Introduction to Categorical Data Analysis July 22, 2004

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Introduction to Categorical
DataAnalysisJuly 22, 2004
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Categorical data

• The t-test, ANOVA, and linear regression all
  assumed outcome variables that were continuous
  (normally distributed).
• Even their non-parametric equivalents assumed at
  least many levels of the outcome (discrete
  quantitative or ordinal).
• We havent discussed the case where the outcome
  variable is categorical.

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Types of Variables a taxonomy
Categorical
Quantitative
continuous
discrete
ordinal
nominal
binary
2 categories more categories
order matters numerical
uninterrupted
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Overview of statistical tests

• Independent variablepredictor
• Dependent variableoutcome
• e.g., BMD pounds age amenorrheic (1/0)

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Difference in proportions

• Example You poll 50 people from random
  districts in Florida as they exit the polls on
  election day 2004. You also poll 50 people from
  random districts in Massachusetts. 49 of
  pollees in Florida say that they voted for Kerry,
  and 53 of pollees in Massachusetts say they
  voted for Kerry. Is there enough evidence to
  reject the null hypothesis that the states voted
  for Kerry in equal proportions?

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Null distribution of a difference in proportions
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Null distribution of a difference in proportions
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Answer to Example

• We saw a difference of 4 between Florida and
  Massachusetts
• Null distribution predicts chance variation
  between the two states of 10.
• P(our data/null distribution)P(Zgt.04/.10.4)gt.05
• Not enough evidence to reject the null.

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Chi-square testfor comparing proportions (of a
categorical variable) between groups
I. Chi-Square Test of Independence When both
your predictor and outcome variables are
categorical, they may be cross-classified in a
contingency table and compared using a chi-square
test of independence.   A contingency table
with R rows and C columns is an R x C contingency
table.
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Example

• Asch, S.E. (1955). Opinions and social pressure.
  Scientific American, 193, 31-35.

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The Experiment

• A Subject volunteers to participate in a visual
  perception study.
• Everyone else in the room is actually a
  conspirator in the study (unbeknownst to the
  Subject).
• The experimenter reveals a pair of cards

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The Task Cards
Standard line
Comparison lines A, B, and C
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The Experiment

• Everyone goes around the room and says which
  comparison line (A, B, or C) is correct the true
  Subject always answers last after hearing all
  the others answers.
• The first few times, the 7 conspirators give
  the correct answer.
• Then, they start purposely giving the (obviously)
  wrong answer.
• 75 of Subjects tested went along with the
  groups consensus at least once.

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Further Results

• In a further experiment, group size (number of
  conspirators) was altered from 2-10.
• Does the group size alter the proportion of
  subjects who conform?

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The Chi-Square test

 
 
 
Apparently, conformity less likely when less or
more group members
 
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• 20 50 75 60 30 235 conformed
• out of 500 experiments.
• Overall likelihood of conforming 235/500 .47

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Expected frequencies if no association between
group size and conformity

 
 
 
 
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• Do observed and expected differ more than
  expected due to chance?

 
 
 
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Chi-Square test
Rule of thumb if the chi-square statistic is
much greater than its degrees of freedom,
indicates statistical significance. Here 85gtgt4.
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The Chi-Square distributionis sum of squared
normal deviates
The expected value and variance of a
chi-square E(x)df   Var(x)2(df)
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Chi-Square test
Rule of thumb if the chi-square statistic is
much greater than its degrees of freedom,
indicates statistical significance. Here 85gtgt4.
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Caveat

• When the sample size is very small in any cell
  (lt5), Fischers exact test is used as an
  alternative to the chi-square test.

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Example of Fishers Exact Test
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Fishers Tea-tasting experiment
Claim Fishers colleague (call her Cathy)
claimed that, when drinking tea, she could
distinguish whether milk or tea was added to the
cup first. To test her claim, Fisher designed
an experiment in which she tasted 8 cups of tea
(4 cups had milk poured first, 4 had tea poured
first). Null hypothesis Cathys guessing
abilities are no better than chance. Alternatives
hypotheses Right-tail She guesses right more
than expected by chance. Left-tail She guesses
wrong more than expected by chance
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Fishers Tea-tasting experiment
Experimental Results
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Fishers Exact Test
Step 1 Identify tables that are as extreme or
more extreme than what actually happened Here
she identified 3 out of 4 of the
milk-poured-first teas correctly. Is that good
luck or real talent? The only way she could have
done better is if she identified 4 of 4 correct.
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Fishers Exact Test
Step 2 Calculate the probability of the tables
(assuming fixed marginals)
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Step 3 to get the left tail and right-tail
p-values, consider the probability mass
function Probability mass function of X, where
X the number of correct identifications of the
cups with milk-poured-first
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SAS code and outputfor generating Fishers Exact
statistics for 2x2 table
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data tea input MilkFirst GuessedMilk
Freq datalines 1 1 3 1 0 1 0 1 1 0 0
3 run data tea Fix quirky reversal of SAS 2x2
tables set tea MilkFirst1-MilkFirst Guessed
Milk1-GuessedMilkrun proc freq
datatea tables MilkFirstGuessedMilk
/exact weight freqrun
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SAS output
Statistics for Table of
MilkFirst by GuessedMilk
Statistic DF Value
Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Chi-Square 1 2.0000
0.1573 Likelihood Ratio
Chi-Square 1 2.0930 0.1480
Continuity Adj. Chi-Square 1
0.5000 0.4795
Mantel-Haenszel Chi-Square 1 1.7500
0.1859 Phi Coefficient
0.5000
Contingency Coefficient 0.4472
Cramer's V
0.5000 WARNING 100
of the cells have expected counts less
than 5. Chi-Square may not be
a valid test.
Fisher's Exact Test
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Cell (1,1) Frequency (F)
3 Left-sided
Pr lt F 0.9857
Right-sided Pr gt F 0.2429
Table Probability (P)
0.2286 Two-sided
Pr lt P 0.4857
Sample Size 8
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Introduction to the 2x2 Table
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Introduction to the 2x2 Table
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Cohort Studies
Disease
Disease-free
Target population
Disease
Disease-free
TIME
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The Risk Ratio, or Relative Risk (RR)
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Hypothetical Data

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Case-Control Studies

• Sample on disease status and ask retrospectively
  about exposures (for rare diseases)
• Marginal probabilities of exposure for cases and
  controls are valid.
• Doesnt require knowledge of the absolute risks
  of disease
• For rare diseases, can approximate relative risk

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Case-Control Studies
Exposed in past

• Disease
• (Cases)

Not exposed
Target population
Exposed
No Disease (Controls)
Not Exposed
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The Odds Ratio (OR)
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The Odds Ratio
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Properties of the OR (simulation)
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Properties of the lnOR
Standard deviation
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Hypothetical Data
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30
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Example Cell phones and brain tumors
(cross-sectional data)
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Same data, but use Chi-square testor Fischers
exact
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Same data, but use Odds Ratio