Growth Curve Episode 1: What Is Growth Curve Modeling?

hi welcome to office hours my name is Patrick Curran and along with my friend and colleague Dan Bauer we make up Curran Bowery analytics today I'd like to spend a few minutes exploring a question that Dan and I often get in our teaching in our workshops and that is what is growth curve modeling on the surface it seems like a simple enough question but it actually gets a bit trickier when you start to delve into it part of the problem is there are a lot of terms that are used to refer to the same set of underlying techniques so you may have heard of growth curve modeling or trajectory analysis or latent curve modeling or latent trajectory analysis all of these are different terms that tend to refer to the same underlying set of analytic approaches and that's what I'd like to talk briefly about today to begin it's helpful to think about traditional methods of analyzing repeated-measures data so for more than a century there have been two main lines of approach for analyzing typically two time point data but it can be expanded to more than two time points the first is raw change score analysis and in raw change score say that you have a pre and a post-test design you take the post-test you subtract the pretest for each individual and the difference between the post and the pre becomes the unit of analysis if you have two time points you can estimate this using the t-test if you have more than two time points you take the difference between 1 & 2 2 & 3 3 & 4 you can model this through repeated measures ANOVA or repeated measures manova framework serves an entire line of models that allow for the analysis of raw change in contrast there's what is called residual eyes change and here instead of taking the difference between the two time points we regress the later time point on the earlier time point and then the residual of that equation becomes the unit of analysis so we would do the say in a regression model we would use our just as a predictor the post-test as a criterion or outcome variable we'd regress the post-test on the pretest and then what's left over as the residual becomes the unit of analysis and we might bring in other predictors or trying to understand that in some other way those models too can be expanded to more than two time points and you often see these as an autoregressive or an autoregressive cross light design that's that can be done in regression or through path analysis using structural equation model the key to understand in these more traditional methods is they tend to be focused on a series of two time point comparisons so if you only have two time points its comparing pre and post-test if you have more than two time points it tends to break it down into what occurred between 1 & 2 2 & 3 3 & 4 and so now we're going to compare that to a growth curve model that takes a different approach it takes conceptually a different approach and analytically it takes a different approach where is in the two time point comparison models we tend to pull over individual went in time point right so if you think about it we can get a mean in the variance of the pretest a mean in the variance of the post-test that's pooling over everyone in our sample then we get the covariance between the two and our models are fitted to those data usually in contrast the growth model is going to flip that on its head is what we're going to do conceptually and in some cases analytically is instead of looking over individuals within time point we're going to look over time points within a given individual so let's use an example to think about this more closely so Dan and I have data and an example set up on the webpage will embed a link here with this video but it examines a set of repeated measures on aggressive behavior and a sample of children so their 405 kids so let's start and say that I'm the first kid all right I'll be the first kid in the sample and we'll look at age so we have six seven eight nine and ten all right so this is age on the x-axis and on the y-axis we have some measure of aggressive anger all right so say that you're the researcher you go into my house I'm six years old you sit down with my mom and you say all right let's think about little Patrick in the prior 90 days and please tell me how many times he endorsed or exhibit at each of these aggressive behaviors and so my mom fills it out and we get some measure that same Q all right then we wait here you go back into the whole mine now seven you asked them all again and maybe it's here and then maybe here is eight and here is nine and here is 10 so we have five repeated measures now the more traditional methods are going to look at those those paired time and Jason comparison so we might have so age six to seven maybe there's not very much difference seven to eight I've got this huge job eight to nine and actually goes down nine to ten is another job and we try to stitch our understanding together looking at these these pairs of six to seven seven days and so on there's a different way of thinking about it and this is core to understanding what the growth model is is notice even though there's jumping up and down that there tends to be an increase over time so we're going to teleport back to eighth grade when we learned about fitting a line and let's say we I'm not terribly good artist but we'll do this all right so let's say this represents the line of best fit right obviously it's not exact because some repeated measures are above somewhere below but notice that it's a data reduction technique we've smoothed over these time points and they can be summarized using just two pieces of information we have some starting point and let's just arbitrarily call that alpha to represent the N of that line and then we have some rate of change we'll just call that beta rise over run it's just a linear slope so what we have is a summary of my 5 repeated measures in a more parsimonious way because there are only two pieces of information now what's interesting about this is remember this is my intercept in my slope this is a summary of my set of damn the term we sometimes encounter with this is called intra-individual change that is what is the change that's happening within the intra-individual now for as much as I'm interested in me rarely do we want to look at one case right in this data that we have on online there are 405 cases so what we want to do is take this concept of intra-individual change and we want to expand it to every case that we have so now imagine that we do what what we just did except we fit a line for each individual observation in the sample okay so some kids are going very fast our flats are lower maybe we've got some kids a few kids going down all right we have an entire set of these trajectories or growth curves you can see where this is going right there individual growth curves now scientists what's the first thing we like to do if we have two or more of anything we want the average so we have a whole set of starting points a whole set rates of change over time we could estimate what is the mean of those all right so again just using some arbitrary notation is we can say mu alpha that's the mean of all the intercepts and then there's some slope we could say all right there's mu beta in growth curve modeling these are often referred to as fixed effects the reason is is they don't vary there's a single mu alpha a single mu beta to the mean of all the individuals in the sample now if the first thing we'd like to do is add everything up and in compute the mean the second thing is to evaluate is their individual variability around them so notice some kids are studying higher right if you're at age 6 and kind of entry in the elementary school some kids are starting higher some kids are lower all right so there's individual variability around the intercepts but there may also be individual variability in the slopes some kids are increasing more steeply some kids are increasing less steeply some kids may not be changing at all this individual variability we're going to capture in the variance the variance of the intercept and the variance of the slope in growth curve modeling we're going to refer to these as random effects they're random because they catch they capture the individual variability and starting point and rate of change and a very cool parameter that we can also estimate is the covariance between the intercept and the slope so one averages does where you start relate to where you go over time so we've got fixed effects we have random effects now here's the second term that we often encounter and growth curve modeling and that is inter-individual differences in intra-individual change so remember each trajectory captures within percent change I have a trajectory you have a trajectory we have 400 trajectories but there's between child differences in to individual differences some kids are higher some kids are lower well the first thing we like to do is get a me and second thing we like to do is get a variance then the third thing that all of us want to do is given individual variability we'd like to know can we bring predictors into the model to understand that that is what kind of child starts higher versus lower what kind of child increases more or less steeply these are sometimes called time invariant covariance what that means is the numerical value that the variable obtains doesn't vary across time so they're they're stable characteristics of the child or the home or the school or they're a construct that we only one assessor and so we want to say well on average are there differences between boys and girls on where they start and where they go over time that would be an example of a predictor a time invariant covariant but the models become really interesting when we expand these to include what are called time varying covariance right these kids are going through elementary school there are age six seven eight nine ten they're changing classrooms they're changing peer groups they may have a new brother or sister there may be a new divorce a parent may lose a job get a job there all of these time dependent influences that are poking and prodding the Candace they traverse through time we can build those in those are called time varying covariance another interesting expansion is we could look at a set of developmental trajectories in aggressive behavior and we could link that to simultaneous developmental trajectories and say reading ability and we could look at behavior problems and school success and how those two constructs travel together over time that's a multi variate growth curve model we can split it on group and we could look at the characteristics of development and the trajectories as a function of gender or some kind of point or constellation or parental diagnosis something like that so there are a whole variety of ways that we can expand these models to test different kinds of hypotheses that we may be interested in becoming full circle to the original question what is growth curve modeling is an initial response is growth curve modeling as a set of analytic techniques that allows us to take the set of information that we did observe on a child the repeated measures on aggressive behavior and based on that make an inference about the existence of something that we believe to exist but we didn't directly observe and that are these right we didn't observe these growth curves we observed the individual measures around them that's why sometimes they're called latent growth curves they're latent in the sense that we believe them to exist but we didn't directly observe them so that's what growth curve modeling neces were using the data that we have to estimate the existence of what we believe to underlie the behavior but that we did not directly observe now the next question is well how do you fit these kinds of models to your own data and I'm going to save that topic for a later video because we could do this case-by-case using ordinary least-squares but we can also fit a whole set of these models within the multi-level framework or within the structural equation modeling framework but I will talk about that in another setting so I hope this has been of some use and thank you for your time

Channel: CenterStat

Views: 29,846

Keywords: growth model, latent curve model, trajectory, change, growth curve, repeated measures, longitudinal

Id: 2hV7MyEX2UA

Length: 13min 52sec (832 seconds)

Published: Thu Mar 09 2017