11.1 Effect Size
Figure 11.1: Sampling distribution of average candy weight under the null hypothesis that average candy weight is 2.8 grams in the population.
We have learned that larger samples have smaller standard errors (Section 4.2.11). Smaller standard errors yield larger test statistic values and larger test statistics have smaller p values. In other words, a test on a larger sample is more often statistically significant.
A larger sample offers more precision, so the difference between our sample outcome and the hypothesized value is more often sufficient to reject the null hypothesis. For example, we would reject the null hypothesis that average candy weight is 2.8 grams in the population if average weight in our sample bag is 2.70 grams and our sample is large. But we may not reject this null hypothesis if we have the same outcome in a small sample bag.
The larger our sample, the more sensitive our test will be, so we will get statistically significant results more often. If we think of our statistical test as a security metal detector, a more sensitive detector will go off more often.
Of course, the size of the difference between our sample outcome and the hypothesized population value matters as well. This difference is called observed effect size. If average candy weight in our sample bag deviates more from the average weight specified in the null hypothesis, we are more likely to reject the null hypothesis. In terms of a security metal detector: Our test will pick up large pieces of metal more easily than small pieces.
The p value and rejection of the null hypothesis based on the p value, then, depend both on sample size and observed effect size.
A larger sample size makes a statistical test more sensitive. The test will pick up (be statistically significant for) smaller effect sizes.
A larger effect size is more easily picked up by a statistical test. Larger effect sizes yield statistically significant results more easily, so they require smaller samples.
Deciding on our sample size, we should ask ourselves this question: What effect size should produce a significant test result? In the security metal detector example, at what minimum quantity of metal should the alert sound? To answer this question, we should consider the practical aims and context of our research.
11.1.1 Practical relevance
Investigating the effects of a new medicine on a person’s health, we may require some minimum level of health improvement to make the new medicine worthwhile medically or economically. If a particular level of improvement is clinically important, it is practically relevant (sometimes called practically significant).
If we have decided on a minimum level of improvement that is relevant to us, we want our test to be statistically significant if the average true health improvement in the population is at least of this size. We want to reject the null hypothesis of no improvement in this situation.
For media interventions such as health, political, or advertisement campaigns, one could think of a minimum change of attitude affected by the campaign in relation to campaign costs. A choice between different campaigns could be based on their efficiency in terms of attitudinal change per cost unit.
Note the important difference between practical relevance and statistical significance. Practical relevance is what we are interested in. If the new medicine is sufficiently effective, we want our statistical test to signal it. In the security metal detector example: If a person carries too much metal, we want the detector to pick it up.
Statistical significance is just a tool that we use to signal practically relevant effects. Statistical significance is not meaningful in itself. For example, we do not want to have a security detector responding to a minimal quantity of metal in a person’s dental filling. Statistical significance is important only if it signals practical relevance. We will return to this topic in Chapter 12.