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In order to properly interpret a statistical analysis you should have an understanding of hypothesis testing.

To put hypothesis testing in lay terms: think of it as a jury trial. In the U.S., a trial is conducted with the assumption that defendant is innocent. Evidence is then presented by a prosecutor to show to the jury evidence of the person’s guilt. If there is a preponderance of evidence of the defendant’s guilt, the jury is instructed to find the defendant guilty (you reject innocence).

That is how hypothesis testing works. The assumption of innocence in a scientific experiment is called a “null” hypothesis (labeled Ho). Typically, in a medical setting at least, this means that there are no treatment differences. The null hypothesis will typically be a statement such as one of these: there is no difference in group means, no linear association between two variables, no difference in distributions, and so on.

In science or industry or marketing an experiment is designed to determine whether there is enough evidence to refute the null hypothesis. Since there is no twelve-person jury, that “decision” is made by a statement of probability (however, the research is still the final judge.) Thus, if the research evidence indicates that the observations made during the trial were improbable under the assumption of the null hypothesis, you would conclude that you have “statistically significant” evidence to reject the null hypothesis. However, if you do not gather sufficient evidence to reject Ho, this does not prove that the null hypothesis is true, only that you did not have enough evidence to “prove the case.” (In a trial, the person may still be guilty even if the evidence was not sufficient to convict.)

Here is the way a typical set of hypotheses are written:

H_o: m₁ = m ₂   (The population means of the two groups are the same.)

H_a: m ₁ ≠ m ₂   (The population means of the two groups are different.)

These particular hypotheses are for a two-sample t-test. The mathematical form of the null hypothesis is m₁ = m ₂ ). It is also common to express the null hypothesis in words, as shown above.

The alternative hypothesis is usually what the investigator wants to show or suspects is true. The alternative in the example above is called a two-tailed alternative (also called a two-sided alternative.) That is, reject Ho if there is sufficient evidence that the null is not true. For a one-tailed alternative (e.g. Ha: m ₁ >  m ₂ ) we would reject Ho only if the evidence against Ho tends to support Ha.

In hypothesis testing, two types of errors can occur as illustrated in table below. The top classification is the “truth” which you do not know. The left categories are your decisions.

For example, if you reject Ho when it is false, you’ve made a correct decision (upper-right cell.) However, if you reject Ho when it is true, you’ve made a “Type I error” (upper left cell.) This error has a particular name, alpha, noted by the Greek character a. In a correctly designed experiment, you make your decision to reject Ho based on a probability statement – how rare you would see the results under the assumption of the null hypothesis (innocence.) If that probability turns out to be small, say at 0.05, you conclude that this is sufficient evidence to reject innocence and proclaim guilt. This means that you are willing to make a Type I error 5% of the time, or 1 in 20 times. (That’s why in some jury trials you only have to show a preponderance of evidence (and the vote of the majority of the jury rules), whereas in something like a capital murder case your alpha has to be very, very small so that you have “not a shadow of doubt.” that you’re making an error – the jury has to be unanimous.) Alpha – the probability at which you will make a decision of guilt -- is referred to the experiment’s significance level.

On the other hand, if Ho is false (the person is indeed guilty) and you do not reject Ho, you commit a Type II error – you let a guilt person off the hook. The probability of committing a Type II error is called beta (b.) The power of the test is defined to be one minus b. When a test has low power it means that you are likely to make a Type II error, (i.e., fail to reject Ho when it is actually false.) Looking at it the other way, the higher the “power”, the better your chance of rejecting Ho when it is false – the better your chance of finding a difference when it in fact exists.

Another important point is that for any given level of significance (a), power can be increased by increasing the sample size (more evidence means a more informed decision.) Thus, sample size should be a consideration when embarking on an experiment. Many negative (non-significant) studies reported in the literature are the result of inadequate sample size (resulting in poor power) (Moher, 1994). Thus, there could have actually BEEN a difference in reality, but not enough evidence was gathers to show the difference. Therefore, the process of selecting a sample size for your analysis should begin early in your study. For more information concerning hypothesis testing see a standard statistical text such as Moore et al (2006). For a good discussion of power and sample size see Keppel & Wickens (2004).


References

1. Elliott, A.C., & Woodward, W.A. (2006) Statistical Analysis Quick Reference Guidebook, Sage.

2. Keppel, G. & Wickens, T.D. (2004). Design and Analysis: A Researcher’s Handbook (4th Ed.), Pearson Prentice Hall.

3. Moher D, Dulberg CS, Wells GA. Statistical power, sample size, and their reporting in randomized controlled trials. JAMA. 1994;272:122-124.

4. Moore, D. and McCabe, G. (2006). Introduction to the Practice of Statistics, Fourth Edition, New York: WH Freeman & Co.
   
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