W8: Cluster Analysis

Question 1

Cluster analysis can be used to classify

Answer 1

individuals and separate them for further study

Question 2

In cluster analysis we find groups of (2)

Answer 2

similar individuals based on their covariate information
These groups are known as clusters

Question 3

Aim to extract a small number of cluster of individuals who share similar characteristics and who have

Answer 3

different characteristics than those in other clusters

Question 4

How can we measure the degree of similarity between individual’s scores across a number of variables?

Answer 4

Using 2 measures: similarity coefficients and dissimilarity coefificents

Question 5

The correlation coefficient ,r, is a measure of

Answer 5

similarlity between 2 variables

Question 6

What does the Pearson’s correlation correlation tell us

Answer 6

whether 1 variable changes the other by a similar amount

Question 7

We could use the Pearson’s correlation coefficient, r, to work out the correlation between 2

Answer 7

individuals

Question 8

We could use the Pearson’s correlation coefficient, r, to work out the correlation between 2

Answer 8

individuals

Question 9

However, although the correlation tells us whether the pattern of responses between people are similar

It does not tell us anything about the

Answer 9

distance between 2 individual profiles

Question 10

An alternative measure compared to Pearson’s correlation coefficient , r, is

Answer 10

Euclidean distance

Question 11

What is Euclidean distance?

Answer 11

Geometric distance between 2 individuals

Question 12

The Euclidean distance for individual i and j formula given below:

Answer 12

Question 13

With Euclidean distance, the smaller the distance

Answer 13

the more similar the individuals

Question 14

Euclidean distances are heavily affected by

Answer 14

variables with large size

Question 15

So if cases are being compared across variables that have different variances, then

Answer 15

Euclidean distances will be inaccurate

Question 16

In such case that Euclidean distances will be inaccurate if cases compared have different variances then (2)

Answer 16

May standardise the scores by subtracting the mean of each variable and dividng by SD

(value - mean)/SD

Question 17

How to calculate SD on your calculator? (8)

Answer 17

Mode
Statistics (2)
1
Input values
OPTN
1 variable (3)

Question 18

Most methods of grouping individuals based on similarlity are done in hierarchical way (2)

Answer 18

1. Begin all individuals treated as one cluster
2. At each subsequent stage clusters merged based on Euclidean distance

Question 19

For example calculating Eucliden distances we first say

Answer 19

We calculate the Eucliden distance between the individuals

Question 20

In Eucliden distance, n represents

Answer 20

number of variables

Question 21

What is n here in calculating Euclidean distance?

Answer 21

4

Question 22

For first individuals lets say A and B we say in the formula before calculating:

Answer 22

For A and B:

Question 23

At the end of Euclidean distance
lets say A and B = 3.26
A and C = 2.13
B and C = 1.23

Ending sentence:

Answer 23

So we would cluster B and C first, then A and C and finally A and B

Question 24

Consider 3 ways of merging clusrers at each step of method (3)

Answer 24

1. Nearest neighbour
2. Furthest neighbour
3. Average linkage

Question 25

What is nearest neighbour?

Answer 25

add individuals to clusters one at a time based on the lowest Euclidean distance to any member of the cluster.

Question 26

What is furthest neighbour?

Answer 26

add individuals to clusters one at a time based on the lowest Euclidean distances to all members of the cluster.

Question 27

What is average linkage?

Answer 27

add individuals to clusters one at a time based on the lowest average Euclidean distance to the cluster.

Question 28

What is a dendogram?

Answer 28

It shows how and when individuals are combined in the clustering algorithm

Question 29

Interpret this denodgram that used nearest neighbour clustering (5)

Answer 29

We see clusters forming: First one is between individuals 1,4,11,7 and 13 Second one is between individuals 10,12,9,15 and 2 Third one is between 6,8,5,3 and 14 First and second clusters are most closely linked than the third cluster

Question 30

In this table, interpret it: Measured trait anxiety, depression, intrusive thoughts, impulsive Patients with same disorder should report similar pattern across scores We asked 2 psychologist to agree on diagnosis of GAD, depression or OCD

Answer 30

We see an exact mapping between the 3 clusters and the diagnosis of the psychologists in spite of clustering algorithm having no knowledge of the diagnoses

Question 31

Interpret the FN as compared to NN

Answer 31

We end up with same 3 clusters although individuals are combined in different order

Question 32

Which of the methods provides a more useful dendogram? Justify your answer (2)

Answer 32

The furthest neighbour clustering provides a more useful dendogram. It forms a more natural set of clusters than the nearest neighbour clustering algorithm, which produces lots of clusters of size 1.

Question 33

Using dendogram you selected (a) what distance should we cut the dendogram? (B) What is the memberships of the resulting clusters? - (5)

Answer 33

Using the dendogram from the furthest neighbour clustering, we would cut the dendogram somwhere between a distance of around 15 and 16. The resulting cluster memberships would be Cluster a: Ernie, Carla, Christa, Ernest, Christopher, Beulah, Linette, Marie, Bo. Cluster b: Tony, Martina, Randolph, Raul, Catalina, Louis, Sunila, Johnson, Mickey. Cluster c: Rosalyn, Lawrence

Question 34

We have collected data on the salary, rank (from 1 to 5, where 5 is most senior), FTE (hours worked, where 1 is full time), number of articles published and years of experience of 20 university staff. (c) Discuss cluter memberships in list of summary statistics and plots (4)

Answer 34

From the boxplots and classification table we can see that: Cluster 1 is made up of individuals with a high salary, a large number of articles and many years of experience, all of whom are professors. It appears that cluster 1 is the most senior academics. This correspondes to cluster c above. Cluster 2 is made up of individuals with a lower salary, fewer articles and less experience than cluster 1. However, most are still professors. Therefore, cluster 2 is made up of a slightly lower level of senior academics. Cluster 3 is made up of individuals who have relatively low salaries, few articles and less experience, and who are not yet full professors. That is, cluster 3 is made up of early career academics.

Question 35

In SPSS what does this table tell us?

Answer 35

- This table tells us which cluster each of my individuals belong to in the cluster membership table

Question 36

In SPSS how many clusters there are in this dendogram?

Answer 36

3 clusters

Question 37

Clusters in R Dendogram how may clusters?

Answer 37

3 clusters

Question 38

Since variables are on the same scale there is no need to

Answer 38

standardise them

Question 39

Interpret this dendogram - (2) dendogram shows different aspect of disgust - FN

Answer 39

Three broad clusters appear, which seem to be distinct types of disgust. We may want to further disaggregate into up to seven clusters.

Question 40

Interpret this dendogram - (2) dendogram shows different aspect of disgust - NN

Answer 40

Clustering of bread, deer, crisp and foxes, but other clustering is hard to distinguish.

Question 41

Which of the two clustering methods used appears to provide the more interpretable clusters? from different aspect of disgust?

Answer 41

Furthest Neighbour

Question 42

Data on 7 different measurements of 41 cities (c) What do you observe from NN dendogram?

Answer 42

The clustering is a little tricky to make out, although Chicago appears different to the other cities.

Question 43

Now perform furthest neighbour clustering on the data. What conclusions can you make this time? Are they different to the nearest neighbour algorithm?

Answer 43

This time, two/three large clusters appear, plus a cluster with just Chicago.

Question 44

Based on the furthest neighbour clustering, find the cluster membership for a 3 cluster solution. Are Wichita and St Louis in the same cluster? Which city is the “odd one out”?

Answer 44

Wichita is in cluster 1 and St Louis in cluster 2. Chicago is the odd one out.

Question 45

Interpret this dendogram - is it similar or different to previous NN/FN

Answer 45

The results are different again. Chicago is clustered with Philadelphia this time.