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When we talk about relative risk in statistical analysis, we're assessing the likelihood of an event happening in one group relative to another. Here, we compare the probability of finding pesticide residues in conventional food to that in organic food.
To calculate relative risk (RR), we divide the probability (or proportion) of the event in our group of interest by the probability of the event in the comparison group. In this exercise, the relative risk was calculated as follows:

• Proportion of conventional foods with pesticides: 0.733
• Proportion of organic foods with pesticides: 0.228

Thus, the RR is: \[ RR = \frac{0.733}{0.228} \approx 3.216 \] This means that conventional foods are about 3.216 times more likely to contain pesticide residues compared to organic foods. This simple metric helps us quickly grasp and compare the risk across these two categories. It’s an essential tool in epidemiology and public health assessments.

An odds ratio (OR) is another extremely valuable metric in statistical analysis. It compares the odds of an event occurring in one group to the odds of it occurring in another. This is particularly useful in case-control studies.
In our current scenario, we want to compare the odds of pesticide residues in conventional versus organic foods. To find the odds for each:

• The odds of pesticides in organic foods: \(\frac{29}{98} = 0.296\)
• The odds of pesticides in conventional foods: \(\frac{19485}{7086} \approx 2.749\)

The odds ratio is calculated by dividing these odds: \[ OR = \frac{2.749}{0.296} \approx 9.283 \] This indicates the odds of finding pesticide residues are approximately 9.283 times higher in conventional foods than in organic foods. Or, put differently, if you select a conventional food sample, the chance it contains pesticide residues is notably greater compared to an organic sample. The odds ratio provides deep insights into the strength of the association between the food type and pesticide presence.

Proportions give us a fundamental understanding of how often an event occurs in a particular group, expressed as a fraction or percentage. In many situations, understanding proportions can help make sense of large datasets in a simplified and quantitative way.
In the context of the pesticide residue data:

• Proportion of organic samples with pesticides: \(\frac{29}{127} \approx 0.228\)
• Proportion of conventional samples with pesticides: \(\frac{19485}{26571} \approx 0.733\)

This means that approximately 22.8% of organic food samples tested contained pesticides, while about 73.3% of conventional samples contained pesticides.
Using proportions is key when interpreting data in epidemiological studies. They allow for a straightforward comparison of the prevalence of events between different groups. However, it is crucial to use these figures correctly within their context. In our exercise, the phrase "The proportion of conventional food samples with pesticide residues present is more than 9 times larger than the proportion of organic food samples with pesticide residues present", is misleading. Such a statement incorrectly applies what is true for the odds ratio (in terms of comparing odds, not proportions), highlighting the importance of distinguishing between probability and odds in statistical communication.