Factors Affecting Variance in ANOVA Models

Random and Mixed Effects ANOVA KNNL Chapter 25 Winner Chapter 10

1-Way Random Effects Model In some settings, the levels of the factor in a 1-Way ANOVA model are a random sample from a larger population of levels Goal is to make statements regarding the population of levels, as opposed to only regarding the levels in the study. We wish to make statements about the variation in the effects of the levels Studies are often interested in measuring reliability of testing/measuring procedures

Statistical Model / Parameters of Interest 1,..., ; 1,..., ij i r j n = + = = Y ij i ( ) ( ) , i 2 2 ~ , independent ~ 0, independent are independent N N i ij ij 2 2 Overall Mean Between Group Variance Within Group Variance E Y Y Y + = + = + = + = 0 E E E ij i ij i ij = + = + = + + = + = 2 2 2 2 2 2 2 2 2 Y 2 , 2(0) Y ij i ij i ij i ij = + + = + + + = 0 0 0 + + + = 2 2 , , , , , , ' Y Y j j ' ' ' ' ij ij i ij i ij i i i ij ij i ij ij = , 0 ' i i ' ' i j ij = = 2, 2: r n 2 Y 2 0 0 0 0 Y Y Y Y 11 2 2 Y Y 12 = = 2 Y 2 Y 2 0 0 0 0 21 2 2 Y 22 , Y Y 2 ' ij ij = = = Intraclass Correlation Coefficient: , Y Y ij Y ' I ij ij 2 Y Y ' ij

Analysis of Variance - I ' ' ' , , 0 ij ij ij i j Y Y Y Y = = E Y Y = = = + 2 2 Y 2 2 2 ' i i ij ij ( ) r n SSE r n 2 ( ) = = = = 2 1 SSE Y Y df r n E MSE E i ( ) ij E 1 = = 1 1 i j ( ) ( ) r n r SSTR r 2 2 = = = = = + 2 2 1 SSTR Y Y n Y Y df r E MSTR E n i i TR 1 = = = 1 1 1 i j i = 2 2 Testing whether al l are equal: i MSTR MSE : 0 : 0 H H 0 A ( ) ( ) = * * Test Statistic: Reject H if 0.95; 1, 1 F F F r r n 0 ( ij E Y Elements of the Derivations of and : E MSTR E MSE E Y Y ) 2 = = + = + + 2 2 2 2 2 2 E Y ij ij 1 n n = = E Y i ij = 1 j ( ) + 2 2 n 1 n n Y 1 n 1 n 1 n 1 n n n n ( ) Y = = + = + + = 2 2 2 2 2 2 2 , 2 Y Y Y n i ' ij ij ij ij 2 2 2 n = = = = + 1 1 1 ' 1 j j j j j ( + 2 2 n ) 2 i 2 = + E Y n ( ) + + 2 2 2 2 n n 1 r r = Y 1 r 1 r 1 r 1 r 1 r r r r r = = = = + 1 + = 2 2 2 2 , 2 (0) E Y E Y Y Y Y Y r ' i i i i i 2 2 2 n nr = = = 1 ' = i i = + 1 1 1 i i i i ( + 2 2 n ) 2 2 = + E Y nr

Analysis of Variance - II E Y Y = = = + = = 2 2 Y 2 2 2 , , 0 ' Y Y Y Y i i ' ' ' i j ij ij ij ij ij ( ) r n r n r 2 ( ) 2 i = = = 2 1 SSE Y Y Y n Y df r n i ij ij E = = = = = 1 1 1 1 1 i j i j i ( ) ( ) r n r r 2 2 2 i 2 = = = = 1 SSTR Y Y n Y Y n Y nrY df r i i TR = = = = 1 1 1 1 + i j i i ( ( + 2 2 2 2 n n ( ij E Y ) ) ) 2 i 2 2 2 2 = + + = + = + 2 2 2 E Y E Y n nr r n r ( ) ( ) ( ) 2 i 2 2 2 2 = + + = + + = + + 2 2 2 2 2 2 2 E Y nr nr nr E n Y nr nr r E nrY nr n ij = = = 1 1 1 i j i ( ) ( ) ( ) 2 2 = + + + + = 2 2 2 2 2 1 E SSE nr nr nr nr nr r r n SSE r n = = 2 E MSE E ( ) 1 E S ( ) ( ) ( ) ( ) 2 2 = + + + + = + 2 2 2 2 2 2 1 1 STR nr nr r nr n n r r SSTR r = = + 2 2 E MSTR E n 1

Estimating Overall Mean E Y Y = = = + = = 2 2 Y 2 2 2 , , 0 ' Y Y Y Y i i ' ' ' i j ij ij ij ij ij + 2 2 n E Y Y 1 n n = = = 2 E Y i i ij n = 1 j s Y s Y MSTR n MSTR n = = 2 i i + 2 2 n 1 r r = = = 2 E Y E Y Y i nr = 1 i MSTR nr MSTR nr = = 2 s Y s Y ) 1 Y ( ( ) ( ) ( ) ~ 1 1 100% C I for : 1 / 2 ; t r Y t r s Y s Y

Estimating Intra-Class Correlation I= 2/ Y2 SSE r n SSE ( ) = = 2 2 ~ 1 E MSE E r n ( ) 2 1 SSTR r SSTR = = + 2 2 2 ~ 1 E MSTR E n r + 2 2 1 n SSTR MSTR ( ) 1 r + + 2 2 2 2 n n ( ) = , independent ~ 1, 1 SSTR SSE F r r n SSE MSE ( ) ( ) 1 r n 2 2 MSTR + 2 2 n ( ) ( ) 1 ( = 2; 1 , 1 2); 1, 1 1 P F r r n F r r n MSE 2 1 n 1 r 1 n 1 MSTR MSE MSTR MSE = = Setting: 1 1 L U ( ) ( ) 1 ( 2); 1, 1 / 2; 1, 1 F r n F r r n 2 + L + U + ( ) = = = * * 1 100% CI for 2: , L U I 2 1 1 L U

Estimating Within Group Variance: 2 SSE ( ) 2 ~ 1 r n 2 SSE ( ) ( r n ) ( ) ( r n ) = 2 2 / 2 ; 1 1 / 2 ; 1 1 P 2 ( ) ( r n SSE ) ( ) ( r n ) 2 2 / 2 ; 1 1 / 2 ; SSE 1 1 = 1 P 2 SSE SSE = 2 1 P ( ) ( r n ) ( ) ( r n ) 2 2 1 / 2 ; 1 / 2 ; 1 SSE / 2 ; SSE ( ) 2 1 100% CI for : , ( ( ( ) ( r n ) ( ) ( r n ) 2 2 1 1 / 2 ; 1 ( ) ) 1 1 r n MSE r n MSE , ( ) ( r n ) ) ( r n ) 2 2 1 / 2 ; 1 / 2 ; 1

Estimating Between Group Variance: 2 = + = 2 2 2 E MSTR n E MSE E MSTR E MSE 1 n 1 n MSTR MSE = = = + 2 E E MSTR E MSE n n MSTR MSE = 2 Point Estimator: Note: This can be negative (usually treated as 0). s n Aside :Satterthwaite approximatio n for Degrees of Freedom ( are constants) c i ^ h h ( ) df L L Approx ^ L = = 2 df ~ L c E MS c MS i i i i = = 1 1 i i + ( ) 2 + ... c MS c MS = 1 1 2 h h where: df ( ) ( ) 2 c MS df c MS df + + 1 1 h h ... 1 h ^ ^ ( ) ( ) df L df L 1 P L ( ) ( ) ( ) 2 2 / 2; df 1 / 2 ; df ^ ^ ( ) ( ) df L df L ( ) 1 100% CI f or : , L ( ) ( ) ( ) 2 2 / 2; df 1 / 2 ; df 1 n 1 n = = = = = 2 Application to estimating : 2, h c c MS MSTR MS MSE 1 2 1 2 ( ) 2 2 2 s MSTR MSE ^ L = = = 2 s df 2 n 1 n 1 n MSTR MSE + ( ) 1 1 r r n

2-Way Random Effects Model Factors and Both Random (Levels are Random Samples from Populations) 1,..., ijk i j ijk ij Y i A B + ( ) = + + + = = = 1,..., 1,..., a j b k n ( , i ) ) 2 2 2 , are independent, normally distributed, with mean 0, and Variances: ( independent , , , i j ij , , j ij ( ) 2 ~ 0, pairwise independent N ijk ijk ijk E Y ( ) = + + + + = E i j ijk ij ijk Y ( ) = + + + + = + + + 2 2 2 2 2 2 i j ijk ij + + + + + = = = = = 2 2 2 2 ', ', ', ', ' ' i i = i j i j i j i j i j j k j k k k 2 2 2 2 = = , Y ' ' ' i j k Y ', ', ', ', j , ' , ' , ' k k i i j j k k k k ijk 2 0 ', ', i

2-Way Mixed Model Unrestricted Case ( ) = + + + + = = = 1,..., ; 1,..., ; 1,..., Y i a j b k n ijk i j ijk ij a ( ) ( ( ) ( ) ) ( ) = 2 2 2 0 ~ 0, ~ 0, ~ 0, NID NID NID i j ijk j ijk ij ij = 1 i ijk E Y ijk Y + = + = + + 2 2 2 2 i + + 0 = = = = = = 2 2 2 ', ', ', ', ', ', ', ' ' i j i j i j i i j i i j k j k j j k k 2 2 2 = , Y ' ' ' i j k Y ijk , ' , ' k k i k k , ', a 2 i bn = + + = + + = + = 2 2 2 2 2 2 2 2 = 1 i E MSA n E MSB n an E MSAB n E MSE 1 a 2 2 2 MSAB MSE MS B MSAB an MSAB bn MSAB abn ^ ^ ^ = = = MSE n n + 2 2 2 2 2 MSAB bn = + = = = 2 Y s Y s Y Y ' i i i i bn b bn + 2 2 2 n 2 = + = = 2 Y s Y abn ab abn

Expected Mean Squares for 2-Way ANOVA Mean Square df Fixed Model Random Model Mixed Model (A Fixed, B Fixed) (A Random, B Random) (A Fixed, B Random) (Unrestricted Case) a a 2 i 2 i bn bn + + + + 2 2 2 2 2 2 = = -1 + 1 1 i i MSA a n bn n 1 1 a a b 2 j an = 1 j + + + + 2 2 2 2 2 2 2 -1 b + MSB n an an n 1 b a b ( ) 2 n ij ( )( ) = = 1 1 i j + + 2 2 2 2 2 1 1 + MSAB a b n n ( )( ) 1 1 a b ( ) 2 2 2 1 MSE ab n

Tests for Main Effects and Interactions Tested Effects Fixed Effects Random Effects Mixed Effects (A Fixed) = = = / = = = = = 2 Factor A Test Statistic Critical Value : ... 0 : 0 / MSA MSAB F : ... / 0 H H H 0 1 0 = 0 = 1 a a F MSA MSE F MSA MSAB A A A ( ( ) ) ( ( )( ) ) ( ( )( ) ) 1 ; 1, 1 1 ; 1, 1 1 1 ; 1, 1 1 F a ab n F a a b F a a b = = = = = = = 2 2 Factor B : ... 0 : 0 : 0 H H H 0 1 0 = 0 b Test Statistic Critical Value / ) ) 1 / ) ) 1 b / F ( 1 MSB MSE F MSB MSAB ( ; 1, b F ( MSB MSAB ( ; 1, b B B ( B 1 ( )( )( ) ) ; 1, 1 1 1 1 F b ab n F a F a b ( ) ( ) = = = = = 2 2 Interaction AB : ... 0 : 0 : = 0 H H = H MSAB MSE 0 0 0 11 MSAB MSE )( 1 b ab = Test Statistic Critical Value / / / F F MSAB MSE F AB ; AB AB F ( ( ) ( ) ) ( ( )( ) ( ) ) ( ( )( ) ( ) ) 1 1 , 1 1 ; 1 1 , 1 1 ; 1 1 , 1 F a ab n F a b ab n a b ab n

Estimating Variance Components Case 1: Random Effects Models: = = 2 2 E MSE s MSE MSAB MSE = + = 2 2 2 E MSAB n s n MSA MSAB bn MSB an = + + = 2 2 2 2 E MSA n bn s MSAB = + + = 2 2 2 2 E MSB n an s Case 1: Mixed Effects Models (B Random): = = 2 2 E MSE s MSE MSAB MSE = + = 2 2 2 E MSAB n s n MSB MSAB an = + + = 2 2 2 2 E MSB n an s

Estimating Fixed Effects in Mixed Model th Effect of Level of Factor A (Fixed Effect) = i i + 2 2 n E MSAB bn MSAB bn ^ ^ ^ = = = 2 2 Y Y s i i i i bn a a = i = Contrast among Levels of Factor A: s.t. 0 L c c i i = = 1 1 i i a a a MSAB bn ^ L ^ ^ L ^ ^ L = = = 2 2 i 2 2 2 i c c s c i i i = = = 1 1 1 i i i ) a ^ L t ^ ( ) ( ) ( )( = i 1 100% CI for : 1 / 2 ; 1 1 L c a b s L i = 1 i Multiple Comparison Procedures (all pairs of levels of Factor A): MSAB bn ( )( ) = ; , a a Tukey: 1 1 1 HSD q b ( ) 1 a a a 2 MSAB bn ( ) ( ; ( ) )( ) = = = Bonferroni: 1 / 2 1 1 MSD t C a b C A A 2 2

Estimating Marginal (Factor A) Means in Mixed Models 1 1 1 1 a nab a nab ^ ^ ^ = = + = + 2 2 Y E MSAB E MSB s MSAB MSB i i i i nab nab Using Satterthwaite's Approximation: 2 1 1 a nab + MSAB MSB nab = df 2 2 1 1 a nab a MSAB MSB nab + ( )( ) 1 1 1 b b ^ ^ ( ) ( ) Approximate 1 100% CI for : 1 / 2 ; t df s i i i

2-Way Mixed Model Restricted Case Factor Fixed, Random (Restricted Model) A B Y = + + + ( ) + = = = 1,..., 1,..., 1,..., i a j b k n ijk i j ijk ij a = 2 are fixed constants with 0 are independent, normal, with mean 0, and Variance: i i j = 1 i 1 1 a a a ( ) ( ) ( ) ( ij ) = = 2 2 ~ 0, s.t. 0 and , N ' ij ij i j a = 1 i ( ) ( ) ij ) 2 ~ 0, independent , , pairwise independent N ijk j ijk ij Y ( = + + + + = + E Y E ijk i j ijk i 1 a ( ) = + + + + = + + 2 2 2 2 2 ijk i j ijk ij a 1 a + + = = = 2 2 2 ', ', ' i i j j k k a 1 a + = = 2 2 ', ', ' i i j j k k a = , Y ' ' ' i j k Y 1 a ijk = , ' k k 2 2 ', ', i i j j , ', , ' i i k k 0 ', j j ( ) ( ) ( ) * * = + + ( + ) + * j 2 Note: Unrestricted Model: ( j j ~ 0, independent Y N ijk i ijk ij ij ) * * ( ) ( ) * = + = * ij ij j j