Hypothesis Testing

What is Statistical Inference?

Statistical inference is the process of using a sample to infer the properties of a population. Statistical procedures use sample data to estimate the characteristics of the whole population from which the sample was drawn.

Scientists typically want to learn about a population. When studying a phenomenon, such as the effects of a new medication or public opinion, understanding the results at a population level is much more valuable than understanding only the comparatively few participants in a study.

Unfortunately, populations are usually too large to measure fully. Consequently, researchers must use a manageable subset of that population to learn about it.

By using procedures that can make statistical inferences, you can estimate the properties and processes of a population. More specifically, sample statistics can estimate population parameters. Learn more about the differences between sample statistics and population parameters.

For example, imagine that you are studying a new medication. As a scientist, you’d like to understand the medicine’s effect in the entire population rather than just a small sample. After all, knowing the effect on a handful of people isn’t very helpful for the larger society!

Consequently, you are interested in making a statistical inference about the medicine’s effect in the population.

Read on to see how to do that! I’ll show you the general process for making a statistical inference and then cover an example using real data.

Related posts: Populations vs. Samples and Descriptive vs. Inferential Statistics

How to Make Statistical Inferences

In its simplest form, the process of making a statistical inference requires you to do the following:

1. Draw a sample that adequately represents the population.
2. Measure your variables of interest.
3. Use appropriate statistical methodology to generalize your sample results to the population while accounting for sampling error.

Of course, that’s the simple version. In real-world experiments, you might need to form treatment and control groups, administer treatments, and reduce other sources of variation. In more complex cases, you might need to create a model of a process. There are many details in the process of making a statistical inference! Learn how to incorporate statistical inference into scientific studies.

Statistical inference requires using specialized sampling methods that tend to produce representative samples. If the sample does not look like the larger population you’re studying, you can’t trust any inferences from the sample. Consequently, using an appropriate method to obtain your sample is crucial. The best sampling methods tend to produce samples that look like the target population. Learn more about Sampling Methods and Representative Samples.

After obtaining a representative sample, you’ll need to use a procedure that can make statistical inferences. While you might have a sample that looks similar to the population, it will never be identical to it. Statisticians refer to the differences between a sample and the population as sampling error. Any effect or relationship you see in your sample might actually be sampling error rather than a true finding. Inferential statistics incorporate sampling error into the results. Learn more about Sampling Error.

Common Inferential Methods

The following are four standard procedures than can make statistical inferences.

• Hypothesis Testing: Uses representative samples to assess two mutually exclusive hypotheses about a population. Statistically significant results suggest that the sample effect or relationship exists in the population after accounting for sampling error.
• Confidence Intervals: A range of values likely containing the population value. This procedure evaluates the sampling error and adds a margin around the estimate, giving an idea of how wrong it might be.
• Margin of Error: Comparable to a confidence interval but usually for survey results.
• Regression Modeling: An estimate of the process that generates the outcomes in the population.

Example Statistical Inference

Let’s look at a real flu vaccine study for an example of making a statistical inference. The scientists for this study want to evaluate whether a flu vaccine effectively reduces flu cases in the general population. However, the general population is much too large to include in their study, so they must use a representative sample to make a statistical inference about the vaccine’s effectiveness.

The Monto et al. study* evaluates the 2007-2008 flu season and follows its participants from January to April. Participants are 18-49 years old. They selected ~1100 participants and randomly assigned them to the vaccine and placebo groups. After tracking them for the flu season, they record the number of flu infections in each group, as shown below.

Treatment	Flu count	Group size	Percent infections
Placebo	35	325	10.8%
Vaccine	28	813	3.4%
Effect			7.4%

Monto Study Findings

From the table above, 10.8% of the unvaccinated got the flu, while only 3.4% of the vaccinated caught it. The apparent effect of the vaccine is 10.8% – 3.4% = 7.4%. While that seems to show a vaccine effect, it might be a fluke due to sampling error. We’re assessing only 1,100 people out of a population of millions. We need to use a hypothesis test and confidence interval (CI) to make a proper statistical inference.

While the details go beyond this introductory post, here are two statistical inferences we can make using a 2-sample proportions test and CI.

1. The p-value of the test is < 0.0005. The evidence strongly favors the hypothesis that the vaccine effectively reduces flu infections in the population after accounting for sampling error.
2. Additionally, the confidence interval for the effect size is 3.7% to 10.9%. Our study found a sample effect of 7.4%, but it is unlikely to equal the population effect exactly due to sampling error. The CI identifies a range that is likely to include the population effect.

For more information about this and other flu vaccine studies, read my post about Flu Vaccine Effectiveness.

In conclusion, by using a representative sample and the proper methodology, we made a statistical inference about vaccine effectiveness in an entire population.

Reference

Monto AS, Ohmit SE, Petrie JG, Johnson E, Truscon R, Teich E, Rotthoff J, Boulton M, Victor JC. Comparative efficacy of inactivated and live attenuated influenza vaccines. N Engl J Med. 2009;361(13):1260-7.