PSYC 612, SPRING 2009Lecture 22: Structural Equation ModelsLecture Week: 4/28/2009Contents1 Preliminary Questions 12 Part I: Introduction to SEM (70 minutes) 12.1 Purpose: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Objectives: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Limitations to the linear model 24 Attenuated Effects 25 SEM in a nutshell 35.1 SEM input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.2 SEM output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.3 Evaluating Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Summary 71 Preliminary Ques tions•Has everyo ne scheduled module 2?•Have you sent me content requests for Lecture X?•Are you ready for module 3?2 Part I: Introduction to SEM (70 minutes)2.1 Purpose:Provide a brief introduction to structural equation mo deling12.2 Objectives:1. Discuss the limitations to linear models2. Explain attenuation effects3. Describe SEM in general4. Show input and output5. Summarize SEM advantages and disadvantages3 Limitations to the linear modelThe GLM offers a very flexible framework to help us understand the relationship among variables.While it is flexible, it has limitations that require some attention so you can appreciate how SEM fillsa need. As you know, the GLM restricts us to a single DV and does not allow us to test mediationin a single model. Multiple DV’s and mediation require separate models that lead to alpha inflation- you are now intimately familiar with those problems. Furthermore, the GLM assumes that allvariables are measured without error; remember I sprang that upon you during the last lecture?Well ’tis true. The GLM makes the crazy assumption that all variables are measured without errorand that assumption leads us to serious questions about the utility of the G L M. Remember thatmany people adopt a very negative view of PCA because that procedure assumes the same error-freemeasurement. To summarize, here are some of the limitations of the GLM:1. Single DV2. Multiple models required for mediation3. Error-free measurement assumed4 Attenuated EffectsWhat happens when we have measures with error? Before I answer that question directly, I wantto walk through some simple logic so yo u can easily appreciate the problems with measurementerror. I mentioned several times this semester that the best conceptualization of error comes fromclassical test theory (CTT) - not because it is correct but rather because it offers a simple heuristicto understand the impor t ance of sound measurement. I begin with the simple formula for CTT:σ2Observed= σ2T rue+ σ2ErrorThe heuristic benefit of knowing this equation cannot be underestimated. Just remember OTE- observed equals true plus error. If you remember this equation then you can easily understandLord’s simple formulation for reliability:ρ =σ2T rueσ2T rue+ σ2Error=σ2T rueσ2ObservedAs error increases, ρ or reliability decreases. Thus, we want to reduce error variance as muchas possible. What happens when we inflate error variance? Rather, what happens when we haveless than perfect measurement? We get attenuated effects. An attenuated effect is an effect that2is decreased due to some artifact. We call it a t t enuated because we assume that there is a true,underlying effect (according to omniscient Jo nes) that is larger. That underlying effect is the latentor population estimate.I pause here to ask you some questions.1. What is the largest correlation observable?2. What is the largest correlation possible between one perfectly measured variable and anotherwith only 5 0% true score variance?3. What is the largest correlation possible between two variables with only 50 % true score vari-ance?I resume once everyone writes down their answers. To keep you from cheating, I intend to tellyou the answer to this question during class. Sneaky of me, huh?5 SEM in a nutshellStructural equation modeling was developed almost 90 years ago by Sewall Wright (1921) when hedeveloped path analysis. Much later (1973) Joresko g wrote a computer program that helped socialscientists estimate complex paths - thus ringing in the modern ag e of SEM. What is SEM then?SEM is a set of linear regressions that get solved for simultaneously. Permit me to show you adiagram so you have a fuller appreciation for what SEM offers.Figure 1: A basic SEM modelRemember this model? Consider the linear equations necessary to test this simple model.V 1 = FAb + ǫV 1V 2 = FAb + ǫV 2V 3 = FAb + ǫV 33V 4 = FCb + ǫV 4V 5 = FCb + ǫV 5V 6 = FCb + ǫV 6FC= FA+ DFCThe model is far more complicated than it appears in the figure but that only becomes apparentwhen we think about the linear models. Now suppose we have some data for that mo del; how dowe go about examing the underlying relationship between FAand FC? Before we do that, I needto explain the model and some of the shapes in the figure. At this point, I intend to talk aboutshapes on the board.Now let me walk you through how I intend to create data. Bear with me because the process ispretty simple and relates back to the CTT equation above.> Base <- rnorm(100)> Fa <- Base + rnorm(100)> Fc <- Base + rnorm(100)> V1 <- Fa + rnorm(100) + rnorm(100)> V2 <- Fa + rnorm(100) + rnorm(100)> V3 <- Fa + rnorm(100) + rnorm(100)> V4 <- Fc + rnorm(100) + rnorm(100)> V5 <- Fc + rnorm(100) + rnorm(100)> V6 <- Fc + rnorm(100) + rnorm(100)> SEMdat <- data.frame(V1, V2, V3, V4, V5, V6)> Mod.1 <- specify.model("./SEMmod1.txt")> Scov <- cov(SEMdat)> sem.1 <- sem(Mod.1, Scov, N = 100)> std.coef(sem.1, digits = 2)Std. Estimate1 0.67 V1 <--- Fa2 lamV2 0.83 V2 <--- Fa3 lamV3 0.58 V3 <--- Fa4 0.83 V4 <--- Fc5 lamV5 0.69 V5 <--- Fc6 lamV6 0.69 V6 <--- Fc7 beta 0.50 Fc <--- Fa> path.diagram(sem.1, out.file = "./SEMpath.dot", min.rank = "V1, V2, V3",+ max.rank = "V4,V5,V6", edge.labels = "values")> system("dot2tex --figonly -f pstricks SEMpath.dot > SEMpath.tex")5.1 SEM inputEvery software package uses it own method of input. The most common format is the RAMspecification that looks like the fo llowing code.4Fa -> V1, NA, 1Fa -> V2, lamV2, NAFa -> V3, lamV3, NAFc -> V4, NA, 1Fc -> V5, lamV5, NAFc -> V6, lamV6, NAV1 <-> V1, the1, NAV2 <-> V2, the2, NAV3 <-> V3, the3, NAV4 <-> V4, the4, NAV5 <-> V5, the5, NAV6 <-> V6, the6, NAFa -> Fc, beta, NAFa <-> Fa, phiFa, NAFc <-> Fc, …