Question

Eating Breakfast Cereal and Conceiving Boys Newscientist.com ran the headline "Breakfast Cereals Boost Chances of Conceiving Boys," based on an article which found that women who eat breakfast cereal before becoming pregnant are significantly more likely to conceive boys. \({ }^{42}\) The study used a significance level of \(\alpha=0.01\). The researchers kept track of 133 foods and, for each food, tested whether there was a difference in the proportion conceiving boys between women who ate the food and women who didn't. Of all the foods, only breakfast cereal showed a significant difference. (a) If none of the 133 foods actually have an effect on the gender of a conceived child, how many (if any) of the individual tests would you expect to show a significant result just by random chance? Explain. (Hint: Pay attention to the significance level.) (b) Do you think the researchers made a Type I error? Why or why not? (c) Even if you could somehow ascertain that the researchers did not make a Type I error, that is, women who eat breakfast cereals are actually more likely to give birth to boys, should you believe the headline "Breakfast Cereals Boost Chances of Conceiving Boys"? Why or why not?

Short answer

In short, it would be expected that by random chance 1-2 of the individual tests of the 133 foods could show a significant result. It is possible that the research made a Type I error - mistakenly rejected the null hypothesis that there was no effect due to breakfast cereals, attributing randomness to a significant effect. Lastly, even if there was a very small actual effect of breakfast cereal consumption on conceiving boys, claiming that 'Breakfast Cereals Boost Chances of Conceiving Boys' might be misleading as a significant statistical correlation doesn't necessarily imply causation or a large practical effect.

Step-by-step solution

Understanding the Scenario

The scenario describes a study examining the likelihood of conceiving boys among women who consume breakfast cereals compared to those who don't. The study involves 133 foods and uses a significance level of \(\alpha=0.01\). It's important to understand that the significance level (\(\alpha\)) is the probability of rejecting the null hypothesis when it is true, i.e., the probability of a Type I error.

Calculating Expected False Positives

Now let's consider (a), which is asking how many of the 133 tests we would expect to show a significant result just by chance if none of the foods had an effect on the gender of a conceived child. This can be done by simply multiplying the total number of tests (133) by the significance level (\(\alpha=0.01\)). \[(133)(0.01) = 1.33\]. This calculation implies that, on average, we would expect about 1 to 2 of the tests to show a significant result purely by chance, even though the food has no actual effect.

Evaluating the Possibility of a Type I Error

In step (b), the question is whether the researchers could have made a Type I error. This would mean that they rejected a true null hypothesis, i.e., they found a significant difference when in fact there was none. Given that only one food (breakfast cereal) showed a significant difference and we expected 1 to 2 foods to show a significant difference by chance, it is entirely possible that the observed significance in the breakfast cereal is merely due to chance. Therefore, it's possible that a Type I error could have occurred. However, without more information, we can't say for certain.

Evaluating the Headline

Looking at question (c), even if the researchers did not make a Type I error and women who eat breakfast cereals really are more likely to conceive boys, the headline 'Breakfast Cereals Boost Chances of Conceiving Boys' could be misleading. A significant result doesn't necessarily mean a sizable effect. In addition, a lot of other confounding variables (like genetics or other dietary habits) may play a role in the gender of the conceived child. Thus, based on this analysis, it would need to be clarified that correlation does not imply causation.

Key concepts

Type I Error

A Type I error occurs in statistical hypothesis testing when a true null hypothesis is falsely rejected. In simpler terms, it's mistakenly concluding that an effect or difference exists when it doesn't. A classic analogy is a false alarm, such as a smoke detector going off when there's no fire.

Imagine a study testing 133 different foods to see if they influence the gender of babies conceived. If none of the foods had an actual effect but one shows a significant difference (like the breakfast cereal in our example), this might just be due to random variation rather than a true effect. Since the significance level was set at \(\alpha=0.01\), we expect that around 1 to 2 foods would show 'significant' results purely by chance. Given that breakfast cereal was the only significant food, we must consider the possibility that this finding was the result of a Type I error.

Recognizing the chance of committing a Type I error is crucial when interpreting research findings. It underlines the importance of replication and robust study design to avoid making conclusions based on false positives.

Significance Level

The significance level in statistics, denoted by \(\alpha\), is a threshold for determining when to reject the null hypothesis. The lower the significance level, the stronger the evidence must be to reject the null hypothesis. Typically, levels of 0.05, 0.01, or 0.001 are used.

In our cereal-eating study, the significance level was 0.01. This means there's a 1% risk of wrongly rejecting the null hypothesis (committing a Type I error) for each individual food tested. With 133 foods, expecting about 1.33 or roughly 1 to 2 false positives is statistically reasonable, which might be precisely what happened with the breakfast cereal claim. Researchers set a significance level to control for Type I errors, but this does not eliminate the risk—especially with multiple comparisons—it only minimizes it.

When researchers conduct multiple tests, as they did with the 133 foods, the concept of 'multiple testing correction' often comes into play. This technique adjusts the significance levels to account for the increased risk of finding false positives due to the sheer volume of tests. Adjustments like the Bonferroni correction are common, although they were not mentioned in relation to the cereal study.

Correlation vs Causation

Correlation and causation are often conflated, leading to misinterpretations of statistical data. Correlation indicates a relationship between two variables—when one changes, so does the other. Causation, on the other hand, implies that one event is the result of the occurrence of the other event; there's a cause-effect relationship.

In the context of the breakfast cereal study, finding that women who ate cereals more frequently conceived boys at a higher rate shows a correlation. However, this does not prove that cereals cause the outcome of conceiving a boy. There could be numerous other factors at play, such as overall diet, health, and lifestyle choices that correlate with cereal consumption and also influence the outcome.

Researchers must be careful not to jump to conclusions about causality without rigorous testing, including controlling for confounding variables and ideally conducting randomized controlled trials. For students and readers alike, anytime you encounter a claim that seems to infer causation from correlation, look deeper into the research methodology to assess the validity of the claim.