with Exam Performance (r=.30), it does not contribute to a multiple regression model (β=.0) that also includes Study Time (β=.40).Either analysis reminds us that the bivariate contribution of a given predictor might not “hold up” when we look at that relationship within a multivariate model!Notice that TQ is “still important” because it seems to have something to do with study time – an indirect effect upon Exam Performance.The “when” of variables and their place in the model …When a variable is “measured”  when we collect the data:• usually concurrent• often postdictive (can be a problem – memory biases, etc.)• sometimes predictive (hypothetical – can really be a problem)When a variable is “manifested”  when the value of thevariable came into being• when it “comes into being for that participant”• may or may not be before the measure was takenE.g., State vs. Trait anxiety• trait anxiety is intended to be “characterological,” “long term”and “context free”  earlier in model• state anxiety is intended to be “short term” & “contextual”  depends when it was measuredSome caveats about the “when” of Path & Mediation Analyses…1. The “Causal Ordering” must be theoretically supported  path analysis can’t “sort out” alternative arrangements -- it can only decide what paths of a specific arrangement can be dropped2. Mediating variables must come after what they are mediatingTxCritβ=.0Sexβ=.4β=.3Looks like a participant’s sex mediates the treatment.But it also looks like treatment causes a participant’s sex ???rCrit,Tx= .4So we run a mediation analysis:E.g. The Treatment is related to the criterion. But the researcher thinks that one’s gender mediates how the treatment has its effect…predictor Motiv St. Time GPA % Pinkr(p) .28(<.01) .45 (<.01) .46 (<.01) .33(<.01)All of these predictors have substantial correlations with Exam grades!!An example  “when” and “operational definition” matter!!!Bivariate & Multivariate