Assessment of the Happiness Index of UP Diliman Undergraduate Students using Multivariate Analysis
Clarisse Rodriguez, Lara Elio, Iana Garcia
library(psych)
library(MVN)
library("GPArotation")
library(MVN)
library(devtools)## Loading required package: usethislibrary(factoextra)## Loading required package: ggplot2##
## Attaching package: 'ggplot2'## The following objects are masked from 'package:psych':
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## %+%, alpha## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBalibrary(cluster)
library(dendextend)##
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## To suppress this message use: suppressPackageStartupMessages(library(dendextend))
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## Attaching package: 'dendextend'## The following object is masked from 'package:stats':
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## cutreelibrary(psych)
library(biotools)## Loading required package: MASS## ---
## biotools version 4.2library(tidyverse)## -- Attaching packages --------------------------------------- tidyverse 1.3.1 --## v tibble 3.1.5 v dplyr 1.0.7
## v tidyr 1.1.4 v stringr 1.4.0
## v readr 2.0.2 v forcats 0.5.1
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## x dplyr::select() masks MASS::select()library(cluster)
library(factoextra)
library(knitr)hpind<-read.csv('hpindex2.csv',fill=TRUE)new_hp2<-hpind[,7:38] #removes columns on demographics
names(new_hp2)<- c('happiness_level', 'vitamin_intake', 'enough_sleep', 'physical_activities', 'physical_appearance', 'sleep_quality', 'degree_prog_satisfaction', 'class_anticipation', 'class_participation', 'degree_program_performance', 'learnings', 'university_satisfaction', 'leisure_time', 'procrastination', 'hobbies', 'multimedia', 'pos_effect_social_med', 'positive_outlook', 'regrets', 'sense_of_meaning_purpose', 'optimism_of_future', 'life_control', 'rewarding_view_of_life', 'happiness_choice', 'family_relationship', 'peer_relationship', 'company_of_animals', 'love_and_affection', 'social_interaction', 'extra_curricular', 'campus_safety', 'safety_going_home')kable(head(new_hp2))| happiness_level | vitamin_intake | enough_sleep | physical_activities | physical_appearance | sleep_quality | degree_prog_satisfaction | class_anticipation | class_participation | degree_program_performance | learnings | university_satisfaction | leisure_time | procrastination | hobbies | multimedia | pos_effect_social_med | positive_outlook | regrets | sense_of_meaning_purpose | optimism_of_future | life_control | rewarding_view_of_life | happiness_choice | family_relationship | peer_relationship | company_of_animals | love_and_affection | social_interaction | extra_curricular | campus_safety | safety_going_home |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 1 | 2 | 3 | 3 | 2 | 4 | 3 | 3 | 3 | 3 | 4 | 2 | 3 | 3 | 3 | 3 | 3 | 2 | 3 | 3 | 3 | 3 | 4 | 3 | 4 | 4 | 4 | 3 | 3 | 3 | 3 |
| 3 | 1 | 3 | 2 | 3 | 3 | 3 | 2 | 2 | 3 | 3 | 4 | 2 | 2 | 2 | 4 | 3 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 3 | 3 | 3 | 2 | 2 | 3 | 3 |
| 3 | 1 | 4 | 3 | 2 | 2 | 4 | 2 | 2 | 3 | 3 | 3 | 2 | 1 | 3 | 3 | 2 | 3 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 1 | 3 | 3 |
| 3 | 1 | 3 | 1 | 3 | 3 | 2 | 2 | 2 | 3 | 2 | 3 | 2 | 2 | 3 | 3 | 3 | 3 | 2 | 3 | 3 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 2 | 1 | 3 | 2 |
| 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 4 | 3 | 2 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 3 | 3 |
| 3 | 3 | 1 | 2 | 3 | 1 | 3 | 2 | 3 | 2 | 2 | 3 | 2 | 2 | 2 | 4 | 3 | 3 | 2 | 4 | 4 | 2 | 4 | 3 | 4 | 3 | 1 | 2 | 3 | 4 | 2 | 2 |
kable(describe(new_hp2))| vars | n | mean | sd | median | trimmed | mad | min | max | range | skew | kurtosis | se | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| happiness_level | 1 | 159 | 2.805031 | 0.6606287 | 3 | 2.806202 | 0.0000 | 1 | 4 | 3 | -0.5560081 | 0.6706330 | 0.0523913 |
| vitamin_intake | 2 | 159 | 2.012579 | 0.9611927 | 2 | 1.922481 | 1.4826 | 1 | 4 | 3 | 0.4850720 | -0.8896365 | 0.0762275 |
| enough_sleep | 3 | 159 | 2.163522 | 0.8181038 | 2 | 2.139535 | 1.4826 | 1 | 4 | 3 | 0.2461292 | -0.5443294 | 0.0648799 |
| physical_activities | 4 | 159 | 2.245283 | 0.7353375 | 2 | 2.271318 | 1.4826 | 1 | 4 | 3 | -0.0350027 | -0.5530330 | 0.0583161 |
| physical_appearance | 5 | 159 | 2.584906 | 0.6871505 | 3 | 2.620155 | 0.0000 | 1 | 4 | 3 | -0.4211245 | -0.0639351 | 0.0544946 |
| sleep_quality | 6 | 159 | 2.157233 | 0.7508189 | 2 | 2.131783 | 0.0000 | 1 | 4 | 3 | 0.4512488 | 0.0990615 | 0.0595438 |
| degree_prog_satisfaction | 7 | 159 | 2.943396 | 0.7895302 | 3 | 2.992248 | 0.0000 | 1 | 4 | 3 | -0.5143077 | -0.0219762 | 0.0626138 |
| class_anticipation | 8 | 159 | 2.364780 | 0.7413756 | 2 | 2.387597 | 1.4826 | 1 | 4 | 3 | 0.0501120 | -0.3536916 | 0.0587949 |
| class_participation | 9 | 159 | 2.540881 | 0.6819753 | 3 | 2.558139 | 1.4826 | 1 | 4 | 3 | -0.2061818 | -0.2116378 | 0.0540842 |
| degree_program_performance | 10 | 159 | 2.477987 | 0.7699784 | 3 | 2.503876 | 1.4826 | 1 | 4 | 3 | -0.1334665 | -0.4267370 | 0.0610633 |
| learnings | 11 | 159 | 3.150943 | 0.6674027 | 3 | 3.201550 | 0.0000 | 1 | 4 | 3 | -0.4319223 | 0.1896944 | 0.0529285 |
| university_satisfaction | 12 | 159 | 3.081761 | 0.6747578 | 3 | 3.131783 | 0.0000 | 1 | 4 | 3 | -0.5881969 | 0.8684358 | 0.0535118 |
| leisure_time | 13 | 159 | 2.119497 | 0.8063418 | 2 | 2.100775 | 1.4826 | 1 | 4 | 3 | 0.2147461 | -0.6254300 | 0.0639471 |
| procrastination | 14 | 159 | 1.823899 | 0.8384328 | 2 | 1.728682 | 1.4826 | 1 | 4 | 3 | 0.7848008 | -0.0602042 | 0.0664920 |
| hobbies | 15 | 159 | 3.056604 | 0.6773485 | 3 | 3.093023 | 0.0000 | 1 | 4 | 3 | -0.4318484 | 0.3593880 | 0.0537172 |
| multimedia | 16 | 159 | 3.433962 | 0.5684567 | 3 | 3.457364 | 0.0000 | 1 | 4 | 3 | -0.5637049 | 0.5083698 | 0.0450816 |
| pos_effect_social_med | 17 | 159 | 2.830189 | 0.7134117 | 3 | 2.844961 | 0.0000 | 1 | 4 | 3 | -0.4720982 | 0.3127539 | 0.0565772 |
| positive_outlook | 18 | 159 | 2.735849 | 0.7750282 | 3 | 2.728682 | 1.4826 | 1 | 4 | 3 | -0.0791834 | -0.5020721 | 0.0614637 |
| regrets | 19 | 159 | 1.918239 | 0.7288673 | 2 | 1.868217 | 0.0000 | 1 | 4 | 3 | 0.5144907 | 0.1195500 | 0.0578029 |
| sense_of_meaning_purpose | 20 | 159 | 2.597484 | 0.7885213 | 3 | 2.612403 | 1.4826 | 1 | 4 | 3 | -0.2437671 | -0.3688615 | 0.0625338 |
| optimism_of_future | 21 | 159 | 2.723270 | 0.8412765 | 3 | 2.751938 | 1.4826 | 1 | 4 | 3 | -0.2091003 | -0.5669791 | 0.0667176 |
| life_control | 22 | 159 | 2.484277 | 0.7618710 | 3 | 2.511628 | 1.4826 | 1 | 4 | 3 | -0.1180194 | -0.4000533 | 0.0604203 |
| rewarding_view_of_life | 23 | 159 | 2.748428 | 0.7792284 | 3 | 2.759690 | 0.0000 | 1 | 4 | 3 | -0.2533972 | -0.3090070 | 0.0617968 |
| happiness_choice | 24 | 159 | 2.893082 | 0.8236319 | 3 | 2.945736 | 0.0000 | 1 | 4 | 3 | -0.4775728 | -0.2190164 | 0.0653183 |
| family_relationship | 25 | 159 | 3.088050 | 0.8142998 | 3 | 3.170543 | 1.4826 | 1 | 4 | 3 | -0.7196450 | 0.1308747 | 0.0645782 |
| peer_relationship | 26 | 159 | 3.194969 | 0.5566399 | 3 | 3.209302 | 0.0000 | 1 | 4 | 3 | -0.1755083 | 0.8885315 | 0.0441444 |
| company_of_animals | 27 | 159 | 3.119497 | 0.9096166 | 3 | 3.240310 | 1.4826 | 1 | 4 | 3 | -0.8367982 | -0.1186993 | 0.0721373 |
| love_and_affection | 28 | 159 | 2.962264 | 0.7537820 | 3 | 2.984496 | 0.0000 | 1 | 4 | 3 | -0.2911746 | -0.3848245 | 0.0597788 |
| social_interaction | 29 | 159 | 2.704403 | 0.7339287 | 3 | 2.689923 | 0.0000 | 1 | 4 | 3 | -0.1498792 | -0.2681676 | 0.0582043 |
| extra_curricular | 30 | 159 | 2.201258 | 0.7695130 | 2 | 2.186046 | 0.0000 | 1 | 4 | 3 | 0.3066198 | -0.2177226 | 0.0610264 |
| campus_safety | 31 | 159 | 2.515723 | 0.6642341 | 3 | 2.573643 | 0.0000 | 1 | 4 | 3 | -0.5078377 | -0.2110440 | 0.0526772 |
| safety_going_home | 32 | 159 | 2.433962 | 0.7674411 | 2 | 2.465116 | 1.4826 | 1 | 4 | 3 | -0.1550558 | -0.4697048 | 0.0608620 |
A. Factor Analysis
cor<-cor(new_hp2)
#lowerCor(new_hp2)
corPlot(new_hp2,numbers=T, MAR=0.5, labels = 1:32)
The correlation matrix above shows that the variables are not that correlated to each other. To evaluate the ‘factorability’ of the data, the Bartlett’s test of Sphericity and Kaiser-Meyer-Olkin measure of sampling adequacy were performed.
A.1. Factorability Tests
Bartlett’s test of Sphericity tests the null hypothesis that the correlation matrix is an identity matrix, which means that the variables are unrelated and not ideal for factor analysis.
cortest.bartlett(new_hp2)## R was not square, finding R from data## $chisq
## [1] 1810.035
##
## $p.value
## [1] 1.167642e-148
##
## $df
## [1] 496Since the p-value is less than 0.05, we reject the null hypothesis that the variables are unrelated.
KMO(new_hp2)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = new_hp2)
## Overall MSA = 0.8
## MSA for each item =
## happiness_level vitamin_intake
## 0.90 0.82
## enough_sleep physical_activities
## 0.64 0.77
## physical_appearance sleep_quality
## 0.89 0.75
## degree_prog_satisfaction class_anticipation
## 0.75 0.84
## class_participation degree_program_performance
## 0.79 0.77
## learnings university_satisfaction
## 0.70 0.72
## leisure_time procrastination
## 0.85 0.68
## hobbies multimedia
## 0.73 0.70
## pos_effect_social_med positive_outlook
## 0.77 0.92
## regrets sense_of_meaning_purpose
## 0.73 0.86
## optimism_of_future life_control
## 0.89 0.87
## rewarding_view_of_life happiness_choice
## 0.90 0.88
## family_relationship peer_relationship
## 0.78 0.86
## company_of_animals love_and_affection
## 0.51 0.85
## social_interaction extra_curricular
## 0.85 0.76
## campus_safety safety_going_home
## 0.58 0.57The overall Measure of Sampling Adequacy (MSA) for the set of variables is 0.8, indicating that correlations between pairs of variables can be explained by the other variables. Moreover, the individual MSAs are all above 0.5 hence, factor analysis is appropriate for the data.
A.2. Factor Analysis Proper
To aid in the selection of the appropriate method to use for factor extraction, the variables were tested for multivariate normality.
mvn(new_hp2,subset=NULL,multivariatePlot = "qq")
## $multivariateNormality
## Test HZ p value MVN
## 1 Henze-Zirkler 1.000335 0 NO
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Anderson-Darling happiness_level 18.0345 <0.001 NO
## 2 Anderson-Darling vitamin_intake 9.9494 <0.001 NO
## 3 Anderson-Darling enough_sleep 9.8890 <0.001 NO
## 4 Anderson-Darling physical_activities 12.1844 <0.001 NO
## 5 Anderson-Darling physical_appearance 15.6425 <0.001 NO
## 6 Anderson-Darling sleep_quality 13.1187 <0.001 NO
## 7 Anderson-Darling degree_prog_satisfaction 11.6865 <0.001 NO
## 8 Anderson-Darling class_anticipation 12.0248 <0.001 NO
## 9 Anderson-Darling class_participation 14.7726 <0.001 NO
## 10 Anderson-Darling degree_program_performance 11.3196 <0.001 NO
## 11 Anderson-Darling learnings 15.4846 <0.001 NO
## 12 Anderson-Darling university_satisfaction 16.4595 <0.001 NO
## 13 Anderson-Darling leisure_time 10.1749 <0.001 NO
## 14 Anderson-Darling procrastination 12.0124 <0.001 NO
## 15 Anderson-Darling hobbies 15.4015 <0.001 NO
## 16 Anderson-Darling multimedia 22.0239 <0.001 NO
## 17 Anderson-Darling pos_effect_social_med 14.8656 <0.001 NO
## 18 Anderson-Darling positive_outlook 10.8890 <0.001 NO
## 19 Anderson-Darling regrets 13.2174 <0.001 NO
## 20 Anderson-Darling sense_of_meaning_purpose 11.1532 <0.001 NO
## 21 Anderson-Darling optimism_of_future 9.3379 <0.001 NO
## 22 Anderson-Darling life_control 11.5135 <0.001 NO
## 23 Anderson-Darling rewarding_view_of_life 11.1536 <0.001 NO
## 24 Anderson-Darling happiness_choice 10.6861 <0.001 NO
## 25 Anderson-Darling family_relationship 11.7514 <0.001 NO
## 26 Anderson-Darling peer_relationship 22.9795 <0.001 NO
## 27 Anderson-Darling company_of_animals 11.6386 <0.001 NO
## 28 Anderson-Darling love_and_affection 11.7200 <0.001 NO
## 29 Anderson-Darling social_interaction 12.4321 <0.001 NO
## 30 Anderson-Darling extra_curricular 11.6117 <0.001 NO
## 31 Anderson-Darling campus_safety 16.9183 <0.001 NO
## 32 Anderson-Darling safety_going_home 11.5048 <0.001 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th 75th
## happiness_level 159 2.805031 0.6606287 3 1 4 2.0 3
## vitamin_intake 159 2.012579 0.9611927 2 1 4 1.0 3
## enough_sleep 159 2.163522 0.8181038 2 1 4 2.0 3
## physical_activities 159 2.245283 0.7353375 2 1 4 2.0 3
## physical_appearance 159 2.584906 0.6871505 3 1 4 2.0 3
## sleep_quality 159 2.157233 0.7508189 2 1 4 2.0 3
## degree_prog_satisfaction 159 2.943396 0.7895302 3 1 4 3.0 3
## class_anticipation 159 2.364780 0.7413756 2 1 4 2.0 3
## class_participation 159 2.540881 0.6819753 3 1 4 2.0 3
## degree_program_performance 159 2.477987 0.7699784 3 1 4 2.0 3
## learnings 159 3.150943 0.6674027 3 1 4 3.0 4
## university_satisfaction 159 3.081761 0.6747578 3 1 4 3.0 3
## leisure_time 159 2.119497 0.8063418 2 1 4 2.0 3
## procrastination 159 1.823899 0.8384328 2 1 4 1.0 2
## hobbies 159 3.056604 0.6773485 3 1 4 3.0 3
## multimedia 159 3.433962 0.5684567 3 1 4 3.0 4
## pos_effect_social_med 159 2.830189 0.7134117 3 1 4 2.0 3
## positive_outlook 159 2.735849 0.7750282 3 1 4 2.0 3
## regrets 159 1.918239 0.7288673 2 1 4 1.0 2
## sense_of_meaning_purpose 159 2.597484 0.7885213 3 1 4 2.0 3
## optimism_of_future 159 2.723270 0.8412765 3 1 4 2.0 3
## life_control 159 2.484277 0.7618710 3 1 4 2.0 3
## rewarding_view_of_life 159 2.748428 0.7792284 3 1 4 2.0 3
## happiness_choice 159 2.893082 0.8236319 3 1 4 2.0 3
## family_relationship 159 3.088050 0.8142998 3 1 4 3.0 4
## peer_relationship 159 3.194969 0.5566399 3 1 4 3.0 4
## company_of_animals 159 3.119497 0.9096166 3 1 4 3.0 4
## love_and_affection 159 2.962264 0.7537820 3 1 4 2.5 3
## social_interaction 159 2.704403 0.7339287 3 1 4 2.0 3
## extra_curricular 159 2.201258 0.7695130 2 1 4 2.0 3
## campus_safety 159 2.515723 0.6642341 3 1 4 2.0 3
## safety_going_home 159 2.433962 0.7674411 2 1 4 2.0 3
## Skew Kurtosis
## happiness_level -0.55600814 0.67063299
## vitamin_intake 0.48507205 -0.88963653
## enough_sleep 0.24612915 -0.54432940
## physical_activities -0.03500273 -0.55303301
## physical_appearance -0.42112447 -0.06393509
## sleep_quality 0.45124884 0.09906149
## degree_prog_satisfaction -0.51430775 -0.02197619
## class_anticipation 0.05011200 -0.35369165
## class_participation -0.20618177 -0.21163785
## degree_program_performance -0.13346652 -0.42673697
## learnings -0.43192227 0.18969437
## university_satisfaction -0.58819686 0.86843580
## leisure_time 0.21474614 -0.62543004
## procrastination 0.78480080 -0.06020418
## hobbies -0.43184835 0.35938804
## multimedia -0.56370494 0.50836981
## pos_effect_social_med -0.47209818 0.31275387
## positive_outlook -0.07918336 -0.50207208
## regrets 0.51449074 0.11954999
## sense_of_meaning_purpose -0.24376706 -0.36886146
## optimism_of_future -0.20910034 -0.56697913
## life_control -0.11801935 -0.40005335
## rewarding_view_of_life -0.25339725 -0.30900700
## happiness_choice -0.47757283 -0.21901640
## family_relationship -0.71964504 0.13087472
## peer_relationship -0.17550834 0.88853149
## company_of_animals -0.83679823 -0.11869927
## love_and_affection -0.29117462 -0.38482445
## social_interaction -0.14987918 -0.26816762
## extra_curricular 0.30661978 -0.21772263
## campus_safety -0.50783774 -0.21104397
## safety_going_home -0.15505581 -0.46970478mvn_hz = mvn(new_hp2, mvnTest = "hz")
mvn_royston = mvn(new_hp2, mvnTest = "royston")
print(mvn_hz$multivariateNormality)## Test HZ p value MVN
## 1 Henze-Zirkler 1.000335 0 NOprint(mvn_royston$multivariateNormality)## Test H p value MVN
## 1 Royston 1634.643 1.482197e-323 NOThe above tests rejected the assumption of multivariate normality. Because of this, the Maximum Likelihood Solution is not applicable. Thus, the Principal Components Solution will be used to estimate the factor scores.
A.2.a. Principal Components Solution
names(new_hp2) <- 1:32
#standardize the data
scaled.hp1<-scale(new_hp2)
# obtain a parallel analysis on the standardized data to determine the number of factors
fa.parallel(scaled.hp1, fa='fa')
## Parallel analysis suggests that the number of factors = 5 and the number of components = NABased on the parallel analysis scree plots, it is possible to extract 5-6 factors.
PC solutions with no rotations would be considered first.
# Extract factors from the standardized data.
pcsolution1<-principal(scaled.hp1,nfactors=6,rotate="none")
pcsolution1## Principal Components Analysis
## Call: principal(r = scaled.hp1, nfactors = 6, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PC1 PC2 PC3 PC4 PC5 PC6 h2 u2 com
## 1 0.62 -0.08 -0.01 -0.35 0.16 0.21 0.59 0.41 2.0
## 2 0.36 0.28 0.07 -0.09 -0.16 -0.20 0.28 0.72 3.3
## 3 0.28 -0.03 0.67 -0.11 0.22 0.33 0.70 0.30 2.2
## 4 0.33 0.23 -0.04 0.18 0.16 0.13 0.24 0.76 3.4
## 5 0.51 0.09 0.16 -0.18 0.01 -0.06 0.33 0.67 1.6
## 6 0.42 0.07 0.64 -0.12 0.03 0.20 0.64 0.36 2.1
## 7 0.40 0.49 0.17 0.28 0.08 -0.25 0.58 0.42 3.5
## 8 0.55 0.37 0.24 0.18 -0.10 0.12 0.55 0.45 2.6
## 9 0.41 0.40 -0.01 0.25 -0.13 -0.05 0.41 0.59 2.9
## 10 0.46 0.41 0.32 0.28 -0.01 -0.19 0.60 0.40 3.9
## 11 0.40 0.38 0.18 0.30 -0.40 -0.11 0.60 0.40 4.4
## 12 0.48 0.04 -0.17 -0.01 0.04 -0.07 0.27 0.73 1.3
## 13 0.49 -0.40 0.11 -0.06 0.22 -0.06 0.46 0.54 2.5
## 14 0.23 0.51 -0.06 -0.26 -0.06 0.30 0.47 0.53 2.7
## 15 0.35 -0.40 0.29 0.13 -0.05 -0.17 0.42 0.58 3.5
## 16 0.22 -0.56 0.15 0.13 -0.04 -0.39 0.55 0.45 2.4
## 17 0.33 -0.44 0.11 0.05 -0.43 -0.04 0.50 0.50 3.0
## 18 0.63 -0.22 0.10 -0.19 0.08 -0.10 0.51 0.49 1.6
## 19 0.27 0.50 -0.23 -0.28 0.08 0.09 0.47 0.53 2.9
## 20 0.63 0.01 -0.23 -0.16 0.14 -0.34 0.62 0.38 2.1
## 21 0.71 0.06 -0.26 -0.23 0.18 -0.24 0.72 0.28 1.9
## 22 0.70 0.05 -0.16 -0.18 0.19 -0.30 0.68 0.32 1.8
## 23 0.75 -0.11 -0.04 -0.27 -0.02 0.02 0.65 0.35 1.3
## 24 0.55 -0.18 0.01 -0.23 -0.20 0.02 0.42 0.58 1.9
## 25 0.51 -0.29 0.00 0.07 -0.38 0.15 0.52 0.48 2.8
## 26 0.51 -0.24 -0.20 0.22 -0.39 0.24 0.62 0.38 3.8
## 27 0.15 0.00 -0.09 0.53 -0.04 -0.12 0.33 0.67 1.3
## 28 0.71 -0.23 -0.18 0.12 -0.01 0.33 0.72 0.28 1.9
## 29 0.51 -0.04 -0.40 0.18 -0.24 0.29 0.59 0.41 3.3
## 30 0.34 0.10 -0.46 0.13 0.06 0.09 0.36 0.64 2.3
## 31 0.35 -0.19 0.00 0.38 0.58 0.30 0.73 0.27 3.4
## 32 0.28 -0.16 -0.10 0.53 0.49 0.01 0.64 0.36 2.8
##
## PC1 PC2 PC3 PC4 PC5 PC6
## SS loadings 7.33 2.71 1.93 1.85 1.64 1.33
## Proportion Var 0.23 0.08 0.06 0.06 0.05 0.04
## Cumulative Var 0.23 0.31 0.37 0.43 0.48 0.52
## Proportion Explained 0.44 0.16 0.11 0.11 0.10 0.08
## Cumulative Proportion 0.44 0.60 0.71 0.82 0.92 1.00
##
## Mean item complexity = 2.6
## Test of the hypothesis that 6 components are sufficient.
##
## The root mean square of the residuals (RMSR) is 0.06
## with the empirical chi square 516.18 with prob < 1.6e-11
##
## Fit based upon off diagonal values = 0.94# Getting the residual matrix: The residual matrix is a measure of how good our model is. Ideally, the residual matrix is close to null.
L<-pcsolution1$loadings
llT<-L%*%t(L)
uniqueness<-cor-llT
m<-matrix(0,32,32)
diag(m)<-diag(uniqueness)
resmatrix<-cor-(llT+m)
#head(resmatrix)
hist(resmatrix)
fa.diagram(pcsolution1)
pcsolution1## Principal Components Analysis
## Call: principal(r = scaled.hp1, nfactors = 6, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
## PC1 PC2 PC3 PC4 PC5 PC6 h2 u2 com
## 1 0.62 -0.08 -0.01 -0.35 0.16 0.21 0.59 0.41 2.0
## 2 0.36 0.28 0.07 -0.09 -0.16 -0.20 0.28 0.72 3.3
## 3 0.28 -0.03 0.67 -0.11 0.22 0.33 0.70 0.30 2.2
## 4 0.33 0.23 -0.04 0.18 0.16 0.13 0.24 0.76 3.4
## 5 0.51 0.09 0.16 -0.18 0.01 -0.06 0.33 0.67 1.6
## 6 0.42 0.07 0.64 -0.12 0.03 0.20 0.64 0.36 2.1
## 7 0.40 0.49 0.17 0.28 0.08 -0.25 0.58 0.42 3.5
## 8 0.55 0.37 0.24 0.18 -0.10 0.12 0.55 0.45 2.6
## 9 0.41 0.40 -0.01 0.25 -0.13 -0.05 0.41 0.59 2.9
## 10 0.46 0.41 0.32 0.28 -0.01 -0.19 0.60 0.40 3.9
## 11 0.40 0.38 0.18 0.30 -0.40 -0.11 0.60 0.40 4.4
## 12 0.48 0.04 -0.17 -0.01 0.04 -0.07 0.27 0.73 1.3
## 13 0.49 -0.40 0.11 -0.06 0.22 -0.06 0.46 0.54 2.5
## 14 0.23 0.51 -0.06 -0.26 -0.06 0.30 0.47 0.53 2.7
## 15 0.35 -0.40 0.29 0.13 -0.05 -0.17 0.42 0.58 3.5
## 16 0.22 -0.56 0.15 0.13 -0.04 -0.39 0.55 0.45 2.4
## 17 0.33 -0.44 0.11 0.05 -0.43 -0.04 0.50 0.50 3.0
## 18 0.63 -0.22 0.10 -0.19 0.08 -0.10 0.51 0.49 1.6
## 19 0.27 0.50 -0.23 -0.28 0.08 0.09 0.47 0.53 2.9
## 20 0.63 0.01 -0.23 -0.16 0.14 -0.34 0.62 0.38 2.1
## 21 0.71 0.06 -0.26 -0.23 0.18 -0.24 0.72 0.28 1.9
## 22 0.70 0.05 -0.16 -0.18 0.19 -0.30 0.68 0.32 1.8
## 23 0.75 -0.11 -0.04 -0.27 -0.02 0.02 0.65 0.35 1.3
## 24 0.55 -0.18 0.01 -0.23 -0.20 0.02 0.42 0.58 1.9
## 25 0.51 -0.29 0.00 0.07 -0.38 0.15 0.52 0.48 2.8
## 26 0.51 -0.24 -0.20 0.22 -0.39 0.24 0.62 0.38 3.8
## 27 0.15 0.00 -0.09 0.53 -0.04 -0.12 0.33 0.67 1.3
## 28 0.71 -0.23 -0.18 0.12 -0.01 0.33 0.72 0.28 1.9
## 29 0.51 -0.04 -0.40 0.18 -0.24 0.29 0.59 0.41 3.3
## 30 0.34 0.10 -0.46 0.13 0.06 0.09 0.36 0.64 2.3
## 31 0.35 -0.19 0.00 0.38 0.58 0.30 0.73 0.27 3.4
## 32 0.28 -0.16 -0.10 0.53 0.49 0.01 0.64 0.36 2.8
##
## PC1 PC2 PC3 PC4 PC5 PC6
## SS loadings 7.33 2.71 1.93 1.85 1.64 1.33
## Proportion Var 0.23 0.08 0.06 0.06 0.05 0.04
## Cumulative Var 0.23 0.31 0.37 0.43 0.48 0.52
## Proportion Explained 0.44 0.16 0.11 0.11 0.10 0.08
## Cumulative Proportion 0.44 0.60 0.71 0.82 0.92 1.00
##
## Mean item complexity = 2.6
## Test of the hypothesis that 6 components are sufficient.
##
## The root mean square of the residuals (RMSR) is 0.06
## with the empirical chi square 516.18 with prob < 1.6e-11
##
## Fit based upon off diagonal values = 0.94pcsolution1$loadings##
## Loadings:
## PC1 PC2 PC3 PC4 PC5 PC6
## 1 0.624 -0.345 0.158 0.210
## 2 0.355 0.278 -0.164 -0.198
## 3 0.284 0.674 -0.113 0.216 0.330
## 4 0.327 0.234 0.176 0.159 0.127
## 5 0.509 0.159 -0.184
## 6 0.423 0.637 -0.116 0.200
## 7 0.398 0.492 0.167 0.280 -0.251
## 8 0.552 0.369 0.235 0.178 0.116
## 9 0.411 0.405 0.249 -0.135
## 10 0.464 0.408 0.320 0.279 -0.192
## 11 0.401 0.383 0.182 0.299 -0.397 -0.109
## 12 0.483 -0.166
## 13 0.487 -0.400 0.106 0.217
## 14 0.229 0.507 -0.255 0.295
## 15 0.353 -0.405 0.293 0.131 -0.166
## 16 0.215 -0.561 0.151 0.131 -0.386
## 17 0.331 -0.442 0.105 -0.428
## 18 0.631 -0.218 -0.195 -0.105
## 19 0.273 0.496 -0.227 -0.280
## 20 0.634 -0.227 -0.165 0.136 -0.344
## 21 0.715 -0.256 -0.228 0.181 -0.239
## 22 0.704 -0.159 -0.185 0.194 -0.297
## 23 0.751 -0.115 -0.274
## 24 0.548 -0.182 -0.228 -0.196
## 25 0.512 -0.289 -0.384 0.148
## 26 0.505 -0.243 -0.197 0.222 -0.393 0.245
## 27 0.150 0.530 -0.119
## 28 0.713 -0.229 -0.176 0.119 0.332
## 29 0.514 -0.396 0.177 -0.238 0.286
## 30 0.342 0.101 -0.456 0.133
## 31 0.348 -0.186 0.381 0.577 0.304
## 32 0.282 -0.159 0.532 0.491
##
## PC1 PC2 PC3 PC4 PC5 PC6
## SS loadings 7.329 2.713 1.928 1.846 1.635 1.325
## Proportion Var 0.229 0.085 0.060 0.058 0.051 0.041
## Cumulative Var 0.229 0.314 0.374 0.432 0.483 0.524pcsolution1$rotation## [1] "none"pcsolution1$communality #variance explained## 1 2 3 4 5 6 7 8
## 0.5853012 0.2827627 0.7037048 0.2355302 0.3293638 0.6429706 0.5759960 0.5501701
## 9 10 11 12 13 14 15 16
## 0.4147633 0.5986296 0.5987999 0.2688509 0.4632723 0.4694280 0.4216330 0.5510083
## 17 18 19 20 21 22 23 24
## 0.5033272 0.5112804 0.4650432 0.6177084 0.7222532 0.6836121 0.6542660 0.4242030
## 25 26 27 28 29 30 31 32
## 0.5196513 0.6174285 0.3272519 0.7162328 0.5927201 0.3642315 0.7264509 0.6387419pcsolution1$uniquenesses## 1 2 3 4 5 6 7 8
## 0.4146988 0.7172373 0.2962952 0.7644698 0.6706362 0.3570294 0.4240040 0.4498299
## 9 10 11 12 13 14 15 16
## 0.5852367 0.4013704 0.4012001 0.7311491 0.5367277 0.5305720 0.5783670 0.4489917
## 17 18 19 20 21 22 23 24
## 0.4966728 0.4887196 0.5349568 0.3822916 0.2777468 0.3163879 0.3457340 0.5757970
## 25 26 27 28 29 30 31 32
## 0.4803487 0.3825715 0.6727481 0.2837672 0.4072799 0.6357685 0.2735491 0.3612581pcsolution1$values## [1] 7.3289851 2.7125623 1.9284264 1.8458333 1.6353971 1.3253830 1.2044231
## [8] 1.1421697 1.0077672 0.9156160 0.9004088 0.8817464 0.8015789 0.7597415
## [15] 0.7056648 0.6834944 0.6359300 0.5966320 0.5505538 0.5426246 0.4668077
## [22] 0.4484879 0.4304351 0.3644519 0.3567825 0.3380279 0.3155461 0.2822585
## [29] 0.2572122 0.2376245 0.2246154 0.1728118pcsolution1$values/length(pcsolution1$values)## [1] 0.229030785 0.084767571 0.060263327 0.057682291 0.051106160 0.041418218
## [7] 0.037638222 0.035692803 0.031492724 0.028613001 0.028137776 0.027554576
## [13] 0.025049342 0.023741923 0.022052025 0.021359199 0.019872812 0.018644749
## [19] 0.017204805 0.016957019 0.014587740 0.014015248 0.013451098 0.011389122
## [25] 0.011149452 0.010563372 0.009860815 0.008820579 0.008037882 0.007425766
## [31] 0.007019233 0.005400368
Components Analysis (No Rotation)
The components analysis of the generated PC solutions with no rotation resulted in PC6 containing no variable. Different rotation methods would be explored.
pcsolution3<-principal(scaled.hp1,nfactors=6,rotate="quartimax")
fa.diagram(pcsolution3)
pcsolution4<-principal(scaled.hp1,nfactors=6,rotate="equamax")
fa.diagram(pcsolution4)
Components Analysis (EQUAMAX and QUARTIMAX)
Variables 2 and 31 do not have corresponding factor assignments using EQUAMAX and QUARTIMAX rotations. This indicates that these rotations may not be appropriate for the data.
# VARIMAX - maximizes the variability of loadings within a factor
pcsolution1v<-principal(new_hp2,nfactors=6,rotate="varimax")
fa.diagram(pcsolution1, main = 'Components Analysis - Varimax')
Components Analysis (VARIMAX)
Compared to the other rotations, VARIMAX has the most dispersed set of variables. Moreover, there is only one variable that has no factor assignment. Because of this, VARIMAX will be used.
pcsolution1v$loadings##
## Loadings:
## RC1 RC4 RC6 RC2 RC3 RC5
## 1 0.593 0.264 -0.163 0.348 0.109
## 2 0.292 0.392 -0.189
## 3 0.823 0.130
## 4 0.141 0.276 -0.193 0.300
## 5 0.437 0.234 0.267
## 6 0.151 0.267 0.736
## 7 0.190 0.701 -0.138 0.156
## 8 0.169 0.592 0.221 -0.155 0.294 0.109
## 9 0.132 0.582 0.163 -0.155
## 10 0.164 0.722 0.192 0.104
## 11 0.721 0.251 -0.117
## 12 0.420 0.184 0.196 0.130
## 13 0.462 0.154 0.324 0.244 0.232
## 14 0.148 0.202 -0.600 0.158 -0.112
## 15 0.197 0.190 0.529 0.232
## 16 0.221 0.700
## 17 0.122 0.509 0.434 -0.176
## 18 0.600 0.199 0.192 0.259
## 19 0.337 0.182 -0.561
## 20 0.743 0.193 -0.123
## 21 0.810 0.178 0.109 0.114
## 22 0.784 0.227 0.117
## 23 0.681 0.362 0.226
## 24 0.459 0.392 0.182 -0.130
## 25 0.205 0.107 0.640 0.196 0.127
## 26 0.130 0.127 0.752
## 27 0.332 0.151 0.212 -0.226 0.297
## 28 0.394 0.634 0.152 0.364
## 29 0.215 0.132 0.663 -0.166 -0.154 0.197
## 30 0.260 0.277 -0.230 -0.280 0.280
## 31 0.124 0.214 0.809
## 32 0.106 0.117 0.202 0.754
##
## RC1 RC4 RC6 RC2 RC3 RC5
## SS loadings 4.634 3.032 2.898 2.130 2.109 1.972
## Proportion Var 0.145 0.095 0.091 0.067 0.066 0.062
## Cumulative Var 0.145 0.240 0.330 0.397 0.463 0.524pcsolution1v$rotation## [1] "varimax"pcsolution1v$communality## 1 2 3 4 5 6 7 8
## 0.5853012 0.2827627 0.7037048 0.2355302 0.3293638 0.6429706 0.5759960 0.5501701
## 9 10 11 12 13 14 15 16
## 0.4147633 0.5986296 0.5987999 0.2688509 0.4632723 0.4694280 0.4216330 0.5510083
## 17 18 19 20 21 22 23 24
## 0.5033272 0.5112804 0.4650432 0.6177084 0.7222532 0.6836121 0.6542660 0.4242030
## 25 26 27 28 29 30 31 32
## 0.5196513 0.6174285 0.3272519 0.7162328 0.5927201 0.3642315 0.7264509 0.6387419pcsolution1v$uniquenesses## 1 2 3 4 5 6 7 8
## 0.4146988 0.7172373 0.2962952 0.7644698 0.6706362 0.3570294 0.4240040 0.4498299
## 9 10 11 12 13 14 15 16
## 0.5852367 0.4013704 0.4012001 0.7311491 0.5367277 0.5305720 0.5783670 0.4489917
## 17 18 19 20 21 22 23 24
## 0.4966728 0.4887196 0.5349568 0.3822916 0.2777468 0.3163879 0.3457340 0.5757970
## 25 26 27 28 29 30 31 32
## 0.4803487 0.3825715 0.6727481 0.2837672 0.4072799 0.6357685 0.2735491 0.3612581pcsolution1v$values## [1] 7.3289851 2.7125623 1.9284264 1.8458333 1.6353971 1.3253830 1.2044231
## [8] 1.1421697 1.0077672 0.9156160 0.9004088 0.8817464 0.8015789 0.7597415
## [15] 0.7056648 0.6834944 0.6359300 0.5966320 0.5505538 0.5426246 0.4668077
## [22] 0.4484879 0.4304351 0.3644519 0.3567825 0.3380279 0.3155461 0.2822585
## [29] 0.2572122 0.2376245 0.2246154 0.1728118pcsolution1v$values/length(pcsolution1$values)## [1] 0.229030785 0.084767571 0.060263327 0.057682291 0.051106160 0.041418218
## [7] 0.037638222 0.035692803 0.031492724 0.028613001 0.028137776 0.027554576
## [13] 0.025049342 0.023741923 0.022052025 0.021359199 0.019872812 0.018644749
## [19] 0.017204805 0.016957019 0.014587740 0.014015248 0.013451098 0.011389122
## [25] 0.011149452 0.010563372 0.009860815 0.008820579 0.008037882 0.007425766
## [31] 0.007019233 0.005400368L<-pcsolution1v$loadings
llT<-L%*%t(L)
uniqueness<-cor-llT
m<-matrix(0,32,32)
diag(m)<-diag(uniqueness)
resmatrix<-cor-(llT+m)
hist(resmatrix)
The residuals from the VARIMAX rotation are distributed along zero, indicating that the current factor model is adequate.
pcsolution1v## Principal Components Analysis
## Call: principal(r = new_hp2, nfactors = 6, rotate = "varimax")
## Standardized loadings (pattern matrix) based upon correlation matrix
## RC1 RC4 RC6 RC2 RC3 RC5 h2 u2 com
## 1 0.59 -0.07 0.26 -0.16 0.35 0.11 0.59 0.41 2.4
## 2 0.29 0.39 0.04 -0.07 0.04 -0.19 0.28 0.72 2.5
## 3 0.05 0.08 -0.02 0.03 0.82 0.13 0.70 0.30 1.1
## 4 0.14 0.28 0.08 -0.19 0.07 0.30 0.24 0.76 3.5
## 5 0.44 0.23 0.10 -0.02 0.27 -0.05 0.33 0.67 2.4
## 6 0.15 0.27 0.07 0.04 0.74 -0.01 0.64 0.36 1.4
## 7 0.19 0.70 -0.14 -0.06 0.03 0.16 0.58 0.42 1.4
## 8 0.17 0.59 0.22 -0.16 0.29 0.11 0.55 0.45 2.3
## 9 0.13 0.58 0.16 -0.15 -0.04 0.08 0.41 0.59 1.5
## 10 0.16 0.72 -0.03 0.03 0.19 0.10 0.60 0.40 1.3
## 11 0.01 0.72 0.25 0.00 0.04 -0.12 0.60 0.40 1.3
## 12 0.42 0.18 0.20 -0.04 -0.05 0.13 0.27 0.73 2.1
## 13 0.46 -0.09 0.15 0.32 0.24 0.23 0.46 0.54 3.4
## 14 0.15 0.20 0.10 -0.60 0.16 -0.11 0.47 0.53 1.7
## 15 0.20 0.09 0.19 0.53 0.23 0.07 0.42 0.58 2.1
## 16 0.22 -0.04 0.10 0.70 0.00 0.03 0.55 0.45 1.2
## 17 0.12 0.03 0.51 0.43 0.10 -0.18 0.50 0.50 2.4
## 18 0.60 0.06 0.20 0.19 0.26 0.06 0.51 0.49 1.9
## 19 0.34 0.18 -0.02 -0.56 -0.03 -0.05 0.47 0.53 1.9
## 20 0.74 0.19 0.07 0.05 -0.12 0.08 0.62 0.38 1.2
## 21 0.81 0.18 0.11 -0.07 -0.06 0.11 0.72 0.28 1.2
## 22 0.78 0.23 0.06 0.01 -0.02 0.12 0.68 0.32 1.2
## 23 0.68 0.09 0.36 -0.01 0.23 0.01 0.65 0.35 1.8
## 24 0.46 0.05 0.39 0.09 0.18 -0.13 0.42 0.58 2.6
## 25 0.21 0.11 0.64 0.20 0.13 -0.04 0.52 0.48 1.6
## 26 0.13 0.13 0.75 0.10 -0.04 0.09 0.62 0.38 1.2
## 27 -0.10 0.33 0.15 0.21 -0.23 0.30 0.33 0.67 4.2
## 28 0.39 0.06 0.63 -0.02 0.15 0.36 0.72 0.28 2.5
## 29 0.22 0.13 0.66 -0.17 -0.15 0.20 0.59 0.41 1.8
## 30 0.26 0.10 0.28 -0.23 -0.28 0.28 0.36 0.64 5.2
## 31 0.12 -0.03 0.09 0.04 0.21 0.81 0.73 0.27 1.2
## 32 0.11 0.12 0.01 0.20 -0.06 0.75 0.64 0.36 1.3
##
## RC1 RC4 RC6 RC2 RC3 RC5
## SS loadings 4.63 3.03 2.90 2.13 2.11 1.97
## Proportion Var 0.14 0.09 0.09 0.07 0.07 0.06
## Cumulative Var 0.14 0.24 0.33 0.40 0.46 0.52
## Proportion Explained 0.28 0.18 0.17 0.13 0.13 0.12
## Cumulative Proportion 0.28 0.46 0.63 0.76 0.88 1.00
##
## Mean item complexity = 2
## Test of the hypothesis that 6 components are sufficient.
##
## The root mean square of the residuals (RMSR) is 0.06
## with the empirical chi square 516.18 with prob < 1.6e-11
##
## Fit based upon off diagonal values = 0.94The cumulative variance explained by the principal components is 52%. The table below summarizes the 6 factors obtained from the analysis and the corresponding variables categorized in each factor.
Summary of Factors
B. Cluster Analysis
#Extract the scores from VARIMAX
pcscores<-factor.scores(new_hp2,f=pcsolution1v,method="Bartlett")
scoress<-pcscores$scores
# Create clusters using the 6 factors from factor analysis
res.agnes_fa <- agnes(scoress, method = "ward")
pltree(res.agnes_fa, cex = 0.6, hang = -1,
main = "Dendrogram")
grp_fa <- cutree(as.hclust(res.agnes_fa), k = 4)
fviz_cluster(list(data = scoress, cluster = grp_fa))
Utilizing all the 6 factors to form clusters, the cumulative variation explained by PC1 and PC2 is only 33.55%. Since this is low and the clusters overlap excessively, it was considered to remove some factors.
# Multivariate Normality test
mvn(scoress,subset=NULL,multivariatePlot = "qq")
## $multivariateNormality
## Test HZ p value MVN
## 1 Henze-Zirkler 1.130275 2.392726e-05 NO
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Anderson-Darling RC1 0.4178 0.3257 YES
## 2 Anderson-Darling RC4 0.3609 0.4424 YES
## 3 Anderson-Darling RC6 0.6891 0.0706 YES
## 4 Anderson-Darling RC2 0.5213 0.1822 YES
## 5 Anderson-Darling RC3 0.9887 0.0128 NO
## 6 Anderson-Darling RC5 1.2690 0.0026 NO
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th
## RC1 159 -1.361431e-16 1.006107 0.022508334 -3.325116 2.394211 -0.6477771
## RC4 159 4.558491e-17 1.004085 0.003587867 -2.618578 2.891897 -0.6486343
## RC6 159 4.474211e-17 1.004408 -0.100346198 -3.061909 2.693130 -0.6114148
## RC2 159 -1.879381e-16 1.004047 -0.082708023 -5.061796 1.990922 -0.6555958
## RC3 159 -4.681571e-17 1.008429 -0.117191375 -2.234044 2.636265 -0.6836079
## RC5 159 1.593095e-17 1.015208 0.216338192 -3.110614 3.044366 -0.6829234
## 75th Skew Kurtosis
## RC1 0.6178239 -0.10104996 0.04435192
## RC4 0.6262030 0.04268047 -0.34623433
## RC6 0.6952380 -0.06060268 0.25546508
## RC2 0.7274664 -0.71101252 2.71681371
## RC3 0.7172917 0.30655986 -0.48175851
## RC5 0.7372391 -0.28023719 0.30390426hz_test = mvn(scoress, mvnTest = "hz")
royston_test = mvn(scoress, mvnTest = "royston")kable(hz_test$univariateNormality)| Test | Variable | Statistic | p value | Normality |
|---|---|---|---|---|
| Anderson-Darling | RC1 | 0.4178 | 0.3257 | YES |
| Anderson-Darling | RC4 | 0.3609 | 0.4424 | YES |
| Anderson-Darling | RC6 | 0.6891 | 0.0706 | YES |
| Anderson-Darling | RC2 | 0.5213 | 0.1822 | YES |
| Anderson-Darling | RC3 | 0.9887 | 0.0128 | NO |
| Anderson-Darling | RC5 | 1.2690 | 0.0026 | NO |
# Extract the factors that are univariate normal to create clusters.
pc_fin <- scoress[, -(4:6)]
mvn(pc_fin,subset=NULL,multivariatePlot = "qq")
## $multivariateNormality
## Test HZ p value MVN
## 1 Henze-Zirkler 1.007624 0.05420563 YES
##
## $univariateNormality
## Test Variable Statistic p value Normality
## 1 Anderson-Darling RC1 0.4178 0.3257 YES
## 2 Anderson-Darling RC4 0.3609 0.4424 YES
## 3 Anderson-Darling RC6 0.6891 0.0706 YES
##
## $Descriptives
## n Mean Std.Dev Median Min Max 25th
## RC1 159 -1.361431e-16 1.006107 0.022508334 -3.325116 2.394211 -0.6477771
## RC4 159 4.558491e-17 1.004085 0.003587867 -2.618578 2.891897 -0.6486343
## RC6 159 4.474211e-17 1.004408 -0.100346198 -3.061909 2.693130 -0.6114148
## 75th Skew Kurtosis
## RC1 0.6178239 -0.10104996 0.04435192
## RC4 0.6262030 0.04268047 -0.34623433
## RC6 0.6952380 -0.06060268 0.25546508Upon the removal of the non-normal factors, the distribution of the data is closer to multivariate normal.
B.1. Hierarchical Clustering
d<-dist(pc_fin, method="euclidean")
res.hc<-hclust(d,method="ward.D2")
plot(res.hc,cex=0.6,hang=-1) 
B.1.a. Agglomerative
res.agnes <- agnes(pc_fin, method = "ward")
# summary(res.agnes)
res.agnes$ac## [1] 0.9588164pltree(res.agnes, cex = 0.6, hang = -1,
main = "Dendrogram of agnes")
plot(as.dendrogram(res.agnes), cex = 0.6,
horiz = TRUE)
grp <- cutree(as.hclust(res.agnes), k = 3)table(grp)## grp
## 1 2 3
## 59 57 43The obtained coefficient from the agglomerative method is 0.9588. Since this is close to 1, we can conclude that the clustering using this method is already sufficient. Based on the dendogram, we can group the observations into three clusters. The first cluster contains 59 observations, the second cluster contains 57 observations, and the third cluster contains 43 observations. These clusters are plotted below.
fviz_cluster(list(data=pc_fin, cluster=grp), main = 'Cluster Plot using Agglomerative Method')
From the cluster plot, PC1 and PC2 explain 66.8% of the total variability in the data. Compared to the previous cluster plot where all factors was utilized, the cluster plot using only 3 factors reduced the problem of excessive overlapping.
B.1.b. Divisive
res.diana <- diana(pc_fin)
pltree(res.diana, cex = 0.6, hang = -1,
main = "Dendrogram of diana")
res.diana$dc ## [1] 0.9129347grp_div <- cutree(as.hclust(res.diana), k = 3)
# Number of members in each cluster
table(grp_div)## grp_div
## 1 2 3
## 102 42 15fviz_cluster(list(data = pc_fin, cluster = grp_div))
The divisive coefficient is 91.29%. This is less than the obtained agglomerative coefficient of 95.88%, indicating that the agglomerative method produced better clustering.
B.2. Non-hierarchical Clustering
B.2.a. K-means clustering
df<-pc_fin
df <- na.omit(df)
df <- scale(df)
head(df)## RC1 RC4 RC6
## [1,] 0.2372729 0.647263867 0.94567040
## [2,] -0.6052084 -0.221401560 -0.07735138
## [3,] 0.2975854 -0.003204691 -1.36302807
## [4,] 0.2222225 -1.264993806 -0.09990577
## [5,] -1.1033870 -1.558145714 -0.16875069
## [6,] 1.7617537 -0.973842322 -0.24187815distance <- get_dist(df)
fviz_dist(distance, gradient = list(low = "#00AFBB", mid = "white", high = "red" )) #"#FC4E07"
func1 <- function(k) {
kmeans(df, k, nstart = 10 )$betweenss/(kmeans(df, k, nstart = 10 )$tot.withinss+kmeans(df, k, nstart = 10 )$betweenss)
}k.values <- 1:10
values <- map_dbl(k.values, func1)
plot(k.values, values,
type="b", pch = 19, frame = FALSE,
xlab="Number of clusters K",
ylab="Metric")
B.3. Optimal Number of Clusters
fviz_nbclust(df, kmeans, method = "wss") 
fviz_nbclust(df, kmeans, method = "silhouette")
fviz_nbclust(df, kmeans, method = "gap_stat")
B.4. Final Clusters and Insights
final <- kmeans(df, 3, nstart = 25)
print(final)## K-means clustering with 3 clusters of sizes 71, 35, 53
##
## Cluster means:
## RC1 RC4 RC6
## 1 0.7932786 -0.04598114 -0.3660569
## 2 -0.9731223 0.77628948 -0.6077800
## 3 -0.4200660 -0.45104662 0.8917423
##
## Clustering vector:
## [1] 3 3 1 1 3 1 1 2 3 1 1 1 1 3 3 2 2 1 3 3 1 1 2 3 3 3 1 2 3 1 1 2 1 1 3 2 3
## [38] 2 3 2 1 2 1 2 1 3 2 1 1 1 1 1 1 1 1 2 2 2 1 1 1 2 1 3 2 1 1 1 1 3 3 1 1 1
## [75] 3 3 1 1 3 2 2 2 3 1 1 2 1 2 3 1 3 1 3 3 3 3 3 3 2 3 1 2 3 1 3 2 3 1 3 2 1
## [112] 2 3 3 1 3 3 3 1 1 3 1 1 2 1 1 1 2 1 3 2 2 2 1 3 3 2 3 1 1 1 3 1 1 1 3 3 3
## [149] 2 3 1 1 3 1 1 3 2 1 1
##
## Within cluster sum of squares by cluster:
## [1] 130.88235 68.00464 91.32434
## (between_SS / total_SS = 38.8 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss" "tot.withinss"
## [6] "betweenss" "size" "iter" "ifault"fviz_cluster(final, data = df)
B.5. Discussion of the Clusters
Cluster Demographics
Cluster 1: Social Relationships Cluster
Mean of the Variables in Cluster 1
This cluster is mainly composed of males, ages 19-21 years old. Majority of them are affiliated with organizations inside the campus. The top three colleges that comprise this cluster are School of Statistics, College of Engineering, and the College of Arts and Letters. Within this cluster, 76.27% have claimed that they are happy.
From the table, most of the variables with high means in Cluster 1 belong to the Social Relationship Factor. These variables are positive effect of social media, family relationships, peer relationships, love and affection, and social interaction (outside comfort zone). From the Degree Program Factor, the variables learnings and company of animals have high means. From the Perspective Factor, university satisfaction; and from the Time Management Factor, Hobbies and Multimedia.
With this, we can conclude that the happiness index of Cluster 1 is heavily influenced by the Social Relationship Factor and slightly influenced by Degree Program, Perspective, and Time Management Factors.
Cluster 2: Perspective Cluster
Mean of the Variables in Cluster 2
This cluster is mainly composed of females, ages 19-21 years old. Majority of them are affiliated with organizations inside the campus. The top three colleges that comprise this cluster are School of Statistics, College of Engineering, and the College of Science. Within this cluster, 88% have claimed that they are happy.
From the table, most of the variables with high means in Cluster 2 belong to the Perspective Factor. These variables are current level of happiness, university satisfaction, positive outlook, optimism about the future, rewarding view on life, and happiness is a choice. From the Social Relationships Factor, the variables family relationships and peer relationships have high means; and from the Time Management Factor, Hobbies and Multimedia. With this, we can conclude that the happiness index of Cluster 2 is heavily influenced by the Perspective Factor and slightly influenced by the Social Relationships and Time Management Factors.
Cluster 3: Degree Program Cluster
This cluster is mainly composed of females, ages 19-21 years old. Majority of them are affiliated with organizations inside the campus. The top three colleges that comprise this cluster are School of Statistics, College of Engineering, and the College of Social Sciences and Philosophy. Within this cluster, 63% have claimed that they are happy.
From the table, most of the variables with high means in Cluster 3 belong to the Degree Program Factor. These variables are degree program satisfaction, class participation, performance in degree program, learnings, and company of animals. From the Time Management Factor, the variables hobbies and multimedia have high means; from the Perspective Factor, university satisfaction; from the Environmental Setting Factor, safety going home at night, and from the Social Relationship Factor, peer relationship.
With this, we can conclude that the happiness index of Cluster 3 is heavily influenced by the Degree Program Factor and slightly influenced by the Time Management, Perspective, Environmental Setting, and Social Relationship Factors.
Based on the characteristics of the clusters discussed above, it can be concluded that cluster 2 is the happiest cluster since it has the highest proportion of respondents who claimed that they are happy. It is followed by cluster 1 and then by cluster 3. It can also be deduced that students from the College of Science are the happiest, followed by those from the College of Arts and Letters, and the College of Social Sciences and Philosophy. The results that came from the School of Statistics and College of Engineering were not primarily considered since they dominated all the clusters. Also, the reason that they were included in the top three colleges per cluster may be attributed to the fact that most of the respondents came from the two colleges respectively, which is why they were omitted from the cluster characteristic interpretation. This may also be due to the fact that the convenience sampling procedure was used.
Moreover, the Happiness Index of students from the College of Arts and Letters are mostly influenced by the Social Relationships Factor. As for the students in the College of Science, their Happiness Index is mostly influenced by the Perspective Factor while for the students in the College of Social Sciences and Philosophy, the Happiness Index is most affected by the Degree Program Factor. It is also important to note that the Time Management Factor and the Social Relationships Factor appear to influence all the three clusters.