1. Biostatistics course Part 14 Analysis of binary paired data
Dr. Sc. Nicolas Padilla Raygoza
Department of Nursing and Obstetrics
Division Health Sciences and Engineering
University of Guanajuato
Campus Celaya-Salvatierra

2. Biosketch Medical Doctor by University Autonomous of Guadalajara.
Pediatrician by the Mexican Council of Certification on Pediatrics.
Postgraduate Diploma on Epidemiology, London School of Hygiene and Tropical Medicine, University of London.
Master Sciences with aim in Epidemiology, Atlantic International University.
Doctorate Sciences with aim in Epidemiology, Atlantic International University.
Associated Professor B, School of Nursing and Obstetrics of Celaya, university of Guanajuato.

3. Competencies The reader will know how show paired binary data.
He (she) will apply hypothesis test for paired binary data – McNemar’s Chi-squared.
He (she) will calculate confidence interval for paired binary data.
He (she) will obtain Odds Ratio and confidence interval for cases-controls paired studies.

4. Introduction
In Parts 12 and 13 of the biostatistics course, we knew, the methods for comparing two proportions estimated from independent samples.
If the observations in a study are not independent, we need to use different methods.
Often we use two types of studies that give rise to observations that are not independent:
Repeated observations in the same individual
Matched case-control studies

5. Example
Tuberculosis can be diagnosed to use a culture media and looking if Mycobacterium tuberculosis is growing.
In a experiment to compare two culture medias for the tuberculosis diagnosis, samples of expectoration from 100 patients were planted in the two medias.
The half of the sample was planted in media A and another half of sample, planted in media B.
In this study, the results of an individual are two observations that are matched to one another. Each result can be + or - to the tuberculosis bacillus.

6. Example
In a study, to examine the relation between breast cancer and oral contraceptives, women with a breast cancer were matched with women without breast cancer, selected from electoral registries.
This is an example of cases-controls study, where each individual with breast cancer is matched with an individual with similar age, for control the potential effect counding, of age.

7. Showing categorical paired data
To use Z test or Chi squared test with paired data is a mistake, because we do not take into account the paired nature of data.
Patient
Culture A
Culture B 1 - 2 3 + 4 5 6 7 8 9 10 11 12 13 14 15
Patient
Culture A
Culture B 16 + 17 18 - 19 20 21 22 23 24 25 26 27 28 29 30
Patient
Culture A
Culture B 31 - 32 + 33 34 35 36 37 38 39 40 41 42 43 44 45
Patient
Culture A
Culture B 46 + 47 - 48 49 50

8. Showing categorical paired data
Patient
Culture A
Culture B 51 - 52 53 + 54 55 56 57 58 59 60 61 62 63 64 65
Patient
Culture A
Culture B 66 + 67 68 - 69 70 71 72 73 74 75 76 77 78 79 80
Patient
CultureA
Culture B 81 - 82 + 83 84 85 86 87 88 89 90 91 92 93 94 95
Patient
Culture A
Culture B 96 + 97 - 98 99
100

9. Showing categorical paired data
The experiment compared the capacity of culture media to detect Mycobacterium tuberculosis.
The results were positive (+) or negative (-).
We have interest in to compare the samples positives of both culture media.
The table summarize the results
Culture media + -
Total A 64 36
100 B 44 56

10. Showing categorical paired data
From this, do you think that media A is better that media B to detect the tuberculosis bacilli? 
To make an adequate analysis, we need to compare the results with both media in each subject.
There are four combinations of results that can occur in each subject:
Combination
Media A
Media B
Pairs 1 + k 2 - r 3 s 4 m
We signalize: 
• The number of times that both media are positive = k
• The number of times that A is positive and B is negative = r
• The number of times that a is negative and B is positive = s
• The number of times that both media are negative = m

11. Showing categorical paired data
To compare the results of each subject, we need count how many times occur each combination.
AN easy form to show the calues is tabulate the results from a sample against another sample.
Media B + -
A + k r
k + r
A - s m
s + m
k + s
r + m N
We signalize: 
• The number of times that both media are positive = k
• The number of times that A is positive and B is negative = r
• The number of times that a is negative and B is positive = s
• The number of times that both media are negative = m

12. Showing categorical paired data
The pairs with same result are pairs with agreement, and they do not give any information on what media is better to detect bacilli.
Of the remaining results were different between the two media:
24 were positive for the A and negative for B.
4 were negative for A and positive for the B.
The pairs whose results were different between both media, are called discordant pairs.
Media B + -
A + 40 24 64
A - 4 32 36 44 56
100
Where N is total number of pairs, N = 100 in this table.
The 72 pairs that agree do not give us any information on what media is better to detect the bacilli.
For this reason, only we take into account pairs that do not agree, in the analysis.
The difference between paired proportions is from 64% - 44% = 20%. This mean that media A had 20% more of positive results than media B.
The next step is to comprobe the hypothesis that ther is not difference between paired proportions, taking into account the paired nature of data.

13. Hypothesis test for binary paired data
If there were no difference between the medias, we should expect similar numbers r and s, r ≈ s
We can use a call McNemar test to assess whether the difference between the numbers of discordant pairs is greater than what you would expect by chance.
To test the null hypothesis that there is no difference between the two proportions, we used the McNemar test:
(|r-s|-1)2
X2paired=
r + s
Subtracting 1 gives us a continuous correction.
The value obtained is refered to X2 distribution table, with one degree of freedom.

14. Hypothesis test for binary paired data
In the study of two culture media for tuberculosis bacilli:
24 were positives in media A and negatives in media B
4 were negatives in media A and positives in media B
(|r-s|-1)2 (|24-4|-1)
X2paired= = = = p<0.05
r + s
Rejected the null hypothesis of non-difference between media.
The obtained value is refered to X2 distribution table with one degree of freedom.

15. Confidence intervals for the difference of two paired proportions
We know thta the difference between proportions of paired data can be calculate by: r – s / N
Where: 
r and s are the number of discordant pairs
N is the total number of pairs
Standard error from the difference between paired proportions is:
√r +s
SE(p1-p2) = N

16. Confidence intervals from difference of two paired proportions
General formula to calculate 95% confidence interval is:
 Estimate ± 1.96 x SE
From the table of results of cultures from expectoration, with medias, A and B, we are using r and s values, and can calculate 95% confidence interval for paired proportions:
r-s / N ± √r +s/N = 24-4/100±1.96 √24+4/100 = 0.2±0.10 = 0.1 a 0.3 = 10% a 30%-
Confidence intervals from 0.1 to 0.3 mean that the percentage of positive cultures for the bacilli could be between 10% and 30% higher in media A than media B, in the population.

17. Odds Ratio for paired data
In case-control studies, usually, we want to evaluate the risk with the exposure at a risk factor; for these studies, we need an effect measure.
In case-control studies, we are using OR, that is a Ratio between odds of the exposure in the cases divided by odds of the exposure in controls.
Calculate of OR with matched data, is based in discordant pairs, the same that the difference between proportions od paired data.

18. Odds Ratios for paired data
Table of exposure in cases against exposure in controls
Controls
Cases Exposed Non-exposed
Exposed k r
Non-exposed s m 
k = number of pairs where the case and control were exposed
r = number of pairs where the case was exposed and the control was not exposed
s = number of pairs where the case was not exposed and the control was exposed.
m = number of pairs where cases and controls were not exposed.
Pairs of cases and controls that agree, they do not give any information on the risk associated with exposure.
This information is given by pairs where exposure between cases and controls,differ.

19. Odds Ratio for paired data
Odds Ratio is calculate as the Ratio of two groups of discordant pairs.
r cases exposed controls not exposed
OR = ---- =
s cases not exposed controls exposed  

20. Odds Ratio for paired data
The table show the results of a matched case-control study, designed to investigate the association between the use of oral contraceptive (OCC) and thromboembolism.
Controls
Cases Use OCC Not use OCC
Use OCC
Not use OCC
OR = 57/13 = 4.38

21. Confidence intervals for paired OR
To calculate confidence intervals is a little more complicated.
It is calculate using square root of the value of McNemar X2 test, instead of standard error.
95% confidence intervals for OR from apired data is:
OR1±1.96/ X

22. Odds Ratio for paired data
X²paired = 26.41 
Xpaired = 5.14
Then, 95% confidence interval is:
From / to / 5.14 to
2.5 to 7.7
In this study, the probability of case exposed to oral contraceptive i 4.4 times higher that the probability of controls exposed to oral contraceptive.
We have 95% of confidence that in the population, the probability of cases exposed is between 2.5 and 7.7 times higher that the probability of controls exposed. If the lower limit was below 1, the data should be consistent with the probability of exposure was higher in controls than in cases.

23. Bibliografía
1.- Last JM. A dictionary of epidemiology. New York, 4ª ed. Oxford University Press, 2001:173.
2.- Kirkwood BR. Essentials of medical ststistics. Oxford, Blackwell Science, 1988: 1-4.
3.- Altman DG. Practical statistics for medical research. Boca Ratón, Chapman & Hall/ CRC; 1991: 1-9.