How Can We Test whether Categorical Variables are Independent?

1
Section 10.2

• How Can We Test whether Categorical Variables are
  Independent?

2
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

3
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

4
How Do We Find the Expected Cell Counts?

• Expected Cell Count
• For a particular cell, the expected cell count
  equals

5
Example Happiness by Family Income
6
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

7
Example Happiness and Family Income
8
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)

9
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

10
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

11
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

12
The Chi-Squared Distribution
13
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)

14
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

15
The Chi-Squared Distribution
16
The Five Steps of the Chi-Squared Test of
Independence

• 1. Assumptions
• Two categorical variables
• Randomization
• Expected counts 5 in all cells

17
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)

18
The Five Steps of the Chi-Squared Test of
Independence

• 3. Test Statistic

19
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

20
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

21
Example Aspirin and Heart Attacks Revisited
22
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

23
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

24
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

25
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

26
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

27
Section 10.3

• How Strong is the Association?

28
The following is a table on Gender and Happiness The following is a table on Gender and Happiness The following is a table on Gender and Happiness The following is a table on Gender and Happiness
Gender Not Pretty Very
Females 163 898 502
Males 130 705 379

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

29
The following is a table on Gender and Happiness The following is a table on Gender and Happiness The following is a table on Gender and 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?
• 898
• 801.5
• 902
• 521

30
The following is a table on Gender and Happiness The following is a table on Gender and Happiness The following is a table on Gender and 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?
• 898
• 801.5
• 902 N(898705)/N(163898502)/N
• 521

31
The following is a table on Gender and Happiness The following is a table on Gender and Happiness The following is a table on Gender and Happiness The following is a table on Gender and Happiness
Gender Not Pretty Very
Females 163 898 502
Males 130 705 379

• Calculate the
• 1.75
• 0.27
• 0.98
• 10.34

32
The following is a table on Gender and Happiness The following is a table on Gender and Happiness The following is a table on Gender and Happiness The following is a table on Gender and Happiness
Gender Not Pretty Very
Females 163 898 502
Males 130 705 379

• 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

33
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

34
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

35
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

36
Measures of Association

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

37
Difference of Proportions

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

38
Difference of Proportions
39
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
40
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
41
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

42
Example Do Student Stress and Depression Depend
on Gender?
43
Example Do Student Stress and Depression Depend
on Gender?

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

44
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

Gender Yes No
Female 35 65
Male 16 84
45
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

Gender Yes No
Female 8 92
Male 6 94
46
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)

47
Example Relative Risk for Seat Belt Use and
Outcome of Auto Accidents
48
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

49
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

50
Section 10.4

• How Can Residuals Reveal the Pattern of
  Association?

51
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

52
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

53
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

54
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

55
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

56
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

57
Example Standardized Residuals for Religiosity
and Gender

• To what extent do you consider yourself a
  religious person?

58
Example Standardized Residuals for Religiosity
and Gender
59
Example Standardized Residuals for Religiosity
and Gender

• Interpret the standardized residuals in the table

60
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

61
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

62
Section 10.5

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

63
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

64
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

65
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

66
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

67
Example Tea Tastes Better with Milk Poured
First?

• Results

68
Example Tea Tastes Better with Milk Poured
First?

• Analysis

69
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?

70
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