SMHS OR RR

Scientific Methods for Health Sciences - Odds Ratio and Relative Risk

HS 550: Fundamentals

Odds Ratio/Relative Risk

1) Overview: The relative risk is measure of dependence which allows us to compare probabilities in terms of their ratio (\(\frac{P_1}{P_2}\) ) rather than their difference (\(p_1 – p_2\)). The relative risk measure is widely used in many studies of public health. Another way to compare two probabilities is in terms of the odds. If an event takes place with probability p, then the odds in favor of the event are p/(1 - p). The odds ratio is the ratio of odds for two probabilities.

Motivation: Suppose we study Brain Cancer in the context of cell phone use. The table below illustrates some (simulated) data. One clear healthcare question in this case-study could be: “Is cell phone use associated with higher incidence of brain cancer?” To address this question, we can look at the relative risk of cell-phone usage.

	Brain cancer (BC)		Total
	Yes (A)	No
Cell Phone (CP) Yes	18	80	98(B)
Cell Phone (CP) No	7	95	102(C)
Total	25	175	200

Computing the (conditional!) probabilities (P) of brain cancer (BC) given either cell-phone use, P1, no cell-phone use, P2, we can form their ratio to determine if the relative risk of brain cancer (BC) is higher in cell-phone users (CP), relative to non-users (NCP).

\(P_1= P(BC|CP) = 18/98= 0.184\) \(P_2= P(BC|NCP) = 7/102 = 0.069\)

So the relative risk is: RR=0.184/0.069 = 2.67.

The risk of having brain cancer is more than 2.5 times greater for cell-phone users when compared to non-cell phone owners.

For the same example, the odds ratio (OR) of brain cancer relative to cell-phone use is\[OR=((P(A|B))/(1-P(A|B)))/((P(A|C))/(1-P(A|C)))=((18/98)/(1-18/98))/((7/102)/(1-7/102))=(0.184/0.816)/(0.069/0.931)=0.225/0.074=3.04\]

Thus, the odds of having brain cancer is about 3 times greater for cell phone owners when compared to non-cell phone owners. We could have compared the odds of owning a cell phone, given that a patient had brain cancer (i.e., the column-wise probabilities), P(CP|BC) = 18/25 = 0.72 versus P(CP|NBC) = 80/175 = 0.457. However this does not seem as important scientifically.

	Factor 1		Total
	Yes (A)	No
Factor 2 Yes	n_1,1	N_1,2	n_1,1+n_1,2
Factor 2 No	n_2,1	n_2,2	n_2,1+n_2,2
Total	n_1,1 + n_2,1	n_1,2 + n_1,2	N

OR=(n_1,1×n_2,2)/(n_1,2×n_2,1 )

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• SOCR Home page: http://www.socr.umich.edu

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