Statistic work
mehett
chi-square_testing.pdf
Agenda
• Introduction to Nonparametric Tests of Significance
• Chi Square Test • One way chi square
• Two way chi square
Parametric versus Nonparametric Tests
• Parametric Test- A test that makes assumptions about the population parameters and the data
• Nonparametric Test- Tests that make fewer assumptions about the population parameters and the data
The One Way Chi Square Test – Terminology
• One way Chi-Square test: Testing for differences between the distribution of one variable among its categories.
• Two way Chi-Square Test: Testing for differences between the distribution of one variable by another variable.
• Observed frequencies: Frequencies that actually occur based on data we collected.
• Expected frequencies: Frequencies that we would expect IF the null hypothesis was true.
The One Way Chi Square Test – Intuition
• Expected frequencies are the frequencies that would occur if the null hypothesis was true
• We want to test the null hypothesis
• Can test it by seeing if the observed frequencies are significantly different from the expected frequencies • If differences are small they might be sampling error retain H0 • If the differences are large enough, however, they probably represent true
differences reject H0
The One Way Chi Square Test – Intuition
• Is the coin fair?
Coin Observed Frequency
Expected Frequency
Head 80 50
Tails 20 50
• Calculate expected frequencies for each category. • Divide total number of observations by K, the number of categories.
• Calculate Chi-Square statistic • Consult textbook page 323
The One Way Chi Square Test – Specific Steps
The One Way Chi Square Test – Specific Steps
• Look up the critical Chi Square value in the table (p.557) • Degrees of Freedom = K-1
• Compare critical Chi Square to calculated Chi square • If your calculated Chi Square is > your critical Chi square you reject the null
hypothesis
• If your calculated Chi Square is < you critical Chi Square you fail to reject the null hypothesis
The One Way Chi Square Test – Example
• The following table summarizes the self-reported ideology of a sample of students from Stony Brook. Test the null hypothesis that ideological self-identification is distributed equally through the student body.
Ideology
Liberal 30 50 -20 400 8
Moderate 75 50 25 625 12.5
Conservative 45 50 -5 25 .5
150
The One Way Chi Square Test – Exercise
• A researcher is interested in studying the voting turnout of university students. She takes a sample of 50 students and asks them whether they voted or not in the last election. She finds that only 15 students reported voting while 35 reported not voting. Test the null hypothesis that voters and non-voters are distributed equally among university students.
The Two Way Chi Square Test
• Are the differences in a cross tabulation statistically significant? • Remember a cross tabulation looks at the distribution of one variable by
categories of another variable
• The null hypothesis is that frequency of one variable does not differ by categories of the other variable
• Procedure is largely the same as the One Way Chi Square Test • Only major difference is how you calculate the expected frequencies and the
degrees of freedom
The Two Way Chi Square Test- Specific Steps
The Two Way Chi Square Test – Specific Steps
The Two Way Chi Square Test – Example
Democrat Republican Total
Obama 20 10 30
Romney 5 65 70
Total 25 75 100
Row Marginal Totals
Column Marginal Totals
N
The Two Way Chi Square Test – Example
Dem. Rep. Total
Obama 20 10 30
Romney 5 65 70
Total 25 75 100
Observed Frequencies
Dem. Rep. Total
Obama 7.5 22.5 30
Romney 17.5 52.5 70
Total 25 75 100
Expected Frequencies
The Two Way Chi Square Test – Example
Cell
Dems for Obama
20 7.5
Dems for Romney
5 17.5
Reps for Obama
10 22.5
Reps for Romney
65 52.5
Total 100 100
12.5
-12.5
-12.5
-12.5
156.25
156.25
156.25
156.25
20.83
8.93
6.94
2.98
39.68
The Two Way Chi Square Test- Example
Computer Skills
Did Not Take Course
Took Course Total
Above Average
15 19 34
Average 25 23 48
Below Average
10 8 18
50 50 100
The Two Way Chi Square Test- Example
Cell
No Course AA 15 17
No Course A 25 24
No Course BA 10 9
Course AA 19 17
Course A 23 24
Course BA 8 9
-2
1
.24
2
-1
-1
4
1
1
1
4
1
1
.04
.11
.24
.04
.11
.78
Requirements/Assumptions for Using Chi Square
• The samples are independent
• The samples are random
• The expected frequency of any one cell should not be too small • Can use Yates Correction if this occurs
Next Class
• Introduction to Correlation
• Homework will be on Blackboard after class • Deadline: April 20th, 5pm.
