Calculating Fitted Values for Given Predictors

From a Fitted Line to a Fitted Response

Once a simple regression line has been estimated, it can be used to calculate a fitted response for any specified value of the predictor. The calculation is short, but its interpretation matters: the result is the response value assigned by the fitted line at that predictor value. It is not automatically the actual response of an individual or object with that predictor value.

For example, suppose a fitted equation relates hours spent studying to a quiz score. Substituting a particular number of study hours gives the score on the fitted line at that value. Reading that number correctly requires keeping track of which quantity is the predictor, which is the response, and what the fitted line represents.

The Model and the Fitted Equation

In simple linear regression, let \(x\) denote a predictor value and \(Y\) a response variable. A standard model for the response at predictor value \(x\) is

Here \(\beta_0\) and \(\beta_1\) are the population intercept and slope, and \(\varepsilon\) represents the part of the response not described by the linear term. For the usual interpretation of the line as describing the mean response, the model assumes that the mean response changes linearly with \(x\), or equivalently that the errors have mean zero at each predictor value: \(E(\varepsilon\mid x)=0\). The estimated intercept and slope are written \(b_0\) and \(b_1\).

Once those estimates have been obtained, the estimated regression equation is

The symbol \(\hat{y}\) denotes the fitted response calculated from the estimated line. It is important to distinguish \(\hat{y}\) from \(Y\), which denotes the response variable, and from a particular observed response, often written \(y\). The fitted equation gives a calculated value; an observed response may differ from it.

To calculate a fitted response, use the coefficients in the estimated equation as given, substitute the specified predictor value for \(x\), and evaluate the expression. The intercept and slope have units determined by the variables: \(b_0\) is in response units, while \(b_1\) is in response units per predictor unit. Thus \(b_1x_0\) has response units and can be added to \(b_0\).

Worked Calculations

Why Substitution Gives a Point on the Line

A line described by \(\hat{y}=b_0+b_1x\) consists of the ordered pairs whose second coordinate equals \(b_0+b_1\) times the first coordinate. When the first coordinate is specified as \(x_0\), the corresponding second coordinate is obtained by replacing \(x\) with \(x_0\). This gives the point on the line without making any claim about whether a particular observed data point is located there.

The justification is simply substitution under the stated condition:

The condition \(x=x_0\) is essential. The equation \(\hat{y}=b_0+b_1x\) describes all points on the line; by itself it does not say that the predictor coordinate equals any particular specified value. Adding that condition makes clear why the calculated pair is the point on the line at \(x_0\).

The arithmetic calculation itself requires only an estimated equation and a specified predictor value. The linear-mean and zero-mean-error assumptions are needed for the usual statistical interpretation of the fitted line as an estimate of the mean response pattern. Normal errors are not required just to perform the substitution.

Interpreting Fitted Values Carefully

A fitted response is a model-based value, not a newly observed measurement. If the response is measured in quiz points, the fitted response is in quiz points; if it is measured in kilowatt-hours, the fitted response is in kilowatt-hours. Keeping those units visible helps prevent confusing the predictor value with the calculated response.

Under the linear-mean model, the fitted value at \(x_0\) can be interpreted as the estimated mean response at that predictor value. It should not automatically be described as the response of a particular individual. Individual outcomes can vary around the mean relationship. To assert that an individual will have exactly the fitted value would overstate what the equation says.

The fitted intercept \(b_0\) is the fitted response when \(x=0\), since \(b_0+b_1(0)=b_0\). Whether that value has a useful practical meaning depends on what zero represents in context. The fitted slope \(b_1\) describes the change in fitted response for a one-unit increase in the predictor. These coefficient interpretations help check the direction and scale of a substitution, but the fitted response still comes from evaluating the full equation at the specified \(x_0\).

Common Mistake: Confusing the Equation With an Observation

A frequent error is to calculate \(\hat{y}\) and then report it as the observed response. For instance, after obtaining a fitted score of 65 points at 5 hours of study, it would be incorrect to say that a student who studied for 5 hours necessarily scored 65. The calculation establishes only that the fitted line assigns 65 points to \(x=5\).

Another common error is to omit a negative sign or to substitute a response value where the predictor value belongs. A useful check is to write the predictor and response with their units before calculating, then verify that the final answer has response units. In the electricity example, \(19\) is in degrees Celsius and is substituted for \(x\); the computed \(36.4\) is in kilowatt-hours and is the fitted response.