rtL_~ . ..,.·

The Analysis of Cross-Over Study with Repeated Measures within Periods by Using Different Covariance Structures

Sanna Hinkka1 and Juhani Tuominen2

1 Department of Biometrics, Farmos Research, Orion-Farmos, Turku, FINLAND 2 Department of Biostatistics, University of Turku, Turku, FINLAND

1. INTRODUCTION

Cross-over designs are used frequently in clinical trials to compare the efficacy of sequential treatments. We are dealing with a two-period and four-treatment cross-over study with repeated measures within periods when missing values are present. Hence, it is carried out as a balanced incomplete block design (BmD). The mixed model techniques are applied in order to analyse the effects of interest. We prefer restricted maximum likelihood (REML) estimation method to ordinary maximum likelihood because of its ability to reduce the problem of biased covariance estimates. Our main aim is to demonstrate how the estimation of the standard errors in the treatment effects can be improved by taking full account of the covariance structure of response variable.

In the next chapter we will represent the general mixed model with spesific attention to the covariance matrices. In chapter three, the general form of the analysis of variance will be displayed and illustrated by a study comparing four formulations of lithium carbonate (Westlake3, 1974). PROC MIXED in SAS2 is used in the analyses. The presentation of the results is organised as follows: Firstly, we compare REML estimates and their standard errors when subject effect is treated as fixed or random. Secondly, because REML estimation method allows the use of different covariance structures for covariance matrix in order to find the most suitable covariance structure for the data, we show how these structures can be fitted to the data when estimating the fixed effects of the mixed model.

2. MIXEU·MODEL

W. J. Westlake presents a study in his article3 where four formulations of lithium carbonate were compared in a BmD in which twelve subjects each received two of four formulations. Blood samples were drawn six times (2, 3, 5, 6, 9 and 12 hours) after administration.

For measured, multivariate normal data, we present the following mixed model for each individual unit i,

Yi =Xia+ZJ3i +ej

where a is a vector of fixed effects including parameters for period,treatment, time, time-by­

treatment and time:...by-period interaction,

Xi is a known nixp design matrix linking a to Yi,

Pi ,.., N(O,D) and D is a kxk positive-definite covariance matrix, the vector J3i of random effects consists of subject, time-by-subject and period-by-treatment-by-subject effect,

1301

Zi is a known nixk design matrix linking ~i to Yi and ej ,..., N(O,Rj ) and Ri is an nixni positive-definite covariance matrix.

The statistical model used by Westlake assumes that the subject and the time-by-subject effect are fixed.

The model equation for the mixed model covariance matrix for each individual unit i is

I:j = ZiDZ; +Ri.

3. ILLUSTRATION

In the analyses we use the log-transformed data throughout the illustration. Five observations were randomly omitted from the analyses in order to make the data incomplete. The mean treatment profiles are plotted in Figure 1.

In Table 1 we compare REML estimates when subject effect is fixed (Westlake's model) with corresponding effects when subject and subject-by-time interaction effects are random and have the usual independence and normal distribution assumptions. All the analyses are carried out by PROC MIXED in SAS2.

Table 1. REML estimates of variance components and treatment effects (standard errors of estimates appear in brackets).

Effect Fixed subject effect Random subject effect Subject Period*treatment*subject Time * subject

0.021034 (0.011245) 0.003227 (0.007425) 0.018500 (0.009368) 0.001368 (0.001532)

Treatment Avs.B Avs. C Avs.D Bvs. C B vs. D Cvs.D

-0.074916 (0.106050) 0.610049 (0.106097)

-0.001501 (0.106066) 0.684965 (0.106256) 0.073416 (0.106136)

-0.611550 (0.106746)

-0.043590 (0.085205) 0.658777 (0.085296) 0.033305 (0.085082) 0.702366 (0.085422) 0.076895 (0.085192)

-0.625471 (0.085365)

We have fitted the following covariance structures for Ri: • Compound symmetry (CS), i.e. Rij = a~ + a 21 (i = j),

1 (i = j) equals 1 when i = j and 0 otherwise, •

•

•

Toeplitz with 2 diagonal bands, (TOEP(2», i.e. Rij = ~i-jl+ll (Ii - jl < 2), l(li - jl < 2) equals 1 when Ii - jl < 2 and 0 otherwise,

Spatial structure (SP(POW», i.e. Rij = a 2pd;j when dij is the Euclidean distance between the coordinates i and j and

First-order autoregressive (AR(l», i.e. Rij = a2pli-jl .

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Table 2. Summary of models fit.

Model Covariance structure Number of covariance REML log likelihood

CS Earameters

1 5 48.1069 2 TOEP(2) 5 56.4375 3 SP(POW) 5 60.2539 4 AR(I) 5 62.5748

The first-order autoregressive covariance structure appears to fit the data best because its REML log likelihood is the highest. No goodness-of-fit test is provided because the number of covariance parameters of all fitted structures is the same. In Table 3 we have the estimates of the effects in the final model with AR( 1) covariance structure.

Table 3. REML estimates of the fmal model.

Effect Subject . Period*treatment*subject Tirne*subject Treatment

Avs.B Avs.C Avs.D Bvs.C

. Bvs.D Cvs.D

Estimate 0.004783 o 0.001396

-0.043732 0.654708 0.033027 0.698439 0.076759 -0.621681

Standard error of estimate 0.007631

0.000807

0.086752 0.086782 0.086677 0.086842 0.086731 0.086878

f REFERENCES ~ 1. Brown, H. K. and Kempton, R. A. 'The application ofREML in clinical trials', Statistics in ~ Medicine, 13, 1601 - 1617 (1994). I 2. SAS Institute Inc., SAS Campus Drive, Cary, NC 27513;. U.S.A. Ii 3. Westlake, W. 1. 'The use of balanced incomplete block designs in comparative bioavailability ~ trials', Biometrics, 30,319 - 327 (1974).

t ~ ~;

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1303

i ~.

\;.

SAS Code for the final model

PROC IUXED; CLASS period treatmntti.e subject; HODEL logserum = treatmnt· period time time*treatmnt time*period / s chisq; RANDOH subject.subject*time subject*period*treatmnt; REPEATED time I subject = subject*period*treatmnt type = AR(l); ESTIMATE 'A vs 0 B' treatmnt ESTIMATE 'A vs. C' treatmnt ESTIMATE 'A ESTIMATE 'B ESTIMA'l'E 'B ESTIMATE 'C

Ouq>ut

vs. vs. vs. vs.

D' treatmnt C' treatmnt D' ~reatmnt

D' treatmnt

1 1 1 0 0 0

-1 0 0; 0 -1 0; 0 0 -1; 1 -1 0; 1 0 -1; 0 1 -1;

Class Level Information

Class Levels Values

PERIOD 2 1 8 TREATMNT 4 1 2 3 4 TIME 6 2 3 5 6 9 12 SUBJECT 12 1 2 3 4 5 6

;1

7 8 9 10 11 12

covariance Parameter Estimates (REML)

cov ParDI Ratio .Estimate Std Error

SUBJECT 0.18170193 0.00478343 0.00763066 TIME*StJBJECT 0.05301811 0.00139574 0.00080653 PERIOD*TREATM*SUBJECT 0.00000000 0.00000000 TIME AR(l) 0.85896962 0.02261297 0.00937347 Residual 1.00000000 0.02632569 0.00935561

Model Pitting Information for LOGSERUM

. Description

Observations Variance Estimate Standard Deviation Estimate REML Log Likelihood Akaike's Information Criterion Schwartz's Bayesian Criterion -2 REML Log Likelihood Null Model LRT Chi-Square Null Model LRT DP Null Model LRT P-Value

1304

value

139.0000 0.0263 0.1623

62.5748 57.5748 50.8464

-125.150 106.0378

4.0000 0.0000

Z

0.63 1.73

2.41 2.81

Pr > IZI

0.5307 0.0835

0.0158 0.0049

I, r·····

- --._-----_.- ----- .

\

\,

Sou roe

'1'REA'1'MN'1'. PERIOD '1'IIIE '1'REA'1'KN'1'*'1'IKE PERIOD*'1'IKE

Parameter

A vs. B A vs. e A vs. D B vs. e B vs. D e vs. D

HOP

3 1 5

15 5

'1'ests of Fixed Effeots

DDP 'l'ype III ehiSq Type III P

8 86.89228181 28.96 8 7.44694832 7.45

55 518.98690536 103.80 35 388.03542984 25.87 35 42.80715282 8.56

ES'1'IKA'l'E Statement Results

Estimate Std Error DDP

-0.04373166 0.08675161 35 0.65470755 0.08678215 35 0.03302696 0.08667736 35 0.69843921 0.08684178 35 0.07675862 0.08673085 35

-0.62168059 0.08687767 35

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Pr > ehiSq Pr > P

0.0000 0.0001 0.0064 0.0259 0.0000 0.0000 0.0000 0 •. 0000 0.0000 0.0000

'1' Pr > 1'1'1

-0.50 0.6173 7.54 0.0000 0.38 0.7055 8.04 0.0000 0.89 0.3822

-7.16 0.0000

,-..

~ 0.0 TreaunenlA

~ 0

• Treaunenl B I.Ll -0.5 " treaunenlE E

:~,-• reaunenl

~ '-" -1.0 til

.~. Q > ~ -1.5 G,- -I E V "-----::l _____ " " :.a V .... -2.0 :.= E 5 -2.5 til

2 3 5 6 9 12

Time (h)

Figure 1. Mean treatment profiles on lithium levels (log mEq/litre).

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The Analysis of Cross-Over Study with Repeated Measures within Periods by Using Different Covariance Structures

Sanna Hinkka1 and Juhani Tuominen2

1 Department of Biometrics, Farmos Research, Orion-Farmos, Turku, FINLAND 2 Department of Biostatistics, University of Turku, Turku, FINLAND

1. INTRODUCTION

Cross-over designs are used frequently in clinical trials to compare the efficacy of sequential treatments. We are dealing with a two-period and four-treatment cross-over study with repeated measures within periods when missing values are present. Hence, it is carried out as a balanced incomplete block design (BIBD). The mixed model techniques are applied in order to analyse the effects of interest. We prefer restricted maximum likelihood (REML) estimation method to ordinary maximum likelihood because of its ability to reduce the problem of biased covariance estimates. Our main aim is to demonstrate how the estimation of the standard errors in the treatment effects can be improved by taking full accouI1t of the covariance structure of response variable.

In the next chapter we will represent the general mixed model with spesific attention to the covariance matrices. In chapter three, the general form of the analysis of variance will be displayed and illustrated by a study comparing four formulations of lithium carbonate (Westlake3, 1974). PROC MIXED in SAS2 is used in the analyses. The presentation of the results is organised as follows: Firstly, we compare REML estimates and their standard errors when subject effect is treated as fixed or random. Secondly, because REML estimation method allows the use of different covariance structures for covariance matrix in order to find the most suitable covariance structure for the data, we show how these structures can be fitted to the data when estimating the fixed effects of the mixed model.

2. MIXED MODEL

W. J. We~t1ake presents a study in his article3 where four formulations of lithium carbonate were compared irt a BmD in which twelve subjects each received two of four formulations. Blood samples were drawn six times (2, 3, 5, 6, 9 and 12 hours) after administration.

For measured, multivariate normal data, we present the following mixed model for each individual unit i,

Yj = Xja + Zj[3j + 8 j where

a is a vector of fixed effects including parameters for period, treatment, time, time-by­treatment and time-by-period interaction,

Xi is a known nixp design matrix linking a to Yi,

[3j ,... N(O~D) and D is a kxk positive-definite covariance matrix, the vector Pi of random effects consists of subject, time-by-subject and period-by-treatment-by-subject effect,

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I ·k

Zi is a known nixk design matrix linking ~i to Yi and ci - N (0, Ri ) and Ri is an nixni positive-definite covariance matrix.

The statistical model used by Westlake assumes that the subject and the time-by-subject effect are fixed.

The model equation for the mixed model covariance matrix for each individual unit i is

I:i = ZiDZ; +Ri·

3. ILLUSTRATION

In the analyses we use the log-transformed data throughout the illustration. Five observations were randomly omitted from the analyses in order to make the data incomplete. The mean treatment profIles are plotted in Figure 1.

In Table 1 we compare REML estimates when subject effect is fixed (Westlake's model) with corresponding effects when subject and subject-by-time interaction effects are random and have the usual independence and normal distribution assumptions. All the analyses are carried out by PROC MIXED in SAS2.

Table 1. REML estimates of variance components and treatment effects (standard errors of estimates appear in brackets).

Effect Fixed subject effect Random subject effect Subject Period*treatment*subject Time*subject

0.021034 (0.011245) 0.003227 (0.007425) 0.018500 (0.009368) 0.001368 (0.001532)

Treatment Avs.B A vs. C A vs. D B vs. C Bvs.D Cvs.D

-0.074916 (0.106050) 0.610049 (0.106097)

-0.001501 (0.106066) 0.684965 (0.106256) 0.073416 (0.106136)

-0.611550 (0.106746)

-0.043590 (0.085205) 0.658777 (0.085296) 0.033305 (0.085082) 0.702366 (0.085422) 0.076895 (0.085192)

-0.625471 (0.085365)

We have fitted the following covariance structures for Ri: • Compound symmetry (CS), i.e. Rij = (J"~ + (J"21 (i = j),

1 (i = j) equals 1 when i = j and 0 otherwise,

•

•

•

Toeplitz with 2 diagonal bands, (TOEP(2», i.e. Rij = ~i_jl+ll (Ii - jl < 2), 1(li - jl < 2) equals 1 when Ii - jl < 2 and 0 otherwise,

Spatial structure (SP(POW», i.e. Rij = (J"2pd;j when diJis the Euclidean distance between the coordinates i and j and .

First-order autoregressive (AR(l», i.e. Rij = a2pli-jl .

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Table 2. Summary of models fit.

Model

1 2 3 4

Covariance structure Number of covariance . parameters 5 5 5 5

REML log likelihood

48.1069 56.4375 60.2539 62.5748

The first-order autoregressive covariance structure appears to fit the data b~st because its REML log likelihood is the highest. No goodness-of-fit test is provided because the number of covariance parameters of all fitted structures is the same. In Table 3 we have the estimates of the effects in the final model with AR(I) covariance structure.

Table 3. REML estimates of the [mal model.

Effect Subject Period*treatment*subject Time*subject Treatment

Avs.B Avs.C Avs.D Bvs. C Bvs.D Cvs.D

REFERENCES'

Estimate 0.004783

·0 0.001396

-0.043732 0.654708

·0.033027 0.698439 0.076759 -0.621681

Standard error of estimate 0.007631

0.000807

0.086752 0.086782 0.086677 0.086842 0.086731 0.086878

1. Brown, H. K. and Kempton, R. A. 'The application ofREML in clinical trials', Statistics in Medicine, 13, 1601 - 1617 (1994).

2. SAS Institute Inc., SAS Campus Drive, Cary, NC 27513, U.S.A. 3. Westlake, W. J. 'The use of balanced incomplete block designs in comparative bioavailability

trials', Biometrics, 30, 319 - 327 (1974).

1309

\ , \. ". "'.

SAS Code for the final model

PROC MIXED; CLASS period treatmnt time_ubject; MODEL logs.rum = tr.atmnt period time time*treatmnt time*period I s chisq; RANDOM sUbject subject*time subject*pe~;lQd*treatmnt;

REPEATED time I subject = subject*period*treatmnt type :II AR ( 1) ; ESTIMATE 'A vs. ESTIMATE 'A vs. ESTIMATE 'A vs. ESTIMATE 'B vs. ESTIMATE ' B vs. ESTIMATE 'C vs.

Output

Class

PERJ:OD TREATMNT TJ:ME SOBJECT

B' treatmnt 1 -1 0 0; C' treatmnt 1 0 -1 0; D' treatmnt 1 0 0 -1; C' treatmnt 0 1 -1 0; D' treatmnt 0 1 0 -1; D' treatmnt 0 0 1 -1;

Class Level J:nformation

Levels Values

2 1 8 4 1 2 3 4 6 2 3 5 6 9 12

12 1 2 3 4 5 6 7

Covariance Parameter Estimates

8 9 10 11 12

(REML)

cov Parm Ratio Estimate Std Brror

SOBJECT 0.18170193 0.00478343 0.00763066 TJ:ME* SUBJECT 0.053ql811 0.00139574 0.00080653 PERJ:OD*TREAT~*SOBJECT 0.00000000 0.00000000 TJ:ME AR(l) 0.85896962 0.02261297 0.00937347 Residual 1.000doooo 0.02632569 0.00935561

Model Fitting J:nformation for LOGSEROM

Description Value

Observations 139.0000 Variance Estimate 0.0263 Standard Deviation Estimate 0.1623 REML Log Likelihood 62.5748 Akaike'. J:nformation Criterion 57.5748 Schwartz's Bayesian Criterion 50.8464 -2 REML Log Likelihood -125.150 Null Model LRT Chi-SQUare 106.0378 Null Model LRT DF '.0000 Null Model LRT P-Value 0.0000

1310

Z Pr > IZI

0.63 0.5307 1.73 0.0835

2.'1 0.0158 2.81 0.0049

Source

TREATMNT PERIOD TIME TREATMNT*TIME PERIOD*TIME

Parameter

A vs. B A vs. e A vs. D B vs. e B vs. D e vs. D

NDF

3 1 5

15 5

Tests of Fixed Effects

DDF Type :III ehiSq Type :III F

8 86.89228181 28.96 8 7.44694832 7.45

55 518.98690536 103.80 35 388.03542984 25.87 35 42.80715282 8.56

ESTIMATE Statement Results

Estimate Std Error DDF

-0.04373166 0.08675161 35 0.65470755 0.08678215 35 0.03302696 0.08667736 35 0.69843921 0.08684178 35 0.07675862 0.08673085 35

-0.62168059 0.086877 67 35

1311

Pr > ehiSq Pr > F

0.0000 0.0001 0.0064 0.0259 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

T Pr > ITI

-0.50 0.6173 7.54 0.0000 0.38 0.7055 8.04 0.0000 0.89 0.3822

-7.16 0.0000

.-... g 0.0

Treatment A ~

0

• Treauncnt B J.Ll -0.5 " freauncnt5 E

:~.-'f reatmcnt

~ '-' -l.0 en

.~,--t;)

~ -1.5 -I E " "----:::s -----"- " :2 " .... -2.0 ;:=

E 5 -2.5

V)

2 3 5 6 9 12

Time (h)

Figure 1. Mean treatment profiles on lithium levels (log mEq/litre).

t \: \ \', '\. \

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