4.1 ONE FACTOR RANDOMISED BLOCK LATIN SQUARE VARIANT Y = C|B|A Analysis of terms: A + B + C Layout: B1 B2 B3 B4 ---------------- C1 | A1 A2 A3 A4 C2 | A2 A4 A1 A3 C3 | A3 A1 A4 A2 C4 | A4 A3 A2 A1 Data: A B C Y 1 1 1 10 2 2 1 8 3 3 1 5 4 4 1 4 2 1 2 11 4 2 2 13 1 3 2 16 3 4 2 12 3 1 3 10 1 2 3 14 4 3 3 9 2 4 3 10 4 1 4 8 3 2 4 6 2 3 4 11 1 4 4 13 Model 4.1(Latin square variant) A and B are fixed factors, C is a random blocking factor (with treatment Order B and Subject C for a crossover design): Source DF Seq SS Adj SS Adj MS F P 1 A 3 63.50 63.50 21.17 7.94 0.016 2 B 3 1.00 1.00 0.33 0.13 0.942 3 C 3 81.50 81.50 27.17 10.19 0.009 4 Error 6 16.00 16.00 2.67 Total 15 162.00 COMMENT: Sequential and adjusted SS are the same for tests. A is a fixed factor, B and C are random blocking factors: Source DF Seq SS Adj SS Adj MS F P 1 A 3 63.50 63.50 21.17 7.94 0.016 2 B 3 1.00 1.00 0.33 0.13 0.942 3 C 3 81.50 81.50 27.17 10.19 0.009 4 Error 6 16.00 16.00 2.67 Total 15 162.00 COMMENT: Sequential and adjusted SS are the same for tests. __________________________________________________________________ LATIN SQUARE REPLICATED Analysis of terms: A + B + C + interactions Layout of a single square with 4 independent observations in each cell: | B1 B2 B3 ------------- C1 | A3 A2 A1 C2 | A1 A3 A2 C3 | A2 A1 A3 Data (taken from Winer et al. 1991): A B C Y 3 1 1 0 2 2 1 0 1 3 1 2 1 1 2 2 3 2 2 0 2 3 2 0 2 1 3 6 1 2 3 9 3 3 3 2 3 1 1 1 2 2 1 2 1 3 1 2 1 1 2 5 3 2 2 1 2 3 2 0 2 1 3 8 1 2 3 10 3 3 3 1 3 1 1 1 2 2 1 2 1 3 1 4 1 1 2 3 3 2 2 1 2 3 2 1 2 1 3 12 1 2 3 12 3 3 3 1 3 1 1 4 2 2 1 5 1 3 1 6 1 1 2 1 3 2 2 4 2 3 2 4 2 1 3 7 1 2 3 12 3 3 3 5 Model 4.1(replicated Latin square variant) A, B and C are fixed factors: Model_1 Source DF Seq SS Adj SS Adj MS F P 1 A 2 92.39 92.39 46.19 12.52 <0.001 2 B 2 40.22 40.22 20.11 5.45 0.010 3 C 2 198.72 198.72 99.36 26.93 <0.001 4 Interactions 2 33.39 33.39 16.70 4.53 0.020 5 Error 27 99.50 99.50 3.69 Total 35 464.22 COMMENT: Two-way interactions are inseparable unless one factor has negligible interactions with both the other two factors. The design has subjects nested in factor combinations. See the stacked Latin square below for a crossover design with repeated measures on subjects. __________________________________________________________________ LATIN SQUARE REPLICATED WITH IDENTICAL SQUARES Analysis of terms: A + B + C + S Layout of 4 identical squares S: S1 | B1 B2 B3 S2 | B1 B2 B3 ------------- ------------- C1 | A3 A2 A1 C1 | A3 A2 A1 C2 | A1 A3 A2 C2 | A1 A3 A2 C3 | A2 A1 A3 C3 | A2 A1 A3 S3 | B1 B2 B3 S4 | B1 B2 B3 ------------- ------------- C1 | A3 A2 A1 C1 | A3 A2 A1 C2 | A1 A3 A2 C2 | A1 A3 A2 C3 | A2 A1 A3 C3 | A2 A1 A3 Data (taken from Winer et al. 1991): A B C S Y 3 1 1 1 0 2 2 1 1 0 1 3 1 1 2 1 1 2 1 2 3 2 2 1 0 2 3 2 1 0 2 1 3 1 6 1 2 3 1 9 3 3 3 1 2 3 1 1 2 1 2 2 1 2 2 1 3 1 2 2 1 1 2 2 5 3 2 2 2 1 2 3 2 2 0 2 1 3 2 8 1 2 3 2 10 3 3 3 2 1 3 1 1 3 1 2 2 1 3 2 1 3 1 3 4 1 1 2 3 3 3 2 2 3 1 2 3 2 3 1 2 1 3 3 12 1 2 3 3 12 3 3 3 3 1 3 1 1 4 4 2 2 1 4 5 1 3 1 4 6 1 1 2 4 1 3 2 2 4 4 2 3 2 4 4 2 1 3 4 7 1 2 3 4 12 3 3 3 4 5 Model 4.1(identically replicated Latin squares variant) A, B and C are fixed or random factors, S is a random block: Model_2 Source DF Seq SS Adj SS Adj MS F P 1 A 2 92.39 92.39 46.19 13.41 <0.001 2 B 2 40.22 40.22 20.11 5.84 0.008 3 C 2 198.72 198.72 99.36 28.85 <0.001 4 S 3 43.33 43.33 14.44 4.19 0.015 5 Error 26 89.56 89.56 3.44 Total 35 464.22 COMMENT: Sequential and adjusted SS are the same for tests. __________________________________________________________________ LATIN SQUARE WITH REPLICATE SQUARES INDEPENDENTLY RANDOMISED Analysis of terms: A + B + C + S + S*A + S*B + S*C (Model 1) or A + B + C + S (Model 2) Layout: S1 | B1 B2 B3 B4 ---------------- C1 | A1 A2 A3 A4 C2 | A2 A4 A1 A3 C3 | A3 A1 A4 A2 C4 | A4 A3 A2 A1 S2 | B1 B2 B3 B4 ---------------- C1 | A3 A4 A2 A1 C2 | A4 A2 A1 A3 C3 | A1 A3 A4 A2 C4 | A2 A1 A3 A4 Data: A B C S Y 1 1 1 1 10 2 2 1 1 8 3 3 1 1 5 4 4 1 1 4 2 1 2 1 11 4 2 2 1 13 1 3 2 1 16 3 4 2 1 12 3 1 3 1 10 1 2 3 1 14 4 3 3 1 9 2 4 3 1 10 4 1 4 1 8 3 2 4 1 6 2 3 4 1 11 1 4 4 1 13 3 1 1 2 5 4 2 1 2 6 2 3 1 2 8 1 4 1 2 9 4 1 2 2 11 2 2 2 2 13 1 3 2 2 16 3 4 2 2 12 1 1 3 2 10 3 2 3 2 9 4 3 3 2 7 2 4 3 2 7 2 1 4 2 11 1 2 4 2 13 3 3 4 2 8 4 4 4 2 9 Model 4.1(independently replicated Latin squares variant) A, B and C are fixed or random, S is a random blocking factor: Model_1 Source DF Seq SS Adj SS Adj MS F P 1 A 3 96.38 96.38 32.13 40.58 0.006 2 B 3 3.38 3.38 1.13 9.00 0.052 3 C 3 151.13 151.13 50.38 11.97 0.035 4 S 1 1.13 1.13 1.13 0.57 0.463* 5 S*A 3 2.38 2.38 0.79 0.40 0.753 6 S*B 3 0.38 0.38 0.13 0.06 0.978 7 S*C 3 12.63 12.63 4.21 2.15 0.147 8 Error 12 23.50 23.50 1.96 Total 31 290.88 * Assumes A, B and C are fixed. COMMENT: Sequential and adjusted SS are the same for tests. Assumes no interactions between factors A, B and C. Post hoc pooling of interactions having P > 0.25 with the Error SS provides a more powerful test of the treatment main effect(s). If independent replicate observations were taken in each cell, this would provide an additional residual error term for the within-cell variation which would permit having S as a fixed factor. Model_2 Source DF Seq SS Adj SS Adj MS F P 1 A 3 96.38 96.38 32.13 17.35 <0.001 2 B 3 3.38 3.38 1.13 0.61 0.617 3 C 3 151.13 151.13 50.38 27.21 <0.001 4 S 1 1.13 1.13 1.13 0.61 0.444 5 Error 12 23.50 23.50 1.96 Total 31 290.88 __________________________________________________________________ LATIN SQUARE WITH INDEPENDENT SQUARES IN A BALANCED SET Analysis of terms: C|B|A + S - C*B*A Layout with a balanced set of all 27 possible combinations of factors A, B and C: S1 | B1 B2 B3 S2 | B1 B2 B3 ------------- ------------- C1 | A1 A3 A2 C1 | A2 A1 A3 C2 | A3 A2 A1 C2 | A1 A3 A2 C3 | A2 A1 A3 C3 | A3 A2 A1 S3 | B1 B2 B3 ------------- C1 | A3 A2 A1 C2 | A2 A1 A3 C3 | A1 A3 A2 Data: A B C S Y 1 1 1 1 2 3 2 1 1 0 2 3 1 1 0 3 1 2 1 0 2 2 2 1 0 1 3 2 1 2 2 1 3 1 6 1 2 3 1 9 3 3 3 1 2 2 1 1 2 8 1 2 1 2 10 3 3 1 2 1 1 1 2 2 5 3 2 2 2 1 2 3 2 2 0 3 1 3 2 1 2 2 3 2 2 1 3 3 2 2 3 1 1 3 1 2 2 1 3 2 1 3 1 3 4 2 1 2 3 12 1 2 2 3 12 3 3 2 3 1 1 1 3 3 3 3 2 3 3 1 2 3 3 3 1 Model 4.1(balanced replicate Latin squares variant) A, B and C are fixed, S is a random blocking factor: Model_1 Source DF Seq SS Adj SS Adj MS F P 1 A 2 93.85 93.85 46.93 42.23 0.000 2 B 2 44.52 44.52 22.26 20.03 0.002 3 C 2 2.30 2.30 1.15 1.03 0.412 4 B*A 4 188.82 188.82 47.20 42.48 0.000 5 C*A 4 4.37 4.37 1.09 0.98 0.482 6 C*B 4 8.37 8.37 2.09 1.88 0.233 7 S 2 14.30 14.30 7.15 6.43 0.032 8 Error 6 6.67 6.67 1.11 Total 26 363.19 COMMENT: Sequential and adjusted SS are the same for tests. Assumes no interactions with S, which would otherwise confound estimates of the main effects and interactions for the other factors. If independent replicate observations were taken in each cell, this would provide an additional residual error term for the within-cell variation which would permit having S as a fixed factor. __________________________________________________________________ LATIN SQUARE WITH STACKED SQUARES HAVING ROWS NESTED IN SQUARE Analysis of terms: A + B + S + S*A + S*B + C(S) (Model 1) or S|B|A + C(S) - S*B*A (alternative Model 1) or A + B + S + C(S) (Model 2) Layout: B1 B2 B3 B4 ---------------- S1 C1 | A1 A2 A3 A4 S1 C2 | A2 A4 A1 A3 S1 C3 | A3 A1 A4 A2 S1 C4 | A4 A3 A2 A1 S2 C5 | A3 A4 A2 A1 S2 C6 | A4 A2 A1 A3 S2 C7 | A1 A3 A4 A2 S2 C8 | A2 A1 A3 A4 Data: A B C S Y 1 1 1 1 10 2 2 1 1 8 3 3 1 1 5 4 4 1 1 4 2 1 2 1 11 4 2 2 1 13 1 3 2 1 16 3 4 2 1 12 3 1 3 1 10 1 2 3 1 14 4 3 3 1 9 2 4 3 1 10 4 1 4 1 8 3 2 4 1 6 2 3 4 1 11 1 4 4 1 13 3 1 5 2 5 4 2 5 2 6 2 3 5 2 8 1 4 5 2 9 4 1 6 2 11 2 2 6 2 13 1 3 6 2 16 3 4 6 2 12 1 1 7 2 10 3 2 7 2 9 4 3 7 2 7 2 4 7 2 7 2 1 8 2 11 1 2 8 2 13 3 3 8 2 8 4 4 8 2 9 Model 4.1(stacked Latin squares variant with rows nested in square) A and B are fixed factors, C and S are random blocking factors (with treatment Order B and Subject C for a crossover design): Model_1 Source DF Seq SS Adj SS Adj MS F P 1 A 3 96.38 96.38 32.13 40.58 0.006 2 B 3 3.38 3.38 1.13 9.00 0.052 3 S 1 1.13 1.13 1.13 0.05 0.847* 4 S*A 3 2.38 2.38 0.79 0.40 0.753 5 S*B 3 0.38 0.38 0.13 0.06 0.978 6 C(S) 6 163.75 163.75 27.92 13.94 <0.001 7 Error 12 23.50 23.50 1.96 Total 31 290.88 * Quasi F-ratio with an unrestricted model. COMMENT: Sequential and adjusted SS are the same for tests. Post hoc pooling of interactions having P > 0.25 with the Error SS provides a more powerful test of the fixed-factor main effects. Alternative Model_1 Source DF Seq SS Adj SS Adj MS F P 1 A 3 96.38 96.38 32.13 40.58 0.006 2 B 3 3.38 3.38 1.13 9.00 0.052 3 B*A 9 96.13 19.29 2.14 1.53 0.400 4 S 1 1.13 1.13 1.13 0.04 0.843* 5 S*A 3 2.38 2.38 0.79 0.56 0.675 6 S*B 3 0.38 0.38 0.13 0.09 0.961 7 C(S) 6 86.93 86.92 14.49 10.33 0.041 8 Error 3 4.21 4.21 1.40 Total 31 290.88 * Quasi F-ratio with an unrestricted model. COMMENT: B*A interaction is partially confounded with C(S); otherwise sequential and adjusted SS are the same for tests. Model_2 Source DF Seq SS Adj SS Adj MS F P 1 A 3 96.38 96.38 32.13 23.03 <0.001 2 B 3 3.38 3.38 1.13 0.77 0.525 3 S 1 1.13 1.13 1.13 0.04 0.846 4 C(S) 6 163.75 163.75 27.29 18.71 <0.001 5 Error 18 26.25 26.25 1.46 Total 31 290.88 COMMENT: Sequential and adjusted SS are the same for tests. In a crossover design, differences between squares may be considered part of a subject effect C, in which case omit S and analyse: Y = A + B + C. A is a fixed factor, B, C and S are random blocking factors: Model_1 Source DF Seq SS Adj SS Adj MS F P 1 A 3 96.38 96.38 32.13 40.58 0.006 2 B 3 3.38 3.38 1.13 9.00 0.052 3 S 1 1.13 1.13 1.13 0.05 0.847* 4 S*A 3 2.38 2.38 0.79 0.40 0.753 5 S*B 3 0.38 0.38 0.13 0.06 0.978 6 C(S) 6 163.75 163.75 27.92 13.94 <0.001 7 Error 12 23.50 23.50 1.96 Total 31 290.88 * Quasi F-ratio with an unrestricted model. COMMENT: Sequential and adjusted SS are the same for tests. Post hoc pooling of interactions having P > 0.25 with the Error SS provides a more powerful test of the treatment effect. Alternative Model_1 Source DF Seq SS Adj SS Adj MS F P 1 A 3 96.38 96.38 32.13 16.88 0.125* 2 B 3 3.38 3.38 1.13 0.91 0.689* 3 B*A 9 96.13 19.29 2.14 1.53 0.400 4 S 1 1.13 1.13 1.13 0.04 0.843* 5 S*A 3 2.38 2.38 0.79 0.56 0.675 6 S*B 3 0.38 0.38 0.13 0.09 0.961 7 C(S) 6 86.92 86.92 14.49 10.33 0.041 8 Error 3 4.21 4.21 1.40 Total 31 290.88 * Quasi F-ratio. COMMENT: B*A interaction is partially confounded with C(S); otherwise sequential and adjusted SS are the same for tests. Model_2 Source DF Seq SS Adj SS Adj MS F P 1 A 3 96.38 96.38 32.13 23.03 <0.001 2 B 3 3.38 3.38 1.13 0.77 0.525 3 S 1 1.13 1.13 1.13 0.04 0.846 4 C(S) 6 163.75 163.75 27.29 18.71 <0.001 5 Error 18 26.25 26.25 1.46 Total 31 290.88 COMMENT: Sequential and adjusted SS are the same for tests. __________________________________________________________________ LATIN SQUARE WITH COMPLETELY CROSS-FACTORED TREATMENTS IN ORTHOGONAL BLOCKS Analysis of terms: A|B + C + D Layout with a balanced set of all 6 combinations of treatment factors A and B with two and three levels respectively: Layout: | C1 C2 C3 C4 C5 C6 ---------------------- D1 | 1 6 4 2 3 5 D2 | 3 1 2 4 5 6 D3 | 6 5 3 1 4 2 D4 | 5 4 1 6 2 3 D5 | 4 2 5 3 6 1 D6 | 2 3 6 5 1 4 1 = A1B1 2 = A1B2 3 = A1B3 4 = A2B1 5 = A2B2 6 = A2B3 Data: AB A B C D Y 1 1 1 1 1 57 6 2 3 2 1 61 4 2 1 3 1 28 2 1 2 4 1 35 3 1 3 5 1 5 5 2 2 6 1 10 3 1 3 1 2 11 1 1 1 2 2 68 2 1 2 3 2 32 4 2 1 4 2 49 5 2 2 5 2 71 6 2 3 6 2 18 6 2 3 1 3 55 5 2 2 2 3 72 3 1 3 3 3 17 1 1 1 4 3 39 4 2 1 5 3 89 2 1 2 6 3 28 5 2 2 1 4 16 4 2 1 2 4 71 1 1 1 3 4 52 6 2 3 4 4 77 2 1 2 5 4 24 3 1 3 6 4 24 4 2 1 1 5 51 2 1 2 2 5 33 5 2 2 3 5 88 3 1 3 4 5 19 6 2 3 5 5 67 1 1 1 6 5 49 2 1 2 1 6 37 3 1 3 2 6 8 6 2 3 3 6 79 5 2 2 4 6 36 1 1 1 5 6 51 4 2 1 6 6 53 Model 4.1(Latin square variant with completely cross-factored treatments) A and B are fixed factors, C and D are random blocking factors: Source DF Seq SS Adj SS Adj MS F P 1 A 1 4489.0 4489.0 4489.0 12.41 0.002 2 B 2 2193.4 2193.4 1096.7 3.03 0.071 3 B*A 2 2675.2 2675.2 1337.6 3.70 0.043 4 C 5 2240.9 2240.9 448.2 1.24 0.328 5 D 5 1331.9 1331.9 266.4 0.74 0.605 6 Error 20 7235.2 7235.2 361.8 Total 35 20165.6 COMMENT: Sequential and adjusted SS are the same for tests. The sum of SS[A|B] = SS[AB]. The column and row factors could alternatively, or additionally, each comprise two orthogonal factors and their interaction. __________________________________________________________________ References Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press. http://www.southampton.ac.uk/~cpd/anovas/datasets/ http://www.soton.ac.uk/~cpd/anovas/datasets/Latin squares.htm Winer, B. J., Brown, D. R. and Michels, K. M. (1991) Statistical Principles in Experimental Design. New York: McGraw-Hill.