Alter, Dang, Kunicki, Counsell Jan 31, 2021
# Packages ----------------------------------------------------------------library(tidyverse)
library(car)
library(psych)
library(mirt) # for multiple IRT
library(GPArotation) # for Promax rotation
library(REdaS) # for assumptions
library(faoutlier) # for efa outliers
library(mice) # for imputations# Initial Data Setup ------------------------------------------------------Uploading raw data
full.data <- readxl::read_xlsx("spssdata.xlsx", col_names = TRUE)
# Remove empty rows
full.data <- full.data[1:181, ]Selecting only SPSS-related columns
# Do not select StudentIDE col bc you do not want it included with EFA
# Num of rows corresponds to StudentIDE
spss.data <- full.data %>%
select(SPSS1E,
SPSS2E,
SPSS3E,
SPSS4E,
SPSS5E,
SPSS6E,
SPSS7E,
SPSS8E,
SPSS9E,
SPSS10E)Reverse code SPSS items 2, 3, 10
spss.data$SPSS2E <- car::recode(spss.data$SPSS2E, "1 = 5; 2 = 4; 3 = 3; 4 = 2; 5 = 1")
spss.data$SPSS3E <- car::recode(spss.data$SPSS3E, "1 = 5; 2 = 4; 3 = 3; 4 = 2; 5 = 1")
spss.data$SPSS10E <- car::recode(spss.data$SPSS10E, "1 = 5; 2 = 4; 3 = 3; 4 = 2; 5 = 1")# Descriptive Stats -------------------------------------------------------
describe(spss.data)## vars n mean sd median trimmed mad min max range skew kurtosis
## SPSS1E 1 180 3.49 1.04 4 3.55 1.48 1 5 4 -0.44 -0.30
## SPSS2E 2 181 3.67 1.11 4 3.76 1.48 1 5 4 -0.58 -0.58
## SPSS3E 3 180 3.51 1.11 4 3.58 1.48 1 5 4 -0.53 -0.43
## SPSS4E 4 181 3.38 0.99 3 3.39 1.48 1 5 4 -0.30 -0.42
## SPSS5E 5 180 3.70 1.05 4 3.79 1.48 1 5 4 -0.52 -0.27
## SPSS6E 6 181 3.52 1.22 4 3.64 1.48 1 5 4 -0.51 -0.60
## SPSS7E 7 181 3.32 1.18 3 3.34 1.48 1 5 4 -0.13 -0.98
## SPSS8E 8 181 3.57 0.98 4 3.63 1.48 1 5 4 -0.40 -0.22
## SPSS9E 9 181 3.43 1.03 3 3.46 1.48 1 5 4 -0.25 -0.44
## SPSS10E 10 181 3.41 1.13 4 3.46 1.48 1 5 4 -0.33 -0.71
## se
## SPSS1E 0.08
## SPSS2E 0.08
## SPSS3E 0.08
## SPSS4E 0.07
## SPSS5E 0.08
## SPSS6E 0.09
## SPSS7E 0.09
## SPSS8E 0.07
## SPSS9E 0.08
## SPSS10E 0.08
Contingency table of the counts
table(spss.data$SPSS1E)##
## 1 2 3 4 5
## 8 21 55 66 30
table(spss.data$SPSS2E)##
## 1 2 3 4 5
## 6 28 31 70 46
table(spss.data$SPSS3E)##
## 1 2 3 4 5
## 10 24 43 70 33
table(spss.data$SPSS4E)##
## 1 2 3 4 5
## 6 28 59 67 21
table(spss.data$SPSS5E)##
## 1 2 3 4 5
## 6 15 52 61 46
table(spss.data$SPSS6E)##
## 1 2 3 4 5
## 16 17 51 50 47
table(spss.data$SPSS7E)##
## 1 2 3 4 5
## 10 40 48 48 35
table(spss.data$SPSS8E)##
## 1 2 3 4 5
## 5 18 58 68 32
table(spss.data$SPSS9E)##
## 1 2 3 4 5
## 7 24 64 57 29
table(spss.data$SPSS10E)##
## 1 2 3 4 5
## 10 30 49 59 33
Missing Data Calculations
table(is.na(spss.data))##
## FALSE TRUE
## 1807 3
# MISSING / (MISSING + NOT MISSING) = PERCENT MISSING
# 3 / (3 + 1807) = 0.001657459
# 0.001657459 * 100 = 0.1657459
# Less than 1% missing data, proceeding with complete case analysesScatterplot Matrix
car::scatterplotMatrix(spss.data, smooth = F, regLine = F, col = 'black')
# Statistical Assumptions -------------------------------------------------Multivariate Normality
mardia(spss.data) # Kurtosis = 15.17 >4. Will not assume mvn.
## Call: mardia(x = spss.data)
##
## Mardia tests of multivariate skew and kurtosis
## Use describe(x) the to get univariate tests
## n.obs = 178 num.vars = 10
## b1p = 19.29 skew = 572.2 with probability = 0
## small sample skew = 583.62 with probability = 0
## b2p = 155.23 kurtosis = 15.17 with probability = 0
EFA Appropriateness
# Barlett's Test of Sphericity tests whether a matrix is significantly different from an identity matrix
bart_spher(spss.data, use = "complete.obs") # p-value < 2.22e-16## Bartlett's Test of Sphericity
##
## Call: bart_spher(x = spss.data, use = "complete.obs")
##
## X2 = 969.939
## df = 45
## p-value < 2.22e-16
## Warning: Used n = 178.
Kaiser-Meyer-Olkin Statistics
KMOS(spss.data, use = "complete.obs")##
## Kaiser-Meyer-Olkin Statistics
##
## Call: KMOS(x = spss.data, use = "complete.obs")
##
## Measures of Sampling Adequacy (MSA):
## SPSS1E SPSS2E SPSS3E SPSS4E SPSS5E SPSS6E SPSS7E SPSS8E
## 0.9098933 0.7138465 0.7359144 0.8758975 0.9281158 0.9543978 0.9378408 0.9022414
## SPSS9E SPSS10E
## 0.8956706 0.6953167
##
## KMO-Criterion: 0.8795382
# Listwise Deletion / Complete Case Analysis ------------------------------
spss.data.withNA <- spss.data # spss data without removing missing values, for imputations
# Previous work suggests using listwise deletion when the missing data rates are extremely low (e.g., < 1%; Flora, 2018; Jakobsen et al., 2017).
spss.data <- spss.data[-c(33, 141, 104), ]
full.data <- full.data[-c(33, 141, 104), ] # needed later for convergent/discriminant validity
spss.data <- data.frame(spss.data)
str(spss.data)## 'data.frame': 178 obs. of 10 variables:
## $ SPSS1E : num 4 4 3 2 4 2 4 2 3 4 ...
## $ SPSS2E : num 4 5 3 4 5 2 3 2 4 5 ...
## $ SPSS3E : num 4 5 3 4 4 2 1 2 5 5 ...
## $ SPSS4E : num 3 4 2 2 3 3 2 3 3 4 ...
## $ SPSS5E : num 3 4 3 2 4 2 4 3 3 3 ...
## $ SPSS6E : num 3 4 2 2 4 1 4 3 1 5 ...
## $ SPSS7E : num 4 4 2 2 3 3 2 3 4 4 ...
## $ SPSS8E : num 4 4 3 2 4 3 3 3 3 2 ...
## $ SPSS9E : num 4 4 3 2 3 3 2 2 2 4 ...
## $ SPSS10E: num 4 5 3 4 5 3 3 3 4 3 ...
# Polychoric Correlations -------------------------------------------------Using polychoric correlations because the data is categorical (5-point Likert scale)
poly.spss.data <- psych::polychoric(spss.data)
# Confidence Intervals for Polychoric Correlations
(cor.ci(spss.data, poly = TRUE, plot = FALSE))$ci## Warning in cor.smooth(mat): Matrix was not positive definite, smoothing was done
## lower low.e upper up.e p
## SPSS1E-SPSS2 0.160276769 0.179082140 0.5579618 0.5530159 1.628678e-03
## SPSS1E-SPSS3 0.186080418 0.184630447 0.5120240 0.5444658 1.832165e-04
## SPSS1E-SPSS4 0.775203933 0.797678851 0.9222432 0.9277895 2.645117e-09
## SPSS1E-SPSS5 0.580141893 0.589024948 0.8224001 0.8240478 1.610307e-08
## SPSS1E-SPSS6 0.427353044 0.431835363 0.6706235 0.6659390 5.996461e-10
## SPSS1E-SPSS7 0.412606224 0.422781221 0.6774286 0.6630702 1.316667e-08
## SPSS1E-SPSS8 0.685931148 0.696267543 0.8662875 0.8576360 7.762413e-11
## SPSS1E-SPSS9 0.721647836 0.729712814 0.8910153 0.8913401 3.795579e-10
## SPSS1E-SPSS10 0.047092447 0.059127935 0.4259574 0.3846658 1.810298e-02
## SPSS2-SPSS3 0.552052739 0.578573511 0.8057753 0.8030813 2.651696e-08
## SPSS2-SPSS4 0.119777389 0.146121883 0.4939452 0.5102536 2.947546e-03
## SPSS2-SPSS5 0.164000796 0.170858729 0.5204012 0.5230084 7.172285e-04
## SPSS2-SPSS6 0.087167125 0.077581367 0.4344254 0.4183002 5.181762e-03
## SPSS2-SPSS7 0.041977249 0.002507876 0.3968045 0.3699906 1.857211e-02
## SPSS2-SPSS8 0.120095295 0.104354778 0.4895279 0.4475953 2.754770e-03
## SPSS2-SPSS9 0.103286553 0.131752779 0.4860850 0.4661219 4.846700e-03
## SPSS2-SPSS10 0.318290143 0.329490648 0.6591298 0.6479111 1.577182e-05
## SPSS3-SPSS4 0.197527389 0.229660636 0.5349691 0.5663862 1.835607e-04
## SPSS3-SPSS5 0.112997669 0.127788937 0.4512748 0.4476987 2.199296e-03
## SPSS3-SPSS6 0.058205914 0.057829075 0.3893122 0.3781991 1.044469e-02
## SPSS3-SPSS7 -0.071439042 -0.085247335 0.3181320 0.2955365 2.099491e-01
## SPSS3-SPSS8 0.193864693 0.201974550 0.5039814 0.5025349 8.656818e-05
## SPSS3-SPSS9 0.185129265 0.184915686 0.5252277 0.5171366 2.784554e-04
## SPSS3-SPSS10 0.321193405 0.327783199 0.6306194 0.6335749 2.565059e-06
## SPSS4-SPSS5 0.596040952 0.587269786 0.8580891 0.8644143 7.576783e-07
## SPSS4-SPSS6 0.421904796 0.445428762 0.6929145 0.6846505 2.624588e-08
## SPSS4-SPSS7 0.402795879 0.442676880 0.6845168 0.6832572 9.153150e-08
## SPSS4-SPSS8 0.637947232 0.646680358 0.8436450 0.8406041 5.206078e-10
## SPSS4-SPSS9 0.766441580 0.793740069 0.9283741 0.9323221 8.054987e-08
## SPSS4-SPSS10 -0.009022745 -0.029685784 0.4177829 0.3869456 6.391011e-02
## SPSS5-SPSS6 0.421109683 0.448616620 0.7157178 0.7084128 3.042553e-07
## SPSS5-SPSS7 0.423719742 0.452072456 0.6617137 0.6596533 2.636571e-10
## SPSS5-SPSS8 0.605744018 0.631474787 0.8386089 0.8315435 1.423733e-08
## SPSS5-SPSS9 0.619221191 0.642141351 0.8064099 0.8016379 4.403145e-13
## SPSS5-SPSS10 -0.060474488 -0.032895802 0.3136078 0.3147487 1.815893e-01
## SPSS6-SPSS7 0.331742750 0.348098598 0.5954962 0.5839777 5.170495e-08
## SPSS6-SPSS8 0.488003199 0.485064685 0.7186286 0.7227732 7.719780e-11
## SPSS6-SPSS9 0.516813949 0.537782086 0.7263604 0.7186939 1.157519e-12
## SPSS6-SPSS10 -0.133533651 -0.112137463 0.2295530 0.2142264 5.969488e-01
## SPSS7-SPSS8 0.466847527 0.482245025 0.7025247 0.7010371 1.610592e-10
## SPSS7-SPSS9 0.434915945 0.445548272 0.7005403 0.6821480 1.347074e-08
## SPSS7-SPSS10 -0.197689451 -0.202532614 0.2231107 0.2116349 9.028841e-01
## SPSS8-SPSS9 0.721015071 0.728336386 0.8694364 0.8643307 5.484502e-14
## SPSS8-SPSS10 0.035973397 0.083158016 0.4160708 0.3930814 2.360287e-02
## SPSS9-SPSS10 -0.090596260 -0.068037623 0.3090340 0.2997900 2.768779e-01
# MAP Test/Parallel Analysis ----------------------------------------------MAP Test
VSS(spss.data, fm = 'minres', cor = 'poly', plot = F)##
## Very Simple Structure
## Call: vss(x = x, n = n, rotate = rotate, diagonal = diagonal, fm = fm,
## n.obs = n.obs, plot = plot, title = title, use = use, cor = cor)
## VSS complexity 1 achieves a maximimum of 0.87 with 1 factors
## VSS complexity 2 achieves a maximimum of 0.95 with 2 factors
##
## The Velicer MAP achieves a minimum of 0.05 with 2 factors
## BIC achieves a minimum of NA with 2 factors
## Sample Size adjusted BIC achieves a minimum of NA with 4 factors
##
## Statistics by number of factors
## vss1 vss2 map dof chisq prob sqresid fit RMSEA BIC SABIC complex
## 1 0.87 0.00 0.057 35 2.3e+02 2.6e-31 4.48 0.87 0.179 53 163.7 1.0
## 2 0.86 0.95 0.049 26 8.7e+01 1.7e-08 1.57 0.95 0.115 -48 34.6 1.1
## 3 0.86 0.95 0.072 18 5.4e+01 1.9e-05 1.27 0.96 0.106 -39 17.6 1.2
## 4 0.86 0.94 0.116 11 1.6e+01 1.3e-01 1.17 0.97 0.052 -41 -5.8 1.3
## 5 0.86 0.89 0.165 5 4.6e+00 4.7e-01 0.98 0.97 0.000 -21 -5.5 1.4
## 6 0.85 0.91 0.240 0 7.6e-01 NA 0.90 0.97 NA NA NA 1.5
## 7 0.86 0.93 0.410 -4 5.1e-07 NA 0.90 0.97 NA NA NA 1.4
## 8 0.86 0.95 0.480 -7 1.5e-12 NA 0.85 0.98 NA NA NA 1.4
## eChisq SRMR eCRMS eBIC
## 1 2.4e+02 1.2e-01 0.138 57
## 2 1.6e+01 3.2e-02 0.042 -119
## 3 6.6e+00 2.0e-02 0.032 -87
## 4 2.7e+00 1.3e-02 0.026 -54
## 5 4.8e-01 5.5e-03 0.016 -25
## 6 6.1e-02 2.0e-03 NA NA
## 7 1.0e-07 2.5e-06 NA NA
## 8 1.3e-13 2.9e-09 NA NA
Parallel Analysis
fa.parallel(spss.data, fm = 'minres', cor = 'poly', fa ='both', n.iter=100)
## Parallel analysis suggests that the number of factors = 2 and the number of components = 2
Both MAP and PA suggest 2F. PA should be interpreted w caution for
polychoric correlations
Next, running 1F, 2F and 3F model. i.e., 1 above and 1 below suggested
num. of factors to help determine which model fits best
# EFA 1 Factor ------------------------------------------------------------
efa1 <- fa(r = spss.data, fm = 'minres', rotate = "oblimin", cor = 'poly', nfactors = 1)EFA 1 Results
# Factor Analysis using method = minres
# Call: fa(r = spss.data, nfactors = 1, rotate = "oblimin", fm = "minres",
# cor = "poly")
# Standardized loadings (pattern matrix) based upon correlation matrix
# MR1 h2 u2 com
# SPSS1E 0.89 0.791 0.21 1
# SPSS2E 0.45 0.203 0.80 1 *
# SPSS3E 0.44 0.197 0.80 1 *
# SPSS4E 0.89 0.801 0.20 1
# SPSS5E 0.80 0.642 0.36 1
# SPSS6E 0.66 0.429 0.57 1
# SPSS7E 0.63 0.393 0.61 1
# SPSS8E 0.88 0.768 0.23 1
# SPSS9E 0.90 0.803 0.20 1
# SPSS10E 0.27 0.071 0.93 1 **
#
# MR1
# SS loadings 5.10
# Proportion Var 0.51
#
# Mean item complexity = 1
# Test of the hypothesis that 1 factor is sufficient.
#
# The degrees of freedom for the null model are 45 and the objective function was 7.36 with Chi Square of 1272.53
# The degrees of freedom for the model are 35 and the objective function was 1.36
#
# The root mean square of the residuals (RMSR) is 0.12
# The df corrected root mean square of the residuals is 0.14
#
# The harmonic number of observations is 178 with the empirical chi square 238.29 with prob < 4.4e-32
# The total number of observations was 178 with Likelihood Chi Square = 234.19 with prob < 2.6e-31
#
# Tucker Lewis Index of factoring reliability = 0.791
# RMSEA index = 0.179 and the 90 % confidence intervals are 0.158 0.201
# BIC = 52.83
# Fit based upon off diagonal values = 0.94
# Measures of factor score adequacy
# MR1
# Correlation of (regression) scores with factors 0.97
# Multiple R square of scores with factors 0.95
# Minimum correlation of possible factor scores 0.90EFA Interpretation
* RMSR = 0.12 = BAD
* Prop. var explained = 0.51
* SPSS10E = BAD factor loading (<.4) and communality (0.07)
* SPSS2E and SPSS3E factor loading almost <.4 and communality almost
<.2
* 1 factor model is BAD
EFA Outliers Check
fS1 <- forward.search(spss.data, 1, criteria = c("mah", "GOF"))
gcdresult1 <- gCD(spss.data, 1)
ldresults1 <- LD(spss.data, 1)plot(gcdresult1)
plot(fS1)
plot(ldresults1)
# EFA 2 Factors -----------------------------------------------------------
efa2 <- fa(r = spss.data, fm = 'minres', cor = 'poly', nfactors = 2)EFA 2 Results
# Factor Analysis using method = minres
# Call: fa(r = spss.data, nfactors = 2, fm = "minres", cor = "poly")
# Standardized loadings (pattern matrix) based upon correlation matrix
# MR1 MR2 h2 u2 com
# SPSS1E 0.85 0.08 0.78 0.22 1.0
# SPSS2E 0.05 0.79 0.66 0.34 1.0
# SPSS3E 0.02 0.82 0.69 0.31 1.0
# SPSS4E 0.88 0.04 0.80 0.20 1.0
# SPSS5E 0.82 -0.02 0.66 0.34 1.0
# SPSS6E 0.69 -0.04 0.45 0.55 1.0
# SPSS7E 0.71 -0.12 0.44 0.56 1.1
# SPSS8E 0.86 0.04 0.77 0.23 1.0
# SPSS9E 0.93 -0.03 0.83 0.17 1.0
# SPSS10E -0.09 0.66 0.39 0.61 1.0
#
# MR1 MR2
# SS loadings 4.74 1.75
# Proportion Var 0.47 0.17 PROP EXPLAINED PER FACTOR
# Cumulative Var 0.47 0.65 *
# Proportion Explained 0.73 0.27
# Cumulative Proportion 0.73 1.00
#
# With factor correlations of
# R1 MR2
# MR1 1.00 0.42
# MR2 0.42 1.00
#
# Mean item complexity = 1
# Test of the hypothesis that 2 factors are sufficient.
#
# The degrees of freedom for the null model are 45 and the objective function was 7.36 with Chi Square of 1272.53
# The degrees of freedom for the model are 26 and the objective function was 0.51
#
# The root mean square of the residuals (RMSR) is 0.03
# The df corrected root mean square of the residuals is 0.04
#
# The harmonic number of observations is 178 with the empirical chi square 16.17 with prob < 0.93
# The total number of observations was 178 with Likelihood Chi Square = 87 with prob < 1.7e-08
#
# Tucker Lewis Index of factoring reliability = 0.913
# RMSEA index = 0.115 and the 90 % confidence intervals are 0.089 0.142
# BIC = -47.72
# Fit based upon off diagonal values = 1
# Measures of factor score adequacy
# MR1 MR2
# Correlation of (regression) scores with factors 0.98 0.91
# Multiple R square of scores with factors 0.95 0.84
# Minimum correlation of possible factor scores 0.90 0.67EFA 2 Interpretation
* RMSR = 0.03 = WOW! huge decrease by adding just 1 more factor
* Prop. var explained = 0.65, 14% raw difference from 1F model
* No poor factor loadings or low communalities
* Column and row parsimony is pretty amazing
* Notice that all negatively worded items load onto factor 2 & all
positively worded items load onto factor 1
* 2F prob wins, but let’s try 3F next anyways
EFA 2 Outliers Check
fS2 <- forward.search(spss.data, 2, criteria = c("mah", "GOF"))
gcdresult2 <- gCD(spss.data, 2)
ldresults2 <- LD(spss.data, 2)plot(gcdresult2)
plot(fS2)
plot(ldresults2)
# EFA 3 Factors -----------------------------------------------------------
efa3 <- fa(r = spss.data, fm = 'minres', cor = 'poly', nfactors = 3)EFA 3 Results
# Factor Analysis using method = minres
# Call: fa(r = spss.data, nfactors = 3, fm = "minres", cor = "poly")
# Standardized loadings (pattern matrix) based upon correlation matrix
# MR1 MR2 MR3 h2 u2 com
# SPSS1E 0.88 0.02 0.11 0.81 0.190 1.0
# SPSS2E -0.01 0.98 -0.09 0.94 0.064 1.0
# SPSS3E 0.12 0.65 0.23 0.60 0.401 1.3
# SPSS4E 0.93 -0.04 0.15 0.85 0.145 1.1
# SPSS5E 0.78 0.07 -0.17 0.68 0.322 1.1
# SPSS6E 0.64 0.07 -0.22 0.49 0.513 1.3
# SPSS7E 0.65 -0.01 -0.22 0.47 0.531 1.2
# SPSS8E 0.86 0.04 0.00 0.77 0.231 1.0
# SPSS9E 0.93 -0.03 -0.01 0.83 0.165 1.0
# SPSS10E 0.01 0.48 0.41 0.47 0.532 1.9
#
# MR1 MR2 MR3
# SS loadings 4.75 1.73 0.43
# Proportion Var 0.47 0.17 0.04
# Cumulative Var 0.47 0.65 0.69 *
# Proportion Explained 0.69 0.25 0.06
# Cumulative Proportion 0.69 0.94 1.00
#
# With factor correlations of
# MR1 MR2 MR3
# MR1 1.00 0.39 0.01
# MR2 0.39 1.00 0.16
# MR3 0.01 0.16 1.00
#
# Mean item complexity = 1.2
# Test of the hypothesis that 3 factors are sufficient.
#
# The degrees of freedom for the null model are 45 and the objective function was 7.36 with Chi Square of 1272.53
# The degrees of freedom for the model are 18 and the objective function was 0.32
#
# The root mean square of the residuals (RMSR) is 0.02
# The df corrected root mean square of the residuals is 0.03
#
# The harmonic number of observations is 178 with the empirical chi square 6.59 with prob < 0.99
# The total number of observations was 178 with Likelihood Chi Square = 53.87 with prob < 1.9e-05
#
# Tucker Lewis Index of factoring reliability = 0.926
# RMSEA index = 0.106 and the 90 % confidence intervals are 0.074 0.139
# BIC = -39.4
# Fit based upon off diagonal values = 1
# Measures of factor score adequacy
# MR1 MR2 MR3
# Correlation of (regression) scores with factors 0.98 0.97 0.72
# Multiple R square of scores with factors 0.96 0.95 0.51
# Minimum correlation of possible factor scores 0.91 0.89 0.03EFA 3 Interpretation
* RMSR = 0.02 = MEH only decreased by 0.01 after adding an additional
factor - not worth it bc RMSR always decreases when adding an additional
factor.
* Prop. var explained = 0.69, 4% raw difference from 2F model
* No low communalities, BUT
* In general, column and row parsimony is not nearly as good as 2F
model
* Concluding that 2F wins bc improvements in model fit isn’t worth it.
Additionally, column and row parsimony for 3F is worse than 2F model
EFA 3 Outliers Check
fS3 <- forward.search(spss.data, 3, criteria = c("mah", "GOF"))
gcdresult3 <- gCD(spss.data, 3)
ldresults3 <- LD(spss.data, 3)plot(gcdresult3)
plot(fS3)
plot(ldresults3)
# Likelihood Ratio Test for EFA 2 vs. EFA 3 -------------------------------
anova(efa2, efa3)## Model 1 = fa(r = spss.data, nfactors = 2, fm = "minres", cor = "poly")
## Model 2 = fa(r = spss.data, nfactors = 3, fm = "minres", cor = "poly")
## ANOVA Test for Difference Between Models
##
## df d.df chiSq d.chiSq PR test empirical d.empirical test.echi BIC d.BIC
## 1 26 87.00 16.17 -47.72
## 2 18 8 53.87 33.13 0 4.14 6.59 9.59 1.2 -39.40 8.32
LRT Interpretation
# Lower BIC indicates better fit, therefore, model 1 (i.e., 2-factor EFA) has a better fit than model 2 (3-factor EFA) # Bifactor Model Attempt --------------------------------------------------psych::schmid(model = poly.spss.data$rho, nfactors = 2, fm = 'minres', rotate = 'oblimin', na.obs = NA, option = 'equal')##
## Three factors are required for identification -- general factor loadings set to be equal.
## Proceed with caution.
## Think about redoing the analysis with alternative values of the 'option' setting.
## Schmid-Leiman analysis
## Call: psych::schmid(model = poly.spss.data$rho, nfactors = 2, fm = "minres",
## rotate = "oblimin", option = "equal", na.obs = NA)
##
## Schmid Leiman Factor loadings greater than 0.2
## g F1* F2* h2 u2 p2
## SPSS1E 0.61 0.64 0.78 0.22 0.47
## SPSS2E 0.54 0.60 0.66 0.34 0.45
## SPSS3E 0.55 0.62 0.69 0.31 0.44
## SPSS4E 0.60 0.67 0.80 0.20 0.45
## SPSS5E 0.52 0.62 0.66 0.34 0.41
## SPSS6E 0.42 0.52 0.45 0.55 0.39
## SPSS7E 0.38 0.54 0.44 0.56 0.33
## SPSS8E 0.59 0.65 0.77 0.23 0.45
## SPSS9E 0.58 0.70 0.83 0.17 0.41
## SPSS10E 0.37 0.50 0.39 0.61 0.35
##
## With eigenvalues of:
## g F1* F2*
## 2.7 2.7 1.0
##
## general/max 1 max/min = 2.7
## mean percent general = 0.42 with sd = 0.05 and cv of 0.11
##
## The orthogonal loadings were
## Unstandardized loadings based upon covariance matrix
## F1 F2 h2 u2 H2 U2
## SPSS1E 0.85 0.26 0.78 0.22 0.78 0.22
## SPSS2E 0.22 0.78 0.66 0.34 0.66 0.34
## SPSS3E 0.20 0.81 0.69 0.31 0.69 0.31
## SPSS4E 0.87 0.23 0.80 0.20 0.80 0.20
## SPSS5E 0.80 0.16 0.66 0.34 0.66 0.34
## SPSS6E 0.66 0.10 0.45 0.55 0.45 0.55
## SPSS7E 0.66 0.03 0.44 0.56 0.44 0.56
## SPSS8E 0.85 0.22 0.77 0.23 0.77 0.23
## SPSS9E 0.90 0.16 0.83 0.17 0.83 0.17
## SPSS10E 0.06 0.62 0.39 0.61 0.39 0.61
##
## F1 F2
## SS loadings 4.61 1.88
## Proportion Var 0.46 0.19
## Cumulative Var 0.46 0.65
##
## The degrees of freedom are 26 and the fit is 0.51
##
## The root mean square of the residuals is 0.03
## The df corrected root mean square of the residuals is 0.04
As indicated by the warning, at least three specific factors required
for identification (in an EFA context).
As a result, the 2 Factor model still fits the data best.
# Reliability -------------------------------------------------------------Since the 2 Factor model is multidimensional, we calculated reliability
for each factor.
We use psych::omega() as recommended by Flora (2020)
# split dataset into each factor
spss.data.f1 <- spss.data %>% select(-c(SPSS2E, SPSS3E, SPSS10E))
spss.data.f2 <- spss.data %>% select(c(SPSS2E, SPSS3E, SPSS10E))
# poly = TRUE because we want to use the polychoric correlation matrix instead of Pearson because of our categorical data
# since nfactors = 1, only omega total is meaningful
omega(m = spss.data.f1, poly = TRUE, plot = F, nfactors = 1) # Omega Total 0.93 ## Omega_h for 1 factor is not meaningful, just omega_t
## Warning in schmid(m, nfactors, fm, digits, rotate = rotate, n.obs = n.obs, :
## Omega_h and Omega_asymptotic are not meaningful with one factor
## Omega
## Call: omegah(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip,
## digits = digits, title = title, sl = sl, labels = labels,
## plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option,
## covar = covar)
## Alpha: 0.93
## G.6: 0.93
## Omega Hierarchical: 0.93
## Omega H asymptotic: 1
## Omega Total 0.93
##
## Schmid Leiman Factor loadings greater than 0.2
## g F1* h2 u2 p2
## SPSS1E 0.88 0.78 0.22 1
## SPSS4E 0.90 0.81 0.19 1
## SPSS5E 0.81 0.66 0.34 1
## SPSS6E 0.67 0.44 0.56 1
## SPSS7E 0.65 0.42 0.58 1
## SPSS8E 0.88 0.77 0.23 1
## SPSS9E 0.91 0.83 0.17 1
##
## With eigenvalues of:
## g F1*
## 4.7 0.0
##
## general/max 8.504964e+16 max/min = 1
## mean percent general = 1 with sd = 0 and cv of 0
## Explained Common Variance of the general factor = 1
##
## The degrees of freedom are 14 and the fit is 0.24
## The number of observations was 178 with Chi Square = 40.74 with prob < 2e-04
## The root mean square of the residuals is 0.03
## The df corrected root mean square of the residuals is 0.03
## RMSEA index = 0.103 and the 10 % confidence intervals are 0.068 0.142
## BIC = -31.81
##
## Compare this with the adequacy of just a general factor and no group factors
## The degrees of freedom for just the general factor are 14 and the fit is 0.24
## The number of observations was 178 with Chi Square = 40.74 with prob < 2e-04
## The root mean square of the residuals is 0.03
## The df corrected root mean square of the residuals is 0.03
##
## RMSEA index = 0.103 and the 10 % confidence intervals are 0.068 0.142
## BIC = -31.81
##
## Measures of factor score adequacy
## g F1*
## Correlation of scores with factors 0.97 0
## Multiple R square of scores with factors 0.95 0
## Minimum correlation of factor score estimates 0.90 -1
##
## Total, General and Subset omega for each subset
## g F1*
## Omega total for total scores and subscales 0.93 0.93
## Omega general for total scores and subscales 0.93 0.93
## Omega group for total scores and subscales 0.00 0.00
omega(m = spss.data.f2, poly = TRUE, plot = F, nfactors = 1) # Omega Total 0.8 ## Omega_h for 1 factor is not meaningful, just omega_t
## Warning in schmid(m, nfactors, fm, digits, rotate = rotate, n.obs = n.obs, :
## Omega_h and Omega_asymptotic are not meaningful with one factor
## Warning in cov2cor(t(w) %*% r %*% w): diag(.) had 0 or NA entries; non-finite
## result is doubtful
## Omega
## Call: omegah(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip,
## digits = digits, title = title, sl = sl, labels = labels,
## plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option,
## covar = covar)
## Alpha: 0.79
## G.6: 0.73
## Omega Hierarchical: 0.8
## Omega H asymptotic: 1
## Omega Total 0.8
##
## Schmid Leiman Factor loadings greater than 0.2
## g F1* h2 u2 p2
## SPSS2E 0.83 0.70 0.30 1
## SPSS3E 0.83 0.68 0.32 1
## SPSS10E 0.60 0.36 0.64 1
##
## With eigenvalues of:
## g F1*
## 1.7 0.0
##
## general/max Inf max/min = NaN
## mean percent general = 1 with sd = 0 and cv of 0
## Explained Common Variance of the general factor = 1
##
## The degrees of freedom are 0 and the fit is 0
## The number of observations was 178 with Chi Square = 0 with prob < NA
## The root mean square of the residuals is 0
## The df corrected root mean square of the residuals is NA
##
## Compare this with the adequacy of just a general factor and no group factors
## The degrees of freedom for just the general factor are 0 and the fit is 0
## The number of observations was 178 with Chi Square = 0 with prob < NA
## The root mean square of the residuals is 0
## The df corrected root mean square of the residuals is NA
##
## Measures of factor score adequacy
## g F1*
## Correlation of scores with factors 0.91 0
## Multiple R square of scores with factors 0.83 0
## Minimum correlation of factor score estimates 0.67 -1
##
## Total, General and Subset omega for each subset
## g F1*
## Omega total for total scores and subscales 0.8 0.8
## Omega general for total scores and subscales 0.8 0.8
## Omega group for total scores and subscales 0.0 0.0
# warning message regarding 'non-finite result is doubtful' refers to the NA or NaN values in the output. They should not be trusted, but exist because the input provided has NA valuesThe following code calculates omega for the scale overall (i.e., treating it as unidimensional).
omega(m = spss.data, poly = TRUE, plot = F, nfactors = 2) # Omega Total for total scores = 0.93, for F1 = 0.94 and for F2 = 0.80 ##
## Three factors are required for identification -- general factor loadings set to be equal.
## Proceed with caution.
## Think about redoing the analysis with alternative values of the 'option' setting.
## Omega
## Call: omegah(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip,
## digits = digits, title = title, sl = sl, labels = labels,
## plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option,
## covar = covar)
## Alpha: 0.9
## G.6: 0.93
## Omega Hierarchical: 0.52
## Omega H asymptotic: 0.55
## Omega Total 0.93
##
## Schmid Leiman Factor loadings greater than 0.2
## g F1* F2* h2 u2 p2
## SPSS1E 0.61 0.64 0.78 0.22 0.47
## SPSS2E 0.54 0.60 0.66 0.34 0.45
## SPSS3E 0.55 0.62 0.69 0.31 0.44
## SPSS4E 0.60 0.67 0.80 0.20 0.45
## SPSS5E 0.52 0.62 0.66 0.34 0.41
## SPSS6E 0.42 0.52 0.45 0.55 0.39
## SPSS7E 0.38 0.54 0.44 0.56 0.33
## SPSS8E 0.59 0.65 0.77 0.23 0.45
## SPSS9E 0.58 0.70 0.83 0.17 0.41
## SPSS10E 0.37 0.50 0.39 0.61 0.35
##
## With eigenvalues of:
## g F1* F2*
## 2.7 2.7 1.0
##
## general/max 1 max/min = 2.7
## mean percent general = 0.42 with sd = 0.05 and cv of 0.11
## Explained Common Variance of the general factor = 0.42
##
## The degrees of freedom are 26 and the fit is 0.51
## The number of observations was 178 with Chi Square = 87 with prob < 1.7e-08
## The root mean square of the residuals is 0.03
## The df corrected root mean square of the residuals is 0.04
## RMSEA index = 0.115 and the 10 % confidence intervals are 0.089 0.142
## BIC = -47.72
##
## Compare this with the adequacy of just a general factor and no group factors
## The degrees of freedom for just the general factor are 35 and the fit is 3.53
## The number of observations was 178 with Chi Square = 608.32 with prob < 9.3e-106
## The root mean square of the residuals is 0.28
## The df corrected root mean square of the residuals is 0.32
##
## RMSEA index = 0.303 and the 10 % confidence intervals are 0.283 0.326
## BIC = 426.95
##
## Measures of factor score adequacy
## g F1* F2*
## Correlation of scores with factors 0.74 0.80 0.74
## Multiple R square of scores with factors 0.55 0.64 0.55
## Minimum correlation of factor score estimates 0.10 0.27 0.11
##
## Total, General and Subset omega for each subset
## g F1* F2*
## Omega total for total scores and subscales 0.93 0.94 0.80
## Omega general for total scores and subscales 0.52 0.39 0.34
## Omega group for total scores and subscales 0.42 0.54 0.47
# Convergent Validity -----------------------------------------------------The following section runs Pearson Correlations between ATSPSS scale and
Quantitative Attitudes scale
to test for convergent validity.
First, the Quantitative Attitudes scale needs to be setup properly
(e.g., reverse code items).
Quantitative Attitudes Setup
qa.data <- full.data %>% select(
MA1E:MA8E
) %>% select(
-MA6E
# removed the item: 'Statistics is a not a worthwhile or necessary subject' based on previous validation paper (Kunicki et al., 2020)
) %>% mutate(
MA2E = car::recode(MA2E, "1 = 5; 2 = 4; 3 = 3; 4 = 2; 5 = 1"),
# reverse code 'Math is one of my most dreaded subjects'
MA3E = car::recode(MA3E, "1 = 5; 2 = 4; 3 = 3; 4 = 2; 5 = 1"),
# reverse code 'I have seldom liked studying mathematics'
MA7E = car::recode(MA7E, "1 = 5; 2 = 4; 3 = 3; 4 = 2; 5 = 1")
# reverse code 'I am not willing to take more than the required amount of statistics courses'
) %>% mutate(
total = rowSums(.[1:ncol(.)], na.rm = TRUE)
)
# Quantitative Attitudes Factor 1
qa.data.f1 <- qa.data %>% select(
MA1E:MA4E
) %>% mutate(
total = rowSums(.[1:ncol(.)], na.rm = TRUE)
)
# Quantitative Attitudes Factor 2
qa.data.f2 <- qa.data %>% select(
MA5E, MA7E, MA8E
) %>% mutate(
total = rowSums(.[1:ncol(.)], na.rm = TRUE)
)ATSPSS Setup
Similarly, the ATSPSS needs to be setup by splitting into its two
factors
# SPSS Attitudes Factor 1
sa.data.f1 <- spss.data.f1 %>% mutate(
total = rowSums(.[1:ncol(.)], na.rm = TRUE)
)
# SPSS Attitudes Factor 2
sa.data.f2 <- spss.data.f2 %>% mutate(
total = rowSums(.[1:ncol(.)], na.rm = TRUE)
)Ok, now we’re ready to run the correlations (and scatterplots) for convergent validity testing.
Correlations
cor.test(qa.data.f1$total, sa.data.f1$total) # r = 0.1528329 95% [0.005880347 0.293323814] t = 2.0517, df = 176, p-value = 0.04168 ##
## Pearson's product-moment correlation
##
## data: qa.data.f1$total and sa.data.f1$total
## t = 2.0517, df = 176, p-value = 0.04168
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.005880347 0.293323814
## sample estimates:
## cor
## 0.1528329
cor.test(qa.data.f1$total, sa.data.f2$total) # r = 0.1854679 95% [0.03945967 0.32372166] t = 2.504, df = 176, p-value = 0.01319 ##
## Pearson's product-moment correlation
##
## data: qa.data.f1$total and sa.data.f2$total
## t = 2.504, df = 176, p-value = 0.01319
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.03945967 0.32372166
## sample estimates:
## cor
## 0.1854679
car::scatterplot(qa.data.f1$total, sa.data.f1$total)
car::scatterplot(qa.data.f1$total, sa.data.f2$total)
cor.test(qa.data.f2$total, sa.data.f1$total) # r = 0.247044 95% [0.1037284 0.3803096] t = 3.3822, df = 176, p-value = 0.000886 ##
## Pearson's product-moment correlation
##
## data: qa.data.f2$total and sa.data.f1$total
## t = 3.3822, df = 176, p-value = 0.000886
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.1037284 0.3803096
## sample estimates:
## cor
## 0.247044
cor.test(qa.data.f2$total, sa.data.f2$total) # r = 0.1250142 95% [-0.02248388 0.26718600] t = 1.6716, df = 176, p-value = 0.09638 ##
## Pearson's product-moment correlation
##
## data: qa.data.f2$total and sa.data.f2$total
## t = 1.6716, df = 176, p-value = 0.09638
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.02248388 0.26718600
## sample estimates:
## cor
## 0.1250142
car::scatterplot(qa.data.f2$total, sa.data.f1$total)
car::scatterplot(qa.data.f2$total, sa.data.f2$total)
The following section runs Pearson Correlations between ATSPSS scale
and Quantitative Anxiety scale / Quantitative Hindrances scale
Quantitative Anxiety
# Set up Quant. Anxiety scale by selecting only the items and calculating total score
qanx.data <- full.data %>% select(
QANX1E:QANX4E
) %>% mutate(
total = rowSums(.[1:ncol(.)], na.rm = TRUE)
)
# Ok, now for the correlations
cor.test(qanx.data$total, sa.data.f1$total) # r = -0.06134227 95% [-0.20656324 0.08652308] t = -0.81533, df = 176, p-value = 0.416 ##
## Pearson's product-moment correlation
##
## data: qanx.data$total and sa.data.f1$total
## t = -0.81533, df = 176, p-value = 0.416
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.20656324 0.08652308
## sample estimates:
## cor
## -0.06134227
cor.test(qanx.data$total, sa.data.f2$total) # r = -0.07568429 95% [-0.22031643 0.07220418] t = -1.007, df = 176, p-value = 0.3153 ##
## Pearson's product-moment correlation
##
## data: qanx.data$total and sa.data.f2$total
## t = -1.007, df = 176, p-value = 0.3153
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.22031643 0.07220418
## sample estimates:
## cor
## -0.07568429
car::scatterplot(qanx.data$total, sa.data.f1$total)
car::scatterplot(qanx.data$total, sa.data.f2$total)
Quantitative Hindrances
Again, first set up Quant. Hindrances scale by selecting only the items
and calculating total score
qh.data <- full.data %>% select(
QHIND1E:QHIND5E
) %>% mutate(
total = rowSums(.[1:ncol(.)], na.rm = TRUE)
)
# Now for the correlations
cor.test(qh.data$total, sa.data.f1$total) # r = -0.03837761 95% [-0.1844213 0.1093242] t = -0.50951, df = 176, p-value = 0.611 ##
## Pearson's product-moment correlation
##
## data: qh.data$total and sa.data.f1$total
## t = -0.50951, df = 176, p-value = 0.611
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.1844213 0.1093242
## sample estimates:
## cor
## -0.03837761
cor.test(qh.data$total, sa.data.f2$total) # r = -0.06702523 95% [-0.21201975 0.08085658] t = -0.89119, df = 176, p-value = 0.374 ##
## Pearson's product-moment correlation
##
## data: qh.data$total and sa.data.f2$total
## t = -0.89119, df = 176, p-value = 0.374
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.21201975 0.08085658
## sample estimates:
## cor
## -0.06702523
car::scatterplot(qh.data$total, sa.data.f1$total)
car::scatterplot(qh.data$total, sa.data.f2$total)
# Correlations between all Measures ---------------------------------------The following correlations are to fill out the remaining cells of the convergent validity table.
ATSPSS Factor 1 x ATSPSS Factor 2
cor.test(sa.data.f1$total, sa.data.f2$total) # r = 0.2943086 95% [0.1538854 0.4230790] t = 4.0854, df = 176, p-value = 6.675e-05##
## Pearson's product-moment correlation
##
## data: sa.data.f1$total and sa.data.f2$total
## t = 4.0854, df = 176, p-value = 6.675e-05
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.1538854 0.4230790
## sample estimates:
## cor
## 0.2943086
Quant. Attitudes Factor 1 x Quant. Attitudes Factor 2
cor.test(qa.data.f1$total, qa.data.f2$total) # r = 0.4117248 95% [0.2816993 0.5269012] t = 5.9937, df = 176, p-value = 1.132e-08##
## Pearson's product-moment correlation
##
## data: qa.data.f1$total and qa.data.f2$total
## t = 5.9937, df = 176, p-value = 1.132e-08
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.2816993 0.5269012
## sample estimates:
## cor
## 0.4117248
Quant. Attitudes Factor 1 x Quant. Anxiety
cor.test(qa.data.f1$total, qanx.data$total) # r = -0.5866839 95% [ -0.6754799 -0.4811158] t = -9.6111, df = 176, p-value < 2.2e-16##
## Pearson's product-moment correlation
##
## data: qa.data.f1$total and qanx.data$total
## t = -9.6111, df = 176, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.6754799 -0.4811158
## sample estimates:
## cor
## -0.5866839
Quant. Attitudes Factor 1 x Quant. Hindrances
cor.test(qa.data.f1$total, qh.data$total) # r = -0.4092008 95% [-0.5247049 -0.2789024] t = -5.9496, df = 176, p-value = 1.418e-08##
## Pearson's product-moment correlation
##
## data: qa.data.f1$total and qh.data$total
## t = -5.9496, df = 176, p-value = 1.418e-08
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.5247049 -0.2789024
## sample estimates:
## cor
## -0.4092008
Quant. Attitudes Factor 2 x Quant. Anxiety
cor.test(qa.data.f2$total, qanx.data$total) # r = -0.3704627 95% [-0.4908038 -0.2362512] t = -5.2912, df = 176, p-value = 3.58e-07##
## Pearson's product-moment correlation
##
## data: qa.data.f2$total and qanx.data$total
## t = -5.2912, df = 176, p-value = 3.58e-07
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.4908038 -0.2362512
## sample estimates:
## cor
## -0.3704627
Quant. Attitudes Factor 2 x Quant. Hindrances
cor.test(qa.data.f2$total, qh.data$total) # r = -0.2618719 95% [-0.3937889 -0.1193856] t = -3.5997, df = 176, p-value = 0.000414##
## Pearson's product-moment correlation
##
## data: qa.data.f2$total and qh.data$total
## t = -3.5997, df = 176, p-value = 0.000414
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.3937889 -0.1193856
## sample estimates:
## cor
## -0.2618719
Quant. Anxiety x Quant. Hindrances
cor.test(qanx.data$total, qh.data$total) # r = 0.6228247 95% [0.5237166 0.7052984] t = 10.561, df = 176, p-value < 2.2e-16##
## Pearson's product-moment correlation
##
## data: qanx.data$total and qh.data$total
## t = 10.561, df = 176, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.5237166 0.7052984
## sample estimates:
## cor
## 0.6228247
# Confirmatory Multidimensional Item Response Theory (MIRT) ---------------irtmodel <- "
F1 = SPSS1E, SPSS4E, SPSS5E, SPSS6E, SPSS7E, SPSS8E, SPSS9E
F2 = SPSS2E, SPSS3E, SPSS10E
COV = F1*F2" #asterisk used to call for covariance
cmirtmod <- mirt.model(irtmodel, itemnames = spss.data)
cmirt <- mirt(data = spss.data, model = cmirtmod, itemtype = 'graded')coef(cmirt , IRTParam = T, simplify = T)## $items
## a1 a2 d1 d2 d3 d4
## SPSS1E 3.997 0.000 6.957 4.064 0.530 -4.052
## SPSS2E 0.000 3.086 6.079 3.087 1.447 -2.197
## SPSS3E 0.000 3.066 5.328 3.155 0.960 -3.106
## SPSS4E 4.467 0.000 8.295 3.983 0.186 -5.773
## SPSS5E 2.774 0.000 5.857 3.631 0.793 -1.996
## SPSS6E 1.639 0.000 3.135 2.038 0.207 -1.466
## SPSS7E 1.588 0.000 3.655 1.407 -0.080 -1.945
## SPSS8E 3.442 0.000 7.028 4.088 0.684 -3.426
## SPSS9E 4.560 0.000 8.014 4.228 -0.078 -4.656
## SPSS10E 0.000 1.573 3.648 1.811 0.212 -2.034
##
## $means
## F1 F2
## 0 0
##
## $cov
## F1 F2
## F1 1.0 0.5
## F2 0.5 1.0
MIRT Model Fit
M2(cmirt, type = "C2")## M2 df p RMSEA RMSEA_5 RMSEA_95 SRMSR
## stats 57.45641 34 0.007196968 0.06243163 0.0324369 0.08944299 0.08820926
## TLI CFI
## stats 0.9811121 0.9857292
MIRT Assumptions
residuals(cmirt)## LD matrix (lower triangle) and standardized values:
##
## SPSS1E SPSS2E SPSS3E SPSS4E SPSS5E SPSS6E SPSS7E SPSS8E SPSS9E
## SPSS1E NA 0.424 0.392 0.226 -0.461 -0.199 -0.224 -0.284 -0.249
## SPSS2E 127.880 NA -0.424 -0.427 0.264 -0.214 -0.289 -0.323 -0.363
## SPSS3E 109.210 128.186 NA -0.324 -0.279 -0.212 -0.258 0.272 -0.257
## SPSS4E 36.465 129.587 74.758 NA -0.806 -0.196 -0.186 -0.309 -0.292
## SPSS5E 151.375 49.721 55.339 462.624 NA 0.271 -0.218 -0.252 -0.197
## SPSS6E 28.309 32.688 31.957 27.333 52.439 NA -0.264 0.166 -0.184
## SPSS7E 35.818 59.299 47.315 24.502 33.901 49.514 NA -0.237 -0.244
## SPSS8E 57.491 74.379 52.782 67.942 45.266 19.632 40.000 NA -0.258
## SPSS9E 44.141 93.740 46.854 60.503 27.716 24.233 42.376 47.429 NA
## SPSS10E 112.082 103.058 87.417 114.234 121.545 64.857 87.976 145.586 163.213
## SPSS10E
## SPSS1E 0.397
## SPSS2E -0.380
## SPSS3E -0.350
## SPSS4E -0.401
## SPSS5E -0.413
## SPSS6E -0.302
## SPSS7E -0.352
## SPSS8E 0.452
## SPSS9E -0.479
## SPSS10E NA
# Sensitivity Analyses ----------------------------------------------------# missing data is removed using listwise deletion
str(spss.data)## 'data.frame': 178 obs. of 10 variables:
## $ SPSS1E : num 4 4 3 2 4 2 4 2 3 4 ...
## $ SPSS2E : num 4 5 3 4 5 2 3 2 4 5 ...
## $ SPSS3E : num 4 5 3 4 4 2 1 2 5 5 ...
## $ SPSS4E : num 3 4 2 2 3 3 2 3 3 4 ...
## $ SPSS5E : num 3 4 3 2 4 2 4 3 3 3 ...
## $ SPSS6E : num 3 4 2 2 4 1 4 3 1 5 ...
## $ SPSS7E : num 4 4 2 2 3 3 2 3 4 4 ...
## $ SPSS8E : num 4 4 3 2 4 3 3 3 3 2 ...
## $ SPSS9E : num 4 4 3 2 3 3 2 2 2 4 ...
## $ SPSS10E: num 4 5 3 4 5 3 3 3 4 3 ...
# missing data is not removed. This data will be used to imputing missing values later.
str(spss.data.withNA)## tibble [181 x 10] (S3: tbl_df/tbl/data.frame)
## $ SPSS1E : num [1:181] 4 4 3 2 4 2 4 2 3 4 ...
## $ SPSS2E : num [1:181] 4 5 3 4 5 2 3 2 4 5 ...
## $ SPSS3E : num [1:181] 4 5 3 4 4 2 1 2 5 5 ...
## $ SPSS4E : num [1:181] 3 4 2 2 3 3 2 3 3 4 ...
## $ SPSS5E : num [1:181] 3 4 3 2 4 2 4 3 3 3 ...
## $ SPSS6E : num [1:181] 3 4 2 2 4 1 4 3 1 5 ...
## $ SPSS7E : num [1:181] 4 4 2 2 3 3 2 3 4 4 ...
## $ SPSS8E : num [1:181] 4 4 3 2 4 3 3 3 3 2 ...
## $ SPSS9E : num [1:181] 4 4 3 2 3 3 2 2 2 4 ...
## $ SPSS10E: num [1:181] 4 5 3 4 5 3 3 3 4 3 ...
Refer to ‘Statistical Assumptions’ section at the beginning
# Multivariate Normality
# Linearitysnstvty_efa2 <- fa(r = spss.data, fm = "ml", nfactors = 2)
# Factor Analysis using method = ml
# Call: fa(r = spss.data, nfactors = 2, fm = "ml")
# Standardized loadings (pattern matrix) based upon correlation matrix
# ML1 ML2 h2 u2 com
# SPSS1E 0.84 0.06 0.74 0.26 1
# SPSS2E 0.02 0.76 0.59 0.41 1
# SPSS3E 0.02 0.77 0.60 0.40 1
# SPSS4E 0.87 0.00 0.76 0.24 1
# SPSS5E 0.76 -0.01 0.58 0.42 1
# SPSS6E 0.62 -0.03 0.38 0.62 1
# SPSS7E 0.63 -0.08 0.36 0.64 1
# SPSS8E 0.81 0.04 0.68 0.32 1
# SPSS9E 0.90 -0.03 0.78 0.22 1
# SPSS10E -0.07 0.59 0.32 0.68 1
#
# ML1 ML2
# SS loadings 4.29 1.51
# Proportion Var 0.43 0.15
# Cumulative Var 0.43 0.58
# Proportion Explained 0.74 0.26
# Cumulative Proportion 0.74 1.00
#
# With factor correlations of
# ML1 ML2
# ML1 1.00 0.38
# ML2 0.38 1.00
#
# Mean item complexity = 1
# Test of the hypothesis that 2 factors are sufficient.
#
# The degrees of freedom for the null model are 45 and the objective function was 5.61 with Chi Square of 969.94
# The degrees of freedom for the model are 26 and the objective function was 0.27
#
# The root mean square of the residuals (RMSR) is 0.03
# The df corrected root mean square of the residuals is 0.04
#
# The harmonic number of observations is 178 with the empirical chi square 15.35 with prob < 0.95
# The total number of observations was 178 with Likelihood Chi Square = 46.19 with prob < 0.0087
#
# Tucker Lewis Index of factoring reliability = 0.962
# RMSEA index = 0.066 and the 90 % confidence intervals are 0.033 0.097
# BIC = -88.53
# Fit based upon off diagonal values = 1
# Measures of factor score adequacy
# ML1 ML2
# Correlation of (regression) scores with factors 0.97 0.88
# Multiple R square of scores with factors 0.94 0.78
# Minimum correlation of possible factor scores 0.87 0.56snstvty_efa3 <- fa(r = spss.data, fm = "ml", nfactors = 3)
# Factor Analysis using method = ml
# Call: fa(r = spss.data, nfactors = 3, fm = "ml")
# Standardized loadings (pattern matrix) based upon correlation matrix
# ML1 ML2 ML3 h2 u2 com
# SPSS1E 0.54 0.09 0.32 0.73 0.274 1.7
# SPSS2E 0.12 0.76 -0.10 0.60 0.401 1.1
# SPSS3E -0.03 0.76 0.06 0.60 0.402 1.0
# SPSS4E 0.06 0.02 0.94 1.00 0.005 1.0
# SPSS5E 0.68 0.01 0.11 0.59 0.408 1.1
# SPSS6E 0.68 -0.02 -0.05 0.41 0.591 1.0
# SPSS7E 0.66 -0.06 -0.02 0.39 0.609 1.0
# SPSS8E 0.89 0.06 -0.07 0.75 0.254 1.0
# SPSS9E 0.64 0.00 0.28 0.76 0.238 1.4
# SPSS10E -0.17 0.58 0.11 0.32 0.676 1.2
#
# ML1 ML2 ML3
# SS loadings 3.18 1.55 1.41
# Proportion Var 0.32 0.15 0.14
# Cumulative Var 0.32 0.47 0.61
# Proportion Explained 0.52 0.25 0.23
# Cumulative Proportion 0.52 0.77 1.00
#
# With factor correlations of
# ML1 ML2 ML3
# ML1 1.00 0.35 0.79
# ML2 0.35 1.00 0.31
# ML3 0.79 0.31 1.00
#
# Mean item complexity = 1.2
# Test of the hypothesis that 3 factors are sufficient.
#
# The degrees of freedom for the null model are 45 and the objective function was 5.61 with Chi Square of 969.94
# The degrees of freedom for the model are 18 and the objective function was 0.14
#
# The root mean square of the residuals (RMSR) is 0.02
# The df corrected root mean square of the residuals is 0.04
#
# The harmonic number of observations is 178 with the empirical chi square 8.75 with prob < 0.97
# The total number of observations was 178 with Likelihood Chi Square = 24.53 with prob < 0.14
#
# Tucker Lewis Index of factoring reliability = 0.982
# RMSEA index = 0.045 and the 90 % confidence intervals are 0 0.086
# BIC = -68.74
# Fit based upon off diagonal values = 1
# Measures of factor score adequacy
# ML1 ML2 ML3
# Correlation of (regression) scores with factors 0.96 0.88 1.00
# Multiple R square of scores with factors 0.91 0.78 0.99
# Minimum correlation of possible factor scores 0.83 0.56 0.99imp <- mice(spss.data.withNA, m = 20)
imp <- complete(imp)
sum(is.na(imp)) # double checking that there is no missing dataefa2imp <- fa(r = imp, fm = "minres", cor = 'poly', nfactors = 2)
# Factor Analysis using method = minres
# Call: fa(r = imp, nfactors = 2, fm = "minres", cor = "poly")
# Standardized loadings (pattern matrix) based upon correlation matrix
# MR1 MR2 h2 u2 com
# SPSS1E 0.85 0.08 0.78 0.22 1.0
# SPSS2E 0.05 0.79 0.66 0.34 1.0
# SPSS3E 0.02 0.82 0.69 0.31 1.0
# SPSS4E 0.88 0.04 0.80 0.20 1.0
# SPSS5E 0.82 -0.02 0.66 0.34 1.0
# SPSS6E 0.68 -0.04 0.45 0.55 1.0
# SPSS7E 0.70 -0.12 0.44 0.56 1.1
# SPSS8E 0.86 0.05 0.77 0.23 1.0
# SPSS9E 0.93 -0.04 0.83 0.17 1.0
# SPSS10E -0.09 0.66 0.39 0.61 1.0
#
# MR1 MR2
# SS loadings 4.72 1.75
# Proportion Var 0.47 0.18
# Cumulative Var 0.47 0.65
# Proportion Explained 0.73 0.27
# Cumulative Proportion 0.73 1.00
#
# With factor correlations of
# MR1 MR2
# MR1 1.00 0.42
# MR2 0.42 1.00
#
# Mean item complexity = 1
# Test of the hypothesis that 2 factors are sufficient.
#
# The degrees of freedom for the null model are 45 and the objective function was 7.31 with Chi Square of 1285.8
# The degrees of freedom for the model are 26 and the objective function was 0.49
#
# The root mean square of the residuals (RMSR) is 0.03
# The df corrected root mean square of the residuals is 0.04
#
# The harmonic number of observations is 181 with the empirical chi square 16.23 with prob < 0.93
# The total number of observations was 181 with Likelihood Chi Square = 84.77 with prob < 3.8e-08
#
# Tucker Lewis Index of factoring reliability = 0.917
# RMSEA index = 0.112 and the 90 % confidence intervals are 0.086 0.139
# BIC = -50.39
# Fit based upon off diagonal values = 1
# Measures of factor score adequacy
# MR1
# Correlation of (regression) scores with factors 0.98
# Multiple R square of scores with factors 0.95
# Minimum correlation of possible factor scores 0.90
# MR2
# Correlation of (regression) scores with factors 0.91
# Multiple R square of scores with factors 0.84
# Minimum correlation of possible factor scores 0.67 efa3imp <- fa(r = imp, fm = "minres", cor = 'poly', nfactors = 3)
# Factor Analysis using method = minres
# Call: fa(r = imp, nfactors = 3, fm = "minres", cor = "poly")
# Standardized loadings (pattern matrix) based upon correlation matrix
# MR1 MR2 MR3 h2 u2 com
# SPSS1E 0.88 0.02 0.11 0.81 0.192 1.0
# SPSS2E -0.01 0.98 -0.08 0.94 0.061 1.0
# SPSS3E 0.12 0.64 0.24 0.60 0.401 1.3
# SPSS4E 0.93 -0.04 0.15 0.85 0.148 1.1
# SPSS5E 0.78 0.07 -0.17 0.68 0.322 1.1
# SPSS6E 0.64 0.07 -0.21 0.48 0.519 1.3
# SPSS7E 0.65 0.00 -0.22 0.47 0.532 1.2
# SPSS8E 0.86 0.04 0.00 0.76 0.236 1.0
# SPSS9E 0.93 -0.04 -0.01 0.83 0.166 1.0
# SPSS10E 0.01 0.47 0.42 0.47 0.528 2.0
#
# MR1 MR2 MR3
# SS loadings 4.74 1.72 0.43
# Proportion Var 0.47 0.17 0.04
# Cumulative Var 0.47 0.65 0.69
# Proportion Explained 0.69 0.25 0.06
# Cumulative Proportion 0.69 0.94 1.00
#
# With factor correlations of
# MR1 MR2 MR3
# MR1 1.00 0.38 0.02
# MR2 0.38 1.00 0.17
# MR3 0.02 0.17 1.00
#
# Mean item complexity = 1.2
# Test of the hypothesis that 3 factors are sufficient.
#
# The degrees of freedom for the null model are 45 and the objective function was 7.31 with Chi Square of 1285.8
# The degrees of freedom for the model are 18 and the objective function was 0.3
#
# The root mean square of the residuals (RMSR) is 0.02
# The df corrected root mean square of the residuals is 0.03
#
# The harmonic number of observations is 181 with the empirical chi square 6.71 with prob < 0.99
# The total number of observations was 181 with Likelihood Chi Square = 52.21 with prob < 3.5e-05
#
# Tucker Lewis Index of factoring reliability = 0.93
# RMSEA index = 0.102 and the 90 % confidence intervals are 0.071 0.136
# BIC = -41.37
# Fit based upon off diagonal values = 1
# Measures of factor score adequacy
# MR1 MR2 MR3
# Correlation of (regression) scores with factors 0.98 0.97 0.72
# Multiple R square of scores with factors 0.96 0.95 0.51
# Minimum correlation of possible factor scores 0.91 0.90 0.03efa2pair <- fa(r = spss.data.withNA, use = "pairwise", fm = "minres", cor = 'poly', nfactors = 2)
# Factor Analysis using method = minres
# Call: fa(r = spss.data.withNA, nfactors = 2, fm = "minres", use = "pairwise",
# cor = "poly")
# Standardized loadings (pattern matrix) based upon correlation matrix
# MR1 MR2 h2 u2 com
# SPSS1E 0.85 0.08 0.79 0.21 1.0
# SPSS2E 0.05 0.79 0.66 0.34 1.0
# SPSS3E 0.02 0.82 0.69 0.31 1.0
# SPSS4E 0.88 0.04 0.80 0.20 1.0
# SPSS5E 0.82 -0.02 0.66 0.34 1.0
# SPSS6E 0.68 -0.04 0.45 0.55 1.0
# SPSS7E 0.70 -0.12 0.44 0.56 1.1
# SPSS8E 0.86 0.05 0.77 0.23 1.0
# SPSS9E 0.93 -0.04 0.83 0.17 1.0
# SPSS10E -0.09 0.66 0.39 0.61 1.0
#
# MR1 MR2
# SS loadings 4.72 1.75
# Proportion Var 0.47 0.17
# Cumulative Var 0.47 0.65
# Proportion Explained 0.73 0.27
# Cumulative Proportion 0.73 1.00
#
# With factor correlations of
# MR1 MR2
# MR1 1.00 0.42
# MR2 0.42 1.00
#
# Mean item complexity = 1
# Test of the hypothesis that 2 factors are sufficient.
#
# The degrees of freedom for the null model are 45 and the objective function was 7.31 with Chi Square of 1284.96
# The degrees of freedom for the model are 26 and the objective function was 0.49
#
# The root mean square of the residuals (RMSR) is 0.03
# The df corrected root mean square of the residuals is 0.04
#
# The harmonic number of observations is 180 with the empirical chi square 16.09 with prob < 0.93
# The total number of observations was 181 with Likelihood Chi Square = 84.89 with prob < 3.6e-08
#
# Tucker Lewis Index of factoring reliability = 0.917
# RMSEA index = 0.112 and the 90 % confidence intervals are 0.086 0.139
# BIC = -50.27
# Fit based upon off diagonal values = 1
# Measures of factor score adequacy
# MR1 MR2
# Correlation of (regression) scores with factors 0.98 0.91
# Multiple R square of scores with factors 0.95 0.83
# Minimum correlation of possible factor scores 0.90 0.67efa3pair <- fa(r = spss.data.withNA, use = "pairwise", fm = "minres", cor = 'poly', nfactors = 3)
# Factor Analysis using method = minres
# Call: fa(r = spss.data.withNA, nfactors = 3, fm = "minres", use = "pairwise",
# cor = "poly")
# Standardized loadings (pattern matrix) based upon correlation matrix
# MR1 MR2 MR3 h2 u2 com
# SPSS1E 0.88 0.02 0.11 0.81 0.191 1.0
# SPSS2E -0.01 0.99 -0.08 0.95 0.051 1.0
# SPSS3E 0.13 0.63 0.24 0.59 0.406 1.4
# SPSS4E 0.93 -0.04 0.14 0.85 0.148 1.1
# SPSS5E 0.78 0.07 -0.17 0.67 0.325 1.1
# SPSS6E 0.64 0.07 -0.21 0.48 0.519 1.3
# SPSS7E 0.65 0.00 -0.22 0.47 0.533 1.2
# SPSS8E 0.86 0.04 0.00 0.77 0.235 1.0
# SPSS9E 0.93 -0.04 -0.01 0.83 0.166 1.0
# SPSS10E 0.02 0.46 0.43 0.47 0.526 2.0
#
# MR1 MR2 MR3
# SS loadings 4.74 1.71 0.45
# Proportion Var 0.47 0.17 0.04
# Cumulative Var 0.47 0.65 0.69
# Proportion Explained 0.69 0.25 0.06
# Cumulative Proportion 0.69 0.94 1.00
#
# With factor correlations of
# MR1 MR2 MR3
# MR1 1.00 0.38 0.02
# MR2 0.38 1.00 0.19
# MR3 0.02 0.19 1.00
#
# Mean item complexity = 1.2
# Test of the hypothesis that 3 factors are sufficient.
#
# The degrees of freedom for the null model are 45 and the objective function was 7.31 with Chi Square of 1284.96
# The degrees of freedom for the model are 18 and the objective function was 0.3
#
# The root mean square of the residuals (RMSR) is 0.02
# The df corrected root mean square of the residuals is 0.03
#
# The harmonic number of observations is 180 with the empirical chi square 6.6 with prob < 0.99
# The total number of observations was 181 with Likelihood Chi Square = 52.43 with prob < 3.2e-05
#
# Tucker Lewis Index of factoring reliability = 0.93
# RMSEA index = 0.103 and the 90 % confidence intervals are 0.071 0.136
# BIC = -41.15
# Fit based upon off diagonal values = 1
# Measures of factor score adequacy
# MR1 MR2 MR3
# Correlation of (regression) scores with factors 0.98 0.98 0.72
# Multiple R square of scores with factors 0.96 0.96 0.52
# Minimum correlation of possible factor scores 0.91 0.92 0.03Since we decided the 2 factor EFA fit the data best, we’re going to try
various oblique rotations
to find the one that provides the best row and column parsimony:
BentlerQ
efa2temp <- fa(r = spss.data, fm = 'minres', cor = 'poly', rotate = 'bentlerQ', nfactors = 2)
# MR1 MR2 h2 u2 com
# SPSS1E 0.85 0.08 0.78 0.22 1.0
# SPSS2E 0.04 0.79 0.66 0.34 1.0
# SPSS3E 0.02 0.82 0.69 0.31 1.0
# SPSS4E 0.88 0.04 0.80 0.20 1.0
# SPSS5E 0.82 -0.02 0.66 0.34 1.0
# SPSS6E 0.69 -0.05 0.45 0.55 1.0
# SPSS7E 0.71 -0.12 0.44 0.56 1.1
# SPSS8E 0.86 0.04 0.77 0.23 1.0
# SPSS9E 0.93 -0.04 0.83 0.17 1.0
# SPSS10E -0.09 0.66 0.39 0.61 1.0GeominQ
efa2temp <- fa(r = spss.data, fm = 'minres', cor = 'poly', rotate = 'geominQ', nfactors = 2)
# MR1 MR2 h2 u2 com
# SPSS1E 0.85 0.09 0.78 0.22 1.0
# SPSS2E 0.05 0.79 0.66 0.34 1.0
# SPSS3E 0.03 0.82 0.69 0.31 1.0
# SPSS4E 0.88 0.05 0.80 0.20 1.0
# SPSS5E 0.82 -0.01 0.66 0.34 1.0
# SPSS6E 0.68 -0.04 0.45 0.55 1.0
# SPSS7E 0.70 -0.11 0.44 0.56 1.1
# SPSS8E 0.86 0.05 0.77 0.23 1.0
# SPSS9E 0.92 -0.03 0.83 0.17 1.0
# SPSS10E -0.08 0.65 0.39 0.61 1.0Quartimin
efa2temp <- fa(r = spss.data, fm = 'minres', cor = 'poly', rotate = "quartimin", nfactors = 2)
# MR1 MR2 h2 u2 com
# SPSS1E 0.85 0.08 0.78 0.22 1.0
# SPSS2E 0.05 0.79 0.66 0.34 1.0
# SPSS3E 0.02 0.82 0.69 0.31 1.0
# SPSS4E 0.88 0.04 0.80 0.20 1.0
# SPSS5E 0.82 -0.02 0.66 0.34 1.0
# SPSS6E 0.69 -0.04 0.45 0.55 1.0
# SPSS7E 0.71 -0.12 0.44 0.56 1.1
# SPSS8E 0.86 0.04 0.77 0.23 1.0
# SPSS9E 0.93 -0.03 0.83 0.17 1.0
# SPSS10E -0.09 0.66 0.39 0.61 1.0Promax
efa2temp <- fa(r = spss.data, fm = 'minres', cor = 'poly', rotate = "Promax", nfactors = 2)
# MR1 MR2 h2 u2 com
# SPSS1E 0.85 0.09 0.78 0.22 1
# SPSS2E 0.08 0.78 0.66 0.34 1
# SPSS3E 0.06 0.81 0.69 0.31 1
# SPSS4E 0.87 0.05 0.80 0.20 1
# SPSS5E 0.81 0.00 0.66 0.34 1
# SPSS6E 0.68 -0.03 0.45 0.55 1
# SPSS7E 0.70 -0.11 0.44 0.56 1
# SPSS8E 0.86 0.05 0.77 0.23 1
# SPSS9E 0.92 -0.02 0.83 0.17 1
# SPSS10E -0.06 0.64 0.39 0.61 1Overall, none of the rotations were significantly better than the
default oblimin rotation.
So, we decided to just stick with oblimin.