As part of a study of consumer consideration of ready-to-eat cereals sponsored by Kellogg Australia, Roberts and Lattin (1991) surveyed consumers regarding their perceptions of their favorite brands of cereals. Each respondent was asked to evaluate three preferred brands on each of 25 different attributes. Respondents used a five point likert scale to indicate the extent to which each brand possessed the given attribute. The data evaluated contains 12 most frequently cirted cereal brands in the sample with the 25 attributes. In total 116 respondents provided 235 observations of the 12 selected brands. How do you characterize the consideration behavior of the 12 selected brands? Analyze and interpret your results using factor analysis.

Read the data

cereal <- read.csv("./data/cereal.csv")
dim(cereal)
## [1] 235  26

Explore the data

Summarize the data to see if the data is as per the expectations.

summary(cereal)
##         Cereals      Filling         Natural          Fibre      
##  CornFlakes :27   Min.   :1.000   Min.   :1.000   Min.   :1.000  
##  Weetabix   :27   1st Qu.:3.000   1st Qu.:3.000   1st Qu.:3.000  
##  Vitabrit   :25   Median :4.000   Median :4.000   Median :4.000  
##  NutriGrain :24   Mean   :3.881   Mean   :3.783   Mean   :3.528  
##  SpecialK   :23   3rd Qu.:4.500   3rd Qu.:4.000   3rd Qu.:4.000  
##  RiceBubbles:21   Max.   :5.000   Max.   :5.000   Max.   :5.000  
##  (Other)    :88                                                  
##      Sweet            Easy            Salt         Satisfying   
##  Min.   :1.000   Min.   :1.000   Min.   :1.000   Min.   :2.000  
##  1st Qu.:2.000   1st Qu.:4.000   1st Qu.:1.000   1st Qu.:3.000  
##  Median :2.000   Median :5.000   Median :2.000   Median :4.000  
##  Mean   :2.506   Mean   :4.532   Mean   :1.991   Mean   :4.004  
##  3rd Qu.:3.000   3rd Qu.:5.000   3rd Qu.:3.000   3rd Qu.:5.000  
##  Max.   :5.000   Max.   :6.000   Max.   :4.000   Max.   :6.000  
##                                                                 
##      Energy           Fun             Kids           Soggy      
##  Min.   :1.000   Min.   :1.000   Min.   :1.000   Min.   :1.000  
##  1st Qu.:3.000   1st Qu.:2.000   1st Qu.:3.000   1st Qu.:1.000  
##  Median :4.000   Median :2.000   Median :4.000   Median :2.000  
##  Mean   :3.643   Mean   :2.617   Mean   :3.843   Mean   :2.255  
##  3rd Qu.:4.000   3rd Qu.:3.000   3rd Qu.:5.000   3rd Qu.:3.000  
##  Max.   :5.000   Max.   :5.000   Max.   :6.000   Max.   :5.000  
##                                                                 
##    Economical        Health          Family         Calories    
##  Min.   :1.000   Min.   :1.000   Min.   :1.000   Min.   :1.000  
##  1st Qu.:3.000   1st Qu.:3.000   1st Qu.:3.000   1st Qu.:2.000  
##  Median :3.000   Median :4.000   Median :4.000   Median :3.000  
##  Mean   :3.217   Mean   :3.809   Mean   :3.877   Mean   :2.702  
##  3rd Qu.:4.000   3rd Qu.:4.000   3rd Qu.:5.000   3rd Qu.:3.000  
##  Max.   :5.000   Max.   :5.000   Max.   :6.000   Max.   :5.000  
##                                                                 
##      Plain           Crisp          Regular          Sugar      
##  Min.   :1.000   Min.   :1.000   Min.   :1.000   Min.   :1.000  
##  1st Qu.:1.000   1st Qu.:2.000   1st Qu.:2.000   1st Qu.:1.000  
##  Median :2.000   Median :3.000   Median :3.000   Median :2.000  
##  Mean   :2.268   Mean   :3.204   Mean   :3.072   Mean   :2.145  
##  3rd Qu.:3.000   3rd Qu.:4.000   3rd Qu.:4.000   3rd Qu.:3.000  
##  Max.   :5.000   Max.   :6.000   Max.   :5.000   Max.   :5.000  
##                                                                 
##      Fruit          Process         Quality          Treat     
##  Min.   :1.000   Min.   :1.000   Min.   :1.000   Min.   :1.00  
##  1st Qu.:1.000   1st Qu.:2.000   1st Qu.:3.000   1st Qu.:2.00  
##  Median :1.000   Median :3.000   Median :4.000   Median :3.00  
##  Mean   :1.694   Mean   :2.936   Mean   :3.694   Mean   :2.63  
##  3rd Qu.:3.000   3rd Qu.:4.000   3rd Qu.:4.000   3rd Qu.:3.00  
##  Max.   :5.000   Max.   :6.000   Max.   :5.000   Max.   :6.00  
##                                                                
##      Boring       Nutritious   
##  Min.   :1.00   Min.   :1.000  
##  1st Qu.:1.00   1st Qu.:3.000  
##  Median :2.00   Median :4.000  
##  Mean   :1.83   Mean   :3.664  
##  3rd Qu.:2.00   3rd Qu.:4.000  
##  Max.   :5.00   Max.   :5.000  
##

We can see that there are values of 6 which is not expected; the max. of the scale is 5. Let’s replace ‘6’ by ‘5’.

cereal[cereal==6] <- 5

Seven 6s replaced by 5.

Recode the scores on negative variables like Soggy, Boring etc.

cereal[,c(12,25)] <- 6 - cereal[,c(12,25)]

Factor Analysis

  1. We need to determine if we have a large enough sample to perform Factor Analysis or Principal Component Analysis and if the dimentionality reduction is a possibility at all(corelation adequacy). We perform the following tests to get the answers:
library(psych)
cerealKMO <- KMO(cereal[,-1])
cerealKMO
## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = cereal[, -1])
## Overall MSA =  0.85
## MSA for each item = 
##    Filling    Natural      Fibre      Sweet       Easy       Salt 
##       0.89       0.90       0.88       0.78       0.83       0.82 
## Satisfying     Energy        Fun       Kids      Soggy Economical 
##       0.91       0.91       0.85       0.67       0.63       0.73 
##     Health     Family   Calories      Plain      Crisp    Regular 
##       0.92       0.73       0.86       0.82       0.83       0.87 
##      Sugar      Fruit    Process    Quality      Treat     Boring 
##       0.78       0.77       0.80       0.91       0.88       0.87 
## Nutritious 
##       0.92
cerealMatrix <- cor(cereal[,-1])
cerealMatrix <- round(cerealMatrix, 2)
cerealBartlett <- cortest.bartlett(cerealMatrix, n = nrow(cereal))
cerealBartlett
## $chisq
## [1] 2878.65
## 
## $p.value
## [1] 0
## 
## $df
## [1] 300
  1. How many Factors are there in the data?
numFactors <- fa.parallel(cereal[,-1], fm="ml", fa="fa")

## Parallel analysis suggests that the number of factors =  4  and the number of components =  NA
sum(numFactors$fa.values>1.0) ##old kaiser crieterion
## [1] 3
sum(numFactors$fa.values>0.7) ##new kaiser crieterion
## [1] 4

Parallel analysis helps you to decide how may factors to retain. The scree plot suggests that 4 factors should be retained.

3.Let’s create a simple structure with a 4 factor model.

fit <- fa(cereal[,-1], nfactors=4, fm="ml", rotate="oblimin")
## Loading required namespace: GPArotation
fit
## Factor Analysis using method =  ml
## Call: fa(r = cereal[, -1], nfactors = 4, rotate = "oblimin", fm = "ml")
## Standardized loadings (pattern matrix) based upon correlation matrix
##              ML1   ML2   ML3   ML4   h2   u2 com
## Filling     0.70  0.17  0.14  0.06 0.56 0.44 1.2
## Natural     0.76 -0.11  0.00 -0.03 0.61 0.39 1.0
## Fibre       0.86  0.02 -0.16 -0.08 0.69 0.31 1.1
## Sweet       0.07  0.71  0.03  0.22 0.65 0.35 1.2
## Easy        0.25  0.09  0.27  0.01 0.15 0.85 2.2
## Salt        0.02  0.73  0.01 -0.22 0.49 0.51 1.2
## Satisfying  0.61  0.12  0.32  0.10 0.57 0.43 1.7
## Energy      0.64  0.13  0.10  0.14 0.51 0.49 1.2
## Fun         0.04  0.11  0.32  0.50 0.47 0.53 1.8
## Kids       -0.04  0.03  0.88 -0.02 0.77 0.23 1.0
## Soggy      -0.15 -0.09 -0.17  0.52 0.23 0.77 1.5
## Economical  0.10 -0.24  0.41 -0.21 0.28 0.72 2.4
## Health      0.84 -0.17 -0.04 -0.01 0.78 0.22 1.1
## Family      0.02 -0.07  0.79  0.08 0.65 0.35 1.0
## Calories   -0.09  0.61 -0.02  0.03 0.41 0.59 1.1
## Plain       0.00  0.01  0.15 -0.69 0.45 0.55 1.1
## Crisp      -0.01  0.10  0.27  0.42 0.32 0.68 1.9
## Regular     0.65  0.02 -0.09 -0.01 0.41 0.59 1.0
## Sugar      -0.14  0.82 -0.07  0.03 0.74 0.26 1.1
## Fruit       0.31  0.18 -0.35  0.41 0.44 0.56 3.3
## Process    -0.18  0.37  0.04 -0.18 0.21 0.79 2.0
## Quality     0.63 -0.18  0.11  0.15 0.56 0.44 1.3
## Treat       0.13  0.16  0.21  0.59 0.58 0.42 1.5
## Boring      0.04 -0.12  0.15  0.53 0.33 0.67 1.3
## Nutritious  0.85 -0.05 -0.02 -0.02 0.73 0.27 1.0
## 
##                        ML1  ML2  ML3  ML4
## SS loadings           5.27 2.61 2.30 2.40
## Proportion Var        0.21 0.10 0.09 0.10
## Cumulative Var        0.21 0.32 0.41 0.50
## Proportion Explained  0.42 0.21 0.18 0.19
## Cumulative Proportion 0.42 0.63 0.81 1.00
## 
##  With factor correlations of 
##       ML1   ML2  ML3  ML4
## ML1  1.00 -0.18 0.11 0.31
## ML2 -0.18  1.00 0.03 0.28
## ML3  0.11  0.03 1.00 0.18
## ML4  0.31  0.28 0.18 1.00
## 
## Mean item complexity =  1.4
## Test of the hypothesis that 4 factors are sufficient.
## 
## The degrees of freedom for the null model are  300  and the objective function was  12.8 with Chi Square of  2877.74
## The degrees of freedom for the model are 206  and the objective function was  1.79 
## 
## The root mean square of the residuals (RMSR) is  0.04 
## The df corrected root mean square of the residuals is  0.05 
## 
## The harmonic number of observations is  235 with the empirical chi square  220.91  with prob <  0.23 
## The total number of observations was  235  with Likelihood Chi Square =  398.19  with prob <  2.3e-14 
## 
## Tucker Lewis Index of factoring reliability =  0.89
## RMSEA index =  0.004  and the 90 % confidence intervals are  0.004 0.072
## BIC =  -726.48
## Fit based upon off diagonal values = 0.98
## Measures of factor score adequacy             
##                                                 ML1  ML2  ML3  ML4
## Correlation of scores with factors             0.97 0.93 0.93 0.90
## Multiple R square of scores with factors       0.94 0.87 0.87 0.81
## Minimum correlation of possible factor scores  0.87 0.74 0.74 0.62
  1. How adequate is the model? We will look at the goodness of fit and residual statistics. We are looking for large values for the former and small ones for the latter. Most of the metrics are given in the output of the fa() function.

  1. Tucker Lewis Index of factoring reliability = 0.906. >0.90 is acceptable. >0.95 is excellent.

  2. Comparative Fit Index(CFI) - calculated manually later.

  1. RMSEA index = 0.004. <0.06 is excellent

  2. The root mean square of the residuals (RMSR) is 0.04. <0.06 is excellent.

We need to calculate comparative fit index (a goodness of fit metric) manually.

1-((fit$STATISTIC-fit$dof)/
           (fit$null.chisq-fit$null.dof))
## [1] 0.9254409

A value of >0.90 for CFI is acceptable while >0.95 is excellent.

Looking at the metrics, we can conclude that we have an acceptable model.

  1. Let’s check the reliability of the factors.
factor1 <- c(2,3,4,8,9,14,19,23,26)
factor2 <- c(5,7,16,20,22)
factor3 <- c(11,13,15)
factor4 <- c(12,17,18,24,25)
factor1alpha <- psych::alpha(cereal[,factor1], check.keys = TRUE)
factor2alpha <- psych::alpha(cereal[,factor2], check.keys = TRUE)
factor3alpha <- psych::alpha(cereal[,factor3], check.keys = TRUE)
factor4alpha <- psych::alpha(cereal[,factor4], check.keys = TRUE)
## Warning in psych::alpha(cereal[, factor4], check.keys = TRUE): Some items were negatively correlated with total scale and were automatically reversed.
##  This is indicated by a negative sign for the variable name.
factor1alpha$total$raw_alpha
## [1] 0.9131105
factor2alpha$total$raw_alpha
## [1] 0.7713322
factor3alpha$total$raw_alpha
## [1] 0.6867598
factor4alpha$total$raw_alpha
## [1] 0.708883

As the alpha values are >0.7, the factors are reliable.

  1. So, what do the factors mean?

-> Filling, Natural, Fibre, Satisfying, Energy, Health, Regular, Quality, Nutritious

-> Sweet, Salt, Calories, Sugar, Process

-> Kids, Economical, Family

-> Fun, Soggy, Plain, Crisp, Treat, Boring

  1. Creating Average Factor Scores grouped by the cereal.
cereal$factor1Score <- apply(cereal[,factor1],1,mean)
cereal$factor2Score <- apply(cereal[,factor2],1,mean)
cereal$factor3Score <- apply(cereal[,factor3],1,mean)
cereal$factor4Score <- apply(cereal[,factor4],1,mean)
colnames(cereal)[27:30] <-c("Health", "Taste", "Family", "Texture/Excitement")
aggregateCereal<-aggregate(cereal[,27:30],  list(cereal[,1]), mean)
format(aggregateCereal, digits = 2)
##        Group.1 Health Taste Family Texture/Excitement
## 1      AllBran    3.9   2.2    3.0                2.9
## 2      CMuesli    4.0   2.8    3.5                3.3
## 3   CornFlakes    3.3   2.7    4.1                3.3
## 4    JustRight    3.6   2.7    3.2                3.2
## 5     Komplete    4.0   2.6    2.6                3.2
## 6   NutriGrain    3.4   3.1    4.0                3.5
## 7      PMuesli    4.1   2.9    3.2                3.5
## 8  RiceBubbles    2.9   2.2    4.2                3.3
## 9     SpecialK    3.5   2.3    3.7                3.4
## 10     Sustain    4.2   2.2    3.3                3.4
## 11    Vitabrit    3.9   1.9    3.9                2.9
## 12    Weetabix    3.9   2.1    3.8                2.7