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Chapter 11: Problem 28

Smelling and mortality A recent study (Pinto et al., Olfactory Dysfunction Predicts 5-Year Mortality in Older Adults. PLoS ONE \(9(10): \mathrm{e} 107541,2014)\) mentions that anosmic (those with almost no sense of smell) older adults had more than three times the odds of death over a 5-year span compared to normosmic (those with normal smell) individuals. Does this imply that anosmic older adults were more than three times as likely to die over the next 5 years than normosmic ones? Explain.

Short Answer

Expert verified
An odds ratio of 3+ means higher odds, not a tripling of death probability.

Step by step solution

01

Understand the Terms

The term 'anosmic' refers to individuals with almost no sense of smell, while 'normosmic' means individuals with a normal sense of smell. The study compares the odds of mortality over 5 years for these two groups.
02

Interpret the Odds Ratio

The study states that anosmic adults had more than three times the odds of death than normosmic individuals. An odds ratio (OR) greater than 1 indicates a higher odds for one group compared to another. Specifically, an OR of more than 3 suggests that the odds of dying are three times higher for anosmic individuals compared to normosmic ones.
03

Difference Between Odds and Probability

Odds and probability are related but different concepts. Odds represent the ratio of the occurrence of an event to its non-occurrence. Probability is the ratio of the occurrence of an event to the total number of possibilities. An odds ratio of 3 does not directly translate to being 'three times as likely' in terms of probability.
04

Calculate Probability from Odds

To understand what an odds ratio means in terms of probability, calculations are needed: if the probability of death for normosmic individuals is P, the odds are P/(1-P). An odds ratio of 3 implies the odds for anosmic individuals are 3 * P/(1-P). The relationship between odds and probability is non-linear, so the interpretation is not straightforward.
05

Conclusion on Interpretation

The statement that anosmic individuals have over three times the odds of death does not directly imply they are three times as likely to die as normosmic individuals. Odds ratios require careful interpretation, often involving statistical analysis, to convert to probability statements.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Odds Ratio
The odds ratio is a vital statistic in many studies, especially in health-related research. Essentially, it compares the odds of an event happening in one group with the odds in another group. If we take the case from the exercise, the study observes the odds of death among anosmic individuals versus normosmic ones.

The term "odds" refers to the likelihood of an event occurring compared to it not occurring. An odds ratio greater than 1 indicates a higher likelihood for the first group in the comparison. Here, with an odds ratio of more than 3, we learn that the odds of death in anosmic individuals are over three times greater than in normosmic individuals. This does not, however, imply a direct tripling of the probability of dying, merely a much higher likelihood in terms of odds.
Probability
Probability is another key concept in understanding statistical outcomes, and it's often confused with odds. Probability measures the chance of an event occurring as a fraction of the total possible outcomes. This is expressed as a number between 0 and 1, where 0 means impossibility and 1 means certainty.

In relation to the study, while the odds ratio for anosmic individuals dying is more than 3 times that of normosmic ones, this doesn't mean the probability of death for anosmic individuals is three times higher. This common error arises because the odds and probability, although related, are not interchangeable without additional calculations. It's important to convert odds to probability correctly to make meaningful interpretations in real-world scenarios.
Mortality Studies
Mortality studies often provide crucial insights into factors affecting survival rates, acting as a guide for public health interventions. These studies examine variables like age, health conditions, and lifestyle that may influence mortality. In the discussed study, the ability to smell is assessed for its predictive value regarding death rates among older adults.

By quantifying risk factors, such as olfactory dysfunction in this case, researchers can better understand which aspects significantly impact mortality rates. Findings such as these support the development of health programs aimed at mitigating risks. Such studies also highlight the necessity of interpreting statistical results, like odds ratios and probabilities, accurately to inform health policies effectively.

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Most popular questions from this chapter

Gender gap? Exercise 11.1 showed a \(2 \times 3\) table relating gender and political party identification, shown again here. The chi-squared statistic for these data equals 10.04 with a P-value of \(0.0066 .\) Conduct all five steps of the chi-squared test. \begin{tabular}{lcccc} \hline & \multicolumn{3}{c} { Political Party Identification } & \\ \cline { 2 - 4 } Gender & Democrat & Independent & Republican & Total \\ \hline Female & 421 & 398 & 244 & 1063 \\ Male & 278 & 367 & 198 & 843 \\ \hline \end{tabular}

True or false: Group 1 becomes Group 2 Interchanging two rows or interchanging two columns in a contingency table has no effect on the value of the \(X^{2}\) statistic.

True or false: Statistical but not practical significance Even when the sample conditional distributions in a contingency table are only slightly different, when the sample size is very large it is possible to have a large \(X^{2}\) statistic and a very small P-value for testing \(\mathrm{H}_{0}\) independence.

Exercise 11.32 showed the association between recreation and happiness. The table shown here gives the standardized residuals for those data in parentheses. a. Explain what a relatively small standardized residual such as -0.5 in the second cell represents. b. Identify the cells in which you would infer that the population has more cases than would occur if recreation and happiness were independent. Pick one of these cells and explain the association relative to independence.

Down syndrome diagnostic test The table shown, from Example 8 in Chapter \(5,\) cross-tabulates whether a fetus has Down syndrome by whether the triple blood diagnostic test for Down syndrome is positive (that is, indicates that the fetus has Down syndrome). a. Tabulate the conditional distributions for the blood test result, given the true Down syndrome status. b. For the Down cases, what percentage was diagnosed as positive by the diagnostic test? For the unaffected cases, what percentage got a negative test result? Does the diagnostic test appear to be a good one? c. Construct the conditional distribution on Down syndrome status for those who have a positive test result. (Hint: You condition on the first column total and find proportions in that column.) Of those cases, what percentage truly have Down syndrome? Is the result surprising? Explain why this probability is small. \begin{tabular}{lrrr} \hline & \multicolumn{2}{c} { Blood Test Result } & \\ \cline { 2 - 3 } Down Syndrome Status & Positive & Negative & Total \\ \hline D (Down) & 48 & 6 & 54 \\ D \(^{c}\) (unaffected) & 1307 & 3921 & 5228 \\ Total & 1355 & 3927 & 5282 \\ \hline \end{tabular}
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