Analysis of Variance and Design of Experiments-II

Transcription

1 Analsis of Variance and Design of Experiments-II MODULE VII LECTURE - 3 CROSS-OVER DESIGNS Dr. Shalabh Department of Mathematics & Statistics Indian Institute of Technolog Kanpur

2 Analsis of variance Now we develop the analsis of variance for higher order cross-over designs when n = n. This is used to test the effects using F - test obtained from an analsis of variance table. The various sums of squares can be derived for the x cross-over design as a simple example of a split-plot design. The subject form the main plots and the periods are treated as the subplots at which repeated measurements are taken. Following this, the total sum of squares is given as SS Total n i Y =, ( ) ooo ijk i= j= k= n+ n the sum of squares between subjects are given b: nn SS = ( ) Carr-over( c- o) oo oo ( n+ n) : Sum of squares due to carr-over effect. SS Residual ( bs - ) Y Y = n ni iok ioo i= k= i= i : Sum of squares due to between-subject residuals, the sum of squares within-subjects are given b nn = ( + ) SSTreat o o o o ( n+ n) SSPeriod o o o o ( n+ n) SS Residual ( w- s) nn = ( + ) Y n i ijo ijk i= j= k= i= j= ni = SS : Sum of squares due to treatments effects. : Sum of squares due to period effects. Residual ( bs - ) : Sum of squares due to within-subject residuals.

3 3 Then the resulting analsis of variance and the construction of F statistics is given in the following Table: Source Sum of squares Degrees of Mean squares F E() freedom () Between subjects Carr-out Residual (between subjects) SScarr over ( c o) SSResidual( bs - ) n + n carr over ( c o) Residual( bs - ) Fc o [( nn ) /( n+ n)]( λ λ) + ( σs + σ ) ( σs + σ ) Within subjects Direct treatment effect SS Treat Treat F Treat ( nn ) /( n + n )[( τ τ ) ( λ λ ) / ] + σ Period effect SS Period Period F Period [( nn ) /( n + n )]( π π ) + σ Residual (withinsubjects) SSResidual( ws - ) SS Total + n Total ( n+ n) n Residual( ws - ) σ

4 4 Under H : λ = λ we have E E and we use the statistic 0 ( carr over ( c o) ) = ( Residual( bs - ) ) F c-o = carr over ( c o) Residual(b-s). Assuming λ = λ and H : τ = τ, we have E( ) = E( ) = σ. Therefore, we get 0 Treat Residual(w-s) F Treat = Treat Residual( ws - ) Testing for period effects does not depend upon the assumption that λ = λ holds. Since ( ) = ( ) = E E σ Period Residual under H : π = π, so the statistic 0 F Period H 0 = Period Residual( ws - ) follows the central F - distribution with and ( n + n ) degrees of freedom.

5 5 Comment on the procedure of testing Usuall, the carr-out effects are tested on a quite high level of significance ( α = 0.) first. If this leads to a significant result, then the test for treatment effects is to be based on the data of the first period onl. If it is not significant, then the treatment effects are tested using the differences between the periods. The hpothesis of no carr-over effect is ver likel to be rejected even if there is a true carr-over effect. Hence, the biased test (biased, because the carr-over was not recognized) is used to test for treatment differences. This test is conservative in the case of a true positive carr-over effect and therefore this is insensitive to potential differences in treatments. On the other hand, this test will exceed the level of significance if there is a true negative carr-over effect (not ver likel in practice, since this refers to a withdrawal effect). If there is no true carr-over effect, the null hpothesis is ver likel to be rejected erroneousl efficient test using first-period data onl is performed. ( α = 0.) and the less This method is not ver useful in testing treatment effects as it depends upon the outcome of the pretest.

6 6 Alternative parameterization in x Cross-Over The model = µ + s + π + τ + λ + ε ijk ik j [, i j] [, i j ] ijk is used in the classical approach and is labeled as parameterization number. A more general parameterization of the x cross-over design, includes a sequence effect and is given b = µ + γ + s + π + τ + λ + ε ijk i ik j t r ijk, γ i with i, j, t, r =, and k =,..., ni.. The data are summarized in a table containing the cell means i.e., ijo, Period Sequence o o o o

7 7 Here Sequence indicates that the treatments are given in the order (AB) and Sequence has the (BA) order. Using the common restrictions γ = γ, π = π, τ = τ, λ = λ and writing γ = γ, π = π, τ = τ, λ = λ for brevit, we get the following equations representing the four expectations: µ = µ + γ + π + τ µ = µ + γ π τ + λ, µ = µ γ + π τ, µ = µ γ π + τ λ. In matrix notation, we can express it as µ µ 0 γ µ = X β = π. µ 0 τ µ λ The X matrix is of order (4 x 5) and has rank 4, so that β is onl estimable if one of the parameters is removed. Various parameterizations are possible depending on which of the five parameters is to be removed and then to be confounded with the remaining ones.

8 8 Parametrization Number The classical approach ignores the sequence parameter. Its expectations ma therefore be represented as a submodel of µ µ 0 γ µ = X β = π µ 0 τ µ λ b dropping the second column of X as 0 µ π Xβ =. 0 τ λ We obtain from this ' E 0 XX =, 0 H where E = 4 I, ' 4 H =, XX = 64, E 0 = 0 H ' ( XX)

9 9 / / with E = I, H =. The least squares estimate of β is obtained as 4 / ˆ µ o ˆ π ˆ ' ' o = = ( XX ) X. ˆ τ o β ˆ λ o Observe that o o = 0 0 ' o o X o o o o o o o o o o o o = o o o + o o o.

10 0 Therefore, the least squares estimates are given as ˆ µ o ˆ π ˆ β ( ) ' ' o = = XX X ˆ τ o ˆ λ o ( o + o + o + o)/4 ( + )/4 o o o o = ( o o)/ ( o + o o o)/ from which we get the following estimators: ˆ µ =, ooo ( ˆ oo oo) ( c o co) π d ˆ π = = = 4 ( ˆ o o) τ d / λd ˆ τ = = ˆ ˆ λd λ = oo oo =.

11 ˆ τ The covariance between and is ˆ / / Cov( ˆ τλ, ) = σh = σ. / ˆ τ ˆλ Hence and are correlated. The correlation coefficient / ˆ ( ˆ ρτ, λ) =. = τˆ ˆλ The estimation of is alwas twice as accurate as the estimation of. Note that uses the data of the first period onl and is confounded with the difference between the two groups (sequences). ˆλ τˆ Remark In fact, parameterization number is a three-factorial design with the main effects, and and with and being πτ λ τ λ correlated. On the other hand, the classical approach uses the split-plot model in addition to the classical parameterization. So it is obvious that we will get different results depending on which of the parameterization is used.