3 Single-factor experiments
We now compare \(t \ge 2\) treatments, the levels of a single factor. In a completely randomised design the \(N\) available experimental units are allocated to the treatments entirely at random, \(n_i\) units to treatment \(i\), with \(N = n_1 + \dots + n_t\). In practice one writes the treatment labels, \(n_1\) copies of label \(1\), \(n_2\) of label \(2\) and so on, in a list of length \(N\), applies a random permutation, and gives the \(k\)th unit the \(k\)th label of the permuted list. When the runs are made in sequence, the run order is randomised in the same way.
Example 3.0.1. A manufacturer of television sets was interested in the effect of five coating types on the conductivity of colour picture tubes. Four tubes were made with each coating, in random order, and the conductivity was measured:
| coating type | total \(y_{i\cdot }\) | mean \(\bar y_{i\cdot }\) | ||||
| \(1\) | \(80\) | \(83\) | \(83\) | \(85\) | \(331\) | \(82.75\) |
| \(2\) | \(75\) | \(75\) | \(79\) | \(79\) | \(308\) | \(77.00\) |
| \(3\) | \(67\) | \(72\) | \(74\) | \(74\) | \(287\) | \(71.75\) |
| \(4\) | \(62\) | \(62\) | \(67\) | \(69\) | \(260\) | \(65.00\) |
| \(5\) | \(60\) | \(61\) | \(64\) | \(66\) | \(251\) | \(62.75\) |
| \(1437\) | \(71.85\) | |||||
In addition, \(\sum _i\sum _jy_{ij}^2 = 104471\). Does the coating type affect conductivity?
The analysis of such data is a special case of the general linear model, which is reviewed first.
3.2 Models for a single-factor experiment
3.3 The analysis of variance
3.4 Estimation
3.5 Model adequacy checking
3.6 Variance-stabilising transformations
3.7 A quantitative factor: the regression model
3.8 Power and sample size
3.9 Practice problems
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