8.3 Reading a model

There are two distinct ways that you can read a model.

  • Read out the model value. Plug in specific values for the explanatory variables and read out the resulting model value. For instance, an input temperature of 35° produces a predicted output gas usage of 125 ccf. Or a single, 30-year old female has a model value of $8.10 per hour. (Remember, the model is based on data from 1985!)

  • Characterize the relationship described by the model. In contrast to reading out a model value for some specific values of the explanatory variables, here interest is in the overall relationship: how gas usage depends on temperature; how wages depend on sex or marital status or age.

Reading out the model value is useful when you want to make a prediction (What would the gas usage be if the temperature were 10° degrees?) or when you want to compare the actual value of the response variable to what the model says is a typical value. (Is the gas usage lower than expected in the 49° month, perhaps due to my new furnace?).

Characterizing the relationship is useful when you want to make statements about broad patterns that go beyond individual cases. Is there really a connection between marital status and wage? Which way does it go?

The shape of the model function tells you about such broad relationships. Reading the shape from a graph of the model is not difficult.

For a quantitative explanatory variable, e.g., temperature or age, the model form is a continuous curve or line. An extremely important aspect of this curve is its slope. For the model of gas usage, the slope is down to the right: a negative slope. This means that as temperature increases, the gas usage goes down. In contrast, for the model of wages, the slope is up to the right: a positive slope. This means that as age increases, the wage goes up.

For categorical variables, slopes don’t apply. Instead, the pattern can be described in terms of differences. In the model where wage is explained only by sex, the difference between typical wages for males and females is 2.12 dollars per hour.

When there is more than one explanatory variable, there will be a distinct slope or difference associated with each.