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1 Introduction to Biostatistics (BIO/EPI 540) Contingency Tables Acknowledgement: Thanks to Professor Pagano (Harvard School of Public Health) for lecture material

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2 Contingency Tables Nominal data that are grouped into categories are often presented in the form of contingency tables Rows denote levels of one variable (e.g. disease) Columns denote the levels of the other variable (e.g. exposure)

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3 Consider whether the rate of caesareans is different for subjects receiving an electronic fetal monitoring (EFM), as compared to those without EMF. Sample 5,824 deliveries: of these 2,850 were EFM exposed and 2,974 were not. 358 of the 2,850 had c-sections as did 229 of the 2,974. Binomial with n huge. Example – Discrete Outcomes

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4 Chi square test Proceed as usual: 1.If there is no difference (null hypothesis) what do we expect to see? 2. How does this compare to what we have observed? (statistic & its distribution) Do the c-section rates differ? Example – Discrete Outcomes

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5 Caesarean Delivery EFM Exposure Total YesNo Yes358229587 No2,4922,7455,237 Total2,8502,9745,824 Data-Contingency table If the c-section rate is the same in both populations, then ignore column classification and go with totals.

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6 2x2 Table – Null Hypothesis Ho: The proportion of C-sections among patents receiving EFM is identical to the proportion of C- sections among patients who do not receive EMF Ha: The proportion of C-sections among patents receiving EFM is different from the proportion of C-sections among patients who do not receive EMF

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7 From the totals we can estimate: Probability of c-section

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8 What do we expect to see if EFM has no effect? EFM exposed (2,850 mothers): No EFM (2,974 mothers) Expected counts under Ho

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9 C-sect EFM Exposure? Total YesNo Yes358287229300587 No24922563274526745237 Total285029745824 Expected, if independence of row and column classification is true, in boxes: Observed and Expected counts – Contingency Table

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10 (Table page A-26) Chi Square Goodness of fit Chi Square Test

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11 In 2x2 tables (only) we apply a continuity correction factor: Continuity correction factor

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12 For the EFM and c-section example, above: Example Note: This is a 2 sided test

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13 Equivalent Tests The above example can be analyzed equivalently using a two sample test of proportions (Chapter 14.6) 2 sample test of proportions (Z test) and Chi-Square test are mathematically equivalent

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14 Assumptions – Chi Square test Chi square test – is an asymptotic test. i.e. Works only when sample size is large Chi Square test – treats the row total and column total of the data as fixed (i.e. not random)

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15 Assumptions – 2 sample test of proportions Z test – is also an asymptotic test. Assumes that the Central Limit Theorem for sample means (i.e. proportions) holds. Thus this test is appropriate only when sample size is large Z test – assumes that the proportions in each group being compared are random variables

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16 e.g. Accuracy of Death Certificates Hospit. Certificate Status Total Conf. Accur. Inacc. No Ch. Incorr. Recode Comm.1571854229 Teach.2684434346 Total4256288575 Extending to multiple categories: r x c Tables

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17 Hospital Certificate Status Total Confirmed Accurate Inaccurate No Change Incorrect Recoded Comm.157169.31824.75435.0229 Teach.268255.74437.33453.0346 Total4256288575 tabi 157 18 54 \ 268 44 34 e.g.

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18 Summary Contingency Tables – –Analysis of 2x2 tables –Analysis of rxc tables Equivalence between Chi square test and two sample test of proportions

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