Growth Curve Episode 4: A Structural Equation Modeling Framework

hi welcome to office hours my name is Patrick Curran and along with Dan Bauer we make up current our analytics in Prior episodes of office hours I've talked about what is growth curve modeling and how do you estimate earth curtain wall' are using a multi-level modeling framework in this episode I'd like to spend a few minutes talking about how do you estimate a growth curve model within a structural equation modeling framework now also in another episode of office hours Dan has talked about what are the advantages of structural equation modeling and where might we want to use those and so very briefly he talked about how a structural equation modeling framework is an extremely powerful and flexible multivariate methodology that subsumes a whole lot of models that were already comfortable with so within a structural equation model we can do a t-test we can do an ANOVA multiple regression but we can expand those and look at tests of complex mediation and multiple group analysis and a whole variety of other things one of the truly unique aspects of the structural equation model is the ability to estimate latent variables and then structure some latent variables with other latent variables or with other measured variables in our data set and this is going to be the key to estimating the growth curve all are using an SEM framework so just to orient let's pretend that we're trying to assess depression so depressive symptomatology and say that we have five items then assess depression and so they might ask an individual in the prior seven days how often did you feel sad and lonely how often did you feel that you couldn't drive fun from activities that you enjoyed before our real salient item is how often have you lonely even in the presence of others and so we get these five lighters all right these are sometimes called indicators on the latent variable so we have Y one two three four and five now all of these five items we believe to arise from some underlying latent propensity to experience depressive symptomatology now one thing we could do is we can add up the five items take the mean and use that as a manifest score but as Dan talked about in a prior video there are a lot of reasons why we wouldn't want to do that it assumes all the items are equally related to the underlying construct and it assumes there's no measurement error but what we can do is say alright the reason that I believe these items exist right they have variances means and they have covariances with the other items the reason that I believe those inner relations to exist is because they share an underlying cause and what that underlying cause is here is a latent depression we didn't directly observe it we believed it to exist but we can't directly observe it and we're going to use the information that we did observe about the symptoms of depression means that we assessed in our scale to infer the existence of this latent depression and so as Dan talked about we can have a variance of that we can have a mean of that so we can call that sy and mu these are called factor loadings they're denoted by lambda these are called residual variances we can call those Sigma squared each item as a residual variance and the point of this is to take information about what we did observe and make an inference about something that we believed to exist but did not directly observe and this is one of the unique strengths of a structural equation model this has drawn out is what we might call a confirmatory factor analysis model we could expand it in any number of ways why I start with this is this is going to be the cornerstone to how we build the growth model so thank you for a moment about what I started with is we believe there to be some underlying propensity to experience depressive feelings and symptoms but we can't directly observe it and we want to infer its existence based on what we did observe well that is exactly what we're trying to do with the growth model if you happen to watch a prior episode where I describe what is a growth model is we can concern sorry my panel is dying st. we have five repeated observations on depression again alright but now here is age all right so say that we have age whatever six seven eight nine ten and we have depression and we observed these five repeated observations we want to fit an individual growth trajectory that may have some kind of starting point let's just call that B and not sub I and some rate of change we could call say beta one sub I that one's done to and notice these dots are what we observed on our individual but that's not what we're interested in what we're interested in is using those repeated observations to identify the underlying latent trajectory that we believe to exist but we didn't directly observe and you can see the direct link to the confirmatory factor analysis model we want to take the information about what we did observe to infer the existence of what we believe to be there but that we have to infer from the data okay so how do we do that well let's go back now and think about those multiple indicators on a latent factor but now instead of five individual items on depression let's say that we're going to examine five repeated measures on depression alright so we have again five boxes the symmetry here is tremendous what we're going to cover so here we have now let's just say it's going to be age six so we'll call it y 6 Y 7 8 9 and 10 so we observe depressive behavior at age 6 7 8 9 and 10 and with some sample of say children that we're studying and what we want to do is use this information to infer the existence of the underlying trajectory well for each component of the trajectory that is the intercept the linear component curvilinear component whole variety of things we can do the general rule is that for each component we're going to estimate a linking factor that's going to capture that part of the trajectory so for example let's say that we want to build a linear trajectory alright so over here we have a dark line that's going to be the fixed effect and go back and watch the broader video on what is growth modeling for more detail on this but we could refer to this as same.you alpha and we can refer to the fixed effect of the slope is mutated these are the mean intercept and the mean slope of pooling over all the individuals in the sample and then we have all the individual trajectories and the random effects were the deviations of each individual from those means and we captured those random effects in the variances of them so we want to build this model well first we have to start with the intercept we have to begin the process all right now also in another episode of office hours and you can go back and look at this if you haven't seen it as well is I talked about rescaling time and where we can put the zero point so let's say that we have time a 0 1 2 3 & 4 which is going to be age minus 6 in this example that sixes are youngest age 6 minus 6 is 0 7 minus 6 is 1 and so on so what we're going to do to define this given that time is going to start at 0 is we're going to estimate a latent variable that has all 6 repeated measures as indicators excuse me 5 repeated measures notice the exact symmetry so far of with the confirmatory factor analysis Mellish are just a couple of minutes ago but here's where the first big difference is moment estimated as a growth model instead of estimating the factor loadings we're going to fix these two values of 11.0 each of these 5 factor loadings is fixed to one we're going to estimate a variance of the weighting factor let's call that sigh alpha and we're going to estimate a mean of the latent factor let's call that mu alpha and let's just call this factor edan at 5 so 8 is the generic notation for latent variable and alpha were used to denote the intercept okay now each of these items will have a residual variance so I'll use the little double-headed arrow to denote residual variance and that's Sigma squared and maybe these Sigma squared values are equal over time maybe they're not that's a testable hypothesis that we could look at lighter the thing we have here though is there are not means estimated in the individual items so these are sometimes called item intercepts we're going to fix all of those item intercepts to 0 and only estimate the mean of this light verb all right so we're going to try to reproduce the means and the variances and the covariances among the repeated measures only through the characteristics of the mean and the variance of the latent factor and the residual variance all right so the base sets of this start of the trajectory so mu alpha is going to be the mean of a latent factor and psy alpha is be the variability in the starting point but now we have to introduce a rate of change if we only have this it would imply all the lines were flat there would be very Movil some would be higher some would be lower but they're all flat but here we want to introduce an increase over time alright or a decrease depending upon the characteristics of the data and so to do that we'll simply estimate a second light inverted we'll call this a tas of beta all right and this is gonna go to Y seven eight nine and ten we won't draw it to Y six because the time coding for the first assessment is zero so there's no influence on the first period but look at what we're going to do this will be one two three and four and there you see one two three and four all right so that's giving that rise over run it's just an increment per unit change as time progresses now this also has a variance this is going to be sine beta it also has a mean it's going to be mu beta and it has a very interesting parameter which is the covariance between these two sine alpha beta and that is gives us a assessment of the linear relation between where you start and where you go over time so for example in this how I've drawn this is on average those who start higher tend to increase more steeply and that's going to be captured in that parameter so this SEM it has all the logins fixed to one these loadings fixed to one two three four only the residual variances are estimate in the repeated measures and then each latent variable has a mean and a variance and then a covariance between the to define this growth model we get an average starting point and mu alpha we get an average rate of change in mu beta we get individual variability and starting point that's captured inside betas the degree of variability and where kids start and we get individual variability and rate of change so how different are the slopes across individual and then this coder is now in a prior episode and I talked about how to estimate this model within the multi-level and this is identical to what I call the unconditional model in the multi level we have a level one equation that have age as a predictor and we have level two equations that were defined by means the fixed effects and then the random effects we call them use and that's exactly what this is the big difference between the two one of many but and a significant one is whereas in the multi-level model age is a numerical predictor in the data set it literally is a column of numerical values in the data file here our data file only consists of the repeated measures and we're incorporating the passage of time through how we fix the factor loadings and in another episode as I as I said in an earlier episode I talked about you could have so Europe the starting point you could have see you wrote the middle point you could have zero at the end point and each of those would define a different intercept all we would do is change how we code these factor learnings so if you wanted zero in the middle point you would have negative 2 negative 1 0 1 2 and you would then just simply have negative 2 negative 1 0 1 2 on the slope factor and then that wouldn't move the intercept so you can take a look at that if you're interested in the coding of time now this is called an unconditional growth model because we have no predictors and often will use this modeling framework to try to identify the optimal functional form that's needed is an intersect appropriate is the model improved when we add a linear slope is it improved further if we had a quadratic that puts in a curvilinear we might want to estimate a piecewise linear where we tie two linear pieces together an interesting model that we can do here is we don't have to fix all the loadings - predefined values we can estimate a subset of them to estimate some nonlinear trajectory that doesn't necessarily follow a parametric for a quadratic or cubic something like that but there are two particular ways that we would like to expand on this model one is to incorporate time invariant covariance those are characteristics of the individual that don't change with time and we might want to expand this to time varying covariance so those are covariant values that can change with the passage of time and so the both of those are very easy to do in the SEM imagine that we have a linear model and now we want to try to understand individual variability in the starting point and change over time and we have some set of covariance say we have x1 I'll just draw x2 + but we can have any number of them maybe it's gender or maybe it's uh alcoholism diagnosis and a parent maybe it's are they an only child or not some characteristic of the child is if they're time invariant covariance so we just have one assessment of them we can just use them to predict the growth factors directly and what this is going to do is allow us to test for a shift in the means of these latent factors as a function of the covariance so for example let's say just as a simple example we have gender and we wanted to know are there meaningful differences in the characteristics of developmental trajectories of depression for boys versus girls well we could regress a to alpha and a to beta on a single binary indicator and what we might find is that there's a unique trajectory for girls now let's make that boys to follow theory sorry about that so we have a unique trajectory for boys and we have a unique trajectory for so this is more consistent with developmental psychopathology especially if the transition to puberty is on average girls tend to accelerate and depressive symptomatology is now boys and girls have their own intercept boys and girls have their own slope and those are derived from these this influence it would be though the predictor of the intercept would separate the two trajectories at the starting point the predictor the predictor of the slope would push the slopes apart so that then we would have one for each and this can be expanded in a whole variety of ways the other way that we can expand this is to bring in what are called time varying covariance so those are predictors in the model that would directly influence the repeated-measures instead of dictating the trajectories themselves so just as one example we might have a time varying calvaria that was say parental divorce was in that 12 months did the child's parents get divorced and that time varying covariant I know this path diagram is getting messy but there are ways we can draw these that are much better is it's looking at at any given time was there a bump to the child's depression above and beyond the trajectory process that could be associated with the parents getting a divorce and so it kind of pokes the individual times specific value and we can expand this and a whole bunch of interesting ways so we can have multiple time enduring covariance we can have multiple time varying covariance we can allow them to interact with one another there are a lot of really interesting things that we could do and finally one last extension that is a particularly powerful characteristic of the structural equation modeling approach to growth modeling is what if you have growth in two constructs at one time so let's say I won't call all the repeated measures I'll focus just on the latent growth factors imagine that we have a de alpha for depression and a de beta for depression so this is just intercept and slope for depression but let's also say that we were interested in a growth trajectory for a de alpha for aggression and a de beta for aggression all right so we have some growth model for aggressive behavior we have some growth model for depressed behavior and we would like to know to what extent are those constructs related over time and we can even test a regression of this initial levels of depression predict changes in aggressive behavior does initial levels of aggressive predict changes in in depressive behavior we can then add covariance here that predict with the growth factors we can have time varying covariance and then there are a whole series of increasingly complicated models where we can simultaneously look at how to to come structure later over time at the level of latent factors but then add on to that also times specific kind of auto regressive structures and there are a variety of ways that we can do that so in summary that is a quick overview of how we can estimate the growth curve model within a structural equation modeling framework it's beyond the scope of what we're talking about here but there are a whole class of growth models that are numerically identical if you do them in a multi-level or a structural equation modeling framework it's arbitrary how you do it they're numerically indistinguishable but then each approach also has unique elements where it expands in a direction that the other one is not as well suited to do and so there are many additional topics we could explore with that so I hope you have found this of some use and thank you for your time

Channel: CenterStat

Views: 9,229

Keywords: SEM, structural equation model, LCM, latent curve, trajectory, growth model, longitudinal

Id: GtYPymUV0K0

Length: 21min 33sec (1293 seconds)

Published: Fri Mar 31 2017