A linear model for proportions?

When we modelled how a numerical explanatory variable affected a numerical response variable, a linear equation was used,

  =  b0 + b1 x

When the response variable is categorical, it is tempting to try a similar linear equation to explain how the proportion in one response category is affected by the explanatory variable,

predicted proportion,   

Unfortunately however, ...

... a linear equation is not appropriate for a proportion since it may result in predicted proportions greater than 1.0 or less than 0.0.

Nonlinear models

To model how a proportion depends on a numerical explanatory variable, X, an equation should give values between 0 and 1 for all possible values of X. This means that the equation must be nonlinear in X.

Quality control for fuses

Some manufactured products are designed to fail under load as a safety precaution. For example, in cars many parts are designed to collapse or break off in accidents. It is important that these items fail within a fairly tight range of loads.

A company manufactures fuses that are designed to blow when a current of 10 amps flows through them. Batches of one hundred fuses were tested at currents of 9 amps, 9.5 amps, ..., 11.5 amps and failures were noted. The bar charts below show the data that were collected.

Drag the vertical red line on the axis to obtain the predicted proportion of fuses failing at different currents.

The linear model is a reasonably close fit to the data between currents 9.5 and 10.5 amps.

However the linear model predicts that more than 100% of fuses will fail if the load is over 11 amps, and a negative proportion will fail under 9 amps. Any linear model will predict proportions outside the range 0-to-1 for extreme enough values of X.

Now select the option Nonlinear model from the pop-up menu. This curve is better than the previous straight line since it remains between 0 and 1 for all ages.

Again drag the vertical red line on the axis to obtain the predicted proportion failing at different currents. Observe that this nonlinear model can provide reasonable predictions at all currents.