Homework assignment 2 at course MVA
Jovana Milić
2025-01-27
mydata <- read.table("~/Desktop/SP.csv",
header = TRUE, sep = ",", dec = ".")
mydata$ID <- 1:nrow(mydata)
head(mydata)## Gender Age
## 1 Male 16
## 2 Male 16
## 3 Male 18
## 4 Male 21
## 5 Female 21
## 6 Male 21
## I.learn.better.by.reading.what.the.teacher.writes.on.the.chalkboard.
## 1 3
## 2 5
## 3 3
## 4 1
## 5 4
## 6 2
## I.learn.better.by.reading.than.by.listening.to.someone.
## 1 3
## 2 4
## 3 3
## 4 4
## 5 5
## 6 3
## When.the.teacher.tells.me.the.instructions.I.understand.better
## 1 4
## 2 3
## 3 2
## 4 2
## 5 4
## 6 3
## I.remember.things.I.have.heard.in.class.better.than.things.I.have.read.
## 1 3
## 2 4
## 3 3
## 4 2
## 5 3
## 6 4
## I.prefer.to.learn.by.doing.something.in.class.
## 1 4
## 2 3
## 3 3
## 4 4
## 5 5
## 6 1
## When.I.do.things.in.class..I.learn.better. ID
## 1 2 1
## 2 4 2
## 3 2 3
## 4 4 4
## 5 5 5
## 6 2 6Data description
Unit of observation: individual student
Sample size: 150 students
Description of variables: Dataset consists of students’ different study preferences on 6 questions/statements. All of them are ranked on a scale form 1-5, where one represents “strongly disagree” and five is “strongly agree”.
Gender- refers to the classification of students based on their identity as male and female
Age- shows how old each student is
I learn better by reading what the teacher writes on the chalkboard- a preference for visual learning, where reading the material written by the teacher helps understand it more effectively
I learn better by reading than by listening to someone- a preference for independent reading
When the teacher tells me the instructions I understand better- verbal explanations from the teacher are an effective way of understanding
I remember things I have heard in class better than things I have read- listening to the teacher or class helps remembering
I prefer to learn by doing something in class- a preference for engaging in activities to increase understanding
When I do things in class, I learn better- active participation helps learn more effectively
ID- ID of each student
The source of data is Kaggle, platform for different datasets.
Data manipulation
colnames(mydata) [1] <- ("gender")
colnames(mydata) [2] <- ("age")
colnames(mydata) [3] <- ("TReading")
colnames(mydata) [4] <- ("Reading")
colnames(mydata) [5] <- ("TListening")
colnames(mydata) [6] <- ("Listening")
colnames(mydata) [7] <- ("Activity")
colnames(mydata) [8] <- ("Participation")
colnames(mydata) [9] <- ("id")
head(mydata)## gender age TReading Reading TListening Listening Activity Participation id
## 1 Male 16 3 3 4 3 4 2 1
## 2 Male 16 5 4 3 4 3 4 2
## 3 Male 18 3 3 2 3 3 2 3
## 4 Male 21 1 4 2 2 4 4 4
## 5 Female 21 4 5 4 3 5 5 5
## 6 Male 21 2 3 3 4 1 2 6# For my own convenience, I changed words to abbreviations.
mydata$gender <- factor(mydata$gender,
levels = c("Female", "Male"),
labels = c("F", "M"))Descriptive statistics
library(psych)
psych:: describe(mydata[ ,-c(1,2,9)])## vars n mean sd median trimmed mad min max range skew
## TReading 1 150 3.29 0.98 3 3.28 1.48 1 5 4 -0.21
## Reading 2 150 3.30 0.98 3 3.30 1.48 1 5 4 -0.16
## TListening 3 150 3.37 0.95 3 3.37 1.48 1 5 4 -0.24
## Listening 4 150 3.19 0.95 3 3.19 1.48 1 5 4 -0.11
## Activity 5 150 3.31 1.04 3 3.29 1.48 1 5 4 0.12
## Participation 6 150 3.46 0.98 4 3.48 1.48 1 5 4 -0.29
## kurtosis se
## TReading -0.49 0.08
## Reading -0.26 0.08
## TListening -0.55 0.08
## Listening -0.41 0.08
## Activity -0.65 0.08
## Participation -0.56 0.08- For each variable, we can see different scores that students assigned. For example, min tells us which was the lowest score, for each question and it was 1 for all of them. On the other side, max tells us the highest scores students assigned, and it was 5 for all the questions, meaning that for each question some of the students choose the highest number. Means are somewhat similar for all the questions, which indicates that the average score on different performance questions were similar, all around 3.
Research question: How can I use PC analysis to combine different study preferences, and summurize them based on passive and active learning?
- For this analysis, I will use correlation matrix, because I will standardize the data, so it has the same effect and contribution to the component.
mydata_PCA <- mydata[, -c(1,2,9)]
R <- cor(mydata_PCA)
round(R,3)## TReading Reading TListening Listening Activity Participation
## TReading 1.000 0.154 0.245 0.157 -0.021 0.106
## Reading 0.154 1.000 0.045 0.132 -0.051 -0.172
## TListening 0.245 0.045 1.000 0.292 0.325 0.253
## Listening 0.157 0.132 0.292 1.000 0.110 0.287
## Activity -0.021 -0.051 0.325 0.110 1.000 0.369
## Participation 0.106 -0.172 0.253 0.287 0.369 1.000library(psych)
corPlot(R)
Here are the elements of correlation matrix and the visualization. It’s dimension is 6x6, as there are 6 variables. It looks like the strongest correlation is between Activity and Participation. And the weakest (or no correlation) between Activity and TReading.
Now, I want to know if the data is suitable for the principal component analysis. For that, I will use Bartlett test. The H0: P=I (population correlation matrix = identity correlation matrix), H1: P≠I.
library(psych)
cortest.bartlett(R, n= nrow(mydata))## $chisq
## [1] 88.97409
##
## $p.value
## [1] 1.540501e-12
##
## $df
## [1] 15- Based on the test, I can reject H0 at p<0.001, meaning that 2 matrices are different, at least some correlation isn’t equal to 0. Number of correlation coefficients is 15.
det(R)## [1] 0.5440486Smaller the determinant is, higher the correlation, which makes PC more suitable, as it can capture a lot of information with few components. In this case, the value is all right, as it is not very close to 1.
Now, I need to check if some of the variables should be removed. I will do that with the help of of Kaiser-Mayer-Olkin statistic as the measure of general adequacy and MSA statistics for measuring adequacy of single variable.
library(psych)
KMO (R)## Kaiser-Meyer-Olkin factor adequacy
## Call: KMO(r = R)
## Overall MSA = 0.58
## MSA for each item =
## TReading Reading TListening Listening Activity
## 0.54 0.43 0.64 0.60 0.56
## Participation
## 0.58KMO is larger than 0.5, meaning that in general, data is appropriate. MSA is all right for every variable, except for Reading, which is a bit less than 0.5. I could drop this variable, but in this case I decided to leave it in the analysis, aware of losing a little bit of data about Reading, when merging variables.
Now, I will standardize the data and get eigenvalues for each variable, which will tell me the importance of each component.
library(FactoMineR)
components <- PCA(mydata_PCA,
scale.unit = TRUE,
graph = FALSE)
library(factoextra)## Loading required package: ggplot2##
## Attaching package: 'ggplot2'## The following objects are masked from 'package:psych':
##
## %+%, alpha## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBaget_eigenvalue(components)## eigenvalue variance.percent cumulative.variance.percent
## Dim.1 1.8928531 31.547551 31.54755
## Dim.2 1.2829543 21.382571 52.93012
## Dim.3 0.8652761 14.421268 67.35139
## Dim.4 0.8193070 13.655116 81.00651
## Dim.5 0.6628751 11.047919 92.05443
## Dim.6 0.4767344 7.945574 100.00000- Based on these, I can see that the first PC can explain 31.5% of the information. Eigenvalues tell how much of variance (information) is contained on a component, for example, variance of second principal component is 1.28. However, eigenvalues can also be used to help determine number of components that should be used. Following Kaiser’s rule, I should retain as many components which have their eigenvalue larger than 1 (for standardized variables). In this case, it would be first 2 components. I will show it as well on a Scree plot. Also, second one does capture at least 5% of total variance and together, they explain over 53% of the information, which is more than enough (40% for soft data- Likert scale).
library(factoextra)
fviz_eig (components,
choice = "eigenvalue",
main = "Scree plot",
ylab = "Eigenvalue",
xlab = "Principal component",
addlabels = TRUE)
From the Scree plot, it is visible that the break is at 0.9, which means that one less component should be taken (based on Elbow rule). In this case it’s 2, as previously found by following Kaiser’s rule.
Now, I will also check how many components should be retained with parallel analysis.
library(psych)
fa.parallel(mydata_PCA,
sim = FALSE,
fa = "pc")
## Parallel analysis suggests that the number of factors = NA and the number of components = 2Again, the answer is 2, because λempirical>λtheoretical only in first 2 components.
Now, I will perform the principal component analysis, and put in number of components that I found (2).
library(FactoMineR)
components <- PCA(mydata_PCA,
ncp = 2,
scale.unit = TRUE,
graph = FALSE)
components## **Results for the Principal Component Analysis (PCA)**
## The analysis was performed on 150 individuals, described by 6 variables
## *The results are available in the following objects:
##
## name description
## 1 "$eig" "eigenvalues"
## 2 "$var" "results for the variables"
## 3 "$var$coord" "coord. for the variables"
## 4 "$var$cor" "correlations variables - dimensions"
## 5 "$var$cos2" "cos2 for the variables"
## 6 "$var$contrib" "contributions of the variables"
## 7 "$ind" "results for the individuals"
## 8 "$ind$coord" "coord. for the individuals"
## 9 "$ind$cos2" "cos2 for the individuals"
## 10 "$ind$contrib" "contributions of the individuals"
## 11 "$call" "summary statistics"
## 12 "$call$centre" "mean of the variables"
## 13 "$call$ecart.type" "standard error of the variables"
## 14 "$call$row.w" "weights for the individuals"
## 15 "$call$col.w" "weights for the variables"- I will analyze a few, number 4 and number 6.
components$var$cor## Dim.1 Dim.2
## TReading 0.37661145 0.5728243
## Reading 0.02134114 0.7564329
## TListening 0.72332496 0.1267760
## Listening 0.60510406 0.2821772
## Activity 0.61530888 -0.3955668
## Participation 0.69469887 -0.3612016Dimension 1 has positive correlation with all 6 variables, meaning that it represents inclination to usage of any type of learning.
Moreover, from those loadings, few different things can be found out, like relationships between variables and components, calculate variance of each component, or calculate how much info is lost, by calculating and using coefficient of determination. For example, linear relationship between Listening and PC1 is positive and moderately strong.
components$var$contrib## Dim.1 Dim.2
## TReading 7.49324861 25.575942
## Reading 0.02406127 44.599468
## TListening 27.64076062 1.252746
## Listening 19.34386358 6.206299
## Activity 20.00181756 12.196311
## Participation 25.49624836 10.169234- Based o these information I can say that, for example, variable Participation has contributed 25.5% of information to the PC1.
fviz_pca_var(components, repel = TRUE)
- This is visual representation of PC analysis, from it, I can see how each variable contributes to principal components. In the first quadrant, there are variables that belong to the “passive learning” group- PC2 and in the fourth “active learning”- PC1.
fviz_pca_biplot(components, repel = TRUE)
- Here, I can see variables and position of each student according to them. I will interpret some extremes. For example, ID 24 is below average and prefers passive learning. ID 37 is above average and prefers passive learning. ID 51 is on average and prefers passive learning.
Conclusion
I used PC analysis and given all the tests and results, I can conclude that the variables can be summarized into two components:
PC1 represents active learning technique, which includes Activity, Participation and contains 31.5% of information.
PC2 represents passive learning technique, which includes Reading, TReading, Listening, TListening and contains 21.4% of information.
Together, they explain 53% of the variance showing how students’ evaluation of study preferences can be summarized, based on passive and active learning.