The document discusses Chi-square tests and their applications. Chi-square tests can be used for goodness of fit, independence, and homogeneity. They are non-parametric tests used to analyze categorical data. The three main types are: 1) goodness of fit tests determine if a sample fits a hypothesized distribution, 2) independence tests determine if two categorical variables are associated, and 3) homogeneity tests determine if a categorical variable is distributed identically across populations. Chi-square tests involve calculating expected frequencies, observed frequencies, and a test statistic to determine if the null hypothesis can be rejected.

1. Chi-square

2. Applications of Chi-Square
• Goodness of fit (適合度檢驗)
• Independence (獨立性檢驗)
• Homogenecity (同質性檢驗)

3. Chi-Square Test on Numerical Data
• The researcher may believe there’s a relationship between X and Y,
but doesn’t want to use regression.
• There are outliers or anomalies that prevent us from assuming that
the data came from a normal population.
• The researcher has numerical data for one variable but not the other.

4. Chi-Square Goodness of Fit Test
• The test is applied when you have one categorical variable from a
single population.
• It is used to determine whether sample data are consistent with a
hypothesized distribution.
• For example, suppose a company printed baseball cards. It claimed
that 30% of its cards were rookies; 60%, veterans; and 10%, All-Stars.
We could gather a random sample of baseball cards and use a chi-
square goodness of fit test to see whether our sample distribution
differed significantly from the distribution claimed by the company.

5. Chi-Square Test for Independence
• The test is applied when you have two categorical variables from a
single population.
• It is used to determine whether there is a significant association
between the two variables.
• For example, in an election survey, voters might be classified by
gender (male or female) and voting preference (Democrat,
Republican, or Independent). We could use a chi-square test for
independence to determine whether gender is related to voting
preference.

6. Chi-Square Test of Homogeneity
• The test is applied to a single categorical variable from two different
populations.
• To determine whether frequency counts are distributed identically
across different populations.
• In a survey of TV viewing preferences, we might ask respondents to
identify their favorite program. We might ask the same question of
two different populations, such as males and females. We could use a
chi-square test for homogeneity to determine whether male viewing
preferences differed significantly from female viewing preferences.

7. Chi-Square Test for Goodness-of-Fit
• The goodness-of-fit (GOF) test helps you decide
whether your sample resembles a particular kind
of population.
• The chi-square test will be used because it is
versatile and easy to understand.
Purpose of the Test

8. Chi-Square Test for Goodness-of-Fit
• A multinomial distribution is defined by any k
probabilities p1, p2, …, pk that sum to unity.
• For example, consider the following “official”
proportions of M&M colors.
Multinomial GOF Test
calc

9. Chi-Square Test for Goodness-of-Fit
• The hypotheses are
H0: p1 = .30, p2 = .20, p3 = .10, p4 = .10, p5 = .10, p6 = .20
H1: At least one of the pj differs from the
hypothesized value
• No parameters are estimated (m = 0) and there are c = 6
classes, so the degrees of freedom are
n = c – m – 1 = 6 – 0 - 1
Multinomial GOF Test

10. Chi-Square Test for Goodness-of-Fit
• The hypotheses are:
H0: The population follows a _____ distribution
H1: The population does not follow a ______
distribution
• The blank may contain the name of any theoretical
distribution (e.g., uniform, Poisson, normal).
Hypotheses for GOF

11. Chi-Square Test for Goodness-of-Fit
• Assuming n observations, the observations are
grouped into c classes and then the chi-square test
statistic is found using:
Test Statistic and Degrees of Freedom for GOF
where fj = the observed frequency of
observations in class j
ej = the expected frequency in class j if
H0 were true
calc

12. Chi-Square Test for Goodness-of-Fit
• If the proposed distribution gives a good fit to the
sample, the test statistic will be near zero.
• The test statistic follows the chi-square distribution
with degrees of freedom
n = c – m – 1
where c is the no. of classes used in the test m is
the no. of parameters estimated
Test Statistic and Degrees of Freedom for GOF

13. Chi-Square Test for Goodness-of-Fit
Test Statistic and Degrees of Freedom for GOF
110  ccmcv
211  ccmcv
312  ccmcv

14. • For example, for n = 6 and a = .05, c2
.05 = 12.59.
Chi-Square Test for Goodness-of-Fit

15. Chi-Square Test for Goodness-of-Fit
• Instead of “fishing” for a good-fitting model,
visualize a priori the characteristics of the
underlying data-generating process.
Data-Generating Situations
• Mixtures occur when more than one data-generating process is superimposed
on top of one another.
Mixtures: A Problem

16. Chi-Square Test for Goodness-of-Fit
• A simple “eyeball” inspection of the histogram or
dot plot may suffice to rule out a hypothesized
population.
Eyeball Tests
• Goodness-of-fit tests may lack power in small samples. As a guideline, a chi-
square goodness-of-fit test should be avoided if n < 25.
Small Expected Frequencies

17. Normal Chi-Square
Goodness-of-Fit Test
• Two parameters, m and s, fully describe the normal
distribution.
• Unless m and s are know a priori, they must be
estimated from a sample by using x and s.
• Using these statistics, the chi-square goodness-of-fit
test can be used.
Normal Data Generating Situations

18. Normal Chi-Square
Goodness-of-Fit Test
• Transform the sample observations x1, x2, …, xn into
standardized values.
• Count the sample observations fj within intervals of
the form and compare them with the
known frequencies ej based on the normal
distribution.
Method 1: Standardizing the Data
x + ks

19. Normal Chi-Square
Goodness-of-Fit Test
Method 1: Standardizing the Data
Advantage is a standardized
scale.
Disadvantage is that data are
no longer in the original units.
Figure 15.14

20. • The chi-square test is unreliable if the expected
frequencies are too small.
• Rules of thumb:
• Cochran’s Rule requires that ejk > 5 for all cells.
• Up to 20% of the cells may have ejk < 5
Small Expected Frequencies
• Most agree that a chi-square test is infeasible if ejk < 1
in any cell.
• If this happens, try combining adjacent rows to enlarge
the expected frequencies.

21. Chi-Square Test for Independence
• A contingency table is a cross-tabulation of n paired
observations into categories.
• Each cell shows the count of observations that fall into
the
category
defined by its
row (r) and
column (c)
heading.
Contingency Tables
A
B

22. Chi-Square Test for Independence
• For example:
Contingency Tables
Table 15.1

23. Chi-Square Test for Independence
• In a test of independence for an r x c contingency table,
the hypotheses are
H0: Variable A is independent of variable B
H1: Variable A is not independent of variable B
• Use the chi-square test for independence to test these
hypotheses.
• This non-parametric test is based on frequencies.
• The n data pairs are classified into c columns and r rows
and then the observed frequency fjk is compared with
the expected frequency ejk.
Chi-Square Test

24. Chi-Square Test for Independence
• In a test of independence for an r x c contingency table, the
hypotheses are
H0: Variable A is independent of variable B
H1: Variable A is not independent of variable B
H0: There is no relationship between the variables.
H1: There is a relationship between the variables.
• If two categorical variables are related, it means the chance that
an individual falls into a particular category for one variable
depends upon the particular category they fall into for the other
variable.
Chi-Square Test

25. Chi-Square Test for Independence
• The critical value comes from the chi-square
probability distribution with n degrees of freedom.
n = degrees of freedom = (r – 1)(c – 1)
where r = number of rows in the table
c = number of columns in the table
• Appendix E contains critical values for right-tail areas of
the chi-square distribution.
• The mean of a chi-square distribution is n with variance
2n.
Chi-Square Distribution

26. Chi-Square Test for Independence
• Assuming that H0 is true, the expected frequency of
row j and column k is:
ejk = RjCk/n
where Rj = total for row j (j = 1, 2, …, r)
Ck = total for column k (k = 1, 2, …, c)
n = sample size
Expected Frequencies

27. Chi-Square Test for Independence
• Step 1: State the Hypotheses
H0: Variable A is independent of variable B
H1: Variable A is not independent of variable B
• Step 2: Specify the Decision Rule
Calculate n = (r – 1)(c – 1)
For a given a, look up the right-tail critical value (c2
R)
from Appendix E or by using Excel.
Reject H0 if c2
R > test statistic.
Steps in Testing the Hypotheses

28. Chi-Square Test for Independence
• For example, for n = 6 and a = .05, c2
.05 = 12.59.
Steps in Testing the Hypotheses

29. Chi-Square Test for Independence
• Here is the rejection region.
Steps in Testing the Hypotheses
Figure 15.3

30. Chi-Square Test for Independence
• Step 3: Calculate the Expected Frequencies
ejk = RjCk/n
• For example,
Steps in Testing the Hypotheses

31. Chi-Square Test for Independence
• The chi-square test is unreliable if the expected
frequencies are too small.
• Rules of thumb:
• Cochran’s Rule requires that ejk > 5 for all cells.
• Up to 20% of the cells may have ejk < 5
Small Expected Frequencies
• Most agree that a chi-square test is infeasible if ejk < 1
in any cell.
• If this happens, try combining adjacent rows or
columns to enlarge the expected frequencies.

32. Chi-Square Test for Independence
• Chi-square tests for independence can also be used to
analyze quantitative variables by coding them into
categories.
Cross-Tabulating Raw Data
For example, the
variables Infant
Deaths per 1,000
and Doctors per
100,000 can
each be coded
into various
categories: