4.2 Completely Randomised Design
It is the simplest type of design in which the treatments are allocated to the experimental units
entirely at random.
The design is completely flexible since it allows missing values. Therefore, any number of replicates
can be used, replicates may vary from treatment to treatment. The analysis of the experiment under
this design is carried out by a one way classification method.
| \(T_1\) | \(T_2\) | \(T_3\) | \(T_4\) |
| \(X_{11}\) | \(X_{12}\) | \(X_{13}\) | \(X_{14}\) |
| \(X_{21}\) | \(X_{22}\) | \(X_{23}\) | \(X_{24}\) |
| \(X_{31}\) | - | \(X_{33}\) | \(X_{34}\) |
| \(X_{14}\) | - | - | \(X_{44}\) |
Example 4.4. In a comparison of detergents, pieces of white cloth were soiled with ink and washed under controlled conditions, some pieces by each detergent. Several pieces were lost in the course of the experiment. The whiteness readings recorded were:
| A | B | C | D |
| 77 | 74 | 73 | 76 |
| 81 | 58 | 57 | 77 |
| 76 | – | 69 | 64 |
| 69 | – | 63 | – |
Test the hypothesis of no difference between the brands of detergent.
Solution. The groups now have unequal sizes – \(4, 2, 4\) and \(3\), so \(N=13\) readings in four groups. Unequal group sizes are no obstacle: the formulas below already divide each group’s total by its own \(n_j\), so nothing needs adjusting.
| Detergent | A | B | C | D |
| \(n_j\) | 4 | 2 | 4 | 3 |
| \(T_j\) | 303 | 132 | 262 | 217 |
| Mean | 75.75 | 66.00 | 65.50 | 72.33 |
With \(T=914\) and \(\sum \sum x^2 = 64\,976\), the correction factor is \[CF=\frac {T^2}{N}=\frac {914^2}{13}=64\,261.23,\] \[SS_T=\sum \sum x^2-CF=64\,976-64\,261.23=714.77,\] \[SS_C=\sum _j \frac {T_j^2}{n_j}-CF =\frac {303^2}{4}+\frac {132^2}{2}+\frac {262^2}{4}+\frac {217^2}{3}-CF=260.35,\] \[SS_E=SS_T-SS_C=714.77-260.35=454.42.\]
| Source | \(SS\) | df | \(MS\) | \(F\) |
| Between detergents | \(260.35\) | \(k-1=3\) | \(86.78\) | \(1.72\) |
| Within (error) | \(454.42\) | \(N-k=9\) | \(50.49\) | |
| Total | \(714.77\) | \(N-1=12\) |
The degrees of freedom add, \(3+9=12\), which is the check to make before going further.
Testing \(H_0:\mu _A=\mu _B=\mu _C=\mu _D\), the critical value is \(F_{3,9,\,0.05}=3.86\) and our \(F=1.72\) falls well short; the \(P\)-value is \(0.23\). We fail to reject \(H_0\): there is no evidence of a difference between the brands.
The means do differ on the face of it – from \(65.5\) to \(75.75\) – but readings within a single detergent range from \(57\) to \(81\). Against variation that large, and with only 13 readings, a spread of ten points between group means is unremarkable.
Remark 4.5. The missing pieces deserve a comment, because the analysis quietly assumes something about them. Dropping them is safe only if they went missing for reasons unconnected with whiteness. If pieces were lost because they washed badly – disintegrating, say – then the surviving readings overstate those detergents, and no amount of correct arithmetic afterwards repairs it. Missing data are a question about the experiment, not about the calculation.
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