discriminant analysis using SPSS video 1

in this video I am going to discuss how to carry out a discriminant analysis using SPSS basically discriminant analysis is a statistical procedure aimed at predicting group membership on a categorical dependent variable based on scores or observations on a set of predictor variables in addition to determining how well I am able to predict group membership I can also describe those combinations of predictor variables that are producing the group separation that might be observed with respect to the predictor variables so in this with respect to the predictor variables it's assumed that they are following in an interval or ratio level of measurement and that they exhibit multivariate normality within your groups and the dependent variable is as I said before it's categorical generally it will follow the nominal or an ordinal level of measurement so in this particular illustration here I've got a set of predictor variables I've got interest in politics dogmatism and external political efficacy and so these three variables are going to be included in a discriminant analysis predicting intention to donate money to a political cause in the next month so a value of zero reflects an intention not to donate the value of one reflects an intention to donate so each row is basically reflecting a person's responses to a set of variables so person one intends not to donate and these are that person's scores on the three predictor variables person three intends to donate and these are the values on the predictor variables so to run the discriminant analysis I'm going to go to analyze classify and go down to discriminant I'll put donate into the grouping variable box so this is for the DV and you can see question marks come up I'm going to click on define range the minimum value of zero that's the lowest value my dependent variable the maximum value is 1 so and these values that I'm putting in here are arbitrary but basically just reflects a dummy coding system that I was using in a previous analysis so now I'm going to click on continue and put my predictor variables into the independence box now under statistics I'm going to click on means I could ask for univariate Innovas which are basically one way ANOVA is comparing my two groups on each of the dependent variables and this would be the same information as the between-subjects test following a one way manova in in manova output next is boxes am this is basically a test of the assumption of equality of variance covariance matrices between your groups in other words the variances of my three predictor variables and their covariances have to be the same across groups and so we can test this out using boxes in the so if the Equality is observed then we would basically assume that our assumption is is met if it's not met and that can introduce various statistical problems so at any rate I'm going to click on continue and under classify what I might be interested in knowing is how well does my statistical model my predictor or how well do my predictors predict group membership so basically what happens is is that based on the relationship that's observed between my predictor variables and my dependent variable a prediction is made in terms of group membership so basically in this particular illustration here individuals would be predicted to following to group 0 or group 1 based on their responses to the predictor variables and so I might be curious to know how well do the predicted and actual group membership actually correspond so I can do that by clicking on summary table down here to get an idea and so this is kind of a another way that you can sort of look at the question of the fit of the model now it happens to be the case of my data set that I do not have equal numbers of observations that are falling into my different groups so the group sizes are quite unequal and so I want to take that into account when I'm looking at the probabilities associated with group membership so I'm gonna click on compute from group sizes and then click on continue you can see if you click under save you could actually get predicted group memberships and then you've also got probabilities of group memberships and so basically the way this the procedure works is it generates a probability that a person will fall into group 0 our Group 1 and then based on that probability it assigns a predicted group membership so basically if there's a high probability that a given case falls into group 1 then the predicted group membership would be 1 and then you know the summary table then that we saw under classify right here would allow us to look at the correspondences between predicted and actual group memberships so I'm not actually going to ask for this information here but this is just to let you know that they are this is information that you can generate and what it would generate is basically additional columns in your SPSS output that would contain predicted group memberships and the probabilities associated with your group memberships so now I'm going to click on OK and here's my output so as I scroll down you can see right here I've got basically my group 0 and Group 1 right here I've got the means associated with each of my DVS and Group 1 and the means for group 2 are group 0 and group 1 excuse me and then standard deviations and so forth down here these are the univariate ANOVA so basically this is comparing the intention to donate and the intention not to donate groups with respect to political interest dogmatism and external political XE so using sort of the conventional 0.05 threshold for rejecting the null hypothesis you know we would say well there's a significant difference between groups on political interests political excuse me dogmatism and external political advocacy so scrolling down a little bit further you can see we now have boxes tests and so this is again this is a test of the assumption of equality of population variance covariance matrices so this is your p-value for this particular test and if when you compare this value against your alpha level or alpha threshold if this indicates that it's non significant that's actually a good thing because you're meeting the assumption of the statistical procedure so the conventional alpha level at 0.05 you can see right here that the p-value is actually greater than 0.05 which again is the conventional threshold so we would maintain the null hypothesis that there is equality of variance covariance matrices and thus we would assume that our assumption is met so it's basically using the same general logic that we see with Levine's test only it's applied to the variance covariance matrices so moving on you'll see that we have the summary of canonical discriminant functions and so basically a discriminant function basically represents a combination a mathematical combination or a linear function in which the information from the predictor variables is combined to form a discriminant function and that discriminant function is essentially serving as a latent variable that the groups are then separated on so in a nutshell the summary is basically describing the dimensions upon which our groups are separated and so the number of discriminant functions that you will observe in discriminant analysis is equal to the number of groups minus one or the number of predictor variables whichever is least okay so in this analysis here we had essentially three predictor variables and we had two groups so two minus one is obviously one so the minimum number or value is would be one and so that would give us one discriminant function so that's basically one way in which the variables can be combined to produce Group separation so the linear function is basically serving as excuse me the discriminant function is essentially serving as sort of a latent variable that represents the combined information from the set of predictors and so when we look at this part of our table right here we'll see that we have function one so there's only one linear combination that can produce group separation with respect to the predictor variables the eigenvalue is basically a variance estimate and then we have percentage of variance and cumulative percentage of variance so basically if we happen to have a situation where we had more than a single discriminant function then we would have say function one function to maybe if there were even more functions each of them would have an eigen value and a percentage of variance associated with them and and the percentage of variance accounted for is actually it would be descending in terms of the amount and then the cumulative percentage would actually sum up to a hundred percent so in this case because we only have a single function a hundred percent of the between-groups variation is accounted for and the cumulative percentage is also 100 percent the canonical correlation is essentially giving you the relationship or it's a correlational index of the relationship between the discriminant function and and our grouping variable and so it's going to range between zero and one and it's interpreted pretty much along the lines of that we would use when interpreting Pearson's correlation so obviously values that are closer to one indicate a greater relationship between our discriminant function and the grouping variable values close to zero obviously would reflect less a relationship so in other words closer to zero there's less discrimination between groups with respect to our discriminant function down here we have a test of our functions and so with only a single discriminant function we have just a test that addresses the question of is there a significant relationship between our discriminant function and and our grouping variable and so using a conventional 0.05 threshold for rejecting the null we can see our p-value right here our p-value is clearly less than 0.05 and so we would reject a null and essentially infer that there is a statistically significant relationship between our discriminant function and our and our grouping variable now if there were more functions for instance let's say we had two functions instead of one then essentially the significance test would would be carried out in a peel-off fashion so in other words there would be a test for the first function of functions one to two so looking at the total the relationship between the set of discriminant functions and the grouping variable and then we would have a test of function two which would it would be a test of that function proper if there were actually three functions then we would have actually a test of 1 and 1/2 1 2 3 then 2 2 3 and then just 3 by itself but like I said in the cut in this case right here we only have one discriminant function so this would actually be a test of that function proper so now scrolling down we so at this point we've we've just determined that yes there is a statistically significant relationship between the discriminant function and our grouping variables so the next question might be well can we describe that function what what does the function really mean and so this is where we can go about the business of naming the discriminant function because right now just saying that hey the groups are different on a discriminant function doesn't tell you a whole lot so what we want to do is we want to name the function to give it some kind of representation that actually means something to people so what we can do is we can look at two pieces of information there's the structure matrix and the matrix of standardized canonical discriminant function coefficients so the structure matrix basically gives you the zero order correlations between each of your predictor variables and the discriminant function so we had function one which was statistically significant so at this point because it was significant we would want to actually name that particular function and so we can do that by looking at the correlations between each of our predictor variables and that function so you can see that political interest correlates at 0.8 4 5 external political efficacy correlates at 0.75 8 and then dogmatism correlates at 0.40 6 so dogmatism is still hanging in there as a pretty good predicting a pretty good relationship with the function 1 but it's not quite as powerful in terms of the zero order correlations so in terms of interpreting the structure matrix this would be very much analogous to interpreting essentially you know a structure matrix in the context of factor analysis where basically you look at the zero order correlations between your measured variables and the factors in factor analysis so based on this I might consider function one I might name this something like political mindedness okay to kind of capture of the information associated with the predictor now I could go even further and I could look at this matrix right here the standardized canonical discriminant function coefficients and get a you know get a little bit more flavor as to what's going on and basically these coefficients right here are analogous to beta waves in the context of regression analysis so you can actually then talk a little bit so when you look at the relationship between a given predictor and the function you're looking at that relationship controlling for or purling out the other predictor variables so like I said this is basically analogous to a beta wait and so when we look at the relationship between political interest and function one controlling for dogmatism and external political efficacy you can see that this value is fairly high followed by dogmatism so dogmatism when you're looking at its relationship to the function controlling for political interests in external political efficacy you can see that actually it kind of weighs in with the the next strongest weight and then external political efficacy kind of weighs in with the the lowest weight nevertheless they're still you know they're still all pretty strongly related to the discriminant function so you know I can still you know again maintain the political mind in this best describes what's going on with this particular function but you know if I wanted to talk about which variables are contributing more to group separation relative to the other variables political interest really is kind of doing most of the work followed by dogmatism followed by external political AXI so you know when I talk about the discriminant function I can talk about in terms of the name or talked about it in terms of you know the relative contributions of the predictors to that particular function so moving on we see that we've got functions at group centroids and so basically this is essentially like a multivariate mean group centroid is just reflecting the intersection of the means your your predictor variables within each group and so essentially you've got group 0 which has a mean of negative point negative point 2 4 5 group 1 has a mean of 0.85 7 so you know essentially if you think about you know we have a single late dimension which is our discriminant function we have you know if we started you know kind of zero right here group zero is falling right around you know negative point two four five right around here this is just relative I don't obviously I don't have a scaling on here so but I'm just kind of giving you a visual representation of where things are located positionally so you know this is the difference that we're really observing right here so when we talk about the groups being different on the discriminant function you know this is basically look talking about the difference in these group centroids or though the multivariate means so that's essentially all it's happening so when I say that the discriminant function is related to the group membership kind of going back and looking at the the the canonical correlation and the significance test and then I go and interpret these two matrices here as political mindedness then that's good that basically tells me then that the groups are differing on political mindedness which is what I named that particular function and that people in the group 0 which was the do not intend to donate group had a lower mean on political mindedness than those people in Group one that were intending to donate so that actually would be consistent with theory so now moving on further we see that we've got our prior probabilities for our groups this is reflecting essentially the the the differences in the number of cases remember that we were not assuming that all groups were equal and we didn't want equal probabilities because of that that that fact so when we look at the classification results you can see right here that basically down here this tells us that a 3.3 percent of our cases were correctly classified based on our statistical model you know like I said in fact let me just kind of go back and just highlight you know if I go to classify discriminate and I'll just save it and and generate this information here you know when looking at my out my SPSS file you can see that up here we've got predicted group memberships and then probabilities of membership and group 0 for analysis one and then probabilities of membership in Group one so you can see you know for this particular case right here the probability of being in group 0 was 0.9 0-5 so basically nine percent chance of being in group zero based on the model and a nine percent probability of being in Group one so because of that this case was predicted to fall into group zero when you look down here you can see that so you know and obviously this person is falling into this is a predicted group membership falling a zero and this is our actual where they were actually falling into group zero when we look at this person here the probability of falling into group zero was 0.49 six eight and the probability of falling into group one was 0.503 so it's a little bit greater than 50% so the predicted group membership was one and so this person actually fell into Group one based on the data so you can see that there's correspondences for those two cases now when you look right here you can see that this person had a probability of falling into group zero point eight six and a probability of falling into Group one 0.14 and so the predicted group membership was group zero but then when you look over here they had their actual group membership was one so that would be a case where that person was incorrectly classified based on the predictors so that's essentially what this particular table is summarizing is the accuracy in terms of prediction so you can see that right here in terms of this this row the total is reflecting the total number of cases that were that actually expressing an intention not to donate so of the 126 individuals who expressed an intention not to donate we had 123 who were correctly classified based on the model to not intend to donate so they were correctly predicted not to express an intention to donate so down here you can see you've got percentages and so you can see that for this particular cell basically ninety roughly ninety-eight percent of the cases were correctly classified so basically ninety-eight percent of those individuals who actually did not indicate the intention to donate 98 percent of them were correctly predicted by the model to express the intention not to donate where you have this sale right here these would be essentially a case where you have a false positive where basically we had three people who were predicted based on the model to express an intention to donate and out of the 126 who did not and so we had 2.4 percent of of our distribution who were incorrectly predicted to express an intention donate and who were actually not who actually did not express the intention to donate then you've got a cell right here and so this would be another hit so these would be hits and hits based on the model so these would be these right here are reflecting correct classifications so in this case we had 36 cases who actually did indicate an intention to donate and of those 36 12 were correctly predicted by the model to in intention to donate so there's the correspondence between the one and the one right here so the hit rate for that is about thirty three point three percent so that's actually pretty low that's not so good and then you can see right here these would be your misses so basically as you kind of cross over here these would be misses and so in this particular case you've got essentially 66.7% are almost roughly sixty-seven percent of the cases who actually were observed to express an intention to donate sixty six point seven percent were incorrectly predicted not to express an intention to donate so the long and short of it is is that when we look at the overall you know kind of fit if you will in terms of the classification results you know we have pretty good prediction based on the model but when we look more closely we can see that the model is doing a much better job of predicting who will not express an intention to donate because basically these this is you know it's a it's correctly classifying at about ninety-eight percent whereas the model is doing really a pretty crappy job of predicting who will express an intention to donate so political mindedness seems to be you know doing a good job of discriminating between are predicting people who won't express who won't intend to donate but it doesn't do a particularly good job of predicting who will express the intention to donate so that pretty much wraps up this particular video on discriminant analysis obviously discriminant analysis can be generalized to cases where you have more than more than two groups but again the purpose of this video was just to illustrate discriminant analysis where there were just two groups involved

Channel: Mike Crowson

Views: 12,480

Rating: 4.8769231 out of 5

Keywords: Discriminant analysis using SPSS, descriminant analysis in spss

Id: LhZ59goALSg

Length: 26min 21sec (1581 seconds)

Published: Wed Nov 02 2016