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Basics Basics Discriminant Function Analysis
East Central University, US has reference to this Academic Journal, Discriminant Function Analysis Basics Psy524 Andrew Ainsworth Basics Used so that 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 be used so that predict group membership. Basics How can continuous variables be linearly combined so that best classify a subject into a group?
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Basics MANOVA in addition to disriminant function analysis are mathematically identical but are different in terms of emphasis discrim is usually concerned alongside actually putting people into groups (classification) in addition to testing how well (or how poorly) subjects are classified Essentially, discrim is interested in exactly how the groups are differentiated not just that they are significantly different (as in MANOVA) Basics Predictors can be given higher priority in a hierarchical analysis giving essentially what would be a discriminate function analysis alongside covariates (a discrim version of MANCOVA) Questions the primary goal is so that find a dimension(s) that groups differ on in addition to create classification functions Can group membership be accurately predicted by a set of predictors? Essentially the same question as MANOVA
Questions Along how many dimensions do groups differ reliably? creates discriminate functions (like canonical correlations) in addition to each is assessed in consideration of significance. Usually the first one or two discriminate functions are worth while in addition to the rest are garbage. Each discrim function is orthogonal so that the previous in addition to the number of dimensions (discriminant functions) is equal so that either the g – 1 or p, which ever is smaller. Questions Are the discriminate functions interpretable or meaningful? Does a discrim function differentiate between groups in some meaningful way or is it just jibberish? How do the discrim functions correlate alongside each predictor? Questions Can we classify new (unclassified) subjects into groups? Given the classification functions how accurate are we? And when we are inaccurate is there some pattern so that the misclassification? What is the strength of association between group membership in addition to the predictors?
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Questions Which 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 so that estimate population parameters? Assumptions The interpretation of discrim results are always taken in the context of the research design. Once again, fancy statistics do not make up in consideration of poor design. Assumptions Usually discrim is used alongside existing groups (e.g. diagnoses, etc.) if classification is your goal you don?t really care If random assignment in addition to you predict if subjects came from the treatment or control group then causal inference can be made. Assumptions are the same as those in consideration of MANOVA
Assumptions Missing data, unequal samples, number of subjects in addition to power Missing data needs so that be handled in the usual ways Since discrim is typically a one-way design unequal samples are not really an issue When classifying subjects you need so that decide if you are going so that weight the classifications by the existing inequality Assumptions You need more cases than predictors in the smallest group small sample may cause something called overfitting. If there are more DVs than cases in any cell the cell will become singular in addition to cannot be inverted. If only a few cases more than DVs equality of covariance matrices is likely so that be rejected. Assumptions Plus, alongside a small cases/DV ratio power is likely so that be very small you can use programs like GANOVA so that calculate power in MANOVA designs or you can estimate it by picking the DV alongside the smallest effect expected in addition to calculate power on that variable in a univariate method
Assumptions Multivariate normality ? assumes that the means of the various DVs in each cell in addition to all linear combinations of them are normally distributed. Difficult so that show explicitly In univariate tests robustness against violation of the assumption is assured when the degrees of freedom in consideration of error is 20 or more in addition to equal samples Assumptions If there is at least 20 cases in the smallest cell the test is robust so that violations of multivariate 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 in addition to not outliers. Absence of outliers ? the test is very sensitive so that outlying cases so univariate in addition to multivariate outliers need so that be assessed in every group Assumptions Homogeneity 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 so that make an error term When inference is the goal discrim is robust so that violations of this assumption
Assumptions When classification is the goal than the analysis is highly influenced by violations because subjects will tend so that be classified into groups alongside the largest dispersion (variance) This can be assessed by plotting the discriminant function scores in consideration of at least the first two functions in addition to comparing them so that see if they are about the same size in addition to spread. If violated you can transform the data, use separate matrices during classification, use quadratic discrim or use non-parametric approaches so that classification. Assumptions Linearity ? Discrim assumes linear relationships between all predictors within each group. Violations tend so that reduce power in addition to not increase alpha. Absence of Multicollinearity/Singularity in each cell of the design. You do not want redundant predictors because they won?t give you anymore info on how so that separate groups. Equations Significance 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 so that MANOVA
Equations Equations Equations
Equations Equations The approximate F ratio is found by: Equations Assessing individual dimensions (discriminant functions) Discriminant functions are identical so that canonical correlations between the groups on one side in addition to the predictors on the other side. The maximum number of functions is equal so that either the number of groups minus 1 or the number of predictors, which ever is smaller
Equations 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 so that calculate correlations between the entries in addition to the discriminant scores (loadings): Equations a standardized discriminant function score ( ) equals the standardized scores times its standardized discriminant function coefficient ( ) where each is chosen so that maximize the differences between groups. You can use a raw score formula as well. Equations Centroids are group means on A canonical correlation is computed in consideration of each discriminant function in addition to it is tested in consideration of significance. Any significant discriminant function can then be interpreted using the loading matrix (later)
Equations = [1.92 -17.56 5.55 .99] Equations These steps are done in consideration of each person in consideration of each group Equations Classification alongside a prior weights from sample sizes (unequal groups problem)
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