Chi Square

1
Chi Square

• Chapter 19

2
Distributional Assumptions

• All of the inferential tests we have talked about
  so far are called parametric tests
• These are tests in which we make assumptions
  about the distribution (e.g. it is normal)
• There are also non-parametric tests that dont
  make assumptions about the distribution
• They are called distribution free tests

3
Chi-Square Test

• A statistical test often used for analyzing
  categorical data
• With categorical data, we have the frequency of
  occurrence of some event in our categories
• One Classification Variable
• The Chi-Square Goodness-of-Fit Test
• Two Classification Variables
• Contingency Table Analysis

4
One Classification Variable

• Goodness-of-fit test
• A test for comparing observed frequencies with
  theoretically predicted frequencies
• Is there a good fit between the data (observed)
  and the theory (expected)
• Hypothesis Tests
• The null hypothesis in this tests whether the
  observed frequency is equal to the expected
• H0 the observed are equal to the expected
• The alternative is that there is a difference
• H1 the observed are not equal to the expected

5
Goodness of Fit Test

• The Chi-Square (x2) Statistic
• ?2 ? (O - E)2
• E
• O Observed frequency in each category
• E Expected frequency in each category
• To get the the expected frequency for each cell,
  divide the total sample size by the number of
  categories. Put that value in each cell
• In the Goodness-of-fit test
• df (k-1) where k is number of categories
• Chi square distribution
• Table E.1, p. 439

6
Example

• Compute the goodness of fit test for the
  following data
• An advertiser wants to know which bookstore in
  Athens students like better. We ask 1200 students
  their favorite. Use ?.05
• Folletts College Specialty
  
• 500 250 450

7
Two Classification Variables

• Asking whether the distribution of one variable
  is contingent upon another
• Is political affiliation contingent on gender?
• Contingency table
• A 2-dimensional table in which each observation
  is classified on the basis of 2 variables
  simultaneously
• Political Party
• Gender Democrat Republican Total
• Male 25 22
  47
• Female 16 7
  23
• Total 41 29 70

8
Contingency Tables

• Expected Frequencies for Contingency tables
• Political Party
• Gender Democrat Republican Total
• Male (E11) (E12) 47
• Female (E21) (E22)
  23
• Total 41 29
  70
• Marginal totals
• Totals for the levels of 1 variable summed across
  the levels of the other variable
• Expected Frequency
• Eij RiCj / N

9
Chi Square Test of Independence

• The test of independence tests whether the one
  variable is contingent on the other
• ?2 ? (O - E)2 df
  (R-1)(C-1)
• E
• R number of rows in table
• C number of columns in the table
• Hypothesis Testing
• Null Hypothesis is that the data are independent
  (e.g. they are not contingent)
• H0 The data are independent
• Alternative hypothesis is that the data are not
  independent (e.g. they are contingent)
• H1 The data are non-independent

10
Example

• Using the flowing data compute the chi square
  test for independence. Use ?.05
• We asked 1000 individuals the following
• Supports Gun Laws
• Parent Yes No
• Yes 400 200
• No 250 150

11
Small Expected Frequencies

• With chi square tests, small expected frequencies
  can be problematic
• To avoid this problem when you have 9 or fewer
  cells all expected to be at least 5
• This problem can result from
• Small sample sizes
• Usually the small expected frequencies will
  results in little statistical power for your test

12
Non-independent Observations

• Chi-Square is based on the assumption that
  observations are taken only once from each
  individual
• A chi square should not be used with repeated
  measures designs

13
Final Example

• Compute the chi square test of independence for
  the following data and use ?.05
• We asked 1200 individuals the following
• Favorite Coffee
  Shop
• Student Perks Brennens Front Room
• Yes 200 150
  350
• No 150 100
  250