Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Multiple-group linear discriminant function• maximum & contributing ldf dimensions• concentrated & diffuse ldf structures• follow-up analyses• evaluating & reporting k-group ldf• evaluating & reporting k-group MANOVALike ANOVA, ldf & MANOVA can be applied to 3+ groups.• When we have multiple groups there may be an advantage tousing multiple discriminant/MANOVA functions to maximally discriminate between the groups. • That is, we must decide whether the multiple groups “line up” on a single dimension (called a concentrated structure), orwhether they are best described by their position in amultidimensional “space” (called a diffuse structure).Maximum # dimensions for a given analysis: the smaller of # groups - 1# predictor variablese.g., 4 groups with 6 predictor variables ? Max # ldfs = _____Concentrated vs. Diffuse Structures – profile differencesBy inspecting the “group profiles,” (means of each group on each of the predictor variables) you can often anticipate whether there will be more than one ldf …•if the groups have similar patterns of differences (similar profiles) for each predictor variable (for which there are differences),then you would expect a single discriminant function.• If the groups have different profiles for different predictor variables, then you would expect more than one ldf Group Var1 Var2 Var3 Var4 Group Var1 Var2 Var3 Var4 1 10 12 6 8 1 10 12 6 14 2 18 12 10 2 2 18 6 6 14 3 18 12 10 2 3 18 6 2 7Concentrated + 0 + - Diffuse 1st + - 0 02nd 0 0 - -Your turn… Group Var1 Var2 Var3 Var4 1 23 35 8 38 2 20 36 7 39 3 11 61 2 40Concentrated + - + 0 Group Var1 Var2 Var3 Var4 1 20 33 28 38 2 18 15 26 37 3 42 13 42 38Diffuse ldf1 + 0 + 0 ldf2 0 + 0 0 Group Var1 Var2 Var3 Var4 1 20 13 28 38 2 18 15 46 36 3 42 43 42 11Diffuse ldf1 + + 0 - ldf2 0 0 + 0 Group Var1 Var2 Var3 Var4 1 21 33 26 68 2 19 34 28 65 3 20 35 12 18Concentrated 0 0 + +Determining the # dimensions (ldfs or MANOVAs)Like other “determinations”, there is a significance test involved• Each ldf/MANOVA variate is tested as to whether it “contributes to the model” using the X²/F-test of the -value.• The first variate will always account for the most between-group variation (have the largest X²/F and Rc) -- subsequent ldfs are “orthogonal” (providing independent information), and will account for successively less between group variation.• If there is a single variate, then the model is said to have a concentrated structure• if there are 2+ variates then the model has a diffuse structure• the distinction between a concentrated and a diffuse structure is considered the “fundamental multivariate question” in a multiple group analysis.Follow-up analyses• Within Psychology, ldf developed in areas of research that traditionally used large samples (e.g., measurement theory & clinical diagnostic research). • With such large samples, “almost everything is significant”. • So, an emphasis on “substantial effects” developed - based on “cutoffs” and “relative size” rather than significance tests• using % variance to determine if additional ldfs “contribute”• .3-.4 cutoff for structure weights when interpreting the ldfs• using % classification to discuss “what the model does” • As ldf was “adopted” into research areas with strong traditions of significance testing, more tests were incorporated into ldf, most commonly “follow-ups”• MANOVA, growing out of ANOVA, had strong ties to the NHST traditions, using both omnibus & follow-up significance testsThere are three major types of follow-up for both ldf & MANOVA• Univariate follow-ups -- abandoning the multivariate analysis, simply describe the results of the ANOVA (with pairwisecomparisons) for each of the predictors (DVs)• ldf follow-ups -- use the ldf(s) as DVs in ANOVA (with pairwise comparisons) to explicate what which ldfs discriminatebetween what groups• this nicely augments the spatial & re-classification depictions of ldf• if you have a concentrated structure, it tells you exactly what groups can be significantly discriminated• if you have a diffuse structure, it tells you whether the second ldf provides discriminatory power the 1st doesn’t• though ldf=MANOVA, differential availability of output led to “ldf as the follow-up of a significant MANOVA”• pairwise ldf/MANOVA follow-ups -- separate ldf/MANOVA analyses for each pair of groups to explicate groups are “multivariately different” from which other groups• for ldf the additional question is which variables maximally discriminate between what groups• this is just what it sounds like• compare groups 1 & 2 then 1 & 3 then 2 & 3• interpret the ldf and tell the discriminatory power for each• might produce pairwise discriminations not provided by the overall analysisDifferent texts/researchers seem to have strong opinions about which of these is the “more appropriate.” I’d suggest that usually one of them is a more direct test of the way you have conceptualized your research question or analysis. Trying them all probably won’t hurt (except for alpha-inflation, of course)!Reporting the Results of a k-group ldf Analysis1. Does the model work -- does each possible ldf contribute? •  for each ldf transformed into either X² to test whether or not that ldf contributes to the model2. How well does the model