Discriminant Function AnalyisCalifornia State University 判别函数分析加利福尼亚州立大学

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1、Discriminant Function AnalysisBasicsPsy524Andrew AinsworthBasicsUsed to predict group membership from a set of continuous predictors Think of it as MANOVA in reverse in MANOVA we asked if groups are significantly different on a set of linearly combined DVs. If this is true, than those same “DVs” can

2、 be used to predict group membership.BasicsHow can continuous variables be linearly combined to best classify a subject into a group?BasicsMANOVA and disriminant function analysis are mathematically identical but are different in terms of emphasisdiscrim is usually concerned with actually putting pe

3、ople into groups (classification) and testing how well (or how poorly) subjects are classifiedEssentially, discrim is interested in exactly how the groups are differentiated not just that they are significantly different (as in MANOVA)BasicsPredictors can be given higher priority in a hierarchical a

4、nalysis giving essentially what would be a discriminate function analysis with covariates (a discrim version of MANCOVA)Questionsthe primary goal is to find a dimension(s) that groups differ on and create classification functions Can group membership be accurately predicted by a set of predictors? E

5、ssentially the same question as MANOVAQuestionsAlong how many dimensions do groups differ reliably? creates discriminate functions (like canonical correlations) and each is assessed for significance. Usually the first one or two discriminate functions are worth while and the rest are garbage. Each d

6、iscrim function is orthogonal to the previous and the number of dimensions (discriminant functions) is equal to either the g - 1 or p, which ever is smaller.QuestionsAre the discriminate functions interpretable or meaningful?Does a discrim function differentiate between groups in some meaningful way

7、 or is it just jibberish?How do the discrim functions correlate with each predictor?QuestionsCan we classify new (unclassified) subjects into groups?Given the classification functions how accurate are we? And when we are inaccurate is there some pattern to the misclassification?What is the strength

8、of association between group membership and the predictors?QuestionsWhich predictors are most important in predicting group membership?Can we predict group membership after removing the effects of one or more covariates?Can we use discriminate function analysis to estimate population parameters?Assu

9、mptionsThe interpretation of discrim results are always taken in the context of the research design. Once again, fancy statistics do not make up for poor design.AssumptionsUsually discrim is used with existing groups (e.g. diagnoses, etc.) if classification is your goal you dont really careIf random

10、 assignment and you predict if subjects came from the treatment or control group then causal inference can be made.Assumptions are the same as those for MANOVAAssumptionsMissing data, unequal samples, number of subjects and powerMissing data needs to be handled in the usual waysSince discrim is typi

11、cally a one-way design unequal samples are not really an issueWhen classifying subjects you need to decide if you are going to weight the classifications by the existing inequalityAssumptionsYou need more cases than predictors in the smallest group small sample may cause something called overfitting

12、.If there are more DVs than cases in any cell the cell will become singular and cannot be inverted. If only a few cases more than DVs equality of covariance matrices is likely to be rejected.AssumptionsPlus, with a small cases/DV ratio power is likely to be very smallyou can use programs like GANOVA

13、 to calculate power in MANOVA designs or you can estimate it by picking the DV with the smallest effect expected and calculate power on that variable in a univariate methodAssumptionsMultivariate normality assumes that the means of the various DVs in each cell and all linear combinations of them are

14、 normally distributed.Difficult to show explicitlyIn univariate tests robustness against violation of the assumption is assured when the degrees of freedom for error is 20 or more and equal samplesAssumptionsIf there is at least 20 cases in the smallest cell the test is robust to violations of multi

15、variate normality even when there is unequal n.If you have smaller unbalanced designs than the assumption is assessed on the basis of judgment; usually OK if violation is caused by skewness and not outliers.Absence of outliers the test is very sensitive to outlying cases so univariate and multivaria

16、te outliers need to be assessed in every groupAssumptionsHomogeneity of Covariance Matrices Assumes that the variance/covariance matrix in each cell of the design is sampled from the same population so they can be reasonably pooled together to make an error termWhen inference is the goal discrim is

17、robust to violations of this assumptionAssumptions When classification is the goal than the analysis is highly influenced by violations because subjects will tend to be classified into groups with the largest dispersion (variance) This can be assessed by plotting the discriminant function scores for

18、 at least the first two functions and comparing them to see if they are about the same size and spread. If violated you can transform the data, use separate matrices during classification, use quadratic discrim or use non-parametric approaches to classification.AssumptionsLinearity Discrim assumes l

19、inear relationships between all predictors within each group. Violations tend to reduce power and not increase alpha.Absence of Multicollinearity/Singularity in each cell of the design. You do not want redundant predictors because they wont give you anymore info on how to separate groups.EquationsSi

20、gnificance of the overall analysis; do the predictors separate the groups?The good news is the fundamental equations that test the significance of a set of discriminant functions are identical to MANOVAEquationstotalbgwgSSSEquations Predictors Group Perf Info Verbexp Age 87 5 31 6.4 97 7 36 8.3 Memo

21、ry 112 9 42 7.2 102 16 45 7 85 10 38 7.6 Perception 76 9 32 6.2 120 12 30 8.4 85 8 28 6.3 Communication 99 9 27 8.2 Equations 314.889 -71.556 -180.000 14.489 -71.556 32.889 8.000 -2.222 Sbg = -180.000 8.000 168.000 -10.400 14.489 -2.222 -10.400 0.736 1286.000 220.000 348.333 50.000 220.000 45.333 73

22、.667 6.367 Swg = 348.333 73.667 150.000 9.733 50.000 6.367 9.733 5.493 Equations13134.70034789 10448.63489 10.010477wgbgwgwgbgwgSSSSSS EquationsThe approximate F ratio is found by:22221/2124,2,6(4) (2)422(4)(2)5(.010477).1023574(2)842 14(2)2(2) 66221 .1023576 (8,6)6.58.1023578bgbgpdfdfsydfdfapproxim

23、ate FEquationsAssessing individual dimensions (discriminant functions)Discriminant functions are identical to canonical correlations between the groups on one side and the predictors on the other side.The maximum number of functions is equal to either the number of groups minus 1 or the number of pr

24、edictors, which ever is smallerEquations If the overall analysis is significant than most likely at least the first discrim function will be significant Once the discrim functions are calculated each subject is given a discriminant function score, these scores are than used to calculate correlations

25、 between the entries and the discriminant scores (loadings):Equations a standardized discriminant function score ( ) equals the standardized scores times its standardized discriminant function coefficient ( ) where each is chosen to maximize the differences between groups. You can use a raw score fo

26、rmula as well.1 122iiiippDd zd zd ziDididEquationsCentroids are group means on A canonical correlation is computed for each discriminant function and it is tested for significance. Any significant discriminant function can then be interpreted using the loading matrix (later)iDEquationsClassification

27、If there are only two groups you can classify based on the discriminant function scores, if they are above 0 they are in one group and if they are below 0 they are in the other.When there are more than two groups use the classification formulaEquations Classification score for group j is found by mu

28、ltiplying the raw score on each predictor (x) by its associated classification function coefficient (cj), summing over all predictors and adding a constant, cj001 1jjjjppCScc xc xEquations The coefficients are found by taking the inverse of the within subjects covariance matrix W and multiplying it

29、by the predictor means:1jjCW MEquations and the intercept is found by:012jjjcC M Equations using the example: 1286.000 220.000 348.333 50.000 220.000 45.333 73.667 6.367 Swg = 348.333 73.667 150.000 9.733 50.000 6.367 9.733 5.493 Swg/dfwg=WEquations 214.333 36.667 58.056 8.333 36.667 7.556 12.278 1.

30、061 W = 58.056 12.278 25.000 1.622 8.333 1.061 1.622 0.916 0.044 -0.202 0.010 -0.180 -0.202 1.630 -0.371 0.606 W-1 = 0.010 -0.371 0.201 -0.013 -0.180 0.606 -0.013 2.050 Equations 0.044 -0.202 0.010 -0.180 98.67 -0.202 1.630 -0.371 0.606 7 C1= 0.010 -0.371 0.201 -0.013 X 36.33 -0.180 0.606 -0.013 2.050 7.30 = 1.92 -17.56 5.55 .99Equations These steps are done for each person for each group1,098.677.00c = (-1/2) 1.92 -17.56 5.55 .9936.337.30EquationsClassification with a prior weights from sample sizes (unequal groups problem)01ln(/)pjjjiijiCcc XnN