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Determine sample size by specifying a confidence interval for a population mean.~ "[Event Procedure]x 6=-J@q<X!%J<&K'LH!(M(h1]`6<a(b !c,,i Command4 5. Determine sample size by specifying the power of a test about the difference in population means.~ "[Event Procedure]x w GaJUs!%J<&K('LH!(MTd5]`6o a\+bc,,  Label7 NJoseph George Caldwell PhD (Statistics)x sr%8NA =Jo K\+LM,d5`6SaL,bc,,  Label8 Z503 Chastine Drive, Spartanburg, SC 29301 USAx snjBu{:JSKL,LMx-d5`6a<-bSc,  Label9 rTelephone (001)(864)541-7324, e-mail jcaldwell9@yahoo.comx g$$L}@;ҟJK<-LM;.h1U`6xad#b@c,@i Command10 Command10~ "[Event Procedure] @( ȤLVAL0c = Quit Appx ]MfV /!%Jx&Kd#'L(M%d5U`6xa.b !c,X Label11 $Copyright (c) 2007-2011 Joseph George Caldwell. All rights reserved. This computer software program may be used for personal, noncommercial use.x )|pJ3JxK.L!M0h1]`6<ab !c,,i Command13 J7. Some notes on use of the programs.~ "[Event Procedure]x I3|^UP!%J<&K'LH!(M h1U`6<a(b !c,,i  CmdSS 1b. Determine confidence interval for population mean, given the sample size.~ "[Event Procedure]x *G*@!%J<&K('LH!(MTh1U`6<aTb !c,,i CmdPower 5b. Determine the power of a test about a difference in population means, given the sample size.~ "[Event Procedure]x @+SL.!%J<&KT'LH!(Md5U7`6<a%b !c, Label16 JThis softLVAL1ware is posted at Internet website http://www.foundationwebsite.org/JGCSampleSizeProgram.mdb. It is programmed in Microsoft Access 2007. For additional information on sample survey design (including course notes), see the "Sample Survey" section of http://www.foundationwebsite.org.x x=$G==6d J<K%LH!M)h1]`6<a!b !c,,i Command18 h8. Determine the optimal within-cluster sample size.~ "[Event Procedure]x oVB%Ld!%J<&K!'LH!(M(#d5`6al*b;c, Label19 2For information, contact:x  -1Gm;0dJKl*LMk+h1U`6<ab !c,,i Command20 2. Determine sample size by specifying a confidence interval for a difference in means.~ "[Event Procedure]x nIENHW!%J<&K'LH!(Mh1U`6<ab !c,,i Command21 2b. Determine confidence interval for a difference in means, given the sample size.~ "[Event Procedure]x . D`O!%J<&K'LH!(M h1]`6<a b !c,,i  Command22 3. Determine sample size by specifying a confidence interval for a double difference in means.~ "[Event Procedure]x "p LhaV!%J<&K 'LH!(M h1U`6<a b !c,,i  Command23 3b. Determine confidence interval for a double difference in means, given the sample size.~ "[Event Procedure]x (evݴFwQ9Y!%J<&K 'LH!(M h1U`6<aLb !c,,i  Command24 4. Determine sample size by specifying the power of a test about the value of a population meLVAL2an.~ "[Event Procedure]x N&BxyeHS`*Q' !%J<&KL'LH!(Mxh1U`6<axb !c,,i  Command25 4b. Determine the power of a test about the value of a population mean.~ "[Event Procedure]x 1dF*A=!%J<&Kx'LH!(Mh1]`6<ab !c,,i  Command26 6. Determine sample size by specifying the power of a test of a mean double difference.~ "[Event Procedure]x q.MAXN !%J<&K'LH!(Mh1U`6<ab !c,,i Command27 6b. Determine the power of a test about a double difference in means, given the sample size.~ "[Event Procedure]x H`FJ^o/!%J<&K'LH!(M h1U`6<ab !c,,i Command28 5c. Determine sample size by specifying the power of a test about the diff. in pop. means, for fixed npsu1.~ "[Event Procedure]x J6Nr[> !%J<&K'LH!(Mh1U`6<a b !c,,i Command29 6c. Determine sample size by specifying the power of a test of a mean double difference, for fixed npsu1.~ "[Event Procedure]x m^Mc d> !%J<&K 'LH!(MLh1U`6<aLb !c,,i Command30 6d. Determine the power function for a test of a double difference in means, given the sample size.~ "[Event Procedure]x bTKE_ f> !%J<&KL'LH!(Mxm45I7UF{`6a81btc,i#k  Date1 =Date()" @ Tahomax j} uLP_s~+.J/K810LLVAL31M72d5U7`68a81bc,d# Label30  Date:x ||XJo<J8K81LTM(2m45I7UF{`6a81btc,i#k  Time1 =Time()" @ Tahomax %zLf%7!p+.J/K810L!1M72d5U7`6a81bc,d# Label32  Time:x }퍹KJK81LM(2h1U`6<ab !c,,i Command32 5d. Determine the power function of a test about a difference in population means, given the sample size.~ "[Event Procedure]x uǚL-L8$b}> !%J<&K'LH!(Mh1U`6<ab !c,,i Command34 4c. Determine sample size by specifying the power of a test about the value of a population mean, for fixed npsu.~ "[Event Procedure]x `K9mEz!%J<&K'LH!(Mh1U`6<ab !c,,i Command35 4d. Determine the power function of a test about the value of a population mean, given the sample size.~ "[Event Procedure]x  ++LюMr I!%J<&K'LH!(Mh1U`6<aTb !c,,i Command36 1c. Determine sample size by specifying a confidence interval for a population mean, for fixed npsu.~ "[Event Procedure]x OOJK6t!%J<&KT'LH!(Mh1`6<a b !c,,i Command37 2c. Determine sample size by specifying a confidence interval for a difference in means, for fixed npsu1.~ "[Event Procedure]x |eH)T(!%J<&K 'LH!(MLVAL h1U`6<a b !c,,i Command38 3c. Determine sample size by specifying a confidence interval for a difference in means, for fixed npsu1.~ "[Event Procedure]x t>لNN!%J<&K 'LH!(MLh1`6<a b !c,;i Command39 L7b. Notes on the design effect (deff).x _&D{P1x33"(,F7bNotesOnDeff0-10!%J<&K 'LH!(M "h1`6<ab !c,;i Command40 4e. Determine the min. det. effect for a test about the value of a pop. mean, given the sample size and power.x +;HKltRT33LPVZF4eDetermineMinDetEffectForTestOfMean0-10!%J<&K'LH!(M7h1`6<ab !c,;i Command41 5e. Determine the min. det. effect for a test about a difference in pop. means, given the sample size and power.x 3?@+f33Z^dhF5eDetermineMinDetEffectForTestOfDiffInMeans0-10!%J<&K'LH!(Mh1`6<axb !c,;i Command42 6e. Determine the min. det. effect for a test of a double difference in means, given the sample size and power.x Qh̳׏I L)33fjptF6eDetermineMinDetEffectForTestOfDoubleDiffInMeans0-10!%J<&Kx'LH!(MLVAL  QrU $` $A` $` nͬ1Uĝ͠MbpCy/Detaild /Label0d /Label3d /Label4d /Label5d /Label6d /Label8m/ txtDd / Label10m/ txtzalphad / Label11m/ txtsigma1d /Label12m/txtdeffd /Label15m/txtn1d /Label17h /cmdDoItd /Label14m/txtsigma2d /Label16m/txtrhod /Label18m/txtratiod /Label20m/txtn2d /Label22h /Command23h /Command24d / Label25m/!txtSSd /"Label27d /#Label28m/txtzbetad /$Label29d /%Label30d /&Label31m/(txtpowerd /)Label33m/*Date1d /+Label34m/,Time1d /-Label35m/.txtAlphad //Label36m/0txtBetad /1Label37d /2Label38d /3Label39ͬ&%[LnGXd/Detailm/txtEd /Label2d /Label3d /Label4d /Label5d /Label6m/txtzd /Label9m/ txtsigmad / Label11m/ txtNd / Label13m/ txtdeffd /Label15m/txtSSd /Label17h /cmdDoItd /Label20d /Label24h /Command25h /Command26d /Label27m/Date1d /Label30m/Time1d /Label32m/txtCCd /Label34ͬ 6R"M8_+/Detaild /Label0h /Command7h / Command8d /Label14d /Label16m/Date1d /Label30m/Time1d /Label32d /Label23d /Label24d /Label25d /Label38d /Label26d /Label27d / Label31ͬ.A6 D1@</Detaild /Label0d /Label2d /Label3d /Label4d /Label5d /Label6d /Label8d /Label9d / Label10d / Label11d /Label12m/txtdeffd /Label15d /Label17h /cmdDoItm/ txtDm/ txtzalphad /Label14m/txtzbetad /Label16m/ txtsigma1m/txtsigma2d /Label18m/txtrhod /Label20m/txtratiom/txtn1d /Label22m/txtn2h / Command23h /!Command24d /"Label25d /$Label27m/#txtSSd /%Label28m/'Date1d /(Label30m/)Time1d /*Label32m/+txtAlphad /,Label34m/-txtBetad /.Label31d //Label33d /0Label35ͬl>Ķ]E3's/Detaild /Label2d /Label3d /Label4d /Label5d /Label6d /Label9d / Label11d / Label13d /Label15d /Label17m/txtEm/ txtsigmam/ txtNm/ txtdeffh /cmdDoItd /Label20m/txtSSm/txtzd /Label24h /Command25h /Command26d /Label27d /Label30m/Date1d /Label32m/Time1d /Label34m/txtCCͬ)Lj<rM/Detaild /Label0d /Label1h /Command2h /Command4d /Label7d /Label8d /Label9h / Command10d / Label11h / Command13h / CmdSSh / CmdPowerd /Label16h /Command18d /Label19h /Command20h /Command21h /Command22h /Command23h /Command24h /Command25h /Command26h /Command27h /Command28h /Command29h /Command30m/Date1d /Label30m/Time1d /Label32h / Command32h /!Command34h /"Command35h /#Command36h /$Command37h /%Command38h /&Command39h /'Command40h /(Command41h /)Command42LVAL儩6u08=5B>a)b*c!e g#h6ij$k #?l9*@x юLHr @ Arial 8 h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVAL7     `,)  Detailx /Xx-I!Mlz@ m7UF{`6@ a|c,, txtE 0.05x &KuLD8"A+.J@ /K|0L1Md5U`6xa|b c,  Label2 DConfidence interfal half-width, E:x AKfTJxK|L" Mld5U`6xab !c,  Label3 This program calculates the sample size (n) required to produce a confidence interval of a prescribed size for a population mean.x #y 3B6JxKL!Md5U`6xab !c,D  Label4 The following values must be specified: (1) half-width (E) of the confidence interval; (2) population standard deviation (sigma); (3) population size (N); (4) confidence coefficient (CC); (5) design effect (deff). The confidence coefficient (CC) is the probability that the confidence interval includes the true population mean. (It is equal to 1 minus alpha, where alpha is the probability that the interval does not include the true population mean.) The design effect (deff) is the ratio of the variance (of the estimated population mean) using the specified sample design to the variance using simple random sampling. It is equal to 1.0 for simple random sampling, and may be larger or smaller than 1.0 for complex survey designs. For multistage sampling, diff is usually greater than 1.0 This program assumes sampling from a normal distribution, but may be used for non-normal sampling (e.g., binomial, poisson, exponential) if the sample size (the program output) exceeds 30.x hqGMB>%JxKL!M\ d5U`6xab !c,,  Label5 The formula for the half-width of the confidence interval LVAL8is: E = z(CC) sigma sqrt((1-n/N) deff/n)x oMXJxKL!Md5U`6xab !c,,  Label6 The formula for the sample size is: n = [z(CC)**2 sigma**2 deff] / [E**2 + z(CC)**2 sigma**2 deff/N]x FSHV%4JxKL!Mm7UF{`6@ a$c,,k txtzx _G)Io+.J@ /K$0L1M%d5U`6xa$bc,  Label9 z(1-alpha/2):x E% bIS.*}JxK$LtM%m7UF{`6@ a c,,k txtsigma 0.5x ML RFI@d(+.J@ /K 0L1MLd5U`6xa bc, Label11 4Standard deviation, sigma:x D7#UELJxK LRMm7UF{`6@ ac,,k txtN 1000000000x LAA@"p;+.J@ /K0L1Md5U`6xab c, Label13 &Population size, N:x QG:JxKL Mm7UF{`6@ a !c,,k txtdeff 1x kYPP>KIU۟,+.J@ /K !0L1M8"d5U`6xa !bc, Label15 (Design effect, deff:x 9[cG2$!JxK !LTM!m7UF{`6@ a4&c,,k  txtSSx 8;NHpO"+.J@ /K4&0L1M`'d5U`6xa4&b c, Label17 Sample size, n:x -tG7H+JxK4&L M$'h1U`6a"b@ c,i cmdDoIt  Do It!~ "[Event Procedure]x "LVAL9GxTe!%J&K"'L\ (MT$d5U`6xa b !c, Label20 Note: Given the value of the confidence coefficient (CC=1-alpha), the program calculates a corresponding z-value (z). The z-value (z(1-alpha/2)) is the value of the standard normal deviate having area (probability) alpha/2 = (1 - CC)/2 to the right of that point. Here is a list of z's corresponding to commonly used CC's: z = 1.9600 for CC = .95 (alpha = .05); z = 1.6449 for CC = .90 (alpha = .1); z = 1.2816 for CC = .80 (alpha = .2); and z = 2.0 for CC = .9545 (alpha = .0455). The values CC = .95 (z = 1.96, alpha = .05) and E = .1 x sigma are sometimes used to determine sample size. For sampling for a proportion, p, (i.e., the population mean is a proportion, p) the value of sigma is sqrt[p(1-p)].x iRO pJJJxK L!Mtd5U`6xa4b !c, Label24 This program assumes that the sampling distribution of the sample mean is approximately normally distributed (this may be assumed if n>30). If sampling is considered to be from an infinite population, specify a large value for N (e.g., N=1,000,000,000).x cFWAJxK4L!Mh1U`6La%b c,i Command25 Close Form~ "[Event Procedure]x 7pB+^z!%JL&K%'LW!(M&h1U`6La`'bi Command26 Quit App~ "[Event Procedure]x @vӺ!%JL&K`''LH!(M(d5U`6xa<b !c,d#  Label27 1. Determine sample size by specifying a confidence interval for a population mean.x nz_$o9J1}oJxK<L!Mm7UF{`6a'btc,k   Date1 =Date()x &EC LVALS cq+.J/K'0LT1M(d5U`6a'bc, Label30  Date:x w06 N^JK'LM(m7UF{`6$a'btc,k   Time1 =Time()x |cG0V+.J$/K'0L1M(d5U`6a'bc, Label32  Time:x BVL GZ.JK'LM(m7UF{`6@ ahc,k   txtCC 0.95x 82KB0ˋ+.J@ /Kh0L1Mg d5U`6xahb c, Label34 @Confidence coefficient, 1-alpha:x RSxBzq+JxKhL MX LVAL儩;u08=5B>a)b*c!e g$h6i,j@k  ?l7,+@x ! UO:Ý @ Arial 8 h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVAL<     `,G  Detailx Ȅu+@Y-d5U`6a b c,  Label0 $This program determines sample size based on specification of the power (1-beta) of a (one-sided) test of the hypothesis about the size of the difference between two population means (e.g., the mean of the population in a base year) and the mean of another (second) population (e.g., the mean of the population in a later year). The null hypothesis is that the difference in means is zero; the alternative hypothesis is that the means differ by an amount D, called the effect size. (The two populations could be two different subpopulations of a larger population, such as two different regions under study, or a "treatment" group and a "comparison" group.) This program calculates the sample size for the first group. The program allows the sample sizes of the two groups to be different: the user specifies the ratio of the sample size of the second group to that of the first group, i.e., ratio = n2/n1. x  /C\8^G[o d5U`6a b c,|   Label2  This program gives the same results as the determination of sample size based on specification of the size of a confidence interval, if (1) N (in the confidence-interval approach) is very large; (2) alpha for this (one-sided) approach is set equal to alpha/2 for that (two-sided) approach (e.g., .025 here, .05 there); (3) beta is set equal to .5 (i.e., zbeta = 0); (4) the standard deviation of the second population is set equal to zero; and D is set equal to E. In typical situations, in which it is desired to detect small differences, this approach may yield sample sizes several times as large as the confidence-interval approach. For detecting large differences, this approach generally produces smaller sample sizes. If the means of the two populatioLVAL=ns are correlated, e.g., rho = .5, then the estimated sample size decreases. The value of rho would be zero for two subpopulations sampled independently, but could be high (e.g., rho = .3 or rho = .5) if the second sample is a resurvey (panel survey) of the first. (Example 1: Using CI approach with CC=.95 (alpha=.05, z(1-alpha/2) = 1.96), sigma=.5, deff=1, E=.05 and N=1,000,000 yields n=384. Using the power approach with alpha=.025 (z(1-alpha)=1.9600), beta=.1 (z(1-beta)=1.2816), sigma1=.5, sigma2=.5, rho=0, deff=1, ratio=1 and D=.05 yields n=4,204 (i.e., 11 times as large). Ex. 2: Same as Ex. 1, but alpha = .1 (z(1-alpha)=1.286) yields n = 2,628 (6.84 times as large). Ex. 3: Same as Ex. 1, but assume that sample two is a resurvey of sample 1, and that sample 1 in fact used a very large sample size. Then, setting sigma1=0 and rho=.5, we obtain n=2,102, which is 1.36 as large as the CI approach.)x #h[O涐bJK L!M$d5U`6ab c,p  Label3 0 The following values must be specified: (1) the effect size, D (size of the difference in the means (D>0)); (2) the standard deviations of the two populations (sigma1 and sigma2); (3) the ratio of the sample sizes of the two groups, ratio = n2/n1 (typically = 1, rarely >2); (4) the coefficient of correlation, rho, between observations of the two groups; (5) alpha, the probability of a making a Type I error (i.e., the significance level of the (one-sided) test (of the mean difference)); (6) beta, the probability of making a Type II error (one minus the power of the test); (7) the design effect (deff). The design effect (deff) is the ratio of the variance (of the estimated difference in means) using the specified sample design to the variance using simple random sampling. It is equal to 1.0 for simple random sampling, and may be larger or smaller than 1.0 for complex survey designs. This program assumes sampling from a normal distribution, but may be used for non-norLVAL>mal sampling (e.g., binomial, poisson, exponential) if the sample size (the program output) exceeds 30. (For two-sided tests, use the z-value corresponding to 1-alpha/2 instead of 1-alpha.) x 3"{FC][JKL!M !d5U`6a\+b c,H  Label4 This program assumes sampling from two infinite populations, since the means of two finite populations are almost always different (so that there is no point to a statistical test of the difference in the population means). Hence, the finite population correction does not apply, and the population sizes (N's) are not specified. In this case, it is possible for the program to specify sample sizes (n's) greater than the finite population sizes (N's).x 6C+sxJK\+LH!M.d5U`6a)b c,  Label5 This program assumes that the sampling distributions of the sample means are approximately normally distributed (may be assumed if n>30, and the infinite populations have finite standard deviations (true for "real" phenomena)).x AgXvC#sJK)L!M*d5U`6a!b c,  Label6 Note: Given the values of alpha (the significance level of the test) and beta (1 - the power of the test), the program calculates corresponding z-values. The z-value for 1-alpha is the value of the standard normal deviate having area (probability) alpha to the right of that point, and the z-value for 1-beta is the value of the standard normal deviate having area beta to the right of that point. Here is a list of z's corresponding to commonly used alpha's / beta's: z = 1.9600 for alpha=.025 or beta = .025; z = 1.6449 for alpha = .05 or beta = .05; z = 1.2816 for alpha = .10 or beta = .10; z = 1.0364 for alpha = .15 or beta = .15; z = .8416 for alpha = .2 or beta = .2; and z = 2.0 for alpha = .02275 or beta = .02275. The values alpha = LVAL?.05 (zalpha = 1.6449), beta = .10 (zbeta = 1.2816), and D = .1 x sigma are sometimes used to determine sample size. For sampling for a proportion, p, (i.e., the population mean is a proportion, p) the value of sigma is sqrt[p(1-p)].x MWI=AO<<pQJK!L!M(d5U`6a/b c,X  Label8 The formula from which the result is derived is Prob([samplemean1 - samplemean2] / sqrt(deff varest/n1) > zalpha | popmean1 - popmean2 = D) =1 - beta, where varest = sigma1**2 + sigma2**2 /ratio -2 rho sigma1 sigma2 /sqrt(ratio).x ʱGDu+JK/LH!Mt1d5U`6a1b c,,  Label9 The formula for the sample size of the first group is n1 = [deff (zalpha + zbeta)**2 varest / D**2.x RI},JK1LH!M3m7UF{`6@ aT3c,, txtD 0.05x &]bSAdqm$+.J@ /KT30L1M4d5U`6aT3bGc, Label10 Effect size, D:x w!rF2ސJKT3LMD4m7UF{`6da(Abc,,k txtzalphax GL,>C+.Jd/K(A0L@ 1MTBd5U`6xa(Abc, Label11 z(1-alpha):x -tԘJhJxK(AL8MBm7UF{`6a$6c,,k txtsigma1 0.5x zI@ ۀ+.J/K$60L@ 1MP7d5U`6a$6bc, Label12 sigma1:x 魤AF#6JK$6L8M7m7UF{`6axb@ c,i cmdDoIt  Do It!~ "[Event Procedure]x F(.K*n<@!%J&K>'L(Mt@m7UF{`6adAc,,k txtzbetax y vTE5]>+.J/KdA0LD1MBd5U`6 adAbc, Label14 z(1-beta):x j/9E4;J KdALMTBm7UF{`6,a$6bTc,,k txtsigma2 0.5x N%Loz +.J,/K$60L1MP7d5U`6 a$6bc, Label16 sigma2:x b<@"J K$6LxM7m7UF{`6xal9b8c,,k  txtrho 0.5x u^/lH?\X+.Jx/Kl90L1M:d5U`6al9bc, Label18 fCorrelation between units in different groups, rho:x ÍU@!\o3JKl9L<M\:m7UF{`6@ a7c,,k  txtratio 1x ϻB:+.J@ /K70L1M8d5U`6a7bDc, Label20 2Sample size ratio, n1/n2:x {+u@H`JK7LM8m7UF{`6 a4Dbc,,k   txtn2x  ^Ed7VN+.J /K4D0L1LVALAM`Ed5U`6pa4Dbc, Label22 n2:x =YO>FLa)b*c!e g h6ij$k [?l.@x *utK}π1J1 @ Arial 8 h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2d# " @ Tahoma    hg#h%" @ Tahoma    mfg4i# LVALD" @Calibri     `,DC  Detailx xPH9GqLvRd5U`6xa<b c,hd#   Label0 J7. Some notes on use of the programs.x #yB2JxK<L !Mh1U`6LaH?b  Command7 Close Form~ "[Event Procedure]x X(}-Hbdη!%JL&KH?'LW!(M@h1U`6LadAbi Command8 Quit App~ "[Event Procedure]x {&_@d}!%JL&KdA'LH!(MBd5U`6xa,.b c,d# Label14 For independent subsamples, the variance of estimated mean single differences or mean double differences is much higher than the variance of the estimated mean (by a factor of 4 or 16, respectively, for the same total sample size, if the groups are independently sampled using simple random sampling). If it is desired to estimate differences, the sample design should be tailored to provide high precision and power for the estimates or tests of hypothesis of interest about differences. The best sample design for estimating a double difference will not be the best sample design for estimating a population mean. The variance of an estimated double difference may be substantially decreased by design features such as matching of treatment and comparison units and and use of a panel survey in which the same households are interviewed in both survey waves.x _ {-G? :JxK,.LH!M5d5U`6xa(b c,Hd# Label16 If a sample involves clustering, the sample-size estimation may be done for the ultimate sample elements (second-stage sample units) or for the clusters. If the sample design invLVALEolves stratification or blocking of the clusters, then it would be preferable, from the standpoint of transparency, to estimate the sample size forthe ultimate sample units (since the effects of the survey design could better be represented in deff for that stage of sampling).x 5Hy}Ba<-JxK(LH!M+m45I7UF{`6aAbtc,i#k  Date1 =Date()" @ Tahomax YPjIU(+.J/KA0LT1MBd5U`6aAbc,d# Label30  Date:x Hm"'MG%JKALMBm45I7UF{`6aAbtc,i#k  Time1 =Time()" @ Tahomax j}ۿ EA=+.J/KA0L\1MBd5U`6aAbc,d# Label32  Time:x ³43dE) nJKALMBd5U`6xaL,b !c,hd#  Label23 rNotes on estimation of differences, and on panel samplingx n*"KNj܏JxKL,L!M-d5U`6xa<b c,,d#  Label24 :General notes on program use.x ɽH +HlJxK<LH!Mhd5U`6xab c,pd# Label25 Note that the formulas for determining sample size based on power level (and for determining sample size based on the size of confidence intervals, with sampling from infinite populations) depend on D and sigma (or E and sigma) only through the ratio, D/sigma (or E/sigma). In many applications, a good estimate of sigma is not available, but good estimates of the coefficient of variation (sigma/mean) are. In this case, it is more useful to use the programs LVALFby specifying values of D (or E) and the sigmas relative to the mean. This is possible because (D/mean / sigma/mean) = D/sigma. With this approach, wherever the program requests a sigma, enter a coefficient of variation (CV = sigma/mean), and wherever the program requests D (or E) enter D/mean (or E/mean). Another advantage of this approach is that for a variable whose standard deviation is proportional to the mean, it is necessary to increase the value of sigma2 by the amount D times the value of sigma1 (i.e., sigma2=(1+D)sigma1 -- the other sigmas are unaffected). When the variation is specified in terms of the coefficient of variation, this adjustment is not neccessary.x ol3yDCec$JxKLH!MP(d5U`6xa5b c,3 d# Label38 H The double-difference estimate arises in evaluation research studies, for a pretest-posttest-comparison-group design. The estimate of program impact is the mean difference, between the treatment and the comparison group, of the mean difference between the posttest and pretest results. In statistical terms, this is the mean interaction between the treatment and time effects. In symbols, the effect of the program intervention is denoted by Impact = (x1 - x2) - (x3 - x4) = x1 -x2 -x3 +x4, where the x's denote the means for the four groups (groups 1 and 3 are the treatment and comparison groups at time 1, and groups 2 and 4 are the treatment and comparison groups at time 2. If simple random sampling were used, the variance of the double-difference estimator would be 16 times as large as the estimator of the overall mean, for the same total (all-group) sample size. By use of a panel survey (e.g., interviewing the same households before and after the program intervention) and matching (applying treatment and non-treatment to matched sample units), the correlations (between the reinterviewed households, and between the matched units) may be high (e.g., .5 for the before and after groupsLVALG and .3 for the treatment and comparison groups (at the same time)), and this factor of 16 would be much reduced.x _C?E$?R<JxK5LH!M>d5U`6xab c,d#  Label26 FNotes on statistical power analysisx :JیJDeL˪JxKL !Md5U`6xa8b c,d# Label27 4This program provides software routines for conducting statistical power analysis for a wide class of sample survey designs, including those associated with estimation of a double-difference measure for a pretest-posttest-comparison group design, often used as the basis for rigorous impact evaluation of socio-economic programs. References on sample survey design for evaluation are presented at the website http://www.foundationwebsite.org. These include http://www.foundationwebsite.org/SampleSurvey3DayCourseDayOne.pdf, http://www.foundationwebsite.org/SampleSurvey3DayCourseDayTwo.pdf, http://www.foundationwebsite.org/SampleSurvey3DayCourseDayThree.pdf, and http://www.foundationwebsite.org/SampleSurveyDesignForEvaluation.pdf. Referencer texts on statistical power analysis include Statistical Power Analysis for the Behavioral Sciences by Jacob Cohen (Academic Press, 1969, 2n ed. Routledge Academic, 1988); Learning More from Social Experiments: Evolving Analytic Approaches by Howard S. Bloom (Russell Sage Foundation, 2005); and Optimal Design for Longitudinal and Multilevel Research: Documentation for the "Optimal Design" Software by Jessaca Spybrook, Stephen W. Raudenbush, Richard Congden and Andres Martinez, July 22, 2009, posted at the William T. Grant Foundation website, http://www.wtgrantfdn.org/resources/overview/research_tools/research_tools . A major problem associated with the latter references is that they do not consider the most popular evaluation design, the pretest-posttest-comparison-group design. They consider mainly the case of a single cross-seciLVALtional survey. To be useful for evaluation research, statistical power analysis must account for the effect of pairwise matching (wheter before or after randomization) and re-interview of the same households in a panel survey.x ǞoHlJxK8L !Md5U`6xab c, d# Label31  Estimation of sample size may be done either by assessing the precision (size of confidence intervals) of estimates of overall population characteristics (means, proportions, totals) or by assessing the power of tests of hypotheses about quantitites of interest, such as a double-difference measure of program impact. The former approach (represented in Forms 1-3 of this software package) is used in descriptive surveys intended to produce estimates of popuation characteristics (such as a monitoring survey). The latter approach (represented in Forms 4-6 of this package) is used in analytical surveys intended to test hypotheses about program impact (such as a survey in support of program evaluation). Descriptive surveys are usually designed using a "design-based" approach, and analytical surveys are usually designed using a "model-based" or "model-assisted" approach. A statistical power analysis must take into full account three things: (1) the mathematical model generating the data; (2) the estimate on which the tests of hypothesis are being based (e.g., a covariate-adjusted single-difference estimator or a double-difference estimator); and (3) the survey design used to collect the data. Unfortunately, the power analysis done in many applications does not take these three aspects into account, and the resulting power and sample-size estimates are very wrong. x +ЇwK)1l-JxKL !MLVAL儩Iu08=5B>a)b*c!e gh6ij$k #?l9*@x ̅CvR7 @ Arial 8 h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVALJ     `,l*  Detailx (|A%PIUm7UF{`6@ a&c,, txtEx # YεJ(j"+.J@ /K&0L1M'd5U`6xa&b c,  Label2 DConfidence interval half-width, E:x _1DU@l SJxK&L" M'd5U`6<ab !c,  Label3 This program calculates the size (half-width, E) of a confidence interval for a population mean, given the sample size (n). (This program is the "inverse" of the program (Form 1) that determines n given E.)x IuaF$rpJ<KLH!Md5U`6<ab !c,  Label4 The following values must be specified: (1) the sample size (n); (2) population standard deviation (sigma); (3) population size (N); (4) confidence coefficient (CC); (5) design effect (deff). The confidence coefficient (CC) is the probability that the confidence interval includes the true population mean. (It is equal to 1 minus alpha, where alpha is the probability that the interval does not include the true population mean.) The design effect (deff) is the ratio of the variance (of the estimated population mean) using the specified sample design to the variance using simple random sampling. It is equal to 1.0 for simple random sampling, and may be larger or smaller than 1.0 for complex survey designs. For multistage designs, deff is usually greater than 1.0. This program assumes sampling from a normal distribution, but may be used for non-normal sampling (e.g., binomial, poisson, exponential) if the sample size (the program output) exceeds 30.x  SIrJDJ<KLH!M d5U`6<atb !c,  Label5 The formula forLVALK the half-width of the confidence interval is: E = z(CC) sigma sqrt((1-n/N) deff/n)x r49IG*J<KtLH!Mdd5U`6<ab !c,,  Label6 The formula for the sample size is: n = [z(CC)**2 sigma**2 deff] / [E**2 + z(CC)**2 sigma**2 deff/N]x (zgM2|[J<KLH!Mm7UF{`6@ a%c,,k txtzx ,z-_D v-+.J@ /K%0L1M4&d5U`6xa%bc,  Label9 z(1-alpha/2):x IMIwUJxK%LtM%m7UF{`6@ ac,,k txtsigma 0.5x  -t3K 70+.J@ /K0L1Md5U`6<abc, Label11 4Standard deviation, sigma:x ("xLP 2g%J<KLMm7UF{`6@ aLc,,k txtN 1000000000x ^NכDλ3+.J@ /KL0L1Mxd5U`6<aLbc, Label13 &Population size, N:x Y^![BMcJ<KLLM<m7UF{`6@ a c,,k txtdeff 1x tC ;K+.J@ /K 0L1M!d5U`6<a bc, Label15 (Design effect, deff:x A'HvvO4J<K LM!m7UF{`6@ ac,,k  txtSS 384x 9xImD+.J@ /K0L1M0d5U`6<abc, Label17 Sample size, n:x OT;IAQJ<KLMh1U`6a"b@ c,i cmdLVALLDoIt  Do It!~ "[Event Procedure]x ܧ#q)2OSɑ$~!%J&K"'L (M$d5U`6<a b !c,d Label20 pNote: Given the value of the confidence coefficient (CC=1-alpha), the program calculates a corresponding z-value (z). The z-value (z(1-alpha/2)) is the value of the standard normal deviate having area (probability) alpha/2 = (1 - CC)/2 to the right of that point. Here is a list of z's corresponding to commonly used CC's: z = 1.9600 for CC = .95 (alpha = .05); z = 1.6449 for CC = .90 (alpha = .1); z = 1.2816 for CC = .80 (alpha = .2); and z = 2.0 for CC = .9545 (alpha = .0455). The value CC = .95 (z = 1.96, alpha = .05) is often used to determine confidence intervals. For sampling for a proportion, p, (i.e., the population mean is a proportion, p) the value of sigma is sqrt[p(1-p)].x Ƕ[qKx¬J<K LH!Md5U`6<ab !c, Label24 This program assumes that the sampling distribution of the sample mean is approximately normally distributed (may be assumed if n>30). If sampling is considered to be from an infinite population, specify a large value for N (e.g., N=1,000,000,000).x ˶lZDŲ)J<KLH!Mh1U`6a4&b c,i Command25 Close Form~ "[Event Procedure]x {sL%!%J&K4&'L!(M'h1U`6a(bi Command26 Quit App~ "[Event Procedure]x "dOp"!%J&K('L !(M)d5U`6<axb !c,d#  Label27 1b. Determine confidence interval for a population mean, given the sample size.x [I PJ<KxLH!M m7UF{`6ha(btc,k   Date1 LVAL  =Date()x .vH~9!rTx+.Jh/K(0L1M)d5U`6a(bc, Label30  Date:x *ak D7wJK(L,M)m7UF{`6a(btc,k   Time1 =Time()x 2]˴^LkIXro+.J/K(0L 1M)d5U`6Ta(bc, Label32  Time:x 6CJa}W*JTK(LpM)m7UF{`6@ a,c,k   txtCC 0.95x "UIGr;+.J@ /K,0L1M+ d5U`6<a,b c, Label34 BConfidence coefficient (1-alpha):x  B|t.J<K,L M LVAL儩Nu08=5B>a)b*c!e g(h6i,j$k Z?l7,+@x APLhUX<ۇ @ Arial 8 h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVALO     `,K  Detailx YygI g d5U`6xab c,d  Label0 This program determines the power (1-beta) of a (one-sided) test of the hypothesis about the size of the difference between the means of two sampled populations (e.g., the mean of the population in a base year) and the mean of another (second) population (e.g., the mean of the population in a later year), given the sample sizes of the two groups. This difference is called the effect size. (The two populations could be two different subpopulations of a larger population, such as two different regions under study, or a "treatment" group and a "comparison" group.) The program allows the sample sizes of the two groups to be different: the user specifies the ratio of the sample size of the second group to that of the first group, i.e., ratio = n2/n1. x VUG~`ǕJxKLH!M$ d5U`6xa b c,p  Label3  The following values must be specified: (1) the effect size, D (size of the difference in the means (D>0)); (2) the sample size, n1, for the first group; (3) the ratio of the sample sizes of the two groups, ratio = n2/n1 (typically = 1, rarely >2); (4) the standard deviations of the two populations (sigma1 and sigma2); (5) the coefficient of correlation, rho, between items of the two groups; (6) alpha, the probability of a making a Type I error (i.e., the significance level of the (one-sided) test (of the mean difference)); (7) the design effect (deff). The program output is beta, the probability of making a Type II error (one minus the power of the test). The design effect (deff) is the ratio of the variance (of the estimated population mean) using the specified sample design to the variance using simple random sampling. It is equal to 1.0 for simple ranLVALPdom sampling, and may be larger or smaller than 1.0 for complex survey designs. This program assumes sampling from a normal distribution, but may be used for non-normal sampling (e.g., binomial, poisson, exponential) if the sample size (the program output) exceeds 30. (For two-sided tests, use the z-value corresponding to 1-alpha/2 instead of 1-alpha.) x m!IHjY@yJxK LH!M(d5U`6xa"b c,H  Label4 This program assumes sampling from two infinite populations, since the means of two finite populations are almost always different (so that there is no point to a statistical test of the difference in the population means). Hence, the finite population correction does not apply, and the population sizes (N's) are not specified. In this case, it is possible for the program to specify sample sizes (n's) greater than the finite population sizes (N's).x %CNtJxK"LH!M%d5U`6xa b c,  Label5 This program assumes that the sampling distributions of the sample means are approximately normally distributed (may be assumed if n>30, and the infinite populations have finite standard deviations (true for "real" phenomena)).x j DFkƆJxK LH!M8"d5U`6xab c,d  Label6 Note: Given the values of alpha (the significance level of the test) and beta (1 - the power of the test), the program calculates corresponding z-values. The z-value for 1-alpha is the value of the standard normal deviate having area (probability) alpha to the right of that point. Here is a list of z's corresponding to commonly used alpha's: z = 1.9600 for alpha=.025; z = 1.6449 for alpha = .05; z = 1.2816 for alpha = .10; z = 1.0364 for alpha = .15; z = .8416 for alpha = .2; and z = 2.0 for alpha = .02275. The values alpha = .05 (zalpha = 1.6449) and D = .1 x sigma are sLVALQometimes used. For sampling for a proportion, p, (i.e., the population mean is a proportion, p) the value of sigma is sqrt[p(1-p)].x D3M2ctG_[JxKLH!Md5U`6xap&b c,X  Label8 The formula from which the result is derived is Prob([samplemean1 - samplemean2] / sqrt(deff varest/n1) > z(1-alpha) | popmean1 - popmean2 = D) =1 - beta, where varest = sigma1**2 + sigma2**2/ratio -2 rho sigma1 sigma2 /sqrt(ratio).x =\@IJxKp&LH!M(m7UF{`6@ a+c,, txtD 0.05x 2uCO "=;+.J@ /K+0L1M,d5U`6xa+bGc, Label10 Effect size, D:x #\d]C(5kyJxK+LM,m7UF{`6a09b(c,,k txtzalphax H>ʗ_IF7|#+.J/K090L$ 1M\:d5U`6xa09bHc, Label11 z(1-alpha):x `d|I:z{2JxK09LM :m7UF{`68at1bc,,k txtsigma1 0.5x iwXCC +.J8/Kt10L 1M2d5U`6xat1bc, Label12 sigma1:x ljL"\bARJxKt1LHMd2m7UF{`6@ a5bc,,k txtdeff 1x ٜ&C,qA;+.J@ /K50LX1M6d5U`6xa5bc, Label15 (Design effect, deff:x mo`tG󕜕.>JxK5L4M6m7UF{`6@ a<-c,,k  txtn1 856x LOˬ+.J@ /K<-0L1Mh.d5U`6xa<-bc,LVALR Label17 @Sample size for first group, n1:x MhGum ړJxK<-Lo M,.h1U`6a7b@ c,i cmdDoIt  Do It!~ "[Event Procedure]x ͶJV %!%J&K7'L,(M8m7UF{`6a:b(c,,k txtzbetax ZTDջ\(+.J/K:0L1M;d5U`6xa:b c, Label14 z(1-beta):x EgnJxK:LM;m7]F{`6a1bTc,,k txtsigma2 0.5x ` @GIe_۝+.J/K10L1M2d5U`6 a1b c, Label16 sigma2:x M[B`DfmJ K1LM2m7WF{`6a2bTc,,k  txtrho 0.5x MtLrN+.J/K20L1M4d5U`6xa2bc, Label18 fCorrelation between units in different groups, rho:x \$ž@Bw? JxK2LM3m7UF{`6@ a.c,,k  txtratio 1x I[/D}:Y-q+.J@ /K.0L1M 0d5U`6xa.b` c, Label20 2Sample size ratio, n1/n2:x R}]d5]`6a!%JL&KF'LW!(M]Hh1U`6LaHbc,i  Command24 Quit App~ "[Event Procedure]x W95Hw!%JL&KH'LH!(MJd5U`6xa b c, Label25 Alpha denotes the probability of making a Type I error (rejecting the null hypothesis when it is true). Beta denotes the probability of making a Type II error (accepting the null hypothesis when it is false). The power is 1 - beta.x B=hxGw#G+JxK LH!M@ m7UF{`6a,=bc,,k   txtSSx 1DiCS)/+.J/K,=0L1MX>d5U`6 a,=bc, Label27 n1 + n2:x )B#Ixth(J K,=LM>d5U`6xaxb c,d#  Label28 5b. Determine the power of a test for the difference in population means, given the sample size.x hesFbJxKxLH!M d5U`6xa|b c, Label29 0The program calculates the z-value (z(1-beta)) corresponding to 1-beta (i.e., the value of the standard normal deviate having area (probability) beta to the right of that point. The power is 1-beta. (To determine the power, consult a table of the cumulative normal distribution function (in the back of any statistics textbook) and read the probability corresponding to the z-value z(1-beta). For example, if z(1-beta) returned by this program is -1.96, then the value of the power (i.e., of 1-beta) is .975. This program does not return the power directly since Microsoft Access does not contain the normal distribution function as a native mathematical function. A later version of this program will incorporate a subroutine to calculate this funLVALTction, and return the power directly.)x zݺc"@c"JxK|LH!M d5U`6xa@)b c, Label30 The formula for z(1-beta) is z(1-beta) = D / sqrt(deff varest/n1) - z(1-alpha), so the power is 1-beta = F(z(1-beta)) where F(.) denotes the cumulative distribution function for the normal distribution.x LezJq \JxK@)L !M*d5U`6xadAb c, Label31 Additional values of the cumulative normal probability function (listed as (z,p)): 5.9978, .999999999; 5.6120, .99999999; 5.1993, .9999999; 4.7534, .999999; 4.2649, .99999; 3.7190, .9999; 3.0902, .999; 2.3263, .99; 1.9600, .975; 1.6449, .95; 1.2816, .90; 1.0364, .85; .8416, .80; .6745, .75; .5244, .70; .3853, .65; .2533, .60; .1257, .55; 0.0, .5; -.1257, .45; -.2533, .40; -.3853, .35; -.5244, .30; -.6745, .25; -.8416, .20; -1.0364, .15; -1.2816, .10; -1.6449, .05; -1.96, .025; -2.3263, .01; -3.0902, .001; -3.7190, .0001; -4.2649, .00001; -4.7534, .000001; -5.1993, .0000001; -5.6120, .00000001; -5.9978, .000000001. x u_cz)LwMÊJxKdALH!MFm7UF{`6@ a?c,,k txtpowerx iN%ђ@H8͚+.J@ /K?0L1M(Ad5U`6xa?bDc, Label33 Power (1-beta):x f%O=\JxK?LM@m7UF{`6,a Ibtc,k  Date1 =Date()x  @8If76+.J,/K I0L1MJd5U`6 a Ibc, Label34  Date:x )WHc*J K ILMJm7UF{`6a\Ibtc,k  Time1 =Time()x @6cْGAp+.J/K\I0L 1M[Jd5U`6Ta\IbLVALc, Label35  Time:x Qs2e;IjVLJTK\ILpMLJm7UF{`6 aD4b c,k txtAlpha 0.05x Sr8L5&+.J /KD40L1MC5d5U`6xaD4b c, Label36 Halpha (significance level of test)):x vnRԘFoP>JxKD4L M45m7UF{`6@ a>c,k txtBetax bwN!r95+.J@ /K>0L1M?d5U`6xa>bDc, Label37 2beta (1 - power of test):x |pOE ͞mVJxK>LM?d5U`6xaH0bP c, Label38 (Standard deviations:x CέN忩ANJxKH0L M0d5`6xaa)b*c!e gh6ij5%k ?l9*@x 9=KJ(xb @ Arial 8 h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVALW     `,!  Detailx qpIK[n* m7UF{`6Tabc,,  txtrho 0.1x _PZK^#0+d5U`6abc,  Label2 rho (icc):x j:K^YBd5U`6ab c,  Label3 This program calculates the optimal size of the second-stage sample in two-stage sampling where the first-stage sample units (clusters, primary sampling units (PSUs)) are of equal size. See Cochran, Sampling Techniques (3rd edition, Wiley, 1977) for details.x c uLDugXd5U`6ab !c,'  Label4 |In cluster sampling with subsampling (e.g., selection of a sample of Census enumeration areas and a sample of households from each), the marginal cost of sampling clusters differs from the marginal cost of sampling elements within clusters. The size of the cluster, M, is often determined by administrative convenience or the structure of the population (e.g., city blocks). The variance of the estimate of the population mean is a function of both the cluster sample size and the subsample (within-cluster) sample size. Since the (marginal) sampling cost is a function of these same two quantities, it is possible to determine an optimal within-cluster sample size, i.e., one that minimizes the variance subject to a constraint on total (marginal) sampling cost, or minimizes the cost subject to a constraint on the variance..x d괔^/>C*qQd5U`6aTb !c,  Label5 The value for n, the number of clusters to select, depends on whether a constraint is being placed on the cost or on the precision (variance of the estimate). The formula for the variance of the estimated population mean (per element) is given by V(ybarbar) = (1/n)(S1**2 -LVALX S2**2/M) + (1/(mn))S2**2 - (1/N)S1**2 . An alternative expression is V(ybarybar) = Vsrs (1 + (m - 1)rho) where Vsrs denotes the variance of the estimated population mean using a simple random sample of size nm. The value of n is determined by solving either the cost equation or the variance equation, depending on which has been constrained.x !~9CMm7UF{`6Ta\bc,,k txtcostratio 100x 3\঍O8q +d5U`6a\bc,  Label9  c1/c2:x 2R?Ium7UF{`6Ta bc,,k txtmoptx e|<ӤHRIF_W+.JT/K 0L@ 1MH!d5U`6a bc, Label17  mopt:x x{cDB5w5ML3JK LtM !h1U`6ab@ c,i cmdDoIt  Do It!~ "[Event Procedure]x tWS_DC !%J&K'L\ (Md5U`6a b !c, Label20 Let n = number of first-stage units (clusters) sampled; m = number of second-stage units (subunits, elements) sampled per first-stage sample unit. Let c1 = the marginal cost of sampling a first-stage unit and c2 = the marginal cost of sampling a second stage unit. Then the total marginal cost is C = c1 n + c2 nm. Let rho denote the intracluster correlation coefficient, S1**2 = variance among primary (cluster) means, and S2))2 = variance amont subunits (elements) within primary units. Then the optimal number of elements to select from each cluster is given by mopt = [S2 sqrt(c1/c2]) / [sqrt (S1**2 - S2**2/M)], which is approximately equal to sqrt ([c1 (1 - rho)] / [c2 rho]. The values of S1 and S2, or or rho, are generally estimated using data from a previous similar survey (e.g., data from a previous Census, for household data). In some cases (if timLVALe and resources permit), a small first-phase pilot survey may be conducted to provide estimates of the survey costs and rho.x nB'2i(FQ*Mh1U`6Lab c,i Command25 Close Form~ "[Event Procedure]x XfL? ~ !h1U`6Labi Command26 Quit App~ "[Event Procedure]x r>p/A'[a!d5U`6a<b !c,d#  Label27 8. Determination of the optimal within-cluster sample size in cluster sampling with clusters of equal size.x ZtTIJKy_AYm7UF{`6aX btc,k  Date1 =Date()x 4I^\w+.J/KX 0L1MW!d5U`6LaX bc, Label30  Date:x {qKIٰJLKX LhMH!m7UF{`6aX btc,k  Time1 =Time()x Guv0Gw)+.J/KX 0L\1MW!d5U`6aX bc, Label32  Time:x /KF=VJKX LMH!LVAL d :ͬ=N;]; AQ0|/Detailm/txtEd /Label2d /Label3d /Label4d /Label5d /Label6m/txtzd /Label9m/ txtsigma1d / Label11m/ txtrho12d / Label13m/ txtdeffd /Label15m/txtn1d /Label17h /cmdDoItd /Label20d /Label24h /Command25h /Command26d /Label27m/txtsigma2d /Label29m/txtn2d /Label31m/txtn3d /Label33m/txtratio2d /Label35m/"txtsigma3d /#Label40m/$txtsigma4d /%Label42m/&txtratio3d /'Label44m/(txtratio4d /)Label46m/*txtrho34d /+Label48m/,txtrho24d /-Label50m/.txtrho23d //Label52m/0txtrho14d /1Label54m/2txtrho13d /3Label56m/4txtn5d /5Label98m/6txtn4d /7Label100d /8Label101m/9Date1d /:Label30m/;Time1d /<Label32m/=txtCCd />Label34d /?Label102ͬ= XAHLZ /Detailm/txtEd /Label2d /Label3d /Label4d /Label5d /Label6m/txtzd /Label9m/ txtsigma1d / Label11d / Label13m/ txtdeffd /Label15m/txtn1d /Label17h /cmdDoItd /Label20d /Label24h /Command25h /Command26d /Label27m/txtsigma2d /Label29m/txtn2d /Label31m/txtn3d /Label33d /Label35d /#Label40d /%Label42d /'Label44d /)Label46m/txtratio2m/&txtratio3m/(txtratio4d /+Label48d /-Label50d //Label52d /1Label54d /3Label56d /5Label98d /7Label100m/ txtrho12m/2txtrho13m/0txtrho14m/.txtrho23m/,txtrho24m/*txtrho34m/"txtsigma3m/$txtsigma4m/6txtn4m/4txtn5d /8Label101m/9Date1d /:Label30m/;Time1d /<Label32m/=txtCCd />Label34d /?Label37ͬ%@IёQ7/Detailm/txtEd /Label2d /Label3d /Label4d /Label5d /Label6m/txtzd /Label9m/ txtsigma1d / Label11m/ txtrhod / Label13m/ txtdeffd /Label15m/txtn1d /Label17h /cmdDoItd /Label20d /Label24h /Command25h /Command26d /Label27m/txtsigma2d /Label29m/txtn2d /Label31m/txtn3d /Label33m/txtratiod /Label35m/ Date1d /!Label30m/"Time1d /#Label32m/$txtCCd /%Label34ͬ(:Caj$LIES1[/Detailm/txtEd /Label2d /Label3d /Label4d /Label5d /Label6m/txtzd /Label9d / Label11d / Label13m/ txtdeffd /Label15d /Label17h /cmdDoItd /Label20d /Label24h /Command25h /Command26d /Label27d /Label29m/ txtsigma1m/txtsigma2m/ txtrhom/txtn1d /Label31m/txtn2d /Label33d /Label35m/txtratiom/txtn3m/ Date1d /!Label30m/"Time1d /#Label32m/$txtCCd /%Label34d /&Label37d /'Label38d /(Label39ͬOٖ Kn J/Detaild J/Label2d J/Label3d J/Label4d J/Label5d J/Label9d J/Label17h J/cmdDoItd J/Label20h J/Command25h J/Command26d J/Label27mJ/txtrhomJ/txtcostratiomJ/txtmoptmJ/Date1d J/Label30mJ/Time1d J/Label32LVAL儩[u08=5B>a)b*c!e g(h6ij(k #?l9*@x q)@a @ Arial 8 h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVAL\     `,2  Detailx `?UD8Nm7UF{`6@ ac,, txtE 0.05x -6]O|(+.J@ /K0L1M d5U`6<ab c,  Label2 DConfidence interval half-width, E:x $JF>_ yJ<KL Md5U`6<aHb !c,X  Label3 jThis program calculates the sample sizes (n1 and n2) required to produce a confidence interval of a prescribed size for a difference in means. For this program, dealing with estimation of differences, it is assumed that the sampling is from infinite populations (i.e., no finite population correction (fpc)).x \.‘MèJ<KHLH!Md5U`6<ab !c,p  Label4  The following values must be specified: (1) half-width of confidence interval (E); (2) standard deviations for the two groups (sigma1 and sigma2); (3) ratio, n2/n1, of the group sample sizes; (4) coefficient of correlation (rho) between observations comprising the difference (i.e., between observations in the two groups); (5) confidence coefficient (CC); (6) design effect (deff). The confidence coefficient (CC) is the probability that the confidence interval includes the true population mean. (It is equal to 1 minus alpha, where alpha is the probability that the interval does not include the true population mean.) The design effect (deff) is the ratio of the variance (of the estimated difference in means) using the specified sample design to the variance using simple random sampling. It is equal to 1.0 for simple random sampling, and may be larger or smaller than 1.0 for complex survey designs. For multistage sampling, it is usually larger than 1.0. This program assumes sampling LVAL]from a normal distribution, but may be used for non-normal sampling (e.g., binomial, poisson, exponential) if the sample size (the program output) exceeds 30.x 'ACC)ҼJ<KLH!Md5U`6<ab !c,  Label5 ZThe formula for the half-width of the confidence interval is: E = z(CC) sqrt(deff varest/n1) where varest = sigma1**2 + sigma2**2 /ratio - 2 rho sigma1 sigma2 / sqrt(ratio).x :*[=Ohw[J<KLH!Md5U`6<a4b !c,,  Label6 The formula for the sample size is: n1 = z(CC)**2 deff varest / E**2.x tM30).x V$ѡFuG>J<KLH!Mh1U`6a.b c,i Command25 Close Form~ "[Event Procedure]x eI@6K;An!%J&K.'L!(Mu0h1U`6a0bi Command26 Quit App~ "[Event Procedure]x d"~Jوp`-)!%J&K0'L !(Md2d5U`6<a<b !c,LVAL_d#  Label27 2. Determine sample size by specifying a confidence interval for a difference in means.x olQBBrJ<K<LH!Mm7UF{`6a!btc,,k  txtsigma2 0.5x eMD$^{+.J/K!0L(1M"d5U`6 a!bc, Label29 sigma2:x =%zpM_J K!L<M"m7UF{`6 a-btc,,k   txtn2x @YkGĸ=H+.J /K-0L1M.d5U`6 a-bc, Label31 n2:x =b;HL|J K-L0 M.m7UF{`6a-btc,,k   txtn3x L&w]?E>%c+.J/K-0L|1M.d5U`68a-bgc, Label33  n1+n2:x !YJ)^ITYJ8K-LM.m7WF{`6@ a(#c,,k  txtratio 1x 1U EiL+.J@ /K(#0L1MT$d5]`6<a(#b c, Label35 2Sample size ratio, n1/n2:x շWJ3aF+J<K(#L@ M$m7UF{`6a81btc,k   Date1 =Date()x 1hԠ)O?++.J/K810Ld1M72d5U`6 a81bc, Label30  Date:x Y$VO麶\GJ K81LM(2m7UF{`6pa81btc,k  Time1 =Time()x @t:0H{.+.Jp/K810L1M72d5U`6a81bc, Label32  Time:x wM.vtDZJKLVAL81L4M(2m7UF{`6 ap&b8c,k  txtCC 0.95x h,mEU+.J /Kp&0LL1Mo'd5U`6<ap&b c, Label34 BConfidence coefficient (1-alpha):x -WܗA| {RJ<Kp&L M`'d5U`6<ab !c, Label37 This program assumes sampling from two infinite populations, since the means of two finite populations are almost always different (so that there is no point to a statistical test of the difference in the population means). Hence, the finite population correction does not apply, and the population sizes (N's) are not specified. In this case, it is possible for the program to specify sample sizes (n's) greater than the finite population sizes (N's).x <>"͓Ft/J<KLH!Mxd5U`6<aX b c, Label38 (Standard deviations:x u@ΌJ<KX L@ M !d5U`6<aL,b c, Label39 Sample sizes:x أC;]J<KL,L@ M<-LVAL儩au08=5B>a)b*c!e g%h6ij(k #?l9*@x tI\: 9N'k탿Lt @ Arial 8 h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVALb     `,X/  Detailx >kA8-tc]3m7UF{`6@ a)c,, txtEx >N~+.J@ /K)0L1M*d5U`6xa)bP c,  Label2 DConfidence interval half-width, E:x >2WKAECJxK)L M*d5U`6xab !c,  Label3 This program calculates the size (half-width, E) of a confidence interval for a difference in means, given the sample sizes (n1 and n2) of the two groups. The program allows the sample sizes of the two groups to be different: the user specifies the ratio of the sample size of the second group to that of the first group, i.e., ratio = n2/n1. It is assumed that the sampling is from infinite populations (i.e., no finite population correction (fpc)). (This program is the "inverse" of the program (Form 2) that determines the sample size from specification of the size of the confidence interval.)x >1Gl;JxKL!Mqd5U`6xab !c,p  Label4  The following values must be specified: (1) the sample size (n1) of the first group; (2) ratio, n2/n1, of the group sample sizes; (3) standard deviations for the two groups (sigma1 and sigma2); (4) coefficient of correlation (rho) between items of the two groups; (5) confidence coefficient (CC); (6) design effect (deff). The program output is E, the half-width of the confidence interval. The confidence coefficient (CC) is the probability that the confidence interval includes the true population mean. (It is equal to 1 minus alpha, where alpha is the probability that the interval does not include the true population mean.) The design effect (deff) is the ratio of the variance (of the estimated population mean) uLVALcsing the specified sample design to the variance using simple random sampling. It is equal to 1.0 for simple random sampling, and may be larger or smaller than 1.0 for complex survey designs. For multistage sampling, it is usually greater than 1.0. This program assumes sampling from a normal distribution, but may be used for non-normal sampling (e.g., binomial, poisson, exponential) if the sample size (the program output) exceeds 30.x  I@'@JxKL!Mhd5U`6xab !c,  Label5 XThe formula for the half-width of the confidence interval is: E = z(CC) sqrt(deff varest/n1) where varest = sigma1**2 + sigma2**2 /ratio - 2 rho sigma1 sigma2 /sqrt(ratio).x MQ K`JxKL!Md5U`6xab !c,,  Label6 The formula for the sample size of the first design group is: n1 = z(CC)**2 deff varest / E**2.x 龻D۳""xJxKL!M@m7UF{`6@ aP(c,,k txtzx   cC?FB+.J@ /KP(0L1M|)d5U`6xaP(bc,  Label9 z(1-alpha/2):x ߯ zH<%JxKP(LtM@)m7UF{`68abc,,k txtsigma1 0.5x YhLɩŰ+.J8/K0L41M !d5U`6xab c, Label11 sigma1:x dsoihED}JxKLM m7UF{`6<a!bc,,k  txtrho 0.5x }Hak M+.J</K!0L1M"d5U`6xa!bc, Label13 fCorrelation between units in different groups, rho:x @0E[=?JxK!LMt"m7UF{LVALd`6l aT$btc,,k txtdeff 1x `\[E.HlHB O+.Jl /KT$0L1M%d5U`6xaT$b c, Label15 (Design effect, deff:x w!B"A]JxKT$L0 MD%m7UF{`6@ ac,,k  txtn1 271x _K Lf-c+.J@ /K0L1Md5U`6xabc, Label17 >Sample size of first group, n1:x eM,TM#JxKL3 Mh1U`6a4&b@ c,i cmdDoIt  Do It!~ "[Event Procedure]x Mq@ ٚ!%J&K4&'L\ (M'd5U`6xab !c, Label20 Note: Given the value of the confidence coefficient (CC=1-alpha), the program calculates a corresponding z-value (z). The z-value (z(1-alpha/2)) is the value of the standard normal deviate having area (probability) alpha/2 = (1 - CC)/2 to the right of that point. Here is a list of z's corresponding to commonly used CC's: z = 1.9600 for CC = .95 (alpha = .05); z = 1.6449 for CC = .90 (alpha = .1); z = 1.2816 for CC = .80 (alpha = .2); and z = 2.0 for CC = .9545 (alpha = .0455). The values CC = .95 (z = 1.96, alpha = .05) and E = .1 x sigma are sometimes used to determine sample size. For sampling for a proportion, p, (i.e., the population mean is a proportion, p) the value of sigma is sqrt[p(1-p)].x RP-JU JxKL!Md5U`6xab !c, Label24  This program assumes that the sampling distribution of the sample mean is approximately normally distributed (may be assumed if n>30).x 5m_F65%zJxKL!M\h1U`6La\+b c,i CommandLVALe25 Close Form~ "[Event Procedure]x TT$PֆC+Sh!%JL&K\+'LW!(M,h1U`6La-bi Command26 Quit App~ "[Event Procedure]x 4Ml p@!%JL&K-'LH!(M/d5U`6xaxb !c,d#  Label27 2b. Determine confidence interval for a difference in means, given the sample size.x !YNsE"1xJxKxL!Mm7UF{`6 a bc,,k  txtsigma2 0.5x ˪DmL':+.J /K 0L1MH!d5U`6a bc, Label29 sigma2:x DDAHk#JK L M !m7UF{`6 a\+btc,,k   txtn2x #(uIC6+.J /K\+0L1M,d5U`6 a\+bc, Label31 n2:x %~DtJ K\+L0 ML,m7UF{`6a\+btc,,k   txtn3x LDeH+.J/K\+0Ll1M,d5U`68a\+bc, Label33  n1+n2:x  eQ n}NQ.NhKJ8K\+LML,m7UF{`6| a<bdc,,k  txtratio 1x dUcB_A9Zf+.J| /K<0L1Mhd5U`6xa<b c, Label35 2Sample size ratio, n1/n2:x i覥B:JxK<L M,m7UF{`6a-btc,k   Date1 =Date()x `ޡ3Isc +.J/K-0Ld1M.d5U`6 a-bc, Label30A LVALQ   Date:x H-JZA[1J K-LM.m7UF{`6pa-btc,k  Time1 =Time()x Si@oCdn{+.Jp/K-0L1M.d5U`6a-bc, Label32  Time:x ɑ3KvJ bJK-L4M.m7UF{`6@ a"c,k  txtCC 0.95x SyH: +.J@ /K"0L1M#d5U`6xa"b c, Label34 BConfidence coefficient (1-alpha):x hiR8EA8}jJxK"L M#LVAL儩gu08=5B>a)b*c!e gfh6Jij$k #?l9*@x h?,BT]a1% @ Arial 8<- Ih `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVALh     `,45  Detailx me  DL?$m7UF{`6 at"c,, txtE 0.05x 2hvIɁI+.J /Kt"0L1M#d5U`6xat"bP c,  Label2 DConfidence interval half-width, E:x QPeC>0O zJxKt"L Md#d5U`6xaHb !c,X  Label3 This program calculates the sample sizes (n1, n2, n3 and n4) required to produce a confidence interval of a prescribed size for a double difference in means. For this program, dealing with estimation of differences, it is assumed that the sampling is from infinite populations (i.e., no finite population correction (fpc)).x ,} O1Kd5U`6xab !c,p  Label4 The following values must be specified: (1) half-width of confidence interval (E); (2) standard deviations for the four groups (sigma1, sigma2, sigma3 and sigma4); (3) ratios of the group sample sizes: ratio2=n2/n1; ratio3=n3/n1; ratio4=n4/n1; (4) coefficients of correlation between observations of different groups, (rho12, rho13, rho14, rho23, rho24); (5) confidence coefficient (CC); (6) design effect (deff). The confidence coefficient (CC) is the probability that the confidence interval includes the true population mean. (It is equal to 1 minus alpha, where alpha is the probability that the interval does not include the true population mean.) The design effect (deff) is the ratio of the variance (of the estimated mean double difference) using the specified sample design to the variance using simple random sampling. It is equal to 1.0 for simple random sampling, and may be larger or smaller than 1.0 for complex survey designs. This program assumes sampling from a normal distribution, but may be used for nonLVALi-normal sampling (e.g., binomial, poisson, exponential) if the sample size (the program output) exceeds 30.x <ȸ*KO M"Jd5U`6xab !c,  Label5 ^The formula for the half-width of the confidence interval is: E = z(CC) sqrt(deff varest/n1) where varest = sigma1**2 + sigma2**2 /ratio2+ sigma3**2 /ratio3 + sigma4**2 /ratio4 -2 rho12 sigma1 sigma2 / sqrt(ratio2) - 2 rho13 sigma1 sigma3 / sqrt(ratio3) + 2 rho14 sigma1 sigma4 / sqrt(ratio4) + 2 rho23 sigma2 sigma3 / sqrt(ratio2 ratio3) - 2 rho24 sigma2 sigma4 / sqrt(ratio2 ratio4) - 2 rho34 sigma3 sigma4 / sqrt(ratio3 ratio4).x H4.H1&+JxKL!M3d5U`6xab !c,,  Label6 The formula for the sample size of the first design group is: n1 = z(CC)**2 deff varest / E**2.x ~Y&^TIw<&̴JxKL!Mm7UF{`6da-bc,,k txtzx vBXH>)z+.Jd/K-0L@ 1M/d5U`6xa-bc,  Label9 z(1-alpha/2):x jO+8I*cJxK-LtM.m7UF{`6a#bc,,k txtsigma1 0.5x ^/M.Ds+.J/K#0Lp1M%d5U`6xa#bc, Label11 sigma1:x OǃNQF`cUJxK#L M$m7UF{`6 a$'bc,,k txtrho12 0.5x NT2WNOּ+.J /K$'0L1MP(d5U`6xa$'b+c, Label13  rho12:x k"[uD:F-JxK$'LM(m7UF{`6a|)b8c,,k txtdeff 1x c@Rrk+.J/K|)0L1MLVALj*d5U`6a|)bc, Label15 (Design effect, deff:x 3La JK|)LMl*m7UF{`6Xa/bHc,,k  txtn1x U)LŧK,+.JX/K/0L1M0d5U`6xa/bhc, Label17 n1:x RB-XK.%~JxK/LM0h1U`6a\+b@ c,i cmdDoIt  Do It!~ "[Event Procedure]x giHKVD!%J&K\+'L (M-d5U`6xab !c, Label20 Note: Given the value of the confidence coefficient (CC=1-alpha), the program calculates a corresponding z-value (z). The z-value (z(1-alpha/2)) is the value of the standard normal deviate having area (probability) alpha/2 = (1 - CC)/2 to the right of that point. Here is a list of z's corresponding to commonly used CC's: z = 1.9600 for CC = .95 (alpha = .05); z = 1.6449 for CC = .90 (alpha = .1); z = 1.2816 for CC = .80 (alpha = .2); and z = 2.0 for CC = .9545 (alpha = .0455). The values CC = .95 (z = 1.96, alpha = .05) and E = .1 x sigma are sometimes used to determine sample size. For sampling for a proportion, p, (i.e., the population mean is a proportion, p) the value of sigma is sqrt[p(1-p)].x f| ؾ@1+fNJxKL!Md5U`6xa|b !c, Label24  This program assumes that the sampling distribution of the sample mean is approximately normally distributed (may be assumed if n>30).x @sL,mJxK|L!M\h1U`6at1b c,i Command25 Close Form~ "[Event Procedure]x 8؞K'F!%J&Kt1'L!(M 3h1U`6a3bi CLVALkommand26 Quit App~ "[Event Procedure]x y!GaP4j!%J&K3'L!(M4d5U`6xa<b !c,d#  Label27 3. Determine sample size by specifying a confidence interval for a double difference in means.x 6XL5s (^m7UF{`6l a#b8c,,k  txtsigma2 0.5x X LH<2@+.Jl /K#0L1M%d5U`6` a#bc, Label29 sigma2:x Ҷn^E ^2'J` K#L M$m7UF{`6a/bc,,k   txtn2x aoLM+ؐ+.J/K/0L| 1M0d5U`6a/b,c, Label31 n2:x ]5;KK*鉦1YJK/LDM0m7UF{`6a/bHc,,k   txtn3x wWoB4(@θ"+.J/K/0LX1M0d5U`60 a/bhc, Label33 n3:x cYbN\?J0 K/L M0m7UF{`6l a%b8c,,k  txtratio2 1x y׿EIB Z+.Jl /K%0L1M&d5U`6` a%bc, Label35 ratio2:x eLBcvlJ` K%L Mp&m7UF{`6a$btc,,k  txtsigma3 0.5x 2&C6+.J/K$0L1MD%d5U`6Xa$bc, Label40 sigma3:x ܱj1URGiMQ5JXK$LM%m7UF{`6a$b8c,,k txtsigma4 0.5x Б͛EUFsv+.J/K$0LLVALlH!1MD%d5U`6a$bc, Label42 sigma4:x 5 j5L#V8JK$LM%m7UF{`6da%bc,,k txtratio3 1x =J1+.Jd/K%0LP1M&d5U`6Xa%bXc, Label44 ratio3:x L"|HR%"JXK%LM&m7UF{`6a%b8c,,k txtratio4 1x N0M|Lf0+.J/K%0L !1M&d5U`6a%bXc, Label46 ratio4:x {lAK_xO5JK%L\M&m7UF{`6,a'bXc,,k txtrho34 0.5x TvވJ8 +.J,/K'0L!1M(d5U`6a'b+c, Label48  rho34:x GDJJQn15qJK'LM(m7UF{`6a`'bXc,,k txtrho24 0.5x 4uMek:+.J/K`'0L 1M(d5U`64a`'b+c, Label50  rho24:x ӣ@OF#}YJ4K`'L_MP(m7UF{`6a`'bc,,k txtrho23 0.5x ezLDs`;lD+.J/K`'0L1M(d5U`6Xa`'b+c, Label52  rho23:x F n@3a3AJXK`'LMP(m7UF{`6a$'bXc,,k txtrho14 0.5x GL;$g+.J/K$'0L1MP(d5U`6 a$'b+c, Label54  rho14:x WDQkJ KLVALm$'LM(m7UF{`6a$'bXc,,k txtrho13 0.5x "EG f O+.J/K$'0L@ 1MP(d5U`6Ta$'b+c, Label56  rho13:x ,BqNKvJTK$'LM(m7WF{`6La/bc,,k  txtn5x Te,eMJ+.JL/K/0L!1M0d5]`6`a/bc, Label98 n1+n2+n3+n4 :x XO%xJ`K/LLM0m7UF{`6(a/bc,,k  txtn4x |yT5MfP>+.J(/K/0L1M0d5U`6 a/bc, Label100 n4:x UE FVJJ K/LM0d5U`6xa b c,, Label101 Note that the rhos may not be arbitrarily specified -- the correlation matrix must be positive definite.x NNb7Z G JxK L !M!m7UF{`6a3btc,k  Date1 =Date()x lj0CJ1Vؓj+.J/K30L1M4d5U`6a3bc, Label30  Date:x ̡HBJ1JK3LM4m7UF{`6`a3btc,k  Time1 =Time()x ș.xfKx<+.J`/K30L1M4d5U`6a3bc, Label32  Time:x LOB1JK3L$M4m7UF{`6P a|)b8c,k  txtCC 0.95x EXCE2ui+.JP /K|)0L1M{*d5U`6xa LVAL( |)b c, Label34 BConfidence coefficient (1-alpha):x NhD4 DnxeߣJxK|)L Ml*d5U`6xab !c, Label37 This program assumes sampling from four infinite populations, since the means of four finite populations are almost always different (so that there is no point to a statistical test of the double difference in the population means). Hence, the finite population correction does not apply, and the population sizes (N's) are not specified. In this case, it is possible for the program to specify sample sizes (n's) greater than the finite population sizes (N's).x >OcES)VJxKL!MX $*6CP s & i  v A L , g"r/RHU x+n {F>>@}Lj+j@PropDataPLH8 >>@}Lj+j@K儩Blob2.*" >>@}Lj+j@20%" w>>@̢j@BlobDelta0,( w>>@̢j@@TypeInfo:62" >>@̢j@PropDataPLH8 >>@̢j@dU儩Blob2.*" >>@̢j@19%" w>>@k@BlobDelta0,( w>>@k@`@TypeInfo:62" >>@k@PropDataPLH8 >>@k@儩Blob2.*" >>@k@18%" w>>@ k@BlobDelta{0,( w>>@ k@~ @TypeInfo{:62" >>@ k@}PropData{PLH8 >>@ k@| 儩Blob{2.*" >>@ k@{17%" w>>@Yk@zBlobDeltav0,( w>>@Yk@y@TypeInfov:62" >>@Yk@xPropDatavPLH8 >>@Yk@wڋ儩Blobv2.*" >>@Yk@v16%" w>>@j@uBlobDeltaq0,( w>>@j@t@TypeInfoq:62" >>@j@sPropDataqPLH8 >>@j@ri儩Blobq2.*" >>@j@q15%" w>>@,k@pBlobDeltal0,( w>>@,k@o@~TypeInfol:62" >>@,k@nPropDatalPLH8 >>@,k@m 儩Blobl2.*" >>@,k@l14%" w>>@)rsk@kBlobDeltag0,( w>>@)rsk@j@~TypeInfog:62" >>@)rsk@iPropDatagPLH8 >>@)rsk@h儩Blobg2.*" >>@)rsk@g13%" w>>@1i@fBlobDeltab0,( w>>@1i@e}@~TypeInfob:62" >>@1i@dPropDatabPLH8 >>@1i@cZ儩Blobb2.*" >>@1i@b12%" w>>@qh@aBlobDelta]0,( w>>@qh@`|@~TypeInfo]:62" >>@qh@_PropData]PLH8 >>@qh@^`w儩Blob]2.*" >>@qh@]11%" w>>@.X@\BlobDeltaX0,( w>>@.X@[+@YTypeInfoX:62" >>@.X@ZPropDataXPLH8 >>@.X@Yyo儩BlobX2.*" >>@.X@X10%" w>>@nX@WBlobDeltaS0,( wLVAL儩pu08=5B>a)b*c!e ggh6i:j$k [?l9*@x gI| @ Arial 8<- Ih `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVALq     `,09  Detailx 95+HCԍo$m7UF{`6@ a1c,, txtEx ."G4A+.J@ /K10L1M2d5U`6xa1b c,  Label2 DConfidence interval half-width, E:x 0RQXH|l9JxK1L2 M2d5U`6xa b !c,d  Label3 This program calculates the size (half-width, E) of a confidence interval for a double difference in means, given the sample sizes (n1, n2, n3 and n4) of the four groups comprising the difference. The program allows the sample sizes of the four groups to be different: the user specifies the ratio of the sample size of the second, third, and fourth groups to that of the first group, i.e., ratio2 = n2/n1, ratio3 = n3/n1, and ratio4 = n4/n1. For this program, dealing with estimationof differences, it is assumed that the sampling is from infinite populations (i.e., no finite population correction (fpc)). (This program is the "inverse" of the program (Form 3) that determines the sample size from specification of the size of the confidence interval.)x `/-@zZp$JxK L!Mpd5U`6xab !c,p  Label4 J The following values must be specified: (1) the sample size (n1) of the first group; (2) ratios of the group sample sizes: ratio2=n2/n1; ratio3=n3/n1; ratio4=n4/n1; (3) standard deviations for the four groups (sigma1, sigma2, sigma3 and sigma4); (4) coefficients of correlation between items in different groups, (rho12, rho13, rho14, rho23, rho24); (5) confidence coefficient (CC); 6) design effect (deff). The program output is E, the half-width of confidence interval. The confidence coefficient (CC) is the probability that the confidence interval includes the LVALrtrue population mean. (It is equal to 1 minus alpha, where alpha is the probability that the interval does not include the true population mean.) The design effect (deff) is the ratio of the variance (of the estimated population mean) using the specified sample design to the variance using simple random sampling. It is equal to 1.0 for simple random sampling, and may be larger or smaller than 1.0 for complex survey designs. This program assumes sampling from a normal distribution, but may be used for non-normal sampling (e.g., binomial, poisson, exponential) if the sample size (the program output) exceeds 30.x '&DڑmL2JxKL!MXd5U`6xab !c,  Label5 \The formula for the half-width of the confidence interval is: E = z(CC) sqrt(deff varest/n1) where varest = sigma1**2 + sigma2**2 /ratio2+ sigma3**2 /ratio3 + sigma4**2 /ratio4 -2 rho12 sigma1 sigma2 / sqrt(ratio2) - 2 rho13 sigma1 sigma3 / sqrt(ratio3) + 2 rho14 sigma1 sigma4 / sqrt(ratio4) + 2 rho23 sigma2 sigma3 / sqrt(ratio2 ratio3) - 2 rho24 sigma2 sigma4 / sqrt(ratio2 ratio) - 2 rho34 sigma3 sigma4 / sqrt(ratio3 ratio4).x UJQS,iJxKL!Md5U`6xab !c,,  Label6 The formula for the sample size of the first design group is: n1 = z(CC)**2 deff varest / E**2.x o"K2@>)aJxKL!Mm7UF{`6@ aH0c,,k txtzx ͽYABڊ/+.J@ /KH00L1Mt1d5U`6xaH0bc,  Label9 z(1-alpha/2):x v)+aOj "JxKH0LtM81m7UF{`6a$'bc,,k txtsigma1 0.5x ̾6%GHD5y+.J/K$'0Lp1MP(d5U`6xa$'bc, Label11 sigmLVALsa1:x \NL|`C8JxK$'L M(m7UF{`6al*bXc,,k txtrho12 0.5x ]tx"J-6^+.J/Kl*0L(1M+d5U`6xal*bc, Label13  rho12:x ajbKAmMi'LJxKl*LXM\+m7F{`6a,btc,,k txtdeff 1x Y> L[+<+.J/K,0L01M-d5]`6,a,bc, Label15 (Design effect, deff:x p 7Mro5J,K,LMx-m7UF{`6@ a%c,,k  txtn1 541x [\5K :+.J@ /K%0L1M&d5U`6xa%bc, Label17 @Sample size for first group, n1:x ' MqbJxK%Lo Mp&h1U`6Xa,.b@ c,i cmdDoIt  Do It!~ "[Event Procedure]x U$tHEu4!%JX&K,.'L (M/d5U`6xab !c, Label20 Note: Given the value of the confidence coefficient (CC=1-alpha), the program calculates a corresponding z-value (z). The z-value (z(1-alpha/2)) is the value of the standard normal deviate having area (probability) alpha/2 = (1 - CC)/2 to the right of that point. Here is a list of z's corresponding to commonly used CC's: z = 1.9600 for CC = .95 (alpha = .05); z = 1.6449 for CC = .90 (alpha = .1); z = 1.2816 for CC = .80 (alpha = .2); and z = 2.0 for CC = .9545 (alpha = .0455). The values CC = .95 (z = 1.96, alpha = .05) and E = .1 x sigma are sometimes used to determine sample size. For sampling for a proportion, p, (i.e., the population mean is a proportion, p) the value of sigma is sqrt[p(1-p)].x %*C\LVALt#JxKL!Mpd5U`6xab !c, Label24  This program assumes that the sampling distribution of the sample mean is approximately normally distributed (may be assumed if n>30).x E#`FٓTM8m|JxKL!Mhh1U`6ap5b c,i Command25 Close Form~ "[Event Procedure]x -F -z!%J&Kp5'L!(M7h1U`6a7bi Command26 Quit App~ "[Event Procedure]x eG߹OT~!%J&K7'L!(M8d5U`6xa<b !c,d#  Label27 3b. Determine confidence interval for a double difference in means, given the sample size.x ; J-6ߜJxK<L!Mm7UF{`60 a$'b8c,,k  txtsigma2 0.5x chxNVp 3\+.J0 /K$'0Lh1MP(d5U`6$ a$'bc, Label29 sigma2:x YʺzK`[4rJ$ K$'L M(m7UF{`6a3bc,,k   txtn2x ||N0.ADZO+.J/K30Ll 1M4d5U`6a3b,c, Label31 n2:x ۖ@}mѼSJK3L4M4m7UF{`6a3bHc,,k   txtn3x US3vGvv+.J/K30LH1M4d5U`6 a3bhc, Label33 n3:x yJgGe/J K3LM4m7UF{`60 a(b8c,,k  txtratio2 1x h]o5AH>nN+.J0 /K(0Lh1M)d5LVALuU`6$ a(bc, Label35 ratio2:x M9CoZRLJ$ K(L M)m7UF{`6da`'btc,,k  txtsigma3 0.5x H"mM9nx_W +.Jd/K`'0L1M(d5U`6a`'bc, Label40 sigma3:x @tH_L]? JK`'LMP(m7UF{`6a`'b8c,,k txtsigma4 0.5x LJif a+.J/K`'0L !1M(d5U`6a`'bc, Label42 sigma4:x ᚠH;Z@JK`'L\MP(m7UF{`6(a)bc,,k txtratio3 1x }nyFK\T+.J(/K)0L1M0*d5U`6a)bXc, Label44 ratio3:x ,μEDV>HJJK)LtM)m7UF{`6a)b8c,,k txtratio4 1x y~93G-}K+.J/K)0L 1M0*d5U`6a)bXc, Label46 ratio4:x o`OJflJK)L M)m7UF{`6a*bXc,,k txtrho34 0.5x I|)Ec+.J/K*0L !1M,d5U`6 a*b+c, Label48  rho34:x ,3N8tJ K*LKM+m7UF{`6Pa*bXc,,k txtrho24 0.5x yċBGQ- +.JP/K*0L1M+d5U`6a*b+c, Label50  rho24:x v=@쫟$JK*LLVALvM+m7UF{`6ta*bc,,k txtrho23 0.5x Ƭ*D iK+.Jt/K*0L1M+d5U`6a*b+c, Label52  rho23:x i8D2k~JK*L M+m7UF{`6al*bXc,,k txtrho14 0.5x ܬI@b:+.J/Kl*0Lh1M+d5U`6| al*b+c, Label54  rho14:x $cť?G=u(J| Kl*L M\+m7UF{`6pal*bXc,,k txtrho13 0.5x >NEm86+.Jp/Kl*0L 1M+d5U`6al*b+c, Label56  rho13:x W+uHj/]KJKl*LM\+m7WF{`6<a3bHc,,k  txtn5x $?Ñ Bی-zU+.J</K30L!1M4d5]`6Pa3bc, Label98 n1+n2+n3+n4 :x _#]XM!#WJPK3L<M4m7UF{`6a3bc,,k  txtn4x  -]J?L2++.J/K30L1M4d5U`6a3bc, Label100 n4:x  &Cn0ΘJK3LM4d5U`6xa#b c,h Label101 Note that the rhos may not be arbitrarily specified -- the correlation matrix must be positive definite.x z|YQFrJxK#LH!MD%m7UF{`6a7btc,k  Date1 =Date()x )ue}J 5&+.J/K70LT1M8d5U`6a7bRLVALbc, Label30  Date:x t')lExJK7LM8m7UF{`6`a7btc,k  Time1 =Time()x l 8oI -8%+.J`/K70L1M8d5U`6a7bc, Label32  Time:x VUJ2a@JK7L$M8m7UF{`6P a,b8c,k  txtCC 0.95x hg <@/>+.JP /K,0L1M-d5U`6xa,b c, Label34 BConfidence coefficient (1-alpha):x J'sKCJxK,L Mx-d5U`6xab c, Label102 This program assumes sampling from two infinite populations, since the means of two finite populations are almost always different (so that there is no point to a statistical test of the difference in the population means). Hence, the finite population correction does not apply, and the population sizes (N's) are not specified. In this case, it is possible for the program to specify sample sizes (n's) greater than the finite population sizes (N's).x wNqDAJxKL !Md#LVAL儩xu08=5B>a)b*c!e g$h6ixj;k + ?l7,+@x ^ڌgNYō @ Arial 8 h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma    mf45I" @ TahomaLVALy     `,@8  Detailx h_!G ׁ}d5U`6xaHb c,  Label0 This program determines sample size based on specification of the power (1-beta) of a (one-sided) test of the hypothesis about the value of a population mean. The null hypothesis is that the population mean is equal to zero. The alternative hypothesis is that the population mean is greater than a specified value, D. D is called the effect size. (Equivalently, the null hypothesis may be that the population mean is m and the alternative hypothesis is that the population mean is m + D.)x HNAZ(Pad5U`6xa b c,  Label2 0This program gives the same results as the determination of sample size based on specification of the size of a confidence interval, if (1) N (in the confidence-interval approach) is very large; (2) alpha for this (one-sided) approach is set equal to alpha/2 for that (two-sided) approach (e.g., .025 here, .05 there); (3) beta is set equal to .5 (i.e., zbeta = 0); and D is set equal to E. In typical situations, in which it is desired to detect a small difference (D), this approach may yield sample sizes several times as large as the confidence-interval approach. For detecting large differences, this approach generally produces smaller sample sizes. (Example 1: Using CI approach with CC=.95 (alpha=.05, z(1-alpha/2) = 1.96), sigma=.5, deff=1, E=.05 and N=1,000,000 yields n=384. Using the power approach with alpha=.025 (z(1-alpha)=1.9600), beta=.1 (z(1-beta)=1.2816), sigma=.5, deff=1, and D=.05 yields n=4,204 (i.e., 11 times as large). Ex. 2: Same as Ex. 1, but alpha = .1 (z(1-alpha)=1.286) yields n = 2,628 (6.84 times as large).x .OQYJxK LH!MXd5U`6xab c,  Label3 LVALzThe following values must be specified: (1) the effect size (size of the difference (D>0)); (2) the standard deviation of the population (sigma); (3) alpha (the probability of a making a Type I error (i.e., the significance level of the (one-sided) test (of difference in means)); (4) beta, the probability of making a Type II error (one minus the power of the test); (5) the design effect (deff). The design effect (deff) is the ratio of the variance (of the estimated population mean) using the specified sample design to the variance using simple random sampling. It is equal to 1.0 for simple random sampling, and may be larger or smaller than 1.0 for complex survey designs. This program assumes sampling from a normal distribution, but may be used for non-normal sampling (e.g., binomial, poisson, exponential) if the sample size (the program output) exceeds 30. (For two-sided tests, use the z-value corresponding to 1-alpha/2 instead of 1-alpha.) x \gY1a{C1Q04JxKLH!Md5U`6xa"b c,  Label4 PThis program assumes sampling from an infinite population (since tests of hypothesis relate primarily to a conceptually infinite population of which the finite population is viewed as a single sample). Hence, the finite population correction does not apply, and the population size (N) is not specified. In this case, it is possible for the program to specify a sample size (n) greater than the finite population size (N).x +{{8A3uAFJxK"L !M&d5U`6xa b c,  Label5 This program assumes that the sampling distribution of the sample mean is approximately normally distributed (may be assumed if n>30, and the infinite populations have finite standard deviations (true for "real" phenomena)).x CK_kCV6IiJxK LH!Mt"d5U`6xaPb c,  Label6 Note: Given tLVAL{he values of alpha (the significance level of the test) and beta (1 - the power of the test), the program calculates corresponding z-values. The z-value for 1-alpha is the value of the standard normal deviate having area (probability) alpha to the right of that point, and the z-value for 1-beta is the value of the standard normal deviate having area beta to the right of that point. Here is a list of z's corresponding to commonly used alpha's / beta's: z = 1.9600 for alpha=.025 or beta = .025; z = 1.6449 for alpha = .05 or beta = .05; z = 1.2816 for alpha = .10 or beta = .10; z = 1.0364 for alpha = .15 or beta = .15; z = .8416 for alpha = .2 or beta = .2; and z = 2.0 for alpha = .02275 or beta = .02275. The values alpha = .05 (zalpha = 1.6449), beta = .10 (zbeta = 1.2816), and D = .1 x sigma are sometimes used to determine sample size. For sampling for a proportion, p, (i.e., the population mean is a proportion, p) the value of sigma is sqrt[p(1-p)].x <&EGSInJxKPLH!MX d5U`6xap&b c,  Label8 The formula from which the result is derived is Prob(samplemean / sqrt(deff sigma**2 /n) > z(1-alpha) | popmean = D) =1 - beta. x ٣NNz<JxKp&L !M(d5U`6xa(b c,,  Label9 The formula for the sample size is n = (z(1-alpha) + z(1-beta))**2 deff sigma**2 / D**2.x '_eG&dJxK(L !M)m7UF{`6@ a0*c,, txtD 0.05x h#Y]A<]M+.J@ /K0*0L1M\+d5U`6xa0*bGc, Label10 Effect size, D:x 3 eHܟJxK0*LM +m7UF{`6da4bc,,k txtzalphax pfxYFCr+.Jd/K40L@ 1M$6d5U`6xa4bc,LVAL| Label11 z(1-alpha):x 63zEnQiJxK4L8M5m7UF{`6@ a+c,,k txtsigma 0.5x d\oLvNl+.J@ /K+0L1M,d5U`6xa+bc, Label12 4Standard deviation, sigma:x ^bOrzBϼJxK+LRM,m7UF{`6@ aX/c,,k txtdeff 1x ˉAјM:x+.J@ /KX/0L1M0d5U`6xaX/bc, Label15 (Design effect, deff:x UEqO91ejJxKX/LTMH0m7UF{`6Ha`6bc,,k  txtn1x ]o%JP`+.JH/K`60L1M7d5U`6xa`6bc, Label17 n:x w-D?OJxK`6LMP7h1U`6a81b@ c,i cmdDoIt  Do It!~ "[Event Procedure]x SL ;FP(!!%J&K81'L(M2m7UF{`6a45c,,k txtzbetax u^H+_8+.J/K450LD1M`6d5U`6 a45bc, Label14 z(1-beta):x ]2JFN*onJ K45LM$6h1U`6a4b c,i Command23 Close Form~ "[Event Procedure]x ZԻ N>x9^!%J&K4'L!(M5h1U`6a$6bc,i Command24 Quit App~ "[Event Procedure]x / rIzS!%J&K$6'L !(M7d5U`6xab c, Label25 Alpha denotes the probabiliLVAL}ty of making a Type I error (rejecting the null hypothesis when it is true). Beta denotes the probability of making a Type II error (accepting the null hypothesis when it is false). The power is 1 - beta.x ~WEseNަJxKLH!M$ d5U`6xaxb c,Xd#  Label28 4. Determine sample size by specifying the power of a test of hypothesis about the value of a population mean.x _ G[H"A_m7UF{`6,a6btc,k   Date1 =Date()x g<$,OsQ+.J,/K60L1M7d5U`6 a6bc, Label30  Date:x r5 /H!./qWAJ K6LM7m7UF{`6a6btc,k   Time1 =Time()x Uf2OGUbޛ+.J/K60L 1M7d5U`6Ta6bc, Label32  Time:x '#P>,@ PJTK6LpM7m7UF{`6@ a-c,k  txtAlpha 0.05x EM?Ni5]+.J@ /K-0L1M-d5U`6xa-b c, Label34 Halpha (signnificance level of test):x Y/4E߬YvqJxK-LA M-m7UF{`6 a3bc,k  txtBeta 0.1x ' w1@|+.J /K30L1M4d5U`6xa3bc, Label31 2beta (1 - power of test):x oPOLܙ<JxK3LM4m7UF{`6@ a,.c,k  txtpower 0.9x xZIn+.J@ /K,.0L1M+/d5U`6xa,.bc, Label35 PowerZLVALj (1-beta):x #yFLN"[JxK,.LM/vLVAL  =?rU $`x $A`| $` nͬIq7KUg/Detaild /Label0d /Label3d /Label4d /Label5d /Label6d /Label8d /Label9m/txtDd / Label10m/ txtzalphad / Label11m/ txtsigma1d / Label12m/txtdeffd /Label15h /cmdDoItm/txtsigma2d /Label16m/txtratio2d /Label20h /Command23h /Command24d /Label25d /Label28m/txtsigma3d /Label31m/txtsigma4d /Label33m/txtratio3d /Label35m/ txtratio4d /!Label37m/"txtrho12d /#Label13m/$txtrho34d /%Label48m/&txtrho24d /'Label50m/(txtrho23d /)Label52m/*txtrho14d /+Label54m/,txtrho13d /-Label56m/.txtn1d //Label39m/0txtn2d /1Label41m/2txtn3d /3Label42m/4txtn5d /5Label98m/6txtn4d /7Label100m/8txtzbetad /9Label14d /:Label43d /;Label44m/<txtpowerd /=Label32m/>Date1d /?Label30m/@Time1d /ALabel45m/BtxtAlphad /CLabel36m/DtxtBetad /ELabel46d /FLabel49d /GLabel51d /HLabel53d /ILabel55ͬF]TDcPu/Detaild /Label0d /Label3d /Label4d /Label5d /Label6d /Label8d /Label9m/txtDd / Label10m/ txtzalphad / Label11m/ txtsigma1d / Label12m/txtdeffd /Label15h /cmdDoItm/txtsigma2d /Label16m/txtratio2d /Label20h /Command23h /Command24d /Label25d /Label28m/txtsigma3d /Label31m/txtsigma4d /Label33m/txtratio3d /Label35m/ txtratio4d /!Label37m/"txtrho12d /#Label13m/$txtrho34d /%Label48m/&txtrho24d /'Label50m/(txtrho23d /)Label52m/*txtrho14d /+Label54m/,txtrho13d /-Label56m/.txtn1d //Label39m/0txtn2d /1Label41m/2txtn3d /3Label42m/4txtn5d /5Label98m/6txtn4d /7Label100m/8txtzbetad /9Label14d /:Label43m/;Date1d /<Label30m/=Time1d />Label32m/?txtAlphad /@Label34m/AtxtBetad /BLabel44d /CLabel47d /DLabel49d /ELabel51d /FLabel53ͬ$:Bk3u/Detaild /Label0d /Label3d /Label4d /Label5d /Label6d /Label8d /Label9m/ txtDd / Label10m/ txtzalphad / Label11m/ txtsigmad /Label12m/txtdeffd /Label15m/txtn1d /Label17h /cmdDoItm/txtzbetad /Label14h /Command23h /Command24d /Label25d /Label28d /Label30d /Label32m/txtpowerm/Date1d /Label33m/Time1d / Label34m/!txtAlphad /"Label35m/#txtBetad /$Label31ͬ$JHkZ5/Detaild /Label0d /Label2d /Label3d /Label4d /Label5d /Label6d /Label8d /Label9m/ txtDd / Label10m/ txtzalphad / Label11m/ txtsigmad /Label12m/txtdeffd /Label15m/txtn1d /Label17h /cmdDoItm/txtzbetad /Label14h /Command23h /Command24d /Label25d /Label28m/Date1d /Label30m/Time1d /Label32m/txtAlphad /Label34m/ txtBetad /!Label31d /#Label35m/"txtpowerLVAL儩u08=5B>a)b*c!e g$h6ij#k =?l7,+@x sҐ'wE;Vqj @ Arial 8!p5h `  ߀ odXLetterPRIV @!''''d(  pX7 '''' ,@ VAIOENGLISHBRSP807A.EXEBRLHLA7A.DLLBRB8LA7A.DLLBO7440N.INI dArialdGeorgeBRLHL07A.DLLEOSC C"<USB0021D/1Fd2" @ Tahoma    hg#h%" @ Tahoma