Agresti/Franklin Statistics, 1 of 90 Section 10.2 How Can We Test whether Categorical Variables are...

Agresti/Franklin Statistics, 1 of 90

Section 10.2

How Can We Test whether Categorical Variables are

Independent?


A Significance Test for Categorical Variables

The hypotheses for the test are:

H0: The two variables are independent

Ha: The two variables are dependent (associated)

• The test assumes random sampling and a large sample size


What Do We Expect for Cell Counts if the Variables Are Independent?

The count in any particular cell is a random variable• Different samples have different values for

the count

The mean of its distribution is called an expected cell count• This is found under the presumption that

H0 is true


How Do We Find the Expected Cell Counts?

Expected Cell Count:

• For a particular cell, the expected cell count equals:

size sample Total

al)Column tot( total)(Rowcount cell Expected


Example: Happiness by Family Income


The Chi-Squared Test Statistic

The chi-squared statistic summarizes how far the observed cell counts in a contingency table fall from the expected cell counts for a null hypothesis

count expected

count) expected -count observed( 2

2


Example: Happiness and Family Income


State the null and alternative hypotheses for this test

H0: Happiness and family income are independent

Ha: Happiness and family income are dependent (associated)




Report the statistic and explain how it was calculated:

To calculate the statistic, for each cell, calculate:

Sum the values for all the cells The value is 73.4

2

2

count expected

count) expected-count (observed 2

2



The larger the value, the greater the evidence against the null hypothesis of independence and in support of the alternative hypothesis that happiness and income are associated

2


The Chi-Squared Distribution

To convert the test statistic to a P-value, we use the sampling distribution of the statistic

For large sample sizes, this sampling distribution is well approximated by the chi-squared probability distribution

2



Main properties of the chi-squared distribution:

• It falls on the positive part of the real number line

• The precise shape of the distribution depends on the degrees of freedom:

df = (r-1)(c-1)



Main properties of the chi-squared distribution:

• The mean of the distribution equals the df value

• It is skewed to the right

• The larger the value, the greater the evidence against H0: independence


The Five Steps of the Chi-Squared Test of Independence

1. Assumptions:

• Two categorical variables

• Randomization

• Expected counts ≥ 5 in all cells



2. Hypotheses:

H0: The two variables are independent

Ha: The two variables are dependent (associated)



3. Test Statistic:

count expected

count) expected -count observed( 2

2



4. P-value: Right-tail probability above the observed value, for the chi-squared distribution with df = (r-1)(c-1)

5. Conclusion: Report P-value and interpret in context• If a decision is needed, reject H0 when P-value ≤

significance level


Chi-Squared is Also Used as a “Test of Homogeneity”

The chi-squared test does not depend on which is the response variable and which is the explanatory variable

When a response variable is identified and the population conditional distributions are identical, they are said to be homogeneous• The test is then referred to as a test of

homogeneity


Example: Aspirin and Heart Attacks Revisited



What are the hypotheses for the chi-squared test for these data?

The null hypothesis is that whether a doctor has a heart attack is independent of whether he takes placebo or aspirin

The alternative hypothesis is that there’s an association



Report the test statistic and P-value for the chi-squared test:

• The test statistic is 25.01 with a P-value of 0.000

This is very strong evidence that the population proportion of heart attacks differed for those taking aspirin and for those taking placebo



The sample proportions indicate that the aspirin group had a lower rate of heart attacks than the placebo group


Limitations of the Chi-Squared Test

If the P-value is very small, strong evidence exists against the null hypothesis of independence

But… The chi-squared statistic and the P-

value tell us nothing about the nature of the strength of the association


Limitations of the Chi-Squared Test

We know that there is statistical significance, but the test alone does not indicate whether there is practical significance as well


Section 10.3

How Strong is the Association?


In a study of the two variables (Gender and Happiness), which one is the response variable?

a. Genderb. Happiness

The following is a table on Gender and Happiness:

Gender: Not Pretty Very

Females 163 898 502

Males 130 705 379


What is the Expected Cell Count for ‘Females’ who are ‘Pretty Happy’?

a. 898b. 801.5c. 902d. 521



Females 163 898 502

Males 130 705 379


What is the Expected Cell Count for ‘Females’ who are ‘Pretty Happy’?

a. 898b. 801.5c. 902 = N*(898+705)/N*(163+898+502)/Nd. 521



Females 163 898 502

Males 130 705 379


Calculate the

a. 1.75b. 0.27c. 0.98d. 10.34

2value



Females 163 898 502

Males 130 705 379


At a significance level of 0.05, what is the correct decision?

a. ‘Gender’ and ‘Happiness’ are independent

b. There is an association between ‘Gender’ and ‘Happiness’



Females 163 898 502

Males 130 705 379


Analyzing Contingency Tables

Is there an association?

• The chi-squared test of independence addresses this

• When the P-value is small, we infer that the variables are associated



How do the cell counts differ from what independence predicts?

To answer this question, we compare each observed cell count to the corresponding expected cell count



How strong is the association?

Analyzing the strength of the association reveals whether the association is an important one, or if it is statistically significant but weak and unimportant in practical terms


Measures of Association

A measure of association is a statistic or a parameter that summarizes the strength of the dependence between two variables


Difference of Proportions

An easily interpretable measure of association is the difference between the proportions making a particular response



Case (a) exhibits the weakest possible association – no association

Accept Credit Card

The difference of proportions is 0

Income No Yes

High 60% 40%

Low 60% 40%



Case (b) exhibits the strongest possible association:

Accept Credit Card

The difference of proportions is 100%

Income No Yes

High 0% 100%

Low 100% 0%



In practice, we don’t expect data to follow either extreme (0% difference or 100% difference), but the stronger the association, the large the absolute value of the difference of proportions


Example: Do Student Stress and Depression Depend on Gender?


Which response variable, stress or depression, has the stronger sample association with gender?




Stress:

The difference of proportions between females and males was 0.35 – 0.16 = 0.19

Gender Yes No

Female 35% 65%

Male 16% 84%



Depression:

The difference of proportions between females and males was 0.08 – 0.06 = 0.02

Gender Yes No

Female 8% 92%

Male 6% 94%



In the sample, stress (with a difference of proportions = 0.19) has a stronger association with gender than depression has (with a difference of proportions = 0.02)



Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents


Treating the auto accident outcome as the response variable, find and interpret the relative risk

Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents


Large Does Not Mean There’s a Strong Association

A large chi-squared value provides strong evidence that the variables are associated

It does not imply that the variables have a strong association

This statistic merely indicates (through its P-value) how certain we can be that the variables are associated, not how strong that association is


Section 10.4

How Can Residuals Reveal the Pattern of Association?


Association Between Categorical Variables

The chi-squared test and measures of association such as (p1 – p2) and p1/p2 are fundamental methods for analyzing contingency tables

The P-value for summarized the strength of evidence against H0: independence


Association Between Categorical Variables

If the P-value is small, then we conclude that somewhere in the contingency table the population cell proportions differ from independence

The chi-squared test does not indicate whether all cells deviate greatly from independence or perhaps only some of them do so


Residual Analysis

A cell-by-cell comparison of the observed counts with the counts that are expected when H0 is true reveals the nature of the evidence against H0

The difference between an observed and expected count in a particular cell is called a residual


Residual Analysis

The residual is negative when fewer subjects are in the cell than expected under H0

The residual is positive when more subjects are in the cell than expected under H0


Residual Analysis

To determine whether a residual is large enough to indicate strong evidence of a deviation from independence in that cell we use a adjusted form of the residual: the standardized residual


Residual Analysis

The standardized residual for a cell:

(observed count – expected count)/se

• A standardized residual reports the number of standard errors that an observed count falls from its expected count

• Its formula is complex

• Software can be used to find its value

• A large value provides evidence against independence in that cell


Example: Standardized Residuals for Religiosity and Gender

“To what extent do you consider yourself a religious person?”



Interpret the standardized residuals in the table



The table exhibits large positive residuals for the cells for females who are very religious and for males who are not at all religious.

In these cells, the observed count is much larger than the expected count

There is strong evidence that the population has more subjects in these cells than if the variables were independent



The table exhibits large negative residuals for the cells for females who are not at all religious and for males who are very religious

In these cells, the observed count is much smaller than the expected count

There is strong evidence that the population has fewer subjects in these cells than if the variables were independent


Section 10.5

What if the Sample Size is Small? Fisher’s Exact Test


Fisher’s Exact Test

The chi-squared test of independence is a large-sample test

When the expected frequencies are small, any of them being less than about 5, small-sample tests are more appropriate

Fisher’s exact test is a small-sample test of independence


Fisher’s Exact Test

The calculations for Fisher’s exact test are complex

Statistical software can be used to obtain the P-value for the test that the two variables are independent

The smaller the P-value, the stronger is the evidence that the variables are associated


Example: Tea Tastes Better with Milk Poured First?

This is an experiment conducted by Sir Ronald Fisher

His colleague, Dr. Muriel Bristol, claimed that when drinking tea she could tell whether the milk or the tea had been added to the cup first



Experiment:• Fisher asked her to taste eight cups of tea:

•Four had the milk added first

•Four had the tea added first

•She was asked to indicate which four had the milk added first

•The order of presenting the cups was randomized



Results:



Analysis:



The one-sided version of the test pertains to the alternative that her predictions are better than random guessing

Does the P-value suggest that she had the ability to predict better than random guessing?



The P-value of 0.243 does not give much evidence against the null hypothesis

The data did not support Dr. Bristol’s claim that she could tell whether the milk or the tea had been added to the cup first

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