PPA 419 – Aging Services Administration Lecture 10c – Public Transportation and Aging.
PPA 415 – Research Methods in Public Administration
19
PPA 415 – Research Methods in Public Administration Lecture 8 – Chi-square
description
PPA 415 – Research Methods in Public Administration. Lecture 8 – Chi-square. Introduction. The chi-square ( 2 ) has probably been the most frequently used test of hypothesis in the social sciences, largely because its assumptions are the easiest to satisfy. - PowerPoint PPT Presentation
Transcript of PPA 415 – Research Methods in Public Administration
PPA 415 – Research Methods in Public Administration
Lecture 8 – Chi-square
Introduction
The chi-square (2) has probably been the most frequently used test of hypothesis in the social sciences, largely because its assumptions are the easiest to satisfy.
It assumes only that the sample is randomly selected and the variable is measured at the nominal level.
No assumptions about shape of distribution or the sampling distribution.
Bivariate Tables
Bivariate tables are used to find if there is a significant relationship between the independent and dependent variable.
Rows (dependent variables), columns (independent variables), and cells (independent by dependent variable).
Bivariate Tables
Table 1. President's disaster recommendation by presidential administration, 1974-1981
37 94 131
29.1% 38.5% 35.3%
18 44 62
14.2% 18.0% 16.7%
72 106 178
56.7% 43.4% 48.0%
127 244 371
100.0% 100.0% 100.0%
Turndown
Emergency Declaration
Major disaster declaration
President'srecommendation
Total
Gerald R. Ford Jimmy Carter
Presidential administration
Total
The Logic of Chi-Square
In two-sample tests, independence means that the two samples are randomly selected.
In chi-square, independence means that the classification of a case on one variable has no influence on how cases are classified on the other variable.
The Logic of Chi-SquareTable 2. President's recommendation by presidential administration, expected
frequencies
44.8 86.2 131.0
35.3% 35.3% 35.3%
21.2 40.8 62.0
16.7% 16.7% 16.7%
60.9 117.1 178.0
48.0% 48.0% 48.0%
127.0 244.0 371.0
100.0% 100.0% 100.0%
Turndown
Emergency Declaration
Major disaster declaration
President'srecommendation
Total
Gerald R.Ford Jimmy Carter
Presidentialadministration
Total
The Computation of Chi-Square
Chi-square is calculated by comparing the observed frequency (fo) each cell to the expected frequency (fe)if there were no relationship.
The larger the difference between observed and expected frequency, the larger is chi-square.
The Computation of Chi-Square
N
marginalColumn x marginal Row
)(2
2
e
e
eo
f
f
ffobtained
The Computation of Chi-Square
Example:
8732.5)(
0522.10232.22510.4830.7058.3580.1)(
1.117
1.117106
9.60
9.6072
8.40
8.4044
2.21
2.2118
2.86
2.8694
8.44
8.4437)(
2
2
2222
2222
obtained
obtained
f
ffobtained
e
eo
Five-Step Hypothesis Test
Step 1. Making assumptions. Independent random samples. Nominal level of measurement.
Step 2. Stating the null hypothesis. H0: The two variables are independent.
H1: The two variables are dependent (related).
Five-Step Hypothesis Test
Step 3. Selecting the sampling distribution and establishing the critical region. Sampling distribution = 2 distribution. Alpha=0.05. Df=(r-1)(c-1)=(3-1)(2-1)=2(1)=2. 2(critical)=Appendix C, p. 466=5.991.
Five-Step Hypothesis Test
Step 4. 2(obtained)=5.87 (from slide 9).
Step 5. Making a decision. 2(obtained) < 2(critical), therefore do not
reject the null hypothesis that there is no relationship between presidential administration and disaster recommendations.
Example 2Have you ever experienced discrimination when renting or buying a house? * What ethnic
background is the head of household? Crosstabulation
46 80 0 126
37.8 85.2 3.0 126.0
90.2% 69.6% .0% 74.1%
5 35 4 44
13.2 29.8 1.0 44.0
9.8% 30.4% 100.0% 25.9%
51 115 4 170
51.0 115.0 4.0 170.0
100.0% 100.0% 100.0% 100.0%
Count
Expected Count
% within What ethnicbackground is thehead of household?
Count
Expected Count
% within What ethnicbackground is thehead of household?
Count
Expected Count
% within What ethnicbackground is thehead of household?
No
Yes
Have you everexperienceddiscriminationwhen renting orbuying a house?
Total
White Black Other
What ethnic background is the headof household?
Total
Example 2
Step 1. Making assumptions. Independent random samples. Nominal level of measurement.
Step 2. Stating the null hypothesis. H0: The two variables are independent.
H1: The two variables are dependent (related).
Example 2
Step 3. Selecting the sampling distribution and establishing the critical region. Sampling distribution = 2 distribution. Alpha=0.05. Df=(r-1)(c-1)=(2-1)(3-1)=1(2)=2. 2(critical)=Appendix C, p. 466=5.991.
Example 2
Step 4.
Chi-Square Tests
19.570a 2 .000
20.367 2 .000
15.378 1 .000
170
Pearson Chi-Square
Likelihood Ratio
Linear-by-LinearAssociation
N of Valid Cases
Value dfAsymp. Sig.
(2-sided)
2 cells (33.3%) have expected count less than 5. Theminimum expected count is 1.04.
a.
Example 2
Step 5. Making a decision. 2(obtained) >2(critical), therefore reject the
null hypothesis that there is no relationship between ethnicity and whether or not Birmingham residents have experienced discrimination. Minorities are significantly more likely to report that they have experienced discrimination than whites.
Limitations of Chi-Square
The chi-square becomes difficult to interpret if the table is larger than 4 x 4.
When sample size is small, expected frequencies can be less than five per cell. If the table is 2 x 2, use the following correction.
e
eoc f
ffobtained
2
2 5.)(
Limitations of Chi-Square
If table is more than 2 x 2, there is no correction for small sample size.
Chi-square is also sensitive to large sample sizes. For example, doubling the sample size, doubles chi-square. For large samples, very small (and meaningless) differences can be significant. You should also calculate a measure of association. See the next lecture.
