**This tutorial shows you**:

- how to specify quadratic terms in regression models
- how to explore nonlinear relationships using lo(w)ess smoothers and generalized additive models
- how to use residuals to interpret model quality

**Note on copying & pasting code from the PDF version of this tutorial**: Please note that you may run into trouble if you copy & paste code from the PDF version of this tutorial into your R script. When the PDF is created, some characters (for instance, quotation marks or indentations) are converted into non-text characters that R won’t recognize. To use code from this tutorial, please type it yourself into your R script or you may copy & paste code from the *source file* for this tutorial which is posted on my website.

**Note on R functions discussed in this tutorial**: I don’t discuss many functions in detail here and therefore I encourage you to look up the help files for these functions or search the web for them before you use them. This will help you understand the functions better. Each of these functions is well-documented either in its help file (which you can access in R by typing `?ifelse`

, for instance) or on the web. The *Companion to Applied Regression* (see our syllabus) also provides many detailed explanations.

As always, please note that this tutorial only accompanies the other materials for Day 12 and that you are expected to have worked through the reading for that day before tackling this tutorial.

So far, we have not encountered serious violations of the assumption of linearity - a linear relationship between predictors and outcome. But this assumption simply means that we impose a linear structure on the relationship between \(x\) and \(y\). Coefficient estimates alone from a regression model will not reveal whether the relationship between \(x\) and \(y\) in your data actually are linear, but a scatterplot will be useful to investigate whether this might be the case.

Theories might often make predictions of the form, “as \(x\) increases, \(y\) first increases, and then drops again”. An example for this is the Kuznets curve in economics, suggesting that as countries developed, income inequality first increased, peaked, and then decreased (summarized, for instance, in Acemoglu and Robinson 2002). This implies a so-called curvilinear relationship between economic development and inequality: both poor and rich countries have low inequality, but middle-income countries should exhibit high levels of inequality.

Take the following example:

```
##
## Call:
## lm(formula = y ~ x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -91.650 -5.757 3.239 9.980 28.822
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -8.1480 0.8235 -9.895 < 2e-16 ***
## x 0.8155 0.2559 3.186 0.00155 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17.02 on 425 degrees of freedom
## Multiple R-squared: 0.02333, Adjusted R-squared: 0.02103
## F-statistic: 10.15 on 1 and 425 DF, p-value: 0.001547
```

Perhaps you might notice the low \(R^2\) value, but that itself is not indicative of problems. Examining the residual plots, however, reveals that the the model produces residuals that are grouped below 0 at low and high values of \(x\):