Question

Exercise 2.19 on page 58 introduces a study examining whether giving antibiotics in infancy increases the likelihood that the child will be overweight. Prescription records were examined to determine whether or not antibiotics were prescribed during the first year of a child's life, and each child was classified as overweight or not at age 12. (Exercise 2.19 looked at the results for age 9.) The researchers compared the proportion overweight in each group. The study concludes that: "Infants receiving antibiotics in the first year of life were more likely to be overweight later in childhood compared with those who were unexposed \((32.4 \%\) versus \(18.2 \%\) at age 12 \(P=0.002) "\) (a) What is the explanatory variable? What is the response variable? Classify each as categorical or quantitative. (b) Is this an experiment or an observational study? (c) State the null and alternative hypotheses and define the parameters. (d) Give notation and the value of the relevant sample statistic. (e) Use the p-value to give the formal conclusion of the test (Reject \(H_{0}\) or Do not reject \(H_{0}\) ) and to give an indication of the strength of evidence for the result. (f) Can we conclude that whether or not children receive antibiotics in infancy causes the difference in proportion classified as overweight?

Short answer

(a) Explanatory variable: intake of antibiotics in infancy (categorical). Response variable: overweight at age 12 (categorical). (b) Observational study. (c) \(H_{0}\): The intake of antibiotics in infancy does not affect chances of being overweight at 12. \(H_{a}\): Infants who received antibiotics are more likely to be overweight at age 12. (d) Notation: \(p_{1}\), \(p_{2}\). Sample statistics: \(p_{1} = 0.324\), \(p_{2} = 0.182\). (e) Reject \(H_{0}\). The p-value provides strong evidence against the null hypothesis. (f) No, we cannot conclude causation based on this study.

Step-by-step solution

Identifying variables

The explanatory variable or independent variable, which is the factor that is presumed to cause or lead to another, is the intake of antibiotics during the first year of a child's life. This is a categorical variable since it can be classified into two categories: 'received antibiotics' and 'did not receive antibiotics'. The response or dependent variable, which is the outcome that is being studied, is whether the child is overweight at age 12. This also is a categorical variable since a child can be classified into 'overweight' or 'not overweight'.

Classifying the study

This is an observational study. In an observational study, the researchers simply observe the subjects without trying to impact or control the outcome. Here, the researchers did not assign infants to receive antibiotics or not, they just observed records. Therefore, it's an observational study, not an experiment.

Hypotheses formulation

Null hypothesis (\(H_{0}\)): There is no difference in the proportion of children classified as overweight at age 12 between those who received antibiotics in their first year of life and those who didn't. Alternative hypothesis (\(H_{a}\)): Infants who received antibiotics in their first year of life are more likely to be overweight at age 12 than those who did not.

Notation and the Value of Relevant Sample Statistic

The notation for the proportion of infants who became overweight after receiving antibiotics during their first year is \(p_{1}\) , and for those who didn't receive such treatment is \(p_{2}\). The values found in the samples were \(p_{1} = 0.324\) and \(p_{2} = 0.182\).

Interpreting the p-value

A p-value of 0.002 is far less than the general α level of 0.05, so we reject \(H_{0}\) and conclude that there is statistically significant evidence to support \(H_{a}\). The p-value is quite small, providing strong evidence against the null hypothesis.

Concluding on the causality

Despite a statistically significant difference being present, we cannot conclude that receiving antibiotics in infancy causes a greater chance of being overweight at age 12. This is because this was an observational study, not a random experiment, so other explanatory variables could be in play.