Statistics for One Variable

In Chapter 2 we examined graphs for one variable, which is the best beginning step for analyses. In Chapter 3, we will examine the follow-up to graphical analysis: statistical procedures for one variable.

Frequencies

As with the last chapter, we will begin with categorical variables. The most common statistics for categorical variables are frequencies and proportions. We will first create a data set based on a recent Google search.

Sample: sample_3_1.R

The rep() function repeats an item for a specified number of times. In the above code, for example, it repeats the word “blue” 3990 times. When the five color words are put together in the object groups, there are 14,645 lines of data. Although it may be easier to work with the data in tabular form—which is what we will do—I prefer to also have a data set with one row per case.

This next command creates a data table:

The data are now in alphabetical order. There may be times when this is helpful, but for now it makes more sense to order them by frequencies.

You may also want to present the data as proportions or percentages instead of frequencies. The function prop.table() will do this:

There are too many decimal places in this table. We can fix this with the round() function. We place the prop.table() function inside it and specify the number of decimal places we want:

That’s an improvement but we can take it further. The leading zeroes and decimals are repetitive. We can remove them by multiplying the results by 100:

Once you have saved your work, you should clear the workspace of unneeded variables and objects with rm(list = ls()).

Descriptive statistics

For categorical variables, there is a limited number of useful descriptive statistics. For quantitative variables, however, there is a much broader range of choices. In this section, we will use both R’s built-in functions and functions from external packages. Together, they give a rich statistical picture of our data.

We will use R’s data set cars, which gives the speed of cars in MPH and the distances taken to stop in feet. The data were recorded in the 1920s, so there are some unusual values, such as a car taking 120 feet to stop from 24 MPH. (See ?cars for more information.)

Sample: sample_3_2.R

The easiest way to get descriptive statistics in R is with the summary() function. For quantitative variables, this function gives the five quartile values—minimum, first quartile, median, third quartile, and maximum—as well as the mean. It can also give some categorical statistics.

To get statistics for a single variable, enter the variable's name:

You can also do an entire data set at once:

Another option for descriptive statistics is the describe function in the psych package. This provides the following:

• n
• mean
• standard deviation
• median
• trimmed mean (10% by default)
• median absolute deviation from median (MAD)
• minimum
• maximum
• range
• skewness
• kurtosis
• standard error

To use the describe function, you must first install and load the psych package, then call the describe() function:

For more information on this package, enter help(package = "psych"). For help on the describe() function, enter ?describe.

Once you have saved your work, you should clear the workspace of unneeded variables, objects, or packages:

Single proportion: Hypothesis test and confidence interval

The simplest inferential procedures are for single proportions. These are dichotomous outcomes: pass or fail, yes or no, left or right, etc. The only data needed for these procedures is the number of trials, n, and the number of positive outcomes, X. For example, if a person flipped a coin 40 times and got 27 heads, then n = 40 and X = 27. R's prop.test() function is able to do both a null hypothesis test and a confidence interval for these proportions. We will use our coin flip data for this example, first with the default settings and then with some options.

Sample: sample_3_3.R

In these results, R repeats the data—27 positive outcomes out of 40 trials—and gives the default null probability of 0.5. “X-squared” is ², the chi-squared value of 4.225. With one degree of freedom, our observed results have a probability value of 0.03983. Because this value is less than the standard .05, we reject the null hypothesis; that is, we conclude that 27 out of 40 heads is statistically significantly greater than 50%. R then states the alternative hypothesis that the true p, or the probability of a positive outcome, is not equal to 0.5. Next is the 95% confidence interval for the proportion, which ranges from 0.51 to 0.81. The output finishes with the observed sample proportion of 0.675, or 67.5%.

R also provides several options for the one-sample proportions test. These options include:

• The choice of a null proportion other than 0.5. non-directional, two-sided hypothesis test or a directional, one-sided test. In the latter case, you must specify whether you hypothesize that the sample proportion is greater than or less than the population value.
• The choice of a confidence level other than the default of 0.95.
• The choice of whether to use Yates' continuity correction; the default is to use it.

Using some of these options, we could revise the previous hypothesis test we conducted to test whether the sample proportion is significantly greater than a population proportion of 0.6 and we could use a confidence level of .90:

While the sample proportion of 67.5% is the same as the default test, we are now using a population of 60%. Even with a directional hypothesis, the sample value does not differ significantly from the null value; with a p-value of 0.21, we cannot reject the null hypothesis. The confidence interval is interesting, too, because it is directional. As a result, the upper limit goes to 1.000, which it would not do for a standard, non-directional test.

Once you have saved your work, you should clear the workspace of unneeded variables and objects with rm(list = ls()).

Single mean: Hypothesis test and confidence interval

For quantitative variables—that is, variables at the interval or ratio level of measurement—the simplest possible test is the one-sample t-test. This test compares the sample mean to a hypothesized population mean.

In this example, we will use R's quakes data set. This data set includes the location, depth, and magnitude of earthquakes, as well as the number of measurement stations that detected the earthquake. It includes data for 1000 earthquakes off the coast of Fiji with a magnitude of at least 4.0 on the Richter scale.

We will examine one variable: mag, for magnitude. Before we conduct the t-test, though, we should at least get a histogram and some basic summary statistics.

Sample: sample_3_4.R

The histogram in Figure 20 shows that the distribution has a strong, positive skew. The distribution is censored, with no values below 4.0, so it is difficult to know what the shape of the entire distribution would be.

20. Histogram of Earthquake Magnitudes

The basic summary statistics for mag are:

The important value here is the mean of 4.62 because that is the value that the t-test compares against a hypothesized population mean.

The default t-test is simple to run:

The output resembles that of the chi-squared test. It gives a t-value of 362.7599, which is enormous. The degrees of freedom are 999, and the probability value is almost zero: 2.2e-16. These strong results are not surprising, though; the hypothesized population mean was 0 but the lowest value in the data was 4. The output also includes the bounds for the 95% confidence interval and the observed sample mean.

Also like the chi-squared test, the t-test provides several options. (See ?t.test for more information.) The following t-test uses some of those options:

The conclusion here is matches the default t-test: the p-value is almost zero and we reject the null hypothesis. This is true even though we chose a hypothesized population mean— or mu—of 4.5, which was much closer to the sample mean of 4.62. The difference between the two means is much smaller than before but, given the large sample size of 1000, even negligible differences would be statistically significant.

Once you have saved your work, you should clear the workspace of unneeded variables, objects, or packages:

Chi-squared goodness-of-fit test

When you have a categorical variable with more than two categories, a chi-squared – ² – test can be useful. In this section of the book we will talk about one variation of the chi-squared test: the goodness-of-fit test. This test compares the proportion of your sample in each category with hypothesized proportions. You can check if an equal number of your observations are in each category. (See ?chisq.test for more information.) That is the version of the test we will first cover. You can also check whether you observations match some other hypothesized distribution. We will cover that afterwards.

The chi-squared test in R uses summary tables as its input. If you have one row of data for each case or if you have a multidimensional table, you may need to restructure your data. In this example, we will use the three-dimensional table HairEyeColor from R's datasets package.

Sample: sample_3_5.R

You can see the full, three-way table by entering ?HairEyeColor. In this analysis, though, we will look only at eye color, so we should create a new data frame to hold those data.

We will first conduct a version of the chi-squared goodness-of-fit test with the hypothesis that the eye colors are evenly distributed. This is the default setting for chisq.test().

The chi-squared test statistic, shown as X-squared in R’s printout, is enormous: 133.473. With three degrees of freedom, the test result has a probability value of nearly zero. Because this probability value is less than the conventional cut-off of .06—much less, in fact—we reject the null hypothesis that the four eye colors are equally common among our sample.

This is, however, not the most appropriate test. Eye colors are not evenly distributed. A better test would be whether our sample proportions differ significantly from the population proportions for eye colors. R does not provide this population data, but the Internet does. One website[9] suggested that the population proportions for brown, blue, hazel, and green eyes were .41, .32, .14, and .12, respectively.[10] We can then combine these values in a vector of probability values using the p attribute in chisq.test() and see if there is a significant difference.

In this case our value of chi-square—again, X-squared in the printout—is much smaller: 6.4717. With three degrees of freedom, that gives a probability value of .09079, which does not exceed the standard cut-off of 0.5. We can therefore conclude that our sample’s eye color proportions do not differ significantly from those of the general population.

At the end, we should clear the workspace of unneeded variables, objects, or packages: