Analysis of Variance and Design of Experiments-I

Transcription

1 Analysis of Variance and Design of Experiments-I MODULE VIII LECTURE - 35 ANALYSIS OF VARIANCE IN RANDOM-EFFECTS MODEL AND MIXED-EFFECTS MODEL Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

2 Confidence intervals for variance components Given the normality and independence assumptions of the random effects model, we can generate various confidence intervals related to the variance components. We consider now several cases. Case 1: For ( n s) MS χ Error Because ~, a 100(1 )% confidence interval for is n s ( n s) MS ( n s) MS χ Error Error /, n s χ1, n s. Case : For There is no closed form for a confidence interval for.

3 3 Case 3: For If all sample sizes are same, i.e., n = r, then because MS and MS are independent MS + r MS /( ) ~ Fs ( 1, n s ). Error / i MS P F1 / ( s 1, n s) F /( s 1, n s) = 1 MSError ( + r ) 1 MSError ( + r ) 1 P 1 F = 1 / ( s 1, n s ) MS F /( s 1, n s ) Error 1 P F 1 /( s 1, n s ) MS Error r = MS F /( s 1, n s) Thus, a 100 (1 )% confidence interval for / is (L, U) where and 1 MS 1 L = 1 k MS Error F /( s 1, n s ) 1 MS 1 U = 1. k MS Error F 1 /( s 1, n s )

4 4 Case 4: For + Note that 1- = P L U = P 1+ L 1+ U + = P 1+ L 1+ U = P L + 1+ U 1 1 = P L + 1+ U L U = P. 1+ L + 1+ U Thus, L U, is a 100 (1 )% confidence interval for which represents the proportion of the total 1+ L 1+ U + variability attribute to the variability among the treatments.

5 5 Analysis of variance in mixed-effects models Suppose that we have a mixed-effects model for a design where one effect is fixed and the other is random. Let us consider a mixed for a general a b factorial treatment structure in a completely randomized design. The model is y ijk = µ + τ i + β j + τβij + ε ijk, where we use the following conditions with the levels of factor A fixed and the levels of factor B randomly selected: µ 1. is the unknown overall mean response. th. τ is a fixed effect corresponding to the i level of factor A with τ = 0. i β j 3. is a random effect due to the level of factor B. The β j ' s have independent normal distributions, with mean 0 and variance τβ. β j th 4. is a random effect due to the interaction of the level of factor A with the level of factor B. The ij i th a j th τβ s ij have independent normal distributions with mean 0 and variance. τβ 5. The β ' s, τβ ' s and ε ' s are mutually independent. j ij ijk Using these assumptions, the analysis of variance table for a fixed, random, or mixed model in a two-factor experiment is shown in following table.

6 6 ANOVA table for an a b factorial treatment structure, with n observations per cell Source Sum Degrees of Mean of freedom squares squares E(MS) Fixed effects Random Effects Mixed Effects A fixed, B Random A SSA a - 1 MSA B SSB b - 1 MSB AB SSAB (a - 1)(b 1) MSAB + bnθ ε ε + nθ ε τ + anθ τβ β + n + bn + n + bnθ ε τβ τ ε τβ τ + n + an + n + anθ ε τβ β ε τβ β + n + n ε τβ ε τβ Error SSE ab(n 1) MSE ε ε ε Total TSS (nab 1)

7 7 τβ The test for is the same in the mixed model as in the random-effects model. That is, to test H : = 0 versus H : > 0, we use the statistic 0 τβ 0 τβ F = MSAB. MSE Reject No matter what the results of our test for A we have H F F 0 if > 1,( a 1)( b 1), ab( n 1). H0 τ1 τ a H a : : =... = = 0 at least one of the τ s τβ, we would proceed to use the following tests for factors A and B. For factor differs from the rest. The test statistic is F = MSA MSAB based on df = ( a 1) and df = ( a 1) ( b 1). 1

8 8 For factor B, we have H H The test statistic is = 0 : β 0 > 1 : β 0. F = MSA MSAB based on df = ( b 1) and df = ( a 1) ( b 1). 1 The analysis of variance procedure outlined for a mixed-effects model for an factorial treatment structure can be used as well for a randomized block design, where treatments are fixed, blocks are assumed to be random, and there are observations for each block and treatment.