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9.3 Interpreting the Parameter Estimates for a Regression Model

Previously, we used the lm() function to fit the Height model of Thumb and saved it as Height_model:

Height_model <- lm(Thumb ~ Height, data = Fingers)

Let’s now look at the parameter estimates for this model and see how to interpret them. Use the code block below to print out the parameter estimates for the height model.

library(coursekata) # saves the Height model Height_model <- lm(Thumb ~ Height, data = Fingers) # print it out # saves the Height model Height_model <- lm(Thumb ~ Height, data = Fingers) # print it out Height_model ex() %>% check_output_expr("Height_model")
CK Code: B5_Code_Regression_01
Call:
lm(formula = Thumb ~ Height, data = Fingers)

Coefficients:
(Intercept)       Height  
    -3.3295       0.9619

The Intercept corresponds to \(b_0\) and the Height coefficient corresponds to \(b_1\). We can write our fitted model as:

\[\text{Thumb}_i=-3.33 + 0.96\text{Height}_i+e_i\]

Or, equivalently, using GLM notation, it can be written:

\[Y_i=-3.33 + 0.96X_i+e_i\]

\(b_0\), which is -3.33, is the y-intercept. It’s the predicted \(Y_i\) (Thumb) when \(X_i\) (Height) equals 0.

Neither a height of 0 inches nor a thumb length of -3.33 mm are possible. Not all predictions from a regression model make sense. We should always be thinking about which values of the predictors, and which predictions, are reasonable.

How Regression Models Make Predictions

We can use the Height model to predict the thumb length of students of different heights (just like we used the Height2Group model to predict the thumb length of short and tall groups of students).

Recall that thumb length (and predicted thumb length) are expressed in millimeters. \(b_0\) (-3.33) is the predicted thumb length in millimeters for a student with a height of 0 inches. If we stretch out the x-axis to include 0, we would expect the regression line to cross the y-axis at -3.33. (Notice, however, that in the plot below that there are no actual students who are 0 inches in height, for obvious reasons!)

A scatterplot of Thumb by Height overlaid with the regression line in red. A vertical arrow points to the part of the regression line where the x-axis equals zero and has the caption: when Height equals zero, predicted Thumb equals -3.33.

The \(b_1\) estimate (0.96) is the slope: for every 1 unit increase in Height, our model predicts a 0.96 unit increase in Thumb. The fact that height is measured in inches and thumb length in millimeters is not a problem; the regression line is a function (the \(b_0 + b_1Height_i\) part) that takes in inches and then makes a prediction in millimeters. This means that students who are 1 inch taller are predicted by our model to have thumbs that are 0.96 millimeters longer (on average). Here’s a visual representation:

default scale zooming in

On the left, a scatterplot of Thumb predicted by Height with the regression model overlaid. The plot is depicted at the default scale, thus, the slope appears very small and difficult to see.

On the right, a scatterplot of Thumb predicted by Height with the regression model overlaid. The plot is depicted at a zoomed in scale, thus, it is easier to point out that the slope, or b-sub-1, of the model can be represented as the vertical distance the regression line rises (a rise of 0.96), for every one unit increase in Height.


The predicted thumb length of a student who is 71 inches tall is 64.83 mm. This is the value of \(Y\) (Thumb) on the regression line when \(X\) (Height) is 71, as visualized below:

A scatterplot of Thumb by Height overlaid with the regression line in red. A dashed line extends from the x-axis where Height equals 71 up to the regression line. It intersects with a point along the regression line where the predicted Thumb equals 64.83

Regression Coefficients are Not Symmetrical

When you fit a regression model, it matters which variable is the outcome and which is the explanatory variable. For example, if you fit the model Thumb ~ Height you won’t get the same y-intercept and slope you would if you fit the model Height ~ Thumb.

Call: lm(formula = Thumb ~ Height, data = Fingers)

Coefficients: (Intercept) Height
-3.3295 0.9619

Call: lm(formula = Height ~ Thumb, data = Fingers)

Coefficients: (Intercept) Thumb
56.391 0.159


The reason for this is that the units, and the distributions of the variables, are different. If the outcome is Thumb, then the slope is the adjustment to predicted thumb length for a one-inch increase in height. But if the outcome is height, then the slope is the adjustment to predicted height length for a one-millimeter increase in thumb length. These are two entirely different things.

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