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# Design of experiments in science

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A design of experiments is the ordered sequence experimental tests, each to acquire new knowledge by controlling one or more input parameters to obtain results validating a model, with a good economy (number of trials as low as possible, for example).

A classic example is the „star-shaped design” when starting from a set of values chosen for the parameters of a central test, complete it by testing where every time a single factor varies, all others being equal.

A type of more comprehensive design is the factorial plan by choosing values for each factor by varying simultaneously all factors (exhaustively or not). The number of tests can become very large (combinatorial explosion).

### Issues

Suppose we wanted to know if the proportion of black balls from an urn is greater than 5%, the urn containing 1000 balls. We start with the idea to draw 100 in the hope of having a good approximation of proportion.

• If during the draw we bring 51 black balls, it can be stopped immediately: to continue would not make sense, since with 51 black balls on 1000 a percentage greater than 5% is now certain.

• It can be refined further by noting that the probability of drawing eg 5 black balls in the first 5 draws makes 0,3 × 10−6 the probability that the proportion of black balls is less than 5%.

• In practice, the calculation provides strict rules according to the results indicating when the draw must stop – with decision one way or the other – or whether it should be continued.

So a design of experiments allows to reduce the number of trials to what is strictly necessary to make a decision, which can save time, money and lives.

It is an experiment design of this type that helped stop an ongoing experiment to determine whether aspirin had a preventive effect on heart attacks, the results clearly establishing that it was the cases (25% reduction in risk). Continuing the experiment would have returned in these circumstances to deny to the date originally scheduled batch-control sufferers access to aspirin, which could have cost the lives of some of them.

There are many processes and properties that a lot is known to depend on external parameters (called factors) but without it we would have analytical models.

When it is desired to know the dependency of an output variable F of such a process, or property, one is faced with several challenges:

• what are the most influential factors?
• there are interactions (correlations) between factors?
• can we linearize the process (or property) depending on these factors and the resulting model is it predictive?
• how to minimize the number of measurement points of the process (or property) to obtain the maximum information?
• there are biases in the measurement results?

The method of experiment design addresses these issues and thus can be applied in many processes/properties that will, by example of clinical trials, evaluating the quality of the most complex industrial processes.

It is thus possible for the industry to ask this new definition: an experiment design is a strictly organized testing suite, to determine, with minimal testing and maximum accuracy, the relative influence of different design parameters or manufacture of a product, in order to optimize performance and cost.