Categorical data analysis

Question 1

When is categorical data analysis used?

Answer 1

When your outcome variable is nominal scale

The predictor variables can be anything, however, these lectures will only have a single predictor variable and the predictor will be a nominal scale

Question 2

What is the chi-square “goodness of fit” test?

Answer 2

A test to determine how good our observed data matches the values expected by theory

Question 3

What value does a chi-square ‘goodness of fit’ test use?

Answer 3

An X² value

Question 4

When should you use a chi-square ‘goodness of fit’ test?

Answer 4

Chi-square goodness of fit test is used for categorical data when you want to compare observed frequencies against some hypothesis about the true probabilities

Question 5

Explain the four principles for a statistical test for the chi-square ‘goodness of fit’ test.

• A diagnostic test statistic T*
• Sampling distribution of T if the null is true*
• The observed T in your data*
• A rule that maps every value of T onto a decision (accept or reject H0)*

Answer 5

Question 6

How do you get the chi-squared distribution (X²)?

Answer 6

X² is what you get when you take normally distributed data, square it and add it.

Question 7

What are the features of the chi-square ( χ2 ) distribution?

Answer 7

Continuous distribution

Has a noticeable positive skew to it

The shape of the distribution depends on the ‘degrees of freedom’

Question 8

What is ‘degrees of freedom’ (DOF)?

Answer 8

The total number of ‘things’ you’re interested in minus the number of known constraints on those ‘things’

• The number of degrees of freedom is the number of quantities of interest in the data - 1 (one constraint on those quantities)*
• E.g. for a chi-square goodeness of fit test involving k categories, the degrees of freedom is equal to k-1*

Question 9

What is another name for the rejection region?

Answer 9

Critical region

Question 10

When can we reject the H0 in a chi-square test?

Answer 10

There is a 5% chance of observing an X² value greater than the significance level

Therefore we can ensure a Type 1 error rate of .05 if we reject H0 only if X² is greater than 95% significance level

Question 11

What are the 3 important outputs in a chi-square ‘goodness of fit’ test?

Answer 11

The test statistic (X²)

The p-value

The degrees of freedom for the test

Question 12

How should you write up a chi-square ‘goodness of fit test’?

Answer 12

1) Report the relevant descriptive statistics (can also do this in a table or figure in your text)
2) Specify the null hypothesis and the statistical test run
3) Give the result of the test

Question 13

Chi-square tests are used for 1)______ data: the outcome variable is 2)_____ scale

Answer 13

1) Categorical
2) Nominal

Question 14

For a chi-square test, describe:

Diagnostic test statistic

Distribution

Degrees of freedom

Answer 14

Question 15

What is the R code for a ‘goodness of fit’ test?

Answer 15

goodnessOfFitTest(x, p)

x = raw nominal data, p = null hypothesis

e.g. polling data, election results

Question 16

What is a chi-square test of association?

(also known as chi-square test of independence or chi-square test of homogeneity)

Answer 16

Very similar to chi-square goodness of fit test but you use it if you aren’t given the expected frequencies

Instead, you estimate them based on the data

E.g. the null hypothesis population parameter for j is given by the total observations for j divided by sample size

Question 17

How is sampling distribution calculated in the chi-square ‘test of association’?

Answer 17

Created by squaring and summing the normally distributed variables

Same as goodness of fit

Question 18

χ²(3) = 11.303, p = 0.0102 is a stat block from what type of test?

Explain what the numbers mean.

Answer 18

Chi-square test

χ²(3) = 11.303, p = 0.0102

χ² = sampling distribution

(3) = degrees of freedom
11. 303 = test statistic

p = 0.0102 = p-value

Question 19

Chi-square ‘goodness of fit’ tests compare observed frequencies of one variable vs what?

Answer 19

A hypothesis about the true probabilities of that variable.

Question 20

What do chi-square ‘tests of association’ / ‘test of independence’ test?

Answer 20

If two nominal scale variables are related to each other

Question 21

What do chi-square ‘tests of association’ / ‘test of independence’ use for its test statistic?

Answer 21

X²

Question 22

How is degrees of freedom calculated for a chi-square test of association?

Answer 22

(r-1)(c-1)

where r=# of categories of one variable and c=# of categories of the other

Question 23

How do you run a chi-square test of association in R?

Answer 23

chisq.test(x)

x = observed frequency contingency table of two nominal variables

Question 24

Describe the

diagnostic test statistic

distribution

degrees of freedom

for chi-square test of association

Answer 24