Advanced High School Statistics: Second Edition, with updates based on AP© Statistics Course Framework

Subsection 8.4.1 Learning objectives

1. See how a log transformation can bring symmetry to an extremely skewed variable.

2. Recognize that data can often be transformed to produce a linear relationship, and that this transformation often involves log of the \(y\)-values and sometimes log of the \(x\)-values.

3. Use residual plots to assess whether a linear model for transformed data is reasonable.

Subsection 8.4.2 Introduction to transformations

There are some standard transformations that may be useful for strongly right skewed data where much of the data is positive but clustered near zero. A transformation is a rescaling of the data using a function. For instance, a plot of the logarithm (base 10) of county populations results in the new histogram in Figure 8.4.2.(b). This data is symmetric, and any potential outliers appear much less extreme than in the original data set. By reigning in the outliers and extreme skew, transformations like this often make it easier to build statistical models against the data.

Transformations can also be applied to one or both variables in a scatterplot. A scatterplot of the population change from 2010 to 2017 against the population in 2010 is shown in Figure 8.4.3.(a). In this first scatterplot, it's hard to decipher any interesting patterns because the population variable is so strongly skewed. However, if we apply a log\(_{10}\) transformation to the population variable, as shown in Figure 8.4.3.(b), a positive association between the variables is revealed. While fitting a line to predict population change (2010 to 2017) from population (in 2010) does not seem reasonable, fitting a line to predict population from log\(_{10}\)(population) does seem reasonable.

Transformations other than the logarithm can be useful, too. For instance, the square root (\(\sqrt{\text{ original observation } }\)) and inverse (\(\frac{1}{\text{ original observation } }\)) are commonly used by data scientists. Common goals in transforming data are to see the data structure differently, reduce skew, assist in modeling, or straighten a nonlinear relationship in a scatterplot.

Subsection 8.4.3 Transformations to achieve linearity

The regression equation is

y = -52.3564 + 2.7842 x

Predictor       Coef   SE Coef         T          P
Constant    -52.3564    7.2757    -7.196      3e-08
x             2.7842    0.1768    15.752    < 2e-16

S = 13.76    R-Sq = 88.26%    R-Sq(adj) = 87.91%

Regression analysis is easier to perform on linear data. When data are nonlinear, we sometimes transform the data in a way that makes the resulting relationship linear. The most common transformation is log of the \(y\) values. Sometimes we also apply a transformation to the \(x\) values. We generally use the residuals as a way to evaluate whether the transformed data are more linear. If so, we can say that a better model has been found.

The regression equation is

log(y) = 1.722540 + 0.052985 x

Predictor         Coef     SE Coef        T          P
Constant      1.722540    0.056731    30.36    < 2e-16
x             0.052985    0.001378    38.45    < 2e-16

S = 0.1073    R-Sq = 97.82%    R-Sq(adj) = 97.75%

Subsection 8.4.4 Section summary

• A transformation is a rescaling of the data using a function. When data are very skewed, a log transformation often results in more symmetric data.

• Regression analysis is easier to perform on linear data. When data are nonlinear, we sometimes transform the data in a way that results in a linear relationship. The most common transformation is log of the \(y\)-values. Sometimes we also apply a transformation to the \(x\)-values.

• To assess the model, we look at the residual plot of the transformed data. If the residual plot of the original data has a pattern, but the residual plot of the transformed data has no pattern, a linear model for the transformed data is reasonable, and the transformed model provides a better fit than the simple linear model.

Subsection 8.4.6 Chapter Highlights

This chapter focused on describing the linear association between two numerical variables and fitting a linear model.

• The correlation coefficient, \(r\text{,}\) measures the strength and direction of the linear association between two variables. However, \(r\) alone cannot tell us whether data follow a linear trend or whether a linear model is appropriate.

• The explained variance, \(R^2\text{,}\) measures the proportion of variation in the \(y\) values explained by a given model. Like \(r\text{,}\) \(R^2\) alone cannot tell us whether data follow a linear trend or whether a linear model is appropriate.

• Every analysis should begin with graphing the data using a scatterplot in order to see the association and any deviations from the trend (outliers or influential values). A residual plot helps us better see patterns in the data.

• When the data show a linear trend, we fit a least squares regression line of the form: \(\hat{y} = a+bx\text{,}\) where \(a\) is the \(y\)-intercept and \(b\) is the slope. It is important to be able to calculate \(a\) and \(b\) using the summary statistics and to interpret them in the context of the data.

• A residual, \(y-\hat{y}\text{,}\) measures the error for an individual point. The standard deviation of the residuals, \(s\text{,}\) measures the typical size of the residuals.

• \(\hat{y} = a+bx\) provides the best fit line for the observed data. To estimate or hypothesize about the slope of the population regression line, first confirm that the residual plot has no pattern and that a linear model is reasonable, then use a \(t\)-interval for the slope or a \(t\)-test for the slope with \(n-2\) degrees of freedom.

In this chapter we focused on simple linear models with one explanatory variable. More complex methods of prediction, such as multiple regression (more than one explanatory variable) and nonlinear regression can be studied in a future course.