Roger E. Millsap

Arizona State University

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*Using Latent Variable Models in Survey Research.* Hi, I'm Roger Millsap from Arizona State University and I’m going to be taking you through a series of slides that explain what latent variable models are, including what latent variables are, what these models might be able to do for you in survey research, and a little bit about how you can use them. So, let's get started.

What is a latent variable in the context of survey research? We think of a latent variable as being an unmeasured variable that is believed to influence responses to a number of survey items. An unmeasured variable simply means that we don't have a perfect measure of it, but it's something that we develop questions to ask about. And, usually we think of this as being something that no one question can fully capture. So, we have multiple questions that attempt to get at this. A good example might be satisfaction with dental services. Suppose you have a survey that's trying to elicit something about consumers’ satisfaction with services. You’ll probably have a number of items that try to get at that and it has a number of different aspects. No one item might capture the full latent variable, so we view satisfaction as a latent variable and something that we kind of target with our survey and we may be more or less successful in measuring it.

Usually we think of some latent variables as being the intended targets of the measurements. So, if we want to measure satisfaction with dental services, this is our target latent variable and we have items that we’ve designed to measure it. We may have more than one target latent variable, so we may consider, for example, satisfaction to be something that has distinct features or aspects, it's multiple dimensions involved in satisfaction, and if so, we’ll develop questions to measure each of those and we’ll have multiple latent targets that we’re interested in.

Inevitably though, there are other latent variables that we don't anticipate but that do contribute to the, or influence respondent’s answers to questions and some of these contribute to error of measurement. We usually think of error of measurement as being of two different types. There's random error, or noise in the measures that is unpredictable and can go in either direction. So, for example, a person might be distracted and not read a question properly and respond in a way that reflects a lack of understanding of the question. If we were to retest that person they may read it more carefully and respond appropriately. So, this is kind of unanticipated or random error.

On the other hand, there’s errors that are viewed as sources of bias. Bias means that the response is skewed in one direction or another consistently. So, for example, if I'm asking people about what they eat, I may get underreporting of certain types of foods because people don't want to say how much they really eat of those foods. So, that would be a systematic bias. Or, if I'm asking about behavior that's maybe not acceptable socially, I tend to get a selective reporting of that behavior. So, these are biases that maybe we do or do not anticipate but they are sort of acting as latent variables although unwanted latent variables influencing responses.

Usually in terms of targeted latent variables, we have multiple items or indicators. Indicators is kind of a generic term to indicate any measured variable that’s supposed to relate to a latent variable. So, for example, here I have a diagram where I have three survey items represented as three little boxes and underlying those three boxes is a latent variable called satisfaction represented by an ellipse, and arrows go from the ellipse to each of the boxes representing the relationship between the latent variable *satisfaction* and the item responses to those three items. This kind of diagram, usually called a path diagram, is a very common representation of the latent variable model. You'll see it in published papers.

We might also consider measurement errors as influences on the item responses. So, here I've amended that diagram to add in three new latent variables represented by the ellipses that float above the boxes. From each ellipse there is an arrow going to each box. That represents the influence of the measurement error on each item, and there's one measurement error latent variable for each item. So, in total then, there’re four latent variables in the system, only one of which we're really interested in. That's our targeted latent variable, which is the satisfaction.

A latent variable model specifies the relationship between the measured variables, whatever they happen to be, and the latent variables. So, the model kind of captures the whole picture. And, in constructing a model, we consider several questions, first of all being, how many latent variables are there going to be. We usually refer to that as the dimensionality problem and we distinguish between unidimensional models where there's only one target latent variable plus maybe some error variables versus multidimensional problems where there's multiple targeted variables. And, we have models appropriate for either situation.

If you're in the multidimensional case, you’ll want to entertain the further question of which measured variables or which items are related to each target latent variable. And, that's something that the model will represent in a certain way, based on whatever theory you have, what you thought about when you designed the questionnaire. But, any of the latent variables is going to address both of these questions: how many target latent variables are there going to be, how many latent variables in general, and how are those latent variables related to each of the measured variables.

Okay, let's talk a little bit about how these models might be helpful to you in survey research. And, there's a number of answers to that question. We’ll start with a case that's quite basic. One traditional use of these models has been to separate measurement error from variance due to the targeted latent variable. In other words, this goes to the issue of the reliability or the information provided by the items about the latent variable. It's hard to get a handle on reliability without some sort of model and there are a range of models that can provide that information for you but this is one of the things we’d want to know, if we have a multiple item scale, for example, that supposed to measure a latent variable, what is the reliability of that scale? We can find that out. If we know the reliability and we have a successful model, we can also use that model to help us estimate relationships between our scale variable, our target latent variable in the scale, and other variables we might be interested in predicting or relating to it. And, we’ll be doing that without the influence of measurement error. So, the latent variable model in effect disattenuates or removes the measurement error, allowing you to look at relations with other variables. And, that's a useful function because measurement error can distort relationships with other variables and prevent you from getting full results from your survey.

In survey research, another frequent goal is to develop a shortened version of a survey. In general, we like our surveys to be short but it can become difficult to just throw out questions to shorten a survey unless you're confident that the resulting information is good. Basic truism is that the shorter the survey usually, the less reliable the survey. So, a latent variable model can help you shorten your survey while maintaining adequate levels of reliability and that's one of the frequent uses of these models. In fact, a collection of models based on item response theory are often used just for that purpose, taking longer surveys and understanding how to shorten them while maintaining reliability standards.

Aside from that application to measurement error, there’s also the goal of trying to understand the targeted latent variables, and their nature and how many there are. So, for example, a latent variable model would help you evaluate whether the items that you wrote to measure a latent variable in fact do measure that latent variable. It might be the case, for example, that some of the items you wrote don't function as anticipated. They don't turn out to be very good measures. You'll see that when you fit the latent variable model to the data. You’ll be able to spot those items and maybe eliminate them or modify them.

And, so, the models are helpful in trying to understand which items are functioning well and which ones maybe are not helping. If you do longitudinal survey work, a latent variable model can also help you understand changes in the item functioning over time in longitudinal surveys. So, for example, in a long-term longitudinal study you might find that the drivers of satisfaction in one age group no longer function that way as those persons age and they’re different. The questions function differently over time within the same population across a long span of time. So, you can use these models to help understand those kinds of phenomena.

Another frequent use of the models is to help evaluate the equivalence of surveys that have been translated into different languages or simply surveys that are used in distinct populations. It's not uncommon nowadays to be surveying populations where there are multiple language groups. And, so, you'll need to have different versions of your survey. But after translation, are the items really functioning the same way? You can use these models to help get a handle on that question.

So, let’s talk about some of the types of latent variable models that are out there. And, I just want to give a broad overview here of what the variety of these things are, and we’ll start with some models for continuous latent variables. In other words, for these models we’re thinking of the latent variable as some continuously-scaled dimension along which we can array people based on their responses to the indicators of the items. And, there are two broad classes of models for these that are very well understood and often used.

First of all, there’s factor analysis, which has been around for a long time and we usually distinguish two variants of this: exploratory factor analysis and confirmatory factor analysis. And, these are linear models, relatively simple, and they allow you to model either item level data, where the responses are discrete, or data in which the indicators are more continuous. Either way you can model them with factor analysis. And, there's a huge literature on this subject and lots of software available for implementing these. We’ll talk about that.

The other class of models are based on item response theory, or IRT, and this is really a family of models, very diverse, depending on the types of items. These attempt to model responses to questionnaire items as a function of one or more latent variables and these are nonlinear models. In the past, the use of these has pretty much been confined to cognitive testing applications, but that's no longer true. In the last decade or two, their uses expanded greatly into noncognitive arenas. And, in particular, health and medical research; these models are used quite a bit now and so they're viable options for survey researchers, especially in the health fields.

A second class of latent variable models assume that the latent variables are not continuous but rather categorical or discrete. So, that, in using these models you're considering the idea that respondents are not arrayed along continuous dimensions but exist in groups. The groups being unknown initially but you might believe that whatever it is you're measuring really defines types of people rather than continuously classified dimensions along which we place people. And, a key issue in any use of these models is going to be how many groups are there, two groups, three groups? How many? The two broad techniques are sets of models for doing this are latent class analysis and latent mixture modeling. And, they're closely related. Both assume this idea that there are groups or types.

Latent class analysis is usually used when the indicators are binary, having two response options, or polytomous, having more than two, but a small number of response options. So, a binary item, an example would be a true/false item, is this true or false. A polytomous item might be five-point Likert scale in which you ask people their strength of agreement with some statement, for example. Latent class models are used for those kinds of indicators.

Latent mixture models are more general. They can be used with discrete or continuous indicators. And, latent mixture modeling is sometimes used even when you don't really consider there to be any latent variable models. It's a very general technique. And, so, latent mixture modeling is exploded in popularity and there's a lot of applications of it. And, we will touch on this again soon.

How do you decide between these two broad classes of models? It hinges on how you think about the latent variables that you want to measure. And, you know, you may use both types of models in the same survey depending on what you're measuring. If you think of your latent variables as best conceptualized as continuous, usually you're going to go to some factor analytic or item response theory type of model. So, satisfaction is a good example of a variable that we usually think of. We don't you just classify people as satisfied/not satisfied, for example. We usually think of this is a degree of variation of some sort. There may be more than one satisfaction dimension, but we think of all of them as continuous. So, you know, that's fairly common. And, there a lot of psychological attributes that we think of this way. If you want to measure someone's level of depression, it’s usually thought of as a continuous scale. You want to measure their strength of opinion about some issue? There again, we usually think of it as a continuous latent variable.

On the other hand, there are cases where we don't necessarily view things as continuous dimensions; but rather, we think of a categorization of people. So, for example, consumer research often thinks of consumers as being defined within market segments. So, you have different types of consumers. And, you want to identify how many types there are and maybe even classify respondents in one or more types. That would be a classic example of a categorical latent variable. And, once you decide how many categories there are, you can also use the model to classify people with varying degrees of success, depending on how well the model fits and the strength of relationship between the items and these classifications.

Another example would be in longitudinal research. Suppose you’re studying growth or change over time. You might hypothesize that not only do people change over time, they change in certain ways and different people follow different patterns of growth. Well, there again, you would have people classified in groups based on their pattern of change over time. And, you might want to identify how many different patterns there are and classify people in one of the different groups based on the data that you have on those people. So, both of these kinds of latent variable definitions come up. We have tools for addressing either case.

It’s also possible to combine several latent variable models in order to get an adequate model for your particular case. So, we were just talking about the example of there being different growth patterns over time. You might have a model that's continuous, that model’s growth over time, as a function of time. But, then you combine that with a latent mixture model that says there are types of growth. So, within each class there’s a growth pattern that's continuous, but the different classes are discrete. So, that would be a growth mixture model.

Or you might have an example in which you say that there's distinct groups of people, say distinct consumer types, and you're looking at something like satisfaction that’s a continuous latent variable also. And, maybe you think that these different consumer types have different sources of satisfaction. So, there are different things that make these people feel satisfied in each case. Well, then you might have a different factor model for each type and you end up with a factor mixture model. So, these things can be combined and they often are, and there’s ways of doing all of those things.

So, let’s shift gears and talk a little bit about what steps you would actually take to incorporate one of these models in your analysis of the data. Obviously the first step might be to decide, at least on a preliminary basis, what type of model you are going to use. So, then you think about some of the things we’ve just talked about, and you think about the questions you want to answer with your survey, and which type of model might be best in helping you to answer those questions.

Next, you would try to locate some appropriate software. I should say that in all cases, no matter which of these models you are using, you’ll need some software to actually implement the model with your data. And, I'm going to talk about a range of available software in a minute, so we’ll have a look at that. But, it is a step that's important because you need to have a software program that's appropriate for your particular application.

So, once you've located the software, then you need to actually use it. And, using it means kind of two aspects. First, you need to specify the model that you think is appropriate for your data. So, if you're using a factor model, that would involve, perhaps, specifying how many factors there are and which variables are related to which factors. And, once you've done that, then you fit the model to the data. This is an important step because we don't just use the model, estimate them, and carry on. We need to make sure that the model fits the data and it may not fit the data. So, there's a step involved in evaluating the fit of the model. If the fit is good then we can use it. If the fit is not so good, then we're going to have to step back and maybe modify the model, or modify our items, or take some other step that will get us to a point where we have a model that fits well and answers the questions that we want to address.

So, the last step might, if your model didn't fit well, might involve revising the model or alternatively possibly changing your items, dropping them or eliminating them from the model or from the survey altogether, depending on what stage you're at in the survey process. So, obviously Steps 3 and 4 require a lot of discussion. We don't have time here to go into all the details of these but I hope that this brief introduction will help you see the utility of these models and maybe spur you to learn a bit more about them.

Let's talk a little bit about software, starting with models for or software for factor analysis. Distinguishing again between exploratory and confirmatory factor analysis, there’s lots of software available for exploratory factor analysis. The major statistical packages, such as SPSS and SAS offer exploratory factor analysis programs as part of their general menu of procedures that they offer. So, it's easy to find software for this technique and, for the most part, the software that is available is quite user-friendly. So, this is relatively easy to do.

Confirmatory factor analysis relies on software that’s a little more specialized. The big packages don't necessarily have this, although SPSS does offer a program called AMOS, which is listed here. On this slide here I show you the websites for each program, where you can go to get more information about each one. AMOS, of the four I list here, AMOS is the one that might be the most user-friendly to beginners, and it is part of the SPSS package now. It didn't used to be. Of the other three, Mplus, LISREL, and EQS, all are well-known programs. All of them can do confirmatory factor analysis and do it well. Mplus is quite versatile and will do a variety of other latent variable models in addition to just factor analysis. LISREL is the oldest one and it's well known, a very reliable program. It will do CFA. And, EQS has also been around for a long time. It can do CFA and is a good program as well. So, any of these would probably suit your needs.

As far as software for Item Response Theory, traditionally the programs available for doing Item Response Theory modeling have been a bit difficult to use, unless you're a specialist. However, the latest addition to the set of programs available, which I list here, called IRTPRO, is user-friendly. So, it's relatively easy to use and it contains the latest bells and whistles and enhancements needed to do IRT modeling of a variety of data types.

The next two, BILOG-MG and PARSCALE. BILOG is oriented toward binary items, items that could be answered only two responses, or scored that way. PARSCALE is more general and can handle polytomous responses. Both of those programs are a little bit difficult to use without some sophistication about IRT, but they're both solid programs.

The last one listed there is called WINSTEPS. It's a relatively easy program to use but it only uses models based on what's called the Rasch model, which is a very commonly used model, but it is a little limited in scope compared to the other three.

For latent class analysis, probably the most frequently used program that's dedicated to latent class analysis is called Latent GOLD, and I have a website listing here for it. Latent GOLD is an excellent program. It does latent class analysis. You can also do latent class analysis in Mplus, the program I mentioned a minute ago with regard to confirmatory factor analysis. And, so, latent class analysis, there are probably, I don't know, half a dozen programs that will do it. I only mention these two here because these are probably the two most frequently used ones, but there are others out there.

If you want to do latent mixture modeling, software for doing this is more specialized. The Mplus program that I mentioned earlier is one that can do mixture modeling and very general types, and can combine the mixture modeling with other forms of latent variable modeling as well. So, I think the Mplus program’s a good choice if you want to do this. But there are other programs that will do latent mixture modeling and some of these will combine mixture models with other types of models as well.

I should mention that there's also a lot of free software available on the web to do all of these models. And, nowadays, the free software that's out there is relatively good quality. So, you can get, although all the other programs I mentioned here cost money, you can get free software that will do most if not all of these techniques. One in particular, the R-Statistical Package, contains programs for doing factor analysis, IRT, latent class analysis, and other techniques. However, it requires some programming skill to use that package. It is free though. It's a wonderful set of statistical tools. So, you know, there is pro and con with these free programs but they are available.

And that ends my presentation. I hope I've given you enough information to make you want to learn more about this. There's lots of resources available on the web. If you Google any of the names of these techniques you can come up with lots of papers and tutorials and so forth, or you can find a workshop somewhere and learn more about a particular one of these that might be interesting to you. Good luck. Thank you.