Bifactor models and hierarchical factor models

by Factor models are very useful when you have scale dimensionality in other words in scenarios where a single dimensional 1 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 buy factor model let's take a look at what buy factor model is so here's the figure of a buy factor model or path diagram and the idea of a buy 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 stay indicators load so it's called buy 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 xxx here the scale is not exactly unanimous now and sometimes the lack of unity nationality is a key feature of your study so you want to study different dimensions of a construct or you want to have our multiple different facets including the scale sometimes you realize afterwards that you have dimensionality that you need tomorrow so what do these are enable you to do by factor models allows you to assess misfit do the skill dimensionality so if your scale if your chi-square test fails the model then quite often that happens because you're you actually have two different dimensions in the scale and that kind of scenarios you can use the boy 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 zero factor and then these minor factors are just like measurement artifacts here are some similar worries in two different parts of the scale and they correlate and you want to eliminate that dimensionality from there now 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 innovations and process innovation and for example service innovation or whatever dimensions your innovation constructors so by factor models can be used for eliminating scale dimensionality if you're interested in the general factor or modeling different sub dimensions of constructs these by factor models can be also understood through the context of a model with correlated factors so here we have F 1 and F 2 and this is an equivalent by factor model so we have M 1 M 2 and G and here the G factor a general factor presents whatever F 1 and F 2 factors have in common and M 1 is whatever F 1 has unique and M 2 is whatever F 2 has unique from F 1 this can be understood also by our looking at a Venn diagram so we have the overlapping path hit 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 on 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 have two doesn't overlap with f1 is what what there are manufacturer two presents in the model so you can take two correlative factors and basically are split the factors in the three so you have unique party with one factor unique part of another factor and then the common part of those factors by factor models can also be applied as an alternative to correlated there's 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 are correlations within the error terms to make the model fit better this is a controversial practice so there are articles that strongly argue this should never be applied post-hoc one of their are some of the reasons why having a correlated or there is that all or not having it not adding it is that all you're going to be capitalizing on trance quite easily and another thing is that if you just add a correlated error then you're not basically addressing what is the reason why the two error terms Coralie you just allow them to do so a by factor model can be applied as an alternative to freeing a correlation so the idea of a by 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 x2 you know the error terms and this is our way of by factor modelling the same thing so what is the advantage of this by factor model over they correlated there's model well there is a couple of advantages one is that by factor more forces in interpretation so whereas you can just simply three a correlation are not thinking about 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 that Factory present and the factor here F is an omitted course of x1 and x2 that is something that F 1 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 tomorrow forces you to think why x1 and x2 are correlated beyond what the factor f1 explains another advantage of the buy factor approach is that it makes your assumptions more explicit so whenever we add a new factor here we can for example or check the modification indices all 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 these are by facture approach to the over the correlate that are smaller 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 by factor models this article by chain 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 um it's very commonly applied in in this silent 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 Marlys the skill dimensionality using a hierarchical factor model so whereas in by factor you you model dimensionality in the items in the hierarchy of the 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 are lower level factors so this is a hard core factor model is like a factor model of factors and in fact feeding a hierarchy like a factor model is pretty much equivalent ability if it's just identified to our running first the 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 down a factor model 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 buy factor model to understand why that is the case let's take a look at an example so here's an example of on an equivalent immoral is equivalent to buy factor model so if we add loadings from the general factor or the hot second-order fracture to the indicators except the first indicator of each scale then this is actually our equivalent to my 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 if whether it's this quality of life or or the cognisant factor that actually explains these correlations so for identification purposes this arm loading must be fixed to zero it is a pretty much the same case as you have when you are estimating an exploratory factor model within a conversion factor analysis framework you always need to fix the factor rotation in a way to constraint one of the loadings of each factor or each one of the loadings of each factor to be 0 to identify how the factors are rotated so let's take a look at why these are equivalent and and what does the e 1 is actually mean so here are the two motors side-by-side and to understand why these are equivalent we need to are let's focus on on this error term so whenever we regress our cognition and 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 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 this minor factor 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 on this higher-order factor simply are differences between the load loadings of the by factor model here and on this on the main factor loadings here so let's that's better understood by looking at the table that the paper presents so this is there are the two models then estimates and we have the by factor model and then we have the second or tomorrow it's 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 indicate factor so you have this zero one pattern here and we can see the equivalence of the factor loadings so the factor loadings on on the the first order condition 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 shows what is the difference between the general factor and the buy factor here and we can do the calculation so that's the difference and this is the relationship between the buy 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 buy factor models applied they may not be called buy factor models but these are actually applied quite commonly in management research for a particular purpose so and the particle purposes are for modeling method values so method variance models typically we have like one general factor called meta factor effects all the indicators and then we have there are the factors that were interested in and those are minor factors in the bio factor model they are the main factors in the middle factor model that moral is a bit problematic because in method variance analysis we allow this manufactures 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 second first-order second-order factor analyst model is quite common its more common than then why 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 there are the hierarchical factor model these models by factor Hart Euler factor because they are nested they can be tested with the chi-square test or nested mono test and then we can use that nested motor test to check if the second-order model is appropriate for the data this is also the foundations for foundation for exploratory structural equation modelling the idea of exploit their structural equation modeling is that are you have a confirmatory factor analysis model and then you specify a structural recursive model between the factors and at the same time you fit an exploratory factor analysis model to the error terms of of that model so you basically fit a comforter factor analyst model and an exploiter factor and are small at the same time and that allows you to model the main factors and also model scale dimensionality that is so kind of like the minor factors without really knowing in advance what kind of specification you want to have for those minor factors so it basically automates some of the data exploration that you may do when you do the acoustics of converter factor analysis models so let's summarize by factor model and hierarchical factor models they are close cousins by factor model has main factors and minor factors in hierarchical model you have first-order factors as they can order or the factors the first-order factors are or their error terms are equivalent to there are minor factors in the by factor model by factor model is more focused on on scale dimensionality so you are interested in in the dimensionality on the level of the scale items in the heart of factor model you are interested in the dimensionality of a construct so if you say that your construct is for main dimensions that depend on on one main dimension then are you want to measure four of those dimensions separately and then model the main factor as a common cause here dimensionality in by factor model typically notions and in here dimensionality typically of theoretical interest by factor Mars also foundations the foundation for hierarchy or reliability coefficients and it's a useful article alternative for freeing the correlations between our error terms in a confirmatory factor analysis

Channel: Mikko Rönkkö

Views: 1,454

Rating: 5 out of 5

Keywords: research methods, statistical analysis, organizational research

Id: Ow1kEu0OjMA

Length: 15min 59sec (959 seconds)

Published: Thu Mar 19 2020