Superiority by a Margin Tests for One Mean

Transcription

1 Chapter 43 uperiority by a Margi ests for Oe Mea Itroductio his module computes power ad sample size for tests i oe-sample desigs with a superiority margi i which the outcome is distributed as a ormal radom variable. his icludes the aalysis of the differeces betwee paired values. he details of sample size calculatio for the oe-sample desig are preseted i the Iequality ests for Oe Mea chapter ad they will ot be duplicated here. his chapter oly discusses those chages ecessary for superiority (or o-zero ull) tests. ample size formulas for o-iferiority ad superiority hypothesis tests of a sigle mea are preseted i Chow et al. (23) page 5. he oe-sample t-test is used to test whether a populatio mea is differet from a specific value. Whe the data are differeces betwee paired values, this test is kow as the paired t-test. his module also calculates the power of the o-parametric aalog of the t-test, the Wilcoxo test. Paired Desigs Paired data may occur because two measuremets are made o the same subject or because measuremets are made o two subjects that have bee matched accordig to other variables. ypothesis tests o paired data ca be aalyzed by cosiderig the differece betwee the paired items as the respose. he distributio of differeces is usually symmetric. I fact, the distributio must be symmetric if the idividual distributios of the two items are idetical. ece, the paired t-test ad the Wilcoxo siged-rak test are appropriate for paired data eve whe the distributios of the idividual items are ot ormal. I paired desigs, the variable of iterest is the differece betwee two idividual measuremets. Although the oiferiority hypothesis refers to the differece betwee two idividual meas, the actual values of those meas are ot eeded. All that is eeded is their differece. he tatistical ypotheses Both o-iferiority ad superiority tests are examples of directioal (oe-sided) tests ad their power ad sample size could be calculated usig the Oe-ample -est procedure. owever, at the urgig of our users, we have developed this module which provides the iput ad output optios that are coveiet for superiority hypothesis tests. his sectio will review the specifics of superiority testig. 43-

2 uperiority by a Margi ests for Oe Mea emember that i the usual t-test settig, the ull () ad alterative () hypotheses for oe-sided tests are defied as X A versus X > A ejectig implies that the mea is larger tha the value A. his test is called a upper-tail test because is rejected i samples i which the sample mea is larger tha A. Followig is a example of a lower-tail test. X A versus X < A uperiority tests are special cases of the above directioal tests. It will be coveiet to adopt the followig specialize otatio for the discussio of these tests. Parameter PA Iput/Output Iterpretatio µ Not used Populatio mea. If the data are paired differeces, this is the mea of those differeces. µ Not used eferece value. Usually, this is the mea of a referece populatio. If the data are paired differeces, this is the hypothesized value of the mea differece. M s M Margi of superiority. his is a tolerace value that defies the magitude of differece that is required for practical importace. his may be thought of as the smallest differece from the referece value that is cosidered to be differet. δ D rue differece. his is the value of µ µ, the differece betwee the mea ad the referece value, at which the power is calculated. Note that the actual values of µ ad µ are ot eeded. Oly their differece is eeded for power ad sample size calculatios. uperiority ests A superiority by a margi test tests that the mea is better tha that of the baselie (referece) populatio by more tha a small superiority margi. he actual directio of the hypothesis depeds o the whether higher values of the respose are good or bad. Case : igh Values Good I this case, higher values are better. he hypotheses are arraged so that rejectig the ull hypothesis implies that the mea is greater tha the referece value by at least the margi of superiority. he value of δ must be greater tha M. Equivalet sets of the ull ad alterative hypotheses are µ + M versus > µ + M µ M versus µ > M :δ M versus :δ > M 43-2

3 uperiority by a Margi ests for Oe Mea Case 2: igh Values Bad I this case, lower values are better. he hypotheses are arraged so that rejectig the ull hypothesis implies that the mea is less tha the referece value by at least the margi of superiority. he value of δ must be less tha. Equivalet sets of the ull ad alterative hypotheses are M < µ M µ M versus µ M versus µ < M :δ M versus :δ < M Example A superiority test example will set the stage for the discussio of the termiology that follows. uppose that a test is to be coducted to determie if a ew cacer treatmet substatially improves the mea boe desity. he adjusted mea boe desity (AMBD) i the populatio of iterest is.23 gm/cm with a stadard deviatio of.3 gm/cm. Cliicias decide that if the treatmet icreases AMBD by more tha 5% (.5 gm/cm), it provides a sigificat health beefit. he hypothesis of iterest is whether the AMBD i the treated group is greater tha =.245. he statistical test will be set up so that if the ull hypothesis that the AMBD is greater tha or equal to. 245 is rejected, the coclusio will be that the ew treatmet is superior, at least i terms of AMBD. he value.5 gm/cm is called the margi of superiority. est tatistics his sectio describes the test statistics that are available i this procedure. Oe-ample -est he oe-sample t-test assumes that the data are a simple radom sample from a populatio of ormallydistributed values that all have the same mea ad variace. his assumptio implies that the data are cotiuous ad their distributio is symmetric. he calculatio of the t-test proceeds as follow where t = X D s X X X i = = i s X = ( X i X ) i= 2 ad D is the value of the mea hypothesized by the ull hypothesis. he sigificace of the test statistic is determied by computig the p-value. If this p-value is less tha a specified level (usually.5), the hypothesis is rejected. Otherwise, o coclusio ca be reached. 43-3

4 uperiority by a Margi ests for Oe Mea Wilcoxo iged-ak est he Wilcoxo siged-rak test is a popular, oparametric substitute for the t-test. It assumes that the data follow a symmetric distributio. he test is computed usig the followig steps.. ubtract the hypothesized mea, D, from each data value. ak the values accordig to their absolute values. 2. Compute the sum of the positive raks p ad the sum of the egative raks. he test statistic, W, is the miimum of p ad. 3. Compute the mea ad stadard deviatio of W usig the formulas ( ) + µ = σ W W = 4 ad where t represets the umber of times the ith value occurs. 4. Compute the z value usig z W ( )( 2 ) t - t W µ W = σ he sigificace of the test statistic is determied by computig the p-value usig the stadard ormal distributio. If this p-value is less tha a specified level (usually.5), the ull hypothesis is rejected i favor of the alterative hypothesis. Otherwise, o coclusio ca be reached. W Computig the Power he power is calculated as follows for a directioal alterative (oe-tailed test) i which D > D. D is the value of the mea at which the power is computed.. Fid t α such that ( tα ) = α, where ( t ) degrees of freedom. 2. Calculate x = D + t a α σ. α D D 3. Calculate the ocetrality parameter λ =. σ a 4. Calculate t a = x - D + λ. σ 5. Calculate the power =,λ ( ta ), where ( x) is the area to the left of x uder a cetral-t curve with,λ is the area to the left of x uder a ocetral-t curve with degrees of freedom ad ocetrality parameter λ. 43-4

5 uperiority by a Margi ests for Oe Mea Procedure Optios his sectio describes the optios that are specific to this procedure. hese are located o the Desig tab. For more iformatio about the optios of other tabs, go to the Procedure Widow chapter. Desig ab he Desig tab cotais most of the parameters ad optios that will be of iterest. olve For olve For his optio specifies the parameter to be calculated from the values of the other parameters. Uder most coditios, you would select either Power or ample ize. elect ample ize whe you wat to determie the sample size eeded to achieve a give power ad alpha error level. elect Power whe you wat to calculate the power. est igher Meas Are his optio defies whether higher values of the respose variable are to be cosidered better or worse. he choice here determies the directio of the test. If igher Meas Are Better the ull hypothesis is Diff M ad the alterative hypothesis is Diff > M. If igher Meas Are Worse the ull hypothesis is Diff -M ad the alterative hypothesis is Diff < -M. Noparametric Adjustmet his optio makes appropriate sample size adjustmets for the Wilcoxo test. esults by Al-uduqchi ad Guether (99) idicate that power calculatios for the Wilcoxo test may be made usig the stadard t test formulatios with a simple adjustmet to the sample size. he size of the adjustmet depeds upo the actual distributio of the data. hey give sample size adjustmet factors for four distributios. hese are for the 2 uiform distributio, 2/3 for the double expoetial distributio, 9 / π for the logistic distributio, ad π / 3 for the ormal distributio. he optios are as follows: Igore Do ot make a Wilcoxo adjustmet. his idicates that you wat to aalyze a t test, ot the Wilcoxo test. Uiform Make the Wilcoxo sample size adjustmet assumig the uiform distributio. ice the factor is oe, this optio performs the same as Igore. It is icluded for completeess. Double Expoetial Make the Wilcoxo sample size adjustmet assumig that the data actually follow the double expoetial distributio. 43-5

6 uperiority by a Margi ests for Oe Mea Logistic Make the Wilcoxo sample size adjustmet assumig that the data actually follow the logistic distributio. Normal Make the Wilcoxo sample size adjustmet assumig that the data actually follow the ormal distributio. Populatio ize his is the umber of subjects i the populatio. Usually, you assume that samples are draw from a very large (ifiite) populatio. Occasioally, however, situatios arise i which the populatio of iterest is of limited size. I these cases, appropriate adjustmets must be made. Whe a fiite populatio size is specified, the stadard deviatio is reduced accordig to the formula 2 2 σ = σ N where is the sample size, N is the populatio size, σ is the origial stadard deviatio, ad σ is the ew stadard deviatio. he quatity /N is ofte called the samplig fractio. he quatity correctio factor. N is called the fiite populatio Power ad Alpha Power his optio specifies oe or more values for power. Power is the probability of rejectig a false ull hypothesis, ad is equal to oe mius Beta. Beta is the probability of a type-ii error, which occurs whe a false ull hypothesis is ot rejected. Values must be betwee zero ad oe. istorically, the value of.8 (Beta =.2) was used for power. Now,.9 (Beta =.) is also commoly used. A sigle value may be etered here or a rage of values such as.8 to.95 by.5 may be etered. Alpha his optio specifies oe or more values for the probability of a type-i error. A type-i error occurs whe a true ull hypothesis is rejected. Values must be betwee zero ad oe. istorically, the value of.5 has bee used for alpha. his meas that about oe test i twety will falsely reject the ull hypothesis. You should pick a value for alpha that represets the risk of a type-i error you are willig to take i your experimetal situatio. You may eter a rage of values such as..5. or. to. by

7 ample ize uperiority by a Margi ests for Oe Mea N (ample ize) his optio specifies oe or more values of the sample size, the umber of idividuals i the study. his value must be a iteger greater tha oe. You may eter a list of values usig the sytax or 5 to 25 by 5. Effect ize Mea Differece M (uperiority Margi) his is the magitude of the margi of superiority. It must be etered as a positive umber. Whe higher meas are better, this value is the distace above the referece value that is required to be cosidered superior. Whe higher meas are worse, this value is the distace below the referece value that is required to be cosidered superior. D (rue Differece) his is the actual differece betwee the mea ad the referece value. Whe higher meas are better, this value should be greater tha M. Whe higher meas are worse, this value should be egative ad greater i magitude tha M. Effect ize tadard Deviatio tadard Deviatio his optio specifies oe or more values of the stadard deviatio. his must be a positive value. PA icludes a special module for estimatig the stadard deviatio. his module may be loaded by pressig the D butto. efer to the tadard Deviatio Estimator chapter for further details. 43-7

8 uperiority by a Margi ests for Oe Mea Example Power Aalysis uppose that a test is to be coducted to determie if a ew cacer treatmet improves the mea boe desity. he adjusted mea boe desity (AMBD) i the populatio of iterest is.23 gm/cm with a stadard deviatio of.3 gm/cm. Cliicias decide that if the treatmet icreases AMBD by more tha 5% (.5 gm/cm), it geerates a sigificat health beefit. hey also wat to cosider what would happe if the margi of superiority is set to 2.5% (.575 gm/cm). he aalysis will be a superiority test usig the t-test at the.25 sigificace level. Power is to be calculated assumig that the ew treatmet has 7.5% improvemet o AMBD. everal sample sizes betwee 2 ad 3 will be aalyzed. he researchers wat to achieve a power of at least 9%. All umbers have bee multiplied by to make the reports ad plots easier to read. etup his sectio presets the values of each of the parameters eeded to ru this example. First, from the PA ome widow, load the uperiority by a Margi ests for Oe Mea procedure widow by expadig Meas, the Oe Mea, the clickig o uperiority by a Margi, ad the clickig o uperiority by a Margi ests for Oe Mea. You may the make the appropriate etries as listed below, or ope Example by goig to the File meu ad choosig Ope Example emplate. Optio Value Desig ab olve For... Power igher Meas Are... Better Noparametric Adjustmet... Igore Populatio ize... Ifiite Alpha N (ample ize) M (uperiority Margi) D (rue Differece) (tadard Deviatio)... 3 Aotated Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric esults for uperiority est (: Diff M; : Diff > M) igher Meas are Better est tatistic: -est uperiority Actual igificace tadard Margi Differece Level Deviatio Power N (M) (D) (Alpha) Beta () (report cotiues) 43-8

9 uperiority by a Margi ests for Oe Mea eport Defiitios (ull hypothesis) is Diff M, where Diff = Mea - eferece Value for the oe-sample case, ad Diff = Mea of Differeces - eferece Value for the paired case. (alterative hypothesis) is Diff > M. Power is the probability of rejectig whe it is false. N is the sample size, the umber of subjects i the study. M is the magitude of the margi of superiority. ice higher meas are better, this value is positive ad is the distace above the referece value that is required to be cosidered superior. D is the mea differece (treatmet - referece value) at which the power is computed. Alpha is the probability of rejectig whe it is true, which is the probability of a false positive. Beta is the probability of acceptig whe it is false, which is the probability of a false egative. is the stadard deviatio of the respose. It measures the variability i the populatio. ummary tatemets A sample size of 2 achieves 37% power to detect superiority usig a oe-sided t-test whe the margi of superiority is.575 ad the true differece betwee the mea ad the referece value is.725. he data are draw from a sigle populatio with a stadard deviatio of 3.. he sigificace level (alpha) of the test is

10 uperiority by a Margi ests for Oe Mea he above report shows that for M =.5, the sample size ecessary to obtai 9% power is just uder 3. owever, if M =.575, the required sample size is oly about

11 uperiority by a Margi ests for Oe Mea Example 2 Fidig the ample ize Cotiuig with Example, the researchers wat to kow the exact sample size for each value of M to achieve 9% power. etup his sectio presets the values of each of the parameters eeded to ru this example. First, from the PA ome widow, load the uperiority by a Margi ests for Oe Mea procedure widow by expadig Meas, the Oe Mea, the clickig o uperiority by a Margi, ad the clickig o uperiority by a Margi ests for Oe Mea. You may the make the appropriate etries as listed below, or ope Example 2 by goig to the File meu ad choosig Ope Example emplate. Optio Value Desig ab olve For... ample ize igher Meas Are... Better Noparametric Adjustmet... Igore Populatio ize... Ifiite Power....9 Alpha M (uperiority Margi) D (rue Differece) (tadard Deviatio)... 3 Output Click the Calculate butto to perform the calculatios ad geerate the followig output. Numeric esults Numeric esults for uperiority est (: Diff M; : Diff > M) igher Meas are Better est tatistic: -est uperiority Actual igificace tadard Margi Differece Level Deviatio Power N (M) (D) (Alpha) Beta () his report shows the exact sample size requiremet for each value of M. Example 3 Validatio his procedure uses the same mechaics as the No-Iferiority ests for Oe Mea procedure. We refer the user to Example 3 of Chapter 45 for the validatio. 43-