DFA {DFA.CANCOR}

Discriminant function analysis

Description

Produces SPSS- and SAS-like output for linear discriminant function analysis.

Usage

DFA(data, groups, variables, plot, predictive, priorprob, covmat_type, CV, verbose)

Arguments

data

A dataframe where the rows are cases & the columns are the variables.

groups

The name of the groups variable in the dataframe,
e.g., groups = 'Group'.

variables

The names of the continuous variables in the dataframe that will be used in the DFA, e.g., variables = c('varA', 'varB', 'varC').

plot

Should a plot of the mean standardized discriminant function scores
for the groups be produced? The options are: TRUE (default) or FALSE.

predictive

Should a predictive DFA be conducted?
The options are: TRUE (default) or FALSE.

priorprob

If predictive = TRUE, how should the prior probabilities of the group sizes be computed? The options are:
'EQUAL' for equal group sizes; or
'SIZES' (default) for the group sizes to be based on the sizes of the groups in the dataframe.

covmat_type

The kind of covariance to be used for a predictive DFA. The options are:
'within' (for the pooled within-groups covariance matrix, which is the default) or
'separate' (for separate-groups covariance matrices).

CV

If predictive = TRUE, should cross-validation (leave-one-out cross-validation) analyses also be conducted? The options are: TRUE (default) or FALSE.

verbose

Should detailed results be displayed in console?
The options are: TRUE (default) or FALSE.

Details

The predictive DFA option using separate-groups covariance matrices (which is often called 'quadratic DFA') is conducted following the procedures described by Rencher (2002). The covariance matrices in this case are based on the scores on the continuous variables. In contrast, the 'separate-groups' option in SPSS involves use of the group scores on the discriminant functions (not the original continuous variables), which can produce different classifications.

When data has many cases (e.g., > 1000), the leave-one-out cross-validation analyses can be time-consuming to run. Set CV = FALSE to bypass the predictive DFA cross-validation analyses.

See the documentation below for the GROUP.DIFFS function for information on the interpretation of the Bayes factors and effect sizes that are produced for the group comparisons.

Value

If verbose = TRUE, the displayed output includes descriptive statistics for the groups, tests of univariate and multivariate normality, the results of tests of the homogeneity of the group variance-covariance matrices, eigenvalues & canonical correlations, Wilks' lambda & peel-down statistics, raw and standardized discriminant function coefficients, structure coefficients, functions at group centroids, one-way ANOVA tests of group differences in scores on each discriminant function, one-way ANOVA tests of group differences in scores on each original DV, significance tests for group differences on the original DVs according to Bird et al. (2014), a plot of the group means on the standardized discriminant functions, and extensive output from predictive discriminant function analyses (if requested).

The returned output is a list with elements

evals

eigenvalues and canonical correlations

mv_Wilks

The Wilks' lambda multivariate test

mv_Pillai

The Pillai-Bartlett multivariate test

mv_Hotelling

The Lawley-Hotelling multivariate test

mv_Roy

Roy's greatest characteristic root multivariate test

coefs_raw

canonical discriminant function coefficients

coefs_structure

structure coefficients

coefs_standardized

standardized coefficients

coefs_standardizedSPSS

standardized coefficients from SPSS

centroids

unstandardized canonical discriminant functions evaluated at the group means

centroidSDs

group standard deviations on the unstandardized functions

centroidsZ

standardized canonical discriminant functions evaluated at the group means

centroidSDsZ

group standard deviations on the standardized functions

dfa_scores

scores on the discriminant functions

anovaDFoutput

One-way ANOVAs using the scores on a discriminant function as the DV

anovaDVoutput

One-way ANOVAs on the original DVs

MFWER1.sigtest

Significance tests when controlling the MFWER by (only) carrying out multiple t tests

MFWER2.sigtest

Significance tests for the two-stage approach to controling the MFWER

classes_PRED

The predicted group classifications

classes_CV

The classifications from leave-one-out cross-validations, if requested

posteriors

The posterior probabilities for the predicted group classifications

grp_post_stats

Group mean posterior classification probabilities

classes_CV

Classifications from leave-one-out cross-validations

freqs_ORIG_PRED

Cross-tabulation of the original and predicted group memberships

chi_square_ORIG_PRED

Chi-square test of independence

PressQ_ORIG_PRED

Press's Q significance test of classifiation accuracy for original vs. predicted group memberships

kappas_ORIG_PRED

Agreement (kappas) between the predicted and original group memberships

PropOrigCorrect

Proportion of original grouped cases correctly classified

freqs_ORIG_CV

Cross-Tabulation of the cross-validated and predicted group memberships

chi_square_ORIG_CV

Chi-square test of indepedence

PressQ_ORIG_CV

Press's Q significance test of classifiation accuracy for cross-validated vs. predicted group memberships

kappas_ORIG_CV

Agreement (kappas) between the cross-validated and original group memberships

PropCrossValCorrect

Proportion of cross-validated grouped cases correctly classified

Author(s)

Brian P. O'Connor

References

Bird, K. D., & Hadzi-Pavlovic, D. (2013). Controlling the maximum familywise Type I error rate in analyses of multivariate experiments. Psychological Methods, 19(2), p. 265-280.

Manly, B. F. J., & Alberto, J. A. (2017). Multivariate statistical methods: A primer (4th Edition). Chapman & Hall/CRC, Boca Raton, FL.

Rencher, A. C. (2002). Methods of Multivariate Analysis (2nd ed.). New York, NY: John Wiley & Sons.

Sherry, A. (2006). Discriminant analysis in counseling research. Counseling Psychologist, 34, 661-683.

Tabachnik, B. G., & Fidell, L. S. (2019). Using multivariate statistics (7th ed.). New York, NY: Pearson.

Examples

# data from Field et al. (2012, Chapter 16 MANOVA)
DFA_Field=DFA(data = data_DFA$Field_2012, 
    groups = 'Group', 
    variables = c('Actions','Thoughts'),
    predictive = TRUE, 
    priorprob = 'EQUAL',   
    covmat_type='within', # altho better to use 'separate' for these data
    verbose = TRUE)



# plots of posterior probabilities by group
# hoping to see correct separations between cases from different groups

# first, display the posterior probabilities
print(cbind(round(DFA_Field$posteriors[1:3],3), DFA_Field$posteriors[4]))

# group NT vs CBT
plot(DFA_Field$posteriors$posterior_NT, DFA_Field$posteriors$posterior_CBT, 
     pch = 16, col = c('red', 'blue', 'green')[DFA_Field$posteriors$Group],
     xlim=c(0,1), ylim=c(0,1),
     main = 'DFA Posterior Probabilities by Original Group Memberships',
     xlab='Posterior Probability of Being in Group NT',
     ylab='Posterior Probability of Being in Group CBT' )
legend(x=.8, y=.99, c('CBT','BT','NT'), cex=1.2, col=c('red', 'blue', 'green'), pch=16, bty='n')

# group NT vs BT
plot(DFA_Field$posteriors$posterior_NT, DFA_Field$posteriors$posterior_BT, 
     pch = 16, col = c('red', 'blue', 'green')[DFA_Field$posteriors$Group],
     xlim=c(0,1), ylim=c(0,1),
     main = 'DFA Posterior Probabilities by Group Membership',
     xlab='Posterior Probability of Being in Group NT',
     ylab='Posterior Probability of Being in Group BT' )
legend(x=.8, y=.99, c('CBT','BT','NT'), cex=1.2,col=c('red', 'blue', 'green'), pch=16, bty='n')

# group CBT vs BT
plot(DFA_Field$posteriors$posterior_CBT, DFA_Field$posteriors$posterior_BT, 
     pch = 16, col = c('red', 'blue', 'green')[DFA_Field$posteriors$Group],
     xlim=c(0,1), ylim=c(0,1),
     main = 'DFA Posterior Probabilities by Group Membership',
     xlab='Posterior Probability of Being in Group CBT',
     ylab='Posterior Probability of Being in Group BT' )
legend(x=.8, y=.99, c('CBT','BT','NT'), cex=1.2, col=c('red', 'blue', 'green'), pch=16, bty='n')


# data from Green & Salkind (2008, Lesson 35)  
DFA(data = data_DFA$Green_2008,
    groups = 'job_cat', 
    variables = c('friendly','gpa','job_hist','job_test'),
    plot=TRUE,
    predictive = TRUE,
    priorprob = 'SIZES',  
    covmat_type='within', 
    CV=TRUE, 
    verbose=TRUE) 


# data from Ho (2014, Chapter 15) 
# with group_1 as numeric 
DFA(data = data_DFA$Ho_2014,
    groups = 'group_1_num',
    variables = c("fast_ris", "disresp", "sen_seek", "danger"),
    plot=TRUE,
    predictive = TRUE,
    priorprob = 'SIZES', 
    covmat_type='within', 
    CV=TRUE, 
    verbose=TRUE) 


# data from Ho (2014, Chapter 15)   
# with group_1 as a factor 
DFA(data = data_DFA$Ho_2014,
    groups = 'group_1_fac',
    variables = c("fast_ris", "disresp", "sen_seek", "danger"),
    plot=TRUE,
    predictive = TRUE,
    priorprob = 'SIZES', 
    covmat_type='within', 
    CV=TRUE, 
    verbose=TRUE) 


# data from Huberty (2006, p 45)
DFA_Huberty=DFA(data = data_DFA$Huberty_2019_p45, 
    groups = 'treatmnt_S', 
    variables = c('Y1','Y2'),
    predictive = TRUE, 
    priorprob = 'SIZES', 
    covmat_type='separate', # altho better to used 'separate' for these data
    verbose = TRUE)


# data from Huberty (2006, p 285)
DFA_Huberty=DFA(data = data_DFA$Huberty_2019_p285, 
    groups = 'Grade', 
    variables = c('counsum','gainsum','learnsum','qelib','qefac','qestacq',
                  'qeamt','qewrite','qesci'),
    predictive = TRUE, 
    priorprob = 'EQUAL', 
    covmat_type='within', 
    verbose = TRUE)


# data from Norusis (2012, Chaper 15)  
DFA_Norusis=DFA(data = data_DFA$Norusis_2012, 
    groups = 'internet', 
    variables = c('age','gender','income','kids','suburban','work','yearsed'),
    predictive = TRUE, 
    priorprob = 'EQUAL', 
    covmat_type='within', 
    verbose = TRUE)


# data from Rencher (2002, p 170)  - rootstock
DFA(data = data_DFA$Rencher_2002_root, 
    groups = 'rootstock', 
    variables = c('girth4','ext4','girth15','weight15'),
    predictive = TRUE, 
    priorprob = 'SIZES',
	covmat_type='within',     
	verbose = TRUE)


# data from Rencher (2002, p 280)  - football
DFA(data = data_DFA$Rencher_2002_football, 
    groups = 'grp', 
    variables = c('WDIM','CIRCUM','FBEYE','EYEHD','EARHD','JAW'),
    predictive = TRUE, 
    priorprob = 'SIZES',
	covmat_type='separate',     
	verbose = TRUE)


# Sherry (2006)   -  with Group as numeric 
DFA_Sherry <- DFA(data = data_DFA$Sherry_2006, 
                  groups = 'Group_num',
                  variables = c('Neuroticism','Extroversion','Openness', 
                                'Agreeableness','Conscientiousness'),
                  predictive = TRUE, 
                  priorprob = 'SIZES', 
                  covmat_type='separate', 
                  verbose = TRUE)


# Sherry (2006)   -  with Group as a factor 
DFA_Sherry <- DFA(data = data_DFA$Sherry_2006, 
                  groups = 'Group_fac',
                  variables = c('Neuroticism','Extroversion','Openness', 
                                'Agreeableness','Conscientiousness'),
                  predictive = TRUE, 
                  priorprob = 'SIZES', 
                  covmat_type='separate', 
                  verbose = TRUE)

# plots of posterior probabilities by group
# hoping to see correct separations between cases from different groups

# first, display the posterior probabilities
print(cbind(round(DFA_Sherry$posteriors[1:3],3), DFA_Sherry$posteriors[4]))

# group 1 vs 2
plot(DFA_Sherry$posteriors$posterior_1, DFA_Sherry$posteriors$posterior_2, 
     pch = 16, cex = 1, col = c('red', 'blue', 'green')[DFA_Sherry$posteriors$Group],
     xlim=c(0,1), ylim=c(0,1),
     main = 'DFA Posterior Probabilities by Original Group Memberships',
     xlab='Posterior Probability of Being in Group 1',
     ylab='Posterior Probability of Being in Group 2' )
legend(x=.8, y=.99, c('1','2','3'), cex=1.2, col=c('red', 'blue', 'green'), pch=16, bty='n')

# group 1 vs 3
plot(DFA_Sherry$posteriors$posterior_1, DFA_Sherry$posteriors$posterior_3, 
     pch = 16, col = c('red', 'blue', 'green')[DFA_Sherry$posteriors$Group],
     xlim=c(0,1), ylim=c(0,1),
     main = 'DFA Posterior Probabilities by Group Membership',
     xlab='Posterior Probability of Being in Group 1',
     ylab='Posterior Probability of Being in Group 3' )
legend(x=.8, y=.99, c('1','2','3'), cex=1.2,col=c('red', 'blue', 'green'), pch=16, bty='n')

# group 2 vs 3
plot(DFA_Sherry$posteriors$posterior_2, DFA_Sherry$posteriors$posterior_3, 
     pch = 16, col = c('red', 'blue', 'green')[DFA_Sherry$posteriors$Group],
     xlim=c(0,1), ylim=c(0,1),
     main = 'DFA Posterior Probabilities by Group Membership',
     xlab='Posterior Probability of Being in Group 2',
     ylab='Posterior Probability of Being in Group 3' )
legend(x=.8, y=.99, c('1','2','3'), cex=1.2, col=c('red', 'blue', 'green'), pch=16, bty='n')
    

# Tabachnik & Fiddel (2019, p 307, 311)   - small - with group as numeric
DFA(data = data_DFA$TabFid_2019_small, 
    groups = 'group_num', 
    variables = c('perf','info','verbexp','age'),
    predictive = TRUE, 
    priorprob = 'SIZES', 
    covmat_type='within', 
    verbose = TRUE)  
    
    
# Tabachnik & Fiddel (2019, p 307, 311)   - small - with group as a factor
DFA(data = data_DFA$TabFid_2019_small, 
    groups = 'group_fac', 
    variables = c('perf','info','verbexp','age'),
    predictive = TRUE, 
    priorprob = 'SIZES', 
    covmat_type='within', 
    verbose = TRUE)  


# Tabachnik & Fiddel (2019, p 324)   - complete  - with WORKSTAT as numeric 
DFA(data = data_DFA$TabFid_2019_complete, 
    groups = 'WORKSTAT_num', 
    variables = c('CONTROL','ATTMAR','ATTROLE','ATTHOUSE'),
    plot=TRUE,
    predictive = TRUE,
    priorprob = 'SIZES',  
    covmat_type='within', 
    CV=TRUE, 
    verbose=TRUE) 

# Tabachnik & Fiddel (2019, p 324)   - complete  -  with WORKSTAT as a factor  
DFA(data = data_DFA$TabFid_2019_complete, 
    groups = 'WORKSTAT_fac', 
    variables = c('CONTROL','ATTMAR','ATTROLE','ATTHOUSE'),
    plot=TRUE,
    predictive = TRUE,
    priorprob = 'SIZES',  
    covmat_type='within', 
    CV=TRUE, 
    verbose=TRUE)