Math 37 - Lecture 30

Math 37 - Lecture 29

Inference for Two-Way Tables (Ch. 9)

One-way Table: Goodness of Fit Tests (M&Ms)

Two-way Tables

Goals: 1) Comparing several population proportions

2) Testing for association between two categorical variables

1) COMPARING PROPORTIONS FOR SEVERAL POPULATIONS

Example 3x2 Two Way Table:

	Child Smokes	Child does not
Both Parents smoke	400	1380	1780
One Parent	416	1823	2239
Neither Parent	188	1168	1356

1004 4371 | 5375

What percent of students smoke?

Does this vary among the 3 groups of parental smoking?

How test if the proportions in the three groups are significantly different?

Graphical - Bar Plot Numerical - Conditional percents

Inference - Could do three separate two-sample hypothesis tests

Problem:

Want an overall test

H₀: H_a:

One-sided or two-sided?

If H_o is true, how many counts would you expect to be in each cell?

Expect about of the children in each group to smoke if the three groups are equal/no difference among them.

	Observed			Expected
	Smoke	No smoke		Smoke	No smoke
Both	400	1380	Both
One	416	1823	One
Neither	188	1168	Neither

Test Statistic Want to compare the observed cell counts to the expected cell counts. If the differences are large, consider that evidence against the null hypothesis.

p-value How often would we expect to find a value of the test statistic this extreme or more extreme?

How does this test statistic behave?

Simulation (Lab 13)
-> Approximate sampling distribution by Chi-Square Distribution

Shape =
Specified by degrees of freedom, for a two-way table:(r-1)(c-1)
p-value is area above observed test statistic value

Table F gives tail probabilities, e.g. df = p-value=
If significant, would examine conditional distributions, larger observed vs. expected deviations etc., to see why.

Checking the validity of using the chi-square procedure

For a 2x2 table: Procedure is valid if all 4 expected cell counts 5

More than 2x2: Procedure is valid if the average of the expected counts 5 and the smallest count 1

2) TESTING FOR RELATIONSHIPS

Example Are smoking habits related to Social Economic Status (SES)?

	SES
Smoking	High	Mid	Low	Total
Current	51	22	43	116
Former	92	21	28	141
Never	68	9	22	99
Total	211	52	93	356

Same idea! Compare the observed counts to expected counts =

(row total*column total/n), see if the differences are large.

H₀: H_a:

In general, use chi-squares tests when

1) Independent SRSs from several populations with each individual classified according to one categorical variable (other variable indicates which group the subject comes from)

2) A single SRS or one population with each individual classified according to both of two categorical variables. Are the categorical variables independent?

Note: When comparing only two proportions can use either a Chi-Square test or a two-sample z-test.

Z-test uses one or two sided, Chi-Square always two-sided alternative
Chi-Square test can be used for more than two populations