Robert I. Kabacoff - R in action

where y is numeric and x is a dichotomous variable, or

wilcox.test(y1, y2)

where y1 and y2 are the outcome variables for each group. The optional data argument refers to a matrix or data frame containing the variables. The default is a two-tailed test. You can add the option exact to produce an exact test, and alternative="less" or alternative="greater" to specify a directional test.

If you apply the Mann–Whitney U test to the question of incarceration rates from the previous section, you’ll get these results:

> with(UScrime, by(Prob, So, median))

> wilcox.test(Prob ~ So, data=UScrime)

Wilcoxon rank sum test

data: Prob by So

W = 81, p-value = 8.488e-05

alternative hypothesis: true location shift is not equal to 0

Again, you can reject the hypothesis that incarceration rates are the same in Southern and non-Southern states (p < .001).

The Wilcoxon signed rank test provides a nonparametric alternative to the dependent sample t-test. It’s appropriate in situations where the groups are paired and the assumption of normality is unwarranted. The format is identical to the Mann–Whitney U test, but you add the paired=TRUE option. Let’s apply it to the unemployment question from the previous section:

>sapply(UScrime[c("U1","U2")], median) U1 U2

92 34

>with(UScrime, wilcox.test(U1, U2, paired=TRUE))

Wilcoxon signed rank test with continuity correction

data: U1 and U2

V = 1128, p-value = 2.464e-09

alternative hypothesis: true location shift is not equal to 0

Again, you’d reach the same conclusion reached with the paired t-test.

In this case, the parametric t-tests and their nonparametric equivalents reach the same conclusions. When the assumptions for the t-tests are reasonable, the

parametric tests will be more powerful (more likely to find a difference if it exists). The nonparametric tests are more appropriate when the assumptions are grossly unreasonable (for example, rank ordered data).

7.5.2Comparing more than two groups

When there are more than two groups to be compared, you must turn to other methods. Consider the state.x77 dataset from section 7.4. It contains population, income, illiteracy rate, life expectancy, murder rate, and high school graduation rate data for US states. What if you want to compare the illiteracy rates in four regions of the country (Northeast, South, North Central, and West)? This is called a one-way design, and there are both parametric and nonparametric approaches available to address the question.

If you can’t meet the assumptions of ANOVA designs, you can use nonparametric methods to evaluate group differences. If the groups are independent, a Kruskal– Wallis test will provide you with a useful approach. If the groups are dependent (for example, repeated measures or randomized block design), the Friedman test is more appropriate.

The format for the Kruskal–Wallis test is

kruskal.test(y ~ A, data)

where y is a numeric outcome variable and A is a grouping variable with two or more levels (if there are two levels, it’s equivalent to the Mann–Whitney U test). For the Friedman test, the format is

friedman.test(y ~ A | B, data)

where y is the numeric outcome variable, A is a grouping variable, and B is a blocking variable that identifies matched observations. In both cases, data is an option argument specifying a matrix or data frame containing the variables.

Let’s apply the Kruskal–Wallis test to the illiteracy question. First, you’ll have to add the region designations to the dataset. These are contained in the dataset state. region distributed with the base installation of R.

states <- as.data.frame(cbind(state.region, state.x77))

Now you can apply the test:

> kruskal.test(Illiteracy ~ state.region, data=states)

Kruskal-Wallis rank sum test

data: states$Illiteracy by states$state.region Kruskal-Wallis chi-squared = 22.7, df = 3, p-value = 4.726e-05

The significance test suggests that the illiteracy rate isn’t the same in each of the four regions of the country (p <.001).

Although you can reject the null hypothesis of no difference, the test doesn’t tell you which regions differ significantly from each other. To answer this question, you

7.6Visualizing group differences

In sections 7.4 and 7.5, we looked at statistical methods for comparing groups. Examining group differences visually is also a crucial part of a comprehensive data analysis strategy. It allows you to assess the magnitude of the differences, identify any distributional characteristics that influence the results (such as skew, bimodality, or outliers), and evaluate the appropriateness of the test assumptions. R provides a wide range of graphical methods for comparing groups, including box plots (simple, notched, and violin), covered in section 6.5; overlapping kernel density plots, covered in section 6.4.1; and graphical methods of assessing test assumptions, discussed in chapter 9.

7.7Summary

In this chapter, we reviewed the functions in R that provide basic statistical summaries and tests. We looked at sample statistics and frequency tables, tests of independence and measures of association for categorical variables, correlations between quantitative variables (and their associated significance tests), and comparisons of two or more groups on a quantitative outcome variable.

In the next chapter, we’ll explore simple and multiple regression, where the focus is on understanding relationships between one (simple) or more than one (multiple) predictor variables and a predicted or criterion variable. Graphical methods will help you diagnose potential problems, evaluate and improve the fit of your models, and uncover unexpected gems of information in your data.

Part 3

Intermediate methods

While part 2 covered basic graphical and statistical methods, section 3 offers coverage of intermediate methods. We move from describing the relationship between two variables, to modeling the relationship between a numerical outcome variable and a set of numeric and/or categorical predictor variables.

Chapter 8 introduces regression methods for modeling the relationship between a numeric outcome variable and a set of one or more predictor variables. Modeling data is typically a complex, multistep, interactive process. Chapter 8 provides step-by-step coverage of the methods available for fitting linear models, evaluating their appropriateness, and interpreting their meaning.

Chapter 9 considers the analysis of basic experimental and quasi-experimental designs through the analysis of variance and its variants. Here we’re interested in how treatment combinations or conditions affect a numerical outcome variable. The chapter introduces the functions in R that are used to perform an analysis of variance, analysis of covariance, repeated measures analysis of variance, multifactor analysis of variance, and multivariate analysis of variance. Methods for assessing the appropriateness of these analyses, and visualizing the results are also discussed.

In designing experimental and quasi-experimental studies, it’s important to determine if the sample size is adequate for detecting the effects of interest (power analysis). Otherwise, why conduct the study? A detailed treatment of power analysis is provided in chapter 10. Starting with a discussion of hypothesis testing, the presentation focuses on how to use R functions to determine the sample size necessary to detect a treatment effect of a given size with a given degree of confidence. This can help you to plan studies that are likely to yield useful results.

Chapter 11 expands on the material in chapter 5 by covering the creation of graphs that help you to visualize relationships among two or more variables. This includes the various types of twoand three-dimensional scatter plots, scatter plot matrices, line plots, and bubble plots. It also introduces the useful, but less well-known, correlograms and mosaic plots.

The linear models described in chapters 8 and 9 assume that the outcome or response variable is not only numeric, but also randomly sampled from a normal distribution. There are situations where this distributional assumption is untenable. Chapter 12 presents analytic methods that work well in cases where data are sampled from unknown or mixed distributions, where sample sizes are small, where outliers are a problem, or where devising an appropriate test based on a theoretical distribution is mathematically intractable. They include both resampling and bootstrapping approaches—computer intensive methods that are powerfully implemented in R. The methods described in this chapter will allow you to devise hypothesis tests for data that do not fit traditional parametric assumptions.

After completing part 3, you’ll have the tools to analyze most common data analytic problems encountered in practice. And you will be able to create some gorgeous graphs!

In this chapter, we’ll focus on regression methods that fall under the rubric of ordinary least squares (OLS) regression, including simple linear regression, polynomial regression, and multiple linear regression. OLS regression is the most common variety of statistical analysis today. Other types of regression models (including logistic regression and Poisson regression) will be covered in chapter 13.

8.1.1Scenarios for using OLS regression

In OLS regression, a quantitative dependent variable is predicted from a weighted sum of predictor variables, where the weights are parameters estimated from the data. Let’s take a look at a concrete example (no pun intended), loosely adapted from Fwa (2006).

An engineer wants to identify the most important factors related to bridge deterioration (such as age, traffic volume, bridge design, construction materials and methods, construction quality, and weather conditions) and determine the