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Chapter 10 Analyzing the Association Between Categorical Variables

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Example: Is There an Association Between Happiness and Family Income? ... In a study of the two variables (Gender and Happiness), which one is the response variable? ... – PowerPoint PPT presentation

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Title: Chapter 10 Analyzing the Association Between Categorical Variables


1
Chapter 10Analyzing the Association Between
Categorical Variables
  • Learn .
  • How to detect and describe associations between
    categorical variables

2
Section 10.1
  • What Is Independence and What is Association?

3
Example Is There an Association Between
Happiness and Family Income?
4
Example Is There an Association Between
Happiness and Family Income?
5
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

6
Example Is There an Association Between
Happiness and Family Income?
7
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

8
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

9
Example Happiness and Gender
10
Example Happiness and Gender
11
Example Belief in Life After Death
12
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

13
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

14
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

15
Section 10.2
  • How Can We Test whether Categorical Variables are
    Independent?

16
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

17
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

18
How Do We Find the Expected Cell Counts?
  • Expected Cell Count
  • For a particular cell, the expected cell count
    equals

19
Example Happiness by Family Income
20
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

21
Example Happiness and Family Income
22
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)

23
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

24
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

25
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

26
The Chi-Squared Distribution
27
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)

28
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

29
The Chi-Squared Distribution
30
The Five Steps of the Chi-Squared Test of
Independence
  • 1. Assumptions
  • Two categorical variables
  • Randomization
  • Expected counts 5 in all cells

31
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)

32
The Five Steps of the Chi-Squared Test of
Independence
  • 3. Test Statistic

33
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

34
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

35
Example Aspirin and Heart Attacks Revisited
36
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

37
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

38
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

39
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

40
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

41
Section 10.3
  • How Strong is the Association?

42
  • In a study of the two variables (Gender and
    Happiness), which one is the response variable?
  • Gender
  • Happiness

43
  • What is the Expected Cell Count for Females who
    are Pretty Happy?
  • 898
  • 801.5
  • 902
  • 521

44
  • Calculate the
  • 1.75
  • 0.27
  • 0.98
  • 10.34

45
  • 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

46
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

47
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

48
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

49
Measures of Association
  • A measure of association is a statistic or a
    parameter that summarizes the strength of the
    dependence between two variables

50
Difference of Proportions
  • An easily interpretable measure of association is
    the difference between the proportions making a
    particular response

51
Difference of Proportions
52
Difference of Proportions
  • Case (a) exhibits the weakest possible
    association no association
  • Accept Credit Card
  • The difference of proportions is 0

53
Difference of Proportions
  • Case (b) exhibits the strongest possible
    association
  • Accept Credit Card
  • The difference of proportions is 100

54
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

55
Example Do Student Stress and Depression Depend
on Gender?
56
Example Do Student Stress and Depression Depend
on Gender?
  • Which response variable, stress or depression,
    has the stronger sample association with gender?

57
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

58
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

59
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)

60
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

61
Example Relative Risk for Seat Belt Use and
Outcome of Auto Accidents
62
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

63
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

64
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

65
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

66
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

67
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

68
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

69
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

70
Section 10.4
  • How Can Residuals Reveal the Pattern of
    Association?

71
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

72
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

73
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

74
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

75
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

76
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

77
Example Standardized Residuals for Religiosity
and Gender
  • To what extent do you consider yourself a
    religious person?

78
Example Standardized Residuals for Religiosity
and Gender
79
Example Standardized Residuals for Religiosity
and Gender
  • Interpret the standardized residuals in the table

80
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

81
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

82
Section 10.5
  • What if the Sample Size is Small? Fishers Exact
    Test

83
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

84
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

85
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

86
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

87
Example Tea Tastes Better with Milk Poured
First?
  • Results

88
Example Tea Tastes Better with Milk Poured
First?
  • Analysis

89
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?

90
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
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