Design of Experiments - Example

Example

This example is attributed to Harold Hotelling. It conveys some of the flavor of those aspects of the subject that involve combinatorial designs.

The weights of eight objects are to be measured using a pan balance and set of standard weights. Each weighing measures the weight difference between objects placed in the left pan vs. any objects placed in the right pan by adding calibrated weights to the lighter pan until the balance is in equilibrium. Each measurement has a random error. The average error is zero; the standard deviations of the probability distribution of the errors is the same number σ on different weighings; and errors on different weighings are independent. Denote the true weights by

We consider two different experiments:

1. Weigh each object in one pan, with the other pan empty. Let X_i be the measured weight of the ith object, for i = 1, ..., 8.
2. Do the eight weighings according to the following schedule and let Y_i be the measured difference for i = 1, ..., 8:

Then the estimated value of the weight θ₁ is

Similar estimates can be found for the weights of the other items. For example

The question of design of experiments is: which experiment is better?

The variance of the estimate X₁ of θ₁ is σ2 if we use the first experiment. But if we use the second experiment, the variance of the estimate given above is σ2/8. Thus the second experiment gives us 8 times as much precision for the estimate of a single item, and estimates all items simultaneously, with the same precision. What is achieved with 8 weighings in the second experiment would require 64 weighings if items are weighed separately. However, note that the estimates for the items obtained in the second experiment have errors which are correlated with each other.

Many problems of the design of experiments involve combinatorial designs, as in this example.