OptinMon 30 - Four Critical Steps in Building Linear Regression Models

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.

15 comments

R² is such a lovely statistic, isn’t it?  Unlike so many of the others, it makes sense–the percentage of variance in Y accounted for by a model.

I mean, you can actually understand that.  So can your grandmother.  And the clinical audience you’re writing the report for.

A big R² is always good and a small one is always bad, right?

Well, maybe. (more…)

57 comments

You may have noticed conflicting advice about whether to leave insignificant effects in a model or take them out in order to simplify the model.

One effect of leaving in insignificant predictors is on p-values–they use up precious df in small samples. But if your sample isn’t small, the effect is negligible.

The bigger effect is  on interpretation, and really the above cases are about whether it aids interpretation to leave them in. Models do get so cluttered it’s hard to figure out what’s going on, and it makes sense to eliminate effects that aren’t serving a purpose, but even insignificant effects can have a purpose. (more…)

15 comments

“Everything should be made as simple as possible, but no simpler” – Albert Einstein*

For some reason, I’ve heard this quotation 3 times in the past 3 days.  Maybe I hear it everyday, but only noticed because I’ve been working with a few clients on model selection, and deciding how much to simplify a model.

And when the quotation fits, use it. (That’s the saying, right?)

*For the record, a quick web search indicated this may be a paraphrase, but it still applies.

The quotation is the general goal of model selection.  You really do want the model to be as simple as possible, but still able to answer the research question of interest.

This applies to many areas of model selection.  Here are a few examples: (more…)

12 comments

Model  Building–choosing predictors–is one of those skills in statistics that is difficult to teach.   It’s hard to lay out the steps, because at each step, you have to evaluate the situation and make decisions on the next step.

If you’re running purely predictive models, and the relationships among the variables aren’t the focus, it’s much easier.  Go ahead and run a stepwise regression model.  Let the data give you the best prediction.

But if the point is to answer a research question that describes relationships, you’re going to have to get your hands dirty.

It’s easy to say “use theory” or “test your research question” but that ignores a lot of practical issues.  Like the fact that you may have 10 different variables that all measure the same theoretical construct, and it’s not clear which one to use. (more…)

9 comments

If you’ve compared two textbooks on linear models, chances are, you’ve seen two different lists of assumptions.

I’ve spent a lot of time trying to get to the bottom of this, and I think it comes down to a few things.

1. There are four assumptions that are explicitly stated along with the model, and some authors stop there.

2. Some authors are writing for introductory classes, and rightfully so, don’t want to confuse students with too many abstract, and sometimes untestable, (more…)

4 comments