Categorical Data

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Categorical Data
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Categorical Data Analysis

• To identify any association between two
  categorical data.

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Chi-Square Test

• Commonly denoted as ?2
• Useful in testing for independence between
  categorical variables (e.g. genetic association
  between cases / controls
• Assumptions
• Sufficiently large data in each cell in the
  cross-tabulation table.

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Small Cell Counts

• In general, require(a) Smallest expected count
  is 1 or more(b) At least 80 of the cells have
  an expected count of 5 or more
• Yates Continuity CorrectionProvides a better
  approximation of the test statistic when the data
  is dichotomous (2 ? 2)

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Goodness-of-fit Test

• Null hypothesis of a hypothesized distribution
  for the data.
• Expected frequencies calculated under the
  hypothesized distribution.
• For example The number of outbreaks of flu
  epidemics is charted over the period 1500 to
  1931, and the number of outbreaks each year is
  tabulated. The variable of interest counts the
  number of outbreaks occurring in each year of
  that 432 year period. E.g. there were 223 years
  with no flu outbreaks.

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Goodness-of-fit Test

• Hypotheses H0 Data follows a Poisson
  distribution with mean 0.692 H1 Data does not
  follow a Poisson distribution with mean 0.692
• Note Mean 0.692 is obtained from the sample
  mean.
• Expected frequency for X 0
• 432 ? P(X 0), where X Poisson(0.692)
• Test Statistic , with df (6 1).
• This yields a p-value of 0.99, indicating that
  we will almost certainly be wrong if we reject
  the null hypothesis.

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Test for Independence

• Most common usage for Pearsons Chi-square
  statistic.
• Expected frequencies calculated by
• Degrees of freedom (r 1) ? (c 1)

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Chi-Square Test
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Chi-Square Test
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Quantification of Effect

• ?2-test identifies whether there is significant
  association between the two categorical
  variables.
• But does not quantify the strength and direction
  of the association.
• Need odds ratio to do this.
• Odds ratio defines how many times more likely
  it is to be in one category compared to the
  other
• Example For the previous example on severe chest
  pain, males are about 1.4 times more likely to
  experience severe chest pains than females.

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Odds Ratio
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Exegesis on Epidemiology

• Case-Control Study
• Compare affected and unaffected individuals
• Usually retrospective in nature
• Temporal sequence cannot be established (timing
  for the onset of the disease)
• No information on population incidence of the
  disease
• Cohort Study
• Usually random sampling of subjects within the
  population
• Prospective, retrospective or both
• Long follow-up loss to follow-up
• Costly to conduct
• Temporal sequence can be established
• Provides information on population incidence of
  the disease

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Confidence Intervals of Odds Ratio

• Not straightforward to obtain confidence
  intervals of odds ratio (due to complexity in
  obtaining the variance)
• Straightforward to obtain the variance of the
  logarithm of odds ratio.
• Odds ratio is always reported together with the
  p-values (obtained from Pearsons Chi-square
  test), and the corresponding confidence
  intervals.

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Case Study on Lung Cancer and Smoking
Odds and Odds Ratio Odds Ratio (OR) (1301/56)/(1
205/152) 2.93 Pearsons Chi-square 47.985,
on df 1? p-value 0 Varlog(OR)
0.026 95 Confidence interval (2.14,
4.02)
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More Examples
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c2 TEST FOR TREND
ORsmoker 1.52 (0.88, 2.63), p
0.180 ORex-smoker 2.11 (1.00, 4.51), p
0.081 with non-smoker as reference category.
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Procedure for Categorical Data Analysis

• Summarise data using cross-tabulation tables,
  with percentages
• Perform a chi-square of independence to test for
  association between the two categorical variables
• Quantify any significant association using odds
  ratios
• Always report odds ratios with corresponding 95
  confidence interval

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Case Study on Lung Cancer and Smoking

• Chi-square test statistic 46.991
• p-value 7.13 ? 10-12
• Odds ratio 2.93
• 95 CI (2.14, 4.02)