 Bi-factor models are very useful when you have scale dimensionality. In other words, in scenarios where a single dimensional one factor model fails to fit your scale. Another commonly applied technique in this context is the hierarchical factor model, which is a special case of the bi-factor model. Let's take a look at what bi-factor model is. So here's a figure of a bi-factor model or path diagram, and the idea of a bi-factor model is that you have one main factor here called the general factor or g-factor, and then you have one or more minor factors on which the indicators load. So it's called a bi-factor model because each indicator loads on two factors, and these factors are constrained to be uncorrelated. Almost always if they are modeled to be correlated, then there is a very high chance that the model is actually not identified. I'll talk about identification of these models in a different video. So why would this kind of models be useful? Quite often when you develop a scale, particularly if you have many items like 30 here, the scale is not exactly unidimensional. And sometimes the lack of unidimationality is a key feature of your study, so you want to study different dimensions of a construct, or you want to have multiple different facets, including the scale. Sometimes you realize afterwards that you have dimensionality that you need to model. So what do these enable you to do? Bi-factor models allows you to assess misfit due to scale dimensionality. So if your chi-square test fails the model, then quite often that happens because you actually have two different dimensions in a scale, and that kind of scenarios you can use the bi-factor model for. There are two different ways you can apply these or interpret results. One is that you eliminate the dimensionality. Quite often we are interested in the general factor, and then these minor factors are just like measurement artifacts. You have some similar awardings in two different parts of the scale, and they correlate, and you want to eliminate that dimensionality from the analysis. In some other scenarios, these could be of theoretical interests. For example, if you have a scale about innovations or innovativeness, you might want to differentiate between product innovation, process innovation, and for example, service innovation or whatever dimensions your innovation construct has. So bi-factor models can be used for eliminating scale dimensionality if you are interested in the general factor or modeling different sub-dimensions of constructs. These bi-factor models can be also understood through the context of a model with correlated factors. So here we have F1 and F2, and this is an equivalent bi-factor model. So we have M1, M2, and G. And here the G factor, the general factor, presents whatever F1 and F2 factors have in common, and M1 is whatever, F1 has unique, and M2 is whatever, F2 has unique from F1. This can be understood also by looking at a Venn diagram. So we have the overlapping path part here. This is the correlation between F1 and F2, how much the factors overlap, and it is quantified by the G factor, the general factor. And then we have this area of F that does not overlap with F2. That is the first minor factor. And then this area of F2 that is unique to F2 does not overlap with F1 is what our minor factor 2 presents in the model. So you can take two correlated factors and basically split the factors in the three. So you have unique part two of one factor, unique part of another factor, and then the common part of those factors. Bi-factor models can also be applied as an alternative to correlated errors. So quite often when you have a model that does not fit the data, at least in published research, then people follow modification indices or they take a look at residuals and then they free these correlations between the error terms to make the model fit better. This is a controversial practice. There are articles that strongly argue that this should never be applied post hoc. Some of the reasons why having a correlated error there is that or not having it, not adding it is that you are going to be capitalizing on chance quite easily. And another thing is that if you just add a correlated error, then you are not basically addressing what is the reason why the two error terms correlate just allow them to do so. Bi-factor model can be applied as an alternative to freeing a correlation. So the idea of a bi-factor as a replacement of this correlated error is that you specify a minor factor which has two loadings and the loadings are constrained to be equal for identification on this x1 and x2. Actually the error should go to the indicators x1 and x2, not the error terms. And this is our way of bi-factor modeling the same thing. So what is the advantage of this bi-factor model over the correlated errors model? Well, there are a couple of advantages. One is that bi-factor model forces an interpretation. So whereas you can just simply free a correlation and not thinking about how to label it, whenever you add a factor to the model, you at least need to give it a name. And when you name a factor, you have to think what does the factor represent. And the factor here f is an omitted cause of x1 and x2. That is something that f1 does not explain. So it's kind of like a secondary dimension in the scale. And by having it here as an explicit part of the model forces you to think why x1 and x2 are correlated beyond what the factor f1 explains. The other advantage of the bi-factor approach is that it makes your assumptions more explicit. So whenever we add a new factor here, we can, for example, check the modification indices on whether this m1 factor should actually be having an effect on, for example, x3 or x4. If we just free a correlation, then we don't get any new diagnostic information. So this bi-factor approach over the correlated errors model is superior because it forces you to think about the problem more and also it provides you more diagnostic information. If you want to learn more about bi-factor models, this article by Chen is perhaps the best source. And this is their example model. They have quality of life, that's a general factor, and then they have four minor factors, cognition, vitality, mental health, and this is worry. I assume that this is applied more to elder people, but it's very commonly applied in this domain. In this paper, we learned that this is actually a special case or a more general case of another model. So we could also model the scale dimensionality using a hierarchical factor model. Whereas in bi-factor, you model dimensionality in the items, in the hierarchical factor model, you assume that these four dimensions are sufficient to explain the items, but they depend on a higher level factor. So if your quality of life is high, then it should be manifested in these lower level factors. So this is a hierarchical factor model, it's like a factor model of factors. And in fact, fitting a hierarchical factor model is pretty much equivalent, if it's just identified to running first factor analysis model to get correlations of these first order factors and then running a factor analysis on that factor correlation matrix to get the second order factor loadings. And so this is just a factor model of factors. Sometimes these are called first order factors and the general factor is called a second order factor. And this is a special case of bi-factor model. To understand why that is the case, let's take a look at an example. So here's an example of a model that is equivalent to a bi-factor model. So if we add loadings from the general factor or the second order factor to the indicators except the first indicator of each scale, then this is actually equivalent to a bi-factor. The first indicator loading here on the quality of life construct or variable must be fixed to zero for identification. If we have a path from quality of life to the first indicator, then we wouldn't know whether it's this quality of life or the cognition factor that actually explains these correlations. So for identification purposes, this loading must be fixed to zero. It is pretty much the same case as you have when you are estimating an exploratory factor model within a converter factor analysis framework. You always need to fix the factor rotation in a way to constrain one of the loadings of each factor or one of the loadings of each factor to be zero to identify how the factors are rotated. So let's take a look at why these are equivalent and what does the equivalence actually mean. So here are the two models side by side and to understand why these are equivalent, we need to focus on this error term. So whenever we regress cognition on quality of life, this cognition has a unique part that is not correlated with quality of life and that unique part is also not correlated with any other of these first order factors and it happens to be that this error term here is actually equivalent or the same as this minor factor of communism here. And this equivalence is nicely shown in this article and the article actually shows an empirical example where these loadings are going to be the same and also these loadings on this higher order factor simply are differences between the loadings of the by factor model here and these main factor loadings here. So let's better understood by looking at the table that the paper presents. So this is the two models, then estimates and we have the by factor model and then we have the second order model which is equivalent. We can see here the identification, the first factor loading must be fixed to zero for the second order factor. This is also the scaling indicator of the first order factor. So you have this zero one pattern here and we can see the equivalence of the factor loadings. The first order cognizant factor is the same as the factor loading for the cognizant minor factor in the by factor model. So these are equivalent. And we can also see that the second order factor here simply shows what is the difference between the general factor and the by factor here and we can do the calculation. So that's the difference and this is the relationship between the by factor model and the general second order factor model where the indicators are allowed to be loading on the second order factor as well. So let's take a look at the broader context. So where are by factor models applied? They may not be called by factor models but these are actually applied quite commonly in management research for a particular purpose. And the particular purpose is for modeling method variance. So method variance models typically we have like one general factor called method factor affects all the indicators and then we have the factors that we're interested in and those are minor factors in the by factor model they are the main factors in the by factor model. That model is a bit problematic because in method variance analysis we allow these minor factors to be correlated and that leads to problems that I'll discuss in another video. By factor models are also applied in the form of a hierarchical factor analysis model. So the first order, second order factor analysis model is quite common. It's more common than a by factor model but they are basically the same thing. This is just a bit more relaxed. See that does not have as many constraints as they are the hierarchical factor model. These models by factor or hierarchical factor because they are nested can be tested with a chi-square test or nested model test and then we can use that nested model test to check if the second order model is appropriate for the data. This is also the foundation for exploratory structural ecosystem modeling. The idea of exploratory structural ecosystem modeling is that you have a confirmatory factor analysis model and then you specify a structural ecosystem model between the factors and at the same time you fit an exploratory factor analysis model to the error terms of that model. So you basically fit a confirmatory factor analysis model and an exploratory factor analysis model at the same time and that allows you to model the main factors and also model scale dimensionality that is kind of like the minor factors without really knowing in advance what kind of specification you want to have for those minor factors. This basically automates some of the data exploration that you may do when you do diagnostics of confirmatory factor analysis models. So let's summarize. Bifactor model and hierarchical factor models they are close cousins. Bifactor model has main factors and minor factors. In hierarchical model you have first order factors and second order factors. The first order factors are or their error terms are equivalent to the minor factors in the bifactor model. Bifactor model is more focused on scale dimensionality. So you are interested in the dimensionality on the level of the scale items. In the hierarchical factor model you are interested in the dimensionality of a construct. So if you say that your construct has four main dimensions that depend on one main dimension then you want to measure four of those dimensions separately and then model the main factor as a common cause. Here dimensionality in bifactor model typically a nuisance and in here dimensionality typically of theoretical interest. Bifactor model is also the foundation for hierarchical reliability coefficients and it's a useful alternative for freeing the correlations between error terms in a confronted factual analysis.