Alpha and Beta Risks

Alpha Risk

Alpha risk (α) is the risk of incorrectly deciding to reject the null hypothesis, H_O. If the chosen confidence level is 95%, then the alpha risk is 5% or 0.05.

For example, there is a 5% chance that a part has been determined defective when it actually is not. One has observed, or made a decision, that a difference exists but there really is none. Or when the data on a control chart indicates the process is out of control but in reality the process is in control. Or the likelihood of detecting an effect when no effect is present.

Alpha risk is also called False Positive, Type I Error, or "Producers Risk".

Confidence Level = 1 - Alpha Risk = 1 - α

Alpha is called the significance level of a test. The level of significance is commonly between 1% or 10% but can be any value depending on your desired level of confidence or need to reduce Type I error.

Selecting 5% signifies that there is a 5% chance that the observed variation is not actually the truth. The most common level for alpha risk is 5% but it varies by application and this value should be agreed upon with your BB/MBB.

In summary, it's the amount of risk you are willing to accept of making a Type I error.

If conducting a 2-sample T test and your conclusion is that the two means are different when they are actually not would represent Type I error:

EXAMPLES:

A carbon monoxide alarm indicating a high-level alert but there is actually not a high level - this is a Type I error.
The probability of scrapping good parts when there is not an actual defect. The "Producer" is taking a risk of losing money due to an incorrect decision, hence the analogy of why alpha-risk is also known as "Producer's Risk".
The probability of convicting an innocent person.

Beta Risk

Beta risk (β) is the risk that the decision will be made that the part is not defective when it really is. In other words, when the decision is made that a difference does not exist when there actually is. Or when the data on a control chart indicates the process is in control but in reality the process is out of control.

If the power desired is 90%, then the beta risk is 10%. There is a 10% chance that the decision will be made that the part is not defective when in reality it is defective.

Power = 1 - Beta risk = 1 - β

Beta risk is also called False Negative, Type II Error, or "Consumer's" Risk.

The Power is the probability of correctly rejecting the Null Hypothesis.

The Null Hypothesis is technically never proven true. It is "failed to reject" or "rejected".

"Failed to reject" does not mean accept the null hypothesis since it is established only to be proven false by testing the sample of data.

Guidelines: If the decision from the hypothesis test is looking for:

Large effects or LOW risk set Beta = 15% (which is Power of 0.85)
Medium effects, MEDIUM risk but not catastrophic, legal or safety related the set Beta = 10%
Small effects, HIGH risk, legal, safety, or critical set beta from 5% to near 0%.

If conducting an F-test and your conclusion is that the variances are the same when they are actually not would represent a Type II error.

Typically, this value is set between 10-20%.

Same note of caution as for alpha risk, the assumption for beta risk should be agreed upon with your BB/MBB.

EXAMPLES:

The risk of passing parts that are actually defective. The "Consumer" is taking a risk of due to the Producer making an incorrect decision and putting these bad parts out for the Consumer (or customer) to use or purchase, hence the analogy of why beta-risk is also known as "Consumer's Risk".

The risk (or probability) of not convicting (releasing) someone that is actually not innocent.

Decision Matrix

Visual Aid One

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Visual Aid Two

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Sampling

The size of the sample must be consciously decided on by the Six Sigma Project Manager based on the allowable alpha and beta-risks (statistical significance) and the magnitude of shift that you need to observe for a change of practical significance.

As the sample size increases, the estimate of the true population parameter gets stronger and can more reliably detect smaller differences.

This seems logical, the closer you get to analyzing all the population, the more accurate your inferences will be about that population.

More information about sampling can be found here.

Hypothesis Testing Steps
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Define the Problem
State the Objectives
Establish the Hypothesis (left tailed, right tailed, or two-tailed)
State the Null Hypothesis (H_O)
State the Alternative Hypothesis (H_A)
Select the appropriate statistical test
State the alpha-risk level
State the beta-risk level
Establish the Effect Size
Create Sampling Plan, determine sample size
Gather samples
Collect and record data
Calculate the test statistic and/or determine the p-value

If p-value is < than alpha-risk, reject H_O and accept the Alternative, H_A

If p-value is > than alpha-risk, fail to reject the Null, H_O

Try to re-run the test (if practical) to further confirm results. The next step is to take the statistical results and translate it to a practical solution.

It is also possible to determine the critical value of the test and use to calculated test statistic to determine the results. Either way, using the p-value approach or critical value provides the same result.

One Tail or Two Tail?

H_o_o: Mean A = Mean B is a two-tailed hypothesis

H_o: Mean A < Mean B is a one-tail hypothesis

H_o: Mean A > Mean B is also a one-tail hypothesis

In a one-tail test, the enter rejection region is on one side of the distribution. In a two-tail test the rejection regions exist on both sides of the distribution, above and below the critical value.

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Confidence INTERVAL vs. Confidence LEVEL

What the difference between a Confidence Level and Confidence Interval?

Confidence Level refers to the percentage of probability, or certainty, that the Confidence Interval would contain the true population parameter when you draw a random sample repeatedly.

Often results are shown for a “95% Confidence Interval” …. this means there is a 95% Confidence Level that the Confidence Interval contains the population parameter and 5% risk (alpha) of the Confidence Interval not containing the population parameter.

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