PASS Sample Sze Software Chapter 70 Probt Analyss Introducton Probt and lot analyss may be used for comparatve LD 50 studes for testn the effcacy of drus desned to prevent lethalty. Ths proram module presents calculates power and sample sze usn the methodoloy outlned n Kodell, Lensn, Landes, Kumar, and Hauer-Jensen (00). Techncal Detals Consder the follown stuaton: suppose two, equal-szed roups of anmals are exposed to the same deathcausn aent such as radaton. The test roup of anmals s exposed to a countermeasure dru, whle the control roup s not. The study s objectve s to test whether the radaton LD 50 of the treatment roup s snfcantly reater than that of the control roup. Probt and lot analyss are often used to study the relatve potency of a test treatment over a control treatment. The probt and lot transformatons are ( P) + lo ( D) Y F 0 0 where F(x) s the cumulatve normal dstrbuton for the probt analyss and the cumulatve lostc dstrbuton for the lot analyss, 0 s the ntercept, s the slope, and D s the dose of the aent (radaton). Defne LD 50 ( T ) as the lethal dose for 50% of the treated populaton and LD 50 ( C) as the lethal dose for 50% of the control populaton. Fnney (978) provdes methodoloy for estmatn the relatve potency (effcacy) usn parallel, lo-dose reresson lnes. Let and The null and alternatve hypotheses are ρ LD LD 50 50 ( T ) ( C) ( ) lo { ( T )} { LD ( C) } lo0 ρ 0 LD50 lo0 50. H 0 H A : ρ : ρ > or H 0 H A :lo :lo 0 0 ( ρ ) 0 ( ρ ) > 0 70- NCSS, LLC. All Rhts Reserved.
PASS Sample Sze Software Let Ths can be estmated by The varance of θˆ s estmated usn θ lo 0 ˆ θ lo Probt Analyss ( ρ ) ˆ 0C 0C 0( ˆ ρ ) ˆ ˆ 0T 0T Vˆ ˆ T, C ( ˆ s ) ( yt yc ) θ + w n where n s the number of anmals n the th dose roup and and w φ w ( Φ ( P )) P ( P ) ( P ) Note that φ( x) s the normal densty functon and Φ ( x) The test statstc for testn H 0 versus H A s ˆ T, C w n for the probt analyss P for a lot analyss. T whch has a t dstrbuton wth f 3 derees of freedom. s the normal c.d.f. ˆ θ Vˆ ( ˆ θ ) ( x x) Usn several smplfcatons and approxmatons, Kodell, et al. (00) show that the sample sze per dose roup s ven by n ( t t ) f, α + f, { lo( ρ )} w The power s ven by t f, n { lo( ρ )} w t f, α 70- NCSS, LLC. All Rhts Reserved.
PASS Sample Sze Software The relatve potency s ven by Probt Analyss A ρ 0 where A ( t + t ) f, α n w f, Procedure Optons Ths secton descrbes the optons that are specfc to ths procedure. These are located on the Desn tab. For more nformaton about the optons of other tabs, o to the Procedure Wndow chapter. Desn Tab The Desn tab contans most of the parameters and optons that you wll be concerned wth. Solve For Solve For Ths opton specfes the parameter to be solved for from the other parameters. Under most stuatons, you wll select ether Power for a power analyss or Sample Sze for sample sze determnaton. Select Sample Sze when you want to calculate the sample sze needed to acheve a ven power and alpha level. Select Power when you want to calculate the power of an experment. Test Model Select whether a Probt or Lot analyss s planned. Ths choce wll result n about the same power and sample sze, but t wll chane the slope. So, f you chane ths value and you have entered Slopes n the Dose Input Opton opton, you must chane the slope values approprately. Power and Alpha Power Ths opton specfes one or more values for power. Power s the probablty of rejectn a false null hypothess, and s equal to one mnus Beta. Beta (consumer s rsk) s the probablty of a type-ii error, whch occurs when a false null hypothess s not rejected. In ths procedure, a type-ii error occurs when you fal to reject the null hypothess of equal thetas when n fact they are dfferent. Values must be between zero and one. Hstorcally, the value of 0.80 (Beta 0.0) was used for power. Now, 0.90 (Beta 0.0) s also commonly used. A snle value may be entered here or a rane of values such as 0.8 to 0.95 by 0.05 may be entered. 70-3 NCSS, LLC. All Rhts Reserved.
PASS Sample Sze Software Probt Analyss Alpha Ths opton specfes one or more values for the probablty of a type-i error. A type-i error occurs when a true null hypothess s rejected. In ths procedure, a type-i error occurs when you reject the null hypothess of equal thetas when n fact they are equal. Values must be between zero and one. Hstorcally, the value of 0.05 has been used for two-sded tests and 0.05 for one-sded tests. You may enter a rane of values such as 0.05 0.05 0.0 or 0.05 to 0.0 by 0.05. Sample Sze n (Sample Sze per Dose-Group) Enter a value (or rane of values) for the sample sze (number of subjects) n each dose-roup. The total sample sze, N, s ths value tmes (the number of roups) tmes the number of doses. These values are nored when you are solvn for n. Effect Sze Taret Response Proportons Enter a set of response proportons, one per dose. Snce these are proportons, they must be between 0 and. It s assumed that the doses of the control roup and the treated roup wll be chosen so that these taret response proportons wll be obtaned. The number of doses,, s set mplctly as the number of values entered here. Popular desns have fve-doses or seven-doses wth proportons that are equally spaced, but you can enter as many as you lke. Expermental desn prncples suest havn some values at each end (near 0 and near ) and some n the mddle. Recommended Values Kodell et al. recommend 0.05, 0.5, 0.5, 0.75, 0.95 for a fve-dose desn and 0.05, 0., 0.35, 0.5, 0.65, 0.8, 0.95 for a seven-dose desn. Input Dose Usn The power calculatons are based on the value of the slope of the reresson equaton. Ths slope can be calculated from a set of doses and response proportons, or t can be entered as a snle value. Doses A set of doses s entered and the slope s calculated from the doses and response proportons. The calculated slope s dfferent dependn on whether a probt analyss or lot analyss s selected. Only one slope may be enerated. Note that the number of doses must match the number of taret response proportons. Slopes One or more slopes may be entered. A separate calculaton s made for each slope. These values depend on the model that s selected. If you chane the model, you need to chane these slopes approprately. Doses (Control Group) Enter a set of doses for the control roup, one for each Taret Response Proporton. The proram calculates the slope usn the Taret Response Proportons entered above. The number of doses must match the number of Taret Response Proportons. 70-4 NCSS, LLC. All Rhts Reserved.
PASS Sample Sze Software Probt Analyss Input Notes. Each control dose should be chosen so that t wll (approxmately) result n the correspondn taret response proporton (lethalty). It s assumed that the treatment doses wll be dfferent from the control doses, but each wll result n the correspondn taret response proporton.. The calculated slopes wll be dfferent dependn on whether you use a probt or a lot model. Slopes Enter a set of slope values. The analyss assumes that the reresson lnes are parallel, that the slopes of the control and treated roups are equal. The slopes depend on the model selected. If you swtch from a probt to lot model, you must chane these slopes approprately. Rho (Relatve Potency) Enter one or more values for rho, the relatve potency or dose reducton factor (DFR), the factor by whch the treatment dru reduces the potency of the dru. Ths desn assumes that the treatment decreases the dose potency, so LD50(Treatment) > LD50(Control). Rane Values must be reater than. Typcal values are.,., etc. You can enter a lst of values such as...3 or. to.5 by 0. Example Power for Several Sample Szes Ths example wll calculate power for several sample szes of a probt analyss study desned to compare the effcacy of a new dru as a countermeasure to radaton-nduced lethalty. Expermenters want to sze the study so that they can detect a relatve potency of.. They also want to study values of.05 and.5. They would lke to study the power at a snfcance level of 0.05 of samples of 5 to 55 subjects. They want to use a fve-dose study wth doses of,, 3, 4, and 5 chosen so that lethaltes of about 0.05, 0.75, 0.5, 0.75, and 0.95 are obtaned. Setup Ths secton presents the values of each of the parameters needed to run ths example. Frst, from the PASS Home wndow, load the Probt Analyss procedure wndow by selectn t from the Survval menu. You may then make the approprate entres as lsted below, or open Example by on to the Fle menu and choosn Open Example Template. Opton Value Desn Tab Solve For... Power Model... Probt Alpha... 0.05 n (Sample Sze per Dose-Group)... 5 to 55 by 0 Taret Response Proportons... 0.05 0.75 0.5 0.75 0.95 Input Dose Usn... Doses Doses (Control Group)... 3 4 5 Rho (Relatve Potency)....05..5 Plot Text Tab Decmal Places Plot Probabltes... 3 70-5 NCSS, LLC. All Rhts Reserved.
PASS Sample Sze Software Probt Analyss Annotated Output Clck the Calculate button to perform the calculatons and enerate the follown output. Numerc Results Numerc Results for a Desn wth 5 Doses Group Total Sample Sample Relatve Sze Sze Potency Slope Power (n) (N) (Rho) () Alpha Beta 0.340 5 50.05 3.07 0.0500 0.86860 0.3583 5 50.05 3.07 0.0500 0.6469 0.57645 5 50.05 3.07 0.0500 0.4355 0.7378 35 350.05 3.07 0.0500 0.68 0.846 45 450.05 3.07 0.0500 0.5874 0.90340 55 550.05 3.07 0.0500 0.09660 0.4538 5 50.0 3.07 0.0500 0.5476 0.9337 5 50.0 3.07 0.0500 0.08663 0.9834 5 50.0 3.07 0.0500 0.0658 0.9954 35 350.0 3.07 0.0500 0.00459 0.99834 45 450.0 3.07 0.0500 0.0066 0.9997 55 550.0 3.07 0.0500 0.00073 0.8064 5 50.5 3.07 0.0500 0.9386 0.99367 5 50.5 3.07 0.0500 0.00633 0.9990 5 50.5 3.07 0.0500 0.00080 0.99980 35 350.5 3.07 0.0500 0.0000 0.99993 45 450.5 3.07 0.0500 0.00007 0.99997 55 550.5 3.07 0.0500 0.00003 Report Defntons Power s the probablty of rejectn a false null hypothess. It should be close to one. Group Sample Sze (n) s the dose-roup sample sze. Total Sample Sze (N) s the total of all dose-roup sample szes. Relatve Potency (Rho) s the rato: (50% Lethal Dose of Treatment) / (50% Lethal Dose of Control). Slope () s the slope of reressn the probts (or lots) on the lo0 doses. Alpha s the probablty of rejectn a true null hypothess. It should be small. Beta s the probablty of acceptn a false null hypothess. It should be small. Summary Statements In a study usn probt analyss to compare the potency of a treatment and a control, 5 subjects are requred from each of 0 dose roups. Ths results n a total sample of 50 subjects. Ths study acheves 3.40% power to detect an ncrease n relatve potency to.05 usn a one-sded t-test wth a 0.0500 snfcance level. The common slope between the probts and the lo doses was assumed to be 3.07. Ths report shows the power for each of the scenaros. Note that the computed slope s about 3. Lethalty Report Lethalty Report Response Group Proporton Weht Number (Lethalty) (w) Dose 0.05000 0.394.00 0.7500 0.55843.00 3 0.50000 0.6366 3.00 4 0.7500 0.55843 4.00 5 0.95000 0.394 5.00 Total.035 Ths report documents the ndvdual values of the lethaltes, wehts, and doses. 70-6 NCSS, LLC. All Rhts Reserved.
PASS Sample Sze Software Plots Secton Probt Analyss 70-7 NCSS, LLC. All Rhts Reserved.
PASS Sample Sze Software Probt Analyss Example Valdaton usn Kodell, et al. We wll valdate ths procedure usn the results of Kodell et al. (00). On pae 43, n Table of ther artcle, Kodell et al. ve the follown example. For a fve-dose example wth taret lethaltes of 0.05, 0.75, 0.5, 0.75, 0.95, a slope of 3.5, rho of., power of 0.9, and alpha of 0.05, they obtan an n of. Setup Ths secton presents the values of each of the parameters needed to run ths example. Frst, from the PASS Home wndow, load the Probt Analyss procedure wndow by selectn t from the Survval menu. You may then make the approprate entres as lsted below, or open Example by on to the Fle menu and choosn Open Example Template. Opton Value Desn Tab Solve For... Sample Sze Model... Probt Power... 0.90 Alpha... 0.05 Taret Response Proportons... 0.05 0.75 0.5 0.75 0.95 Input Dose Usn... Slopes Slopes... 3.5 Rho (Relatve Potency).... Output Clck the Calculate button to perform the calculatons and enerate the follown output. Numerc Results for a Desn wth 5 Doses Group Total Sample Sample Relatve Sze Sze Potency Slope Power (n) (N) (Rho) () Alpha Beta 0.90538 0.0 3.5 0.05000 0.0946 Note that PASS has also calculated the requred sample sze at, or a total sample sze of 0. 70-8 NCSS, LLC. All Rhts Reserved.