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9
Power and Sample Size ■
Power and Sample Size Overview, 9-2
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Z-Test and t-Tests, 9-4
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Tests of Proportions, 9-7
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One-Way Analysis Of Variance, 9-10
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Two-Level Factorial and Plackett-Burman Designs, 9-13
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Chapter 9
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Power and Sample Size Overview
Power and Sample Size Overview Use MINITAB’s power and sample size capabilities to evaluate power and sample size before you design and run an experiment (prospective) or after you perform an experiment (retrospective). ■
A prospective study is used before collecting data to consider design sensitivity. You want to be sure that you have enough power to detect differences (effects) that you have determined to be important. For example, you can increase the design sensitivity by increasing the sample size or by taking measures to decrease the error variance.
■
A retrospective study is used after collecting data to help understand the power of the tests that you have performed. For example, suppose you conduct an experiment and the data analysis does not reveal any statistically significant results. You can then calculate power based on the minimum difference (effect) you wish to detect. If the power to detect this difference is low, you may want to modify your experimental design to increase the power and continue to evaluate the same problem. However, if the power is high, you may want to conclude that there is no meaningful difference (effect) and discontinue experimentation.
MINITAB provides power, sample size, and difference (effect) calculations (also the number of center points for factorial and Plackett-Burman designs) for the following procedures: ■
one-sample Z
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one-sample proportion
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two-level factorial designs
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one-sample t
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two-sample proportion
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Plackett-Burman designs
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two-sample t
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one-way analysis of variance
What is power? There are four possible outcomes for a hypothesis test. The outcomes depend on whether the null hypothesis (H0) is true or false and whether you decide to “reject” or “fail to reject” H0. The power of a test is the probability of correctly rejecting H0 when it is false. In other words, power is the likelihood that you will identify a significant difference (effect) when one exists. The four possible outcomes are summarized below: Null hypothesis Decision
True
False
fail to reject H0
correct decision p=1−α
Type II error p=β
reject H0
Type I error p=α
correct decision p=1−β
power
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Power and Sample Size Overview
HOW TO USE Power and Sample Size
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When H0 is true and you reject it, you make a type I error. The probability (p) of making a type I error is called alpha (α) and is sometimes referred to as the level of significance for the test.
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When H0 is false and you fail to reject it, you make a type II error. The probability (p) of making a type II error is called beta (β).
Choosing probability levels When you are determining the α and β values for your test, you should consider the ■
■
severity of making an error—The more serious the error, the less often you should be willing to allow it to occur. Therefore, you should assign smaller probability values to more serious errors. magnitude of effect you want to detect—Power is the probability (p = 1 − β) of correctly rejecting H0 when it is false. Ideally, you want to have high power to detect a difference that you care about, and low power for a meaningless difference. For example, suppose you want to claim that children in your school scored higher than the general population on a standardized achievement test. You need to decide how much higher than the general population your test scores need to be so you are not making claims that are misleading. If your mean test score is only .7 points higher than the general population on a 100 point test, do you really want to detect a difference? Probably not. Therefore, you should choose your sample size so that you only have power to detect differences that you consider meaningful.
Factors that influence power A number of factors influence power: ■
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α, the probability of a type I error (level of significance). As the probability of a type I error (α) increases, the probability of a type II error (β) decreases. Hence, as α increases, power = 1 − β also increases. σ, the variability in the population. As σ increases, power decreases.
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the size of the population difference (effect). As the size of population difference (effect) decreases, power decreases.
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sample size. As sample size increases, power increases.
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Chapter 9
HOW TO USE Z-Test and t-Tests
Z-Test and t-Tests Z- and t-tests are used to perform hypothesis tests of the mean (one-sample) or the difference in means (two-sample). For these tests, you can calculate the ■
power
■
sample size
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minimum difference (effect)
You need to determine what are acceptable values for any two of these parameters and MINITAB will solve for the third. For example, to calculate the sample size, you need to determine the power and the minimum difference that you consider to be acceptable. Then, MINITAB solves for the sample size you need to be able to reject the null hypothesis when the true value differs from the hypothesized value by this minimum difference. See Defining the minimum difference on page 9-5. h To calculate power, sample size, or minimum difference 1 Choose Stat ➤ Power and Sample Size ➤ 1-Sample Z, 1-Sample t, or 2-Sample t.
This dialog box is for a one-sample Z-test. The dialog boxes for the 1- and 2-Sample t are identical.
2 Do one of the following: ■
Solve for power 1 In Sample sizes, enter one or more numbers. For a two-sample test, the
number you enter is considered the sample size for each group. For example, if you want to determine power for an analysis with 10 observations in each group for a total of 20, you would enter 10. 2 In Differences, enter one or more numbers. ■
Solve for sample size 1 In Differences, enter one or more numbers. 2 In Power values, enter one or more numbers.
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HOW TO USE Power and Sample Size
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Solve for the minimum difference 1 In Sample sizes, enter one or more numbers. For a two-sample test, the
number you enter is considered the sample size for each group. 2 In Power values, enter one or more numbers.
MINITAB will solve for all combinations of the specified values. For example, if you enter 3 values in Sample sizes and 2 values in Differences, MINITAB will compute the power for all 6 combinations of sample sizes and differences. For a discussion of the value needed in Differences, see Defining the minimum difference on page 9-5. 3 In Sigma, enter an estimate of the population standard deviation (σ) for your data.
See Estimating σ on page 9-6.
4 If you like, use one or more of the options listed below, then click OK.
Options Options subdialog box ■
define the alternative hypothesis by choosing less than (lower-tailed), not equal (two-tailed), or greater than (upper-tailed). The default is a two-tailed test.
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specify the significance level (α). The default is 0.05.
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store the sample sizes, differences (effects), and power values. When calculating sample size, MINITAB stores the power value that will generate the nearest integer sample size.
Defining the minimum difference When calculating sample size or power, you need to specify the minimum difference you are interested in detecting. The manner in which you express this difference depends on whether you are performing a one- or two-sample test: ■
For a one-sample Z- or t-test, express the difference in of the null hypothesis. For example, suppose you are testing whether or not your students’ mean test score is different from the population mean. You would like to detect a difference of three points. In the dialog box, you would enter 3 in Differences.
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For a two-sample t-test, express the difference as the difference between the population means that you would like to be able to detect. For example, suppose you are investigating the effects of water acidity on the growth of two groups of tadpoles. You decide that any difference in growth between the two groups that is smaller than 4 mm is not important. In the dialog box, you would enter 4 in Differences.
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Chapter 9
HOW TO USE Z-Test and t-Tests
Estimating σ For power or minimum difference calculations, the estimate of σ depends on whether or not you have already collected data. ■
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Prospective studies are done before collecting data so σ has to be estimated. You can use related research, pilot studies, or subject-matter knowledge to estimate σ. Retrospective studies are done after data have been collected so you can use the sample standard deviation to estimate σ. You could also use related research, pilot studies, or subject-matter knowledge. Use Display Descriptive Statistics (page 1-6) to calculate the sample standard deviation.
For sample size calculations, the data have not been collected yet so the population standard deviation (σ) has to be estimated. You can use related research, pilot studies, or subject-matter knowledge to estimate σ. Note
By default, MINITAB sets σ to 1.0. This is fine if the differences (effects) are standardized, but will present erroneous results if they are not. When the differences (effects) are not standardized, be sure to enter an estimate of σ.
e Example of calculating sample size for a one-sample t-test
Suppose you are the production manager at a dairy plant. In order to meet state requirements, you must maintain strict control over the packaging of ice cream. The volume cannot vary more than 3 oz for a half gallon (64 oz) container. The packaging machine tolerances are set so the process σ is 1. How many samples must be taken to estimate the mean package volume at a confidence level of 99% (α = .01) for power values of 0.7, 0.8, and 0.9? 1 Choose Stat ➤ Power and Sample Size ➤ 1-Sample t. 2 In Differences, enter 3. In Power values, enter 0.7 0.8 0.9. 3 In Sigma, enter 1. 4 Click Options. In Significance level, enter 0.01. Click OK in each dialog box.
Session window output
Power and Sample Size 1-Sample t Test Testing mean = null (versus not = null) Calculating power for mean = null + difference Alpha = 0.01 Sigma = 1 Difference 3 3 3
Sample Size 5 5 6
Target Power 0.7000 0.8000 0.9000
Actual Power 0.8947 0.8947 0.9827
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Tests of Proportions
HOW TO USE Power and Sample Size
Interpreting the results MINITAB displays the sample size required to obtain the requested power values. Because the target power values would result in non-integer sample sizes, MINITAB displays the power (Actual Power) that you would have to detect differences in volume greater than three ounces using the nearest integer value for sample size. If you take a sample of five cartons, power for your test is 0.895; for a sample of six cartons, power is 0.983.
Tests of Proportions Proportion tests are used to perform hypothesis tests of a proportion (one-sample) or the difference in proportions (two-sample). For these tests, you can calculate the ■
power
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sample size
■
minimum difference (effect)
You need to determine what are acceptable values for any two of these parameters and MINITAB will solve for the third. For example, to calculate the sample size, you need to determine the power and the minimum difference that you consider to be acceptable. Then, MINITAB solves for the sample size you need to be able to reject the null hypothesis when the true value differs from the hypothesized value by this minimum difference. See Defining the minimum difference on page 9-9. h To calculate power, sample size, or minimum difference 1 Choose Stat ➤ Power and Sample Size ➤ 1 Proportion or 2 Proportions.
1 Proportion
2 Proportions
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Chapter 9
Tests of Proportions 2 Do one of the following: ■
Solve for power 1 In Sample sizes, enter one or more numbers. For two proportion test, the
number you enter is considered the sample size for each group. For example, if you want to determine power for an analysis with 10 observations in each group for a total of 20, you would enter 10. 2 In Alternative values of p or Proportion 1 values, enter one or more proportions. ■
Solve for sample size 1 In Alternative values of p or Proportion 1 values, enter one or more
proportions. 2 In Power values, enter one or more numbers. ■
Solve for the minimum difference 1 In Sample sizes, enter one or more numbers. For a two proportion test, the
number you enter is considered the sample size for each group, not the total number for the experiment. 2 In Power values, enter one or more numbers.
MINITAB will solve for all combinations of the specified values. For example, if you enter 3 values in Sample sizes and 2 values in Alternative values of p, MINITAB will compute the power for all 6 combinations of sample sizes and alternative proportions. For a discussion of the values needed in Alternative values of p and Proportion 1 values, see Defining the minimum difference on page 9-9. 3 Do one of the following: ■
For a one-sample test, enter the expected proportion under the null hypothesis in Hypothesized p. The default is 0.5.
■
For a two-sample test, enter the second proportion in Proportion 2. The default is 0.5.
For a discussion of the values needed in Hypothesized p and Proportion 2, see Defining the minimum difference on page 9-9. 4 If you like, use one or more of the options listed below, then click OK.
Options Options subdialog box ■
■
define the alternative hypothesis by choosing less than (lower-tailed), not equal (two-tailed), or greater than (upper-tailed). The default is a two-tailed test. specify the significance level of the test. The default is α = 0.05.
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Tests of Proportions
HOW TO USE Power and Sample Size
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store the sample sizes, alternative values of p or proportion 1 values, and power values. When calculating sample size, MINITAB stores the power value that will generate the nearest integer sample size.
Defining the minimum difference MINITAB uses two proportions to determine the minimum difference. The manner in which you express these proportions depends on whether you are performing a one- or two-sample proportion test. ■
For a one-sample test of proportion, enter the expected proportion under the null hypothesis for Hypothesized p in the dialog box. Suppose you are testing whether the data are consistent with the following null hypothesis and would like to detect any differences where the true proportion is greater than .73. H0: p = .7
H1: p > .7
where p is the population proportion
In MINITAB, enter .73 in Alternative values of p; enter .7 in Hypothesized p. (The alternative proportion is not the value of the alternative hypothesis, but the value at which you want to evaluate power.) ■
For a two-sample test of proportion, enter the expected proportions under the null hypothesis for Proportion 2 in the dialog box. Suppose a biologist wants to test whether or not there is a difference in the proportion of fish that have been affected by pollution in two lakes. Previous research suggests that approximately 25% of fish have been affected. The biologist would like to detect a difference in proportions of 0.03. H0: p1 = p2
H1: p1 ≠ p2
In MINITAB, enter 0.22 and 0.28 in Proportion 1 values; enter 0.25 in Proportion 2. e Example of calculating power for a two-sample test of proportion
As a political advisor, you want to determine whether there is a difference between the proportion of men and the proportion of women who a tax reform bill. Results of a previous survey of ed voters indicate that 30% (p = .30) of the voters the tax bill. If you mail 1000 surveys, what is the power to detect differences greater than .05 between the proportions of men and women who the tax bill? 1 Choose Stat ➤ Power and Sample Size ➤ 2 Proportions. 2 In Sample sizes, enter 1000. 3 In Proportion 1 values, enter 0.25 and 0.35. 4 In Proportion 2, enter 0.30. Click OK.
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Session window output
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One-Way Analysis Of Variance Power and Sample Size Test for Two Proportions Testing proportion 1 = proportion 2 (versus not =) Calculating power for proportion 2 = 0.3 Alpha = 0.05 Proportion 1 0.250000 0.350000
Sample Size 1000 1000
Power 0.7071 0.6656
Interpreting the results If you mail 1000 surveys, you will have about a 71% chance of detecting a difference of −0.05 and a 67% chance of detecting a difference of + 0.05 in the proportions of males and females who the tax bill.
One-Way Analysis Of Variance A one-way ANOVA is used to test the equality of population means. For this test, you can calculate the ■
power
■
sample size
■
minimum detectable difference between the smallest and largest factor means (maximum difference)
You need to determine what are acceptable values for any two of these parameters and MINITAB will solve for the third. For example, to calculate sample size, you need to determine the power and the maximum difference between the factor level means that you consider to be meaningful. Then, MINITAB solves for the sample size you need to be able to reject the null hypothesis when the true value differs from the hypothesized value by the specified maximum difference. See Defining the maximum difference on page 9-12.
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One-Way Analysis Of Variance
HOW TO USE Power and Sample Size
h To calculate power, sample size, or maximum difference 1 Choose Stat ➤ Power and Sample Size ➤ One-way ANOVA.
2 In Number of levels, enter the number of factor levels (treatment conditions). 3 Do one of the following: ■
Solve for power 1 In Sample sizes, enter one or more numbers. Each number you enter is
considered the number of observations in every factor level. For example, if you have 3 factor levels with 5 observations each, you would enter 5. 2 In Values of the maximum difference between means, enter one or more numbers. ■
Solve for sample size 1 In Values of the maximum difference between means, enter one or more
numbers. 2 In Power values, enter one or more numbers. ■
Solve for the maximum difference 1 In Sample sizes, enter one or more numbers. Each number you enter is
considered the number of observations in every factor level. 2 In Power values, enter one or more numbers.
MINITAB will solve for all combinations of the specified values. For example, if you enter 3 values in Sample sizes and 2 values in Values of the maximum difference between means, MINITAB will compute the power for all 6 combinations of sample sizes and maximum differences. See Defining the maximum difference on page 9-12. 3 In Sigma, enter an estimate of the population standard deviation (σ) for your data.
See Estimating σ on page 9-6.
4 If you like, use one or more of the options listed below, then click OK.
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One-Way Analysis Of Variance
Options Options subdialog box ■ ■
specify the significance level of the test. The default is α = 0.05. store the sample sizes, sums of squares, and power values. When calculating sample size, MINITAB stores the power value that will generate the nearest integer sample size.
Defining the maximum difference In order to calculate power or sample size, you need to estimate the maximum difference between the smallest and largest actual factor level means. For example, suppose you are planning an experiment with four treatment conditions (four factor levels). You want to find a difference between a control group mean of 10 and a level mean that is 15. In this case, the maximum difference between the means is 5. e Example of calculating power for a one-way ANOVA
Suppose you are about to undertake an investigation to determine whether or not 4 treatments affect the yield of a product using 5 observations per treatment. You know that the mean of the control group should be around 8, and you would like to find significant differences of +4. Thus, the maximum difference you are considering is 4 units. Previous research suggests the population σ is 1.64. 1 Choose Stat ➤ Power and Sample Size ➤ One-way ANOVA. 2 In Number of levels, enter 4. 3 In Sample sizes, enter 5. 4 In Values of the maximum difference between means, enter 4. 5 In Sigma, enter 1.64. Click OK.
Session window output
Power and Sample Size One-way ANOVA Sigma = 1.64 Alpha = 0.05 Number of Levels = 4 SS Means 8
Sample Maximum Size Power Difference 5 0.8269 4
Interpreting the results If you assign five observations to each treatment level, you have power of 0.83 to detect differences of up to 4 units between the treatment means.
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Two-Level Factorial and Plackett-Burman Designs
HOW TO USE Power and Sample Size
Two-Level Factorial and Plackett-Burman Designs For two-level full and fractional factorial designs and Plackett-Burman designs, you can calculate ■
number of replicates
■
power
■
minimum effect
■
number of center points
You need to determine what are acceptable values for any three of these parameters and MINITAB will solve for the fourth. For example, to calculate the number of replicates, you need to specify the minimum effect, power, and the number of center points that you consider to be acceptable. Then, MINITAB solves for the number of replicates you need to be able to reject the null hypothesis when the true value differs from the hypothesized value by the specified minimum effect. See Defining the effect on page 9-15. h To calculate power, replicates, minimum effect, or number of center points 1 Choose Stat ➤ Power and Sample Size ➤ 2-Level Factorial Design or
Plackett-Burman Design.
This dialog box is for a two-level factorial design. The dialog box for a Plackett-Burman design is identical.
2 In Number of factors, enter the number of factors (input variables). 3 In Number of corner points, enter a number. See Determining the number of corner
points on page 9-14. 4 Do one of the following: ■
Solve for power 1 In Replicates, enter one or more numbers. 2 In Effects, enter one or more numbers. 3 In Number of center points, enter one or more numbers.
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Two-Level Factorial and Plackett-Burman Designs ■
Solve for the number of replicates 1 In Effects, enter one or more numbers. 2 In Power values, enter one or more numbers. 3 In Number of center points, enter one or more numbers.
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Solve for the minimum effect 1 In Replicates, enter one or more numbers. 2 In Power values, enter one or more numbers. 3 In Number of center points, enter one or more numbers.
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Solve the number of center points 1 In Replicates, enter one or more numbers. 1 In Effects, enter one or more numbers. 2 In Power values, enter one or more numbers.
For information on the value needed in Effects, see Defining the effect on page 9-15. 5 In Sigma, enter an estimate of the population standard deviation (σ) for your data.
See Estimating σ on page 9-6.
6 If you like, use one or more of the options listed below, then click OK.
Options Designs subdialog box ■
include blocks (two-level factorial designs only)
■
omit from the model
■
include the center points as a term in the model
Options subdialog box ■ ■
specify the significance level of the test. The default is α = 0.05. store the number of replicates, effects, power values, and center points. When calculating the number of replicates, MINITAB stores the power value that will generate the nearest integer number of replicates.
Determining the number of corner points For all designs, you need to specify the appropriate number of corner points given the number of factors. For example, for a 6 factor full factorial design you would have 64 corner points. However, for a 6 factor fractional factorial design, you can have either 8, 16, or 32 corner points. Use the information provided in Summary of Two-Level Designs on page 19-28 to determine the correct number of corner points for your design. 9-14
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Two-Level Factorial and Plackett-Burman Designs
HOW TO USE Power and Sample Size
Defining the effect When calculating power or number of replicates, you need to specify the minimum effect you are interested in detecting. You express this effect as the difference between the low and high factor level means. For example, suppose you are trying to determine the effect of column temperature on the purity of your product. You are only interested in detecting a difference in purity that is greater than 0.007 between the low and high levels of temperature. In the dialog box, enter 0.007 in Effects.
Determining the number of replicates Rather than using sample size to indicate the number of observations you need, factorial designs are expressed in of the number of replicates. A replicate is a repeat of each of the design points (experimental conditions) in the base design. For example, if you are doing a full factorial with three factors, one replicate would require eight runs. The set of experimental conditions would include all combinations of the low and high levels for all factors. Each time you replicate the design eight runs are added to the design; these runs are duplicates of the original eight runs. For a discussion of replication, see Replicating the design on page 19-12. For a discussion of two-level factorial and Plackett-Burman designs, see Chapter 19, Factorial Designs. e Example of calculating power for a two-level fractional factorial design
As a quality engineer, you need to determine the “best” settings for 4 input variables (factors) to improve the transparency of a plastic part. You have determined that a 4 factor, 8 run design (½ fraction) with 3 center points will allow you to estimate the effects you are interested in. Although you would like to perform as few replicates as possible, you must be able to detect effects of 5 or more. Previous experimentation suggests that 4.5 is a reasonable estimate of σ. 1 Choose Stat ➤ Power and Sample Size ➤ 2-Level Factorial Design. 2 In Number of factors, enter 4. 3 In Number of corner points, enter 8. 4 In Replicates, enter 1 2 3 4. 5 In Effects, enter 5. 6 In Number of center points, enter 3. 7 In Sigma, enter 4.5. Click OK.
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Two-Level Factorial and Plackett-Burman Designs Power and Sample Size 2-Level Factorial Design Sigma = 4.5
Alpha = 0.05
Factors: Blocks:
4 none
Base Design:
4, 8
Including a term for center points in model. Center Points Per Block 3 3 3 3
Effect 5 5 5 5
Reps 1 2 3 4
Power 0.1577 0.5189 0.7305 0.8565
Interpreting the results If you do not replicate your design (Reps = 1), you will only have a 16% chance of detecting effects that you have determined are important. If you use four replicates of your ½ fraction design for a total 32 runs, you will have an 86% chance of finding important effects.
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9
Power and Sample Size ■
Power and Sample Size Overview, 9-2
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Z-Test and t-Tests, 9-4
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Tests of Proportions, 9-7
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One-Way Analysis Of Variance, 9-10
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Two-Level Factorial and Plackett-Burman Designs, 9-13
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Chapter 9
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Power and Sample Size Overview
Power and Sample Size Overview Use MINITAB’s power and sample size capabilities to evaluate power and sample size before you design and run an experiment (prospective) or after you perform an experiment (retrospective). ■
A prospective study is used before collecting data to consider design sensitivity. You want to be sure that you have enough power to detect differences (effects) that you have determined to be important. For example, you can increase the design sensitivity by increasing the sample size or by taking measures to decrease the error variance.
■
A retrospective study is used after collecting data to help understand the power of the tests that you have performed. For example, suppose you conduct an experiment and the data analysis does not reveal any statistically significant results. You can then calculate power based on the minimum difference (effect) you wish to detect. If the power to detect this difference is low, you may want to modify your experimental design to increase the power and continue to evaluate the same problem. However, if the power is high, you may want to conclude that there is no meaningful difference (effect) and discontinue experimentation.
MINITAB provides power, sample size, and difference (effect) calculations (also the number of center points for factorial and Plackett-Burman designs) for the following procedures: ■
one-sample Z
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one-sample proportion
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two-level factorial designs
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one-sample t
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two-sample proportion
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Plackett-Burman designs
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two-sample t
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one-way analysis of variance
What is power? There are four possible outcomes for a hypothesis test. The outcomes depend on whether the null hypothesis (H0) is true or false and whether you decide to “reject” or “fail to reject” H0. The power of a test is the probability of correctly rejecting H0 when it is false. In other words, power is the likelihood that you will identify a significant difference (effect) when one exists. The four possible outcomes are summarized below: Null hypothesis Decision
True
False
fail to reject H0
correct decision p=1−α
Type II error p=β
reject H0
Type I error p=α
correct decision p=1−β
power
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Power and Sample Size Overview
HOW TO USE Power and Sample Size
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When H0 is true and you reject it, you make a type I error. The probability (p) of making a type I error is called alpha (α) and is sometimes referred to as the level of significance for the test.
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When H0 is false and you fail to reject it, you make a type II error. The probability (p) of making a type II error is called beta (β).
Choosing probability levels When you are determining the α and β values for your test, you should consider the ■
■
severity of making an error—The more serious the error, the less often you should be willing to allow it to occur. Therefore, you should assign smaller probability values to more serious errors. magnitude of effect you want to detect—Power is the probability (p = 1 − β) of correctly rejecting H0 when it is false. Ideally, you want to have high power to detect a difference that you care about, and low power for a meaningless difference. For example, suppose you want to claim that children in your school scored higher than the general population on a standardized achievement test. You need to decide how much higher than the general population your test scores need to be so you are not making claims that are misleading. If your mean test score is only .7 points higher than the general population on a 100 point test, do you really want to detect a difference? Probably not. Therefore, you should choose your sample size so that you only have power to detect differences that you consider meaningful.
Factors that influence power A number of factors influence power: ■
■
α, the probability of a type I error (level of significance). As the probability of a type I error (α) increases, the probability of a type II error (β) decreases. Hence, as α increases, power = 1 − β also increases. σ, the variability in the population. As σ increases, power decreases.
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the size of the population difference (effect). As the size of population difference (effect) decreases, power decreases.
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sample size. As sample size increases, power increases.
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Chapter 9
HOW TO USE Z-Test and t-Tests
Z-Test and t-Tests Z- and t-tests are used to perform hypothesis tests of the mean (one-sample) or the difference in means (two-sample). For these tests, you can calculate the ■
power
■
sample size
■
minimum difference (effect)
You need to determine what are acceptable values for any two of these parameters and MINITAB will solve for the third. For example, to calculate the sample size, you need to determine the power and the minimum difference that you consider to be acceptable. Then, MINITAB solves for the sample size you need to be able to reject the null hypothesis when the true value differs from the hypothesized value by this minimum difference. See Defining the minimum difference on page 9-5. h To calculate power, sample size, or minimum difference 1 Choose Stat ➤ Power and Sample Size ➤ 1-Sample Z, 1-Sample t, or 2-Sample t.
This dialog box is for a one-sample Z-test. The dialog boxes for the 1- and 2-Sample t are identical.
2 Do one of the following: ■
Solve for power 1 In Sample sizes, enter one or more numbers. For a two-sample test, the
number you enter is considered the sample size for each group. For example, if you want to determine power for an analysis with 10 observations in each group for a total of 20, you would enter 10. 2 In Differences, enter one or more numbers. ■
Solve for sample size 1 In Differences, enter one or more numbers. 2 In Power values, enter one or more numbers.
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Z-Test and t-Tests
HOW TO USE Power and Sample Size
■
Solve for the minimum difference 1 In Sample sizes, enter one or more numbers. For a two-sample test, the
number you enter is considered the sample size for each group. 2 In Power values, enter one or more numbers.
MINITAB will solve for all combinations of the specified values. For example, if you enter 3 values in Sample sizes and 2 values in Differences, MINITAB will compute the power for all 6 combinations of sample sizes and differences. For a discussion of the value needed in Differences, see Defining the minimum difference on page 9-5. 3 In Sigma, enter an estimate of the population standard deviation (σ) for your data.
See Estimating σ on page 9-6.
4 If you like, use one or more of the options listed below, then click OK.
Options Options subdialog box ■
define the alternative hypothesis by choosing less than (lower-tailed), not equal (two-tailed), or greater than (upper-tailed). The default is a two-tailed test.
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specify the significance level (α). The default is 0.05.
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store the sample sizes, differences (effects), and power values. When calculating sample size, MINITAB stores the power value that will generate the nearest integer sample size.
Defining the minimum difference When calculating sample size or power, you need to specify the minimum difference you are interested in detecting. The manner in which you express this difference depends on whether you are performing a one- or two-sample test: ■
For a one-sample Z- or t-test, express the difference in of the null hypothesis. For example, suppose you are testing whether or not your students’ mean test score is different from the population mean. You would like to detect a difference of three points. In the dialog box, you would enter 3 in Differences.
■
For a two-sample t-test, express the difference as the difference between the population means that you would like to be able to detect. For example, suppose you are investigating the effects of water acidity on the growth of two groups of tadpoles. You decide that any difference in growth between the two groups that is smaller than 4 mm is not important. In the dialog box, you would enter 4 in Differences.
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Chapter 9
HOW TO USE Z-Test and t-Tests
Estimating σ For power or minimum difference calculations, the estimate of σ depends on whether or not you have already collected data. ■
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Prospective studies are done before collecting data so σ has to be estimated. You can use related research, pilot studies, or subject-matter knowledge to estimate σ. Retrospective studies are done after data have been collected so you can use the sample standard deviation to estimate σ. You could also use related research, pilot studies, or subject-matter knowledge. Use Display Descriptive Statistics (page 1-6) to calculate the sample standard deviation.
For sample size calculations, the data have not been collected yet so the population standard deviation (σ) has to be estimated. You can use related research, pilot studies, or subject-matter knowledge to estimate σ. Note
By default, MINITAB sets σ to 1.0. This is fine if the differences (effects) are standardized, but will present erroneous results if they are not. When the differences (effects) are not standardized, be sure to enter an estimate of σ.
e Example of calculating sample size for a one-sample t-test
Suppose you are the production manager at a dairy plant. In order to meet state requirements, you must maintain strict control over the packaging of ice cream. The volume cannot vary more than 3 oz for a half gallon (64 oz) container. The packaging machine tolerances are set so the process σ is 1. How many samples must be taken to estimate the mean package volume at a confidence level of 99% (α = .01) for power values of 0.7, 0.8, and 0.9? 1 Choose Stat ➤ Power and Sample Size ➤ 1-Sample t. 2 In Differences, enter 3. In Power values, enter 0.7 0.8 0.9. 3 In Sigma, enter 1. 4 Click Options. In Significance level, enter 0.01. Click OK in each dialog box.
Session window output
Power and Sample Size 1-Sample t Test Testing mean = null (versus not = null) Calculating power for mean = null + difference Alpha = 0.01 Sigma = 1 Difference 3 3 3
Sample Size 5 5 6
Target Power 0.7000 0.8000 0.9000
Actual Power 0.8947 0.8947 0.9827
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Tests of Proportions
HOW TO USE Power and Sample Size
Interpreting the results MINITAB displays the sample size required to obtain the requested power values. Because the target power values would result in non-integer sample sizes, MINITAB displays the power (Actual Power) that you would have to detect differences in volume greater than three ounces using the nearest integer value for sample size. If you take a sample of five cartons, power for your test is 0.895; for a sample of six cartons, power is 0.983.
Tests of Proportions Proportion tests are used to perform hypothesis tests of a proportion (one-sample) or the difference in proportions (two-sample). For these tests, you can calculate the ■
power
■
sample size
■
minimum difference (effect)
You need to determine what are acceptable values for any two of these parameters and MINITAB will solve for the third. For example, to calculate the sample size, you need to determine the power and the minimum difference that you consider to be acceptable. Then, MINITAB solves for the sample size you need to be able to reject the null hypothesis when the true value differs from the hypothesized value by this minimum difference. See Defining the minimum difference on page 9-9. h To calculate power, sample size, or minimum difference 1 Choose Stat ➤ Power and Sample Size ➤ 1 Proportion or 2 Proportions.
1 Proportion
2 Proportions
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Chapter 9
Tests of Proportions 2 Do one of the following: ■
Solve for power 1 In Sample sizes, enter one or more numbers. For two proportion test, the
number you enter is considered the sample size for each group. For example, if you want to determine power for an analysis with 10 observations in each group for a total of 20, you would enter 10. 2 In Alternative values of p or Proportion 1 values, enter one or more proportions. ■
Solve for sample size 1 In Alternative values of p or Proportion 1 values, enter one or more
proportions. 2 In Power values, enter one or more numbers. ■
Solve for the minimum difference 1 In Sample sizes, enter one or more numbers. For a two proportion test, the
number you enter is considered the sample size for each group, not the total number for the experiment. 2 In Power values, enter one or more numbers.
MINITAB will solve for all combinations of the specified values. For example, if you enter 3 values in Sample sizes and 2 values in Alternative values of p, MINITAB will compute the power for all 6 combinations of sample sizes and alternative proportions. For a discussion of the values needed in Alternative values of p and Proportion 1 values, see Defining the minimum difference on page 9-9. 3 Do one of the following: ■
For a one-sample test, enter the expected proportion under the null hypothesis in Hypothesized p. The default is 0.5.
■
For a two-sample test, enter the second proportion in Proportion 2. The default is 0.5.
For a discussion of the values needed in Hypothesized p and Proportion 2, see Defining the minimum difference on page 9-9. 4 If you like, use one or more of the options listed below, then click OK.
Options Options subdialog box ■
■
define the alternative hypothesis by choosing less than (lower-tailed), not equal (two-tailed), or greater than (upper-tailed). The default is a two-tailed test. specify the significance level of the test. The default is α = 0.05.
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Tests of Proportions
HOW TO USE Power and Sample Size
■
store the sample sizes, alternative values of p or proportion 1 values, and power values. When calculating sample size, MINITAB stores the power value that will generate the nearest integer sample size.
Defining the minimum difference MINITAB uses two proportions to determine the minimum difference. The manner in which you express these proportions depends on whether you are performing a one- or two-sample proportion test. ■
For a one-sample test of proportion, enter the expected proportion under the null hypothesis for Hypothesized p in the dialog box. Suppose you are testing whether the data are consistent with the following null hypothesis and would like to detect any differences where the true proportion is greater than .73. H0: p = .7
H1: p > .7
where p is the population proportion
In MINITAB, enter .73 in Alternative values of p; enter .7 in Hypothesized p. (The alternative proportion is not the value of the alternative hypothesis, but the value at which you want to evaluate power.) ■
For a two-sample test of proportion, enter the expected proportions under the null hypothesis for Proportion 2 in the dialog box. Suppose a biologist wants to test whether or not there is a difference in the proportion of fish that have been affected by pollution in two lakes. Previous research suggests that approximately 25% of fish have been affected. The biologist would like to detect a difference in proportions of 0.03. H0: p1 = p2
H1: p1 ≠ p2
In MINITAB, enter 0.22 and 0.28 in Proportion 1 values; enter 0.25 in Proportion 2. e Example of calculating power for a two-sample test of proportion
As a political advisor, you want to determine whether there is a difference between the proportion of men and the proportion of women who a tax reform bill. Results of a previous survey of ed voters indicate that 30% (p = .30) of the voters the tax bill. If you mail 1000 surveys, what is the power to detect differences greater than .05 between the proportions of men and women who the tax bill? 1 Choose Stat ➤ Power and Sample Size ➤ 2 Proportions. 2 In Sample sizes, enter 1000. 3 In Proportion 1 values, enter 0.25 and 0.35. 4 In Proportion 2, enter 0.30. Click OK.
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Session window output
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One-Way Analysis Of Variance Power and Sample Size Test for Two Proportions Testing proportion 1 = proportion 2 (versus not =) Calculating power for proportion 2 = 0.3 Alpha = 0.05 Proportion 1 0.250000 0.350000
Sample Size 1000 1000
Power 0.7071 0.6656
Interpreting the results If you mail 1000 surveys, you will have about a 71% chance of detecting a difference of −0.05 and a 67% chance of detecting a difference of + 0.05 in the proportions of males and females who the tax bill.
One-Way Analysis Of Variance A one-way ANOVA is used to test the equality of population means. For this test, you can calculate the ■
power
■
sample size
■
minimum detectable difference between the smallest and largest factor means (maximum difference)
You need to determine what are acceptable values for any two of these parameters and MINITAB will solve for the third. For example, to calculate sample size, you need to determine the power and the maximum difference between the factor level means that you consider to be meaningful. Then, MINITAB solves for the sample size you need to be able to reject the null hypothesis when the true value differs from the hypothesized value by the specified maximum difference. See Defining the maximum difference on page 9-12.
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One-Way Analysis Of Variance
HOW TO USE Power and Sample Size
h To calculate power, sample size, or maximum difference 1 Choose Stat ➤ Power and Sample Size ➤ One-way ANOVA.
2 In Number of levels, enter the number of factor levels (treatment conditions). 3 Do one of the following: ■
Solve for power 1 In Sample sizes, enter one or more numbers. Each number you enter is
considered the number of observations in every factor level. For example, if you have 3 factor levels with 5 observations each, you would enter 5. 2 In Values of the maximum difference between means, enter one or more numbers. ■
Solve for sample size 1 In Values of the maximum difference between means, enter one or more
numbers. 2 In Power values, enter one or more numbers. ■
Solve for the maximum difference 1 In Sample sizes, enter one or more numbers. Each number you enter is
considered the number of observations in every factor level. 2 In Power values, enter one or more numbers.
MINITAB will solve for all combinations of the specified values. For example, if you enter 3 values in Sample sizes and 2 values in Values of the maximum difference between means, MINITAB will compute the power for all 6 combinations of sample sizes and maximum differences. See Defining the maximum difference on page 9-12. 3 In Sigma, enter an estimate of the population standard deviation (σ) for your data.
See Estimating σ on page 9-6.
4 If you like, use one or more of the options listed below, then click OK.
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One-Way Analysis Of Variance
Options Options subdialog box ■ ■
specify the significance level of the test. The default is α = 0.05. store the sample sizes, sums of squares, and power values. When calculating sample size, MINITAB stores the power value that will generate the nearest integer sample size.
Defining the maximum difference In order to calculate power or sample size, you need to estimate the maximum difference between the smallest and largest actual factor level means. For example, suppose you are planning an experiment with four treatment conditions (four factor levels). You want to find a difference between a control group mean of 10 and a level mean that is 15. In this case, the maximum difference between the means is 5. e Example of calculating power for a one-way ANOVA
Suppose you are about to undertake an investigation to determine whether or not 4 treatments affect the yield of a product using 5 observations per treatment. You know that the mean of the control group should be around 8, and you would like to find significant differences of +4. Thus, the maximum difference you are considering is 4 units. Previous research suggests the population σ is 1.64. 1 Choose Stat ➤ Power and Sample Size ➤ One-way ANOVA. 2 In Number of levels, enter 4. 3 In Sample sizes, enter 5. 4 In Values of the maximum difference between means, enter 4. 5 In Sigma, enter 1.64. Click OK.
Session window output
Power and Sample Size One-way ANOVA Sigma = 1.64 Alpha = 0.05 Number of Levels = 4 SS Means 8
Sample Maximum Size Power Difference 5 0.8269 4
Interpreting the results If you assign five observations to each treatment level, you have power of 0.83 to detect differences of up to 4 units between the treatment means.
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Two-Level Factorial and Plackett-Burman Designs
HOW TO USE Power and Sample Size
Two-Level Factorial and Plackett-Burman Designs For two-level full and fractional factorial designs and Plackett-Burman designs, you can calculate ■
number of replicates
■
power
■
minimum effect
■
number of center points
You need to determine what are acceptable values for any three of these parameters and MINITAB will solve for the fourth. For example, to calculate the number of replicates, you need to specify the minimum effect, power, and the number of center points that you consider to be acceptable. Then, MINITAB solves for the number of replicates you need to be able to reject the null hypothesis when the true value differs from the hypothesized value by the specified minimum effect. See Defining the effect on page 9-15. h To calculate power, replicates, minimum effect, or number of center points 1 Choose Stat ➤ Power and Sample Size ➤ 2-Level Factorial Design or
Plackett-Burman Design.
This dialog box is for a two-level factorial design. The dialog box for a Plackett-Burman design is identical.
2 In Number of factors, enter the number of factors (input variables). 3 In Number of corner points, enter a number. See Determining the number of corner
points on page 9-14. 4 Do one of the following: ■
Solve for power 1 In Replicates, enter one or more numbers. 2 In Effects, enter one or more numbers. 3 In Number of center points, enter one or more numbers.
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Two-Level Factorial and Plackett-Burman Designs ■
Solve for the number of replicates 1 In Effects, enter one or more numbers. 2 In Power values, enter one or more numbers. 3 In Number of center points, enter one or more numbers.
■
Solve for the minimum effect 1 In Replicates, enter one or more numbers. 2 In Power values, enter one or more numbers. 3 In Number of center points, enter one or more numbers.
■
Solve the number of center points 1 In Replicates, enter one or more numbers. 1 In Effects, enter one or more numbers. 2 In Power values, enter one or more numbers.
For information on the value needed in Effects, see Defining the effect on page 9-15. 5 In Sigma, enter an estimate of the population standard deviation (σ) for your data.
See Estimating σ on page 9-6.
6 If you like, use one or more of the options listed below, then click OK.
Options Designs subdialog box ■
include blocks (two-level factorial designs only)
■
omit from the model
■
include the center points as a term in the model
Options subdialog box ■ ■
specify the significance level of the test. The default is α = 0.05. store the number of replicates, effects, power values, and center points. When calculating the number of replicates, MINITAB stores the power value that will generate the nearest integer number of replicates.
Determining the number of corner points For all designs, you need to specify the appropriate number of corner points given the number of factors. For example, for a 6 factor full factorial design you would have 64 corner points. However, for a 6 factor fractional factorial design, you can have either 8, 16, or 32 corner points. Use the information provided in Summary of Two-Level Designs on page 19-28 to determine the correct number of corner points for your design. 9-14
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Two-Level Factorial and Plackett-Burman Designs
HOW TO USE Power and Sample Size
Defining the effect When calculating power or number of replicates, you need to specify the minimum effect you are interested in detecting. You express this effect as the difference between the low and high factor level means. For example, suppose you are trying to determine the effect of column temperature on the purity of your product. You are only interested in detecting a difference in purity that is greater than 0.007 between the low and high levels of temperature. In the dialog box, enter 0.007 in Effects.
Determining the number of replicates Rather than using sample size to indicate the number of observations you need, factorial designs are expressed in of the number of replicates. A replicate is a repeat of each of the design points (experimental conditions) in the base design. For example, if you are doing a full factorial with three factors, one replicate would require eight runs. The set of experimental conditions would include all combinations of the low and high levels for all factors. Each time you replicate the design eight runs are added to the design; these runs are duplicates of the original eight runs. For a discussion of replication, see Replicating the design on page 19-12. For a discussion of two-level factorial and Plackett-Burman designs, see Chapter 19, Factorial Designs. e Example of calculating power for a two-level fractional factorial design
As a quality engineer, you need to determine the “best” settings for 4 input variables (factors) to improve the transparency of a plastic part. You have determined that a 4 factor, 8 run design (½ fraction) with 3 center points will allow you to estimate the effects you are interested in. Although you would like to perform as few replicates as possible, you must be able to detect effects of 5 or more. Previous experimentation suggests that 4.5 is a reasonable estimate of σ. 1 Choose Stat ➤ Power and Sample Size ➤ 2-Level Factorial Design. 2 In Number of factors, enter 4. 3 In Number of corner points, enter 8. 4 In Replicates, enter 1 2 3 4. 5 In Effects, enter 5. 6 In Number of center points, enter 3. 7 In Sigma, enter 4.5. Click OK.
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Session window output
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Two-Level Factorial and Plackett-Burman Designs Power and Sample Size 2-Level Factorial Design Sigma = 4.5
Alpha = 0.05
Factors: Blocks:
4 none
Base Design:
4, 8
Including a term for center points in model. Center Points Per Block 3 3 3 3
Effect 5 5 5 5
Reps 1 2 3 4
Power 0.1577 0.5189 0.7305 0.8565
Interpreting the results If you do not replicate your design (Reps = 1), you will only have a 16% chance of detecting effects that you have determined are important. If you use four replicates of your ½ fraction design for a total 32 runs, you will have an 86% chance of finding important effects.
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