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Chapter 10 Analyzing the Association Between Categorical Variables. Learn …. How to detect and describe associations between categorical variables. Section 10.1. What Is Independence and What is Association?. Example: Is There an Association Between Happiness and Family Income?. - PowerPoint PPT Presentation
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Agresti/Franklin Statistics, 1 of 90
Chapter 10Analyzing the Association
Between Categorical Variables
Learn ….
How to detect and describe associations between categorical variables
Agresti/Franklin Statistics, 2 of 90
Section 10.1
What Is Independence and What is Association?
Agresti/Franklin Statistics, 3 of 90
Example: Is There an Association Between Happiness and Family Income?
Agresti/Franklin Statistics, 4 of 90
Example: Is There an Association Between Happiness and Family Income?
Agresti/Franklin Statistics, 5 of 90
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
Example: Is There an Association Between Happiness and Family Income?
Agresti/Franklin Statistics, 6 of 90
Example: Is There an Association Between Happiness and Family Income?
Agresti/Franklin Statistics, 7 of 90
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
Example: Is There an Association Between Happiness and Family Income?
Agresti/Franklin Statistics, 8 of 90
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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Example: Happiness and Gender
Agresti/Franklin Statistics, 10 of 90
Example: Happiness and Gender
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Example: Belief in Life After Death
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Example: Belief in Life After Death
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
Agresti/Franklin Statistics, 13 of 90
Example: Belief in Life After Death
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
Agresti/Franklin Statistics, 14 of 90
Independence vs Dependence
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
Agresti/Franklin Statistics, 15 of 90
Section 10.2
How Can We Test whether Categorical Variables are
Independent?
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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
Agresti/Franklin Statistics, 17 of 90
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
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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
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Example: Happiness by Family Income
Agresti/Franklin Statistics, 20 of 90
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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Example: Happiness and Family Income
Agresti/Franklin Statistics, 22 of 90
State the null and alternative hypotheses for this test
H0: Happiness and family income are independent
Ha: Happiness and family income are dependent (associated)
Example: Happiness and Family Income
Agresti/Franklin Statistics, 23 of 90
Example: Happiness and Family Income
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
Agresti/Franklin Statistics, 24 of 90
Example: Happiness and Family Income
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
Agresti/Franklin Statistics, 25 of 90
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
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The Chi-Squared Distribution
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The Chi-Squared Distribution
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)
Agresti/Franklin Statistics, 28 of 90
The Chi-Squared Distribution
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
Agresti/Franklin Statistics, 29 of 90
The Chi-Squared Distribution
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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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The Five Steps of the Chi-Squared Test of Independence
2. Hypotheses:
H0: The two variables are independent
Ha: The two variables are dependent (associated)
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The Five Steps of the Chi-Squared Test of Independence
3. Test Statistic:
count expected
count) expected -count observed( 2
2
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The Five Steps of the Chi-Squared Test of Independence
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
Agresti/Franklin Statistics, 34 of 90
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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Example: Aspirin and Heart Attacks Revisited
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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
Agresti/Franklin Statistics, 37 of 90
Example: Aspirin and Heart Attacks Revisited
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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Example: Aspirin and Heart Attacks Revisited
The sample proportions indicate that the aspirin group had a lower rate of heart attacks than the placebo group
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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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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
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Section 10.3
How Strong is the Association?
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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
Agresti/Franklin Statistics, 43 of 90
What is the Expected Cell Count for ‘Females’ who are ‘Pretty Happy’?
a. 898b. 801.5c. 902d. 521
The following is a table on Gender and Happiness:
Gender: Not Pretty Very
Females 163 898 502
Males 130 705 379
Agresti/Franklin Statistics, 44 of 90
Calculate the
a. 1.75b. 0.27c. 0.98d. 10.34
2value
The following is a table on Gender and Happiness:
Gender: Not Pretty Very
Females 163 898 502
Males 130 705 379
Agresti/Franklin Statistics, 45 of 90
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’
The following is a table on Gender and Happiness:
Gender: Not Pretty Very
Females 163 898 502
Males 130 705 379
Agresti/Franklin Statistics, 46 of 90
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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Analyzing Contingency Tables
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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Analyzing Contingency Tables
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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Measures of Association
A measure of association is a statistic or a parameter that summarizes the strength of the dependence between two variables
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Difference of Proportions
An easily interpretable measure of association is the difference between the proportions making a particular response
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Difference of Proportions
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Difference of Proportions
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%
Agresti/Franklin Statistics, 53 of 90
Difference of Proportions
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%
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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 large the absolute value of the difference of proportions
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Example: Do Student Stress and Depression Depend on Gender?
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Which response variable, stress or depression, has the stronger sample association with gender?
Example: Do Student Stress and Depression Depend on Gender?
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Example: Do Student Stress and Depression Depend on 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%
Example: Do Student Stress and Depression Depend on Gender?
Agresti/Franklin Statistics, 58 of 90
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%
Example: Do Student Stress and Depression Depend on Gender?
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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)
Example: Do Student Stress and Depression Depend on Gender?
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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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Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents
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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
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Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents
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
Agresti/Franklin Statistics, 64 of 90
Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents
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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Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents
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
Agresti/Franklin Statistics, 66 of 90
Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents
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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Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents
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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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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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 10.4
How Can Residuals Reveal the Pattern of Association?
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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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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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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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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
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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
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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
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Example: Standardized Residuals for Religiosity and Gender
“To what extent do you consider yourself a religious person?”
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Example: Standardized Residuals for Religiosity and Gender
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Example: Standardized Residuals for Religiosity and Gender
Interpret the standardized residuals in the table
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Example: Standardized Residuals for Religiosity and Gender
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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Example: Standardized Residuals for Religiosity and Gender
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 10.5
What if the Sample Size is Small? Fisher’s Exact Test
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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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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
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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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Example: Tea Tastes Better with Milk Poured 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
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Example: Tea Tastes Better with Milk Poured First?
Results:
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Example: Tea Tastes Better with Milk Poured First?
Analysis:
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Example: Tea Tastes Better with Milk Poured First?
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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Example: Tea Tastes Better with Milk Poured First?
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