The aim of this study was to examine the perception of households towards their residential location considering several land use and accessibility factors as well as household socioeconomic and attitudinal characteristics.
Reference: Martinez, L. G., de Abreu e Silva, J., & Viegas, J. M. (2010). Assessment of residential location satisfaction in the Lisbon metropolitan area, TRB (No. 10-1161).
Your task: Analyze the data and create meaningful latent factors.
DWELCLAS: Classification of the dwelling;INCOME: Income of the household;CHILD13: Number of children under 13 years old;H18: Number of household members above 18 years old;HEMPLOY: Number of household members employed;HSIZE: Household size;IAGE: Age of the respondent;ISEX: Sex of the respondent;NCARS: Number of cars in the household;AREA: Area of the dwelling;BEDROOM: Number of bedrooms in the dwelling;PARK: Number of parking spaces in the dwelling;BEDSIZE: BEDROOM/HSIZE;PARKSIZE: PARK/NCARS;RAGE10: 1 if Dwelling age <= 10;TCBD: Private car distance in time to CBD;DISTTC: Euclidean distance to heavy public transport system stops;TWCBD: Private car distance in time of workplace to CBD;TDWWK: Private car distance in time of dwelling to work place;HEADH: 1 if Head of the Household;POPDENS: Population density per hectare;EQUINDEX: Number of undergraduate students/Population over 20 years old (500m)
- At least 10 variables
- n < 50 (Unacceptable); n > 200 (recommended)
- It is recommended to use continuous variables. If your data contains categorical variables, you should transform them to dummy variables.
- Normality;
- linearity;
- Homogeneity;
- Homoscedasticity (some multicollinearity is desirable);
- Correlations between variables < 0.3 (not appropriate to use Factor Analysis)
library(foreign) # Library used to read SPSS files
library(nFactors) # Library used for factor analysis
library(tidyverse) # Library used in data science to perform exploratory data analysis
library(summarytools) # Library used for checking the summary of the dataset
library(psych) # Library used for factor analysis
library(GPArotation) # Library used for factor analysisdf <- read.spss("Data/example_fact.sav", to.data.frame = T) #transforms a list into a data.frame directlyclass(df) #type of data## [1] "data.frame"
str(df)## 'data.frame': 470 obs. of 32 variables:
## $ RespondentID: num 7.99e+08 7.98e+08 7.98e+08 7.98e+08 7.98e+08 ...
## $ DWELCLAS : num 5 6 6 5 6 6 4 2 6 5 ...
## $ INCOME : num 7500 4750 4750 7500 2750 1500 12500 1500 1500 1500 ...
## $ CHILD13 : num 1 0 2 0 1 0 0 0 0 0 ...
## $ H18 : num 2 1 2 3 1 3 3 4 1 1 ...
## $ HEMPLOY : num 2 1 2 2 1 2 0 2 1 1 ...
## $ HSIZE : num 3 1 4 4 2 3 3 4 1 1 ...
## $ AVADUAGE : num 32 31 41.5 44.7 33 ...
## $ IAGE : num 32 31 42 52 33 47 62 21 34 25 ...
## $ ISEX : num 1 1 0 1 0 1 1 0 0 0 ...
## $ NCARS : num 2 1 2 3 1 1 2 3 1 1 ...
## $ AREA : num 100 90 220 120 90 100 178 180 80 50 ...
## $ BEDROOM : num 2 2 4 3 2 2 5 3 2 1 ...
## $ PARK : num 1 1 2 0 0 0 2 0 0 1 ...
## $ BEDSIZE : num 0.667 2 1 0.75 1 ...
## $ PARKSIZE : num 0.5 1 1 0 0 0 1 0 0 1 ...
## $ RAGE10 : num 1 0 1 0 0 0 0 0 1 1 ...
## $ TCBD : num 36.79 15.47 24.1 28.72 7.28 ...
## $ DISTHTC : num 629 551 548 2351 698 ...
## $ TWCBD : num 10 15.5 12.71 3.17 5.36 ...
## $ TDWWK : num 31.1 0 20.4 32.9 13 ...
## $ HEADH : num 1 1 1 1 1 1 1 0 1 1 ...
## $ POPDENS : num 85.7 146.4 106.6 36.8 181.6 ...
## $ EDUINDEX : num 0.0641 0.2672 0.1 0.0867 0.1309 ...
## $ GRAVCPC : num 0.249 0.329 0.24 0.273 0.285 ...
## $ GRAVCPT : num 0.249 0.31 0.29 0.249 0.291 ...
## $ GRAVPCPT : num 1 1.062 0.826 1.099 0.98 ...
## $ NSTRTC : num 38 34 33 6 31 45 12 6 4 22 ...
## $ DISTHW : num 2036 748 2279 1196 3507 ...
## $ DIVIDX : num 0.323 0.348 0.324 0.327 0.355 ...
## $ ACTDENS : num 0.672 2.486 1.625 1.766 11.325 ...
## $ DISTCBD : num 9776 3524 11036 6257 1265 ...
## - attr(*, "variable.labels")= Named chr(0)
## ..- attr(*, "names")= chr(0)
## - attr(*, "codepage")= int 1252
descriptive_stats <- dfSummary(df)
view(descriptive_stats)Dimensions: 470 x 32
Duplicates: 0
| No | Variable | Stats / Values | Freqs (% of Valid) | Graph | Missing |
|---|---|---|---|---|---|
| 1 | RespondentID [numeric] |
Mean (sd) : 784082878 (8627501) min < med < max: 773001005 < 780248283 < 808234671 IQR (CV) : 16634785 (0) |
470 distinct values |  |
0 (0.0%) |
| 2 | DWELCLAS [numeric] |
Mean (sd) : 5.1 (1.3) min < med < max: 1 < 5 < 7 IQR (CV) : 2 (0.2) |
1 : 5 ( 1.1%) 2 : 14 ( 3.0%) 3 : 31 ( 6.6%) 4 : 75 (16.0%) 5 : 130 (27.7%) 6 : 162 (34.5%) 7 : 53 (11.3%) |
 |
0 (0.0%) |
| 3 | INCOME [numeric] |
Mean (sd) : 4259.6 (3001.8) min < med < max: 700 < 2750 < 12500 IQR (CV) : 2000 (0.7) |
700 : 20 ( 4.3%) 1500 : 96 (20.4%) 2750 : 142 (30.2%) 4750 : 106 (22.6%) 7500 : 75 (16.0%) 12500 : 31 ( 6.6%) |
 |
0 (0.0%) |
| 4 | CHILD13 [numeric] |
Mean (sd) : 0.4 (0.8) min < med < max: 0 < 0 < 4 IQR (CV) : 0 (2) |
0 : 353 (75.1%) 1 : 62 (13.2%) 2 : 41 ( 8.7%) 3 : 13 ( 2.8%) 4 : 1 ( 0.2%) |
 |
0 (0.0%) |
| 5 | H18 [numeric] |
Mean (sd) : 2.1 (0.9) min < med < max: 0 < 2 < 6 IQR (CV) : 0.8 (0.4) |
0 : 1 ( 0.2%) 1 : 112 (23.8%) 2 : 239 (50.9%) 3 : 77 (16.4%) 4 : 35 ( 7.4%) 5 : 3 ( 0.6%) 6 : 3 ( 0.6%) |
 |
0 (0.0%) |
| 6 | HEMPLOY [numeric] |
Mean (sd) : 1.5 (0.7) min < med < max: 0 < 2 < 5 IQR (CV) : 1 (0.5) |
0 : 39 ( 8.3%) 1 : 171 (36.4%) 2 : 237 (50.4%) 3 : 21 ( 4.5%) 4 : 1 ( 0.2%) 5 : 1 ( 0.2%) |
 |
0 (0.0%) |
| 7 | HSIZE [numeric] |
Mean (sd) : 2.6 (1.3) min < med < max: 1 < 2 < 7 IQR (CV) : 2 (0.5) |
1 : 104 (22.1%) 2 : 147 (31.3%) 3 : 96 (20.4%) 4 : 96 (20.4%) 5 : 20 ( 4.3%) 6 : 5 ( 1.1%) 7 : 2 ( 0.4%) |
 |
0 (0.0%) |
| 8 | AVADUAGE [numeric] |
Mean (sd) : 37.8 (9.9) min < med < max: 0 < 36 < 78 IQR (CV) : 12.7 (0.3) |
126 distinct values |  |
0 (0.0%) |
| 9 | IAGE [numeric] |
Mean (sd) : 36.9 (11.6) min < med < max: 0 < 34 < 78 IQR (CV) : 15 (0.3) |
53 distinct values |  |
0 (0.0%) |
| 10 | ISEX [numeric] |
Min : 0 Mean : 0.5 Max : 1 |
0 : 214 (45.5%) 1 : 256 (54.5%) |
 |
0 (0.0%) |
| 11 | NCARS [numeric] |
Mean (sd) : 1.7 (0.9) min < med < max: 0 < 2 < 5 IQR (CV) : 1 (0.5) |
0 : 23 ( 4.9%) 1 : 182 (38.7%) 2 : 193 (41.1%) 3 : 56 (11.9%) 4 : 13 ( 2.8%) 5 : 3 ( 0.6%) |
 |
0 (0.0%) |
| 12 | AREA [numeric] |
Mean (sd) : 133 (121.5) min < med < max: 30 < 110 < 2250 IQR (CV) : 60 (0.9) |
76 distinct values |  |
0 (0.0%) |
| 13 | BEDROOM [numeric] |
Mean (sd) : 2.9 (1.1) min < med < max: 0 < 3 < 7 IQR (CV) : 1 (0.4) |
0 : 1 ( 0.2%) 1 : 28 ( 6.0%) 2 : 153 (32.6%) 3 : 180 (38.3%) 4 : 73 (15.5%) 5 : 26 ( 5.5%) 6 : 7 ( 1.5%) 7 : 2 ( 0.4%) |
 |
0 (0.0%) |
| 14 | PARK [numeric] |
Mean (sd) : 0.8 (1) min < med < max: 0 < 1 < 4 IQR (CV) : 1 (1.2) |
0 : 224 (47.7%) 1 : 136 (28.9%) 2 : 84 (17.9%) 3 : 18 ( 3.8%) 4 : 8 ( 1.7%) |
 |
0 (0.0%) |
| 15 | BEDSIZE [numeric] |
Mean (sd) : 1.4 (0.8) min < med < max: 0 < 1 < 5 IQR (CV) : 0.7 (0.6) |
22 distinct values |  |
0 (0.0%) |
| 16 | PARKSIZE [numeric] |
Mean (sd) : 0.5 (0.6) min < med < max: 0 < 0.2 < 3 IQR (CV) : 1 (1.2) |
13 distinct values |  |
0 (0.0%) |
| 17 | RAGE10 [numeric] |
Min : 0 Mean : 0.2 Max : 1 |
0 : 356 (75.7%) 1 : 114 (24.3%) |
 |
0 (0.0%) |
| 18 | TCBD [numeric] |
Mean (sd) : 24.7 (16.2) min < med < max: 0.8 < 23.8 < 73.3 IQR (CV) : 25.7 (0.7) |
434 distinct values |  |
0 (0.0%) |
| 19 | DISTHTC [numeric] |
Mean (sd) : 1347 (1815.8) min < med < max: 49 < 719 < 17732.7 IQR (CV) : 1125 (1.3) |
434 distinct values |  |
0 (0.0%) |
| 20 | TWCBD [numeric] |
Mean (sd) : 17 (16.2) min < med < max: 0.3 < 9.9 < 67.8 IQR (CV) : 20 (1) |
439 distinct values |  |
0 (0.0%) |
| 21 | TDWWK [numeric] |
Mean (sd) : 23.5 (17.1) min < med < max: 0 < 22.2 < 80.7 IQR (CV) : 23.6 (0.7) |
414 distinct values |  |
0 (0.0%) |
| 22 | HEADH [numeric] |
Min : 0 Mean : 0.9 Max : 1 |
0 : 64 (13.6%) 1 : 406 (86.4%) |
 |
0 (0.0%) |
| 23 | POPDENS [numeric] |
Mean (sd) : 92 (58.2) min < med < max: 0 < 83.2 < 255.6 IQR (CV) : 89.2 (0.6) |
431 distinct values |  |
0 (0.0%) |
| 24 | EDUINDEX [numeric] |
Mean (sd) : 0.2 (0.1) min < med < max: 0 < 0.2 < 0.7 IQR (CV) : 0.2 (0.6) |
434 distinct values |  |
0 (0.0%) |
| 25 | GRAVCPC [numeric] |
Mean (sd) : 0.3 (0.1) min < med < max: 0.1 < 0.3 < 0.4 IQR (CV) : 0.1 (0.2) |
433 distinct values |  |
0 (0.0%) |
| 26 | GRAVCPT [numeric] |
Mean (sd) : 0.3 (0.1) min < med < max: 0 < 0.3 < 0.4 IQR (CV) : 0.1 (0.2) |
434 distinct values |  |
0 (0.0%) |
| 27 | GRAVPCPT [numeric] |
Mean (sd) : 1.2 (0.3) min < med < max: 0.5 < 1.1 < 2.9 IQR (CV) : 0.2 (0.3) |
434 distinct values |  |
0 (0.0%) |
| 28 | NSTRTC [numeric] |
Mean (sd) : 22.2 (12.6) min < med < max: 0 < 22 < 84 IQR (CV) : 16 (0.6) |
59 distinct values |  |
0 (0.0%) |
| 29 | DISTHW [numeric] |
Mean (sd) : 1883.4 (1748.3) min < med < max: 74.7 < 1338.7 < 16590.1 IQR (CV) : 1820.5 (0.9) |
434 distinct values |  |
0 (0.0%) |
| 30 | DIVIDX [numeric] |
Mean (sd) : 0.4 (0.1) min < med < max: 0.3 < 0.4 < 0.6 IQR (CV) : 0.1 (0.2) |
144 distinct values |  |
0 (0.0%) |
| 31 | ACTDENS [numeric] |
Mean (sd) : 5.8 (8.5) min < med < max: 0 < 2.5 < 63.2 IQR (CV) : 3.5 (1.5) |
144 distinct values |  |
0 (0.0%) |
| 32 | DISTCBD [numeric] |
Mean (sd) : 7967.4 (7442.9) min < med < max: 148.9 < 5542.3 < 44004.6 IQR (CV) : 9777.9 (0.9) |
434 distinct values |  |
0 (0.0%) |
Note: I used a different library of the MLR chapter for perfoming the summary statistics. “R” allows you to do the same or similar tasks with different packages.
head(df,5)## RespondentID DWELCLAS INCOME CHILD13 H18 HEMPLOY HSIZE AVADUAGE IAGE ISEX
## 1 799161661 5 7500 1 2 2 3 32.00000 32 1
## 2 798399409 6 4750 0 1 1 1 31.00000 31 1
## 3 798374392 6 4750 2 2 2 4 41.50000 42 0
## 4 798275277 5 7500 0 3 2 4 44.66667 52 1
## 5 798264250 6 2750 1 1 1 2 33.00000 33 0
## NCARS AREA BEDROOM PARK BEDSIZE PARKSIZE RAGE10 TCBD DISTHTC
## 1 2 100 2 1 0.6666667 0.5 1 36.791237 629.1120
## 2 1 90 2 1 2.0000000 1.0 0 15.472989 550.5769
## 3 2 220 4 2 1.0000000 1.0 1 24.098125 547.8633
## 4 3 120 3 0 0.7500000 0.0 0 28.724796 2350.5782
## 5 1 90 2 0 1.0000000 0.0 0 7.283384 698.3000
## TWCBD TDWWK HEADH POPDENS EDUINDEX GRAVCPC GRAVCPT GRAVPCPT
## 1 10.003945 31.14282 1 85.70155 0.06406279 0.2492962 0.2492607 1.0001423
## 2 15.502989 0.00000 1 146.43494 0.26723192 0.3293831 0.3102800 1.0615674
## 3 12.709374 20.38427 1 106.60810 0.09996816 0.2396229 0.2899865 0.8263245
## 4 3.168599 32.94246 1 36.78380 0.08671065 0.2734539 0.2487830 1.0991661
## 5 5.364160 13.04013 1 181.62720 0.13091674 0.2854017 0.2913676 0.9795244
## NSTRTC DISTHW DIVIDX ACTDENS DISTCBD
## 1 38 2036.4661 0.3225354 0.6722406 9776.142
## 2 34 747.7683 0.3484588 2.4860345 3523.994
## 3 33 2279.0577 0.3237884 1.6249059 11036.407
## 4 6 1196.4665 0.3272149 1.7664923 6257.262
## 5 31 3507.2402 0.3545181 11.3249309 1265.239
df<-data.frame(df, row.names = 1)Let’s run a random regression model in order to evaluate some assumptions
random = rchisq(nrow(df), 32)
fake = lm(random ~ ., data = df)
standardized = rstudent(fake)
fitted = scale(fake$fitted.values)- Normality
hist(standardized)
- Linearity
qqnorm(standardized)
abline(0,1)
- Homogeneity
plot(fitted, standardized)
abline(h=0,v=0)
- Correlations between variables Correlation matrix
corr_matrix <- cor(df, method = "pearson")The Bartlett test examines if there is equal variance (homogeneity) between variables. Thus, it evaluates if there is any pattern between variables. Check for correlation adequacy - Bartlett’s Test.
cortest.bartlett(corr_matrix, n = nrow(df))## $chisq
## [1] 9880.074
##
## $p.value
## [1] 0
##
## $df
## [1] 465
Note: The null hypothesis is that there is no correlation between variables. Therefore, in factor analysis you want to reject the null hypothesis.
- Check for sampling adequacy - KMO test
KMO(corr_matrix)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = corr_matrix)
## Overall MSA = 0.68
## MSA for each item =
## DWELCLAS INCOME CHILD13 H18 HEMPLOY HSIZE AVADUAGE IAGE
## 0.70 0.85 0.33 0.58 0.88 0.59 0.38 0.40
## ISEX NCARS AREA BEDROOM PARK BEDSIZE PARKSIZE RAGE10
## 0.71 0.74 0.60 0.53 0.62 0.58 0.57 0.84
## TCBD DISTHTC TWCBD TDWWK HEADH POPDENS EDUINDEX GRAVCPC
## 0.88 0.88 0.82 0.89 0.47 0.82 0.85 0.76
## GRAVCPT GRAVPCPT NSTRTC DISTHW DIVIDX ACTDENS DISTCBD
## 0.71 0.31 0.83 0.76 0.63 0.70 0.86
Note: We want at least 0.7 of the overall Mean Sample Adequacy (MSA). If, 0.6 < MSA < 0.7, it is not a good value, but acceptable in some cases.
1. Parallel Analysis
num_factors = fa.parallel(df, fm = "ml", fa = "fa")
## Parallel analysis suggests that the number of factors = 9 and the number of components = NA
Note:
fm= factor math;ml= maximum likelihood;fa= factor analysis
The selection of the number of factors in the Parallel analysis can be threefold:
- Detect where there is an “elbow” in the graph;
- Detect the intersection between the “FA Actual Data” and the “FA Simulated Data”;
- Consider the number of factors with eigenvalue > 1.
2. Kaiser Criterion
sum(num_factors$fa.values > 1) #Determines the number of factors with eigenvalue > 1## [1] 4
You can also consider factors with eigenvalue > 0.7, since some of the literature indicate that this value does not overestimate the number of factors as much as considering an eigenvalue = 1.
3. Principal Component Analysis (PCA)
- Print variance that explains the components
df_pca <- princomp(df, cor=TRUE) #cor = TRUE, standardizes your dataset before running a PCA
summary(df_pca) ## Importance of components:
## Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
## Standard deviation 2.450253 1.9587909 1.61305418 1.43367870 1.27628545
## Proportion of Variance 0.193669 0.1237697 0.08393367 0.06630434 0.05254531
## Cumulative Proportion 0.193669 0.3174388 0.40137245 0.46767679 0.52022210
## Comp.6 Comp.7 Comp.8 Comp.9 Comp.10
## Standard deviation 1.26612033 1.22242045 1.11550534 1.0304937 0.99888665
## Proportion of Variance 0.05171164 0.04820361 0.04014039 0.0342554 0.03218628
## Cumulative Proportion 0.57193374 0.62013734 0.66027774 0.6945331 0.72671941
## Comp.11 Comp.12 Comp.13 Comp.14 Comp.15
## Standard deviation 0.97639701 0.92221635 0.9042314 0.85909928 0.80853555
## Proportion of Variance 0.03075326 0.02743494 0.0263753 0.02380812 0.02108806
## Cumulative Proportion 0.75747267 0.78490761 0.8112829 0.83509102 0.85617908
## Comp.16 Comp.17 Comp.18 Comp.19 Comp.20
## Standard deviation 0.7877571 0.74436225 0.72574751 0.69380677 0.67269732
## Proportion of Variance 0.0200181 0.01787339 0.01699063 0.01552799 0.01459747
## Cumulative Proportion 0.8761972 0.89407058 0.91106120 0.92658920 0.94118667
## Comp.21 Comp.22 Comp.23 Comp.24 Comp.25
## Standard deviation 0.63466979 0.61328635 0.55192724 0.397467153 0.384354087
## Proportion of Variance 0.01299373 0.01213291 0.00982657 0.005096133 0.004765421
## Cumulative Proportion 0.95418041 0.96631331 0.97613988 0.981236017 0.986001438
## Comp.26 Comp.27 Comp.28 Comp.29
## Standard deviation 0.364232811 0.322026864 0.276201256 0.262018088
## Proportion of Variance 0.004279534 0.003345203 0.002460875 0.002214628
## Cumulative Proportion 0.990280972 0.993626175 0.996087050 0.998301679
## Comp.30 Comp.31
## Standard deviation 0.1712372644 0.1527277294
## Proportion of Variance 0.0009458774 0.0007524438
## Cumulative Proportion 0.9992475562 1.0000000000
- Scree Plot
plot(df_pca,type="lines", npcs = 31) 
Note: Check the cummulative variance of the first components and the scree plot, and see if the PCA is a good approach to detect the number of factors in this case.
PCA is not the same thing as Factor Analysis! PCA only considers the common information (variance) of the variables, while factor analysis takes into account also the unique variance of the variable. Both approaches are often mixed up. In this example we use PCA as only a first criteria for choosing the number of factors. PCA is very used in image recognition and data reduction of big data.
- Model 1: No rotation
- Model 2: Rotation Varimax
- Model 3: Rotation Oblimin
# No rotation
df_factor <- factanal(df, factors = 4, rotation = "none", scores=c("regression"), fm = "ml")
# Rotation Varimax
df_factor_var <- factanal(df, factors = 4, rotation = "varimax", scores=c("regression"), fm = "ml")
# Rotiation Oblimin
df_factor_obl <- factanal(df, factors = 4, rotation = "oblimin", scores=c("regression"), fm = "ml")Let’s print out the results of df_factor_obl, and have a look.
print(df_factor, digits=2, cutoff=0.3, sort=TRUE) #cutoff of 0.3 due to the sample size is higher than 350 observations.##
## Call:
## factanal(x = df, factors = 4, scores = c("regression"), rotation = "none", fm = "ml")
##
## Uniquenesses:
## DWELCLAS INCOME CHILD13 H18 HEMPLOY HSIZE AVADUAGE IAGE
## 0.98 0.82 0.09 0.01 0.73 0.01 0.98 0.99
## ISEX NCARS AREA BEDROOM PARK BEDSIZE PARKSIZE RAGE10
## 0.98 0.64 0.93 0.79 0.89 0.62 0.92 0.91
## TCBD DISTHTC TWCBD TDWWK HEADH POPDENS EDUINDEX GRAVCPC
## 0.14 0.54 0.80 0.74 0.67 0.78 0.71 0.04
## GRAVCPT GRAVPCPT NSTRTC DISTHW DIVIDX ACTDENS DISTCBD
## 0.05 0.01 0.85 0.66 0.84 0.80 0.31
##
## Loadings:
## Factor1 Factor2 Factor3 Factor4
## TCBD 0.92
## DISTHTC 0.60 0.31
## EDUINDEX -0.53
## GRAVCPC -0.97
## GRAVCPT -0.70 -0.67
## DISTHW 0.56
## DISTCBD 0.79
## H18 0.93 -0.36
## HEMPLOY 0.51
## HSIZE 0.93 0.35
## NCARS 0.58
## BEDSIZE -0.61
## GRAVPCPT 0.98
## CHILD13 0.31 0.90
## DWELCLAS
## INCOME 0.40
## AVADUAGE
## IAGE
## ISEX
## AREA
## BEDROOM 0.44
## PARK
## PARKSIZE
## RAGE10
## TWCBD 0.43
## TDWWK 0.47
## HEADH -0.45 0.35
## POPDENS -0.39
## NSTRTC -0.36
## DIVIDX -0.40
## ACTDENS -0.41
##
## Factor1 Factor2 Factor3 Factor4
## SS loadings 5.04 3.54 1.90 1.32
## Proportion Var 0.16 0.11 0.06 0.04
## Cumulative Var 0.16 0.28 0.34 0.38
##
## Test of the hypothesis that 4 factors are sufficient.
## The chi square statistic is 3628.43 on 347 degrees of freedom.
## The p-value is 0
Note: The variability contained in the factors = Communality + Uniqueness.
Varimax assigns orthogonal rotation, and oblimin assigns oblique rotation.
Plot factor 1 against factor 2, and compare the results of different rotations
- No Rotation
plot(
df_factor$loadings[, 1],
df_factor$loadings[, 2],
xlab = "Factor 1",
ylab = "Factor 2",
ylim = c(-1, 1),
xlim = c(-1, 1),
main = "No rotation"
)
abline(h = 0, v = 0)
load <- df_factor$loadings[, 1:2]
text(
load,
names(df),
cex = .7,
col = "blue"
)
- Varimax rotation
plot(
df_factor_var$loadings[, 1],
df_factor_var$loadings[, 2],
xlab = "Factor 1",
ylab = "Factor 2",
ylim = c(-1, 1),
xlim = c(-1, 1),
main = "Varimax rotation"
)
abline(h = 0, v = 0)
load <- df_factor_var$loadings[, 1:2]
text(
load,
labels = names(df),
cex = .7,
col = "red"
)
- Oblimin Rotation
plot(
df_factor_obl$loadings[, 1],
df_factor_obl$loadings[, 2],
xlab = "Factor 1",
ylab = "Factor 2",
ylim = c(-1, 1),
xlim = c(-1, 1),
main = "Oblimin rotation"
)
abline(h = 0, v = 0)
load <- df_factor_obl$loadings[, 1:2]
text(
load,
labels = names(df),
cex = .7,
col = "darkgreen"
)
When you have more than two factors it is difficult to analyze the factors by the plots. Variables that have low explaining variance in the two factors analyzed, could be highly explained by the other factors not present in the graph. However, try comparing the plots with the factor loadings and plot the other graphs to get more familiar with exploratory factor analysis.