Chapter 10 Analyzing the Association Between Categorical Variables

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Chapter 10Analyzing the Association Between
Categorical Variables

• Learn .
• How to detect and describe associations between
  categorical variables

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Section 10.1

• What Is Independence and What is Association?

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Example Is There an Association Between
Happiness and Family Income?
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Example Is There an Association Between
Happiness and Family Income?
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Example Is There an Association Between
Happiness and Family Income?

• 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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Example Is There an Association Between
Happiness and Family Income?
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Example Is There an Association Between
Happiness and Family Income?

• 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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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
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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

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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

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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

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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

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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

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Example Happiness by Family Income
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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

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Example Happiness and Family Income
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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)

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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

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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

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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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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)

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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

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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

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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

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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 theres an
  association

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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?
• Gender
• Happiness

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• What is the Expected Cell Count for Females who
  are Pretty Happy?
• 898
• 801.5
• 902
• 521

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• Calculate the
• 1.75
• 0.27
• 0.98
• 10.34

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• At a significance level of 0.05, what is the
  correct decision?
• Gender and Happiness are independent
• There is an association between Gender and
  Happiness

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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
  

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Difference of Proportions

• Case (b) exhibits the strongest possible
  association
• Accept Credit Card
• The difference of proportions is 100
  

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Difference of Proportions

• In practice, we dont 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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Example Do Student Stress and Depression Depend
on Gender?

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

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Example Do Student Stress and Depression Depend
on Gender?
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

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Example Do Student Stress and Depression Depend
on Gender?

• Depression
• The difference of proportions between females and
  males was 0.08 0.06 0.02

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Example Do Student Stress and Depression Depend
on Gender?

• 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)

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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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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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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

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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

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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 Theres 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? Fishers Exact
  Test

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Fishers 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
• Fishers exact test is a small-sample test of
  independence

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Fishers Exact Test

• The calculations for Fishers 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. Bristols claim that
  she could tell whether the milk or the tea had
  been added to the cup first