Chapter 448 Superiority by a Margin Tet for Two Mean uing Difference Introduction Thi procedure compute power and ample ize for uperiority by a margin tet in two-ample deign in which the outcome i a continuou normal random variable. Meaurement are made on individual that have been randomly aigned to one of two group. Thi i ometime referred to a a parallel-group deign. Thi deign i ued in ituation uch a the comparion of the income level of two region, the nitrogen content of two lake, or the effectivene of two drug. The two-ample t-tet i commonly ued with thi ituation. When the variance of the two group are unequal, Welch t-tet may be ued. When the data are not normally ditributed, the Mann-Whitney (Wilcoxon igned-rank) U tet may be ued. The detail of ample ize calculation for the two-ample deign are preented in the Two-Sample T-Tet chapter and they will not be duplicated here. Thi chapter only dicue thoe change neceary for non-inferiority and uperiority (or non-zero null) tet. Sample ize formula for non-inferiority and uperiority tet of two mean are preented in Chow et al. (003) page 57-59. The Statitical Hypothee Both non-inferiority and uperiority tet are example of directional (one-ided) tet and their power and ample ize could be calculated uing the Two-Sample T-Tet procedure. However, at the urging of our uer, we have developed thi module, which provide the input and output in format that are convenient for thee type of tet. Thi ection will review the pecific of non-inferiority and uperiority teting. Remember that in the uual t-tet etting, the null (H0) and alternative (H) hypothee for one-ided tet are defined a H 0 :µ µ D veru H :µ µ Rejecting thi tet implie that the mean difference i larger than the value D. Thi tet i called an upper-tailed tet becaue it i rejected in ample in which the difference between the ample mean i larger than D. Following i an example of a lower-tailed tet. H 0 :µ µ D veru H :µ µ on-inferiority and uperiority tet are pecial cae of the above directional tet. It will be convenient to adopt the following pecialized notation for the dicuion of thee tet. > D < D 448-
Parameter PASS Input/Output Interpretation µ ot ued Mean of population. Population i aumed to conit of thoe who have received the new treatment. µ ot ued Mean of population. Population i aumed to conit of thoe who have received the reference treatment. M SM Margin of uperiority. Thi i a tolerance value that define the magnitude of difference that i required for practical importance. Thi may be thought of a the mallet difference from the reference that i conidered to be different. δ D True difference. Thi i the value of µ µ, the difference between the mean. Thi i the value at which the power i calculated. ote that the actual value of µ and µ are not needed. Only their difference i needed for power and ample ize calculation. Superiority Tet A uperiority by a margin tet tet that the treatment mean i better than the reference mean by more than the uperiority margin. The actual direction of the hypothei depend on the repone variable being tudied. Cae : High Value Good In thi cae, higher value are better. The hypothee are arranged o that rejecting the null hypothei implie that the treatment mean i greater than the reference mean by at leat the margin of uperiority. The value of δ mut be greater than M. The following are equivalent et of hypothee. S H 0 : µ µ + M S veru H : µ µ + M S > H 0 : µ µ M S veru H : µ µ > M S H 0 :δ M S veru H :δ > M S Cae : High Value Bad In thi cae, lower value are better. The hypothee are arranged o that rejecting the null hypothei implie that the treatment mean i le than the reference mean by at leat the margin of uperiority. The value of δ mut be le than ε. The following are equivalent et of hypothee. H : µ µ veru H : µ < µ M S 0 M S H 0 : µ µ M S veru H : µ µ < M S H :δ M S veru H :δ < M S 0 448-
Example A uperiority tet example will et the tage for the dicuion of the terminology that follow. Suppoe that a tet i to be conducted to determine if a new cancer treatment ubtantially improve mean bone denity. The adjuted mean bone denity (AMBD) in the population of interet i 0.00300 gm/cm with a tandard deviation of 0.000300 gm/cm. Clinician decide that if the treatment increae AMBD by more than 5% (0.0005 gm/cm), it provide a ignificant health benefit. The hypothei of interet i whether the mean AMBD in the treated group i more than 0.0005 above that of the reference group. The tatitical tet will be et up o that if the null hypothei i rejected, the concluion will be that the new treatment i uperior. The value 0.0005 gm/cm i called the margin of uperiority. Tet Statitic Thi ection decribe the tet tatitic that are available in thi procedure. Two-Sample T-Tet Under the null hypothei, thi tet aume that the two group of data are imple random ample from a ingle population of normally-ditributed value that all have the ame mean and variance. Thi aumption implie that the data are continuou and their ditribution i ymmetric. The calculation of the tet tatitic for the cae when higher repone value are good i a follow. where t df ( X X) X X ε X k k X i k ki X X ( X i X) + ( X i X ) i i + + df + The null hypothei i rejected if the computed p-value i le than a pecified level (uually 0.05). Otherwie, no concluion can be reached. Welch T-Tet Welch (938) propoed the following tet when the two variance are not aumed to be equal. t * f ( X X ) * X X ε 448-3
where * X X ( X i X) ( ) i + i ( X i X ) ( ) f + 4 + ( ) ( ) 4 ( X i X) i, ( X i X ) i Mann-Whitney U Tet Thi tet i the nonparametric ubtitute for the equal-variance t-tet. Two key aumption are that the ditribution are at leat ordinal and that they are identical under H0. Thi mean that tie (repeated value) are not acceptable. When tie are preent, you can ue approximation, but the theoretic reult no longer hold. The Mann-Whitney tet tatitic i defined a follow in Gibbon (985). where W ( ) Rank X k k W ( + + ) + C z The rank are determined after combining the two ample. The tandard deviation i calculated a W W 3 ( ti ti ) ( + + ) i ( + )( + ) where t i i the number of obervation tied at value one, t i the number of obervation tied at ome value two, and o forth. The correction factor, C, i 0.5 if the ret of the numerator i negative or -0.5 otherwie. The value of z i then compared to the normal ditribution. 448-4
Computing the Power Standard Deviation Equal When σ σ σ, the power of the t tet i calculated a follow.. Find t α uch that Tdf ( tα ) α, where Tdf ( tα ) df +.. Calculate: σ σ x + ε δ 3. Calculate the noncentrality parameter: λ σ 4. Calculate: Power Tdf,λ ( tα ), where T ( x) x i the area under a central-t curve to the left of x and df,λ i the area to the left of x under a noncentral-t curve with degree of freedom df and noncentrality parameter λ. Standard Deviation Unequal Thi cae often recommend Welch tet. When σ σ, the power i calculated a follow. σ σ. Calculate: σ +.. Calculate: f x 4 σ x - σ 4 ( +) + σ 4 ( +) which i the adjuted degree of freedom. Often, thi i rounded to the next highet integer. ote that thi i not the value of f ued in the computation of the actual tet. Intead, thi i the expected value of f. 3. Find t α uch that Tf ( tα ) α, where Tf ( tα ) degree of freedom. ε 4. Calculate: λ, the noncentrality parameter. σ x 5. Calculate: Power Tf,λ ( tα ) ( ), where T x degree of freedom f and noncentrality parameter λ. i the area to the left of x under a central-t curve with f f,λ i the area to the left of x under a noncentral-t curve with onparametric Adjutment When uing the Mann-Whitney tet rather than the t tet, reult by Al-Sunduqchi and Guenther (990) indicate that power calculation for the Mann-Whitney tet may be made uing the tandard t tet formulation with a imple adjutment to the ample ize. The ize of the adjutment depend on the actual ditribution of the data. They give ample ize adjutment factor for four ditribution. Thee are for uniform, /3 for double exponential, 9 / π for logitic, and π / 3 for normal ditribution. 448-5
Procedure Option Thi ection decribe the option that are pecific to thi procedure. Thee are located on the Deign tab. For more information about the option of other tab, go to the Procedure Window chapter. Deign Tab The Deign tab contain mot of the parameter and option that you will be concerned with. Solve For Solve For Thi option pecifie the parameter to be calculated from the value of the other parameter. Under mot condition, you would elect either Power or Sample Size (). Select Sample Size () when you want to determine the ample ize needed to achieve a given power and alpha. Select Power when you want to calculate the power of an experiment that ha already been run. Tet Higher Mean Are Thi option define whether higher value of the repone variable are to be conidered better or wore. The choice here determine the direction of the tet. If Higher Mean Are Better the null hypothei i Diff SM and the alternative hypothei i Diff > SM. If Higher Mean Are Wore the null hypothei i Diff -SM and the alternative hypothei i Diff < -SM. onparametric Adjutment (Mann-Whitney Tet) Thi option make appropriate ample ize adjutment for the Mann-Whitney tet. Reult by Al-Sunduqchi and Guenther (990) indicate that power calculation for the Mann-Whitney tet may be made uing the tandard t tet formulation with a imple adjutment to the ample ize. The ize of the adjutment depend upon the actual ditribution of the data. They give ample ize adjutment factor for four ditribution. Thee are for the uniform ditribution, /3 for the double exponential ditribution, 9 / π for the logitic ditribution, and π / 3 for the normal ditribution. The option are a follow: Ignore Do not make a Mann-Whitney adjutment. Thi indicate that you want to analyze a t tet, not the Wilcoxon tet. Uniform Make the Mann-Whitney ample ize adjutment auming the uniform ditribution. Since the factor i one, thi option perform the ame function a Ignore. It i included for completene. Double Exponential Make the Mann-Whitney ample ize adjutment auming that the data actually follow the double exponential ditribution. 448-6
Logitic Make the Mann-Whitney ample ize adjutment auming that the data actually follow the logitic ditribution. ormal Make the Mann-Whitney ample ize adjutment auming that the data actually follow the normal ditribution. Power and Alpha Power Thi option pecifie one or more value for power. Power i the probability of rejecting a fale null hypothei, and i equal to one minu Beta. Beta i the probability of a type-ii error, which occur when a fale null hypothei i not rejected. In thi procedure, a type-ii error occur when you fail to reject the null hypothei of inferiority when the null hypothei hould be rejected. Value mut be between zero and one. Hitorically, the value of 0.80 (Beta 0.0) wa ued for power. ow, 0.90 (Beta 0.0) i alo commonly ued. A ingle value may be entered here or a range of value uch a 0.8 to 0.95 by 0.05 may be entered. Alpha Thi option pecifie one or more value for the probability of a type-i error. A type-i error occur when a true null hypothei i rejected. In thi procedure, a type-i error occur when you reject the null hypothei of inferiority when in fact the mean i not non-inferior. Value mut be between zero and one. Hitorically, the value of 0.05 ha been ued for alpha. Thi mean that about one tet in twenty will falely reject the null hypothei. You hould pick a value for alpha that repreent the rik of a type-i error you are willing to take in your experimental ituation. You may enter a range of value uch a 0.0 0.05 0.0 or 0.0 to 0.0 by 0.0. Sample Size (When Solving for Sample Size) Group Allocation Select the option that decribe the contraint on or or both. The option are Equal ( ) Thi election i ued when you wih to have equal ample ize in each group. Since you are olving for both ample ize at once, no additional ample ize parameter need to be entered. Enter, olve for Select thi option when you wih to fix at ome value (or value), and then olve only for. Pleae note that for ome value of, there may not be a value of that i large enough to obtain the deired power. Enter, olve for Select thi option when you wih to fix at ome value (or value), and then olve only for. Pleae note that for ome value of, there may not be a value of that i large enough to obtain the deired power. Enter R /, olve for and For thi choice, you et a value for the ratio of to, and then PASS determine the needed and, with thi ratio, to obtain the deired power. An equivalent repreentation of the ratio, R, i 448-7
R *. Enter percentage in Group, olve for and For thi choice, you et a value for the percentage of the total ample ize that i in Group, and then PASS determine the needed and with thi percentage to obtain the deired power. (Sample Size, Group ) Thi option i diplayed if Group Allocation Enter, olve for i the number of item or individual ampled from the Group population. mut be. You can enter a ingle value or a erie of value. (Sample Size, Group ) Thi option i diplayed if Group Allocation Enter, olve for i the number of item or individual ampled from the Group population. mut be. You can enter a ingle value or a erie of value. R (Group Sample Size Ratio) Thi option i diplayed only if Group Allocation Enter R /, olve for and. R i the ratio of to. That i, R /. Ue thi value to fix the ratio of to while olving for and. Only ample ize combination with thi ratio are conidered. i related to by the formula: where the value [Y] i the next integer Y. [R ], For example, etting R.0 reult in a Group ample ize that i double the ample ize in Group (e.g., 0 and 0, or 50 and 00). R mut be greater than 0. If R <, then will be le than ; if R >, then will be greater than. You can enter a ingle or a erie of value. Percent in Group Thi option i diplayed only if Group Allocation Enter percentage in Group, olve for and. Ue thi value to fix the percentage of the total ample ize allocated to Group while olving for and. Only ample ize combination with thi Group percentage are conidered. Small variation from the pecified percentage may occur due to the dicrete nature of ample ize. The Percent in Group mut be greater than 0 and le than 00. You can enter a ingle or a erie of value. 448-8
Sample Size (When ot Solving for Sample Size) Group Allocation Select the option that decribe how individual in the tudy will be allocated to Group and to Group. The option are Equal ( ) Thi election i ued when you wih to have equal ample ize in each group. A ingle per group ample ize will be entered. Enter and individually Thi choice permit you to enter different value for and. Enter and R, where R * Chooe thi option to pecify a value (or value) for, and obtain a a ratio (multiple) of. Enter total ample ize and percentage in Group Chooe thi option to pecify a value (or value) for the total ample ize (), obtain a a percentage of, and then a -. Sample Size Per Group Thi option i diplayed only if Group Allocation Equal ( ). The Sample Size Per Group i the number of item or individual ampled from each of the Group and Group population. Since the ample ize are the ame in each group, thi value i the value for, and alo the value for. The Sample Size Per Group mut be. You can enter a ingle value or a erie of value. (Sample Size, Group ) Thi option i diplayed if Group Allocation Enter and individually or Enter and R, where R *. i the number of item or individual ampled from the Group population. mut be. You can enter a ingle value or a erie of value. (Sample Size, Group ) Thi option i diplayed only if Group Allocation Enter and individually. i the number of item or individual ampled from the Group population. mut be. You can enter a ingle value or a erie of value. R (Group Sample Size Ratio) Thi option i diplayed only if Group Allocation Enter and R, where R *. R i the ratio of to. That i, R / Ue thi value to obtain a a multiple (or proportion) of. i calculated from uing the formula: where the value [Y] i the next integer Y. [R x ], 448-9
For example, etting R.0 reult in a Group ample ize that i double the ample ize in Group. R mut be greater than 0. If R <, then will be le than ; if R >, then will be greater than. You can enter a ingle value or a erie of value. Total Sample Size () Thi option i diplayed only if Group Allocation Enter total ample ize and percentage in Group. Thi i the total ample ize, or the um of the two group ample ize. Thi value, along with the percentage of the total ample ize in Group, implicitly define and. The total ample ize mut be greater than one, but practically, mut be greater than 3, ince each group ample ize need to be at leat. You can enter a ingle value or a erie of value. Percent in Group Thi option i diplayed only if Group Allocation Enter total ample ize and percentage in Group. Thi value fixe the percentage of the total ample ize allocated to Group. Small variation from the pecified percentage may occur due to the dicrete nature of ample ize. The Percent in Group mut be greater than 0 and le than 00. You can enter a ingle value or a erie of value. Effect Size Mean Difference SM (Superiority Margin) Thi i the magnitude of the margin of uperiority. It mut be entered a a poitive number. When higher mean are better, thi value i the ditance above the reference mean that i required to be conidered uperior. When higher mean are wore, thi value i the ditance below the reference mean that i required to be conidered uperior. D (True Difference, Trt Mean Ref Mean) Thi i the actual difference between the treatment mean and the reference mean at which the power i calculated. When higher mean are better, thi value hould be greater than SM. When higher mean are wore, thi value hould be negative and greater in magnitude than SM. Effect Size Standard Deviation S and S (Standard Deviation) Thee option pecify the value of the tandard deviation for each group. When the S i et to S, the EQUAL VARIACE tet i ued and only S need to be pecified. The value of S will be ued for S. Otherwie, the UEQUAL VARIACE tet i ued (even if the value entered for S equal S). When thee value are not known, you mut upply etimate of them. Pre the SD button to diplay the Standard Deviation Etimator window. Thi procedure will help you find appropriate value for the tandard deviation. 448-0
Example Power Analyi Suppoe that a tet i to be conducted to determine if a new cancer treatment improve bone denity. The adjuted mean bone denity (AMBD) in the population of interet i 0.00300 gm/cm with a tandard deviation of 0.000300 gm/cm. Clinician decide that if the treatment increae AMBD by more than 5% (0.0005 gm/cm), it generate a ignificant health benefit. They alo want to conider what would happen if the margin of uperiority i et to.5% (0.0000575 gm/cm). The analyi will be a non- zero null tet uing the t-tet at the 0.05 ignificance level. Power to be calculated auming that the new treatment ha 7.5% improvement on AMBD. Several ample ize between 0 and 800 will be analyzed. The reearcher want to achieve a power of at leat 90%. All number have been multiplied by 0000 to make the report and plot eaier to read. Setup Thi ection preent the value of each of the parameter needed to run thi example. Firt, from the PASS Home window, load the procedure window by expanding Mean, then Two Independent Mean, then clicking on Superiority by a Margin, and then clicking on. You may then make the appropriate entrie a lited below, or open Example by going to the File menu and chooing Open Example Template. Option Value Deign Tab Solve For... Power Higher Mean Are... Better onparametric Adjutment... Ignore Alpha... 0.05 Group Allocation... Equal ( ) Sample Size Per Group... 0 50 00 00 300 500 600 800 SM (Superiority Margin)... 0.575.5 D (True Difference)....75 S (Standard Deviation Group )... 3 S (Standard Deviation Group )... S Annotated Output Click the Calculate button to perform the calculation and generate the following output. umeric Reult and Plot umeric Reult for Superiority Tet (H0: Diff SM; H: Diff > SM) Higher Mean are Better Tet Statitic: T-Tet 0.553 0 0 0 0.575.75 3.0 3.0 0.05 0.4754 50 50 00 0.575.75 3.0 3.0 0.05 0.76957 00 00 00 0.575.75 3.0 3.0 0.05 0.96885 00 00 400 0.575.75 3.0 3.0 0.05 0.9968 300 300 600 0.575.75 3.0 3.0 0.05 0.99998 500 500 000 0.575.75 3.0 3.0 0.05.00000 600 600 00 0.575.75 3.0 3.0 0.05.00000 800 800 600 0.575.75 3.0 3.0 0.05 0.0603 0 0 0.50.75 3.0 3.0 0.05 0.560 50 50 00.50.75 3.0 3.0 0.05 (report continue) 448-
Report Definition Power i the probability of rejecting a fale null hypothei. and are the number of item ampled from each population. i the total ample ize, +. SM i the magnitude of the margin of uperiority. Since higher mean are better, thi value i poitive and i the ditance above the reference mean that i required to be conidered uperior. D i the mean difference at which the power i computed. D Mean - Mean, or Treatment Mean - Reference Mean. S and S are the aumed population tandard deviation for group and, repectively. Alpha i the probability of rejecting a true null hypothei. Summary Statement Group ample ize of 0 and 0 achieve 3% power to detect uperiority uing a one-ided, two-ample t-tet. The margin of uperiority i 0.575. The true difference between the mean i aumed to be.75. The ignificance level (alpha) of the tet i 0.0500. The data are drawn from population with tandard deviation of 3.000 and 3.000. Chart Section 448-
The above report how that for SM.5, the ample ize neceary to obtain 90% power i about 600 per group. However, if SM 0.575, the required ample ize i only about 80 per group. 448-3
Example Finding the Sample Size Continuing with Example, the reearcher want to know the exact ample ize for each value of SM to achieve 90% power. Setup Thi ection preent the value of each of the parameter needed to run thi example. Firt, from the PASS Home window, load the procedure window by expanding Mean, then Two Independent Mean, then clicking on Superiority by a Margin, and then clicking on. You may then make the appropriate entrie a lited below, or open Example by going to the File menu and chooing Open Example Template. Option Value Deign Tab Solve For... Sample Size Higher Mean Are... Better onparametric Adjutment... Ignore Power... 0.90 Alpha... 0.05 Group Allocation... Equal ( ) SM (Superiority Margin)... 0.575.5 D (True Difference)....75 S (Standard Deviation Group )... 3 S (Standard Deviation Group )... S Output Click the Calculate button to perform the calculation and generate the following output. umeric Reult umeric Reult for Superiority Tet (H0: Diff SM; H: Diff > SM) Higher Mean are Better Tet Statitic: T-Tet Target Actual Power Power SM D S S Alpha 0.90 0.90004 44 44 88 0.6.7 3.0 3.0 0.05 0.90 0.90036 573 573 46..7 3.0 3.0 0.05 Thi report how the exact ample ize requirement for each value of SM. Example 3 Validation Thi procedure ue the ame mechanic a the on-inferiority Tet for Two Mean uing Difference procedure. We refer the uer to Example 3 and 4 of Chapter 450 for the validation. 448-4