 Se on common reason for convergence problems in maximal likelihood estimates. In any other statistical estimates that use numerical optimization. Let's take a look at an example and how printing out the starting values... ...might help us to troublesuit a problematic model. The idea of maximal likelihood estimation is that you are trying to find a population... ...that maximizes the likelihood of observations. Joten, jos meillä on esimerkiksi nämä, on kolme valoja 2, 3 ja 4, ja olemme haluat löydä normaaleita, joka vaikuttaa tällaiset kolme valoja, olemme tarkemmin uudestaan, että olemme haluat löydä normaaleita minua 0 ja standard deviation 1, olemme vähän yksinkertaisemme, sitten olemme vähän yksinkertaisemme, ja olemme nähneet, miten tällaiset osallistuvat muuttavat. Ja voimme nähdä, että jos olemme tullut tällaiset osallistuvat vähemmän, niin osallistuvat on tullut yksinkertaisemme, ja olemme enemmän hyvää, koska osallistuvat osallistuvat on enemmän, niin kuin se on yksinkertaisemme. Joten olemme tullut tullut vähemmän yksinkertaisemme, ja olemme nähneet, että se on yksi yksinkertaisemme, että 3 ja standard deviation on vähän yksinkertaisemme. Joten, mitä on yksi yksinkertaisemme? Yksi yksinkertaisemme nähdään, jos osallistuvat on täällä, ja se on 22, 23 ja 24 tämän kesken, ja olemme yksi yksinkertaisemme. Joten, jos olemme tullut tullut yksinkertaisemme, ja yksi yksinkertaisemme, niin yksi yksinkertaisemme on yksi yksinkertaisemme, ja yksi yksinkertaisemme on joten vähemmän yksinkertaisemme. Nämä asioiden osallistuvat ovat ymmärtäneet kai agrees, jotka ovat nähdä tullut yksinkertaisemme. Voi asioiden osallistuvat on yksi yksinkertaisemme, likelihood of these observations will be zero. Let's take a look with a practical example on how what does this actually look like on a computer and how do we solve the problem. I'm using this example from UCLA so they have these nine six-indicators, y1 to y6 and we're feeding an exploratory factor analysis using confirmatory factor analysis. So, they explain the procedure here, and I'm using Stata, so the model is here, so we have two factors, we have F1 that is loaded on which all the indicators except Y5 load on, and then we have F2 on which all the indicators except Y2 load on, so this is, they are called reference indicators or something like that, so this is how you do an EFA using the CFA. And we run that, we have, all the variables are standardized, so the variances of all the indicators are ones, and we have the factor variances set to ones, so this is comparable with what exploratory factor analysis would do. And we know that in an exploratory factor analysis, if it's unrotated, the factor loadings will be between plus one and minus one, and in a rotated solution like this, they are roughly in that ballpark, they can exceed plus one, they can be smaller than minus one, but typically they tend to be between the one and minus one. So, we run it, convergence not achieved, so Stata tells us that this does not work, and how do we then know that we're to look at, well one symptom of non-identification is, or a symptom of a computational problem related to specific parameter is missing standard error. We can see that the variance of the indicator 6 is missing, or its standard error of the variance is missing. The reason here is that Stata does not convert to Haywood cases, you can get it to convert to Haywood cases with Haywood options, but it wouldn't help here. Another thing that we notice is that the standard errors for the factor loadings of this indicator Y6 are missing, and the factor loadings are also pretty large, and this variance is very close to zero. How do we then know what is the problem? Well, we can start by printing out the starting values, and we do that using the no estimate options in Stata, and then Stata prints us the starting values. So, these are the initial guesses that Stata tried, we can see already that there are a couple of problems here. First the variance, the error variance of Y6 is about 13, and that shouldn't be possible, because the variance of Y6 is 1, because the data are standardized, and error variance cannot exceed the variance of the indicator. Another thing is that we can see that these loadings of the indicator are plus 13 and minus 13, so that looks very weird, given that the loadings are normally in the plus 1 minus 1 board part. So, how do we fix this problem? First of all, we need to document the problem into our RDO file, and this applies to any other statistical software. So, when you do some changes to your model based on diagnostics, always summarize the diagnostics, and this is the way I like to do it. So, I'll just copy paste the Stata output relevant lines, and then I say that, okay, so these lines cost me to rethink the starting values in this case. Then we set some reasonable estimates, so the factor loadings are, we know based on theory that indicator Y6 should not load highly on the first factor, so we set the factor loadings in the RDO. We know based on theory that it load reasonably on the second factor, so we set the factor loadings to 0.1, and then we don't actually need to set the error variance because this is sufficient to get the model to work. So, whenever you have a model that doesn't work, printing out the starting values can be a useful thing to do. And if you see some extreme or really unreasonable starting values, then give some more reasonable estimates based on prior empirical research, based on your own intuition, based on theory. But just, it doesn't really matter where those numbers come from as long as they are more reasonable than the ones that the statistical software tries to use. If the model is identified and it converges, then the actual final estimates should not depend on the starting values. So, adjusting the starting values can be done pretty freely, and it is only to make the model to converge, it will not affect actual estimates after convergence. Another thing that we can do is to estimate another model that works and use those other model estimates as starting values. So, what I'm doing here is estimating another model where I use indicator Y6 as the reference indicator for the first factor, and then I estimate that model. So, instead of using indicator 5 as the reference factor, I use indicator 6, which was the problematic as reference indicator. That works, and then I use estimates from this model as starting values for the other model, which was problematic. That also works. When I use this as an assignment for the students on my course, I tell them to use an exploratory factor analysis and then take the values from that exploratory factor analysis, use those as starting values. It works as well. But this is a bit more elegant. So, running the EFA, you would have to type in the values or use some more exotic status syntax to get the estimates directly from the EFA results without typing them. Now, let's take a look at the bigger picture. Here is the list of all different ways that I can think of that an estimation technique might fail, and here are some of the solutions. If you take a look at the different problems that I've identified, looking at starting values, fixing starting values can be used to address all kinds of optimization, calculation, comparison, implementation, and iteration issues. Starting values, just inspecting the starting values, will not address model identification issues. Some of the empirical checks for identification use starting values as a tool, but simply printing out the starting values will not help you identifying these other problems. But because starting values can identify many or be a cause of these four classes of problems, it's generally a good practice to always include printing out the starting values early on in your workflow for dealing with convergence problems. . . . . . . . . . . . . .