1. Chapter 18 Cross-Tabulated Counts Part A
11/24/2018
Chapter 18 Cross-Tabulated Counts Part A
November 18
Basic Biostat

2. Chapter 18, Part A: 18.1 Types of Samples
18.2 Naturalistic and Cohort Samples
18.3 Chi-Square Test of Association
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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) relationship via a naturalistic sample?
A population was identified and sampled
Sample classified as CMV+ and CMV−
Disease occurrence (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 we 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.
Disease occurrence (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, N = 585
Cross-classify education level (categorical exposure) and smoking status (categorical disease)
Talley R-by-C table “cross-tab”
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10. Cross-tabulation (cont.)
Educ.
Smoke? + −
Tot HS 12 38 50 JC 18 67 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. Cross-tabulation of counts
For uniformity, we will always:
put the exposure variable in rows
put the disease variable in columns
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12. Exposure / Disease relationship
Use conditional proportions to describe relationships between exposure and disease
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13. Conditional Proportions Exposure / Disease Relationship
In naturalistic and cohort samples  row percents!
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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14. Example Prevalence of smoking by education:
Lower education associated with higher prevalence (negative association between education and smoking)
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15. Let group 1 represent the least exposed group
Relative Risks
Let group 1 represent the least exposed group
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16. Illustration: RRs
Note trend
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17. k Levels of Disease
Efficacy of Echinacea example. Randomized controlled clinical trial: echinacea vs. placebo in treatment of URI
Exposure ≡ Echinacea vs. placebo
Disease ≡ severity of illness
Source: JAMA 2003, 290(21),
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18. Row Percents for Echinacea Example
Echinacea group fared slightly worse than placebo group
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19. Chi-Square Test of Association
A. H0: no association in population Ha: association in population
B. Test statistic
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20. Observed Degree Smoke + 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
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21. Expected
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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22. Continuity Corrected Chi-Square
Pearson’s (“uncorrected”) chi-square
Yates’ continuity-corrected chi-square:
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23. Chi-Square Hand Calc.
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24. Chi-Square  P-value X2stat= 13.20 with 4 df
Table E  4 df row  bracket chi-square statistic  look up right tail (P-value) regions
Example bracket X2stat between (P = .025) and (P = .01)
.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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25. Illustration: X2stat= 13.20 with 4 df
The P-value = AUC in the tail beyond X2stat
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26. WinPEPI > Compare2 > F1
Input screen
row 5 not visible
Output
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27. 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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28. Chi-Square, cont.
Supplement chi-squares with descriptive stat. Chi-square statistics do not quantify effects
For 2-by-2 tables, chi-square and z tests produce identical P-values.
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29. Discussion and demo on power and sample size
For estimation
For testing
Power
Sample size
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