09 Power & Sample Size

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MINITAB Users Guide 2

MEET MTB UGUIDE 1 SC QREFUGUIDE 2INDEXCONTENTS HOW TO USE


9Power andSample Size

Power and Sample Size Overview , 9-2

Z-Test and t-Tests , 9-4

Tests of Proportions , 9-7One-Way Analysis Of Variance , 9-10

Two-Level Factorial and Plackett-Burman Designs , 9-13
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Chapter 9 Power and Sample Size Overvi

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Null hypothesis

Decision True False fail to reject H 0 correct decision

p = 1 Type II error p =

reject H 0 Type I error p =

correct decisionp = 1

Power and Sample Size OverviewUse M INITABs power and sample size capabilities to evaluate power and sample sizebefore you design and run an experiment (prospective) or after you perform anexperiment (retrospective).

A prospective study is used before collecting data to consider design sensitivity. Youwant to be sure that you have enough power to detect differences (effects) that youhave determined to be important. For example, you can increase the designsensitivity by increasing the sample size or by taking measures to decrease the errorvariance.

A retrospective study is used after collecting data to help understand the power ofthe tests that you have performed. For example, suppose you conduct an experimentand the data analysis does not reveal any statistically significant results. You can thencalculate power based on the minimum difference (effect) you wish to detect. If thepower to detect this difference is low, you may want to modify your experimentaldesign 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 thenumber of center points for factorial and Plackett-Burman designs) for the followingprocedures:

What is power?There are four possible outcomes for a hypothesis test. The outcomes depend onwhether the null hypothesis (H 0) is true or false and whether you decide to reject orfail to reject H 0. The power of a test is the probability of correctly rejecting H 0 wheis false. In other words, power is the likelihood that you will identify a significantdifference (effect) when one exists.

The four possible outcomes are summarized below:

one-sample Zone-sample ttwo-sample t

one-sample proportiontwo-sample proportionone-way analysis of variance

two-level factorial designsPlackett-Burman designs

power
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Power and Sample Size Overview Power and Sample Si




When H 0 is true and you reject it, you make a type I error. The probability (p) ofmaking a type I error is called alpha ( ) and is sometimes referred to as the levelsignificance for the test.

When H 0 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 ().

Choosing probability levels

When you are determining the and values for your test, you should consider the

severity of making an error The more serious the error, the less often you shouldbe willing to allow it to occur. Therefore, you should assign smaller probability valuesto more serious errors.

magnitude of effect you want to detect Power is the probability (p = 1 ) ocorrectly rejecting H 0 when it is false. Ideally, you want to have high power to detecta 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 higherthan the general population on a standardized achievement test. You need to decidehow much higher than the general population your test scores need to be so you arenot making claims that are misleading. If your mean test score is only .7 pointshigher than the general population on a 100 point test, do you really want to detecta 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:

, the probability of a type I error (level of significance). As the probability of atype I error ( ) increases, the probability of a type II error ( ) decreases. Hence, asincreases, power = 1 also increases.

, the variability in the population. As increases, power decreases.

the size of the population difference (effect). As the size of population difference(effect) decreases, power decreases.

sample size. As sample size increases, power increases.


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Chapter 9 Z-Test and t-T

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Z-Test and t-TestsZ- and t-tests are used to perform hypothesis tests of the mean (one-sample) or thedifference 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 andMINITAB will solve for the third.

For example, to calculate the sample size, you need to determine the power and theminimum difference that you consider to be acceptable. Then, M INITAB solves for thsample 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 minimumdifference 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

2 Do one of the following:

Solve for power1 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 eachgroup for a total of 20, you would enter 10.

2 In Differences , enter one or more numbers.

Solve for sample size1 In Differences , enter one or more numbers.2 In Power values, enter one or more numbers.

This dialog box is for a one-sampleZ-test. The dialog boxes for the1- and 2-Sample t are identical.


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Z-Test and t-Tests Power and Sample S




Solve for the minimum difference1 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 , M INITAB will compute tpower for all 6 combinations of sample sizes and differences.

For a discussion of the value needed in Differences , see Defining the minimumdifference on page 9-5 .

3 In Sigma , enter an estimate of the population standard deviation ( ) for your datSee Estimating 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.

specify the significance level ( ). The default is 0.05.

store the sample sizes, differences (effects), and power values. When calculatingsample size, M INITAB stores the power value that will generate the nearest integersample size.

Defining the minimum difference

When calculating sample size or power, you need to specify the minimum differenceare interested in detecting. The manner in which you express this difference dependson whether you are performing a one- or two-sample test:

For a one-sample Z- or t-test, express the difference in terms of the null hypothesis.

For example, suppose you are testing whether or not your students mean test scoreis different from the population mean. You would like to detect a difference of threepoints. In the dialog box, you would enter 3 in Differences .

For a two-sample t-test, express the difference as the difference between thepopulation means that you would like to be able to detect.

For example, suppose you are investigating the effects of water acidity on the growthof two groups of tadpoles. You decide that any difference in growth between the twogroups that is smaller than 4 mm is not important. In the dialog box, you wouldenter 4 in Differences .


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Chapter 9 Z-Test and t-T

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Estimating

For power or minimum difference calculations, the estimate of depends on whetheror not you have already collected data.

Prospective studies are done before collecting data so has to be estimated. You canuse related research, pilot studies, or subject-matter knowledge to estimate .

Retrospective studies are done after data have been collected so you can use thesample standard deviation to estimate . You could also use related research, pilotstudies, or subject-matter knowledge. Use Display Descriptive Statistics (page 1-6calculate the sample standard deviation.

For sample size calculations, the data have not been collected yet so the populationstandard deviation ( ) has to be estimated. You can use related research, pilot studies,or subject-matter knowledge to estimate .

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 staterequirements, you must maintain strict control over the packaging of ice cream. Thevolume cannot vary more than 3 oz for a half gallon (64 oz) container. The packagingmachine tolerances are set so the process is 1. How many samples must be taken toestimate the mean package volume at a confidence level of 99% ( = .01) for powervalues 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.

Sessionwindow output

Power and Sample Size

1-Sample t Test

Testing mean = null (versus not = null)Calculating power for mean = null + differenceAlpha = 0.01 Sigma = 1

Sample Target ActualDifference Size Power Power 3 5 0.7000 0.8947 3 5 0.8000 0.8947 3 6 0.9000 0.9827

Note By default, M INITAB sets 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 .
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Tests of Proportions Power and Sample S




Interpreting the results

MINITAB displays the sample size required to obtain the requested power values. Becausethe target power values would result in non-integer sample sizes, M INITAB displays thpower (Actual Power) that you would have to detect differences in volume greater thanthree ounces using the nearest integer value for sample size. If you take a sample of fivecartons, power for your test is 0.895; for a sample of six cartons, power is 0.983.

Tests of ProportionsProportion tests are used to perform hypothesis tests of a proportion (one-sample) orthe 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 andMINITAB will solve for the third.

For example, to calculate the sample size, you need to determine the power and theminimum difference that you consider to be acceptable. Then, M INITAB solves for thsample 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 minimumdifference 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 Proporti

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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 size1 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 difference1 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 totalnumber 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 youenter 3 values in Sample sizes and 2 values in Alternative values of p , M INITAB compute the power for all 6 combinations of sample sizes and alternativeproportions.

For a discussion of the values needed in Alternative values of p and Proportion 1values , see Defining the minimum difference on page 9-9 .


For a one-sample test, enter the expected proportion under the null hypothesis inHypothesized p . The default is 0.5.

For a two-sample test, enter the second proportion in Proportion 2 . The defau0.5.

For a discussion of the values needed in Hypothesized p and Proportion 2 , seeDefining the minimum difference on page 9-9.


Options


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 = 0.05.


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Tests of Proportions Power and Sample S




store the sample sizes, alternative values of p or proportion 1 values, and powervalues. When calculating sample size, M INITAB stores the power value that willgenerate the nearest integer sample size.

Defining the minimum difference

MINITAB uses two proportions to determine the minimum difference. The manner inwhich you express these proportions depends on whether you are performing a one- ortwo-sample proportion test.

For a one-sample test of proportion, enter the expected proportion under the nullhypothesis for Hypothesized p in the dialog box.

Suppose you are testing whether the data are consistent with the following nullhypothesis and would like to detect any differences where the true proportion isgreater than .73.

H0: p = .7 H1: p > .7 where p is the population proportion

In M INITAB, enter .73 in Alternative values of p ; enter .7 in Hypothesized p . (Thalternative proportion is not the value of the alternative hypothesis, but the value atwhich you want to evaluate power.)

For a two-sample test of proportion, enter the expected proportions under the nullhypothesis for Proportion 2 in the dialog box.

Suppose a biologist wants to test whether or not there is a difference in theproportion of fish that have been affected by pollution in two lakes. Previous researchsuggests that approximately 25% of fish have been affected. The biologist would liketo detect a difference in proportions of 0.03.

H0: p1 = p2 H1: p1 p2In M INITAB, enter 0.22 and 0.28 in Proportion 1 values ; enter 0.25 in Proportion

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 theproportion of men and the proportion of women who support a tax reform bill. Resultsof a previous survey of registered voters indicate that 30% (p = .30) of the voterssupport the tax bill. If you mail 1000 surveys, what is the power to detect differencesgreater than .05 between the proportions of men and women who support 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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Chapter 9 One-Way Analysis Of Varia

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Test for Two Proportions

Testing proportion 1 = proportion 2 (versus not =)Calculating power for proportion 2 = 0.3Alpha = 0.05

SampleProportion 1 Size Power 0.250000 1000 0.7071 0.350000 1000 0.6656


If you mail 1000 surveys, you will have about a 71% chance of detecting a difference of0.05 and a 67% chance of detecting a difference of + 0.05 in the proportions of malesand females who support the tax bill.

One-Way Analysis Of Variance A one-way ANOVA is used to test the equality of population means. For this test, youcan 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 andMINITAB will solve for the third.

For example, to calculate sample size, you need to determine the power and themaximum difference between the factor level means that you consider to bemeaningful. Then, M INITAB solves for the sample size you need to be able to reject thenull hypothesis when the true value differs from the hypothesized value by the specifiedmaximum difference. See Defining the maximum difference on page 9-12.


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One-Way Analysis Of Variance Power and Sample S




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).


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 youhave 3 factor levels with 5 observations each, you would enter 5.

2 In Values of the maximum difference between means , enter one or more numbe

Solve for sample size1 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 difference1 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 youenter 3 values in Sample sizes and 2 values in Values of the maximum differencebetween means , M INITAB will compute the power for all 6 combinations of samplesizes and maximum differences. See Defining the maximum difference on page 9-12




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Chapter 9 One-Way Analysis Of Varia




Options



store the sample sizes, sums of squares, and power values. When calculating sample

size, M INITAB stores the power value that will generate the nearest integer samplesize.

Defining the maximum difference

In order to calculate power or sample size, you need to estimate the maximumdifference between the smallest and largest actual factor level means. For example,suppose you are planning an experiment with four treatment conditions (four factorlevels). You want to find a difference between a control group mean of 10 and a levelmean that is 15. In this case, the maximum difference between the means is 5.

e Example of calculating power for a one-way ANOVASuppose you are about to undertake an investigation to determine whether or not 4treatments affect the yield of a product using 5 observations per treatment. You knowthat the mean of the control group should be around 8, and you would like to findsignificant differences of +4. Thus, the maximum difference you are considering is 4units. Previous research suggests the population 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.



One-way ANOVA

Sigma = 1.64 Alpha = 0.05 Number of Levels = 4

Sample MaximumSS Means Size Power Difference 8 5 0.8269 4


If you assign five observations to each treatment level, you have power of 0.83 to detectdifferences of up to 4 units between the treatment means.


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Two-Level Factorial and Plackett-Burman Designs Power and Sample Siz




Two-Level Factorial and Plackett-Burman DesignsFor two-level full and fractional factorial designs and Plackett-Burman designs, you cancalculate

number of replicates

powerminimum effectnumber of center points

You need to determine what are acceptable values for any three of these parametersand M INITAB will solve for the fourth.

For example, to calculate the number of replicates, you need to specify the minimumeffect, power, and the number of center points that you consider to be acceptable.Then, M INITAB solves for the number of replicates you need to be able to reject the nullhypothesis when the true value differs from the hypothesized value by the specifiedminimum 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 .

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 cornpoints on page 9-14.


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.

This dialog box is for a two-level factorial design. The dialog box

for a Plackett-Burman design isidentical.


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Chapter 9 Two-Level Factorial and Plackett-Burman Desig




Solve for the number of replicates1 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 effect1 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 points1 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 9-1



Options

Designs subdialog box

include blocks (two-level factorial designs only)

omit terms from the modelinclude the center points as a term in the model



store the number of replicates, effects, power values, and center points. Whencalculating the number of replicates, M INITAB stores the power value that willgenerate the nearest integer number of replicates.

Determining the number of corner pointsFor all designs, you need to specify the appropriate number of corner points given thenumber of factors. For example, for a 6 factor full factorial design you would have 64corner 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 Deson page 19-28 to determine the correct number of corner points for your design.
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Two-Level Factorial and Plackett-Burman Designs Power and Sample Siz




Defining the effect

When calculating power or number of replicates, you need to specify the minimumeffect you are interested in detecting. You express this effect as the difference betweenthe low and high factor level means. For example, suppose you are trying to determinethe 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 highlevels 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 terms of the number of replicates. A replicate is arepeat of each of the design points (experimental conditions) in the base design. Forexample, if you are doing a full factorial with three factors, one replicate would requireeight runs. The set of experimental conditions would include all combinations of thelow 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 discussof two-level factorial and Plackett-Burman designs, see Chapter 19, Factorial Design

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 ( fraction) with 3 center points will allow you to estimate theeffects you are interested in. Although you would like to perform as few replicates aspossible, you must be able to detect effects of 5 or more. Previous experimentation

suggests that 4.5 is a reasonable estimate of .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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Chapter 9 Two-Level Factorial and Plackett-Burman Desig





2-Level Factorial Design

Sigma = 4.5 Alpha = 0.05

Factors: 4 Base Design: 4, 8

Blocks: noneIncluding a term for center points in model.

CenterPointsPer Block Effect Reps Power 3 5 1 0.1577 3 5 2 0.5189 3 5 3 0.7305 3 5 4 0.8565


If you do not replicate your design (Reps = 1), you will only have a 16% chance ofdetecting effects that you have determined are important. If you use four replicates of

your fraction design for a total 32 runs, you will have an 86% chance of findingimportant effects.
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09 Power & Sample Size