13.3. Interpreting the slope and intercept, prediction and residuals#

Next, we’ll practice how to interpret the regression coefficients, and how to use the regression equation to get a predicted value of $y$.

Example 1: Weather patterns and crops#

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This is a real example from an old study by Pitt and Heady (1978) in the journal of Ecology. The study, on the “Response of Annual Vegetation to Temperature and Rainfall Patterns in Northern California”, found that the impact of weather patterns on standing crop (the total number of crops in a particular area at a given time), could be predicted by the following predictive equation: $\hat{y}=59.21+2.34x$, where

$y$ = June standing crop (in g/m2) and

$x$ = mean minimum temperature in November (in °C).

  • Use words to interpret the y-intercept and the slope.

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The y-intercept is the value of $y$ when $x$ equals zero. In this example, $x$ represents “mean minimum temperature in November”. The value of the $y$-intercept (in this case 59.21), represents the predicted June standing crop (in g/m2) rate when the temperature in November was zero.

The slope then tells us how much we can expect $y$ to change as $x$ increases/decreases. We need to consider the units of $x$ and of $y$ to understand the connection between these two variables. The unit of a slope is the unit of the $y$ variable, per units of the $x$ variable. For this example, we can say that the June standing crop rate is expected to increase by 2.34 units (g/m2), on average, per 1°C increase, in temperature.

  • Find the predicted June standing crop rates for a mean minimum temperature of 7.7°C. (Again, for now, use a calculator, Excel, or do a rough calculation on paper)

Click to reveal answer

$x = 7.7°C, \hat{y}̂ = 59.21+2.34*7.7 = 77.228 g/m^2$

  • Imagine we changed the units of temperature (which is our $x$-variable) so that they are in Fahrenheit rather than Celsius. Without doing any calculations, what do you think will happen to the regression coefficients? Will the slope value change, the intercept, both, or neither?

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Both the slope will change (because it is in different units), and the intercept (as the line will cross the y-axis at a different point)

Example 2: GDP vs CO2#

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For recent UN data from 39 countries on $y$ (per capita carbon dioxide emissions in metric tons per capita) and $x$ (per capita gross domestic product in thousands of dollars), the prediction equation is $\hat{y}̂=1.26+0.346x$.

Use your calculator or excel for the next couple of questions.

  • Predict $y$ at the (i) minimum $x$-value of 0.8, and (ii) maximum $x$ of 34.3.

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(i) $\hat{y} = 1.26 + 0.346*0.8 = 1.54$

(ii) $\hat{y} = 1.26 + 0.346*34.3$ = 13.1.

For the U.S., $x$ = 34.3 and $y$ = 19.7. Find the predicted carbon dioxide value. Find the residual and interpret.

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The residual is the difference between the observed and predicted value of ($y-\hat{y}$̂).

For the U.S., we know from the first question that, for $x$ = 34.3, $\hat{y}̂$=13.1. The observed value for $y$ = 19.7, then the residual $y - \hat{y}$ = 19.7 – 13.1 = 6.6. This indicates that the U.S. is producing 6.6 metric tons per capita, more CO2 emissions, than predicted by the regression line.

If the residual was a negative number, then it would suggest that that country is producing less CO2 emissions than predicted by the regression line.