Analysis of two-factor unreplicated experiments with one factor random

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1 Bostatstcs (2013), 0, 0, pp do: /bostatstcs/ms V4 Analyss of two-factor unreplcated experments wth one factor random Ka Wang Department of Bostatstcs, College of Publc Health, Unversty of Iowa, Iowa Cty, IA 52242, USA ka-wang@uowa.edu Summary Ths work concerns two-factor unreplcated experments n whch one factor s random. The model s presented n a columns-lnear form. After applyng dentfablty condtons, model parameters are estmated by the maxmum lkelhood and the restrcted/resdual maxmum lkelhood. For ether lkelhood, lkelhood rato tests and score tests for nonaddtvty of the two factors as well as the noncentralty parameters are presented. The performance of these tests s compared to Mandel s test (whch s desgned for the same null) usng smulaton studes. Applcaton of these tests are llustrated by an example. Key words: two-way ANOVA; addtvty; random effect; Tukey test; Mandel test 1. Introducton Research n studes of addtvty n two-factor experments that have no replcaton has a long hstory. The most famous one s Tukey s test of addtvty (Tukey, 1949). Ths 1-df test s desgned for a specfc non-addtvty structure. A more general test s the Mendel s test (Mandel, 1961). Recent developments nclude Mandel (1971), Johnson and Graybll (1972), Bok (1993), Tusell (1990), and Franck and others (2013). A recent revew of ths research on ths topc s presented n Aln and Kurt (2006). We focus on a non-addtvty structure that s columns-lnear. Ths structure s equvalent to the rows-lnear structure consdered by Mandel (1961). In partcular, we allow the effect of the row factor to be random. For nstance, n studes of comparng dfferent methods of measurement, the effect of subjects can be regarded as random. However, tests for mxed effects n ths context seem to be non-exstant. Rasch and others (2009) smply appled the tests desgned for fxed effects and studed ther performance n mxed effects settng. Ths research s motvated by studes comparng methods of measurement. So another focus s parameter estmaton. The estmated parameters are useful n calbratng dfferent methods. Ths s n contrast to exstng studes on testng non-addtvty. Ths paper s organzed as follows. We frst defne the model wth specal attenton gven to Ka Wang, Department of Bostatstcs, N322 CPHB, College of Publc Health, Unversty of Iowa, Iowa Cty, IA 52242, USA. E-mal: ka-wang@uowa.edu, phone: (319) , fax: (319) c The Author Publshed by Oxford Unversty Press. All rghts reserved. For permssons, please e-mal: journals.permssons@oup.com

2 2 K. Wang dentfablty of model parameters. Model parameters are estmated by the maxmum lkelhood and then by the restrcted maxmum lkelhood. Lkelhood rato tests and score tests are ntroduced for testng non-addtvty. These tests and the Mandel s test are compared n smulaton studes and an emprcal study. 2. Methods Consder a two-way unreplcated experment wth factor A and factor B. There are n levels for factor A and m levels for factor B. The response y j correspondng to level of A and level j of level B s assumed to follow the followng model y j = α j + β j u + e j, e j N(0, σ 2 ), = 1,..., n, j = 1,..., m, (2.1) where α j, β j, u, and σ 2 are model parameters. In ths notaton, s the row ndex and j the column ndex. Ths model assumes the response s lnear n each column wth column-specfc ntercept α j and slope β j. It s equvalent to the rows-lnear model consdered n Mandel (1961). However, t appears convenent to lay out data ths way n studes comparng methods of measurements: the columns correspond to dfferent methods whle the rows correspond to subjects. Model (2.1) s popular n studes comparng methods of measurements (Carstensen, 2010). It extends the tradtonal Bland-Altman method (Bland and Altman, 1986). Let u = n 1 u. Equatons n (2.1) can be wrtten y j = (α j + β j u ) + β j (u u ) + e j. That s, wthout loss of generalty, t can be assumed that u = 0. α j represents the mean response of column j on an average subject whose value s u. Furthermore, the term β j u remans the same when β j s multpled by a factor whle u s dvded by the same factor and so does when β j and u swtch ther sgns. So β j s normalzed such that u2 = 1 and β 1 0. Overall, the total number of parameters n (2.1) s 2m + n 1. Takng each row of matrx {y j } as a response from a subject, t s natural to treat the row effect as random. From now on t s assumed that u follows a normal dstrbuton wth mean 0 and varance 1: u N(0, 1). The varance 1 reflects the scale normalzaton on u. Let y be a column vector consstng of the th row of matrx {y j } and α = α 1 α 2.. α m, β = Wth these vector notatons, model (2.1) assumes the followng more concse form: β 1 β 2. β m. y u MV N(α + u β, σ 2 I), u N(0, 1). Ths s a specal case of mxed effects model. It s specal because the varance of the random effect u s fxed at 1. It can be shown that ths model s equvalent to (detals omtted) The null hypothess of nterest s y MV N(α, Σ), where Σ = ββ t + σ 2 I. (2.2) H 0 : β 1 = β 2 = = β j = β

3 Nonaddtvty test for two-way ANOVA 3 for a common β. Under ths hypothess, the effect of the row factor and that of the column factor are addtve. There s no nteracton between these two factors. The alternatve s that H 0 does not hold. 2.1 MLE of model parameters The log-lkelhood functon for model (2.2) s l(α, β, σ 2 ) = nm 2 log(2π) n 2 log Σ 1 2 tr Σ 1 (y α)(y α) t ]. Here tr(a) means the sum of the dagonal elements of matrx A,.e., the trace of A. The frstorder dervatves are (Appendx) l α = Σ 1 (y α), l β = nσ 1 β n ] 1 (y α)(y α) t Σ 1 β, ( l σ 2 = n tr(σ 1 ) tr Σ 2 n )] 1 (y α)(y α) t. 2 Defne S = n 1 (y y )(y y ) t and y = n 1 y. Settng each of the three frst-order dervatves to 0 and solvng them smultaneously for α, β, and σ 2, we obtan the followng relatonshps ther maxmum lkelhood estmates (MLE) obey (Appendx): ˆα = y ˆβ = (ˆβ t ˆβ + ˆσ 2 ) 1 Sˆβ, ˆσ 2 = m 1 (tr(s) ˆβ t ˆβ). There s an explct soluton for ˆα but not for ˆσ 2 and ˆβ. However, the last two expressons can be used teratvely to fnd approxmate solutons for ˆσ 2 and ˆβ. Under the null hypothess H 0, the MLEs of β and σ 2 are drectly avalable. The MLE of β s equal to the square root of ˆβ 2, where ˆβ 2 = 1t S1 tr(s) m(m 1) s the average of the off-dagonal elements of S. The MLE of σ 2 s equal to ˆσ 2 0 = tr(s) m ˆβ 2, whch s the dfference between the averages of the dagonal elements of S and ts off-dagonal elements. These results are easly nterpretable.

4 4 K. Wang Havng the MLEs of the model parameters under both the null and the alternatve, the lkelhood rato test can be formed: Λ = 2log l( ˆα, ˆβ, ˆσ 2 ) log l( ˆα, ˆβ1, ˆσ 2 0)]. Note that the MLE of α s the same under the null as under the alternatve. When H 0 holds, Λ asymptotcally follows a ch-square dstrbuton wth degrees of freedom equal to m 1. The Fsher nformaton matrx s block-dagonal (Appendx): ( ) nσ 1 0 F =, (2.3) 0 F 1 where ( (β t Σ 1 β)σ 1 + Σ 1 ββ t Σ 1 Σ 2 β F 1 = n β t Σ 2 0.5tr(Σ 2 ) Accordng to standard asymptotc theory, a score statstc s defned by T = ( l α t, l β t, l σ 2)F 1 ( l α t, l β t, l σ 2) t It s easy to verfy that l α = 0, therefore ). evaluated at the null. T = ( l β t, l σ 2)F 1 1 ( l β t, l σ 2) t evaluated at the null. It s also straghtforward to verfy that l σ 2 = 0 under the null. Usng block-matrx nverse formula, ] 1 T = n 1 lβ t (β t Σ 1 β)σ 1 + Σ 1 ββ t Σ 1 2 tr(σ 2 ) Σ 2 ββ t Σ 2 lβ. Snce Σ 1 β = 1 ) 1 (I σ 2 β t β + σ 2 ββt β 1 = β t β + σ β, 2 each of the last two terms between the brackets s proportonal to ββ t. Usng the Sherman- Morrson formula, T = n 1 lβ tσ l β (β t Σ 1 β) + terms dependng on (βt lβ ) 2. The terms dependng on (β t lβ ) 2 are 0 because β = β1 under the H 0 and β t lβ 1 t (1 ˆΣ 1 S1) = 0. Pluggng n the MLE estmates of β and σ 2 0 under H 0 and smplfyng (detals omtted), the score statstc for H 0 s T = n 1 βt β + σ 2 l β t β tσ l β β n = ˆσ 0 2 1t S1 1t S 2 1 m 1 (1 t S1) 2 ]

5 Nonaddtvty test for two-way ANOVA 5 Snce 1 t S 2 1 = (S1) t (S1), statstc T depends on the varaton of n the row sums of S. If H 0 holds, no varaton s expected. The non-centralty parameter (NCP) of T s lm n n 1 T = 1t Σ 2 1 m 1 (1 t Σ1) 2 σ 2 (1 t Σ1) = m 2 Ave(β)] 2 σ 2 (mave(β)] 2 + σ 2 ) D(β), where Ave(β) = m 1 j β j s the average of the β j s and D(β) = m 1 j β 2 j Ave(β)] 2 measures the dvergence n β j s. The power of T at sgnfcance level α s Pr(X > χ 2 1 α,m 1) where X follows a ch-square dstrbuton wth df = m 1 and non-centralty parameter NCP and χ 2 1 α,m 1 s the crtcal value from a ch-square dstrbuton wth df = m 1 and non-centralty parameter 0. Snce the lkelhood rato statstc Λ s asymptotcally equvalent to the score statstc T, they share the same NCP. 2.2 Restrcted maxmum lkelhood estmaton The maxmum lkelhood method s known to generate based estmate of varance components such as σ 2. A restrcted maxmum lkelhood s useful n reducng the bas. We consder n m lnear combnatons of the responses y j such that the dstrbuton of these lnear combnatons are free of the parameters n the mean structure (.e., the α parameter). Defne y = y 1 y 2. y n 1 (n 1)m 1 where y = y y n, = 1, 2,..., n 1. The dstrbuton of y s multvarate normal wth mean vector 0 and covarance matrx Ω equal to 2Σ Σ... Σ Σ 2Σ... Σ Ω =.... Σ Σ... 2Σ = (I + 11 t ) (n 1) (n 1) Σ m m.

6 6 K. Wang Here denotes the Kronecker product of two matrces. In ths formulaton, subject n s used as the reference. It turns out that the choce of the reference subject does not affect the restrcted lkelhood. The lkelhood for y s the restrcted lkelhood for the untransformed, or the orgnal, data. The logarthm of ths lkelhood s Snce l (β, σ 2 (n 1)m ) = log(2π) log Ω 1 2 tr Ω 1 y (y ) t]. Ω = I + 11 t m Σ n 1 = n m Σ n 1, Ω 1 = (I + 11 t ) 1 Σ 1 = (I n 1 11 t ) Σ 1, and tr ( n 1 Ω 1 y (y ) t] n 1 = tr Σ 1 y (y ) t n 1 = tr =1 ( n Σ 1 y (y ) t n 1 =1 =1 y =1 =1 n n y =1 =1 )] n 1 (y ) t (y ) t )] =1 (defne y n = y n y n = 0) ( n n )] n = tr Σ 1 (y y n )(y y n ) t n 1 (y y n ) (y y n ) t = tr ( n Σ 1 y y t n 1 =1 = ntr ( Σ 1 S ), n =1 the lkelhood functon l (β, σ 2 ) can be wrtten y =1 y t )] l (β, σ 2 (n 1)m ) = log(2π) m 2 2 log(n) n 1 2 =1 log Σ n 1 tr ( Σ 1 S ) 2 wth S = n n 1 S. Ths lkelhood does not depend on α. Smlar to lkelhood l(α, β, σ2 ), the restrcted maxmum lkelhood estmates (REMLEs) of β and Σ satsfy the followng relatonshp: β = ( β t β + σ 2 ) 1 S β, σ 2 = m 1 (tr(s ) β t β). Under H 0, the REMLE of β and σ 2 can be solved explctly: β 2 = 1t S 1 tr(s ), m(m 1) σ 2 = tr(s ) m β 2. These REMLEs relate to those from l(α, β, σ 2 ) n the followng way: n β = n 1 ˆβ, σ 2 = n n n 1 σ2, β = n 1 ˆβ, σ 0 2 = n n 1 σ2 0. (2.4)

7 Nonaddtvty test for two-way ANOVA 7 The lkelhood rato statstc s Λ = (n 1) max β,σ 2 ( log Σ tr(σ 1 S ) max β,σ 2 ( log Σ tr(σ 1 S )]. It s straghtforward to show that It s also straghtforward to verfy that Λ = n 1 n Λ. T = n 1 n T. 2.3 A Predcton problem A predcton problem n studes of methods of measurement s how to predct the measurement of method j, denoted by y j, gven the measurements of the others, denoted by y j, on a subject. Let u be the unknown true value of the subject. The jont dstrbuton of (y t j, u)t s ( y j u ) N (( α j 0 ) ( Σ 1 j, j, β t j 1 β j )). Here a subscrpt j means the jth row (or the jth column for Σ 1 ) s removed. We have E(u y j ) = β t jσ 1 j, j (y j α j ) = βt j(y j α j ) β t jβ j + σ 2, V ar(u y j ) = 1 β t jσ 1 j, j β j σ 2 = β t jβ j + σ. 2 A predcton of the measurement by method j s y p j = α j + β j E(u y j ) ( = α j β j β t jα j β t jβ j + σ 2 Ths s a lnear functon n y j. The predcton varance s ) + V ar(y p j ) = V ar(u y j) + σ 2 = σ 2 + β j β t jβ j + σ 2 βt jy j. (2.5) σ 2 β t jβ j + σ 2. (2.6) A 95% predcton lmts for y j s y p j ± 1.96 V ar(y p j ). In ths calculaton, the unknown parameters α, β, and σ 2 are to be substtuted by ther respectve MLEs or REMLEs.

8 8 K. Wang Table 1. Smulated rejecton rate under the null and the alternatve. The frst two values for β correspond to the null and the the last two values correspond to the alternatve. Nomnal Statstc β Level Λ Λ T T Mendel (1, 1) t (1, 1, 1, 1) t (1.1, 1.2, 1.3, 1.4) t (1.1, 1.2, 1.3, 1.4, 1.5, 1.6) t Smulaton Studes The goal of ths smulaton study s to nvestgate the performance of the lkelhood rato statstcs and score statstcs as well as the Mandel s test (Mandel, 1961). The Mandel s test statstc s j Mandel = (b j 1) 2 (y y ) 2 j (y (n 2), (3.7) j y j ) b j (y y )] 2 where y = m 1 y j, y j = n 1 y j, y = (nm) 1,j y j, and b j = y j(y y ) (y y ) 2. Under H 0, Mandel follows an F dstrbuton wth m 1 and (m 1)(n 2) degrees of freedom. Data were generated from model (2.2) wth α = 0 and σ 2 = 1. For the study of type I error rate, two stuatons wth β = (1, 1) t and β = (1, 1, 1, 1) t, respectvely, are consdered. Another two β vectors were consdered for the study of power: β = (1.1, 1.2, 1.3, 1.4) t and β = (1.1, 1.2, 1.3, 1.4, 1.5, 1.6) t. The number of subjects n s fxed at 100. The smulated rejecton rates over 1000 replcates are reported n table 1. All the tests have satsfactory type I error rate. Ther power are also smlar to sach other. 4. An Example The data frame named hba1c n the R package MethComp contans measurements of HbA1c (glycosylated haemoglobn) on blood samples from 38 ndvduals. A venous blood sample and a capllary blood sample were obtaned from each ndvdual. Three analyzers were then used to determne the level of HbA1c n each sample. So the total number of methods s m = 2 3 = 6. Each sample was analyzed on fve dfferent days. Ths data were analyzed n Carstensen (2004).

9 Nonaddtvty test for two-way ANOVA 9 Fg. 1. Plot of the HbA1c measurement averaged over 5 dfferent sample analyss days. One lne corresponds to one measurement method y Subject ID

10 10 K. Wang Table 2. MLEs of the model parameter for the HbA1c data. Each method conssts of two parts: analyzer (BR.V2, BR.VC, or Tosoh) and type of blood sample (Cap or Ven). REMLEs can be obtaned easly by usng formulae n (2.4). Method Parameter BR.V2.Cap BR.VC.Cap Tosoh.Cap BR.V2.Ven BR.VC.Ven Tosoh.Ven α β σ Table 3. P -value of varous tests for H 0 : β 1 = β 2 = β 3 = β 4 = β 5 = β 6. Λ Λ T T Mandel In our analyss, measurements from each method over these 5 days are averaged so there s no replcated measurement per method per ndvdual. The data s graphcally presented n fgure 1. The MLEs of the model parameters are presented n table 2. There s no sgnfcant dfference among the 6 β coeffcents (table 3). Based on the results n table 2, predcton equatons for each method gven measurements from all others can be computed usng equatons (2.5) and (2.6). These predcton equatons are shown n table 4. For nstance, the predcton equaton for BR.V2.Cap s BR.V2.Cap = BR.VC.Cap Tosoh.Cap BR.V2.Ven BR.VC.Ven Tosoh.Ven. A salent feature of these predcton equatons s that they use all other methods smultaneously. In contrast, the equatons reported n Carstensen (2004) are for par-wse predcton only one predctng method s used n each equaton. The predcton standard error s about 10 tmes as large as reported here. 5. Dscusson We have presented analyss methods for two-factor unreplcated experments where one factor s random. As ths research s motvated by studes of comparng methods of measurement, ts foc nclude parameter estmaton, tests of addtvty, and predcton of one method gven measurements of other methods. In ths context, treatng one factor (.e., subject) as random s more reasonable than treatng t as fxed effect. In addton, t reduces the number of nusance parameters and s expected to result n more powerful tests (although the smulated power of the proposed tests are smlar to that of the Mandel s test). The programmng of the method proposed n ths work s straghtforward. The author s R code used for the analyss n ths paper s avalable upon request.

11 REFERENCES 11 Table 4. Predctng Predcted method Method BR.V2.Cap BR.VC.Cap Tosoh.Cap BR.V2.Ven BR.VC.Ven Tosoh.Ven Intercept BR.V2.Cap BR.VC.Cap Tosoh.Cap BR.V2.Ven BR.VC.Ven Tosoh.Ven Predcton Standard Error References Aln, Ayln and Kurt, S. (2006). Testng non-addtvty (nteracton) n two-way anova tables wth no replcaton. Statstcal Methods n Medcal Research 15, Bland, J. M. and Altman, D. G. (1986). Statstcal methods for assessng agreement between two methods of clncal measurement. Lancet, Bok, Robert J. (1993). Testng addtvty n two-way classfcatons wth no replcatons:the locally best nvarant test. Journal of Appled Statstcs 20(1), Carstensen, Bendx. (2004). Comparng and predctng between several methods of measurement. Bostatstcs 5(3), Carstensen, Bendx. (2010). Comparng methods of measurement: Extendng the LoA by regresson. Stat Med 29(3), Franck, Chrstopher T., Nelsen, Dahla M. and Osborne, Jason A. (2013). A method for detectng hdden addtvty n two-factor unreplcated experments. Computatonal Statstcs and Data Analyss 67, Johnson, Dallas E. and Graybll, Frankln A. (1972). An analyss of a two-way model wth nteracton and no replcaton. Journal of The Amercan Statstcal Assocaton 67(340), Mandel, John. (1961). Non-addtvty n two-way analyss of varance. Journal of the Amercan Statstcal Assocaton 56(296), Mandel, John. (1971). A new analyss of varance model for non-addtve data. Technometrcs 13(1), Rasch, Deter, Rusch, Thomas, Šmečková, Mare, Kubnger, Klaus D., Moder, Karl and Šmeček, Petr. (2009). Tests of addtvty n mxed and fxed effect two-way anova models wth sngle sub-class numbers. Stat Papers 50,

12 12 REFERENCES Tukey, John. (1949). One degree of freedom for non-addtvty. Bometrcs 5(3), Tusell, Fernando. (1990). Testng for nteracton n two-way anova tables wth no replcaton. Computatonal Statstcs and Data Analyss 10(1),

13 Appendx Because of the general relatonshps that and REFERENCES 13 Dervatons related to maxmum lkelhood estmates log Σ θ Σ 1 ( ) = tr Σ 1 Σ θ = Σ 1 Σ θ θ Σ 1, t s straghtforward to compute the frst-order dervatves. l α = Σ 1 l βm = n 2 (y α), (5.8) log Σ β m 1 2 tr Σ 1 = n 2 tr(σ 1 Σ β m ) tr β m ] (y α)(y α) t Σ 1 Σ β m Σ 1 (y α)(y α) t ] = n 2 tr(σ 1 (β1 t m + 1 m β t )) tr Σ 1 (β1 t m + 1 m β t )Σ ] 1 (y α)(y α) t = n 2 tr(σ 1 (β1 t m + 1 m β t )) + 1 (y α) t Σ 1 (β1 t m + 1 m β t )Σ 1 (y α) 2 = 1 t m nσ 1 β + Σ 1 (y α)(y α) t Σ 1 β], (5.9) l β = nσ 1 β + Σ 1 n ] 1 (y α)(y α) t Σ 1 β, ] l σ 2 = n log Σ 2 σ tr Σ 1 σ 2 (y α)(y α) t = n Σ 2 tr(σ 1 σ 2 ) tr Σ 1 Σ ] σ 2 Σ 1 (y α)(y α) t = n 2 tr(σ 1 ) + n 2 tr Σ 2 n ] 1 (y α)(y α) t. (5.10) From (5.8), the α s equal to ˆα = y. Substtutng ˆα nto (5.9) and (5.10), whch s equvalent to replacng n 1 (y 1 α)(y α) t by S, β = SΣ 1 β 1 = β t Sβ, (5.11) β + σ2 tr(σ 1 ) = tr(σ 2 S). (5.12)

14 14 REFERENCES From (5.11), β t Sβ = β t β(β t β + σ 2 ) (5.13) we have from (5.12) and (5.13) σ 2 = tr(s) βt β. (5.14) m The Fsher nformaton s computed usng standard formulae for matrx expectaton. E( l α lα t) = nσ 1, E( l α lβ t) = E(Σ 1 (y α)] nβ t Σ 1 + β t Σ 1 (y α)(y α) t Σ 1 ] = nσ 1 E(y α)β t Σ 1 (y α)(y α) t ]Σ 1 = 0, E( l α lσ 2) = n 2 EΣ 1 (y α) (y α) t Σ 1 Σ 1 (y α)] = 0, E( l β lβ t) = E( nσ 1 + Σ 1 (y α)(y α) t Σ 1 )ββ t ( nσ 1 + Σ 1 (y α)(y α) t Σ 1 ) = nσ 1 ββ t Σ 1 + nσ 1 E(y α)(y α) t Σ 1 ββ t Σ 1 (y α)(y α) t ]Σ 1 = nσ 1 ββ t Σ 1 + n(β t Σ 1 β)σ 1 E( l β lσ 2) = 1 2 E ntr(σ 1 ) + (y α) t Σ 1 Σ 1 (y α)] nσ 1 + Σ 1 (y α)(y α) t Σ 1 ]β = 1 2 E ntr(σ 1 ) + (y α) t Σ 1 Σ 1 (y α)] Σ 1 (y α)(y α) t Σ 1 ]β = 1 2 E n2 tr(σ 1 )Σ 1 + n(n 1)tr(Σ 1 )Σ 1 + n(y α) t Σ 2 (y α)σ 1 (y α)(y α) t Σ 1 ]β = n 2 E tr(σ 1 ) + (y α) t Σ 2 (y α)σ 1 (y α)(y α) t ]Σ 1 β = n 2 tr(σ 1 )Σ 1 β + n 2 Σ 1 E(y α)(y α) t Σ 2 (y α)(y α) t ]Σ 1 β = n 2 tr(σ 1 )Σ 1 β + n 2 Σ 1 2I + tr(σ 1 )Σ]Σ 1 β = nσ 2 β E( l σ 2) 2 = 1 4 ( n2 tr(σ 1 )] 2 + n(n 1)tr(Σ 1 )] 2 + ne(y α) t Σ 2 (y α)] 2 ) = 1 4 ( ntr(σ 1 )] 2 + ne(y α) t Σ 2 (y α)] 2 ) = n 2 tr(σ 2 ) Receved August 1, 2010; revsed October 1, 2010; accepted for publcaton November 1, 2010 ]