Robert I. Kabacoff - R in action

The graph in figure 6.16 allows you to see the mpg for each make of car on the same horizontal axis. Dot plots typically become most interesting when they’re sorted and grouping factors are distinguished by symbol and color. An example is given in the following listing.

Listing 6.11 Dot plot grouped, sorted, and colored

x <- mtcars[order(mtcars$mpg),] x$cyl <- factor(x$cyl) x$color[x$cyl==4] <- "red" x$color[x$cyl==6] <- "blue" x$color[x$cyl==8] <- "darkgreen" dotchart(x$mpg,

labels = row.names(x), cex=.7,

groups = x$cyl, gcolor = "black", color = x$color, pch=19,

main = "Gas Mileage for Car Models\ngrouped by cylinder", xlab = "Miles Per Gallon")

In this example, the data frame mtcars is sorted by mpg (lowest to highest) and saved as data frame x. The numeric vector cyl is transformed into a factor. A character vector (color) is added to data frame x and contains the values "red", "blue", or "darkgreen" depending on the value of cyl. In addition, the labels for the data points are taken from the row names of the data frame (car makes). Data points are grouped by number of cylinders. The numbers 4, 6, and 8 are printed in black. The color of the points and labels are derived from the color vector, and points are represented by filled circles. The code produces the graph in figure 6.17.

In figure 6.17, a number of features become evident for the first time. Again, you see an increase in gas mileage as the number of cylinders decrease. But you also see exceptions. For example, the Pontiac Firebird, with eight cylinders, gets higher gas mileage than the Mercury 280C and the Valiant, each with six cylinders. The Hornet 4 Drive, with six cylinders, gets the same miles per gallon as the Volvo 142E, which has four cylinders. It’s also clear that the Toyota Corolla gets the best gas mileage by far, whereas the Lincoln Continental and Cadillac Fleetwood are outliers on the low end.

You can gain significant insight from a dot plot in this example because each point is labeled, the value of each point is inherently meaningful, and the points are arranged in a manner that promotes comparisons. But as the number of data points increase, the utility of the dot plot decreases.

NOTE There are many variations of the dot plot. Jacoby (2006) provides a very informative discussion of the dot plot and provides R code for innovative applications. Additionally, the Hmisc package offers a dot plot function (aptly named dotchart2) with a number of additional features.

Gas Milage for Car Models grouped by cylinder

4

Toyota Corolla Fiat 128 Lotus Europa Honda Civic Fiat X1−9

Porsche 914−2 Merc 240D Merc 230 Datsun 710 Toyota Corona Volvo 142E

6

Hornet 4 Drive

Mazda RX4 Wag Mazda RX4 Ferrari Dino Merc 280 Valiant

Merc 280C

8

Pontiac Firebird Hornet Sportabout Merc 450SL

Merc 450SE

Ford Pantera L Dodge Challenger AMC Javelin Merc 450SLC Maserati Bora Chrysler Imperial Duster 360 Camaro Z28

Lincoln Continental Cadillac Fleetwood

Miles Per Gallon

Figure 6.17 Dot plot of mpg for car models grouped by number of cylinders

6.7Summary

In this chapter, we learned how to describe continuous and categorical variables. We saw how bar plots and (to a lesser extent) pie charts can be used to gain insight into the distribution of a categorical variable, and how stacked and grouped bar charts can help us understand how groups differ on a categorical outcome. We also explored how histograms, kernel density plots, box plots, rug plots, and dot plots can help us visualize the distribution of continuous variables. Finally, we explored how overlapping kernel density plots, parallel box plots, and grouped dot plots can help you visualize group differences on a continuous outcome variable.

In later chapters, we’ll extend this univariate focus to include bivariate and multivariate graphical methods. You’ll see how to visually depict relationships among many variables at once, using such methods as scatter plots, multigroup line plots, mosaic plots, correlograms, lattice graphs, and more.

In the next chapter, we’ll look at basic statistical methods for describing distributions and bivariate relationships numerically, as well as inferential methods for evaluating whether relationships among variables exist or are due to sampling error.

■What’s the correlation between income and life expectancy? Is it significantly different from zero?

■Are you more likely to receive imprisonment for a crime in different regions of the United States? Are the differences between regions statistically significant?

In this chapter we’ll review R functions for generating basic descriptive and inferential statistics. First we’ll look at measures of location and scale for quantitative variables. Then we’ll learn how to generate frequency and contingency tables (and associated chi-square tests) for categorical variables. Next, we’ll examine the various forms of correlation coefficients available for continuous and ordinal variables. Finally, we’ll turn to the study of group differences through parametric (t-tests) and nonparametric (Mann–Whitney U test, Kruskal–Wallis test) methods. Although our focus is on numerical results, we’ll refer to graphical methods for visualizing these results throughout.

The statistical methods covered in this chapter are typically taught in a first-year undergraduate statistics course. If these methodologies are unfamiliar to you, two excellent references are McCall (2000) and Snedecor & Cochran (1989). Alternatively, there are many informative online resources available (such as Wikipedia) for each of the topics covered.

7.1Descriptive statistics

In this section, we’ll look at measures of central tendency, variability, and distribution shape for continuous variables. For illustrative purposes, we’ll use several of the variables from the Motor Trend Car Road Tests (mtcars) dataset you first saw in chapter 1. Our focus will be on miles per gallon (mpg), horsepower (hp), and weight (wt).

>vars <- c("mpg", "hp", "wt")

>head(mtcars[vars])

First we’ll look at descriptive statistics for all 32 cars. Then we’ll examine descriptive statistics by transmission type (am) and number of cylinders (cyl). Transmission type is a dichotomous variable coded 0=automatic, 1=manual, and the number of cylinders can be 4, 5, or 6.

7.1.1A menagerie of methods

When it comes to calculating descriptive statistics, R has an embarrassment of riches. Let’s start with functions that are included in the base installation. Then we’ll look at extensions that are available through the use of user-contributed packages.

In the base installation, you can use the summary() function to obtain descriptive statistics. An example is presented in the following listing.

The summary() function provides the minimum, maximum, quartiles, and the mean for numerical variables and frequencies for factors and logical vectors. You can use the apply() or sapply() function from chapter 5 to provide any descriptive statistics you choose. For the sapply() function, the format is

sapply(x, FUN, options)

where x is your data frame (or matrix) and FUN is an arbitrary function. If options are present, they’re passed to FUN. Typical functions that you can plug in here are mean, sd, var, min, max, median, length, range, and quantile. The function fivenum() returns Tukey’s five-number summary (minimum, lower-hinge, median, upper-hinge, and maximum).

Surprisingly, the base installation doesn’t provide functions for skew and kurtosis, but you can add your own. The example in the next listing provides several descriptive statistics, including skew and kurtosis.

Listing 7.2 Descriptive statistics via sapply()

> mystats <- function(x, na.omit=FALSE){

For cars in this sample, the mean mpg is 20.1, with a standard deviation of 6.0. The distribution is skewed to the right (+0.61) and somewhat flatter than a normal distribution (–0.37). This will be most evident if you graph the data. Note that if

you’d wanted to omit missing values, you would have used sapply(mtcars[vars], mystats, na.omit=TRUE).

EXTENSIONS

Several user-contributed packages offer functions for descriptive statistics, including Hmisc, pastecs, and psych. Because these packages aren’t included in the base distribution, you’ll need to install them on first use (see chapter 1, section 1.4).

The describe() function in the Hmisc package returns the number of variables and observations, the number of missing and unique values, the mean, quantiles, and the five highest and lowest values. An example is provided in the following listing.

Listing 7.3 Descriptive statistics via describe() in the Hmisc package()

>library(Hmisc)

>describe(mtcars[vars])

3 Variables 32 Observations

---------------------------------------------------------------------------

---------------------------------------------------------------------------

---------------------------------------------------------------------------

lowest : 1.513 1.615 1.835 1.935 2.140, highest: 3.845 4.070 5.250 5.345 5.424

---------------------------------------------------------------------------

The pastecs package includes a function named stat.desc() that provides a wide range of descriptive statistics. The format is

stat.desc(x, basic=TRUE, desc=TRUE, norm=FALSE, p=0.95)

where x is a data frame or time series. If basic=TRUE (the default), the number of values, null values, missing values, minimum, maximum, range, and sum are provided. If desc=TRUE (also the default), the median, mean, standard error of the mean, 95 percent confidence interval for the mean, variance, standard deviation, and coefficient of variation are also provided. Finally, if norm=TRUE (not the default), normal distribution statistics are returned, including skewness and kurtosis (and their statistical significance), and the Shapiro–Wilk test of normality. A p-value option is used to calculate the confidence interval for the mean (.95 by default). Listing 7.4 gives an example.

Listing 7.4 Descriptive statistics via stat.desc() in the pastecs package

>library(pastecs)

>stat.desc(mtcars[vars])

As if this isn’t enough, the psych package also has a function called describe() that provides the number of nonmissing observations, mean, standard deviation, median, trimmed mean, median absolute deviation, minimum, maximum, range, skew, kurtosis, and standard error of the mean. You can see an example in the following listing.

I told you that it was an embarrassment of riches!

NOTE In the previous examples, the packages psych and Hmisc both provided functions named describe(). How does R know which one to use? Simply put, the package last loaded takes precedence, as seen in listing 7.5. Here, psych is loaded after Hmisc, and a message is printed indicating that the describe() function in Hmisc is masked by the function in psych. When you type in the describe() function and R searches for it, R comes to the psych package first and executes it. If you want the Hmisc version instead, you can type Hmisc::describe(mt). The function is still there.