Recovering the Cell Probabilities of a 2-by-2 Contingency Table

A 2-by-2 contingency table is Control on one axis and Experimental on the other. Control and Experimental are further dichotomized as Event and Non-event. The odds ratio (OR), number needed to harm (NNH), and absolute risk increase (ARI) are functions of the cell probabilities of a 2-by-2 contingency table, and conversely, the cell probabilities can be recovered given knowledge of these parameters. Recover the contingency table cell probabilities EE, EN, CE, and CN using OR, NNH, and ARI as knowns and the following definitions as necessary:

Experimental group (E), Control group (C), Events (E), Non-events (N), Total subjects (S), ES = EE + EN, CS = CE + CN, Event rate (ER), EER = EE / ES, CER = CE / CS, ARI = EER β CER, NNH = 1 / (EER β CER), and OR = (EE / EN) / (CE / CN). Also NNH = 1 + [CER x (OR-1)]/(1-CER) x (CER) x (OR-1).

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A 2-by-2 contingency table is Control on one axis and Experimental on the other. Control and Experimental are further dichotomized as Event and Non-event. The odds ratio (OR), number needed to harm (NNH), and absolute risk increase (ARI) are functions of the cell probabilities of a 2-by-2 contingency table, and conversely, the cell probabilities can be recovered given knowledge of these parameters. Recover the contingency table cell probabilities EE, EN, CE, and CN using OR, NNH, and ARI as knowns and the following definitions as necessary:

Experimental group (E), Control group (C), Events (E), Non-events (N),
Total subjects (S), ES = EE + EN, CS = CE + CN,
Event rate (ER), EER = EE / ES, CER = CE / CS,
ARI = EER β CER, NNH = 1 / (EER βCER), and
OR = (EE / EN) / (CE / CN). ...

Solution Summary

The solution examines recovering the cell probabilities of a 2-by-2 contingency tables.

Given the following contingencytable:
C D Total
A 10 30 40
B 20 40 60
Total 30 70 100
Find the probability of A and C.
a) 33.3%
b) 10%
c) 25%
Find the probability of A or C
a) 60%
b) 70%
c) 10%
Find the probability of B given C
a) 20%
b) 30%
c) 66.67%

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Event A Event B
Event C 9 6
Event D 4 21
Event E 7 3
Determine the following probabilities:
a) P (A and C)
b) P (A and D)
c) P B and E)
d) P (A and B)

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