You might be surprised to hear that not only can linear regression fit lines between a response variable Y and one or more predictor variables, X, it can fit curves too. There are many ways to do this, but the simplest is by adding a polynomial term.
So what is a polynomial term and how do you know you need one?
The linear parameters in a regression model
A linear regression model has a few key parameters. These include the intercept coefficient, the slope coefficient, and the residual variance.
That intercept defines the height of the regression line. It does so by measuring the height of the line at one specific point: when all X = 0.
The slope defines how much Y differs, on average, for each one unit difference in X. In other words, it measures the constant relationship between X and Y. Yes, there can be multiple Xs and each one has its own slope.
A polynomial term–a quadratic (squared) or cubic (cubed) term turns a linear regression model into a curve.
(more…)
In Part 3 we used the lm() command to perform least squares regressions. In Part 4 we will look at more advanced aspects of regression models and see what R has to offer.
One way of checking for non-linearity in your data is to fit a polynomial model and check whether the polynomial model fits the data better than a linear model. However, you may also wish to fit a quadratic or higher model because you have reason to believe that the relationship between the variables is inherently polynomial in nature. Let’s see how to fit a quadratic model in R.
We will use a data set of counts of a variable that is decreasing over time. Cut and paste the following data into your R workspace.
A <- structure(list(Time = c(0, 1, 2, 4, 6, 8, 9, 10, 11, 12, 13,
14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30),
Counts = c(126.6, 101.8, 71.6, 101.6, 68.1, 62.9, 45.5, 41.9,
46.3, 34.1, 38.2, 41.7, 24.7, 41.5, 36.6, 19.6,
22.8, 29.6, 23.5, 15.3, 13.4, 26.8, 9.8, 18.8, 25.9, 19.3)), .Names = c("Time", "Counts"),
row.names = c(1L, 2L, 3L, 5L, 7L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 19L, 20L, 21L, 22L, 23L, 25L, 26L, 27L, 28L, 29L, 30L, 31L),
class = "data.frame")
Let’s attach the entire dataset so that we can refer to all variables directly by name.
attach(A)
names(A)
First, let’s set up a linear model, though really we should plot first and only then perform the regression.
linear.model <-lm(Counts ~ Time)
We now obtain detailed information on our regression through the summary() command.
summary(linear.model)
Call:
lm(formula = Counts ~ Time)
Residuals:
Min 1Q Median 3Q Max
-20.084 -9.875 -1.882 8.494 39.445
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 87.1550 6.0186 14.481 2.33e-13 ***
Time -2.8247 0.3318 -8.513 1.03e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 15.16 on 24 degrees of freedom
Multiple R-squared: 0.7512, Adjusted R-squared: 0.7408
F-statistic: 72.47 on 1 and 24 DF, p-value: 1.033e-08
The model explains over 74% of the variance and has highly significant coefficients for the intercept and the independent variable and also a highly significant overall model p-value. However, let’s plot the counts over time and superpose our linear model.
plot(Time, Counts, pch=16, ylab = "Counts ", cex.lab = 1.3, col = "red" )
abline(lm(Counts ~ Time), col = "blue")
Here the syntax cex.lab = 1.3 produced axis labels of a nice size.
The model looks good, but we can see that the plot has curvature that is not explained well by a linear model. Now we fit a model that is quadratic in time. We create a variable called Time2 which is the square of the variable Time.
Time2 <- Time^2
quadratic.model <-lm(Counts ~ Time + Time2)
Note the syntax involved in fitting a linear model with two or more predictors. We include each predictor and put a plus sign between them.
summary(quadratic.model)
Call:
lm(formula = Counts ~ Time + Time2)
Residuals:
Min 1Q Median 3Q Max
-24.2649 -4.9206 -0.9519 5.5860 18.7728
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 110.10749 5.48026 20.092 4.38e-16 ***
Time -7.42253 0.80583 -9.211 3.52e-09 ***
Time2 0.15061 0.02545 5.917 4.95e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 9.754 on 23 degrees of freedom
Multiple R-squared: 0.9014, Adjusted R-squared: 0.8928
F-statistic: 105.1 on 2 and 23 DF, p-value: 2.701e-12
Our quadratic model is essentially a linear model in two variables, one of which is the square of the other. We see that however good the linear model was, a quadratic model performs even better, explaining an additional 15% of the variance. Now let’s plot the quadratic model by setting up a grid of time values running from 0 to 30 seconds in increments of 0.1s.
timevalues <- seq(0, 30, 0.1)
predictedcounts <- predict(quadratic.model,list(Time=timevalues, Time2=timevalues^2))
plot(Time, Counts, pch=16, xlab = "Time (s)", ylab = "Counts", cex.lab = 1.3, col = "blue")
Now we include the quadratic model to the plot using the lines() command.
lines(timevalues, predictedcounts, col = "darkgreen", lwd = 3)
The quadratic model appears to fit the data better than the linear model. We will look again at fitting curved models in our next blog post.
About the Author: David Lillis has taught R to many researchers and statisticians. His company, Sigma Statistics and Research Limited, provides both on-line instruction and face-to-face workshops on R, and coding services in R. David holds a doctorate in applied statistics.
See our full R Tutorial Series and other blog posts regarding R programming.
