The Model
One problem is to find a model that both best fits the data and
performs well in calculating the marginal effects of a change in
response time. For an example of this, let us look at the relationship
between response time and deaths in traffic accidents.
Because there seems to be no change in proportions of deaths
after about 25 minutes, one choice of a model is to restrict the
data to only those dispatches in which the response time is less
than or equal to 25 minutes. The difficulty with such a model is
that it will predict a much higher proportion of deaths above 25
minutes than is reasonable according to the data. We can see
that about 5% deaths is a reasonable figure for a response time of
40 minutes (Fig. 1). A logistic regression model that is restricted to
less than 25 minutes, however, would predict this to be about
40% (Fig. 2). Another suggestion is to choose something such as a
moving average logistic regression model, in which the first
model includes data for only 1 to 5 minutes, the second from 2
to 6 minutes, the third from 3 to 7 minutes, and so forth.
Predictions and marginal effects are then calculated for 3
minutes for the first model, 4 minutes for the second model,
and so forth. Such a model fits the data much better, but it is not
very general because it has different parameter values for each
minute of response time. Yet another alternative is to try to
include as many data points as possible. This is the approach
used here, and all response times up to median time þ one SD are
included. What we are after is a value for a change of 1 minute in
response time for an average dispatch, and we use this model
even if it does not fit the data perfectly. The models thus contain
response times up to 249 minutes and operational times up to
313 minutes. (The maximum time is chosen according to mean þ
one SD.) Because the relationship between the outcome and the
response time seems to be somewhat different, depending on the
case of the emergency, we have chosen to perform different
statistical analyses for each case of emergency (traffic accidents,
medical emergency, physical trauma, and others)