Most of us have heard the advice, “Change only one variable at a time to understand how that variable affects your system.”

Sometimes this advice is correct, but only in a very local sense. For example, if we want to estimate how sensitive a system is to a change in variable A, then we can hold all other variables constant and change variable A very slightly. The change in the system response divided by the change in variable A is an estimate of the sensitivity derivative at the original design point.

But, the key word in the previous sentence is “point,” because *derivatives are defined at a point*. If we select a different starting design point, and then repeat the above exercise, we would expect to get a different value for the sensitivity derivative.

Let’s examine this idea further. If we hold variable B constant at value B1, changing variable A will have a certain influence on the system response. But if we hold variable B constant at value B2, the effect of variable A might be very different than before. If so, then the effect of variable A depends on the value of variable B. When this occurs, we say that there is an interaction between variables A and B. We can easily generalize this argument to many variables. Continue reading →