Solutions HW 10 1a) The following SAS will give you the needed output. proc glm; class block trt; model numheads=block trt block*trt; run; Source DF Type III SS Mean Square F Value Pr > F block*trt 8 1969.066667 246.133333 5.94 0.0016 F = 5.94 with 8 and 15 df. p = 0.0016 There is significant evidence of a treatment-by-block interaction. 1b) The following SAS will give you the needed output for parts b and c. proc glm; class block trt; model numheads=block trt block*trt; lsmeans block*trt / slice=block; run; The GLM Procedure Least Squares Means numheads block trt LSMEAN 1 0 109.000000 1 50 132.000000 1 100 144.000000 1 150 153.500000 1 200 139.500000 2 0 116.500000 2 50 159.000000 2 100 154.000000 2 150 161.500000 2 200 158.500000 3 0 156.000000 3 50 156.000000 3 100 160.500000 3 150 152.000000 3 200 167.500000 The main reason for the interaction being significant is as follows. In blocks 1 and 2, there is a large difference between the yield of the 0 treatment and the other treatment means. In block 3, the difference between the 0 treatment and the others disappears. Thus the differences among the treatment means are not the same for all blocks, and we have a significant interaction. 1c) The GLM Procedure Least Squares Means block*trt Effect Sliced by block for numheads Sum of block DF Squares Mean Square F Value Pr > F 1 4 2253.400000 563.350000 13.60 <.0001 2 4 2847.400000 711.850000 17.18 <.0001 3 4 279.400000 69.850000 1.69 0.2054 Block 1: F = 13.6 with 4 and 15 df. p < 0.0001 There is significant evidence of a difference among treatment means. Block 2: F = 17.18 with 4 and 15 df. p < 0.0001 There is significant evidence of a difference among treatment means. Block 3: F = 1.69 with 4 and 15 df. p = 0.2054 There is no evidence of a difference among treatment means. 1d) The code in 1(a) provides the output. Source DF Type III SS Mean Square F Value Pr > F block 2 2655.266667 1327.633333 32.04 <.0001 trt 4 3411.133333 852.783333 20.58 <.0001 block*trt 8 1969.066667 246.133333 5.94 0.0016 F = 20.58 with 4 and 15 df. p < 0.0001 There is significant evidence of a difference in treatment means. 1e) Use the following code. proc mixed method=type3 data=one; class block trt; model numheads=trt; random block block*trt; run; Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt 4 8 3.46 0.0634 F = 3.46 with 4 and 8 df. p = 0.0634 There is inconclusive evidence of difference in treatment means. There is significant difference at the 0.1 level but not at the 0.05 level. Note that the proper error term for testing treatment effects becomes block*trt interaction when block*trt interaction is treated as a random term. 1f) The following code will provide the necessary output. proc means data=one; var numheads; by block trt; output out=two mean=y; run; proc print; run; proc glm data=two; class block trt; model y=block trt; run; Source DF Type III SS Mean Square F Value Pr > F block 2 1327.633333 663.816667 5.39 0.0329 trt 4 1705.566667 426.391667 3.46 0.0634 F = 3.46 with 4 and 8 df. p = 0.0634 There is inconclusive evidence of difference in treatment means. There is significant difference at the 0.1 level but not at the 0.05 level. This is the same test as the one conducted in part 1(e). 2a) Source df Diet 3 – 1 = 2 Drug 2 – 1 = 1 Diet*Drug 2 * 1 = 2 Dam(Diet Drug) (4-1)*6 = 18 Gender 2 – 1 = 1 Gender*Diet 1 * 2 = 2 Gender*Drug 1 * 1 = 1 Gender*Diet*Drug 1 * 2 = 2 Error 24 – 6 = 18 c. Total (4*6*2)-1 = 47 Note the term labeled "Error" can be written as Gender*Dam(Diet Drug) with d.f.=(2-1)*(4-1)*6=18. 2b) For each of the 6 dams mated to any one sire, randomly assign the 6 treatments. This is a RCBD with sires as blocks and the dams mated to sires as the experimental units. 2c) Source df Sire 4 – 1 = 3 Diet 3 – 1 = 2 Drug 2 – 1 = 1 Diet*Drug 2 * 1 = 2 Error(1)=Sire*Trt (4-1)*(6-1) = 15 Gender 2 – 1 = 1 Gender*Diet 1 * 2 = 2 Gender*Drug 1 * 1 = 1 Gender*Diet*Drug 1 * 2 = 2 Error(2) 47-sum above = 18 C. Total (4*6*2)-1 = 47 Note that the six combinations of Diet and Drug correspond to what is called "Trt" in the table above. The second error term called Error(2) consists of Gender*Sire + Gender*Trt*Sire for a total of (2-1)*(4-1)+(2-1)*(6-1)*(4-1)=18 d.f. Note that we would not usually include separate lines for Sire*Diet, Sire*Drug, Sire*Gender, Sire*Diet*Gender, or Sire*Drug*Gender because these interaction terms do not correspond to either a whole-plot or a split-plot experimental unit. To fit this model in SAS we would write proc mixed method=type3; class sire diet drug gender; model y=diet|drug|gender; random sire sire*diet*drug; run; The sire*diet*drug term corresponds to Error(1). SAS will automatically add Error(2) in this case. 3) Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F temp 1 4 19.21 0.0118 humidity 1 4 14.10 0.0199 temp*humidity 1 4 4.31 0.1064 variety 4 32 3.72 0.0135 temp*variety 4 32 0.75 0.5681 humidity*variety 4 32 1.72 0.1700 temp*humidit*variety 4 32 0.18 0.9449 The output shows that none of the interactions are significant, thus we can consider the main effects across all factors. All three main effects are significant (see p-values for temp, humidity, and variety in the table above. Based on the lsmeans for temp and the p-value of 0.0118, we can say that the seed quality mean was significantly higher for 50 degrees than 70 degrees. Based on the lsmeans for humidity and the p-value of 0.0199, we can see that seed quality mean was significantly higher for low humidity than for high humidity. (See the temp and humidity lsmeans below.) Least Squares Means Effect humidity variety temp Estimate temp 50 48.1333 temp 70 39.9667 humidity high 41.0667 humidity low 47.0333 Differences of Least Squares Means Effect variety _variety Pr > |t| Adjustment Adj P variety A B 0.6598 Tukey-Kramer 0.9915 variety A C 0.0335 Tukey-Kramer 0.1979 variety A D 0.0631 Tukey-Kramer 0.3252 variety A E 0.0170 Tukey-Kramer 0.1118 variety B C 0.0119 Tukey-Kramer 0.0820 variety B D 0.0240 Tukey-Kramer 0.1500 variety B E 0.0057 Tukey-Kramer 0.0423 variety C D 0.7690 Tukey-Kramer 0.9982 variety C E 0.7690 Tukey-Kramer 0.9982 variety D E 0.5578 Tukey-Kramer 0.9753 There was a significant difference between varieties B and E (p = 0.0423). There was weak evidence of a difference between varieties B and C (p = 0.082). All other varieties were not significantly different from each other. Least Squares Means Standard Effect variety Estimate Error DF variety A 44.7500 0.9974 32 variety B 45.0000 0.9974 32 variety C 43.5000 0.9974 32 variety D 43.6667 0.9974 32 variety E 43.3333 0.9974 32 In conclusion, the lower temperature and humidity significantly raised the seed quality for all varieties. Variety B has a significantly higher seed quality than variety E. No other differences among varieties were detected at the 0.05 level.