Comparing Predicted Responses Across Two Values

Comparing Two Predictions

A fitted regression line can provide a predicted response at a chosen explanatory value. Sometimes the question is not simply “What response does the line predict at this value?” but “How different are the responses predicted at these two values?” The slope gives a direct answer: multiply the change in the explanatory value by the fitted slope.

This comparison concerns two predictions from the same fitted line. It describes how the line’s predicted response changes as the explanatory value changes. It does not say that a particular individual’s response will change by that amount, nor does it compare two observed responses.

The Fitted Difference Formula

Write the fitted regression line as \(\hat{y}=b_0+b_1x\). Here, \(x\) is a value of the explanatory variable, \(\hat{y}\) is the response predicted by the line at that value, \(b_0\) is the fitted intercept, and \(b_1\) is the fitted slope. The slope gives the line’s predicted response change for a one-unit increase in \(x\), measured in response units per explanatory-variable unit.

The order matters. The expression \(\hat{y}(x_2)-\hat{y}(x_1)\) compares the prediction at \(x_2\) with the prediction at \(x_1\). If the slope is positive and \(x_2>x_1\), the difference is positive. If the slope is negative and \(x_2>x_1\), it is negative. Reversing the order of the two explanatory values reverses the sign.

You may calculate both fitted responses and subtract, or use the slope formula directly. The direct method is often quicker; calculating both predictions can make the comparison easier to check and explain.

Why the Intercept Cancels

The direct formula follows from substituting each explanatory value into the same fitted line and subtracting. This is an algebraic property of the line, not a separate statistical assumption.

The intercept cancels because the same fitted line is used for both predictions. Thus, to find their difference, you do not need to know the intercept. The slope and the difference between the two explanatory values are enough. If the two explanatory values are equal, their predicted responses are equal and the difference is zero.

This result applies to any two values substituted into the same fitted linear equation. It does not itself establish that the line is an appropriate description of the relationship in a particular application. The calculations below take the fitted line as given; their interpretations are specifically about the predictions made by that line.

Direction, Units, and Meaning

A slope-based comparison should be reported with its direction and units. The sign tells whether the prediction at the second value is higher or lower than the prediction at the first. The magnitude gives the size of the difference in response units. Keep the explanatory-variable units in the multiplication as well: the slope’s units cancel the explanatory units, leaving response units.

For example, if a fitted slope is \(-0.6\) kilograms per week and \(x\) is measured in weeks, a positive increase of 4 weeks gives a fitted difference of \((-0.6)(4)=-2.4\) kilograms. The negative sign means the predicted response at the later value is 2.4 kilograms lower. The line does not guarantee that every individual’s response decreases by that amount.

Changing the Explanatory-Value Units

The slope’s numerical value depends on the units used for the explanatory variable, but the predicted-response difference does not change when the same explanatory values are correctly converted. The slope and the size of the explanatory-value change adjust in opposite ways.

Difference Between a Prediction and a Predicted Difference

A prediction answers a question about one explanatory value: “What response does the line predict at \(x=17\)?” A predicted difference answers a question about two values: “How far apart are the line’s predictions at \(x=17\) and \(x=12\)?” These are different quantities, even though both come from the same equation.

The predicted difference also differs from a difference between observed responses. The fitted line supplies \(\hat{y}(x_1)\) and \(\hat{y}(x_2)\), not the actual observed outcomes for two individuals. Actual outcomes can differ from the fitted predictions. The comparison formula does not remove that individual variation or state what either person’s observed response will be.

Nor does a positive or negative slope, by itself, show that changing \(x\) causes a change in \(y\). The calculation describes the fitted relationship between the explanatory and response variables. A causal claim requires support beyond the arithmetic of a regression line.

Common Mistake: Reporting the Difference as an Individual Change

A common error is to calculate a predicted difference correctly and then say, “An individual who increases \(x\) by five units will change their response by this amount.” That conclusion treats the fitted comparison as a guaranteed within-person outcome. The line only gives two predicted responses; it does not guarantee that either observed response equals its prediction.

A better interpretation names both values and keeps the result attached to the line: “The fitted line predicts a response that is 9.0 points higher at 17 hours than at 12 hours.” Also check the subtraction order and the units before writing the sentence. If the calculated difference is negative, state that the second prediction is lower, rather than dropping the sign and leaving the direction unclear.