An Introduction to Structural Equation Modeling

NICK ALLUM: My name's Professor Nick Allum.And I'm from the Department of Sociologyat the University of Essex.And I teach statistics and survey methodology.In this tutorial, I provide an introductionto structural equation modeling.Structural equation modeling is reallya perspective and a framework thatincorporates a number of other techniqueswith which you might be familiar.

NICK ALLUM [continued]: I'll cover latent variables, factor analysis, and whatwe call a measurement model.I'll cover path analysis, what we call the structural model.I'll talk a little bit about model fit,how to tell if your model fits the data well or not,and what to look for in terms of interpretingthe estimates and the output that we get from our analysis.

NICK ALLUM [continued]: Structural equation modeling is the frameworkwhich includes various methods of statistical analysis,but chiefly, regression analysis and factor analysis.The idea of structural equation modelingis to be able to simultaneously model lotsof outcomes at the same time.So it's very useful for operationalizingin a statistical model, complex theories about howand why the world works.

NICK ALLUM [continued]: It's used widely in sociology and political scienceand in psychology.And it's even used in hard sciences, neuroscience,physics, and other areas.One of the key concepts in structural equation modelingis the idea, what we call latent variable.

NICK ALLUM [continued]: A latent variable is a variable that is unobserved,in other words, we don't directly measure it,yet we think it's there and it's relevant for our research.So the classic example of latent variable would be an attitude.Attitudes are psychological phenomenathat we think existent and that can explain behavior,but we can't measure them directly.

NICK ALLUM [continued]: There's no light on someone's foreheadthat tells you whether they're positive or negativeabout a particular thing.All we can do is to ask questionswhich we think tap into those attitudes in termsof specifics.And then from that, we can determinewhat their attitude might be from the combinationof answers.

NICK ALLUM [continued]: So for instance, if you're asking someone,you want to find out how left-wing or right-wing someoneis, there's no direct measure of that,but you can ask them questions about,do you think that strong trade unions are important?Do you think you should redistribute wealth?Do you think rich people should beallowed to get richer and poor to get poorer?

NICK ALLUM [continued]: And from combining and inferring from all of those answerssomething about people's overall perspective,we can generate what we might call a latent variable.And structural equation modeling gives us statistical modelsto do this.

NICK ALLUM [continued]: Latent variable modeling has a long history,and that is the history of factual analysis,which is really around 100 years old now.And structural equation modeling which was pretty much inventedin the 1970s and came to the fore in the '80s and '90s,combines factor analysis with regression analysisto form an overarching statistical frameworkfor the testing of theories to dowith substantive sociological or other social science topics.

NICK ALLUM [continued]: But also as ways of measuring latent variables, attitudes,beliefs, and so forth.Structural equation modeling allowsus to estimate latent variables and derivemeasures of unobserved concepts or constructs.

NICK ALLUM [continued]: But what it also does is to allowus look at the relationship between those constructs.So for instance, we might be interested in doinga study about intelligence and job satisfaction.We would measure intelligence using ability measures,cognitive ability indices, and combinethose in our structural equation model.

NICK ALLUM [continued]: And we might then also measure job satisfactionalong a range of dimensions.Perhaps how successful someone is,how much like their co-workers, whether theyfelt there was prospects for promotion, and so forth.In structural equation modeling, we don't need to stop there.We can then look at the correlationbetween those two concepts measured at the latent level.

NICK ALLUM [continued]: And we don't have to stop there.Path analysis is the idea of combiningsimultaneous regression equationsin what can be quite a complex picture which representsa causal model of a process or situation in the social worldthat you're trying to investigate.

NICK ALLUM [continued]: So we might think that parental resources mightlead to educational success.Educational success might itself leadto a particular type of job, a particular type of jobmight then lead to a particular kind of satisfaction.And this might differ for men and women.All of those things that I've just mentionedcan be operationalized into a structural equation modeland estimated simultaneously.

NICK ALLUM [continued]: So path analysis is this idea of estimatingsimultaneous regression equations,looking at the relationship between onevariable and another, or multiple variables,and multiple other variables.In standard regression analysis, wehave independent variables, which are our predictors.And we have one dependent variable,which is the variable for which we want to get estimatesof its level of variation.

NICK ALLUM [continued]: In path analysis, we can have as many predictorsor independent variables and as many outcome or dependentvariables as we like.So we don't use that terminology anymore.And we talk instead about exogenous variablesand endogenous variables.An endogenous variable is somethingwhich is predicted by another variable in the model.

NICK ALLUM [continued]: An exogenous variable is one thatis not predicted by any other variable in the model.And it's easy to understand this if we look at a path diagram.So in this path diagram, we're interested in understandingexamination performance.And we're looking at the idea of exam stress.

NICK ALLUM [continued]: Looking at the path diagram, we cansee that there are some of these ovals whichare variables, which have arrows pointing into them.And one of them which has no arrows pointing into it.So that one is exam stress.And that's an exogenous variable.The others are endogenous.And the idea here is that exam stresscauses people both to increase their exam preparation, whichhas a positive effect on exam performance.

NICK ALLUM [continued]: But the more they stress, the more they get anxious,and that has a negative effect on their exam performance.So rather than thinking about these processesseparately, as in separate regression analyses,in structural equation modeling, wecan model the whole process simultaneously,and look at the total effect of exam stress on performance.

NICK ALLUM [continued]: And also the indirect effects.In other words, the effects that exam stresshas through intervening variables, like exampreparation, and anxiety.So to recap, we have the idea of factor analysis,which captures the latent variables of interest.

NICK ALLUM [continued]: In other words, it may be attitudes,beliefs, maybe social class.It may be intelligence.And we've got our path analysis, or our path model,which specifies the relationshipsbetween the latent variables.And in structural equation modeling,we call the first of these, the factor analysis,the measurement model.

NICK ALLUM [continued]: And we call the path analytic part, the structural model,because we refer to structural relationships between latentvariables as measured through our factor analysis.So if we add some detail to our conceptual pathmodel of exam stress and exam performance,we can think about measuring exam stressin terms of perhaps some questions about how muchsomeone is thinking about their impending exam,how worried they are about it, how concerned theyare that it'll affect their career if they don't pass.

NICK ALLUM [continued]: We could think about measuring anxietythrough a combination of perhaps increasedheart rate, fixation on the exam,and through lack of sleep.It could be any number of indicators we could use there.And in terms of preparation, we couldhave questions about how much revision do you do each day,how many books have you read on the subject, and so forth.

NICK ALLUM [continued]: These are all the observed variables,which would go into our measure of models.One model for each latent variable, which would thenbe combined in our path model, as you can see,which is now predicting a single variable, whichis just the exam score.And that's just an observed variable,what we call in structural equation modeling, a manifestvariable.So structural equation models can contain a combinationof latent variables, and observed variables,and linked together with structural paths,in other words, with these arrowsthat you can see on the diagram.

NICK ALLUM [continued]: When we're fitting a structural equation model,what are we actually estimating?And what are we looking for in the results?Well, each of the arrows in the path diagram,or in our structural equation model, more generally,represents a regression coefficient.So we'll have estimated a regression coefficientfor each of those.And that regression coefficient symbolizes the relationshipbetween the two variables involved.

NICK ALLUM [continued]: An important aspect of structural equation modelingthough is it's deductive nature.We're also, as well as estimating observedrelationships, our path diagrams represent constraintson what we're estimating.So for instance, in the path diagram that we looked at,there's no path between anxiety and exam preparation.

NICK ALLUM [continued]: That's our theory.So our theories are represented by the pathsthat we draw into our model, but thosecritically, that we leave out.And that's part of evaluation of how well the model fits.Because if it was really the case that there was absolutelyno relationship between anxiety and exam preparation,our model wouldn't really fit the data very well.

NICK ALLUM [continued]: So how do we actually go about looking at that?Well in terms of model fit, we have some standard measures.But the general idea is that thereis a correlational or covariance matrix whichdefines all the correlations between all of the observedvariables.And that's something, that's our data.We can look at that, and we can calculate those correlations.

NICK ALLUM [continued]: If we look at our path diagram that we'vegenerated through theoretical and any other kind of planning,then there is also an implied correlation matrix,or covariance matrix, which would represent and reproducethe relationships which we have specified alongwith the parameter estimates that we've made.

NICK ALLUM [continued]: And what we do in structural equation modelingis to compare the observed covariance matrix with the oneimplied by our model.And if the one implied by the modelis a long way from the one that we observed,then we say the model doesn't fit very well.If it's very close, in other words,if our model with all its constraints,theoretical relationships actually represents the datapretty well, then we're going to getwhat we call a good fit, or a reasonable fit.

NICK ALLUM [continued]: There are lots of measures that we use to do that.And more than it's possible to go into in detailin this tutorial.But we use a combination of what we call exactand approximate fit indices.So we have the results now from fitting our structural equationmodel.And what you can see is that the more stress, examstress there is, that leads to a high level of exam preparation.

NICK ALLUM [continued]: And also, high level of anxiety.The effect of anxiety on exam performanceis negative, negative point five.Whereas the effect of preparation on exam performanceis as we might expect, positive.Now we originally thought there wouldbe a direct path from stress to exam performance.

NICK ALLUM [continued]: But as we can see, this is at zero, or very, very near zero.And it's actually not statistically significant.So there is no direct effect in our results from exam stresson exam performance.But the effect comes through the twin phenomena of anxiety,which has a negative effect, and preparation,which has a positive effect.

NICK ALLUM [continued]: So you can see that we have a combination of direct effects,or hypothesized direct effects.Indirect effects which can work in different directions,and the idea here is that we wouldwant to interpret this in terms of a process model.So we start with stress, and thatleads people to do certain things, feel certain ways.

NICK ALLUM [continued]: And end up with a particular kindof accomplishment on an exam.Now, as with any statistical model,we wouldn't expect this to explainall of the exam performance.We can at least explain some of the variationin exam performance with a structural equationmodel of this kind.In this short tutorial, I've justscratched the surface of what structural equation modeling isall about.

NICK ALLUM [continued]: But I hope I've given you a flavor of the kinds of thingsit can used for, and the way in which it works,and the relationship it has with other statistical techniques.I've looked at the idea of latent variables, factoranalysis, and the measurement model.We've looked at path analysis, and the structural model.

NICK ALLUM [continued]: I've looked at how to very broadly assessthe fit of the model.How well does our model fit the data?And also looked at what kinds of estimateswe get from our model, and how to interpret these.