1
A randomized block example
A farmer wishes to compare the growth times of four different varieties of daffodil
under a range of different conditions. She decides to use a randomized block design
where six fields are split into four equally sized plots and then the varieties are
randomly assigned to these plots. The response is the number of days until half of
the daffodils are ready for cutting.
Field
Var 1 2 3 4 5 6 Mean
A 43 46 46 42 40 37 42.33
B 34 30 33 25 32 22 29.33
C 36 35 38 27 29 28 30.50
D 37 43 40 33 39 35 37.83
Mean 37.5 38.5 36.75 31.75 35.0 30.5 35.0
The data were entered into MINITAB and the twoway ANOVA program was run.
MTB > set c1
DATA> 43 34 36 37 46 30 35 43 46 33 28 40 42 25 27 33
DATA> 40 32 29 39 37 22 28 35
DATA> end
MTB > set c2
DATA> 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
DATA> end
MTB > set c3
DATA> 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6
DATA> end
MTB > Name c4 “RESI1” c5 “FITS1”
MTB > Twoway c1 c2 c3;
SUBC> Residuals ’RESI1’;
SUBC> Fits ’FITS1’;
SUBC> Additive;
SUBC> Means c2 c3.
Two-way ANOVA: C1 versus C2, C3
Source DF SS MS F P
C2 3 685.0 228.333 31.28 0.000
C3 5 209.5 41.900 5.74 0.004
Error 15 109.5 7.300
Total 23 1004.0
S = 2.702 R-Sq = 89.09% R-Sq(adj) = 83.28%
• There are significant differences among the varieties (P ≈ 0)
• There are also significant differences among the fields (P = .004)
• The variety effect estimates are α̂1 = 7.33, α̂2 = −5.67,α̂3 = −4.5 and α̂4 = 2.83
• There are 6 possible comparisons among the varieties, and the Bonferroni correction for simul-
taneous inference uses α∗ = .05/6 = .0083.
• The width of confidence intervals for the variety differences αi − αk is
2
t15.05/12
√
MSE
√
2
6
= 3.0363(2.702)(.5774) = 4.74
• The confidence intervals are
i k Interval
1 2 8.26 17.74
1 3 7.10 16.57
1 4 -.24 9.24
2 3 -5.91 3.57
2 4 -13.24 -3.76
3 4 -12.07 -2.59
• We conclude that variety A is not significantly different from variety D, and variety B is not
significantly different from variety C. The other comparisons are significant at the α = .05 level.