1. Chapter 18 Cross-Tabulated Counts
11/15/2018
Chapter 18 Cross-Tabulated Counts
November 18
Basic Biostat

2. In Chapter 18: 18.1 Types of Samples
18.2 Naturalistic and Cohort Samples
18.3 Chi-Square Test of Association
18.4 Test for Trend
18.5 Case-Control
18.6 Matched Pairs
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3. Types of Samples
I. Naturalistic Samples ≡ simple random sample or complete enumeration of the population
II. Purposive Cohorts ≡ select fixed number of individuals in each exposure group
III. Case-Control ≡ select fixed number of diseased and non-diseased individuals
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4. Naturalistic (Type I) Sample
Random sample of study base
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5. Naturalistic (Type I) Sample
Random sample of study base
How did we study CMV (the exposure) and restenosis (the disease) with a naturalistic sample?
A population was identified and sampled
The sample was classified as CMV+ and CMV−
The outcome (restenosis) was studied and compared in the groups.
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6. Purposive Cohorts (Type II sample)
Fixed numbers in exposure groups
How would I do study CMV and restenosis with a purposive cohort design?
A population of CMV+ individuals would be identified.
From this population, select, say 38, individuals.
A population of CMV− individuals would be identified.
From this population, select, say, 38 individuals.
The outcome (restenosis) would be studied and compared among the groups.
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7. Case-control (Type III sample)
Set number of cases and non-cases
How would I do study CMV and restenosis with a case-control design?
A population of patents who experienced restenosis (cases) would be identified.
From this population, select, say 38, individuals.
A population of patients who did not restenose (controls) would be identified.
From this population, select, say, 38 individuals.
The exposure (CMV) would be studied and compared among the groups.
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8. Case-Control (Type III sample)
Set number of cases and non-cases
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9. Naturalistic Sample Illustrative Example
Edu.
Smoke? + −
Tot HS 12 38 50 JC 18 67 85
JC+ 27 95
122 UG 32
239
271
Grad 5 52 57
Total 94
491
585
SRS of 585
Cross-classify education level (categorical exposure) and smoking status (categorical disease)
Talley R rows by C columns “cross-tab”
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10. Table Margins Row margins Total Educ. Smoke? + − Tot HS 12 38 50 JC 1867 85
Some 27 95
122 UG 32
239
271
Grad 5 52 57
Total 94
491
585
Row margins
Total
Column margins
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11. Naturalistic & Cohort Samples
R-by-2 Table + −
Total
Grp 1 a1 b1 n1
Grp 2 a2 b2 n2 ↓
Grp R aR bR nR m1 m2 N
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12. Example Prevalence of smoking by education:
Example, prevalence group 1:
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13. Let group 1 represent the least exposed group
Relative Risks
Let group 1 represent the least exposed group
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14. Illustration: RRs
Note trend
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15. k Levels of Response
Efficacy of Echinacea. Randomized controlled clinical trial: echinacea vs. placebo in treatment of URI in children.
Response variable ≡ severity of illness
Source: JAMA 2003, 290(21),
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16. Echinacea Example Purposive cohorts  row percents
% severe, echinacea = 48 / 329 = .146 = 14.6%
% severe, placebo = 40 / 367 = .109 = 10.9%
Echinacea group fared worse than placebo
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17. §18.3 Chi-Square Test of Association
A. Hypotheses. H0: no association in population Ha: association in population
B. Test statistic – by hand or computer
C. P-value. Via Table E or software
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18. Chi-Square Example H0: no association in the population
Ha: association in the population
Data
Degree
Smoke +
Smoke −
Tot
HighS 12 38 50 JC 18 67 85
Some 27 95
122 UG 32
239
271
Grad 5 52 57
Total 94
491
585
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19. Expected Frequencies (under H0)
Smoke +
Smoke −
Total
HighS
(50 × 94) ÷ 585 = 8.034
(50 × 491) ÷ 585 = 50 JC
13.658
71.342 85
Some
19.603
122 UG
43.545
271
Grad
9.159
47.841 57 94
491
585
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20. Chi-Square Hand Calc.
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21. Chi-Square  P-value X2stat= 13.20 with 4 df
Table E  4 df row  bracket chi-square statistic  look up tail regions (approx P-value)
Example (below) shows bracketing values for example are (P = .025) and (P = .01)  thus .01 < P < .025
Right tail
0.975
0.25
0.20
0.15
0.10
0.05
0.025
0.01
df =4
0.48
5.39
5.99
6.74
7.78
9.49
11.14
13.28
14.86
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22. Illustration: X2stat= 13.20 with 4 df
The P-value = AUC in the tail beyond X2stat
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23. WinPEPI > Compare2 > F1
Input screen
row 5 not visible
Output
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24. Continuity Corrected Chi-Square
Two different chi-square statistics
Both used in practice
Pearson’s (“uncorrected”) chi-square
Yates’ continuity-corrected chi-square:
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25. Chi-Square, cont.
How the chi-square works. When observed values = expected values, the chi-square statistic is 0. When the observed minus expected values gets large  evidence against H0 mounts
Avoid chi-square tests in small samples. Do not use a chi-square test when more than 20% of the cells have expected values that are less than 5.
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26. Chi-Square, cont.
3. Supplement chi-squares with measures of association. Chi-square statistics do not quantify effects (need RR, RD, or OR)
4. Chi-square and z tests (Ch 17) produce identical P-values. The relationship between the statistics is:
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27. 18.4 Test for Trend
See pp. 431 – 436
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28. §18.5 Case-Control Sampling
Identify all cases in source population
Randomly select non-cases (controls) from source population
Ascertain exposure status of subjects
Cross-tabulate
Efficient way to study rare outcomes
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29. Case-Control Sampling
Select non-case at random when case occurs
Miettinen. Am J Epidemiol 1976; 103,
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30. Odds Ratio OR stochastically = RR
Cross-tabulate exposure (E) & disease (D) D+ D− E+ a1 b1 E− a2 b2
Calculate
cross-product ratio
OR stochastically = RR
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31. Relative risk associated with exposure
BD1 Data
Cases: esophageal cancer
Controls: noncases selected at random from electoral lists
Exposure: alcohol consumption dichotomized at 80 gms/day
Relative risk associated with exposure
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32. (1– α)100% CI for the OR
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33. 90% CI for OR – Example D+ D− E+ 96
109 E−
104
666
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34. WinPEPI > Compare2 > A.
Data entry
Output
WinPEPI’s Mid-P interval similar to ours
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35. Ordinal Exposure
Break data up into multiple tables, using the least exposed level as baseline each time
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36. Ordinal Exposure
Dose-response
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37. 18.6 Matched Pairs
Cohort matched pairs: each exposed individual uniquely matched to non-exposed individual
Case-control matched pairs: each case uniquely matched to a control
Controls for matching (confounding) factor
Requires special matched-pair analysis
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38. Matched-Pairs, Cohort Exposed D+ Exposed D− Non-exp D+ a b Non-exp D−
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39. Matched-Pairs, Case-Control
Case E+
Case E−
Control E+ a b
Control E− c d
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40. Matched-Pairs Case-Cntl Example
Cases = colon polyps; Controls = no polyps
Exposure = low fruit & veg consumption
Case E+
Case E−
Cntl E+ ? 24
Cntl E− 45
88% higher risk w/ low fruit/veg consumption
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41. Matched-Pairs - Example
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42. WinPEPI > PairEtc > A.
Input
Output
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43. Hypothesis Test Matched Pairs
A. H0: OR = 1
B. McNemar’s test (z or chi-square)
C. P-value from z stat
Avoid if fewer than 5 discordancies expected
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44. Twins Mortality Example
Smoker D+
Smoker D−
Non-smoker D+ 5
Non-smoker D− 17
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