1. Analysis of Categorical Data
Dr Siti Azrin Binti Ab Hamid
Unit Biostatistics and Research Methodology

2. Outline
Types of categorical analysis
Steps to analysis

3. Overview univariable analysis
Dependent variable
Independent variable
Number of groups in independent variable
Parametric test
Non parametric test
Numerical (one) -
One sample t
Sign test
Categorical
2 groups (independent)
Independent t
Mann Whitney
2 groups (dependent)
Paired t
Signed rank test
> 2 groups (independent)
One way ANOVA
Kruskal Wallis
(2 groups)
Chi square test
Fisher exact test
McNemar test

4. Introduction
Categorical data analysis deals with discrete data that can be organized into categories.
The data are organized into a contingency table.

5. Types of categorical data analysis
Statistical tests
One proportion
Chi-square goodness of fit
Two proportion
Independent sample
Pearson chi-square / Fisher exact
Dependent sample
McNemar test
Stratified sampling to control confounder
Mantel-Haenszel test

6. Hypothesis testing Step 1: State the hypotheses Step 2:
Set the significance level
Step 3:
Check the assumptions
Step 4:
Perform the statistical analysis
Step 5:
Make interpretation
Step 6:
Draw conclusion

7. Contingency table Consists of two columns and two rows.
Cells are labeled A through D.
Columns and rows are added for labels.
Row: independent variable / exposure / risk factors
Column: dependent variable / outcome

8. Example of contingency table
CHD
present
CHD absent
Total
Smoker
138 32
170
Non-smoker
263
105
368
137
401
538

9. Pearson Chi-square
To test the association between two categorical variables
Independent sample
Result of test:
Not significant: no association
Significant: an association

10. Research Question
Does estrogen receptor associated with breast cancer status?
Data: Breast cancer.sav

11. Step 1: State the hypothesis
HO: There is no association between estrogen receptor and breast cancer status.
HA: There is an association between estrogen receptor and breast cancer status.

12. Step 2: Set the significance level
α = 0.05

13. Step 3: Check the assumption
Two variables are independent
Two variables are categorical
Expected count of < 5
- > 20%: Fisher exact test
- < 20%: Pearson Chi-square
Expected count = Row total x Column total
Grand total
Variable
Breast Ca
Total
Died
Alive
ER - ve
310 28
338
ER + ve
508 23
531
818 51
869

14. Step 3: Check the assumption
Variable
Breast Ca
Total
Died
Alive
ER - ve
310
E = 318.2 28
E = 19.8
338
ER + ve
508
E = 499.8 23
E = 31.2
531
818 51
869

15. Step 4: Statistical test
Calculate the Chi-square value
x2 = ∑((O – E)2/ E)
= 5.897
df = (R-1)(C-1)
= (2-1)(2-1)
= 1
Between 0.01 – 0.02

16. Step 4: Statistical test1 5 3 7 2 6 8 10 9

17. Step 5: Interpretation p value = 0.016
< 0.05 – reject HO, accept HA

18. Step 6: Conclusion
There is significant association between estrogen receptor and breast cancer status using Pearson Chi-square test (p = 0.016).

19. Fisher’s Exact Test
To test the association between two categorical variables
Independent sample
Sample sizes are small

20. Research Question Does gender associated with coronary heart disease?
Data: CHD data.sav

21. Step 1: State the hypothesis
HO: There is no association between gender and coronary heart disease.
HA: There is an association between gender and coronary heart disease.

22. Step 2: Set the significance level
α = 0.05

23. Step 3: Check the assumption
Two variables are independent
Two variables are categorical
Expected count of < 5
- > 20%: Fisher exact test
- < 20%: Pearson Chi-square
Expected count = Row total x Column total
Grand total
Variable
Coronary Heart Disease
Total
Presence
Absent
Male 15 5 20
Female 10 25 30

24. Step 3: Check the assumption
Variable
Coronary Heart Disease
Total
Presence
Absent
Male 15
E = 16.7 5
E = 3.3 20
Female 10
E = 8.3
E = 1.7 25 30
2 cells (50%) – expected count < 5

25. Step 4: Statistical test
Calculate the Chi-square value
x2 = ∑((O – E)2/ E) =
df = (R-1)(C-1)
= (2-1)(2-1)
= 1
Between 0.1 – 0.05

26. Step 4: Statistical test1 5 3 7 6 2 8 10 9

27. Step 5: Interpretation
p value = 0.140
> 0.05 – accept HO

28. Step 6: Conclusion
There is no significant association between gender and coronary heart disease using Fisher’s Exact test (p = 0.140).

29. McNemar Test Categorical data Dependent sample - Matched sample
- Cross over design
- Before & after (same subject)
To determine whether the row and column marginal frequencies are equal (marginal homogeneity)

30. Hypotheses
Null hypothesis of marginal homogeneity states the two marginal probabilities for each outcome are the same
HO : PB = PC
HA : PB ≠ PC
A & D = concordant pair
B & C = discordant pair
Discordant pair is pair of different outcome

31. Research Question
Does type of mastectomy associated with 5-year survival proportion in patients with breast cancer?
The sample were breast cancer patients
- matched for age (same decade of age)
- same clinical condition
Data: breast ca.sav

32. Step 1: State the hypothesis
HO: There is no association between type of mastectomy and 5-year survival proportion in patients with breast cancer.
HA: There is an association between type of mastectomy and 5-year survival proportion in patients with breast cancer.

33. Step 2: Set the significance level
α = 0.05

34. Step 3: Check the assumption
Two variables are dependent
Two variables are categorical

35. Step 4: Statistical test
x2 = (|b-c|-1)2/(b + c)
= (|0 – 8| - 1)2 / (0 +8)
=6.125
df = (R-1)(C-1)
= (2-1)(2-1)
= 1
Calculated x2 > tabulated x2
*x2 = (|b-c|-0.5)2/(b + c)

36. Step 4: Statistical test3 6 2 1 9 7 4 5 8

37. Step 5: Interpretation p value = 0.008
< 0.05 – reject HO, accept HA

38. Step 6: Conclusion
There is an association between type of mastectomy and 5-year survival proportion in patients with breast cancer using McNemar test (p = 0.008).

39. Cochran Mantel-Haenszel Test
Test is a method to compare the probability of an event among independent groups in stratified samples.
The stratification factor can be study center, gender, race, age groups, obesity status or disease severity.
Gives a stratified statistical analysis of the relationship between exposure and disease, after controlling for a confounder (strata variables).
The data are arranged in a series of associated 2 × 2 contingency tables.

40. Research Question
Does the type of treatment associated with response of treatment among migraine patients after controlling for gender?
Confounder: gender
Active
Placebo
Female
No of patients 27 25
No of better
response 16 5
Male 28 26 12 7

41. Step 1: 2x2 contingency table
Better
Same
Total
Reasons of failure
Strata 1
Female
Active 16 11 27
Placebo 5 20 25
Strata 2
Male 12 28 7 19 26

42. Step 2: Check the assumption
Random sampling
Stratified sampling

43. Step 3: State the hypothesis
HO: There is no association between type of treatment and response of treatment among female and male migraine patients.
HA: There is an association between type of treatment and response of treatment among female and male migraine patients.

44. Step 4: Statistical test
Compute the expected frequency from each stratum
ei = (ai + bi)(ai + ci) ni
Compute each stratum
vi = (ai +bi)(ci +di)(ai +ci)(bi + di)
ni2(ni -1)
Compute Mantel-Haenszel statistics
x2MH = ∑(ai –ei)2
∑vi

45. Step 4: Statistical test
Compute the expected frequency from each stratum
ei = (ai + bi)(ai + ci) ni
e1 = (16 +11)(16+ 5) 52 =
e2 = (12 +16)(12+ 7) 54 =

46. Step 4: Statistical test
Compute each stratum
vi = (ai +bi)(ci +di)(ai +ci)(bi + di)
ni2(ni -1)
v1 = ( )(5 + 20)(16 + 5)(11+20)
(52)2(52-1) =
v2 = ( )(7 + 19)(12 + 7)(16+19)
(54)2(54-1) =

47. Step 4: Statistical test
Compute Mantel-Haenszel statistics
x2MH = (∑ai –∑ei)2
∑vi
= (( ) - ( ))2 =
= 8.31

48. Step 4: Statistical test
Compute odd ratio
ORMH = ∑(ai di/ ni)
∑(bi ci/ ni)
= (16 x 20/ 52) + (12 x 19 / 54)
(11 x 5/ 52) + (16 x 7/ 54
= 3.313

49. Step 4: Statistical test
Data: Migraine.sav 1 3 2 4 6 5

50. Step 5: Interpretation Compute Mantel-Haenszel statistics
x2MH = (∑ai –∑ei)2
∑vi
= (( ) - ( ))2 =
= 8.31
Calculated value > tabulated value
Reject HO

51. Step 5: Interpretation
HO = OR1 = OR2
Association homogenous
*Tarone’s - adjusted
HO = OR1 = 1
HO = OR2 = 1
Conditionally independent
The large p-value for the Breslow-Day test (p = 0.222) indicates no significant gender difference in the odds ratios.

52. Step 6: Conclusion
There is significant association between type of treatment and response of treatment among female and male migraine patients (p = 0.004).
We estimate that female patients and male patients who receive active treatment are 3.33 times more likely to have better symptoms in migraine for any reason than patients who receive placebo.