6 Sample Size Calculations

Transcription

1 6 Sample Size Calculatios Oe of the major resposibilities of a cliical trial statisticia is to aid the ivestigators i determiig the sample size required to coduct a study The most commo procedure for determiig the sample size ivolves computig the miimum sample size ecessary i order that importat treatmet differeces be determied with sufficiet accuracy We will focus primarily o hypothesis testig 6 Hypothesis Testig I a hypothesis testig framework, the questio is geerally posed as a decisio problem regardig a parameter i a statistical model: Suppose a populatio parameter correspodig to a measure of treatmet differece usig the primary edpoit is defied for the study This parameter will be deoted by For example, if we are cosiderig the mea respose of some cotiuous variable betwee two treatmets, we ca deote by µ, the populatio mea respose for treatmet ad µ 2, the mea respose o treatmet 2 We the deote by = µ µ 2 the measure of treatmet differece A cliical trial will be coducted i order to make iferece o this populatio parameter If we take a hypothesis testig poit of view, the we would cosider the followig decisio problem: Suppose treatmet 2 is curretly the stadard of care ad treatmet is a ew treatmet that has show promise i prelimiary testig What we wat to decide is whether we should recommed the ew treatmet or stay with the stadard treatmet As a startig poit we might say that if, i truth, 0 the we would wat to stay with the stadard treatmet; whereas, if, i truth, > 0, the we would recommed that future patiets go o the ew treatmet We refer to 0 as the ull hypothesis H 0 ad > 0 as the alterative hypothesis H A The above is a example of a oe-sided hypothesis test I some cases, we may be iterested i a two-sided hypothesis test where we test the ull hypothesis H 0 : = 0 versus the alterative H A : 0 I order to make a decisio o whether to choose the ull hypothesis or the alterative hypothesis, PAGE 7

2 we coduct a cliical trial ad collect data from a sample of idividuals The data from idividuals i our sample will be deoted geerically as (z,,z ) ad represet realizatios of radom vectors (Z,,Z ) The Z i may represet a vector of radom variables for idividual i; eg respose, treatmet assigmet, other covariate iformatio As statisticias, we posit a probability model that describes the distributio of (Z,,Z ) i terms of populatio parameters which icludes (treatmet differece) as well as other parameters ecessary to describe the probability distributio These other parameters are referred to as uisace parameters We will deote the uisace parameters by the vector θ As a simple example, let the data for the i-th idividual i our sample be deoted by Z i = (Y i,a i ), where Y i deotes the respose (take to be some cotiuous measuremet) ad A i deotes treatmet assigmet, where A i ca take o the value of or 2 depedig o the treatmet that patiet i was assiged We assume the followig statistical model: let (Y i A i = 2) N(µ 2,σ 2 ) ad (Y i A i = ) N(µ 2 +,σ 2 ), ie sice = µ µ 2, the µ = µ 2 + The parameter is the test parameter (treatmet differece of primary iterest) ad θ = (µ 2,σ 2 ) are the uisace parameters Suppose we are iterested i testig H 0 : 0 versus H A : > 0 The way we geerally proceed is to combie the data ito a summary test statistic that is used to discrimiate betwee the ull ad alterative hypotheses based o the magitude of its value We refer to this test statistic by T (Z,,Z ) Note: We write T (Z,,Z ) to emphasize the fact that this statistic is a fuctio of all the data Z,,Z ad hece is itself a radom variable However, for simplicity, we will most ofte refer to this test statistic as T or possibly eve T The statistic T should be costructed i such a way that (a) Larger values of T are evidece agaist the ull hypothesis i favor of the alterative PAGE 72

3 (b) The probability distributio of T ca be evaluated (or at least approximated) at the border betwee the ull ad alterative hypotheses; ie at = 0 After we coduct the cliical trial ad obtai the data, ie the realizatio (z,,z ) of (Z,,Z ), we ca compute t = T (z,,z ) ad the gauge this observed value agaist the distributio of possible values that T ca take uder = 0 to assess the stregth of evidece for or agaist the ull hypothesis This is doe by computig the p-value P =0 (T t ) If the p-value is small, say, less tha 05 or 025, the we use this as evidece agaist the ull hypothesis Note: Most test statistics used i practice have the property that P (T x) icreases as icreases, for all x I particular, this would mea that if the p-value P =0 (T t ) were less tha α at = 0, the the probability P (T t ) would also be less tha α for all correspodig to the ull hypothesis H 0 : 0 2 Also, most test statistics are computed i such a way that the distributio of the test statistic, whe = 0, is approximately a stadard ormal; ie T ( =0) N(0,), regardless of the values of the uisace parameters θ For the problem where we were comparig the mea respose betwee two treatmets, where respose was assumed ormally distributed with equal variace by treatmet, but possibly differece meas, we would use the two-sample t-test; amely, T = Ȳ Ȳ2 s Y + 2 ) /2, where Ȳ deotes the sample average respose amog the idividuals assiged to treatmet, Ȳ 2 deotes the sample average respose amog the 2 idividuals assiged to treatmet 2, PAGE 73

4 = + 2 ad the sample variace is { s 2 Y = j=(y j Ȳ) ( + 2 2) j=(y 2j Ȳ2) 2 } Uder = 0, the statistic T follows a cetral t distributio with degrees of freedom If is large (as it geerally is for phase III cliical trials), this distributio is well approximated by the stadard ormal distributio If the decisio to reject the ull hypothesis is based o the p-value beig less that α (05 or 025 geerally), the this is equivalet to rejectig H 0 wheever T Z α, where Z α deotes the α-th quatile of a stadard ormal distributio; eg Z 05 = 64 ad Z 025 = 96 We say that such a test has level α Remark o two-sided tests: If we are testig the ull hypothesis H 0 : = 0 versus the alterative hypothesis H A : 0 the we would reject H 0 whe the absolute value of the test statistic T is sufficietly large The p-value for a two-sided test is defied as P =0 ( T t ), whichequalsp =0 (T t )+P =0 (T t ) IftheteststatisticT isdistributedasastadard ormal whe = 0, the a level α two-sided test would reject the ull hypothesis wheever the p-value is less tha α; or equivaletly T Z α/2 The rejectio regio of a test is defied as the set of data poits i the sample space that would lead to rejectig the ull hypothesis For oe sided level α tests, the rejectio regio is {(z,,z ) : T (z,,z ) Z α }, ad for two-sided level α tests, the rejectio regio is {(z,,z ) : T (z,,z ) Z α/2 } Power I hypothesis testig, the sesitivity of a decisio(ie level-α test) is evaluated by the probability of rejectig the ull hypothesis whe, i truth, there is a cliically importat differece This is PAGE 74

5 referred to as the power of the test We wat power to be large; geerally power is chose to be 80, 90, 95 Let us deote by A the cliically importat differece This is the miimum value of the populatio parameter that is deemed importat to detect If we are cosiderig a oesided hypothesis test, H 0 : 0 versus H A : > 0, the by defiig the cliically importat differece A, we are essetially sayig that the regio i the parameter space = (0, A ) is a idifferece regio That is, if, i truth, 0, the we would wat to coclude that the ull hypothesis is true with high probability (this is guarateed to be greater tha or equal to ( α) by the defiitio of a level-α test) However, if, i truth, A, where A > 0 is the cliically importat differece, the we wat to reject the ull hypothesis i favor of the alterative hypothesis with high probability (probability greater tha or equal to the power) These set of costraits imply that if, i truth, 0 < < A, the either the decisio to reject or ot reject the ull hypothesis may plausibly occur ad for such values of i this idifferece regio we would be satisfied by either decisio Thus the level of a oe-sided test is ad the power of the test is P =0 (fallig ito the rejectio regio) = P =0 (T Z α ), P = A (fallig ito the rejectio regio) = P = A (T Z α ) I order to evaluate the power of the test we eed to kow the distributio of the test statistic uder the alterative hypothesis Agai, i most problems, the distributio of the test statistic T ca be well approximated by a ormal distributio Uder the alterative hypothesis H A =( A,θ) T N(φ(, A,θ),σ 2 ( A,θ)) I other words, the distributio of T uder the alterative hypothesis H A follows a ormal distributio with o zero mea which depeds o the sample size, the alterative A ad the uisace parameters θ We deote this mea by φ(, A,θ) The stadard deviatio σ ( A,θ) may also deped o the parameters A ad θ Remarks Ulike the ull hypothesis, the distributio of the test statistic uder the alterative hypothesis also depeds o the uisace parameters Thus durig the desig stage, i order to determie the PAGE 75

6 power of a test ad to compute sample sizes, we eed to ot oly specify the cliically importat differece A, but also plausible values of the uisace parameters 2 It is ofte the case that uder the alterative hypothesis the stadard deviatio σ ( A,θ) will be equal to (or approximately equal) to oe If this is the case, the the mea uder the alterative φ(, A,θ) is referred to as the o-cetrality parameter For example, whe testig the equality i mea respose betwee two treatmets with ormally distributed cotiuous data, we ofte use the t-test T = Ȳ Ȳ2 s Y + 2 ) /2 Ȳ Ȳ2 σ Y + 2 ) /2, which is approximately distributed as a stadard ormal uder the ull hypothesis Uder the alterative hypothesis H A : µ µ 2 = = A, the distributio of T will also be approximately ormally distributed with mea Ȳ Ȳ2 E HA (T ) E ( ) /2 σ Y + 2 = µ µ 2 σ Y + 2 ) /2 = A σ Y + 2 ) /2, ad variace ( var HA (T ) = {var(ȳ)+var(ȳ2)} ( σy 2 + ) = σ2 Y + 2) σ 2 2 Y + = 2) Hece H T A A N ( ) /2, σ Y + 2 Thus A φ(, A,θ) = ( ) /2, (6) σ Y + 2 ad σ ( A,θ) = (62) Hece ( A ) /2 is the o-cetrality parameter σ Y + 2 Note: I actuality, the distributio of T is a o-cetral t distributio with degrees of freedom ad o-cetrality parameter ( A ) /2 However, with large this is well σ Y + 2 approximated by the ormal distributio give above PAGE 76

7 62 Derivig sample size to achieve desired power We are ow i a positio to derive the sample size ecessary to detect a cliically importat differece with some desired power Suppose we wat a level-α test (oe or two-sided) to have power at least equal to β to detect a cliically importat differece = A The how large a sample size is ecessary? For a oe-sided test cosider the figure below Figure 6: Distributios of T uder H 0 ad H A Desity x It is clear from this figure that φ(, A,θ) = {Z α +Z β σ ( A,θ)} (63) Therefore, if we specify the level of sigificace (type I error) α the power ( - type II error) β the cliically importat differece A the uisace parameters θ PAGE 77

8 the we ca fid the value which satisfies (63) Cosider the previous example of ormally distributed respose data where we use the t-test to test for treatmet differeces i the mea respose If we radomize patiets with equal probability to the two treatmets so that = 2 /2, the substitutig (6) ad (62) ito (63), we get or A σ Y ( 4 /2 = ) /2 = (Z α +Z β ), { } (Zα +Z β )σ Y 2 A { (Zα +Z β ) 2 σ 2 } Y 4 = 2 A Note: For two-sided tests we use Z α/2 istead of Z α Example Suppose we wated to fid the sample size ecessary to detect a differece i mea respose of 20 uits betwee two treatmets with 90% power usig a t-test (two-sided) at the 05 level of sigificace We expect the populatio stadard deviatio of respose σ Y to be about 60 uits I this example α = 05, β = 0, A = 20 ad σ Y = 60 Also, Z α/2 = Z 025 = 96, ad Z β = Z 0 = 28 Therefore, or about 89 patiets per treatmet group = (96+28)2 (60) 2 4 (20) (roudig up), 63 Comparig two respose rates We will ow cosider the case where the primary outcome is a dichotomous respose; ie each patiet either respods or does t respod to treatmet Let π ad π 2 deote the populatio resposeratesfortreatmetsad2respectively Treatmetdiffereceisdeotedby = π π 2 We wish to test the ull hypothesis H 0 : 0 (π π 2 ) versus H A : > 0 (π > π 2 ) I some cases we may wat to test the ull hypothesis H 0 : = 0 agaist the two-sided alterative H A : 0 PAGE 78

9 A cliical trial is coducted where patiets are assiged treatmet ad 2 patiets are assiged treatmet 2 ad the umber of patiets who respod to treatmets ad 2 are deoted by X ad X 2 respectively As usual, we assume X b(,π ) ad X 2 b( 2,π 2 ), ad that X ad X 2 are statistically idepedet If we let π = π 2 +, the the distributio of X ad X 2 is characterized by the test parameter ad the uisace parameter π 2 If we deote the sample proportios by p = X / ad p 2 = X 2 / 2, the we kow from the properties of a biomial distributio that This motivates the test statistic E(p ) = π, var(p ) = π ( π ), E(p 2 ) = π 2, var(p 2 ) = π 2( π 2 ) 2 T = p p 2 { p( p) + 2)} /2, where p is the combied sample proportio for both treatmets; ie p = (X +X 2 )/( + 2 ) Note: The statistic T 2 is the usual chi-square test used to test equality of proportios We ca also write p = p ( ) ( ) +p = p +p As such, p is a approximatio (cosistet estimator) for ( ) ( ) 2 π +π 2 = π, where π is a weighted average of π ad π 2 Thus T p p 2 { π( π) + 2)} /2 PAGE 79

10 The mea ad variace of this test statistic uder the ull hypothesis = 0 (border of the ull ad alterative hypotheses for a oe-sided test) are p p 2 E =0 (T ) E =0 ( { π( π) + /2 2)} var =0 (T ) {var =0(p )+var =0 (p 2 )} { π( π) + 2)} = But sice π = π 2 = π, we get var =0 (T ) = = E =0 (p p 2 ) ( { π( π) + /2 = 0, 2)} { π ( π ) + π } 2( π 2 ) 2 { π( π) + 2)} Because the distributio of sample proportios are approximately ormally distributed, this will imply that the distributio of the test statistic, which is roughly a liear combiatio of idepedet sample proportios, will also be ormally distributed Sice the ormal distributio is determied by its mea ad variace, this implies that, T ( =0) N(0,) For the alterative hypothesis H A : = π π 2 = A, E HA (T ) (π π 2 ) { π( π) + 2)} /2 = A { π( π) + 2)} /2, ad usig the same calculatios for the variace as we did above for the ull hypothesis we get { π ( π ) var HA (T ) + π } 2( π 2 ) ( 2 { π( π) + 2)} Whe = 2 = /2, we get some simplificatio; amely ad π = (π +π 2 )/2 = (π 2 + A /2) var HA (T ) = π ( π )+π 2 ( π 2 ) 2 π( π) Note: The variace uder the alterative is ot exactly equal to oe, although, if π ad π 2 are ot very differet, the it is close to oe Cosequetly, with equal treatmet allocatio, H T A N A { π( π) 4 } /2, π ( π )+π 2 ( π 2 ) 2 π( π) PAGE 80

11 Therefore, ad where π = π 2 + A φ(, A,θ) = A } /2, { π( π) 4 σ 2 = π ( π )+π 2 ( π 2 ), 2 π( π) Usig formula (63), the sample size ecessary to have power at least ( β) to detect a icrease of A, or greater, i the populatio respose rate of treatmet above the populatio respose rate for treatmet 2, usig a oe-sided test at the α level of sigificace is /2 A {4 π( π)} /2 = Z α +Z β { } /2 π ( π )+π 2 ( π 2 ) 2 π( π) Hece = { { } } Z α +Z π ( π )+π 2 ( π 2 ) /2 β 2 π( π) 24 π( π) (64) 2 A Note: For two-sided tests we replace Z α by Z α/2 Example: Suppose the stadard treatmet of care(treatmet 2) has a respose rate of about35 (best guess) After collaboratios with your cliical colleagues, it is determied that a cliically importat differece for a ew treatmet is a icrease i 0 i the respose rate That is, a resposerateof45orlarger Ifwearetocoductacliicaltrialwherewewillradomizepatiets with equal allocatio to either the ew treatmet (treatmet ) or the stadard treatmet, the how large a sample size is ecessary to detect a cliically importat differece with 90% power usig a oe-sided test at the 025 level of sigificace? Note for this problem α = 025, Z α = 96 β = 0 (power = 9), Z β = 28 A = 0 π 2 = 35, π = 45, π = 40 PAGE 8

12 Substitutig these values ito (64) we get = { { } } / ,004, (0) 2 or about 502 patiets o each treatmet arm 63 Arcsi square root trasformatio Sice the biomial distributio may ot be well approximated by a ormal distributio, especially whe is small (ot a problem i most phase III cliical trials) or π is ear zero or oe, other approximatios have bee suggested for derivig test statistics that are closer to a ormal distributio We will cosider the arcsi square root trasformatio which is a variace stabilizig trasformatio Before describig this trasformatio, I first will give a quick review of the delta method for derivig distributios of trasformatios of estimators that are approximately ormally distributed Delta Method Cosider a estimator ˆγ of a populatio parameter γ such that ˆγ N(γ, σ2 γ ) Roughly speakig, this meas that E(ˆγ ) γ ad var(ˆγ ) σ2 γ Cosider the variable f(ˆγ ), where f( ) is a smooth mootoic fuctio, as a estimator for f(γ) Usig a simple Taylor series expasio of f(ˆγ ) about f(γ), we get f(ˆγ ) = f(γ)+f (γ)(ˆγ γ)+(small remaider term), where f (γ) deotes the derivative df(γ) dγ The E{f(ˆγ )} E{f(γ)+f (γ)(ˆγ γ)} PAGE 82

13 = f(γ)+f (γ)e(ˆγ γ) = f(γ) ad Thus var{f(ˆγ )} var{f(γ)+f (γ)(ˆγ γ)} ( ) σ 2 = {f (γ)} 2 var(ˆγ ) = {f (γ)} 2 γ f(ˆγ ) N ( f(γ),{f (γ)} 2 ( σ 2 γ )) Take the fuctio f( ) to be the arcsi square root trasformatio; ie f(x) = si (x) /2 If y = si (x) /2, the si(y) = x /2 The derivative dy dx That is, dsi(y) dx = dx/2 dx, cos(y) dy dx = 2 x /2 is foud usig straightforward calculus Sice cos 2 (y)+si 2 (y) =, this implies that cos(y) = { si 2 (y)} /2 = ( x) /2 Therefore ( x) /2dy dx = 2 x /2, or dy dx = 2 {x( x)} /2 = f (x) If p = X/ is the sample proportio, where X b(,π), the var(p) = π( π) Usig the delta method, we get that Cosequetly, { } π( π) var{si (p) /2 } {f (π)} 2, [ ] 2 { } π( π) = 2 {π( π)} /2, { }{ } π( π) = = 4π( π) 4 si (p) /2 N ( ) si (π) /2, 4 Note: The variace of si (p) /2 does ot ivolve the parameter π, thus the term variace stabilizig PAGE 83

14 The ull hypothesis H 0 : π = π 2 is equivalet to H 0 : si (π ) /2 = si (π 2 ) /2 This suggests that aother test statistic which could be used to test H 0 is give by The expected value of T is T = si (p ) /2 si (p 2 ) /2 ( ) / E(T ) = E{si (p ) /2 } E{si (p 2 ) /2 } ( ) / ad the variace of T is si (π ) /2 si (π 2 ) /2 ( ) / var(t ) = var{si (p ) /2 si (p 2 ) /2 } ( ) = var{si (p )/2}+var{si (p 2 )/2 } ( ) ( ) ( 2 ) = I additio to the variace stabilizig property of the arcsi square root trasformatio for the sample proportio of a biomial distributio, this trasformed sample proportio also has distributio which is closer to a ormal distributio Sice the test statistic T is a liear combiatio of idepedet arcsi square root trasformatios of sample proportios, the distributio of T will also be well approximated by a ormal distributio Specifically, H T 0 N(0,) H T A N si (π ) /2 si (π 2 ) /2 ( ) /2, If we take = 2 = /2, the the o-cetrality parameter equals φ(, A,θ) = /2 A, where A, the cliically importat treatmet differece, is parameterized as A = si (π ) /2 si (π 2 ) /2 Cosequetly, if we parameterize the problem by cosiderig the arcsi square root trasformatio, ad use the test statistic above, the with equal treatmet allocatio, the sample size ecessary to detect a cliically importat treatmet differece of A i the arcsi square root PAGE 84

15 of the populatio proportios with power ( β) usig a test (oe-sided) at the α level of sigificace, is derived by usig (63); yieldig /2 A = (Z α +Z β ), = (Z α +Z β ) 2 2 A Remark: Remember to use radias rather tha degrees i computig the arcsi or iverse of the sie fuctio Some calculators give the result i degrees where π = 3459 radias is equal to 80 degrees; ie radias= degrees If we retur to the previous example where we computed sample size based o the proportios test, but ow istead use the formula for the arc si square root trasformatio we would get = (96+28) 2 {si (45) /2 si (35) /2 } 2 = (96+28)2 ( ) 2 = 004, or 502 patiets per treatmet arm This aswer is virtually idetical to that obtaied usig the proportios test This is probably due to the relatively large sample sizes together with probabilities beig bouded away from zero or oe where the ormal approximatio of either the sample proportio of the arcsi square root of the sample proportio is good PAGE 85