1.4 Blocking: the tyre example
Example 1.4.1. Suppose we want to compare two brands of tyres, \(A\) and \(B\), by the tread wear after a fixed distance. Possible complicating factors include
- (i)
- the type of car;
- (ii)
- road conditions;
- (iii)
- the type of driver;
and so on. How do we control for them? Known factors are handled by blocking, and unknown ones by randomisation. Consider the possible experimental strategies.
- (a)
- \(2n\) cars. Choose \(n\) cars at random for brand \(A\) and fit the other \(n\) with brand \(B\). Then \(\mu _A - \mu _B\) is estimated by \(\bar y_A - \bar y_B\), with \[ E(\bar Y_A - \bar Y_B) = \mu _A - \mu _B , \qquad \Var (\bar Y_A - \bar Y_B) = \frac {2\sigma ^2}{n} , \] where \(\sigma ^2\) is the variance of the wear on a single car. The only way to reduce this variance is to increase \(n\).
- (b)
- \(n\) good drivers and \(n\) rough drivers. There are now two possible strategies.
- (i)
- Randomly assign \(n\) of the \(2n\) drivers to brand \(A\) and the other \(n\) to brand \(B\), ignoring the type of driver. The variance \(\sigma _1^2\) of \(\bar y_A - \bar y_B\) then includes variation between brands and among drivers.
- (ii)
- Randomly assign half (\(n/2\)) of the good drivers to brand \(A\) and the other half to brand \(B\), and do the same for the rough drivers. Writing \(\bar y_{1A}\) for the mean of the good drivers on brand \(A\), and so on, estimate \(\mu _A - \mu _B\) by \[ \frac {\bar y_{1A} + \bar y_{2A} - (\bar y_{1B} + \bar y_{2B})}{2} , \] with variance \(\sigma _2^2\).
The second strategy gives the higher precision, \(\sigma _2^2 < \sigma _1^2\). Proposition 1.4.2 below says by how much.
- (c)
- \(n\) cars, both brands on each. Fit brand \(A\) on one side of each car and brand \(B\) on the other, choosing the side at random. The car is now a block of two units, and each car gives a difference \(d_k = y_{kA} - y_{kB}\) in which the car’s own effect cancels.
To compare the strategies we need a model for the wear. Write the wear of a tyre of brand \(T\) on car (or driver) \(k\) as \[ y = \mu _T + \beta _k + \varepsilon , \] where \(\beta _k\) is the effect of the car or driver and \(\varepsilon \), with variance \(\sigma ^2\), is the remaining variation of a tyre about its car’s level.
Proposition 1.4.2. In strategy (b), let good drivers have \(\beta = -\delta \) and rough drivers \(\beta = +\delta \), and let the errors be independent with variance \(\sigma ^2\). Then \[ \sigma _1^2 = \frac {2\sigma ^2}{n} + \frac {4\delta ^2}{2n - 1} , \qquad \sigma _2^2 = \frac {2\sigma ^2}{n} . \]
Proof. For (i), apply Theorem 1.3.2 conditionally on the errors, with unit values \(u_k = \beta _k + \varepsilon _k\). The driver effects are \(n\) values \(-\delta \) and \(n\) values \(+\delta \), with mean \(0\) and \(\sum \beta _k^2 = 2n\delta ^2\), so taking expectations over the errors, \[ E(S^2) = \sigma ^2 + \frac {2n\delta ^2}{2n - 1} , \qquad \text {and}\qquad \sigma _1^2 = \frac 2nE(S^2) = \frac {2\sigma ^2}{n} + \frac {4\delta ^2}{2n - 1} . \] For (ii), each of the four group means is over \(n/2\) drivers of the same type. In \(\bar y_{1A} - \bar y_{1B}\) the good drivers’ \(-\delta \) cancels, and in \(\bar y_{2A} - \bar y_{2B}\) the rough drivers’ \(+\delta \) cancels, so only the errors remain: \[ \sigma _2^2 = \frac 14\cdot 4\cdot \frac {\sigma ^2}{n/2} = \frac {2\sigma ^2}{n} . \] □
The driver effect has not been reduced by strategy (ii); it has been removed from the comparison entirely, by balancing it across the brands. Strategy (c) goes further. Each \(d_k = \mu _A - \mu _B + \varepsilon _{kA} - \varepsilon _{kB}\) is free of every car effect, known or unknown, so with \(n\) cars \[ \Var (\bar d) = \frac {2\sigma ^2}{n} . \] In strategy (a), on the other hand, the car-to-car variation \(\sigma _\beta ^2\) enters each observation, and \(\Var (\bar y_A - \bar y_B) = 2(\sigma _\beta ^2 + \sigma ^2)/n\) with \(2n\) cars. Blocking on cars therefore gives the same number of comparisons from half the cars, with the variance reduced by the factor \(\sigma ^2/(\sigma _\beta ^2 + \sigma ^2)\). When cars differ a great deal, which in a wear study they do, the gain is large. Strategy (c) is the paired comparison design of the next section, the simplest case of the randomised block design.
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