 Seuraavaksi statistiikan analysointi, jossa olisinkin pitäisi tehdä yhdessä diagnostiikkoja, jossa olisinkin pitäisi tuntua. Konfrontaarifaktio analysointi, jossa on tärkeintä diagnostiikan informaation, on kai-squarestatistikkoa. Kun on kai-squarestatistikkoa, joka on tärkeintä, se tarkoittaa, että modelissa ei yhdistetty yhdessä yhdessä yhdessä yhdessä. Se tarkoittaa, että modelissa ei yhdessä yhdessä yhdistetty olemaan. Nämä ovat ei kai-squarestatistikkoa, mutta on siirrytä, että mietit Keskara korrelationen ovat yhdistetty yhdessä. Kai-squareta entää, että korrelationen ei yhdessä yhdessä yhdessä, jossa se puhuu modelissaan. Yksi hyvin yksi, että kaistakuvastokseniä ei ole tai saa yksityisestä. Joten voisi koko ajan koko ajan koko ajan koko ajan. Se on tärkeä, että yksi haluaa yksityisestä diagnostiikkaan. Yksi huomio on koko ajan diagnostiikkaan yksityisestä koko ajan yksityisestä yksityisestä. Yksi yksityisestä yksityisestä on, että ei ole eri hyvätiisiä, mitä on korrektu. The first approach is modificators and indices. I said earlier that your software could indicate that if you add a correlation between two error terms, then that will improve the fit of the model. It will make the chi-square smaller and we hope non-significant. The idea of modification indices is that the computer calculates things that you could add to your model to make it better. That should not be done mindlessly. Meskuita, Lasarina gives a good example of how to report these modification indices. First of all, they report what's the purpose of these indices. So the purpose of these indices is that you can make the model reproduce the correlation matrix better by adding something to the model. Then you explain what you do. So they add some stuff and they add some other stuff. Is that justified? Well, every time when you do a change to your model, it has to be justified based on your theory. For example, if we have these six indicators and we have a modification index that indicates that these error terms should be correlated, then we have to explain what the correlation means. For example, if we have indicators about innovativeness, indicators about productivity, we could say that this indicator also measures something about personnel and this measures something about personnel as well. So these indicators have this personnel dimension and therefore we say that their error should be correlated. The first structural ecosystem model course that I took, the instructor totally little told us that when you see a modification index, then unless it gives you this kind of aha moment, then you shouldn't add anything to your model. So the modification index is only something that tells you that this is a part that you should consider. Then it's up to you to decide whether it makes sense. The idea of factor analysis model is not to reproduce the data perfectly. The idea is to have a theoretical representation of the process that could have caused your data. And it's also possible that factor analysis simply says that no, your data don't measure the things that you say they do measure. And that's a result. So every modification must be done based on theory. Another way of doing this is looking at the residuals. So we have the residual correlations, which is the difference with the implied matrix and the observed correlation matrix or covariance matrix. And here are the residuals for the full model. So there are two things that we need to check. First is the overall distribution of these residuals. Turns out that if the model is correctly specified, these residual correlations are normally distributed with the mean at zero. And we can see here that we have this bump here on the right hand side on the tail. So that indicates misspecification. And this tail also indicates because there's a bump, it indicates that there's local misspecification. So there is some part of the model that is incorrectly specified. It's mostly okay. So most of these correlations are close to zero. But there are some parts, this bump here, big bump and smaller bump, then indicate that there are parts where the model doesn't reproduce the data. Then it's up to us to look at the residuals and see where are the high values. So we can see here that one block of items here, the vertical governance, horizontal governance indicators correlate much more than what the model implies. Then we have to look at the model and then think, okay, so we have an implied correlation of let's say zero. So why is it zero in the implied correlation matrix? That relates back to the tracing rules. So what in the model predicts the correlation? In this case, I constrain these two factors to be uncorrelated. And that caused these errors, the residuals to go up. And it indicates the model is misspecified because they are horizontal and vertical are actually quite highly correlated. Another thing is that we can see that these high values also are a single indicator factor. I constrain that to be uncorrelated with other factors as well. So that way you can look at the residuals, look at which correlation the model doesn't explain well. And then you think, okay, so why, what influence is that correlation in your model, is that part of your model correct? This requires a bit more expertise than just doing the modification indices. But the problem with the modification indices is that sometimes the modification indices don't make any sense at all. And it's easier to do nonsensical decisions using the modification indices than it's using the residuals. So my, the way I do diagnostics is that I usually quickly check the modification indices if my model doesn't fit well and then I print out the residuals. Also it may make sense to print out a part of these residuals. So after this is a big matrix, so going through it one by one is difficult. But once you have identified a segment of the matrix where you have large values, then you could fit a sub-submodel. So for example, we could only fit the model with horizontal governors, vertical governors, and then maybe one other factor. So the way to do diagnostics is that if a full model doesn't work, then you start doing sub models. So can you get the smaller model work, drop something from the model, and then if it works, then you know that something that you dropped from the model was the reason why it didn't work. Then you can look at the part that you dropped or split the model into two and then do diagnostics for the first part. Once you're happy with that, then do it for the second part. Once you're happy with that, then do that for the full model. It's a good idea, like a good engineering principle is that once you have a big system and it doesn't work, start looking at individual parts and then figure out which of those parts don't work and whether they can be fixed. And only after you've verified all the parts, then you look at the whole because looking at the big correlation matrix is very difficult to do.