Guidelines for calculating sample size in 2x2 crossover trials : a simulation study


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1 J.Natn.Sci.Foundation Sri Lanka (1): RESEARCH ARTICLE Guidelines for calculating saple size in x crossover trials : a siulation study N.M. Siyasinghe and M.R. Sooriyarachchi Departent of Statistics, Faculty of Science, University of Colobo, Colobo 03. Revised: 04 August 010 ; Accepted: 1 January 011 Abstract: In crossover trials, patients receive two or ore treatents in a rando order in different periods. The saple size deterination is often an iportant step in planning a crossover study. This paper concerns saple size calculations in x crossover trials, with rando patient effects and no interaction between the treatent and the patient under two scenarios, naely the exact and the large saple size approaches. Siulation was carried out for deterining the saple size for both scenarios. For varying paraeter values, siulation was used for generating saples of the required size and exaining whether the significance level and power of the tests are aintained. The results indicate that when the saple size was 5, neither ethod aintained error rates and when the saple size was >5 and < 1 only the exact approach aintained error rates. However, when the saple size is approxiately > 1 both ethods aintained error rates. In addition it was found that a saving in saple size can be achieved depending on the extent of the correlation between the observations on the sae patient. The siulation results indicate that crossover studies should not be conducted when the anticipated saple size is 5 and when a saple size of >5 and < 1 is anticipated, the exact ethod of deterining saple size should be used. When larger saple sizes are anticipated either ethod can be used but the ethod based on large saple size approxiation is sipler. Keywords: Crossover trial, exact ethod, large saple ethod, saple size calculation, siulation. INTRODUCTION Crossover trials are clinical trials in which patients are given all the edications to be studied in a rando order. According to Grizzle(1965) these studies are generally conducted on patients with chronic diseases to control their syptos. The data are analyzed according to the original intention to treat. Ideally, clinical trials should be large enough to reliably detect the sallest possible difference in the priary outcoe with treatents that are considered clinically worthwhile. According to Lee et al. (005), it is not uncoon for studies to be underpowered, failing to detect even large treatent effects because of inadequate saple size. It is considered unethical to recruit patients for a study that does not have a large enough saple size for the trial to deliver eaningful inforation on the tested intervention. Thus, saple size should be based on scientific considerations. Several approaches are discussed (Pocock, 1983; Julious & Patterson, 004) for calculating saple size including the power approach and the confidence interval approach. According to previous studies (Chow et al. 003;Woodward, 199), these approaches require the specification of several paraeters such as between treatent and within treatent variances for the treatents under consideration, the correlation within patients and the reference iproveent, which is required to be detected. Chow et al. (003) explain the two different approaches for x crossover designs but do not give guidelines on when to use specific approaches. In this study siulation is extensively used to exaine the proble of setting guidelines. In this study, two situations are considered in the calculation of saple size of a crossover study as explained in Chow et al. (003). These are, (i) The exact approach (ii) The large saple ( approxiate ) approach. Further, the study gives guidelines for when to use the exact approach and the large saple approach and to study how uch saving in saple size can be achieved when observations on the sae patient are correlated. * Corresponding author
2 78 N. M. Siyasinghe & M.R. Sooriyarachchi METHODS AND MATERIALS Mixed odel used: In clinical trials, it is coon to assue that the patients respond consistently to treatents. However, the assuption is invalid if the patients vary randoly in their responses to the drug. For this type of situation, a rando subject effects odel where the subject effect is considered to be rando and the treatent and period effects are considered to be fixed has to be considered (Brown & Prescott, 006). Chow et al. (003) have explained how to calculate the saple size in a crossover design using either of the two approaches, naely the exact and the approxiate. In this paper a siilar crossover design coparing ean responses for two groups is considered. In the first approach the test statistic is based on the Student s t distribution, whereas in the second approach the test statistic is based on the noral distribution. In the exact approach, the saple size depends on the degrees of freedo. The calculation of saple size is therefore not straightforward. The sae calculation can be done without difficulty if the approxiate approach, which is based on the noral approxiation, is used. Values of the inverse t distribution function need to be deterined for calculating the saple sizes for the exact approach. This is done by using the approxiation given in Cooke et al. (198). The criteria used for deterining the ethod to be used for calculating saple size is based on which ethod aintains power and significance level. Let Y ijk be the response observed fro the j th (j= 1,,..,n) subject in the i th sequence (i = 1,) under k th treatent (k=1,). The odel considered is Y ijk =µ + tk + pi + sijk + eijk, where µ is the overall ean, tk is the k th treatent effect, pi is the i th sequence (period) effect, s ijk is the rando effect of the j th subject in the i th sequence under k th treatent and e ijk is the error ter corresponding to the j th subject in the i th sequence under k th treatent. The following ixed odel is used. Y = µ + t + p + s + e...(1) ijk k i ijk ijk Here, treatent effect and period effect are considered as fixed effects and subject effect as rando. In this study equal allocation of patients to treatent groups are assued and no replication is considered. Then define the following notation. Since µ is a constant, we can take new odel is; µ =µ+ t. So the k k Yijk = µ. k+ pi + sijk + eijk...() Here it is assued that there is no treatent by period interaction, since a siple hypothesis test can be used only under this assuption. The subject effects S ij1, S ij are assued to be independent and identically distributed as bivariate noral rando variables with ean 0 and covariance atrix σ BT = ρσ BTσ ρσ σ BT BR BR σ BR where σ is the variance between patients for the BT treated group, σ is the variance between patients for BR the reference group and ρ is the correlation between subjects in the treated and reference groups. So, S ij1 and S ij have a bivariate noral distribution with ean 0 and variance covariance atrix. It is assued that the errors e ij1 and e ij are such that e ij1 ~ iid N(0, σwt ) e iid N σ ij ~ (0, WR ) where, σ is the within patient variation for the treated WT group and σ is the within patient variation for the WR reference group. Consider a group, which gets treatent 1 in the first period and treatent in the second period, then the odel can be written as follows, Y = + p + s + e...(3). 1j1 µ 1 1 1j1 1j1 and Y = µ + p + s + e...(4). j j j Estiation of the ean and variance of the treatent difference: The ethod of Chow et al. (003) explains a procedure to easure the treatent difference of a crossover trial and this section discusses that procedure. Let ε be the easure of treatent difference, then ε = µ 1 µ (testreference). Now take dij = yij1 yij An unbiased estiate for ε is given by, 1 ε n ˆ = d i n and V [ ˆ ] i = 1 j = 1 where σ BT BR σ ε =, n = σ + σ ρσ σ σ + σ BT BR+ WT WR In practice, there is no prior inforation regarding the value of σ and it is deterined by the assued values of σ, BT σ, ρ, BR σ and σ and used for saple size WT WR March 011 Journal of the National Science Foundation of Sri Lanka 39 (1)
3 Saple size for crossover trials 79 calculation at the design stage. Thus it is required to find an unbiased estiate for use in the test statistic at the analysis stage. An unbiased estiate for σ can be given by, 1 n Var ( ˆ ε ) = σˆ = ( dij di. ). ( n 1) i= 1 j= 1 where d i. 1 n = d n j= 1 ij...(5) Estiating saple size: The null hypothesis (H o ): ε = 0 and the alternative hypothesis (H 1 ) : ε 0 are used to test whether the effect of two treatents is equal or not. Under H 0, the test statistic ˆ ε ˆ σ follows a t distribution n with n degrees of freedo (Chow et al., 003). The null hypothesis is rejected at α level of significance if σˆ εˆ n > t α, n...(6) The above entioned hypothesis test will satisfy a power of = 1 β if ε^  ˆε Pr > tα, n 1 β σˆ = ε= ε R n...(7) Fro equations (6) and (7) the corresponding saple size can be obtained by n = T 1 [Here T ( a) ( 1 ) + T ( 1 ) α β σˆ εr...(8) 1 1 n n n indicates the a th ordinate of the t distribution, with n degrees of freedo] When considering the large saple approach, instead of the t distribution, the standard noral distribution is used. Then the forulae for the saple size calculation can be obtained as, ( 1 ) + Z ( 1 ) Z 1 α 1 β σˆ n =...(9) ε R 1 [Here Z ( a) indicates the a th ordinate of the standard noral distribution] Siulation studies: (a) Description: In order to satisfy the above entioned objectives, a siulation study was carried out. For the exact approach, the bisection ethod was used as the root finding technique for deterining the saple size, as described in Press et al.(00). The siulation study was also used for deterining whether the type 1 error and the power are aintained, for both approaches. Saple sizes were deterined for varying correlations for both approaches, and thereby the saving in saple size with increasing correlation was studied. Finally, based on the results of the siulation study carried out, guidelines are provided for saple size calculation in crossover trials. A C prograe was written for perforing the siulation study. The C language was selected since it is efficient in doing large scale siulations. The first step of the siulation study was to set soe practically plausible values for the paraeters required (Sooriyarachchi & Whitehead, 1998 ; Whitehead et al., 008). Usually crossover trials are associated with a sall saple size, due to coparison of treatents being within patient rather than between patient, and variances within patient being usually saller than the between variances. The between treatent standard deviation for the treated group (σ BT ) was exained over two values naely, 3 and 4 and the between treatent standard deviation for the reference group (σ BR ) was set equal to σ BT, which is often assued in crossover trials. Two values were assigned for the within patient standard deviation for the treated group (σ WT ) naely, 0.3 and 0.5 and again the within patient standard deviation for the reference group (σ WR ) set equal to σ WT. The within subject correlation coefficient was indicated by the variable ρ. The values of ρ were exained over 0, 0.3, 0.6, and 0.9. i.e. considering there is no correlation at all, soe correlation, high correlation and very high correlation, respectively so that we can see and copare the outcoes for various situations. Note that although it was not considered in this study it is also possible to consider situations where σ BT does not equal σ BR does not equal σ WR. The reference iproveent is indicated by the variable naed ε R. The values of ε R that were exained are 1.5, and 3. For each of these cobinations 1000 siulations were carried out under the null and the alternative hypotheses. Under the null hypothesis, the ean difference between treatents (ε) is set to zero and under the alternative Journal of the National Science Foundation of Sri Lanka 39 (1) March 011
4 80 N. M. Siyasinghe & M.R. Sooriyarachchi hypothesis, the ean difference between treatents (ε) is set to ε R. As explained in the introduction, calculation of saple size is not straightforward for the exact case as the saple size is dependent on the degrees of freedo in this case and thus the saple size deterination requires solving of a nonlinear equation in n; hence a root finding technique is needed. The ethod used here is the Bisection Method explained in Press et al. (00). After obtaining an estiate for the saple size, it was of interest to deterine the proportion of rejections under the null and alternative hypotheses out of thousand siulations to see whether the power and the significance level are aintained. That is to siulate each saple size 1000 ties and get the proportion of rejections. In order to do that, we need to siulate the odel explained in the introduction. For that we need to generate s ijk s and e ijk s for each saple size. (b) Rando nuber generation: In siulating uncorrelated variables, BoxMuller transforation was used (Golder & Settle, 1976) and the ethod described in AlSubaihi (004) is used for siulating correlated variables. RESULTS AND DISCUSSION Checking whether the significance level and the power of the test are aintained was a ajor objective of this study. In order to do that, two probability intervals were calculated (Sooriyarachchi & Whitehead, 1998) for the true values of significance level and power. The 95% probability interval for a significance level of size α can be obtained by, α ± Thus for ( 1 α) α 1000 α = 5% the probability interval is, ( ) = 0.05± 1000 = [0.015, 0.035] The 95% probability interval for a power of 90% can be obtained by, ( ) ± = [0.881, 0.919] 1000 of the test can be obtained by the proportion of rejections of the null hypothesis when the alternative hypothesis is true. If the corresponding proportions are within the above probability intervals, it can be concluded that the significance level/power is well aintained by the test. Table 1 gives the proportion of rejections of the null hypothesis under the null and alternative hypothesis for the exact approach and Table corresponds to the siilar table for the large saple approach. The proportions which are out of the confidence liits are highlighted in the tables. In these tables the values taken by all the nuisance paraeters ( σ BT, σ BR, σ WT, σ WR, ρ) and reference iproveent (ε R ) are given including whether the siulation was done under the null hypothesis (g =1) or under the alternative hypothesis (g =). Then for each cobination, the calculated saple size, the proportion of rejections of the null hypothesis, σ and the ean value of ˆ σ are given. This is useful in deciding how close the ean of ˆ σ is to σ and hence the unbiasedness σ. of ˆ When considering Table 1 corresponding to the exact ethod, based on the t statistic, it can be observed that usually when the saple size is very sall (less than or equal to five), the estiated power is outside the probability liits and higher than the upper liit. This is because for very sall saple sizes the approxiation to the inverse of the t distribution, which is described in Cooke et al. (198), is an overestiate resulting in a too large saple size. But when the saple size is soewhat larger (greater than 5) the power is generally well aintained. Except for a very few cases (row nuber 34), the significance level is usually aintained by the test. When considering Table corresponding to the large saple approxiation ethod based on the z statistic, it can be observed that when the saple size is less than or equal to 10 in ost of the cases the test is underpowered, it is worse when the saple size gets saller. But when the saple size is equal to 11 in row nubers 6 and 7, the power is well aintained, but in row nuber 1 the test is under  powered. This is because the noral ordinate is an underestiate of the t ordinate for saple sizes up to about 0 (ss=0 iplying ss=11). It can be said that when the saple size is approxiately less than 1, the test is under  powered when the large saple approxiation is used. When the saple size is approxiately greater than or equal to 1 the power is generally well aintained. Except for a few cases (row nubers 14, 64) for alost all the cases the significance level is well aintained. An estiate for the significance level can be obtained by the proportion of rejections of the null hypothesis when the null hypothesis is true, and an estiate for the power Tables 1 and show that for ost of the cobinations, values of the σ and the ean of ˆ σ are close to each other for both situations. March 011 Journal of the National Science Foundation of Sri Lanka 39 (1)
5 Saple size for crossover trials 81 In order to illustrate the results ore clearly, several graphs have been plotted in addition to the two tables. Figure 1 is drawn to illustrate the variation of saple size with respect to del for different cobinations of ρ, σ BT, σ BW for the exact approach. Figure 1 shows how the values of nuisance paraeters and the reference iproveent effect the calculation of saple size for the Exact ethod. It can be seen that when ρ increases, the saple size required rapidly decreases, irrespective of the situation. Here ρ represents the within Table 1: Proportion of rejections of the null hypotheses under the null and the alternative hypotheses for the exact ethod No σ BT σ BR ρ σ WT σ WR del (ε R ) Hypothesis Saple size Proportion σ Mean of ˆ Journal of the National Science Foundation of Sri Lanka 39 (1) March 011
6 8 N. M. Siyasinghe & M.R. Sooriyarachchi Table 1 continued No σ BT σ BR ρ σ WT σ WR del (ε R ) Hypothesis Saple size Proportion σ Mean of ˆ March 011 Journal of the National Science Foundation of Sri Lanka 39 (1)
7 Saple size for crossover trials 83 Table : Proportion of rejections of the null hypotheses under the null and the alternative hypotheses for the large saple approxiation No σ BT σ BR ρ σ WT σ WR del (ε R ) Hypothesis Saple size Proportion σ Mean of ˆ Journal of the National Science Foundation of Sri Lanka 39 (1) March 011
8 84 N. M. Siyasinghe & M.R. Sooriyarachchi Table continued No σ BT σ BR ρ σ WT σ WR del (ε R ) Hypothesis Saple size Proportion σ Mean of ˆ March 011 Journal of the National Science Foundation of Sri Lanka 39 (1)
9 Saple size for crossover trials 85 patient correlation coefficient. The higher the correlation between patients, the higher the gain in saple size. Also it can be observed that, the within patient variance (σ WT ) has less effect than the between patient variance (σ BT ) on the calculation of saple size because the latter is usually uch greater than the forer. When the reference iproveent increases the saple size becoes saller, because the difference we want to detect is larger. Also the gain in saple size due to increasing correlation is higher for saller reference iproveent. Figure is drawn in order to see whether the significance level is aintained by the test for the exact approach. It shows a graph of the proportion of rejections of the null hypothesis, when the null hypothesis is true versus del for different cobinations of ρ, σ BT for the exact approach. Two coloured lines represent the band within which the proportion should lie, in order to aintain the significance level. The corresponding saple size is shown near the points, which are out of the bands. Siilar results as shown by Table 1 are illustrated here. Figure 3 is drawn in order to see how well the power is aintained by the test for the exact approach. It gives a graph of the proportion of rejections of the null hypothesis, when the alternative hypothesis is true versus del for different cobinations of ρ, σ BT for the exact approach. When considering Figure 3 it can be seen that ost of the saple sizes which lie outside the bands are very sall nubers except for 1, which is very close to the upper liit. When the saple size is five or less, any points lie outside the band. The reason is the iprecision of the approxiation used in calculating the inverse t distribution values when the saple size is less than or equal to five. When ρ is very high ( 0.9) and the reference iproveent is large ( 3 ), there is a higher tendency in obtaining a sall saple size, hence a higher nuber of points can be observed outside the bands in those cobinations. Fro Figures and 3, it can be seen that siilar results are obtained as per the table for the exact approach. Figures 4 and 5 are the corresponding graphs to Figures 1 and, for the large saple approach respectively. The conclusions drawn fro Figure 4 are sae as those drawn fro Figure 1. Figure 5 illustrates siilar results as given in Table. Figure 6 is the corresponding graph to Figure 3, for the large saple approach and illustrates siilar results as in Table. When ρ is very high (0.9) and the reference iproveent is large (3), there is a higher tendency in obtaining a sall saple size. CONCLUSION This study deals with the saple size calculation of crossover trials under two situations in the power approach, naely the exact and large saple ethods (Chow et al. 003). Fro the results of the siulation study the following guidelines can be given. It was seen that when the saple size is very sall (less than or equal to five), neither ethod aintains error rates. i.e. even the exact ethod which was based on the t distribution failed for very sall saple sizes. This is because the approxiation used to calculate the inverse of the t distribution which is described in Cooke et al. (198), in deterining the saple size, is not accurate for very sall saple sizes. Also it is evident that when the saple size is fairly large in ters of crossover studies (5 < saple size < 1), only the exact approach has aintained error rates. This is because the noral ordinate is an underestiate of the t ordinate for saple sizes within that range. That eans within the specified saple liits the exact approach should be used. When the saple size is large in ters of crossover studies ( 1) both ethods have aintained error rates. i.e. for the saple sizes greater than about eleven, the large saple approach, which is uch sipler than the exact approach, can be used for saple size deterination instead of the exact approach, which needs nuerical ethods. A higher reduction in saple size can be achieved when the within patient correlation is high. A better approxiation for the inverse distribution of the t distribution should be found when calculating saple sizes, which are believed to be very sall. The study is done for x crossover trials, which consider only two treatents and two periods. As an extension one can consider ore treatents and periods. Also in this study there are no replications of treatents, i.e. a treatent is given to a set of patients (subjects) only once. One can extend this study to have replications. An assuption used in this study is that of equality of within and between patient variances for both treated and reference groups (i.e. σ BT = σ BR = σ WR ) and equal allocation of patients to both groups. Thus further investigation can be carried out taking different values for these paraeters. Journal of the National Science Foundation of Sri Lanka 39 (1) March 011
10 86 N. M. Siyasinghe & M.R. Sooriyarachchi Saple size vs vs delta for different cobinations of of rho, rho, sigabt sigabt and and sigawt sigawt Saple Salpe size Delta Figure 1: Graph of saple size versus delta for different cobinations of ρ, σ BT for the exact approach Proportion of of rejections of of the the null hypothesis when tha the null hypothesis is is true true vs delta vs delta for different for different cobinations cobinations of of rho, rho, sigabt, and sigawt sigawt 4 Significance level Level 4 Delta Figure : Graph of proportion of rejections of the null hypothesis, when the null hypothesis is true versus delta for different cobinations of ρ, σ BT for the exact approach March 011 Journal of the National Science Foundation of Sri Lanka 39 (1)
11 Saple size for crossover trials 87 Proportion of rejections of of the the null null hypothesis hypothesis when when tha alternative the null hypothesis is is true true vs delta vs delta for different for different cobinations of of rho, sigabt, sigawt and sigawt Power Power Delta Figure 3: Graph of proportion of rejections of the null hypothesis, when the alternative hypothesis is true versus delta for different cobinations of ρ, σ BT for the exact approach Saple Saple size vs size delta vs delta for for different cobinations of rho, sigabt, sigawt and sigawt Saple size size Delta Figure 4: Graph of saple size versus delta for different cobinations of ρ, σ BT for the large saple approach Journal of the National Science Foundation of Sri Lanka 39 (1) March 011
12 88 N. M. Siyasinghe & M.R. Sooriyarachchi Proportion of of rejections of the null hypothesis when when tha the null null hypothesis hypothesis true is true vs vs delta delta for for different cobinations of of rho, rho, sigabt sigabt and and sigawt sigawt Significance level Delta Figure 5 : Graph of proportion of rejections of the null hypothesis, when the null hypothesis is true versus delta for different cobinations of ρ, σ BT for the large saple approach Proportion of of rejections rejections of the of the null null hypothesis hypothesis when when the alternative the null hypothesis is is true vs vs delta delta, for for different different cobinations of of rho, rho, sigabt sigabt and and sigawt sigawt Power Delta Figure 6: Graph of proportion of rejections of the null hypothesis, when the alternative hypothesis is true versus delta for different cobinations of ρ, σ BT for the large saple approach March 011 Journal of the National Science Foundation of Sri Lanka 39 (1)
13 Saple size for crossover trials 89 In the odel specified, it was assued that there is no treatent by period interaction (that is, the effect of the treatent reains consistent over the two periods). If such an interaction is present, a siple t test cannot be used in testing the hypothesis and a odelling approach will have to be used (Jones & Kenward, 003). This requires a new investigation to be carried out. References 1. AlSubaihi A.A. (004). Siulating correlated ultivariate pseudorando nubers. Journal of Statistical Software 9(i04) Brown H. & Prescott R. (006). Applied Mixed Models in Medicine. John Wiley & Sons, Inc., Chichester, UK. 3. Chow S.C., Shao J. & Wang H. (003). Saple Size Calculations in Clinical Research. Chapan and Hall/ CRC Press, New York, USA. 4. Cooke D., Craven A.H. & Clarke G.M. (198). Basic Statistical Coputing. Edward Arnold Press, London, UK. 5. Golder E.R. & Settle J.G. (1976). The BoxMuller ethod for generating pseudorando noral deviates. Applied Statistics 5(1): Grizzle J.E. (1965). The twoperiod changeover design and its use in clinical trials. Bioetrics 1(): Jones B. & Kenward M. (003). The Analysis of Crossover Trials. Chapan and Hall / CRC Press, New York, USA. 8. Julious S.A. & Patterson S.D. (004). Saple sizes for estiation in clinical trials. Pharaceutical Statistics 3(3): Lee C., Lee L.H., Christoper L.W., Chen M. & Benjain R. (005). Clinical Trials of Drugs and Biopharaceuticals. Chapan and Hall /CRC Press, New York, USA. 10. Pocock S.J. (1983). Clinical Trials: A Practical Approach. John Wiley & Sons, Inc., Chichester, UK. 11. Press W.H., Teukolsky S.A., Vetterling W.T. & Flannery B.P. (00). Nuerical Recipes in C++, pp Cabridge University Press, Cabridge, UK. 1. Sooriyarachchi M.R. & Whitehead J. (1998). A ethod for sequential analysis of survival data with nonproportional hazards. Bioetrics 54(3): Whitehead A., Sooriyarachchi M.R., Whitehead J. & Bolland K. (008). Incorporating interediate binary responses into interi analyses of clinical trials: a coparison of four ethods. Statistics in Medicine 7(10): Woodward M. (199). Forulae for saple size, power and iniu detectable risk in edical studies. The Statistician 41(): Journal of the National Science Foundation of Sri Lanka 39 (1) March 011
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