hi everyone in this video I will be giving a tutorial on exploratory factor analysis using SPSS and our example is based in part off of a study that is published at the plus one website the article is entitled three Factor structure for epistemic belief inventory across validation study and um with this article comes the supporting data actually the raw data file and we're going to be using just a sub-sample of the data that was provided so in the data file that the authors provide they actually have a single data file with both sub-samples in in there and I'm I've basically created a separate data file just with subsample one so you can access a copy of that data file by following the link that's provided underneath the video description and I'll also include a link to this article if you want to read more on it but basically in the art in the uh article uh the authors uh collect data on the 28 item epistemic belief inventory from 1785 Chilean high school students and they perform an exploratory factor analysis and ultimately decide uh to stick with a three-factor model so for our demonstration here we're not going to be using all 28 items just to kind of keep things a little bit more manageable we're just going to perform a factor analysis based on the 17 items that the authors actually ended up going with as part of their three-factor model and so those 17 items are actually shown in table two and just to kind of give you a feeling or a flavor of what uh the content is you can see the first item it says most things worth knowing are easy to understand the second one is what is true is a matter of opinion the third one students who learn things quickly are the most successful and so forth so these items are basically kind of reflecting beliefs about the nature of knowledge and knowing and also there's beliefs about you know the nature of learning and you know the nature of kind of intelligence is being fixed versus kind of more malleable if you will so that's the content that we will be working with okay so now uh I've opened up the SPSS data file uh that we'll be working with again you can find this linked underneath the video description so you can download it and um again there's 28 items associated with the original measure these 28 items were administered to the high school students but we're only going to be analyzing those 17 that were found in that table so when it comes to carrying out factor analysis there are multiple steps and sometimes it can become rather complex and so I'm kind of I'm going to kind of distill a few basic steps uh in this particular conversation there's other things that that theoretically we might do like uh you know checking for outliers or our lack of normality things like that we're not going to be going into that we're just going to be focusing in on the main steps that you might see in the context of carrying out the analysis using this interface so uh or at the SPSS interface so the first thing that we're going to be kind of paying attention to is just the question of whether it makes sense to carry out the factor analysis so that's really you know if you're you're reading a textbook or something uh where they refer to uh the question of you know is the Matrix factorable essentially what we end up doing when we carry out our factor analysis is that we we essentially generate a correlation Matrix for a set of measured variables and this particular demonstration right here our measured variables are the items themselves and the responses uh the data on those variables are um as as what we're analyzing so we have 17 measured variables or items that we're going to be carrying out the factor analysis on and we want to know whether or not it makes sense because you know in some situations if the Matrix is not factorable what that really reflects is a you know a situation where maybe the relationships among the measured variables are items in this case uh are so small that really what you end up with is uh is um you know maybe over factoring uh having less reliable factors that sort of thing and so we don't want to have a situation where we can't really trust our results very well other situations might result in uh inadmissible Solutions or situations where maybe we just can't carry out the Matrix operations and that would be in those situations where you might have uh high levels of collinearity among your variables or items in this case or Singularity uh in our in our correlation Matrix so that's kind of the first step in the process then we move on to if we decide that we have a matrix that is factorable and that makes sense to carry out the analysis then we proceed to determining the number of factors that may account for the intercorrelation among our measured variables and so in that particular case um you know there are various methods that can be used to make that determination by default SPSS uses the Kaiser Criterion basically that's the eigenvalue cutoff rule um in a nutshell it just means that you retain the number of factors uh that have an eigenvalue that's greater than one and typically that eigenvalue or sub you're supposed to apply that eigenvalue cutoff rule to the unreduced correlation Matrix and essentially that's following a principal components analysis so that's one approach that's actually a pretty lousy approach but it is the default in SPSS so I'm going to show you some other Alternatives as well which will include parallel analysis and using the map test and also looking at the screen plot so that's kind of the second thing and then when we did make that determination regarding the number of factors then we proceed on to um uh you know kind of forcing that final Factor solution forcing extraction of the number of factors that we decide upon uh and then rotating those factors for uh us to be able to interpret the meaning of those factors so we'll go ahead and get started here by just going up to the analyze button we'll go down to data reduction we'll click on factor which you see right here and we will select our items for inclusion uh in our factor analysis so we have items one through five ce1 through ce5 right there then we have item eight then we have item nine and then number 11 we have item 14. and then 15. then we have 17 20 22 uh 20 and then 24 through 27 so these are the items that were found in that original table so next we'll click on descriptives right here and this is where we get information that is going to bear on the factorability um of our correlation Matrix so uh things that we want to click we want to be sure to click on univariate descriptive so we could get means standard deviations um and so forth associated with the items you'll see that with the correlation Matrix we can click on coefficients here this is going to give us the correlations the zero order correlations among all of our measured variables then we'll click on determinate kmo and Bartlett's tests of sphericity and then we'll click on anti-image right here so we'll click on continue and then on OK and so now we get our output now a lot of folks when they are carrying out um their Factor analyzes they tend to kind of do a One-Stop shopping of everything so if you go back and you look through uh various options there are other options in here like if you go under the extraction tab uh you've got you know your your method the default is principal components this method is oftentimes used but incorrectly during factor analysis so we're not really going to be focusing on uh this particular issue right now but it let's say you want to carry out say principal axis factoring or maximum likelihood or whatever you could theoretically click on that but I my preference is not just not to do all this and to kind of approach this sequentially because of that eigenvey cutoff rule uh the way that it forces the a particular solution and it just kind of muddies things up so my preference is just to leave the other defaults in place and then go back look at the factorability question first and then proceed to The Next Step so um at any rate we'll uh go ahead and click out of this and so we have our descriptive statistics you'll see we have the mean standard deviations and Sample sizes associated with our items so the sample size is 1039 so that's actually the first sub sample from that original data set when you scroll down a little bit further you'll see that we have the zero order correlations among our measured variables or our items and as you're looking at this just you know it looks like any other correlation Matrix along the principal diagonal you have the ones so there's our ones that you see right there all the way down um I'll just kind of note over here we'll just kind of highlight that as well and just keep in mind that when you're looking at this Matrix the values that you see in the lower triangle below the principle diagonal are reproduced above so they're exactly the same values it's just basically a reversal of the rows and columns but that's they're the same value so you only need to look at one triangle if you will I tend to still gravitate towards looking at the lower triangle and so that's where we're at so what we're looking for as we're examining this correlation Matrix we're looking for items our correlations between the items that at least are 0.30 or greater so we're looking for at least some uh correlations that are you know at 0.30 or greater and so as I kind of have gone through this I've already kind of scanned this and and looked at it so I've looked I've found that basically we have have about 10 correlations that would meet that minimum criteria but what you don't want to have is a situation where the correlation Matrix where with the correlation Matrix where the correlations are all very near zero because if you have a very near zero correlations in your Matrix as a overall then there's not really any factors that you can extract that would meaningfully account for relationships because there are no relationships to be had so we want there to be at least a reasonable number of correlations that would that would put you in the 0.30 or above range by the same token we also don't want to have a situation where we have a complete redundancy in the information within the correlation Matrix as well so we don't want to have a situation where we have correlations that are in the 0.90s um where essentially you've got high levels of collinearity among the uh the items and you certainly don't want to have a problems with Singularity as well where basically you would have uh you know one item being sort of a linear function of all the other items you don't want to have that situation either in the case where you have a multi-collinearity or Singularity you'll either have situations where uh maybe you end up with inadmissible values in the case of multi-collinearity or you may end up with Singularity which would essentially preclude certain Matrix operations when you are carrying out your exploratory factor analysis so as we kind of looked at this correlation Matrix right here you'll see that that really they're all pretty low correlations but we do have about 10 correlations that are above 0.30 so that's one uh thing that you can do in order to screen for the appropriateness of carrying out the factor analysis on our set of items another index that we might rely on is the kmo measure of sampling adequacy so this is uh the the value that we have right here so if you go back and you think about Kaiser's Criterion you know basically kind of provides this sort of um a sort of a a framework if you will for judging the adequacy of the items for inclusion in your factor analysis so it uses a lot of creative names so these are the criteria that uh that he provided um so you can see right here he's describing uh kmo values in the 0.90s as marvelous values in the 0.80 as meritorious values in the 0.7s as middling values in the 0.6s is mediocre values in the 0.5s is miserable and then less than 0.5 are considered unacceptable so like I said these are very creative ways of describing uh the factorability of the Matrix but in general you know if you happen to have kmo values that are putting you uh definitely below 0.50 the Matrix would be considered unacceptable and then obviously you have better matrices that you're factoring if you have those kmo values that would put you you know higher than the the 0.50 so going back to our output then the kmo that we have for our set of items is 0.778 which would put us in that middling range but uh base clearly uh you know falling uh at an acceptable level for carrying out the factor analysis then we have uh Bartlett's test next you'll see that in this output this is basically a test of whether the correlation Matrix that we saw above does that deviate significantly from an identity Matrix and an identity Matrix is one that contains ones on the principal diagonal but zeros on all the off diagonal elements so zeros in a correlation Matrix uh basically signal a lack of correlation between uh variables and so if your correlation Matrix does not deviate or is not significantly different from an identity Matrix that's basically the same as saying that there are no associations among your variables in which case it makes no sense to carry out the factor analysis so with Bartlett's test right here this is essentially a chi-square test and so if we find that this is indicating statistical significance then we would reject the null hypothesis of an identity Matrix and essentially go with the alternative which is that our correlation Matrix is not equivalent to an identity Matrix all that basically means is that our matrix it's reasonable to carry out the factor analysis because our Matrix we we won't assume that it's uh coming from a population Matrix that's an identity Matrix so uh in this case you can see our P value that's given is less than .001 right here for our chi-squared tests so basically that's that's a good sign in terms of the factorability of our Matrix also keep in mind that the this bartletts test is impacted by sample size and so you know we have a really huge sample size of 1039 students who had responded to this particular within this particular sub sample so you can have very uh minor um uh uh you know minor discrepancies between the correlation Matrix and uh an identity Matrix but it but because of the the power due to the large sample size you might end up kind of rejecting that null so that's why you know it's it's a useful supplement but I wouldn't you know uh put all my uh weight on that particular index I would again go back and look at the correlations in The Matrix look at uh the kmo the Kaiser mire okin measure of sampling adequacy that we just looked at a second ago so these are useful for looking at the overall factorability of the scale another thing that you might do early on too is to maybe identify potentially problematic items and so the way that you can do this is to refer to the anti-image correlation Matrix so that's what this next table is and you'll see it's got anti-image covariances and anti-image correlations and we're going to focus our attention mainly on uh this part of our Matrix down here where that we have the anti-image correlations and in particular we're looking at the elements in the principal diagonal of this Matrix right here so these values right here are also measures of sampling adequacy they're kmo values at the item level and so what we're lit so if you go back and you think about Kaiser's descriptions then what we're really shooting for are kmo values for the items that would be greater than 0.5 and ideally not in the 0.5 but you know maybe even greater than 0.6 I would say but nevertheless as you're looking at all of these values right here these values are all um you know falling in the 0.6s point sevens and some point eights so basically all of these items would be considered reasonable candidates for inclusion in our exploratory factor analysis and those items you know if it's the case that we had items that are poor uh for inclusion in the factory analysis and that we might consider removing that item and then performing the factor analysis on the remaining items so that's basically what we have right there let me kind of move this over just a bit there was one additional item down here at the bottom that I didn't highlight so that's the last one right there okay so at this point we've uh you know we've we've got some some good indication of the factorability of the Matrix uh one other thing that we want to pay attention to in terms of making that decision about factorability is the determinant of the Matrix or the determinant of our correlation Matrix so the determinant is basically like a generalized variance for our Matrix and ideally what we want are values for the determinant that would be greater than .0001 and so you can find that information if you go back up to the correlation Matrix we've got it says determinant uh for our Matrix is 0.113 so that's clearly greater than the point zero zero zero zero one and the thing is is that if the determinant of the correlation Matrix if it's less than this or or effectively zero what that's going to translate into is a situation where when it comes to performing uh various Matrix operations during the the factor analysis it's not going to be able to do that and so you'll end up with uh sort of the the program crashing or issuing warning messages and so forth so based on all of this Collective information we look like we're in pretty good shape in terms of carrying out the exploratory factor analysis uh on the correlations among these items so we met we then move on to step two which is basically um you know determining or coming up with an estimate of the number of factors that may explain or account for the intercorrelations among our items and so there are various criteria that are utilized as I said you know basically in uh in SPSS the default Criterion is the eigenvalue cutoff rule basically Kaiser's Criterion and so uh you know when we ran our initial analysis by default it actually goes ahead and does that for us so if you scroll down here at this little uh box that you see right here you'll see that in the first uh part of the output right here you've got initial eigenvalues and in this column you've got these are eigenvalues from a principal components solution sorry about the drawing right there in a nutshell what we've done is we've extracted um you know the number of components that's e equal to the number of measured variables but the components are basically repackaging the information such that you know each component is accounting for a certain amount of variation in the original set of our items and so with that repackaging uh you'll see that basically the components are extracted such that you know the first component will explain the most variation uh this the next component will explain the next greatest amount of variation the third one the next greatest amount of variation and so forth so the components are extracted such that the eigenvalues which kind of summarizes the variation accounted for uh uh are found in sort of a decreasing order you know essentially what we we start with the largest eigenvalue and then we proceed to the smallest eigenvalue from the first component to the 17th component and the basic idea then is to um as I said before the eigenvisor basically just kind of summarizing the variation accounted for with the first one accounting for the greatest amount of variation the last one obviously accounting for the least amount of the total variation and so the eigenvalue cutoff rule or the Kaiser Criterion uh that's that's issued by default essentially says all right um we basically are going to retain as many items um or as many components that explain as much variation as at least one single measured variable um so in a nutshell uh you know as you're looking at this the first eigenvalue right here is 3.059 the first component is accounting for about the same amount of variation as 3.059 of the original measured variables the next component accounts for as much variation as about a little bit over two measured variables then the third one is accounting for another 1.4 of the of the original measured variables and so forth and then if you add it up all the eigenvalues in this column right here it would actually sum up to the the total number of measured variables which is 17. and with respect to the eigenvac cutoff rule then you basically have our first three of our components all of those have eigenvalues that are greater than one and so by default then it sort of shoots that information over here and says we're only going to retain uh those three components and that's why you see a reproduction of those same eigenvalues over here so as we think about that particular rule uh being in place it's it's suggesting then uh if we go back and think in terms of how many factors we should retain uh in our factor analysis or ultimately should extract in our final factor analysis uh based on this rule we would it would suggest that we retain or extract uh three factors okay but now the Kaiser Criterion is a really crummy rule uh historically it is tended towards over factoring and um it's just generally not regarded very favorably uh nowadays in particular there are other uh procedures that do a better job of determining the number of factors so some of those include the use of parallel analysis uh also there is uh the use of the map tests and and there's some other options that are available through other programs but they're not necessarily available uh through SPSS which is a little bit frustrating thing because it seems like SPSS would do a better job of updating um you know according to what is considered best practice so right now it's still stuck in not so good practice I will also say while I'm here I'll go ahead and show you one other approach to determining the number of factors we can also look at the screen test if we go back under Factor right here and click on extraction if we leave this all um the same if we if I click on screen plot right here then it will also give us a plot of the eigenvalues that you saw in the First Column from that table against the component numbers so basically all you have to do to obtain that is click on screen plot right here so just keep in mind though that the eigenvalues are based off of essentially a PCA solution and whether you change the method to a more common factor analytic approach or not the screen plot will still be reflecting that PCA solution just to show you too that default I was telling you about this is where it says right here it says extract and it says based on eigenvalues greater than one that's the default right there and so what it does is that even if we made other Selections in terms of methods for extraction using more of a standard common factor analytic approach it's still going to use this rule to make that final determination of the number of factors to extract and so it kind of steals the decision making away from the analyst if you're not careful so what we're going to do is consider the screen plot in addition to what we just looked at with the the Kaiser Criterion and then we're going to move on to two better approaches for determining the number of factors which are parallel analysis in the map test so we'll go ahead and click on continue right here and then on OK just taking a look now when we scroll down we have our screen plot and as you're looking at this just kind of think about the side of a mountain okay so you have the main slope if you will and then you have sort of the rubble at the base of the mountain so as you're looking at the screen plot I'll kind of move it up here just a bit the eigenvalues that we had up here in this table are being plotted so these are the eigenvalues that are being plotted in this in this plot right here so the first eigenvalue of 3.059 that's this value right here the second one the 2.042 is this one then the third one is 1.427 which is this one uh and then so forth and so you'll see that really what you're looking for in this plot is kind of an elbow and so the elbow basically marks the start of trivial more trivial factors so you could think of it as you know these three factors right here are you know apparently they're not uh we would not consider those to be trivial but major more but more major factors and then here's our elbow right here and essentially then we see a trailing off of those eigenvalues as we move uh from uh you know component four through component 17 and so that would be essentially our break point and we would retain those factors with an eigenvalue or those we would basically retain the three major factors that we see in this plot this is is a great visualization tool it helps to kind of better understand um what you know uh you know the difference between major and minor factors as you can see right here it looks pretty clean uh one of the downsides of the screen plot is that subjectivity even though we have a fairly clean picture right here of supporting a three Factor approach or three Factor solution if you will that we would want to um uh force in some screen plots things can become a lot more ambiguous sometimes it looks like there are multiple elbows if you will multiple breaks and it can become a lot uh trickier to interpret so fortunately in this case it's uh pretty straightforward but the downside of the screen plot is its subjectivity and sometimes it can become much more tricky in order to determine the number of factors okay so now we're going to move on to parallel analysis and the map test so underneath the video description you'll find a link to a syntax file that I've kind of put together based on two separate syntax files that was created by Brian O'Connor and is associated with this 2000 article um in a behavioral research methods but what I wanted to do because uh you know basically he had two separate files one for parallel analysis one for map test I wanted to kind of put it all in one place uh where you might be able to uh just kind of generate all this output at the same time so if you follow that link underneath the video description and open it open up um the syntax file as we have our data opened up this is basically what it looks like right here so the first part of the file really focuses in on the parallel analysis the second part is on the um using the map test and so really all you have to do there's only you know a couple of things that you have to do in order to use this particular file and I will encourage you really my suggestion is to not leave open multiple uh SPSS data files when you're using this particular um syntax okay so my suggestion is leave open only the one file that you want to be analyzing so just because sometimes if you're not careful if you have multiple files open I think there might be some issues that can emerge so the first thing that you need to do is to kind of make your specifications with respect to parallel analysis and with parallel analysis the idea is um you know when you're looking at this when we kind of go back and we think about that screen plot and we think about the eigenvalues that were generated during our initial PCA that's that's based on the actual raw data and what parallel analysis is kind of doing is trying to account for possible sampling error that might also factor into the computations of of eigenvalues and so that's kind of where Kaiser's Criterion kind of falls down so the the idea with uh parallel analysis is to generate simulated eigenvalues and then compare that compare those eigenvalues against our the eigenvalues from our data and then make our determination of the number of factors based on the number of uh eigenvalues from our data that exceed the number of simulated eigenvalues so what we're going to do is go to line 15 right here and really we're going to leave everything up to this point the same do not touch anything else the only thing that we need to do is following the equal sign is to include the items that we are that are referred to the items in our open uh SPSS data file that we are subjecting to factor analysis so you'll see right here I already had like C1 to C5 but basically you know we can type in ce1 uh ce2 ce3 those are three of the items all the way to ce5 if you want to make life a little bit easier because these uh these variables are all adjacent to each other in the data file you can just say ce1 to ce5 and then we have ce8 ce9 CE 11 ce14 ce15 CE 17 ce20 ce22 and then ce24 2ce 27. so that's all you have to do okay and make sure that that period is at the end of the line so if it's not there you may have problems so I'm going to actually copy all of this rather than typing it in later but basically all I need to do now is scroll down uh leave everything else the same do not touch any of the other uh syntax go all the way down to the end right here and you can see that we've got our VAR um set equal to and then we're just going to put those uh names in here as well so this is uh you know the specifications for the map test so we have our variables in both of these sections the other thing I will draw your attention to as well when you are performing the fact the parallel analysis you need you need to determine the number of simulated data sets that you're going to be working with so by default I have 1000 right here um so uh just basically is compute in data set set equal to one thousand in the original syntax he had I believe he might have had a 100 or something like that I just added a zero to make it one thousand you have right here uh desired percentile the default is uh 95. uh basically what happens during the context in the context of parallel analysis is that simulated data sets where essentially the the relationships among the variables are all zero we generate simulated data sets like in this case it would be 1 000 of those simulated data sets uh from those you would have correlation matrices that would be uh generated from those correlation matrices the eigenvalues would be computed and then if with this particular setting right here it would take the 95th percentile of the eigenvalues for all of the components that are extracted from those and then that would be the criteria that you're using um in comparison to the eigenvalues from your data uh there the other approach is to take the mean of the those eigenvalues and so that's also provided in the output by default the down here where it says um you have actually a couple of options you've got a setting of one or two for uh the first one being principal components analysis two for common factor analysis basically principal access common factor analysis so you've got this compute kind equals the the default setting as one for principal components analysis this is actually the historical uh way that you uh would tend to do this but you've got other authors particularly like Humphreys and some of those uh authors who suggested we shouldn't be carrying out the parallel analysis on uh using principal components analysis we ought to be carrying out the analysis using a reduced correlation Matrix uh such as in the context of principal axis uh factor analysis the historical you know the tendency historically has been to rely on principal components analysis it does tend to tend to behave pretty well in terms of determining the number of factors um and so we're just going to stick with that particular approach um if you if you disagree with that approach that's perfectly fine you can still if you wanted to you can come down here and type A 2 in it's very simple then you've got down below you've got just you know whether you want to simulate data from um you know a normal distribution or if you want to use permutations of the raw data set we're just going to stick with the one option which is for normally distributed random data generation so at this point we've made our specifications I'm going to right click and then select run all so when I click on run all I get my output so you'll see that first off we have the results related to our parallel analysis and so as you're looking at this you can see these are the eigenvalues right here the raw data eigenvalues those are from the initial PCA or principal components analysis uh basically it's just the extraction of components from an unreduced correlation Matrix so then we've got this column here contains the mean eigenvalues from those simulated correlation matrices and the 95th percentile so depending on which criteria you want to use you could use either the means column or the the the 95th percentile column that's fine uh the basic idea though is that you are making this comparison between the random eigenvalues in either of these two columns and the eigenvalues from your raw data and you want to maintain the number of factors that ex with eigenvalues that exceed the randomly generated eigenvalue so I'm going to go with the 95th percentile uh column right here I'm going to use these as my basis and so in this particular case you can see that the the random eigenvalue for the first component is um less than the the eigenvalue for the first component uh from our data then the eigenvey from our second component is also less than the eigenvalue from our raw data then the eigenvalue for the third component of the randomly generated eigenvalue is less than that for the data for our third component and then you'll notice a kind of a reversal here so the eigenvalue the random eigenvalue of 1.1406 is now greater than the eigenvalue from our raw data which is 0.997997 and so at this point the random eigenvalues are now going to start to exceed the eigenvalues from our raw data so basically what that conveys to us is that we do not want to retain any more factors than three so we're going to stick with three factors and we're just going to kind of um you know ignore all of the other factors so in this case right here we this would suggest a three Factor solution so then if we scroll down a little bit further you'll see that we've got the map test so the basic idea behind the map test is that we're essentially Computing uh the average of squared partial correlations for different uh based on different component models and then we are taking the the uh the uh the smallest average of those squared partial correlations um as sort of the determining Factor when it comes to making a decision about which which number of factors we should uh stick with so you know basically with the first um the first part of our output right here you'll see it says 0 for this is basically this column is kind of a component number if you will and you'll see the zero basically there's no components that are being extracted all you're doing in this case is taking the uh the average of the squared zero order correlations I'm among the variables and you'll see right here that is .0247 then there is the extraction of a single component and what we're doing in this case is we are Computing the correlations among all of our items uh after but controlling for that first principle component so we're sort of partialing that principal component out of the associations among the original set of items and then we're squaring those correlations and then uh and then averaging them and so that that's what this value is right here and so all subsequent um values in that second column basically are reflecting the partialing of more and more components from those associations and so what we're looking for in the second column right here is the smallest map value and so it just so happens that you'll see right here that if we added if we kind of uh partialed two components you'll see that this value actually starts to increase again so you'll see it's decreasing right here and then increasing as we move from a one component model to a two component model and so with that basically that would suggest uh retaining a one component model and so that's why down here it just says the number of components according to our map test is one so now you can see that you know kind of looking at the overall set of our uh information the Kaiser Criterion suggested a three-factor solution uh the screen plot gave us a good indication of a three-factor solution would be a good idea uh the uh more well-established uh technique of parallel analysis uh also suggests a three-factor solution but our map test is actually suggesting a one-factor model so there you know when it comes to thinking about these different um approaches to determining a number of factors keep in mind that they don't all have to agree they can disagree and so then you have to kind of make other you know have other considerations when it comes to um determining the number of factors one other approach that you could you could use is to carry out then your factor analysis based on the uh the number of factors that are suggested by these different uh procedures and then kind of look to see which which of those is most interpretable basically kind of force the factor solution associated with the get with with some of these different procedures rotate the factors and then interpret them and then kind of make that determination based on you know the number of items per factor and also the interpretability of those factors but for this particular purpose right here I'm I'm familiar enough with the the literature on uh the ebi that I'm pretty comfortable with sticking with a three Factor solution um rather than a one-factor solution because the evidence in this this particular area has been that uh epistemological and learning beliefs and so forth are multi-dimensional not a unit dimensional so at this point we're going to then transition to carrying out the final factor analysis so we'll go to analyze we'll go down to Dimension reduction and Factor right here we'll click on extraction and so with respect to the method the default is principal components analysis and there is a long history of debate in the literature on whether it's reasonable to use principal components analysis or should we use um a common factor analytic approach so the basic idea with principal components analysis is that you are extracting components out of the correlation Matrix with that correlation Matrix being a a a full unreduced correlation Matrix okay so that Matrix basically contains once on the principal diagonal it's our standard correlation Matrix in the context of common factor analysis uh we we basically are oftentimes substituting uh estimates of communality into the the correlation Matrix kind of as an initial starting point and then sort of iterating through various Solutions until we find are iterating through several Cycles to come up with final estimates of communality and the idea is to capture that common variation that's what we're really interested in factoring principal components analysis is basically an analysis of all the total variation in a set of measured variables whereas um common factor analytic approaches are are focused on analyzing the common variation among those those variables so as I said there's a long debate uh in the literature about whether it's reasonable to use principal components analysis or a common factor analytic approach I tend to go with the common factor analytic approach as a stronger basis uh for uh this type of analysis so I'm going to select one of those other methods and the method that we're going to stick with in this video is uh principal access factoring so we'll click on that and then because we determined that we want three factors I'm going to select fixed number of factors factors to extract is equal to 3. so we're going to force a three Factor solution now as I said before you know if we had left the default right here on the eigenvalue cutoff rule the program would have uh essentially ended up uh going with a three-factor solution uh and then using the principal axis factoring uh with with determining the three factors but you know just because the Kaiser Criterion says it's one thing you know other approaches to determining the number of factors can can suggest other numbers of factors so I'm kind of trying to give you a little bit more of a realistic take of you know what to do if you decide on a different number of factors than what the program is going to give you based on that Kaiser Criterion so we're going to select this right here again we've typed in three and we'll click on continue so we're forcing a three-factor solution we'll click on rotation and then we've got you know various options so the basic idea behind rotation is this that when we extract our factors um you know at that point they are unrotated and basically when it comes to interpreting the factor pattern matrices and so forth they can yield uh really not very much Clarity when it comes to kind of naming and defining those factors so oftentimes or really most of the time researchers will will end up performing a rotation in Factor space in order to uh provide meaning or naming uh or definition to the factors so there are two general classes of rotation that one might use one is called orthogonal rotation the other is called oblique rotation so with our orthogonal rotation basically uh the idea is to maintain the original orthogonality of the factors but just perform a rotation in space and so you know when when it comes to the initial extraction like our three factors when we extract them they are going to um they're basically going to be orthogonal to each other in other words each factor is going to account for um you know additive amounts of of information or variation in the original set of measured variables so if we perform that orthogonal rotation it's going to do it's going to maintain that that property and so a common you know probably the most common of orthogonal rotation that people utilize is varimax rotation which is this one right here now another the other class of rotations is oblique rotation so the idea behind that is that you know when it comes to real you know real life so to speak um you know things are not usually uh completely uncorrelated with each other uh and so what we might want to do is to relax that constraint of orthogonality and allow the factors to correlate which might be more a realistic assumption in terms of the relationships um involving our factors so oblique rotation basically relaxes that assumption allows for correlation to become manifest um if there if there is one and so uh the approach in that particular case will want you know a very common approach uh is to use like the direct oblumen or Promax we're going to stick in this particular demonstration on using Pro Max rotation so as I proceed I'm going to really just kind of show both of these uh approaches to rotation just to kind of uh let you see the differences uh in in the output so we'll start off actually with Pro Max rotation and we'll click on continue and then on okay to generate our output so all of the other stuff that we had actually like the kmo bartletts Tassie anti-image correlations and so forth all those are exactly the same as what we had before now in terms of the first table right here you'll see it says communalities and this is where I was talking about with respect to the reduced correlation Matrix so you'll see right here this initial column this is basically just a an initial set of estimates of communality that are placed into the principal diagonal of the uh of the correlation Matrix that's being subjected to factor analysis and so these estimates are in a nutshell you can easily compute them uh just if you take take each item and regress it on the remaining items the R square value in that particular case is the squared multiple correlation and that is the value that you see for each of these so that's literally where it starts and then the extraction commonalities these are final estimates of the variation in each of the items it's accounted for by the set of extracted factors so in other words you'll see right here for the first item ce1 the value is 0.207 so we would say then that uh the the three factors that we have extracted accounts collectively for about roughly 21 of the variation in ce1 the second um I the second communality right here uh is .309 and so we can say then that the set of our factors is accounting for roughly 31 percent of the variation in ce2 you'll see down here for ce4 the set of variables account for roughly 41 of the variation in ce4 and so forth so that's the way that you can think about uh the meaning behind the commonalities it's essentially reflecting the proportion of variation in each of our indicators that's accounted for by the full set of factors that have been extracted we'll scroll down a little bit further and so we have our table and just kind of keep in mind that again all of this information right over here is exactly the same as what we had before when we ran the principal components analysis notice that the language has changed a little bit they're using the term Factor here um but you know in a nutshell uh the eigenvalues that you see in this column right here these are all essentially from a principal components analysis and uh so just kind of keep that in mind as as we go through this one other thing I do want to mention too is that these eigenvalues if we divide the eigenvalues by the total number of measured variables then that's what gets us a proportion if we multiply that proportion by 100 that's the percentage of variance accounted for by each in this case each of these components right here so you know simply put if we kind of go back over here if I divide 3.059 by the total number of items which was 17 and multiply that by 100 percent we would say then that that first component was accounting for almost 18 percent of the variation in the items cumulatively obviously that's going to be the same for the second component 2.042 if I divide that by the number of items and multiply by 100 you can say then that uh fat the component two is accounting for an additional 12 percent of the variation so cumulatively factors one and two account for roughly 30 point or basically 30 percent of the variation and so forth so when we look over here in the extraction sum of squared loadings basically the eigenvalues that you see right here are based off of the final extracted solution so using the principal axis factoring method and so when it comes to reporting uh the um the eigenvalues and percentage of variance accounted for you'll want to make sure that you're reporting information over here in these columns right here as opposed to what we had in the initial eigenvalues so uh you know basically the first eigenvalue that you see right here just take this number divide by the 17 the number of variables multiplied by 100 you can see them that factor one accounts for about 13.7 percent of the variation in the items Factor two the 1.351 accounts for about uh uh 7.946 percent of the variation or basically roughly eight percent of the variation and so forth foreign factor factor three accounts uh the eigenvalue is 0.703 and it accounts for about four percent of the variation so cumulatively uh the fact the three factors are accounting for about 25 roughly 26 percent of the variation in our uh indicator variables or measured variables so then you'll see over here we've got the rotated sums of squared loadings these are basically also um you know kind of eigenvalues if you will uh but in this particular case you'll notice that that the percentages that we saw pre prior to rotation all of this is up here this is pre-rotation okay so basically uh you know the the factors are counting for um uh distinct percentages of variation in the original set of measured variables but then post rotation over here I'll just call this post rotation um well we have the factors are accounting for variation but because the factors are allowed to correlate remember we uh Promax rotation is an oblique rotation it allows for correlations among the factors then we can't talk about the factors as accounting for uh additive proportions of variation in the original set of measured variables and so that's the reason why we don't have the additional columns for percentage of variance the cumulative percentage of variance that we had previously so when you're reporting and you're you're running the analysis using um you know uh Pro Max rotation you know you'll you'll want to make sure that you report on the pre-rotation eigenvales percentage of variants accounted for and so forth and then the post-rotated uh post rotation uh sums of squared loadings that you see here but you're not you're still accounting for com as a as a set you're still accounting for the same total amount of variation but you have a redistribution of the variation uh uh accounted for by the factors and also because the factors are correlated we can't talk about each factor is accounting for a distinct amount of variation so next we'll scroll down there's our screen plot again you'll see that here we have our Factor pattern Matrix right here and so this is it right here and basically the values in this Matrix these are zero order correlations between each item and the three extracted factors so the correlation between ce1 and Factor one is .095 the correlation between fact C1 and Factor two is .33 and then the correlation between C1 and factor three is point two nine nine now the idea uh following uh or during your analysis is is that you not only want to extract the factors but you want to give meaning to them give you know name them or Define them or you know give some kind of um uh some kind of clarity as to what is being measured by the set of your um your items and so that's where you use uh Factor loading matrices in order to do that now the factor Matrix that you see right here this is actually reflecting or this is uh these loadings are reflecting the fact that the factors have not been rotated so in other words this Factor Matrix that you see right here is basically corresponding to the information that you see in this part of our table basically pre-rotation so because of that they're oftentimes not going to be particularly useful when it comes to interpretation of those factors based on the loadings that are presented within so what we want to do is to hang our interpretation on rotated loading so the way that we can do that is we'll scroll down a little bit further and we'll go to the pattern Matrix that you see right here so in this particular case this Matrix is following rotation okay so in other words I'll just say following rotation okay and so as you're looking at this just kind of keep in mind that as you know when we rotated uh the uh the um the previous loadings or the previous factors if you will then the loadings in this particular Matrix um reflect the association between each measured variable and the factor is still the same idea but now because we performed an oblique rotation uh essentially through our Pro Max rotation uh now the factors are correlated so then when it comes to talking about their relationships or looking at the loadings and then talking about the relationships between the items and the factors we can't talk about them using the language that we discussed above which was in terms of the zero order correlations here the loadings are more analogous to standardized regression coefficients so I'll just say uh analogous to standardized regression coefficients okay so you know think about it this way that each item the variation in each item we're trying to account for that variation uh by the respective factors so if you think about it in uh regression terminology think about the three factors as as predictor variables in a regression where you are predicting variation in the individual items and because the factors are intercorrelated with each other then the regression uh the standardized regression uh coefficients um are essentially kind of controlling for the correlations um involving the the other um the other predictors in the model so in other words as we're looking at this first loading right here 0.032 for ce1 on Factor one think about it this way that basically that's the relationship between the first factor and C1 controlling for the other two factors then we've got the .027 right here that's the relationship between ce1 and Factor two controlling for you know factors one and three and then we've got the loading here 0.445 that's the relationship between C1 and factor three controlling four factors one and two so that's the basic idea um as we're looking at this particular Matrix and that's why I say they're analogous to standardized regression coefficients now what we want to do is we want to give naming or meaning to each of the factors and so the way that we do that is we look at which items are most associated with those factors now ideally what we want is a situation where we have the items are near zero on on most of the factors and then you know non-trivially loaded onto a single Factor that's kind of the ideal and so that's part of the purpose too of Performing the rotation so what we typically do then is we we make a sort of a we come up with a criteria for deciding on um you know whether an item is loading non-trivially onto a factor so that is what is referred to as a loading criteria okay and you know various authors have provided you know various rules of thumb about what constitutes a non-trivial loading so you'll see in the literature some folks suggest a minimum loading of you know let's say uh 0.30 in absolute value uh some you might see 0.32 as a minimum loading some I would suggest 0.40 as a minimum loading criteria I know that's in the the pitic and Stevens book that's mostly where I kind of land uh but you know others other authors would suggest uh maybe these other rules of thumb so in general the loading criteria uh that are proposed in the literature you'll see them kind of ranging but you know somewhere in the neighborhood of 0.3 and 0.4 but the basic idea is to identify which of the items um you know uh or which of the which of the items are associated with the factors uh at some minimum threshold so for this particular vid uh demonstration right here I I am actually going to just stick with a 0.30 as the minimum criteria so if you look at the pattern Matrix right here you'll see that and me uh ce1 meets the minimum loading criteria on factor three and obviously it's well below that criteria on factors one and two then you've got for ce2 uh it's meeting the minimum loading criteria on factor three if we just kind of uh but not on factors one and two um if we just kind of continue on down using the minimum loading criteria we'll just stick with three right now you'll see that uh item 11 meets that minimum loading criteria uh as well as item 17 and then item 22 right here so these are the items that we might use to define or name uh factor three when it comes to uh Factor uh one we'll just kind of go ahead and go back over here we'll go down uh column one you'll see item three uh loads onto that particular Factor then we've got item five and item eight and also item nine all three of those meet that minimum loading criteria then you've got item 14 and item 15 right here and then item 20 right here and then item 24 and item 27 right there so those items would be used uh we just are essentially going to look for thematic content in the items it helps us to give a name or representation to that factor then for Factor two we'll scroll down kind of look down here we've got uh item four right there when we kind of keep going down you'll see that it actually turns out there's only three items that um meet that minimum loaded criteria on that factor so and keep in mind too that when it comes to uh you know determining or naming or giving representation to the factors we need to have at least three indicators on those factors if you happen to have two indicators um that's not really um uh that's not that's not something we want to have happen we need to have at least three indicators so you know if it happens to be the case where you end up you know performing an analysis and you get some factors that maybe have one or two indicators that could be a symptom of over uh of over factoring and you might need to adopt a factor model that has fewer factors so and if it happens to be the case where you have really uh all of the items we're loading very closely together you know loading onto a single factor and they should be sort of breaking out different factors that would be a symptom of possible under factoring so in this particular case these are the items that seem to meet the minimum loading criteria you'll also see that for each of those items that are loading onto a single factor that you know they exhibit lower loadings on the other factors and that's ideally what we would have that's that you know that's the best case scenario in some situations though when you carry out your analysis you might have an item that loads um that meets the minimum loading criteria across multiple factors maybe two or perhaps even more factors but and on top of that there's not a much distinction uh or or difference in the in the loadings themselves in that particular case in terms of scale development that item might not necessarily be a useful thing to include in your final scale because it's not doing a good job of discriminating between the different factors so that is something to kind of keep in mind if you are you know working on developing a scale for use so at any rate we've we've sort of identified those items that meet the minimum loading criteria so now we need to name them and give them some meaning so this is the table that I created just with the item content and also uh the loadings that we saw in our um our Factor pattern Matrix our rotated Factor pattern Matrix so as you're looking at this you know we have to think about you know what I have right here are the items in bold are the ones that are meeting that loading criteria so um you'll see that the first item says most things worth knowing are easy to understand the second item is what is true as a matter of opinion and that's actually a negatively worded item in that scale so if you're responding to the item prior to reversal uh basically that would indicate more of a complex view of knowledge um and uh you know basically sort of a view of knowledge as uh less certain if you will so if we reverse coded it's actually reflecting sort of more of a naive belief in certain knowledge and the first item most things knowing are easy to understand that's also reflecting that sort of naive belief so you can see that both of these items are loading here uh positively you'll see we've got the best ideas are often the most simple then you've got things are simpler than most professors would have you believe then item 22 that's another reverse coded item that you see right there so after reversal which is what what's already been done uh then you know a higher value on that reversed uh item would reflect more of a naive belief so in the end the content of those items uh are reflecting sort of a belief in simple and certain knowledge the belief that knowledge is simple and again certain uh if we look in terms of factor two right here you'll see that we've got people should always obey the law then we have the items when someone in Authority tells me what to do I usually do it and then we also have people shouldn't question authority so that's really kind of reflecting the belief uh that uh you know knowledge and really the way that we should do things comes from authority figures that's you know that's the the basic belief that's represented right there then if we look at Factor one we've got students who learn things quickly are the most successful we've got people's intellectual birth of potential is fixed at Birth uh really smart students don't have to work as hard in school and also if a person tries to understand a problem they will most likely end up being confused then we have how well do you how well you do in school depends on how smart you are so again you know some of these items are reflecting kind of the belief that uh intelligence uh is fixed um and something that's not malleable and then also some of these items are blue are reflecting sort of uh a belief that learning should be quick and not at all and maybe even a blend of sort of you know smarter people are going to learn faster that sort of idea so uh at any rate all of these items right here are loaded on that first Factor so we're going to call this uh fixed ability and click learning or the belief in fixed ability and click learning so that's the basic uh idea that we're adopting here we're using the loadings to help give meaning or representation to the fact uh to the factors that we have extracted so now what we're going to do is move to the next part of our output and we'll scroll down and you'll see we have a structure Matrix so this Matrix is basically just containing correlations the zero order correlations between each item and each factor now unlike the pattern Matrix that we saw Above This Matrix does not control for the inner correlations between uh the items and the fact the inner correlations among the factors and so um so that's really kind of an important distinction between these two matrices and so sometimes you'll find that there will be some some differences between these two matrices um but keep in mind the main naming and and description of of the factors really should pivot off of the pattern Matrix but the structure Matrix can still offer some additional information that you might find useful then finally when we look at the very bottom we've got the the uh Factor correlation Matrix right here so you'll see uh you know if we were to extract the extract our factors and compute Factor scores this these would be the correlations among the factor scores so you'll see Factor one is correlated with Factor two at about 0.144 with factor three about .009 so largely the belief in fixed ability and quick learning doesn't seem to be very highly correlated with the belief in omniscient Authority or simple and certain knowledge but when we look at the correlation between omniscient Authority and uh the simple uncertain knowledge you see a positive correlation right there a modest positive correlation between those two factors that the correlation being about .308 okay so now let's go ahead and run our analysis and in this case though we're going to use varimax rotation so I'll go back rerun the analysis go back to extraction we actually all that's going to be exactly the same as before we'll go to rotation and click on varimax and then continue and then on OK so all of this stuff that we had before is exactly the same the commonalities the initial and extraction will be exactly the same if we look in our table containing the breakdown of factors the pre-rotation eigenvalues pre-rotation sorry about that the pre-rotation eigenvalues and variance account 4 is all going to be the same now you'll see though that we have post rotation I'll just say post rotation eigenvalues and percentage of variants accounted for and remember what I said earlier that basically the the pre-rotated factors are count are going to account for additive amounts of information in the original set of measured variables if we use an oblique rotation we can't talk about it that way in terms of the rotated factors but if we have an orthogonal rotation we maintain that assumption that the factors are uncorrelated and so now you've got the eigenvalues right here uh post rotation for the varimax rotated factors and then you've got the percentage of variance accounted for and the cumulative percentage of variance accounted for here and notice you know obviously that you know the pre-rotation and post rotation cumulative percentage of variance accounted for is um is going to be exactly the same but uh you know when you perform the rotation the variance accounted for by each of the factors will also uh change as well and so that's why the eigenvalues and percentage of variance accounted for pre-rotation to post rotation look a little bit different But be sure when you're reporting uh on on your factor analysis results be sure to report on the eigenvalues percentage of variance account for pre-rotation and then also post rotation um so that you you're giving enough information for the reader to to you know really better understand um you know the differences uh between pre and post rotation results so we'll scroll down we've got our Factor Matrix right here the um it's going to be exactly the same as what we had before because this is pre-rotation when we scroll down a little bit further now we've got essentially the the the factor pattern kind of structure Matrix if you will which is given right here and this is the Matrix that we use to interpret um our factors so if we still stick with the same loading criteria that we had before you're actually going to find that the the factors are going to be exactly the same in terms of their names uh probably the one difference that you're going to see is this this loading right here is 0.296 so you know if you don't round it off then that that a particular item 17 would not be included uh in defining that factor as we had seen before so it just kind of depends on you know how how closely you went around um but basically that's that's really all there is to it in terms of interpreting that Matrix essentially you know when when you're dealing with as we saw before when we used oblique rotation we saw we had a factor pattern Matrix that accounted for the interrelations or inner correlations among uh the factors and so the loadings were essentially the relationship between each item and the factors controlling for the other factors uh the structure Matrix was the zero order correlations between the items and the factors in this case because the factor the factors are still orthogonal to each other then basically the the pattern Matrix and the structure Matrix are exactly the same and that's the reason why we only have a single Matrix that's given right here so here we have our varimax rotated loadings associated with each of the items as I said before the items are um basically loading in this you know uh meeting the the same loading criteria on the same factors as we had before it's just that one item appear to fall just just a little bit below uh the 0.30 threshold that we had talked about I will go ahead and say that had I adopted a more uh rigorous standard a minimum loading criteria of 0.4 what we would have seen in this particular case you would still had with Factor one you would have had uh one two three items uh right there four or five items uh six seven items that would uh and eight items that would have met the minimum loading criteria but this item would not have been included on that factor uh if for the omniscient Authority Factor right here uh basically all three of them would have met the minimum loading criteria but then with factor three you would have seen that these two items would have met the loading criteria but then the remainder would not have in which case we would have had a factor with two items um defining that factor and so in that particular case then that would suggest remember that basically we need to have at least three indicators that meet our loading criteria on that Factor so in that particular case that would suggest and that factor three could be a kind of a problematic Factor we might need to consider either removing the items or consider maybe re refactoring maybe even suggesting a two-factor solution so uh you know that's just kind of the nature of factor analysis there are a lot of decisions that that can be made there are a lot of decision points where people agree and disagree um so this was again just uh designed to kind of give you an idea about some of those decisions using SPSS so at any rate that's going to wrap up this video presentation and I appreciate you watching