1. Categorical Data Analysis
Chapter 10
Categorical Data Analysis

2. Inference for a Single Proportion (p)
Goal: Estimate proportion of individuals in a population with a certain characteristic (p). This is equivalent to estimating a binomial probability
Sample: Take a SRS of n individuals from the population and observe y that have the characteristic. The sample proportion is y/n and has the following sampling properties:

3. Large-Sample Confidence Interval for p
Take SRS of size n from population where p is true (unknown) proportion of successes.
Observe y successes
Set confidence level (1-a) and obtain za/2 from z-table

4. Example - Ginkgo and Azet for AMS
Study Goal: Measure effect of Ginkgo and Acetazolamide on occurrence of Acute Mountain Sickness (AMS) in Himalayan Trackers
Parameter: p = True proportion of all trekkers receiving Ginkgo&Acetaz who would suffer from AMS.
Sample Data: n=126 trekkers received G&A, y=18 suffered from AMS

5. Wilson-Agresti-Coull Method
For moderate to small sample sizes, large-sample methods may not work well wrt coverage probabilities
Simple approach that works well in practice:
Adjust observed number of Successes (y) and sample size (n)

6. Example: Lister’s Tests with Antiseptic
Experiments with antiseptic in patients with upper limb amputations (John Lister, circa 1870)
n=12 patients received antiseptic y=1 died

7. Sample Size for Margin of Error = E
Goal: Estimate p within E with 100(1-a)% Confidence
Confidence Interval will have width of 2E

8. Significance Test for a Proportion
Goal test whether a proportion (p) equals some null value p0 H0: p=p0
Large-sample test works well when np0 and n(1-p0)  5

9. Ginkgo and Acetaz for AMS
Can we claim that the incidence rate of AMS is less than 25% for trekkers receiving G&A?
H0: p= Ha: p < 0.25
Strong evidence that incidence rate is below 25% (p < 0.25)

10. Comparing Two Population Proportions
Goal: Compare two populations/treatments wrt a nominal (binary) outcome
Sampling Design: Independent vs Dependent Samples
Methods based on large vs small samples
Contingency tables used to summarize data
Measures of Association: Absolute Risk, Relative Risk, Odds Ratio

11. Contingency Tables
Tables representing all combinations of levels of explanatory and response variables
Numbers in table represent Counts of the number of cases in each cell
Row and column totals are called Marginal counts

12. 2x2 Tables - Notation n1+n2 (n1+n2)-(y1+y2) y1+y2 Outcome Total n2
Group 2 n1
n1-y1 y1
Group 1
Group
Absent
Present

13. Example - Firm Type/Product Quality
172
134 38
Outcome
Total 84 79 5
Vertically
Integrated 88 55 33
Not
Group
Low
Quality
High
Groups: Not Integrated (Weave only) vs Vertically integrated (Spin and Weave) Cotton Textile Producers
Outcomes: High Quality (High Count) vs Low Quality (Count)
Source: Temin (1988)

14. Notation
Proportion in Population 1 with the characteristic of interest: p1
Sample size from Population 1: n1
Number of individuals in Sample 1 with the characteristic of interest: y1
Sample proportion from Sample 1 with the characteristic of interest:
Similar notation for Population/Sample 2

15. Example - Cotton Textile Producers
p1 - True proportion of all Non-integretated firms that would produce High quality
p2 - True proportion of all vertically integretated firms that would produce High quality

16. Notation (Continued)
Parameter of Primary Interest: p1-p2, the difference in the 2 population proportions with the characteristic (2 other measures given below)
Estimator:
Standard Error (and its estimate):
Pooled Estimated Standard Error when p1=p2=p:

17. Cotton Textile Producers (Continued)
Parameter of Primary Interest: p1-p2, the difference in the 2 population proportions that produce High quality output
Estimator:
Standard Error (and its estimate):
Pooled Estimated Standard Error when p1=p2=p:

18. Significance Tests for p1-p2
Deciding whether p1=p2 can be done by interpreting “plausible values” of p1-p2 from the confidence interval:
If entire interval is positive, conclude p1 > p2 (p1-p2 > 0)
If entire interval is negative, conclude p1 < p2 (p1-p2 < 0)
If interval contains 0, do not conclude that p1  p2
Alternatively, we can conduct a significance test:
H0: p1 = p2 Ha: p1  p2 (2-sided) Ha: p1 > p2 (1-sided)
Test Statistic:
RR: |zobs|  za/2 (2-sided) zobs  za (1-sided)
P-value: 2P(Z|zobs|) (2-sided) P(Z zobs) (1-sided)

19. Example - Cotton Textile Production
Again, there is strong evidence that non-integrated performs are more likely to produce high quality output than integrated firms

20. Fisher’s Exact Test
Method of testing for testing whether p2=p1 when one or both of the group sample sizes is small
Measures (conditional on the group sizes and number of cases with and without the characteristic) the chances we would see differences of this magnitude or larger in the sample proportions, if there were no differences in the populations

21. Example – Echinacea Purpurea for Colds
Healthy adults randomized to receive EP (n1=24) or placebo (n2=22, two were dropped)
Among EP subjects, 14 of 24 developed cold after exposure to RV-39 (58%)
Among Placebo subjects, 18 of 22 developed cold after exposure to RV-39 (82%)
Out of a total of 46 subjects, 32 developed cold
Out of a total of 46 subjects, 24 received EP
Source: Sperber, et al (2004)

22. Example – Echinacea Purpurea for Colds
Conditional on 32 people developing colds and 24 receiving EP and 22 receiving placebo, the following table gives the outcomes that would have been as strong or stronger evidence that EP reduced risk of developing cold (1-sided test). P-value from SPSS is .079 (next slide).

23. Example - SPSS Output

24. McNemar’s Test for Paired Samples
Common subjects (or matched pairs) being observed under 2 conditions (2 treatments, before/after, 2 diagnostic tests) in a crossover setting
Two possible outcomes (Presence/Absence of Characteristic) on each measurement
Four possibilities for each subject/pair wrt outcome:
Present in both conditions
Absent in both conditions
Present in Condition 1, Absent in Condition 2
Absent in Condition 1, Present in Condition 2

25. McNemar’s Test for Paired Samples

26. McNemar’s Test for Paired Samples
Data: n12 = # of pairs where the characteristic is present in condition 1 and not 2 and n21 # where present in 2 and not 1
H0: Probability the outcome is Present is same for the 2 conditions (p1 = p2)
HA: Probabilities differ for the 2 conditions (p1 ≠ p2)

27. Example - Reporting of Silicone Breast Implant Leakage in Revision Surgery
Subjects women having revision surgery involving silicone gel breast implants
Conditions (Each being observed on all women)
Self Report of Presence/Absence of Rupture/Leak
Surgical Record of Presence/Absence of Rupture/Leak
Source: Brown and Pennello (2002), “Replacement Surgery and Silicone Gel Breast Implant Rupture”, Journal of Women’s Health & Gender-Based Medicine, Vol. 11, pp

28. Example - Reporting of Silicone Breast Implant Leakage in Revision Surgery
H0: Tendency to report ruptures/leaks is the same for self reports and surgical records
HA: Tendencies differ

29. Multinomial Experiment / Distribution
Extension of Binomial Distribution to experiments where each trial can end in exactly one of k categories
n independent trials
Probability a trial results in category i is pi
ni is the number of trials resulting in category I
p1+…+pk = 1
n1+…+nk = n

30. Multinomial Distribution / Test for Cell Probabilities

31. Goodness of Fit Test for a Probability Distribution
Data are collected and wish to be determined whether it comes from a particular probability distribution (e.g. Poisson, Normal, Gamma)
Estimate any unknown model parameters (p estimates)
Break down the range of data values into k > p intervals (typically where ≥ 80% have expected counts ≥ 5) obtain observed (n) and expected (E) values for each interval

32. Associations Between Categorical Variables
Case where both explanatory (independent) variable and response (dependent) variable are qualitative
Association: The distributions of responses differ among the levels of the explanatory variable (e.g. Party affiliation by gender)

33. Contingency Tables
Cross-tabulations of frequency counts where the rows (typically) represent the levels of the explanatory variable and the columns represent the levels of the response variable.
Numbers within the table represent the numbers of individuals falling in the corresponding combination of levels of the two variables
Row and column totals are called the marginal distributions for the two variables

34. Example - Cyclones Near Antarctica
Period of Study: September,1973-May,1975
Explanatory Variable: Region (40-49,50-59,60-79) (Degrees South Latitude)
Response: Season (Aut(4),Wtr(5),Spr(4),Sum(8)) (Number of months in parentheses)
Units: Cyclones in the study area
Treating the observed cyclones as a “random sample” of all cyclones that could have occurred
Source: Howarth(1983), “An Analysis of the Variability of Cyclones around Antarctica and Their Relation to Sea-Ice Extent”, Annals of the Association of American Geographers, Vol.73,pp

35. Example - Cyclones Near Antarctica
For each region (row) we can compute the percentage of storms occuring during each season, the conditional distribution. Of the 1517 cyclones in the band, 370 occurred in Autumn, a proportion of 370/1517=.244, or 24.4% as a percentage.

36. Example - Cyclones Near Antarctica
Graphical Conditional Distributions for Regions

37. Guidelines for Contingency Tables
Compute percentages for the response (column) variable within the categories of the explanatory (row) variable. Note that in journal articles, rows and columns may be interchanged.
Divide the cell totals by the row (explanatory category) total and multiply by 100 to obtain a percent, the row percents will add to 100
Give title and clearly define variables and categories.
Include row (explanatory) total sample sizes

38. Independence & Dependence
Statistically Independent: Population conditional distributions of one variable are the same across all levels of the other variable
Statistically Dependent: Conditional Distributions are not all equal
When testing, researchers typically wish to demonstrate dependence (alternative hypothesis), and wish to refute independence (null hypothesis)

39. Pearson’s Chi-Square Test
Can be used for nominal or ordinal explanatory and response variables
Variables can have any number of distinct levels
Tests whether the distribution of the response variable is the same for each level of the explanatory variable (H0: No association between the variables
r = # of levels of explanatory variable
c = # of levels of response variable

40. Pearson’s Chi-Square Test
Intuition behind test statistic
Obtain marginal distribution of outcomes for the response variable
Apply this common distribution to all levels of the explanatory variable, by multiplying each proportion by the corresponding sample size
Measure the difference between actual cell counts and the expected cell counts in the previous step

41. Pearson’s Chi-Square Test
Notation to obtain test statistic
Rows represent explanatory variable (r levels)
Cols represent response variable (c levels)
n..
n.c …
n.2
n.1
Total
nr.
nrc
nr2
nr1 r
n2.
n2c
n22
n21 2
n1.
n1c
n12
n11 1 c

42. Pearson’s Chi-Square Test
Observed frequency (nij): The number of individuals falling in a particular cell
Expected frequency (Eij): The number we would expect in that cell, given the sample sizes observed in study and the assumption of independence.
Computed by multiplying the row total and the column total, and dividing by the overall sample size.
Applies the overall marginal probability of the response category to the sample size of explanatory category

43. Pearson’s Chi-Square Test
Large-sample test (at least 80% of Eij > 5)
H0: Variables are statistically independent (No association between variables)
Ha: Variables are statistically dependent (Association exists between variables)
Test Statistic:
P-value: Area above in the chi-squared distribution with (r-1)(c-1) degrees of freedom. (Critical values in Table 8)

44. Example - Cyclones Near Antarctica
Observed Cell Counts (nij):
Note that overall: (1876/9165)100%=20.5% of all cyclones occurred in Autumn. If we apply that percentage to the 1517 that occurred in the 40-49S band, we would expect (0.205)(1517)=310.5 to have occurred in the first cell of the table. The full table of Eij:

45. Example - Cyclones Near Antarctica
Computation of

46. Example - Cyclones Near Antarctica
H0: Seasonal distribution of cyclone occurences is independent of latitude band
Ha: Seasonal occurences of cyclone occurences differ among latitude bands
Test Statistic:
RR: cobs2  c.05,62 = 12.59
P-value: Area in chi-squared distribution with (3-1)(4-1)=6 degrees of freedom above 71.2
From Table 8, P(c222.46)=.001  P< .001

47. Likelihood Ratio Statistic
Note: The formula on page 512 of textbook is
incorrect

48. SPSS Output - Cyclone Example
P-value

49. Misuses of chi-squared Test
Expected frequencies too small (at least 80% of expected counts should be above 5, not necessary for the observed counts)
Dependent samples (the same individuals are in each row, see McNemar’s test)
Can be used for nominal or ordinal variables, but more powerful methods exist for when both variables are ordinal and a directional association is hypothesized

50. Residual Analysis
Once dependence has been determined from a chi-squared test, often interested in determining which cells contributed
Residual: fo-fe measures the difference between the observed and expected counts
Positive implies observed more than expected
Residual’s practical importance depends on level of fe
Adjusted Residual (computed for each cell):
Adjusted residuals above 3 in absolute value give strong evidence against independence in that cell

51. Example - Cyclones Near Antarctica
Adjusted residuals are computed in the following table.
Row proportion for Region 40-49S: 1517/9165=0.1655
Column Proportion for Season Autumn is: 1876/9165=0.2047

52. Ordinal Explanatory and Response Variables
Pearson’s Chi-square test can be used to test associations among ordinal variables, but more powerful methods exist
When theories exist that the association is directional (positive or negative), measures exist to describe and test for these specific alternatives from independence:
Gamma
Kendall’s tb

53. Concordant and Discordant Pairs
Concordant Pairs - Pairs of individuals where one individual scores “higher” on both ordered variables than the other individual
Discordant Pairs - Pairs of individuals where one individual scores “higher” on one ordered variable and the other individual scores “higher” on the other
C = # Concordant Pairs D = # Discordant Pairs
Under Positive association, expect C > D
Under Negative association, expect C < D
Under No association, expect C  D

54. Example - Alcohol Use and Sick Days
Alcohol Risk (Without Risk, Hardly any Risk, Some to Considerable Risk)
Sick Days (0, 1-6, 7)
Concordant Pairs - Pairs of respondents where one scores higher on both alcohol risk and sick days than the other
Discordant Pairs - Pairs of respondents where one scores higher on alcohol risk and the other scores higher on sick days
Source: Hermansson, et al (2003)

55. Example - Alcohol Use and Sick Days
Concordant Pairs: Each individual in a given cell is concordant with each individual in cells “Southeast” of theirs
Discordant Pairs: Each individual in a given cell is discordant with each individual in cells “Southwest” of theirs

56. Example - Alcohol Use and Sick Days

57. Measures of Association
Goodman and Kruskal’s Gamma:
Kendall’s tb:
When there’s no association between the ordinal variables, the population based values of these measures are 0. Statistical software packages provide these tests.

58. Example - Alcohol Use and Sick Days

59. Measures of Association
Absolute Risk (AR): p1-p2
Relative Risk (RR): p1 / p2
Odds Ratio (OR): o1 / o2 (o = p/(1-p))
Note that if p1 = p2 (No association between outcome and grouping variables):
AR=0
RR=1
OR=1

60. Relative Risk
Ratio of the probability that the outcome characteristic is present for one group, relative to the other
Sample proportions with characteristic from groups 1 and 2:

61. Relative Risk Estimated Relative Risk:
95% Confidence Interval for Population Relative Risk:

62. Relative Risk Interpretation
Conclude that the probability that the outcome is present is higher (in the population) for group 1 if the entire interval is above 1
Conclude that the probability that the outcome is present is lower (in the population) for group 1 if the entire interval is below 1
Do not conclude that the probability of the outcome differs for the two groups if the interval contains 1

63. Example - Concussions in NCAA Athletes
Units: Game exposures among college socer players
Outcome: Presence/Absence of a Concussion
Group Variable: Gender (Female vs Male)
Contingency Table of case outcomes:
Source: Covassin, et al (2003)

64. Example - Concussions in NCAA Athletes
There is strong evidence that females have a higher risk of concussion

65. Odds Ratio
Odds of an event is the probability it occurs divided by the probability it does not occur
Odds ratio is the odds of the event for group 1 divided by the odds of the event for group 2
Sample odds of the outcome for each group:

66. Odds Ratio Estimated Odds Ratio:
95% Confidence Interval for Population Odds Ratio

67. Odds Ratio Interpretation
Conclude that the probability that the outcome is present is higher (in the population) for group 1 if the entire interval is above 1
Conclude that the probability that the outcome is present is lower (in the population) for group 1 if the entire interval is below 1
Do not conclude that the probability of the outcome differs for the two groups if the interval contains 1

68. Osteoarthritis in Former Soccer Players
Units: 68 Former British professional football players and 136 age/sex matched controls
Outcome: Presence/Absence of Osteoathritis (OA)
Data:
Of n1= 68 former professionals, y1 =9 had OA, n1-y1=59 did not
Of n2= 136 controls, y2 =2 had OA, n2-y2=134 did not
Interval > 1
Source: Shepard, et al (2003)

69. Mantel-Haenszel Test / CI for Multiple Tables
Data collected from q studies or strata in 2x2 contingency tables with common groupings/outcomes
Each table has 4 cells: nh11, nh12, nh21, nh21 h=1,…,q
They can be combined for an overall Chi-square statistic or odds ratio and confidence Interval

70. Mantel-Haenszel Computations

71. Inter-Rater Agreement – Cohen’s Kappa
Two Raters rate the same items, typically on an ordinal scale
Goal: Measure Strength of their agreement above “chance”

72. Agreement Among Movie Reviewers
Reviews by Gene Siskel and Roger Ebert (160 movies between April, 1995 through September 1996)