As can be seen in Table 4, you studied 200 children for five years for a total of 1000 person-years and had 50 Asthma cases. This resulted in an incidence rate of 5 per 100 person-years (do the math yourself), and for the non-exposed group, you had 300 children followed for five years for a total of 1500 person-years and in the five years, you found 15 children with Asthma. This resulted in an incidence rate of 1 per 100 person-years. The ratio between the two rates was 5.0 and is referred to as Relative Risk or Rate Ratio (RR). 

Concepts of Attributable Risk and Population Attributable Risk

In this same study, you found that there were 4 per 100 person-year more cases of Asthma in the group that was exposed to environmental tobacco smoke. This figure is referred to as "Attributable Risk" or the excess risk of Asthma in a population exposed to ETS. If ETS were to be completely or 100% eradicated from the exposed group, by how much percent would the Asthma incidence be reduced? This would depend on two factors: the relative risk of association between ETS and Asthma but also on the existing prevalence of exposure to ETS. Let us imagine that this was a real study and that we found that the RR of the relationship between ETS and Asthma was 5.0. Soliman et.al. (2004) estimated that in the 2000s, the prevalence of ETS in the US was about 25% \cite{Soliman_2004}. They also found that this figure dropped from 35% in the 1990s to 25% by 2000. If we were to use the 25% as the estimated prevalence of exposure to ETS, then we can use these two figures to estimate the population attributable risk percent or PAR% -- this figure would indicate the proportion by which childhood asthma would decrease if ETS were to be completely removed. The formula is given as follows:
PAR = P* (RR- 1) / [1 + P* (RR-1)]
Where, 
PAR = Population Attributable Risk
P= Prevalence of Exposure
RR= Relative Risk of the association between Exposure and Disease
If we were to plug this formula to our own situation where RRe = 5.00 and P= 0.25, then we would have:
PAR = 0.2 or in terms of percents, PAR% = 20%
This states that if we were to eliminate the ETS altogether in a population where the prevalence of ETS was 25%, it would lead to a 20% reduction in Asthma in the children.

Call to action

By how much would childhood asthma be reduced if the prevalence was 35%? What does this tell us?
You can see that if the prevalence of the exposure would be high, the corresponding decrease in the disease condition as a result of the exposure would be high as well. This is the strength of evidence of the association between a risk factor and a disease. 

Concepts of Odds Ratio

Another common measure for the association between an exposure and a disease is Odds Ratio. Odds refers to a ratio of probabilities. Suppose we toss a coin. The probability of the head is 0.5, and the probability of the coin not falling head is (1 - 0.5) = 0.5. Odds refers to the ratio of probability of an event happening and the complementary probability of an event not occurring. Here, in the coin example, the ratio of probability of heads versus not head is 1:1, or even. In other situations, you will have other odds. 

Call to Action

Take the Rugby matches between Crusaders and Hurricanes (for stats, check this site).  According to the site, for the 28 matches that the Crusaders played against the Hurricanes, Crusaders won 16 of those 28 matches, and three of those matches were drawn, so let's say Crusaders won 16 out of the remaining 25 matches. What is the Odds of Crusaders Win Against Hurricanes? Can you figure out?
Answer: Odds are 16:9 in favour of the Crusaders
We use this concept of Odds and comparison of Odds in Epidemiology. In order to study rare diseases, the exposure and outcome associations between rare diseases such as cancers. Torres-Duran and colleagues conducted a case control study (we will learn about different types of study designs in the next section) of exposure to Radon gas in the residential area and risk of lung cancer among non-smokers in Spain \cite{Torres_Duran_2014}. Before we dive in to the results of that study, let's see how odds ratios work.