One of the most interesting introductions to the idea of statistical power is given in the 'OJ' Page which was created by Rob Becker to illustrate how the decision a jury has to reach (guilty vs. not guilt) is similar to the decision a researcher makes when assessing a relationship. The OJ Page uses the infamous OJ Simpson murder trial to introduce the idea of statistical power and illustrate how manipulating various factors (e.g., the amount of evidence, the "effect size", and the level of risk) affects the validity of the verdict.
There are four interrelated components that influence the conclusions you might reach from a statistical test in a research project. The logic of statistical inference with respect to these components is often difficult to understand and explain. This paper attempts to clarify the four components and describe their interrelationships.
The four components are:
- sample size, or the number of units (e.g., people) accessible to the study
- effect size, or the salience of the treatment relative to the noise in measurement
- alpha level (a, or significance level), or the odds that the observed result is due to chance
- power, or the odds that you will observe a treatment effect when it occurs
Given values for any three of these components, it is possible to compute the value of the fourth. For instance, you might want to determine what a reasonable sample size would be for a study. If you could make reasonable estimates of the effect size, alpha level and power, it would be simple to compute (or, more likely, look up in a table) the sample size.
Some of these components will be more manipulable than others depending on the circumstances of the project. For example, if the project is an evaluation of an educational program or counseling program with a specific number of available consumers, the sample size is set or predetermined. Or, if the drug dosage in a program has to be small due to its potential negative side effects, the effect size may consequently be small. The goal is to achieve a balance of the four components that allows the maximum level of power to detect an effect if one exists, given programmatic, logistical or financial constraints on the other components.
Figure 1 shows the basic decision matrix involved in a statistical conclusion. All statistical conclusions involve constructing two mutually exclusive hypotheses, termed the null (labeled H0) and alternative (labeled H1) hypothesis. Together, the hypotheses describe all possible outcomes with respect to the inference. The central decision involves determining which hypothesis to accept and which to reject. For instance, in the typical case, the null hypothesis might be:
H0: Program Effect = 0
while the alternative might be
H1: Program Effect <> 0
The null hypothesis is so termed because it usually refers to the "no difference" or "no effect" case. Usually in social research we expect that our treatments and programs will make a difference. So, typically, our theory is described in the alternative hypothesis.
Figure 1 below is a complex figure that you should take some time studying. First, look at the header row (the shaded area). This row depicts reality -- whether there really is a program effect, difference, or gain. Of course, the problem is that you never know for sure what is really happening (unless youre God). Nevertheless, because we have set up mutually exclusive hypotheses, one must be right and one must be wrong. Therefore, consider this the view from Gods position, knowing which hypothesis is correct. The first column of the 2x2 table shows the case where our program does not have an effect; the second column shows where it does have an effect or make a difference.
The left header column describes the world we mortals live in. Regardless of whats true, we have to make decisions about which of our hypotheses is correct. This header column describes the two decisions we can reach -- that our program had no effect (the first row of the 2x2 table) or that it did have an effect (the second row).
Now, lets examine the cells of the 2x2 table. Each cell shows the Greek symbol for that cell. Notice that the columns sum to 1 (i.e., a + (1-a) = 1 and b + (1-b) = 1). Why can we sum down the columns, but not across the rows? Because if one column is true, the other is irrelevant -- if the program has a real effect (the right column) it cant at the same time not have one. Therefore, the odds or probabilities have to sum to 1 for each column because the two rows in each column describe the only possible decisions (accept or reject the null/alternative) for each possible reality.
Below the Greek symbol is a typical value for that cell. You should especially note the values in the bottom two cells. The value of a is typically set at .05 in the social sciences. A newer, but growing, tradition is to try to achieve a statistical power of at least .80. Below the typical values is the name typically given for that cell (in caps). If you havent already, you should note that two of the cells describe errors -- you reach the wrong conclusion -- and in the other two you reach the correct conclusion. Sometimes its hard to remember which error is Type I and which is Type II. If you keep in mind that Type I is the same as the a or significance level, it might help you to remember that it is the odds of finding a difference or effect by chance alone. People are more likely to be susceptible to a Type I error, because they almost always want to conclude that their program works. If they find a statistical effect, they tend to advertise it loudly. On the other hand, people probably check more thoroughly for Type II errors because when you find that the program was not demonstrably effective, you immediately start looking for why (in this case, you might hope to show that you had low power and high b -- that the odds of saying there was no treatment effect even when there was were too high). Following the capitalized common name are several different ways of describing the value of each cell, one in terms of outcomes and one in terms of theory-testing. In italics, we give an example of how to express the numerical value in words.
To better understand the strange relationships between the two columns, think about what happens if you want to increase your power in a study. As you increase power, you increase the chances that you are going to find an effect if its there (wind up in the bottom row). But, if you increase the chances that you wind up in the bottom row, you must at the same time be increasing the chances of making a Type I error! Although we cant sum to 1 across rows, there is clearly a relationship. Since we usually want high power and low Type I Error, you should be able to appreciate that we have a built-in tension here.
H1 (alternative hypothesis) false
|H0 (null hypothesis)
H1 (alternative hypothesis) true
|We accept the null hypothesis (H0)
We reject the alternative hypothesis (H1)
THE CONFIDENCE LEVEL
The odds of saying there is no relationship, difference, gain, when in fact there is none
The odds of correctly not confirming our theory
95 times out of 100 when there is no effect, well say there is none
TYPE II ERROR
The odds of saying there is no relationship, difference, gain, when in fact there is one
The odds of not confirming our theory when its true
20 times out of 100, when there is an effect, well say there isnt
|We reject the null hypothesis (H0)
We accept the alternative hypothesis (H1)
TYPE I ERROR
The odds of saying there is an relationship, difference, gain, when in fact there is not
The odds of confirming our theory incorrectly
5 times out of 100, when there is no effect, well say there is on
We should keep this small when we cant afford/risk wrongly concluding that our program works
The odds of saying that there is an relationship, difference, gain, when in fact there is one
The odds of confirming our theory correctly
80 times out of 100, when there is an effect, well say there is
We generally want this to be as large as possible
Figure 1. The Statistical Inference Decision Matrix
We often talk about alpha (a) and beta (b) using the language of "higher" and "lower." For instance, we might talk about the advantages of a higher or lower a-level in a study. You have to be careful about interpreting the meaning of these terms. When we talk about higher a-levels, we mean that we are increasing the chance of a Type I Error. Therefore, a lower a-level actually means that you are conducting a more rigorous test. With all of this in mind, lets consider a few common associations evident in the table. You should convince yourself of the following:
- the lower the a, the lower the power; the higher the a, the higher the power
- the lower the a, the less likely it is that you will make a Type I Error (i.e., reject the null when its true)
- the lower the a, the more "rigorous" the test
- an a of .01 (compared with .05 or .10) means the researcher is being relatively careful, s/he is only willing to risk being wrong 1 in a 100 times in rejecting the null when its true (i.e., saying theres an effect when there really isnt)
- an a of .01 (compared with .05 or .10) limits ones chances of ending up in the bottom row, of concluding that the program has an effect. This means that both your statistical power and the chances of making a Type I Error are lower.
- an a of .01 means you have a 99% chance of saying there is no difference when there in fact is no difference (being in the upper left box)
- increasing a (e.g., from .01 to .05 or .10) increases the chances of making a Type I Error (i.e., saying there is a difference when there is not), decreases the chances of making a Type II Error (i.e., saying there is no difference when there is) and decreases the rigor of the test
- increasing a (e.g., from .01 to .05 or .10) increases power because one will be rejecting the null more often (i.e., accepting the alternative) and, consequently, when the alternative is true, there is a greater chance of accepting it (i.e., power)
Copyright ©2006, William M.K. Trochim, All Rights Reserved
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Last Revised: 10/20/2006