Real-Life Math

You are an epidemiologist working with epidemiologist Linda Cowan. You are beginning a case-control study. It is one of the types of study designs used in epidemiology. It will allow you to investigate whether or not a particular exposure may be a cause of a disease.

Since you are a new member of Cowan's team, she will walk you through the study with specific instructions for the math she wants you to do.

In the case-control design, newly diagnosed cases of the disease (cases) are identified. A comparison group of similar individuals without diseases are identified (controls). Both groups are assessed for their past exposure to the suspected causal agent (the thing that may have caused the disease).

You need to calculate the odds ratio. That is the ratio of the odds of past exposure in cases with the disease divided by the odds of past exposure in controls without the disease. Data are often presented in a 2 x 2 table which shows the division of all cases (A+C) and all controls (B+D) into their respective exposure categories as follows:

The odds ratio (OR) is: [(A/(A+C)) ÷ (C/( A+C))] ÷ [(B/(B+D)) ÷ (D/( B+D))]

This reduces to OR = AD ÷ BC

The odds ratio (OR) is interpreted as follows:

An OR = 1.0 indicates the odds of prior exposure are the same in cases and controls. In that case there is no association.

ORs > 1.0 occur when exposures are more common in cases than controls (exposure may be causing the disease).

ORs < 1.0 occur when exposure histories are less common in cases than in controls (exposure may be protective).

"The usual way to examine the odds ratio for statistical significance is to calculate what is called a 'confidence interval,' typically set at 95 percent. This means that if I sampled the same basic population 100 times, using the same study methods, at least 95 times I would get an OR that fell within the confidence interval range." says Cowan. For example the confidence interval could be expressed as 1.50 ± 0.20, meaning the measured OR will fall between 1.30 and 1.70 at least 95 percent of the time.

An approximate 95% CI for an odds ratio measure is calculated using the following formulae: where Zα = 1.96, e is the mathematical constant (base natural log), and lnOR is the natural log of the OR.

Lower limit 95% CI = e^{ln OR - Zα √ (1/A + 1/B + 1/C + 1/D)}

Upper limit 95% CI = e^{ln OR + zα √ (1/ A + 1/B + 1/C + 1/D)}

If the interval includes an OR value = 1.0 then it is not considered "statistically significant" at the p=0.05 level because a possible value in the interval is that associated with NO relationship (i.e., OR=1.0).

With that information in mind, Cowan has the following problem for her newest epidemiologist. You are working on a case-control study. You want to find out whether past use of over-the-counter anti-inflammatory medicines (such as aspirin, naproxen, and ibuprofen) increases the risk of acne. You are looking at those who used the medicines for at least 10 days within the past two months. Cases were teenagers aged 13 to 16 newly diagnosed with acne. Controls were teenagers aged 13 to 16 who had never experienced acne.

1. Calculate the OR and 95% CI for the relation between prior exposure to OTC anti-inflammatory medicines and development of acne.
2. What do you conclude about whether or not there is an association based on this OR value?
3. Is it a statistically significant result?

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