Chapter 11: Analyzing the Association Between Categorical Variables

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Chapter 11: Analyzing the Association Between Categorical Variables

Section 11.1: What is Independence and What is Association?

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Learning Objectives

1. Comparing Percentages

2. Independence vs. Dependence

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Learning Objective 1: Example: Is There an Association Between Happiness and Family Income?

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The percentages in a particular row of a table are called conditional percentages

They form the conditional distribution for happiness, given a particular income level


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Guidelines when constructing tables with conditional distributions: Make the response variable the column

variable

Compute conditional proportions for the response variable within each row

Include the total sample sizes


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Learning Objective 2:Independence vs. Dependence

For two variables to be independent, the population percentage in any category of one variable is the same for all categories of the other variable

For two variables to be dependent (or associated), the population percentages in the categories are not all the same

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Are race and belief in life after death independent or dependent?

The conditional distributions in the table are similar but not exactly identical

It is tempting to conclude that the variables are dependent

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Are race and belief in life after death independent or dependent?

The definition of independence between variables refers to a population

The table is a sample, not a population

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Even if variables are independent, we would not expect the sample conditional distributions to be identical

Because of sampling variability, each sample percentage typically differs somewhat from the true population percentage


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Section 11.2: How Can We Test Whether Categorical Variables Are Independent?

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Learning Objectives

1. A Significance Test for Categorical Variables

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

3. How Do We Find the Expected Cell Counts?

4. The Chi-Squared Test Statistic

5. The Chi-Squared Distribution

6. The Five Steps of the Chi-Squared Test of Independence

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Learning Objectives

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

8. Chi-Squared and the Test Comparing Proportions in 2x2 Tables

9. Limitations of the Chi-Squared Test

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Learning Objective 1:A Significance Test for Categorical Variables

Create a table of frequencies divided into the categories of the two 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 (cell counts in the frequency table of at least 5)

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Learning Objective 2: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 count

values The mean of its distribution is called an

expected cell count This is found under the presumption that

H0 is true

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Learning Objective 3:How Do We Find the Expected Cell Counts?

Expected Cell Count:

For a particular cell,

The expected frequencies are values that have the same row and column totals as the observed counts, but for which the conditional distributions are identical (this is the assumption of the null hypothesis).

size sample Total

al)Column tot( total)(Rowcount cell Expected

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Learning Objective 3:How Do We Find the Expected Cell Counts?Example

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Learning Objective 4: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

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State the null and alternative hypotheses for this test

H0: Happiness and family income are independent

Ha: Happiness and family income are dependent (associated)

Learning Objective 4:Example: Happiness and Family Income

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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

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count expected

count) expected-count (observed 2

2


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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

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Learning Objective 4:The Chi-Squared Test Statistic

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Learning Objective 5: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

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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)


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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


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Learning Objective 6:The Five Steps of the Chi-Squared Test of Independence

1. Assumptions:

Two categorical variables

Randomization

Expected counts ≥ 5 in all cells

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2. Hypotheses:

H0: The two variables are independent

Ha: The two variables are dependent (associated)

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3. Test Statistic:

count expected

count) expected -count observed( 2

2

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Learning Objective 6:The Five Steps of the Chi-Squared Test of Independence4. 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

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Learning Objective 7: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

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Learning Objective 8:Chi-Squared and the Test Comparing Proportions in 2x2 Tables In practice, contingency tables of size 2x2 are very

common. They often occur in summarizing the responses of two groups on a binary response variable. Denote the population proportion of success by p1 in

group 1 and p2 in group 2 If the response variable is independent of the group,

p1=p2, so the conditional distributions are equal H0: p1=p2 is equivalent to H0: independence

z2 2 where

z ˆ p 1 ˆ p 2 se0

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Learning Objective 8:Example: Aspirin and Heart Attacks Revisited

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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

Learning Objective 8: Example: Aspirin and Heart Attacks Revisited

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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


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


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Learning Objective 9: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

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

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The chi-squared test is often misused. Some examples are: when some of the expected frequencies are

too small when separate rows or columns are

dependent samples data are not random quantitative data are classified into categories

- results in loss of information

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Learning Objective 10:“Goodness of Fit” Chi-Squared Tests

The Chi-Squared test can also be used for testing particular proportion values for a categorical variable. The null hypothesis is that the distribution of the

variable follows a given probability distribution; the alternative is that it does not

The test statistic is calculated in the same manner where the expected counts are what would be expected in a random sample from the hypothesized probability distribution

For this particular case, the test statistic is referred to as a goodness-of-fit statistic.

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Section 11.3: How Strong is the Association?

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Learning Objectives

1. Analyzing Contingency Tables

2. Measures of Association

3. Difference of Proportions

4. The Ratio of Proportions: Relative Risk

5. Properties of the Relative Risk

6. Large Chi-square Does Not Mean There’s a Strong Association

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Learning Objective 1: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

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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

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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

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Learning Objective 2:Measures of Association

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

comparing associations

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Learning Objective 3: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. The difference of proportions is 0

Case (b) exhibits the strongest possible association: The difference of proportions is 1

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Learning Objective 3:Difference of Proportions

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

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Learning Objective 3: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?

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

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

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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)

Learning Objective 3:Difference of Proportions Example: Do Student Stress and Depression Depend on Gender?

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Learning Objective 4:The Ratio of Proportions: Relative Risk

Another measure of association, is the ratio of two proportions: p1/p2

In medical applications in which the proportion refers to an adverse outcome, it is called the relative risk

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Learning Objective 4: 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

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The adverse outcome is death

The relative risk is formed for that outcome

For those who wore a seat belt, the proportion who died equaled 510/412,878 = 0.00124

For those who did not wear a seat belt, the proportion who died equaled 1601/164,128 = 0.00975


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The relative risk is the ratio:

0.00124/0.00975 = 0.127

The proportion of subjects wearing a seat belt who died was 0.127 times the proportion of subjects not wearing a seat belt who died


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Many find it easier to interpret the relative risk but reordering the rows of data so that the relative risk has value above 1.0


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Reversing the order of the rows, we calculate the ratio: 0.00975/0.00124 = 7.9

The proportion of subjects not wearing a seat belt who died was 7.9 times the proportion of subjects wearing a seat belt who died


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A relative risk of 7.9 represents a strong association

This is far from the value of 1.0 that would occur if the proportion of deaths were the same for each group

Wearing a set belt has a practically significant effect in enhancing the chance of surviving an auto accident


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Learning Objective 5:Properties of the Relative Risk

The relative risk can equal any nonnegative number

When p1= p2, the variables are independent and relative risk = 1.0

Values farther from 1.0 (in either direction) represent stronger associations

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Learning Objective 6: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

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Section 11.4: How Can Residuals Reveal The Pattern of Association?

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Learning Objectives

1. Association Between Categorical Variables

2. Residual Analysis

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Learning Objective 1: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

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Learning Objective 1: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

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Learning Objective 2: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

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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

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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

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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

The se describes how much the difference would tend to vary in repeated sampling if the variables were independent Its formula is complex Software can be used to find its value

A large standardized residual value provides evidence against independence in that cell

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Learning Objective 2: Example: Standardized Residuals for Religiosity and Gender

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

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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


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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


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Section 11.5: What if the Sample Size is Small?

Fisher’s Exact Test

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Learning Objectives

1. Fisher’s Exact Test

2. Example using Fisher’s Exact Test

3. Summary of Fisher’s Exact Test of Independence for 2x2 Tables

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Learning Objective 1: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

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Learning Objective 1: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 the evidence that the variables are associated

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Learning Objective 2:Fisher’s Exact Test 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

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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


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Results:


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Analysis:


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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?


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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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Learning Objective 3:Summary of Fisher’s Exact Test of Independence for 2x2 Tables Assumptions:

Two binary categorical variables Data are random

Hypotheses: H0: the two variables are independent (p1=p2)

Ha: the two variables are associated

(p1≠p2 or p1>p2 or p1<p2)

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Learning Objective 3:Summary of Fisher’s Exact Test of Independence for 2x2 Tables Test Statistic:

First cell count (this determines the others given the margin totals)

P-value: Probability that the first cell count equals the

observed value or a value even more extreme as predicted by Ha

Conclusion: Report the P-value and interpret in context. If

a decision is required, reject H0 when P-value ≤ significance level

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Chapter 11: Analyzing the Association Between Categorical Variables