 Linear-late and growth model is one of the simplest models for modeling effects of time than you can use in SCM and estimate from wide data. Let's take a look at the late and growth model. Our example comes from status user manual, the SCM manual example 18 in status 15 and the model contains data or it's estimated from data from different cities and we have crime rates over time. The research question is how does the crime rate of these different cities evolve over time? We have repeated observations structured as wide from data, so each repeated observation is a different variable. What we are modeling here is the effect of intercept, which is constant. So the idea here is that intercept is always added, so intercept always has a coefficient of one. And then we have a slope random variable which has effect zero, one, two and three. So we add intercept once to every observation. Slope to the initial observation is added zero times. To the first observation, which is one year from the initial observation, we add slope once. The third observation gets the slope two times because it's two years from the initial observation. And the third observation gets the slope three times, so fourth observation at time three because it's three years from the initial observation. This same model can be, so how do we interpret this model? It's the effect of change here, the slope latent variable and this intercept latent variable is the initial difference. We can estimate the exact same model using long form data. So we can estimate the same model using mixed modeling. The idea of mixed modeling would be that instead of having each observation as a separate variable, we have the observations as different entries in the data set and we have a time index variable. So we would model year observations as function of initial difference and change over time. We can of course add predictors, so we can add predictors x variable that explain the initial differences, explain the different directions of change over time. We can do the same here for the latent growth model. Typically we are not simply interested in describing the different trajectories that cases take Instead we are interested in explaining what determines that some cases go up, others go down, what determines the initial differences. To do that we add predictors here that are observed and that explain the latent variables. These variables here would be level two variables, so they are variables that don't vary over time. So for example, where the city is located, that kind of thing. To understand why these latent growth model and mixed effect model of time are the same and how they are related to other models. Let's take a look at the equations. So we have multivariate perspective, latent growth or SCM approach. We have our data here comes from police employee hard who study the job satisfaction in the army and they have, one of the variables they have is age. So we are saying here that job satisfaction is time one is intercept plus an error term. Job satisfaction of time two is intercept plus one times the slope because we go one year from the initial period and for third observation is the same but we take the slope two times. We explain intercept and slope with age, so this is a latent variable model where the intercept and slope are latent variables. Age is observed variable, job satisfaction one, two and three are observed variables. We can take a look at the same in the mixed model format. So we say job satisfaction is intercept plus time times slope plus e where intercept and slope are latent variables. Intercept here is determined by age and this slope here is determined by age and they both have random effects. So what is there? If we write this out we can see that it's an interaction model. So what does this math tell us? It tells us that adding a predictor of the intercept is simply the same as adding the level two predictor directly as a predictor of all the observed variables. Adding a predictor of slope is basically the same thing as doing a cross-level interaction model. So if we have a mixed model of time as the main independent variable and we have a cross-level interaction that is equivalent, we get the exact same likelihoods if we have balanced data that we would get from the cross-locked model. So these models are basically the same. Of course we can also do more complex trends like in the mixed model we can have a u-shape effect. We can have the same u-shape effect in the latent growth model formulation as well. For example we would have the curvature effect here. That's the squared effect of time. So time zero, it's zero, time one, it's one, time two, it's four, time three, it's nine. So we have this effect that goes down and it compensates for the slope that goes up. Or it goes up steeper. So these are basically just two different ways of modeling the same data. Which one of these is applied the mixed form model using long form data or the latent chains model using the white form data depends mostly on the tradition of a particular discipline. For example in strategy management we use mixed models mostly, not these kind of models. In applied psychology these kind of models are more common. The latent growth model is a bit more complicated to specify. So if you have long time series like let's say 30 year observations then doing this kind of model is a bit tedious and perhaps we would go for the mixed model where we don't have to specify a model for each time observation. On the other hand this model gives us more flexibility. We can free some of these constraints. We can estimate for example a heteroscedasticity by figuring the errors and so on. And it also allows us to do some kind of model testing in some configurations. But it's mostly about matter of personal preference which one of these techniques applies.