Categorical data analysis (I)

Recap

Last week we covered

Means and medians
Quantiles
Range, IQR, variance, and standard deviations
Histograms and boxplots
Scatter plots (briefly)

Check in: Was there anything from last week that was confusing or unclear?

Plan for today

Today we will learn how to:

Represent categorical data using a contingency table
Describe marginal and conditional distributions
Visualise the relationship between two categorical variables
Understand independence
Calculate expected counts
Quantify the difference between observed and expected counts using the \(\chi^2\) statistic
Use a \(\chi^2\) test to assess evidence of an association

Research question

Imagine we surveyed 100 airline passengers.

We recorded:

Travel class
- Business
- Economy
Satisfaction
- Very poor
- Fair
- Good
- Very good

Research question: Is satisfaction related to travel class?

Airline satisfaction data

Class	Satisfaction
B	Very Good
B	Fair
B	Very Good
B	Very Good
E	Good
B	Good

Both of these variables are categorical

How would we analyse such data?

Contingency tables

A contingency table shows the frequency of observations for every combination of categorical variables.

contingency.r

table(airlines)

Class	Poor	Fair	Good	Very Good
B	3	5	14	28
E	21	17	10	2

A contingency table lets us see the joint distribution of two categorical variables.

Contingency tables

A contingency table shows the frequency of observations for every combination of categorical variables.

contingency.r

table(airlines)

Class	Poor	Fair	Good	Very Good
B	3	5	14	28
E	21	17	10	2

A contingency table lets us see the joint distribution of two categorical variables.

Marginal distributions

Marginal distributions tell us:

How common is each category, ignoring the other category?

What proportion of all passengers reported “Very Good” satisfaction?

What proportion of passengers travelled in Business class?

marginal.R

addmargins(table(airlines))

Class	Poor	Fair	Good	Very Good	Sum
B	3	5	14	28	50
E	21	17	10	2	50
Sum	23	22	24	30	100

Conditional distributions

Conditional distributions tell us:

What proportion of each class gave each satisfaction rating?

Out of all the people in Class B, how many people were very satisfied?

Out of all the people in Class E, how many people were fairly satisfied?

Essentially, we are ignoring the other group

Marginal distribution

\[ \frac{\text{Very good passengers}}{\text{All passengers}} \]

Conditional distribution

\[ \frac{\text{Very good business passengers}}{\text{All business passengers}} \]

Conditional distributions

conditional.r

# There are two ways to achieve this

# METHOD 1 - CALCULATING MANUALLY ----------------------------------------------
tab <- table(airlines)

# The probability of satisfaction conditional on Business class
# The first row divided by the sum of the first row
cond_prob_B <- tab[1, ] / sum(tab[1, ])

# The probability of satisfaction conditional on Economy class
# The second row divided by the sum of the second row
cond_prob_E <- tab[2, ] / sum(tab[2, ])

# METHOD 2 - LET R DO IT FOR YOU -----------------------------------------------
tab <- table(airlines)
prop.table(tab, margin = 1)

Class	Poor	Fair	Good	Very Good
B	0.06	0.10	0.28	0.56
E	0.42	0.34	0.20	0.04

Visualising conditional distributions

Which plot does a better job of answering the research question

Does satisfaction differs between classes?

Visualising conditional distributions

barplots.R

# How to make the bar plots (colours are omitted as I used a slightly different method)

tab <- table(airline) # Make a contingency table

# STACKED BARPLOT
barplot(
  t(tab),
  legend = TRUE,
  args.legend = list(x = "top", ncol = 2),
  ylim = c(0,95)
)

# GROUPED BARPLOT
barplot(
  t(tab),
  beside = TRUE, # This argument is the key difference
  legend = TRUE,
  args.legend = list(x = "topleft", ncol = 1),
  ylim = c(0,40)
  )

What would we see if class didn’t matter?

Let’s say that 15% of all passengers report “Poor” satisfaction.

If travel class has nothing to do with satisfaction, what percentage of Business passengers should we expect to report poor satisfaction?

15%

And what about in Economy class?

15%

Independence

Two categorical variables are independent when knowing the category of one variable gives us no information about the category of the other.

If Class and Satisfaction were independent, the distribution of satisfaction should be the same in Business and Economy.

Expected counts

Class	Poor	Fair	Good	Very Good
B	3	5	14	28
E	21	17	10	2

What would out airline data look like if class had no effect on satisfaction

\[ E_{ij} = \frac{(\text{Row}_i \; \text{total}) \times \text{(Column}_j \; \text{total)}}{\text{Grand total}} \]

Class	Poor	Fair	Good	Very Good
B	12	11	12	15
E	12	11	12	15

Observed versus expected counts

What actually happened

Class	Poor	Fair	Good	Very Good
B	3	5	14	28
E	21	17	10	2

If independence was assumed

Class	Poor	Fair	Good	Very Good
B	12	11	12	15
E	12	11	12	15

For each cell, we can calculate \(O_{ij} - E_{ij}\)
If \(O_{ij} = E_{ij}\) then that cell is exactly what independence predicts
If \(O_{ij} \ne E_{ij}\) then that cell contributes evidence that the variables may not be independent.

The \(\chi^2\) statistic

How different are observed counts and expected counts?

We need a single number that summarises how different our observed counts are from the counts we’d expect under independence.

\[ \chi^2 = \sum_i \sum_j \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \]

Small \(\chi^2\): Observed counts are relatively close to expected counts.
Large \(\chi^2\): Observed counts are far from expected counts.

Larger \(\chi^2\) values indicate greater disagreement between observed and expected counts.

How do we determine what is a large \(\chi^2\) value?

The \(\chi^2\) distribution

The \(\chi^2\) test statistic has a mathematical distribution called the
\(\chi^2\) distribution.

The important specification to make in describing the \(\chi^2\) distribution is something called degrees of freedom.
Different \(\chi^2\) distributions correspond to different degrees of freedom

Degrees of freedom

How many cells can vary freely once the totals are fixed?

Once you know the row \((R)\) and column \((C)\) totals, not every cell can vary independently.

\[ DF = (R - 1)(C - 1) \]

Looking back at our airline example, we had 2 rows and 4 columns

\[ DF = (2 - 1)(4 - 1) = 3 \]

Hypothesis testing

Once we have a research question, we can do a hypothesis test.

Is airline class associated with satisfaction?

Null hypothesis

Class and satisfaction are independent.

Alternative hypothesis

Class and satisfaction are associated.

If the null hypothesis were true, how surprising would our observed table be?

chi-square.R

tab <- table(airlines)
chisq.test(tab)


    Pearson's Chi-squared test

data:  tab
X-squared = 43.245, df = 3, p-value = 2.183e-09

So … Does travel class matter

We found evidence of an association between travel class and satisfaction.

Satisfaction differs between Business and Economy passengers.

Association does not automatically imply causation.

We have not demonstrated that changing someone’s travel class would cause their satisfaction to change.

Wrapping up

We learned how to represent categorical data using a contingency table.
We learned the difference between marginal and conditional frequencies.
Expected versus observed counts
\(\chi^2\) statistic

Recommended reading

Chapters 4 and 5 of Introduction to Statistical and Data Analysis by Christian Heimann & Michael Schomaker Shalabh