Sample Size Determination (Two or More Samples)

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Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie for estimatig or testig hypotheses cocerig ay of the followig:. the differece betwee the meas ad of two ormal distributios.. the ratio of the stadard deviatios ad of two ormal distributios. 3. the differece betwee the proportios ad of two biomial distributios. 4. the differece betwee the rates ad of two Poisso distributios. 5. the pairwise differeces betwee the meas of more tha two ormal distributios. It fids a sample sie that achieves either of two goals:. geerates a cofidece iterval for the differece or ratio of specified width.. yields the desired power i a test of hypotheses cocerig the differece or ratio. Sample StatFolio: samsie.sgp Sample Data: Noe. 3 by StatPoit Techologies, Ic. Sample Sie Determiatio (Two Samples) -

STATGRAPHICS Rev. 963 Data Iput The first dialog displayed by this procedure is used to specify the problem of iterest to the aalyst. Compare: the problem of iterest. It is assumed that radom samples of sie j will be take from j populatios that follow the specified distributio ad used to estimate or test the value of the idicated parameters. The procedure will determie suitable values for j. Hypothesied Differece or Ratio: the aticipated value of the differece or ratio. If performig a hypothesis test, this value forms the ull hypothesis (usually ). If costructig a cofidece iterval, this value is oly used if the desired width of the iterval is specified i relative (percetage) terms. Hypothesied Withi-Group Sigmas: the aticipated value of the stadard deviatio withi each of the j populatios sampled, assumed to be the same for all populatios. Whe comparig or more meas, this value is a critical part of the calculatio ad should either be kow exactly or be a reliable estimate from previous data. Hypothesied Meas: a approximate value for the meas j. This value is ot used i the calculatios. Hypothesied Proportios: a approximate value for the biomial proportios. This value is used to determie the likely stadard error of the differece betwee the two proportios. Hypothesied Rates: a approximate value for the Poisso rates. This value is used to determie the likely stadard error of the differece betwee the two rates. 3 by StatPoit Techologies, Ic. Sample Sie Determiatio (Two Samples) -

STATGRAPHICS Rev. 963 Number of Meas: the umber of samples k whe comparig more tha meas. Percet of Data i First Sample: whe comparig two samples, the percet of data i the first sample: % () Except i rare cases, the percetage should be set to 5%. For example, the above dialog box idicates a desire to compare the meas of ormal distributios, thought to be aroud = with stadard deviatios of = 3. The ull hypothesis is that the differece betwee the meas ( - ) equals. Equal sample sies for the samples are desired. The secod dialog box elicits iformatio about the goal of the aalysis: Cotrol: specifies the goal from amog the followig choices:. Absolute error: a cofidece iterval for the differece or ratio is to be costructed. That iterval should ot deviate from the poit estimate of the differece or ratio i either directio by more tha the absolute distace W idicated. Note: whe comparig more tha meas, the iterval used is based o Tukey s multiple compariso method.. Relative error: a cofidece iterval for the differece or ratio is to be costructed. That iterval should ot deviate from the poit estimate of the differece or ratio i either 3 by StatPoit Techologies, Ic. Sample Sie Determiatio (Two Samples) - 3

STATGRAPHICS Rev. 963 directio by more tha the relative percetage P idicated. This is idetical to the absolute error case with W set equal to P times the specified differece or ratio. 3. Power: a hypothesis test is to be performed. The power of the test (-) % should equal the percetage specified whe the true value of the differece or ratio deviates from the ull hypothesis by the idicated = Differece to Detect. Power is defied as the probability of rejectig the ull hypothesis whe it is false. If a two-sided test is to be performed, the that power must be achieved both above ad below the value specified by the ull hypothesis. Note: whe comparig more tha meas, power refers to the F test for betwee group differeces i the ANOVA table ad refers to the largest differece betwee ay meas. 4. Sample Sie: the predetermied sample sie, assumed to be the same for all samples. This optio is used to plot the power curve for a sample sie that was ot calculated by this procedure. Cofidece Level: the level of cofidece (-)% used whe costructig cofidece itervals. The value is also used as the level of Type I error whe testig hypotheses. A Type I error occurs whe the ull hypothesis is falsely rejected. Alterative Hypothesis: select Not Equal for a two-sided hypothesis test, Less Tha if the alterative hypothesis is that the parameter is less tha the value specified by the ull hypothesis, or Greater Tha if the alterative hypothesis is that the parameter is greater tha the value specified by the ull hypothesis. Sigma: whe comparig or testig ormal meas, whether the stadard deviatio is assumed to be kow ( test) or if it will be estimated from the data (t test). For example, the dialog box above idicates that the followig test is to be performed: Null hypothesis H : = Alt. hypothesis H A : The probability of a Type I error (rejectig a true ull hypothesis) is set to = 5%, while the probability of a Type II error (ot rejectig a false ull hypothesis) is set to = % whe the true absolute differece betwee the meas equals 3. The followig table may be helpful i rememberig how to set the error probabilities. Do Not Reject H Reject H H is true Correct decisio Type I error risk = H is false Type II error risk = Correct decisio power = - 3 by StatPoit Techologies, Ic. Sample Sie Determiatio (Two Samples) - 4

Power STATGRAPHICS Rev. 963 Aalysis Summary The Aalysis Summary displays the desired goal ad the sample sies that will achieve it: Sample-Sie Determiatio Parameter to be estimated: differece betwee two ormal meas Desired power: 9.% for differece =. versus differece = 3. Type of alterative: ot equal Alpha risk: 5.% Sigma: 3. (to be estimated) The required sample sie is 3 observatios from sample ad 3 observatios from sample. I the curret example, samples of = 3 observatios from each populatio are required to achieve the power requested. Power Curve The Power Curve shows the power of the specified test of hypotheses for the derived sample sies. Power Curve alpha =.5, sigma = 3., =3, =3.8.6.4. -4-4 True Differece Betwee Meas It ca be see that the power (probability of rejectig the ull) is oly aroud whe the true differece is close to ero, by it rises to - whe the differece varies i either directio by the specified Differece to Detect. 3 by StatPoit Techologies, Ic. Sample Sie Determiatio (Two Samples) - 5

STATGRAPHICS Rev. 963 Calculatios Normal Mea Cofidece Iterval If is assumed to be kow, fid the smallest ad such that W () If will be estimated from the data, fid the smallest ad such that t, W (3) Normal Mea Hypothesis Test If is assumed to be kow, fid the smallest ad such that If is to be estimated from the data, fid the smallest ad such that t, t, (4) (5) Normal Sigma Cofidece Iterval Fid the smallest ad such that F,, W (6) ad F W (7),, Normal Sigma Hypothesis Test Fid the smallest or such that 3 by StatPoit Techologies, Ic. Sample Sie Determiatio (Two Samples) - 6

STATGRAPHICS Rev. 963 3 by StatPoit Techologies, Ic. Sample Sie Determiatio (Two Samples) - 7 ) l( if > (8) or ) l( if < (9) Biomial Proportio Cofidece Itervals Fid the smallest ad such that W () Biomial Proportio Hypothesis Tests Fid the smallest ad such that si si () Poisso Rate Cofidece Itervals Fid the smallest ad such that W () Poisso Rate Hypothesis Tests

Fid the smallest ad such that STATGRAPHICS Rev. 963 (3) More Tha Normal Meas Cofidece Itervals Usig Tukey s T, fid the smallest commo sample sie such that: T, k, k( ) W (4) More Tha Normal Meas Hypothesis Test Fid the smallest commo sample sie for which the power of the betwee group F-test i the aalysis of variace table equals or exceeds that specified whe the largest differece betwee ay two meas equals, based o a o-cetral F distributio with o-cetrality parameter k (5) Note: for all oe-sided tests, replace by i the equatios for the hypothesis tests. 3 by StatPoit Techologies, Ic. Sample Sie Determiatio (Two Samples) - 8

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