1. Contingency Tables
Prepared by Yu-Fen Li

2. contingency table
When working with nominal data that have been grouped into categories, we often arrange the counts in a tabular format known as a contingency table (or cross-tabulation)
A r × c table (r rows and c columns)
In the simplest case, two dichotomous random variables are involved; the rows of the table represent the outcomes of one variable, and the columns represent the outcomes of the other.

3. Example
To examine the effectiveness of bicycle safety helmets, we wish to know whether these is an association between the incidence of head injury and the use of helmets among individuals who have been involved in accidents.

4. What are the hypotheses?
H0: The proportion of persons suffering head injuries among the population of individuals wearing safety helmets at the accident is equal to the proportion of persons sustaining head injuries among those not wearing helmets versus HA: The proportions of persons suffering head injuries are not identical in the two populations

5. The Chi-Square Test
The first step in carrying out the test is to calculate the expected count for each cell of the contingency table, given that H0 is true
The chi-square test compares the observed frequencies in each category of the contingency table (represented by O) with the expected frequencies in each category of the contingency table (represented by E) given the null hypothesis is true.

6. The Chi-Square Test
It is used to determine whether the deviations between the observed and the expected counts, O−E, are too large to be attributed to chance
where rc is the number of cells in the table.
To ensure that the sample size is large enough to make this approximation valid,
no cell in the table should have an expected count less than 1, and
no more than 20% of the cells should have an expected count less than 5.

7. How to compute the expected values?
Observed (O)
Expected (E)

8. Chi-square distributions
A chi-square random variable cannot be negative; it assumes values from zero to infinity and is skewed to the right.
Chi-square distributions with 2, 4, and 10 degrees of freedom

9. Yates correction
We are using discrete observations to estimate χ2, a continuous distribution.
The approximation is quite good when the degrees of freedom are big.
We can apply a continuity correction (Yates correction) for a 2 × 2 table as

10. TS formula for a 2 × 2 table
For a 2 × 2 table in the general format shown below

11. Another TS formula for a 2 × 2 table
the test statistic (TS) χ2 without continuity correction can be express as
the test statistic (TS) χ2 with continuity correction can also be express as

12. Example
For the bicycle example we talked about earlier, if we apply the Yates correction, we would get
the p-value is smaller than 0.001

13. Fisher’s exact test
When the sample size is small, one can use Fisher’s exact test to obtain the exact probability of the observed frequencies in the contingency table, given that there is no association between the rows and columns and that the marginal totals are fixed.
The details of this test is not presented here because the computations involved can be arduous.

14. McNemar’s Test
We cannot use the regular chi-square test for the matched data, as the previous chi-square test disregard the paired nature of the data.
We must take the pairing into account in our analysis.
Consider a 2 × 2 table of observed cell counts about exposure status for a sample of n matched case-control pairs as follows

15. McNemar’s Test
If the data of interest in the contingency table are paired rather than independent, we use McNemar’s test to evaluate hypotheses about the data
To conduct McNemar’s test for matched pairs, we calculate the test statistic
where b and c are the number of discordant pairs

16. Independent vs dependent data
Independent Data
Dependent Data

17. Example: matched pairs
Consider the following data taken from a study investigating acute myocardial infarction (MI) among Navajos in the US.
In this study, 144 MI cases were age- and gender-matched with 144 individuals free of heart disease
Independent Data
Dependent Data

18. Example: matched pairs
The test statistic is
with < p < Since p is less than α = 0.05, we reject the null hypothesis.
For the given population of Navajos, we conclude that if there is a difference between individuals who experience infarction and those who do not, victims of acute MI are more likely to suffer from diabetes than the individuals free from heart disease who have been matched on age and gender

19. Strength of the association
The chi-square test allows us to determine whether an association exists between two independent nominal variables
McNemar’s test does the same thing for paired dichotomous variables
However, neither test provides us a measure of the strength of the association

20. The Odds Ratio
If an event occurs with probability p, the odds in favor of the event are p/(1−p) to 1.
We can express an estimator of the OR as

21. The Confidence Interval (CI) for Odds Ratio
The cross-product ratio is simply a point estimate of the strength of association between two dichotomous variables.
To gauge the uncertainty in this estimate, we must calculate a confidence interval (CI) as well; the width of the interval reflects the amount of variability in the estimate of OR

22. The Confidence Interval (CI) for Odds Ratio
When computing a CI for the OR, we must make the assumption of normality.
However, the probability distribution of the OR is skewed to the right, and the relative odds can be any positive value between 0 and infinity.
In contrast, the probability distribution of the natural logarithm of the OR, i.e. ln(OR), is more symmetric and approximately normal.
Therefore, when calculating a CI for the OR, we typically work in the log scale.

23. The Confidence Interval (CI) for Odds Ratio
Besides, to ensure that the sample size is large enough, the expected value of each cell in the contingency table should be at least 5.
a 95% CI for the natural logarithm of the OR is
where
a 95% CI for the OR itself is

24. Bicycle Example
we reject the null hypothesis and conclude that wearing a safety helmet at the accident is protective to the head injury

25. The Odds Ratio and 95% CI for matched pairs
An OR can be calculated to estimate the strength of association between two paired dichotomous variables
a 95% CI for the OR itself is

26. MI and DM example

27. Berkson’s Fallacy
Berkson’s Fallacy is a common type of bias in case-control studies in particular hospital-based and practice-based studies.
It occurs due to differential admission rates between cases and controls.
This leads to positive (and spurious) associations between exposure and the case control status with the lowest admission rate.

28. Example : Berkson’s Fallacy
Hospitalized patients + nonhospitalized subjects
Hospitalized patients
individuals who have a disease of
the circulatory system are more likely to suffer from respiratory illness than individuals who do not
there is no association between the
two diseases

29. What happened?
Why do the conclusions drawn from these two samples differ so drastically?
To answer this question, we must consider the rates of hospitalization that occur within each of the four disease subgroups:

30. What happened?
individuals with both circulatory and respiratory disease are more likely to be hospitalized than individuals in any of the three other
subjects with circulatory disease are more likely to be hospitalized than those with respiratory illness.
Therefore, the conclusions will be biased if we only sample patients who are hospitalized

31. What’s the lesson?
We observe an association that does not actually exist.
This kind of spurious relationship among variables – which is evident only because of the way in which the sample was chosen – is known as Berkson’s fallacy
the sample must be representative