Statistical Analysis with R For Dummies

Introduction

So you’re holding a statistics book. In my humble (and absolutely biased) opinion, it's not just another statistics book. It’s also not just another R book. I say this for two reasons.

First, many statistics books teach you the concepts but don't give you an easy way to apply them. That often leads to a lack of understanding. Because R is ready-made for statistics, it’s a tool for applying (and learning) statistics concepts.

Second, let’s look at it from the opposite direction: Before I tell you about one of R’s features, I give you the statistical foundation it's based on. That way, you understand that feature when you use it — and you use it more effectively.

I didn’t want to write a book that only covers the details of R and introduces some clever coding techniques. Some of that is necessary, of course, in any book that shows you how to use a software tool like R. My goal was to go way beyond that.

Neither did I want to write a statistics cookbook: when-faced-with-problem-category-#152-use-statistical-procedure-#346. My goal was to go way beyond that, too.

Bottom line: This book isn't just about statistics or just about R — it’s firmly at the intersection of the two. In the proper context, R can be a great tool for teaching and learning statistics, and I’ve tried to supply the proper context.

About This Book

Although the field of statistics proceeds in a logical way, I’ve organized this book so that you can open it up in any chapter and start reading. The idea is for you to find the information you're looking for in a hurry and use it immediately — whether it's a statistical concept or an R-related one.

On the other hand, reading from cover to cover is okay if you're so inclined. If you're a statistics newbie and you have to use R to analyze data, I recommend that you begin at the beginning.

Similarity with This Other For Dummies Book

You might be aware that I’ve written another book: Statistical Analysis with Excel For Dummies (Wiley). This is not a shameless plug for that book. (I do that elsewhere.)

I’m just letting you know that the sections in this book that explain statistical concepts are much like the corresponding sections in that one. I use (mostly) the same examples and, in many cases, the same words. I’ve developed that material during decades of teaching statistics and found it to be very effective. (Reviewers seem to like it, too.) Also, if you happen to have read the other book and you’re transitioning to R, the common material might just help you make the switch.

And, you know: If it ain’t broke… .

What You Can Safely Skip

Any reference book throws a lot of information at you, and this one is no exception. I intended for it all to be useful, but I didn't aim it all at the same level. So if you're not deeply into the subject matter, you can avoid paragraphs marked with the Technical Stuff icon.

As you read, you'll run into sidebars. They provide information that elaborates on a topic, but they're not part of the main path. If you're in a hurry, you can breeze past them.

Foolish Assumptions

I'm assuming this much about you:

You know how to work with Windows or the Mac. I don’t describe the details of pointing, clicking, selecting, and other actions.

You’re able to install R and RStudio (I show you how in Chapter 2) and follow along with the examples. I use the Windows version of RStudio, but you should have no problem if you’re working on a Mac.

How This Book Is Organized

I’ve organized this book into five parts and three appendixes (which you can find on this book’s companion website at www.dummies.com/go/statisticalanalysiswithr).

Part 1: Getting Started with Statistical Analysis with R

In Part 1, I provide a general introduction to statistics and to R. I discuss important statistical concepts and describe useful R techniques. If it’s been a long time since your last course in statistics or if you’ve never even had a statistics course, start with Part 1. If you have never worked with R, definitely start with Part 1.

Part 2: Describing Data

Part of working with statistics is to summarize data in meaningful ways. In Part 2, you find out how to do that. Most people know about averages and how to compute them. But that's not the whole story. In Part 2, I tell you about additional statistics that fill in the gaps, and I show you how to use R to work with those statistics. I also introduce R graphics in this part.

Part 3: Drawing Conclusions from Data

Part 3 addresses the fundamental aim of statistical analysis: to go beyond the data and help you make decisions. Usually, the data are measurements of a sample taken from a large population. The goal is to use these data to figure out what's going on in the population.

This opens a wide range of questions: What does an average mean? What does the difference between two averages mean? Are two things associated? These are only a few of the questions I address in Part 3, and I discuss the R functions that help you answer them.

Part 4: Working with Probability

Probability is the basis for statistical analysis and decision-making. In Part 4, I tell you all about it. I show you how to apply probability, particularly in the area of modeling. R provides a rich set of capabilities that deal with probability. Here's where you find them.

Part 5: The Part of Tens

Part V has two chapters. In the first, I give Excel users ten tips for moving to R. In the second, I cover ten statistical- and R-related topics that wouldn't fit in any other chapter.

Online Appendix A: More on Probability

This online appendix continues what I start in Part 4. The material is a bit on the esoteric side, so I’ve stashed it in an appendix.

Online Appendix B: Non-Parametric Statistics

Non-parametric statistics are based on concepts that differ somewhat from most of the rest of the book. In this appendix, you learn these concepts and see how to use R to apply them.

Online Appendix C: Ten Topics That Just Didn't Fit in Any Other Chapter

This is the Grab Bag appendix, where I cover ten statistical- and R-related topics that wouldn't fit in any other chapter.

Icons Used in This Book

Icons appear all over For Dummies books, and this one is no exception. Each one is a little picture in the margin that lets you know something special about the paragraph it sits next to.

tip This icon points out a hint or a shortcut that can help you in your work (and perhaps make you a finer, kinder, and more insightful human being).

remember This one points out timeless wisdom to take with you on your continuing quest for statistics knowledge.

warning Pay attention to the information accompanied by this icon. It's a reminder to avoid something that might gum up the works for you.

technicalstuff As I mention in the earlier section "What You Can Safely Skip," this icon indicates material you can blow past if it’s just too technical. (I’ve kept this to a minimum.)

Where to Go from Here

You can start reading this book anywhere, but here are a couple of hints. Want to learn the foundations of statistics? Turn the page. Introduce yourself to R? That's Chapter 2. Want to start with graphics? Hit Chapter 3. For anything else, find it in the table of contents or the index and go for it.

In addition to what you’re reading right now, this product comes with a free access-anywhere Cheat Sheet that presents a selected list of R functions and describes what they do. To get this Cheat Sheet, visit www.dummies.com and type Statistical Analysis with R For Dummies Cheat Sheet in the search box.

Part 1

Getting Started with Statistical Analysis with R

IN THIS PART …

Find out about R’s statistical capabilities

Explore how to work with populations and samples

Test your hypotheses

Understand errors in decision-making

Determine independent and dependent variables

Chapter 1

Data, Statistics, and Decisions

IN THIS CHAPTER

check Introducing statistical concepts

check Generalizing from samples to populations

check Getting into probability

check Testing hypotheses

check Two types of error

Statistics? That’s all about crunching numbers into arcane-looking formulas, right? Not really. Statistics, first and foremost, is about decision-making. Some number-crunching is involved, of course, but the primary goal is to use numbers to make decisions. Statisticians look at data and wonder what the numbers are saying. What kinds of trends are in the data? What kinds of predictions are possible? What conclusions can we make?

To make sense of data and answer these questions, statisticians have developed a wide variety of analytical tools.

About the number-crunching part: If you had to do it via pencil-and-paper (or with the aid of a pocket calculator), you’d soon get discouraged with the amount of computation involved and the errors that might creep in. Software like R helps you crunch the data and compute the numbers. As a bonus, R can also help you comprehend statistical concepts.

Developed specifically for statistical analysis, R is a computer language that implements many of the analytical tools statisticians have developed for decision-making. I wrote this book to show how to use these tools in your work.

The Statistical (and Related) Notions You Just Have to Know

The analytical tools that that R provides are based on statistical concepts I help you explore in the remainder of this chapter. As you’ll see, these concepts are based on common sense.

Samples and populations

If you watch TV on election night, you know that one of the main events is the prediction of the outcome immediately after the polls close (and before all the votes are counted). How is it that pundits almost always get it right?

The idea is to talk to a sample of voters right after they vote. If they’re truthful about how they marked their ballots, and if the sample is representative of the population of voters, analysts can use the sample data to draw conclusions about the population.

That, in a nutshell, is what statistics is all about — using the data from samples to draw conclusions about populations.

Here’s another example. Imagine that your job is to find the average height of 10-year-old children in the United States. Because you probably wouldn’t have the time or the resources to measure every child, you’d measure the heights of a representative sample. Then you’d average those heights and use that average as the estimate of the population average.

Estimating the population average is one kind of inference that statisticians make from sample data. I discuss inference in more detail in the upcoming section "Inferential Statistics: Testing Hypotheses."

remember Here’s some important terminology: Properties of a population (like the population average) are called parameters, and properties of a sample (like the sample average) are called statistics. If your only concern is the sample properties (like the heights of the children in your sample), the statistics you calculate are descriptive. If you’re concerned about estimating the population properties, your statistics are inferential.

remember Now for an important convention about notation: Statisticians use Greek letters (μ, σ, ρ) to stand for parameters, and English letters ( , s, r) to stand for statistics. Figure 1-1 summarizes the relationship between populations and samples, and between parameters and statistics.

FIGURE 1-1: The relationship between populations, samples, parameters, and statistics.

Variables: Dependent and independent

A variable is something that can take on more than one value — like your age, the value of the dollar against other currencies, or the number of games your favorite sports team wins. Something that can have only one value is a constant. Scientists tell us that the speed of light is a constant, and we use the constant π to calculate the area of a circle.

Statisticians work with independent variables and dependent variables. In any study or experiment, you’ll find both kinds. Statisticians assess the relationship between them.

For example, imagine a computerized training method designed to increase a person’s IQ. How would a researcher find out if this method does what it’s supposed to do? First, he would randomly assign a sample of people to one of two groups. One group would receive the training method, and the other would complete another kind of computer-based activity — like reading text on a website. Before and after each group completes its activities, the researcher measures each person’s IQ. What happens next? I discuss that topic in the upcoming section "Inferential Statistics: Testing Hypotheses."

For now, understand that the independent variable here is Type of Activity. The two possible values of this variable are IQ Training and Reading Text. The dependent variable is the change in IQ from Before to After.

remember A dependent variable is what a researcher measures. In an experiment, an independent variable is what a researcher manipulates. In other contexts, a researcher can’t manipulate an independent variable. Instead, he notes naturally occurring values of the independent variable and how they affect a dependent variable.

remember In general, the objective is to find out whether changes in an independent variable are associated with changes in a dependent variable.

remember In the examples that appear throughout this book, I show you how to use R to calculate characteristics of groups of scores, or to compare groups of scores. Whenever I show you a group of scores, I'm talking about the values of a dependent variable.

Types of data

When you do statistical work, you can run into four kinds of data. And when you work with a variable, the way you work with it depends on what kind of data it is. The first kind is nominal data. If a set of numbers happens to be nominal data, the numbers are labels – their values don’t signify anything. On a sports team, the jersey numbers are nominal. They just identify the players.

The next kind is ordinal data. In this data-type, the numbers are more than just labels. As the name ordinal might tell you, the order of the numbers is important. If I ask you to rank ten foods from the one you like best (one), to the one you like least (ten), we’d have a set of ordinal data.

But the difference between your third-favorite food and your fourth-favorite food might not be the same as the difference between your ninth-favorite and your tenth-favorite. So this type of data lacks equal intervals and equal differences.

Interval data gives us equal differences. The Fahrenheit scale of temperature is a good example. The difference between 30o and 40o is the same as the difference between 90o and 100o. So each degree is an interval.

People are sometimes surprised to find out that on the Fahrenheit scale, a temperature of 80o is not twice as hot as 40o. For ratio statements (twice as much as, half as much as) to make sense, zero has to mean the complete absence of the thing you’re measuring. A temperature of 0o F doesn’t mean the complete absence of heat – it’s just an arbitrary point on the Fahrenheit scale. (The same holds true for Celsius.)

The fourth kind of data, ratio, provides a meaningful zero point. On the Kelvin Scale of temperature, zero means absolute zero, where all molecular motion (the basis of heat) stops. So 200o Kelvin is twice as hot as 100o Kelvin. Another example is length. Eight inches is twice as long as four inches. Zero inches means a complete absence of length.

remember An independent variable or a dependent variable can be either nominal, ordinal, interval, or ratio. The analytical tools you use depend on the type of data you work with.

A little probability

When statisticians make decisions, they use probability to express their confidence about those decisions. They can never be absolutely certain about what they decide. They can only tell you how probable their conclusions are.

What do we mean by probability? Mathematicians and philosophers might give you complex definitions. In my experience, however, the best way to understand probability is in terms of examples.

Here’s a simple example: If you toss a coin, what’s the probability that it turns up heads? If the coin is fair, you might figure that you have a 50-50 chance of heads and a 50-50 chance of tails. And you’d be right. In terms of the kinds of numbers associated with probability, that’s .

Think about rolling a fair die (one member of a pair of dice). What’s the probability that you roll a 4? Well, a die has six faces and one of them is 4, so that’s . Still another example: Select one card at random from a standard deck of 52 cards. What’s the probability that it’s a diamond? A deck of cards has four suits, so that’s .

These examples tell you that if you want to know the probability that an event occurs, count how many ways that event can happen and divide by the total number of events that can happen. In the first two examples (heads, 4), the event you’re interested in happens only one way. For the coin, we divide one by two. For the die, we divide one by six. In the third example (diamond), the event can happen 13 ways (Ace through King), so we divide 13 by 52 (to get ).

Now for a slightly more complicated example. Toss a coin and roll a die at the same time. What’s the probability of tails and a 4? Think about all the possible events that can happen when you toss a coin and roll a die at the same time. You could have tails and 1 through 6, or heads and 1 through 6. That adds up to 12 possibilities. The tails-and-4 combination can happen only one way. So the probability is .

In general, the formula for the probability that a particular event occurs is

At the beginning of this section, I say that statisticians express their confidence about their conclusions in terms of probability, which is why I brought all this up in the first place. This line of thinking leads to conditional probability — the probability that an event occurs given that some other event occurs. Suppose that I roll a die, look at it (so that you don’t see it), and tell you that I rolled an odd number. What’s the probability that I’ve rolled a 5? Ordinarily, the probability of a 5 is , but I rolled an odd number narrows it down. That piece of information eliminates the three even numbers (2, 4, 6) as possibilities. Only the three odd numbers (1,3, 5) are possible, so the probability is .

What’s the big deal about conditional probability? What role does it play in statistical analysis? Read on.

Inferential Statistics: Testing Hypotheses

Before a statistician does a study, he draws up a tentative explanation — a hypothesis that tells why the data might come out a certain way. After gathering all the data, the statistician has to decide whether or not to reject the hypothesis.

That decision is the answer to a conditional probability question — what’s the probability of obtaining the data, given that this hypothesis is correct? Statisticians have tools that calculate the probability. If the probability turns out to be low, the statistician rejects the hypothesis.

Back to coin-tossing for an example: Imagine that you’re interested in whether a particular coin is fair — whether it has an equal chance of heads or tails on any toss. Let’s start with The coin is fair as the hypothesis.

To test the hypothesis, you’d toss the coin a number of times — let’s say, a hundred. These 100 tosses are the sample data. If the coin is fair (as per the hypothesis), you’d expect 50 heads and 50 tails.

If it’s 99 heads and 1 tail, you’d surely reject the fair-coin hypothesis: The conditional probability of 99 heads and 1 tail given a fair coin is very low. Of course, the coin could still be fair and you could, quite by chance, get a 99-1 split, right? Sure. You never really know. You have to gather the sample data (the 100 toss-results) and then decide. Your decision might be right, or it might not.

Juries make these types of decisions. In the United States, the starting hypothesis is that the defendant is not guilty (innocent until proven guilty). Think of the evidence as data. Jury-members consider the evidence and answer a conditional probability question: What’s the probability of the evidence, given that the defendant is not guilty? Their answer determines the verdict.

Null and alternative hypotheses

Think again about that coin-tossing study I just mentioned. The sample data are the results from the 100 tosses. I said that we can start with the hypothesis that the coin is fair. This starting point is called the null hypothesis. The statistical notation for the null hypothesis is H0. According to this hypothesis, any heads-tails split in the data is consistent with a fair coin. Think of it as the idea that nothing in the sample data is out of the ordinary.

An alternative hypothesis is possible — that the coin isn't a fair one and it's loaded to produce an unequal number of heads and tails. This hypothesis says that any heads-tails split is consistent with an unfair coin. This alternative hypothesis is called, believe it or not, the alternative hypothesis. The statistical notation for the alternative hypothesis is H1.

Now toss the coin 100 times and note the number of heads and tails. If the results are something like 90 heads and 10 tails, it's a good idea to reject H0. If the results are around 50 heads and 50 tails, don't reject H0.

Similar ideas apply to the IQ example I gave earlier. One sample receives the computer-based IQ training method, and the other participates in a different computer-based activity — like reading text on a website. Before and after each group completes its activities, the researcher measures each person’s IQ. The null hypothesis, H0, is that one group’s improvement isn't different from the other. If the improvements are greater with the IQ training than with the other activity — so much greater that it's unlikely that the two aren't different from one another — reject H0. If they're not, don't reject H0.

remember Notice that I did not say "accept H0." The way the logic works, you never accept a hypothesis. You either reject H0 or don't reject H0. In a jury trial, the verdict is either guilty (reject the null hypothesis of not guilty) or not guilty (don’t reject H0). Innocent (acceptance of the null hypothesis) is not a possible verdict.

Notice also that in the coin-tossing example I said around 50 heads and 50 tails. What does around mean? Also, I said that if it's 90-10, reject H0. What about 85-15? 80-20? 70-30? Exactly how much different from 50-50 does the split have to be for you to reject H0? In the IQ training example, how much greater does the IQ improvement have to be to reject H0?

I won't answer these questions now. Statisticians have formulated decision rules for situations like this, and we'll explore those rules throughout the book.

Two types of error

Whenever you evaluate data and decide to reject H0 or to not reject H0, you can never be absolutely sure. You never really know the true state of the world. In the coin-tossing example, that means you can’t be certain if the coin is fair or not. All you can do is make a decision based on the sample data. If you want to know for sure about the coin, you have to have the data for the entire population of tosses — which means you have to keep tossing the coin until the end of time.

Because you're never certain about your decisions, you can make an error either way you decide. As I mention earlier, the coin could be fair and you just happen to get 99 heads in 100 tosses. That's not likely, and that's why you reject H0 if that happens. It's also possible that the coin is biased, yet you just happen to toss 50 heads in 100 tosses. Again, that’s not likely and you don’t reject H0 in that case.

Although those errors are not likely, they are possible. They lurk in every study that involves inferential statistics. Statisticians have named them Type I errors and Type II errors.

If you reject H0 and you shouldn't, that's a Type I error. In the coin example, that's rejecting the hypothesis that the coin is fair, when in reality it is a fair coin.

If you don't reject H0 and you should have, that's a Type II error. It happens if you don't reject the hypothesis that the coin is fair, and in reality it's biased.

How do you know if you've made either type of error? You don’t — at least not right after you make the decision to reject or not reject H0. (If it's possible to know, you wouldn't make the error in the first place!) All you can do is gather more data and see if the additional data is consistent with your decision.

If you think of H0 as a tendency to maintain the status quo and not interpret anything as being out of the ordinary (no matter how it looks), a Type II error means you’ve missed out on something big. In fact, some iconic mistakes are Type II errors.

Here’s what I mean. On New Year’s day in 1962, a rock group consisting of three guitarists and a drummer auditioned in the London studio of a major recording company. Legend has it that the recording executives didn’t like what they heard, didn’t like what they saw, and believed that guitar groups were on the way out. Although the musicians played their hearts out, the group failed the audition.

Who was that group? The Beatles!

And that’s a Type II error.

Chapter 2

R: What It Does and How It Does It

IN THIS CHAPTER

check Getting R and RStudio

check Working with RStudio

check Learning R functions

check Learning R structures

check Working with packages

check Forming R formulas

check Reading and writing files

R is a computer language. It’s a tool for doing the computation and number-crunching that set the stage for statistical analysis and decision-making. An important aspect of statistical analysis is to present the results in a comprehensible way. For this reason, graphics is a major component of R.

Ross Ihaka and Robert Gentleman developed R in the 1990s at the University of Auckland, New Zealand. Supported by the Foundation for Statistical Computing, R is getting more and more popular by the day.

RStudio is an open source integrated development environment (IDE) for creating and running R code. It’s available in versions for Windows, Mac, and Linux. Although you don’t need an IDE in order to work with R, RStudio makes life a lot easier.

Downloading R and RStudio

First things first. Download R from the Comprehensive R Archive Network (CRAN). In your browser, type this address if you work in Windows:

cran.r-project.org/bin/windows/base/

Type this one if you work on the Mac:

cran.r-project.org/bin/macosx/

Click the link to download R. This puts the win.exe file in your Windows computer, or the .pkg file in your Mac. In either case, follow the usual installation procedures. When installation is complete, Windows users see an R icon on their desktop, Mac users see it in their Application folder.

tip Both URLs provides helpful links to FAQs. The Windows-related URL also links to Installation and other instructions.

Now for RStudio. Here’s the URL:

www.rstudio.com/products/rstudio/download

Click the link for the installer for your computer, and again follow the usual installation procedures.

After the RStudio installation is finished, click the RStudio icon to open the window shown in Figure 2-1.

FIGURE 2-1: RStudio, immediately after you install it and click on its icon.

tip If you already have an older version of RStudio