# Errors and Power

CO-6: Apply basic concepts of probability, random variation, and commonly used statistical probability distributions.
LO 6.28: Define a Type I and Type II error in general and in the context of specific scenarios.
LO 6.29: Explain the concept of the power of a statistical test including the relationship between power, sample size, and effect size.
Video: Errors and Power (12:03)

## Type I and Type II Errors in Hypothesis Tests

We have not yet discussed the fact that we are not guaranteed to make the correct decision by this process of hypothesis testing. Maybe you are beginning to see that there is always some level of uncertainty in statistics.

Let’s think about what we know already and define the possible errors we can make in hypothesis testing. When we conduct a hypothesis test, we choose one of two possible conclusions based upon our data.

If the p-value is smaller than your pre-specified significance level (α, alpha), you reject the null hypothesis and either

• You have made the correct decision since the null hypothesis is false

OR

• You have made an error (Type I) and rejected Ho when in fact Ho is true (your data happened to be a RARE EVENT under Ho)

If the p-value is greater than (or equal to) your chosen significance level (α, alpha), you fail to reject the null hypothesis and either

• You have made the correct decision since the null hypothesis is true

OR

• You have made an error (Type II) and failed to reject Ho when in fact Ho is false (the alternative hypothesis, Ha, is true)

The following summarizes the four possible results which can be obtained from a hypothesis test. Notice the rows represent the decision made in the hypothesis test and the columns represent the (usually unknown) truth in reality. Although the truth is unknown in practice – or we would not be conducting the test – we know it must be the case that either the null hypothesis is true or the null hypothesis is false. It is also the case that either decision we make in a hypothesis test can result in an incorrect conclusion!

TYPE I Error occurs when we Reject Ho when, in fact, Ho is True. In this case, we mistakenly reject a true null hypothesis.

• P(TYPE I Error) = P(Reject Ho | Ho is True) = α = alpha = Significance Level

TYPE II Error occurs when we fail to Reject Ho when, in fact, Ho is False. In this case we fail to reject a false null hypothesis.

• P(TYPE II Error) = P(Fail to Reject Ho | Ho is False) = β = beta

When our significance level is 5%, we are saying that we will allow ourselves to make a Type I error less than 5% of the time. In the long run, if we repeat the process, 5% of the time we will find a p-value < 0.05 when in fact the null hypothesis was true.

In this case, our data represent a rare occurrence which is unlikely to happen but is still possible. For example, suppose we toss a coin 10 times and obtain 10 heads, this is unlikely for a fair coin but not impossible. We might conclude the coin is unfair when in fact we simply saw a very rare event for this fair coin.

Our testing procedure CONTROLS for the Type I error when we set a pre-determined value for the significance level.

Notice that these probabilities are conditional probabilities. This is one more reason why conditional probability is an important concept in statistics.

Unfortunately, calculating the probability of a Type II error requires us to know the truth about the population. In practice we can only calculate this probability using a series of “what if” calculations which depend upon the type of problem.

Comment: As you initially read through the examples below, focus on the broad concepts instead of the small details. It is not important to understand how to calculate these values yourself at this point.

• Try to understand the pictures we present.  Which pictures represent an assumed null hypothesis and which represent an alternative?
• It may be useful to come back to this page (and the activities here) after you have reviewed the rest of the section on hypothesis testing and have worked a few problems yourself.
Interactive Applet: Statistical Significance

Here are two examples of using an older version of this applet. It looks slightly different but the same settings and options are available in the version above.

In both cases we will consider IQ scores.

Our null hypothesis is that the true mean is 100. Assume the standard deviation is 16 and we will specify a significance level of 5%.

## EXAMPLE:

In this example we will specify that the true mean is indeed 100 so that the null hypothesis is true. Most of the time (95%), when we generate a sample, we should fail to reject the null hypothesis since the null hypothesis is indeed true.

Here is one sample that results in a correct decision: In the sample above, we obtain an x-bar of 105, which is drawn on the distribution which assumes μ (mu) = 100 (the null hypothesis is true). Notice the sample is shown as blue dots along the x-axis and the shaded region shows for which values of x-bar we would reject the null hypothesis. In other words, we would reject Ho whenever the x-bar falls in the shaded region.

Enter the same values and generate samples until you obtain a Type I error (you falsely reject the null hypothesis). You should see something like this: If you were to generate 100 samples, you should have around 5% where you rejected Ho. These would be samples which would result in a Type I error.

The previous example illustrates a correct decision and a Type I error when the null hypothesis is true. The next example illustrates a correct decision and Type II error when the null hypothesis is false. In this case, we must specify the true population mean.

## EXAMPLE:

Let’s suppose we are sampling from an honors program and that the true mean IQ for this population is 110. We do not know the probability of a Type II error without more detailed calculations. In the sample above, we obtain an x-bar of 111, which is drawn on the distribution which assumes μ (mu) = 100 (the null hypothesis is true).

Enter the same values and generate samples until you obtain a Type II error (you fail to reject the null hypothesis). You should see something like this: You should notice that in this case (when Ho is false), it is easier to obtain an incorrect decision (a Type II error) than it was in the case where Ho is true. If you generate 100 samples, you can approximate the probability of a Type II error.

We can find the probability of a Type II error by visualizing both the assumed distribution and the true distribution together. The image below is adapted from an applet we will use when we discuss the power of a statistical test. There is a 37.4% chance that, in the long run, we will make a Type II error and fail to reject the null hypothesis when in fact the true mean IQ is 110 in the population from which we sample our 10 individuals.

Can you visualize what will happen if the true population mean is really 115 or 108? When will the Type II error increase? When will it decrease? We will look at this idea again when we discuss the concept of power in hypothesis tests.

• It is important to note that there is a trade-off between the probability of a Type I and a Type II error. If we decrease the probability of one of these errors, the probability of the other will increase! The practical result of this is that if we require stronger evidence to reject the null hypothesis (smaller significance level = probability of a Type I error), we will increase the chance that we will be unable to reject the null hypothesis when in fact Ho is false (increases the probability of a Type II error).
• When α (alpha) = 0.05 we obtained a Type II error probability of 0.374 = β = beta • When α (alpha) = 0.01 (smaller than before) we obtain a Type II error probability of 0.644 = β = beta (larger than before) • As the blue line in the picture moves farther right, the significance level (α, alpha) is decreasing and the Type II error probability is increasing.
• As the blue line in the picture moves farther left, the significance level (α, alpha) is increasing and the Type II error probability is decreasing

Let’s return to our very first example and define these two errors in context.

## EXAMPLE:

A case of suspected cheating on an exam is brought in front of the disciplinary committee at a certain university.

There are two opposing claims in this case:

• Ho = The student’s claim: I did not cheat on the exam.
• Ha = The instructor’s claim: The student did cheat on the exam.

Adhering to the principle “innocent until proven guilty,” the committee asks the instructor for evidence to support his claim.

There are four possible outcomes of this process. There are two possible correct decisions:

• The student did cheat on the exam and the instructor brings enough evidence to reject Ho and conclude the student did cheat on the exam. This is a CORRECT decision!
• The student did not cheat on the exam and the instructor fails to provide enough evidence that the student did cheat on the exam. This is a CORRECT decision!

Both the correct decisions and the possible errors are fairly easy to understand but with the errors, you must be careful to identify and define the two types correctly.

TYPE I Error: Reject Ho when Ho is True

• The student did not cheat on the exam but the instructor brings enough evidence to reject Ho and conclude the student cheated on the exam. This is a Type I Error.

TYPE II Error: Fail to Reject Ho when Ho is False

• The student did cheat on the exam but the instructor fails to provide enough evidence that the student cheated on the exam. This is a Type II Error.

In most situations, including this one, it is more “acceptable” to have a Type II error than a Type I error. Although allowing a student who cheats to go unpunished might be considered a very bad problem, punishing a student for something he or she did not do is usually considered to be a more severe error. This is one reason we control for our Type I error in the process of hypothesis testing.

Comment:

• The probabilities of Type I and Type II errors are closely related to the concepts of sensitivity and specificity that we discussed previously. Consider the following hypotheses:

Ho: The individual does not have diabetes (status quo, nothing special happening)

Ha: The individual does have diabetes (something is going on here)

In this setting:

When someone tests positive for diabetes we would reject the null hypothesis and conclude the person has diabetes (we may or may not be correct!).

When someone tests negative for diabetes we would fail to reject the null hypothesis so that we fail to conclude the person has diabetes (we may or may not be correct!)

Let’s take it one step further:

Sensitivity = P(Test + | Have Disease) which in this setting equals
P(Reject Ho | Ho is False) = 1 – P(Fail to Reject Ho | Ho is False) = 1 – β = 1 – beta

Specificity = P(Test – | No Disease) which in this setting equals
P(Fail to Reject Ho | Ho is True) = 1 – P(Reject Ho | Ho is True) = 1 – α = 1 – alpha

Notice that sensitivity and specificity relate to the probability of making a correct decision whereas α (alpha) and β (beta) relate to the probability of making an incorrect decision.

Usually α (alpha) = 0.05 so that the specificity listed above is 0.95 or 95%.

Next, we will see that the sensitivity listed above is the power of the hypothesis test!

## Reasons for a Type I Error in Practice

Assuming that you have obtained a quality sample:

• The reason for a Type I error is random chance.
• When a Type I error occurs, our observed data represented a rare event which indicated evidence in favor of the alternative hypothesis even though the null hypothesis was actually true.

## Reasons for a Type II Error in Practice

Again, assuming that you have obtained a quality sample, now we have a few possibilities depending upon the true difference that exists.

• The sample size is too small to detect an important difference. This is the worst case, you should have obtained a larger sample. In this situation, you may notice that the effect seen in the sample seems PRACTICALLY significant and yet the p-value is not small enough to reject the null hypothesis.
• The sample size is reasonable for the important difference but the true difference (which might be somewhat meaningful or interesting) is smaller than your test was capable of detecting. This is tolerable as you were not interested in being able to detect this difference when you began your study. In this situation, you may notice that the effect seen in the sample seems to have some potential for practical significance.
• The sample size is more than adequate, the difference that was not detected is meaningless in practice. This is not a problem at all and is in effect a “correct decision” since the difference you did not detect would have no practical meaning.
• Note: We will discuss the idea of practical significance later in more detail.

## Power of a Hypothesis Test

It is often the case that we truly wish to prove the alternative hypothesis. It is reasonable that we would be interested in the probability of correctly rejecting the null hypothesis. In other words, the probability of rejecting the null hypothesis, when in fact the null hypothesis is false. This can also be thought of as the probability of being able to detect a (pre-specified) difference of interest to the researcher.

Let’s begin with a realistic example of how power can be described in a study.

## EXAMPLE:

In a clinical trial to study two medications for weight loss, we have an 80% chance to detect a difference in the weight loss between the two medications of 10 pounds. In other words, the power of the hypothesis test we will conduct is 80%.

In other words, if one medication comes from a population with an average weight loss of 25 pounds and the other comes from a population with an average weight loss of 15 pounds, we will have an 80% chance to detect that difference using the sample we have in our trial.

If we were to repeat this trial many times, 80% of the time we will be able to reject the null hypothesis (that there is no difference between the medications) and 20% of the time we will fail to reject the null hypothesis (and make a Type II error!).

The difference of 10 pounds in the previous example, is often called the effect size. The measure of the effect differs depending on the particular test you are conducting but is always some measure related to the true effect in the population. In this example, it is the difference between two population means.

Recall the definition of a Type II error:

A TYPE II Error occurs when we fail to Reject Ho when, in fact, Ho is False. In this case we fail to reject a false null hypothesis.

P(TYPE II Error) = P(Fail to Reject Ho | Ho is False) = β = beta

Notice that P(Reject Ho | Ho is False) = 1 – P(Fail to Reject Ho | Ho is False) = 1 – β = 1- beta.

The POWER of a hypothesis test is the probability of rejecting the null hypothesis when the null hypothesis is false. This can also be stated as the probability of correctly rejecting the null hypothesis.

POWER = P(Reject Ho | Ho is False) = 1 – β = 1 – beta

Power is the test’s ability to correctly reject the null hypothesis. A test with high power has a good chance of being able to detect the difference of interest to us, if it exists.

As we mentioned on the bottom of the previous page, this can be thought of as the sensitivity of the hypothesis test if you imagine Ho = No disease and Ha = Disease.

## Factors Affecting the Power of a Hypothesis Test

The power of a hypothesis test is affected by numerous quantities (similar to the margin of error in a confidence interval).

Assume that the null hypothesis is false for a given hypothesis test. All else being equal, we have the following:

• Larger samples result in a greater chance to reject the null hypothesis which means an increase in the power of the hypothesis test.
• If the effect size is larger, it will become easier for us to detect. This results in a greater chance to reject the null hypothesis which means an increase in the power of the hypothesis test. The effect size varies for each test and is usually closely related to the difference between the hypothesized value and the true value of the parameter under study.
• From the relationship between the probability of a Type I and a Type II error (as α (alpha) decreases, β (beta) increases), we can see that as α (alpha) decreases, Power = 1 – β = 1 – beta also decreases.
• There are other mathematical ways to change the power of a hypothesis test, such as changing the population standard deviation; however, these are not quantities that we can usually control so we will not discuss them here.
In practice, we specify a significance level and a desired power to detect a difference which will have practical meaning to us and this determines the sample size required for the experiment or study.

For most grants involving statistical analysis, power calculations must be completed to illustrate that the study will have a reasonable chance to detect an important effect. Otherwise, the money spent on the study could be wasted. The goal is usually to have a power close to 80%.

For example, if there is only a 5% chance to detect an important difference between two treatments in a clinical trial, this would result in a waste of time, effort, and money on the study since, when the alternative hypothesis is true, the chance a treatment effect can be found is very small.

Comment:

• In order to calculate the power of a hypothesis test, we must specify the “truth.” As we mentioned previously when discussing Type II errors, in practice we can only calculate this probability using a series of “what if” calculations which depend upon the type of problem.

The following activity involves working with an interactive applet to study power more carefully.

The following reading is an excellent discussion about Type I and Type II errors.

(Optional) Outside Reading: A Good Discussion of Power  (≈ 2500 words)

We will not be asking you to perform power calculations manually. You may be asked to use online calculators and applets. Most statistical software packages offer some ability to complete power calculations. There are also many online calculators for power and sample size on the internet, for example, Russ Lenth’s power and sample-size page.