Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Transcription

1 Analyss of Varance and Desgn of Experments- MODULE LECTURE - 6 EXPERMENTAL DESGN MODELS Dr. Shalabh Department of Mathematcs and Statstcs ndan nsttute of Technology Kanpur

2 Two-way classfcaton wth nteractons Consder the two-way classfcaton wth an equal number, say K observatons per cell. Let : k th observaton n (, th cell,.e., recevng the treatments th level of factor A and th level of factor B, and y are ndependently drawn from N ( μ, σ so that the lnear model under consderaton s y k yk = μ + ε k ε k where are dentcally and ndependently dstrbuted followng N (0, σ. Thus E( yk = μ = μoo + ( μo μoo + ( μo μoo + ( μ μo μo + μoo = μ+ α + β + γ where wth μ = μoo α = μo μoo β = μo μoo γ = μ μ μ + μ o o oo J J α = 0, β = 0, γ = 0, γ = 0. = = = = y k =,,..., ; =,,..., J; k =,,..., K Assume that the desgn matrx X s of full rank so that all the parametrc functons of μ are estmable.

3 3 The null hypothess are : α = α =... = α = 0 0α 0β 0γ : β = β =... = β = 0 : All γ = 0 for all,. J The correspondng alternatve hypothess s : At least one α α, for α : At least one β β, for β :,. γ At least one γ γk for k Mnmzng the error sum of squares J K ( k, = = k= E = y μ α β γ the normal equatons are obtaned as E = 0, μ E = 0 for all, α E β E γ = 0 for all and = 0 for all and.

4 4 The least squares estmates are obtaned as ˆ μ = y = JK J K = = k= y k ˆ α = y y = y y oo k JK = J ˆ β = y y = y y ˆ γ oo k K = = y y y + y o oo oo J yk yoo yoo y K =. = = + The error sum of square s J K SSE = Mn ( y μ α β γ wth SSE σ ˆ μ, ˆ α, ˆ β, ˆ γ = = k= J K k = ( y ˆ μ ˆ α ˆ β ˆ γ = = k= J K = ( y y = = k= k k ~ χ ( J ( K. o

5 5 0 α = α = α =... = α = 0 Now mnmzng the error sum of squares under,.e., mnmzng E J K = = k= = ( y μ β γ k wth respect to μ, β γ and and solvng the normal equatons E E E = 0, = 0 for all and = 0 for all and μ β γ gves the least squares estmates as ˆ μ = yˆ ˆ β = y y oo ˆ γ = y y y + y. o oo oo The sum of squares due to α s 0 Mn μβ,, γ J K = = k= J K = = k= ( y μ β γ k = ( y ˆ μ ˆ β ˆ γ k J K ( y k y o JK ( y oo y = = k= = = + ( oo. = = SSE + JK y y

6 6 Thus the sum of squares due to devaton from 0 α or the sum of squares due to effect A s wth SSA = Sum of squares due to SSE = JK ( y y SSA σ χ ~ (. 0α = oo Mnmzng the error sum of squares under 0 β : β = β =... = β J = 0.e., mnmzng E = ( y μ α γ J K = = k= k and solvng the normal equatons E E E = 0, = 0 for all and = 0 for all and μ α γ yelds the least squares estmators ˆ μ = y ˆ α = y y ˆ γ = y y y + y. o oo oo The mnmum error sum of squares s J K J ( y ˆ ˆ ˆ k μ α γ = SSE + K( yoo y = = k= =

7 7 and the sum of squares due to devaton from 0 β or the sum of squares due to effect B s SSB = Sum of squares due to SSE wth SSB σ = K ( y y = ~ χ ( J. J oo 0β Next, mnmzng the error sum of squares under E = ( y μ α β 3 J K = = k= k 0 γ : all γ = 0 for all,,.e., mnmzng wth respect to μ, α and β and solvng the normal equatons E3 E3 E3 = 0, = 0 = 0 μ α β for all and for all yelds the least squares estmators as ˆ μ = y ˆ α = y y oo ˆ β = y y. oo

8 8 The sum of squares due to 0 γ s Mn μα,, β J K = = k= J K ( y μ α β k = ( y ˆ μ ˆ α ˆ β = = k= k J ( o oo oo. = = = SSE + K y y y + y Thus the sum of squares due to devaton from SSAB = Sum of squares due to SSE 0γ 0 γ or the sum of squares due to nteracton effect AB s wth SSAB σ J = K ( y y y + y = = J ~ χ ((. o oo oo The total sum of squares can be parttoned as TSS = SSA + SSB + SSAB + SSE where SSA, SSB, SSAB and SSE are mutually orthogonal. So ether usng the ndependence of SSA, SSB, SSAB and SSE as well as ther respectve α χ dstrbutons or usng the lkelhood rato test approach, the decson rules for the null hypothess at level of sgnfcance are based on F - statstcs as follows:

9 9 J ( K SSA F =. ~ F[ (, J ( K ] under 0 α, SSE and So Reect Reect Reect J ( K SSB F =. ~ F [ ( J, J( K ] under 0 β, J SSE J ( K SSAB F3 =. ~ F[ ( ( J, J( K ] under 0γ. ( ( J SSE 0α α [(, ( ] f F > F J K 0 β f F F α > [( J, J( K ] [ ] 0γ f F3 > F α ( ( J, J( K. 0 α 0β f or s reected, one can use t -test or multple comparson test to fnd whch pars of α ' s or β ' s are sgnfcantly dfferent. f 0 γ s reected, one would not usually explore t further but theoretcally t- test or multple comparson tests can be used.

10 0 JK t can also be shown that ESSA ( = σ + α K ESSB ( = σ + ESSAB ( = σ + ESSE ( = σ. = J β J = K J γ ( ( J = = The analyss of varance table s as follows: Source of Degrees of Sum of squares Mean squares F - value varaton freedom Factor A ( - SSA MSA MSA MSA = F = MSE λ C = F α ( p, n p Factor B (J - SSB MSB = MSB J F = MSB MSE 0 nteracton AB ( -(J - : β = β 0 SSAB SSAB MSAB = ( ( J F 3 = MSAB MSE Error J (K - SSE Total ( J K TSS MSE = SSE J ( K