 Statistika-mallista must be identified before we can meaningfully estimate it. However, establishing the identification status of a model can be challenging to do. The bulletproof way of establishing that your model is identified is to simply prove that you can solve the model parameters from known population covariances. Tässä on todella tykköinen tehdä ja on kuten on kokoisena, jos en ole tehnyt. For a Convertor Factor Analysis model, with let's say three factors, and seven indicators each, doing the proof might take a better part of a day, even if you know what you're doing. So that's not very feasible, particularly for more complicated models. In practice we use two different strategies for identification analysis. One is there are different heuristics and rules that you can apply. The simplest one is that you check if decrease of freedom is a non-negative number. It must be zero or positive. If it's negative then you know that the model is not identified for sure. A positive decrease of freedom on the other hand does not guarantee identification. So there are these rules and once you know the rules then you can develop this kind of like intuition on whether a model is not identified or not. So you start to recognize certain patterns in models that are associated with identification and certain other patterns that are associated with lack of identification. But then there is another set of techniques called these empirical checks. And there are two empirical checks that your software does or that you can check right from the software output. Warnings from software but that's not bulletproof and missing of extremely large standard errors. That indicates a non identification. Then there are also techniques that require a bit of programming or a bit of reanalyzing the data. And I'm going to address two of these empirical checks estimating the same model from two different starting values. The idea is that if a model is identified and if you get convergence then you should get the same result regardless of your starting values. If it's not identified then you might get different results. So try different starting values. If you get the same result then you have identification. If you don't then it's not identified for sure. Another technique is to take the model implied covariance matrix from a set of estimates and use that model implied covariance matrix as your new sample and then fit the model to that model implied matrix. If you get the same result then the model might be identified. If you get a different result then it's not identified for sure. And then the third strategy is simulating data from a representative model without any of the empirical data. But that's not challenging but it's kind of a different strategy. So I'll just show these two strategies because they are quick to implement using an empirical data set. So our example model will be this. So we have two factors F1 and F2 and there is a bidirectional regression path. So we have F1, F2, F2, F1. This is not identified because there's only one latent correlation and we can't estimate two directional paths from just one correlation. And our data comes from UCLA. So this is a data set, an example of how you can do exploratory factor analysis using converter factor analysis tools but we'll just be using it to estimate this converter factor analysis model. Let's start first with an identified model. So this is a latent regression model, structure regression model. We have F2 predicted by F1. Both are measured with three indicators and we'll try different starting values and re-estimating from the model implied covariance matrix. You can do different starting values by hand but you can also do it programmatically. I'll do it programmatically because it's kind of like more of my style and also then I don't have to think about what would be feasible starting values. So I'll just use random normal variables and this is the original data and then this is the identification check and this is the check for the estimated from the same populace. Okay, so these are our results and they look fine and they look the exact same regards of which starting values we use. So I'll just skip the output of that to make the presentation a bit shorter but let's take a look at how we estimate the same model using the model implied covariance matrix as indicators as the population or as a sample. So we do estimate framework in this data and that gives us the model matrices or it gives us this SCM output in matrix format and then from the return we get R and the return values we get sigma which is the model implied covariance matrix and then we take the first six rows and first six columns because this is first absolute values, absolute indicators then the related variables. We are only concerned about the absolute indicators here and then we use SSD to set up summary data and we use this covariance matrix as our summary data. We set sample size number of observations to 500. Using a large value here would be preferable but I'll just set it to 500 because that was my original sample size and I put it there just without thinking. So put a large sample instead of 500 and then we run the same SCM and then we compare the results. What we will see is this is the sigma there are the variance covariance matrix of the estimates we will rerun the same thing using the implied covariance matrix and we'll see that the results are identical. The only small difference is that these variance estimates differ a little tiny bit. The reason here relates to the differences between the sample covariance and the population covariance and how maximum likelihood estimates for variances are slightly biased because of the difference. That's a technicality. You can get the estimates to be identical to use a large sample size in the SSD command 10,000 or 100,000 It doesn't make a difference for computational speed because you're working with covariances but it just makes these estimates to be identical. So we can use the model implied covariance matrix from data estimates, re-estimate the model using the model implied covariance data and those results should be identical if the model is identified. Now let's take a look at this non-identified model so it's not identified because we can't estimate a bidirectional path from one covariance. Now we're going to work through the full set of diagnostics. This is the first model just with our original data. Interestingly, one of the checks for identification is warnings, we get none. These checks are not bulletproof. We do get large covariance large than at errors which indicates a potential identification issue with the model but it doesn't really tell us the nature of the identification issue. Let's go and proceed with the diagnostics. So we'll first do this loop here so I will estimate six more models so the first model is one and then models seven, two through seven, six more are using random starting values. I set seed in the beginning of the file to make this reproducible to ensure that I will always get the same results if I rerun this too far. And then I tabulate these models and the likelihoods. What we will see is that we get different sets of estimates for both recursion paths depending on the starting values. The likelihood of these models are the same it appears to be missing from my slides but you would see that it's actually identical. And these estimates are also highly correlated so when one goes up, another one goes down and that indicates there's a trade-off so they are correlated over repeated samples that means that there's an identification issue involving these two parameters. In other words, we know that these parameters together are responsible for the correlation with F1 and F2 but we don't know which one it is. That's the identification problem. All these measurement parameters, the factor loadings and indicator error variances are identical so there are no identification issues there. Another way that we can show the lack of identification of this model is to estimate the model from the model-implied covariances from the estimates. So we do this thing, we take the framework we take the sigma, the model-implied correlation matrix or covariances matrix of indicators we use status data using SSD we have a large sample size, larger than 500 would be better I just happened to use 500 in the example and then we re-estimate the model. Then we compare the two models side by side we can see that the models are similar but they are not the same so you should get the same result so these differ in the third decimal you should get exactly the same set of estimates because we don't that is an indication of model, non-identification. So these are two simple techniques that you can use so when you program it once in state or once in R then you can just reuse your state or R code from one project to the next and you can also do this with other SCM software but I use these two the most so this is the empirical checks different starting values re-estimating model for model-implied covariance matrix how does this relate to the bigger picture so these are a number of different ways that a model cannot convert number of issue reasons for that and these identification checks are useful for diagnosing model identification or empirical under identification but not really useful for diagnosing purely computational problems so for that for example inspecting starting values will be more useful