Question

Consider the following quote from the article "Review Finds No Link Between Vaccine and Autism" (San Luis Obispo Tribune, October 19,2005 ): " 'We found no evidence that giving MMR causes Crohn's disease and/or autism in the children that get the MMR,' said Tom Jefferson, one of the authors of The Cochrane Review. 'That does not mean it doesn't cause it. It means we could find no evidence of it." (MMR is a measles-mumps-rubella vaccine.) In the context of a hypothesis test with the null hypothesis being that MMR does not cause autism, explain why the author could not just conclude that the MMR vaccine does not cause autism.

Short answer

The author couldn't conclusively say the MMR vaccine doesn't cause autism because hypothesis testing is about disproving the null hypothesis, not proving it. They found no evidence to reject the null hypothesis (that the MMR vaccine doesn't cause autism), but that doesn't mean the null hypothesis is definitively true. It's a distinction of 'failing to show' vs. 'showing a failure'.

Step-by-step solution

Understanding the Null and Alternative Hypotheses

In a hypothesis test, two competing hypotheses are presented: the null hypothesis (often denoted as H0) and the alternative hypothesis (often denoted as Ha). In this scenario, the null hypothesis is that the MMR vaccine does not cause autism. The alternative hypothesis, therefore, would be that the MMR vaccine does indeed cause autism.

Explaining the outcome of the Hypothesis Test

The purpose of a hypothesis test is to gather evidence against the null hypothesis, trying to disprove it. In this case, the study was trying to find evidence that the MMR vaccine causes autism. Their statement means that they found no such evidence. Therefore, they failed to reject the null hypothesis. However, failing to reject the null hypothesis and proving the null hypothesis are not the same thing.

Discussing the Limitations of Hypothesis Testing

A significant limitation of hypothesis testing is the inability to 'prove' a hypothesis, null or alternative. Hypothesis tests can only provide statistical evidence to either accept or reject the null hypothesis. Failing to reject the null hypothesis means that, given the data and assuming the null hypothesis is true, the observed outcome happens with a high probability. It does not mean the null hypothesis is definitively correct. Therefore, the researcher correctly notes that their finding does not mean the vaccine doesn't cause autism; it just means they couldn't find any evidence to support that it does.

Key concepts

Null Hypothesis

In the realm of statistics, the null hypothesis represents a default or baseline statement that there is no effect or no difference. It's denoted as H0 and serves as a starting point for hypothesis testing. For instance, in a study investigating a potential link between a vaccine and autism, the null hypothesis would be that there is no association between the two—that is, the vaccine does not cause autism.

If you think of a legal trial, the null hypothesis is like the presumption of innocence: the defendant is presumed innocent (null hypothesis) until proven guilty (alternative hypothesis). This presumption remains until enough evidence is gathered to challenge it. Consequently, when researchers embark on a study, they gather data and perform statistical tests to see if there’s sufficient evidence to reject the null hypothesis, not to prove it directly.

Alternative Hypothesis

Contrasting the null hypothesis is the alternative hypothesis (Ha or H1), which asserts that there is an effect or a difference. Referring to our vaccine study, the alternative hypothesis posits that the MMR vaccine does cause autism, directly contradicting the null hypothesis.

The underlying goal of hypothesis testing is to determine whether the data collected provides strong enough evidence to support the alternative hypothesis. Researchers aim to refute the null hypothesis in favor of the alternative. However, they must do so with a rigorous statistical analysis to ensure that any conclusion drawn is not due simply to chance. It’s important to understand that the alternative hypothesis is not confirmed by default when the null is rejected; it just becomes more plausible based on the statistical evidence at hand.

Statistical Evidence

Statistical evidence in hypothesis testing refers to the data and analysis that statisticians use to support or refute the null hypothesis. The evidence comes from the result of statistical tests, which provide a p-value—a probability that measures the strength of the evidence against the null hypothesis. A low p-value indicates that the observed data would be very unlikely if the null hypothesis were true, suggesting that the null hypothesis may not be a good fit for the data.

In our vaccine example, the researchers did not find statistical evidence linking the MMR vaccine to autism, meaning their p-value was likely above a predefined threshold (often 0.05). This does not prove the null hypothesis (that the vaccine does not cause autism); it simply means that the available evidence is insufficient to contradict it. Such statistical conclusions are always made with a degree of uncertainty, as the evidence is based on sample data that might not perfectly represent the entire population.

Limitations of Hypothesis Testing

Hypothesis testing is a valuable tool in statistics, but it comes with limitations. One major limitation is that hypothesis tests cannot provide absolute proof for a null or alternative hypothesis; they can only suggest whether evidence exists to reject the null hypothesis. This means that even if a test fails to reject the null hypothesis, we shouldn't take it as proof that the null hypothesis is true; there might simply not be enough data or the study may have insufficient power to detect an actual effect.

Another limitation is related to the risk of errors. Type I error occurs when the null hypothesis is wrongly rejected, while Type II error happens when a false null hypothesis is not rejected. Finally, results can be influenced by biases in data collection or sample selection. Researchers and statisticians must therefore interpret the outcomes of hypothesis tests with caution, considering these limitations and making decisions based on the balance of evidence rather than definitive proof.