18: Cross-Tabulated Counts

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Chapter 18Cross-Tabulated Counts
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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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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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Naturalistic (Type I) Sample
Random sample of study base
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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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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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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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Case-Control (Type III sample)
Set number of cases and non-cases
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Naturalistic Sample Illustrative Example

• 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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Table Margins
Row margins
Total
Column margins
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Exposure and disease relationship
Use these conditional proportions to describe
relationships exposure and disease

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Naturalistic Cohort Samples
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Example
Prevalence of smoking by education
Example, prevalence group 1
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Relative Risks
Let group 1 represent the least exposed group
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Illustration RRs
Note trend
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Odds Ratios (optional)

• Odds ratio of successes to failures
• Odds ratios associated with exposure level i
• Interpretation. OR1 implies no association

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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), 2824-30
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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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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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Chi-Square Example

• H0 no association in the population
• Ha association in the population
• Data

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Expected Frequencies (under H0)
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Chi-Square Hand Calc.
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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 11.14 (P .025) and 13.28 (P .01)
  ? thus .01 lt P lt .025

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Illustration X2stat 13.20 with 4 df
The P-value AUC in the tail beyond X2stat
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WinPEPI gt Compare2 gt F1
Input screen row 5 not visible
Output
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Continuity Corrected Chi-Square

• Two different chi-square statistics
• Both used in practice
• Pearsons (uncorrected) chi-square
• Yates continuity-corrected chi-square

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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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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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18.4 Test for Trend

• See pp. 431 436

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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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Case-Control Sampling
Select non-case at random when case occurs
Miettinen. Am J Epidemiol 1976 103, 226-235.
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Odds Ratio
Cross-tabulate exposure (E) disease (D)
Calculate
cross-product ratio
OR stochastically RR
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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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(1 a)100 CI for the OR
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90 CI for OR Example
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WinPEPI gt Compare2 gt A.
Data entry
Output
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Ordinal Exposure
Break data up into multiple tables, using the
least exposed level as baseline each time
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Ordinal Exposure
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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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Matched-Pairs, Cohort

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Matched-Pairs, Case-Control

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Matched-Pairs Case-Cntl Example
Cases colon polyps Controls no
polyps Exposure low fruit veg consumption
88 higher risk w/ low fruit/veg consumption
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Matched-Pairs - Example
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WinPEPI gt PairEtc gt A.
Input
Output
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Hypothesis TestMatched Pairs

• A. H0 OR 1
• B. McNemars test (z or chi-square)
• C. P-value from z stat

Avoid if fewer than 5 discordancies expected
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Twins Mortality Example