12.1: Introduction to NonParametric Statistics
Learning Objectives
 Understand situations in which nonparametric analytical methods should be used and the advantages and disadvantages of each of these methods.
 Understand situations in which the sign test can be used and calculate
z scores for evaluating a hypothesis using matched pair data sets.  Use the sign test to evaluate a hypothesis about the median of a population.
 Examine a categorical data set to evaluate a hypothesis using the sign test.
 Understand the signedranks test as a more precise alternative to the sign test when evaluating a hypothesis.
Introduction
In previous lessons, we discussed the use of the normal distribution, Student's
These tests include tests such as the sign test, the signranks test, the rankssum test, the KruskalWallis test, and the runs test. While parametric tests are preferred, since they are more powerful, they are not always applicable. The following sections will examine situations in which we would use nonparametric methods and the advantages and disadvantages of using these methods.
Situations Where We Use NonParametric Tests
If nonparametric tests have fewer assumptions and can be used with a broader range of data types, why don’t we use them all the time? The reason is because there are several advantages of using parametric tests. They are more robust and have greater power, which means that they have a greater chance of rejecting the null hypothesis relative to the sample size when the null hypothesis is false.
However, parametric tests demand that the data meet stringent requirements, such as normality and homogeneity of variance. For example, a onesample
As mentioned, an advantage of nonparametric tests is that they do not require the data to be normally distributed. In addition, although they test the same concepts, nonparametric tests sometimes have fewer calculations than their parametric counterparts. Nonparametric tests are often used to test different types of questions and allow us to perform analysis with categorical and rank data. The table below lists the parametric tests, their nonparametric counterparts, and the purpose of each test.
Commonly Used Parametric and Nonparametric Tests
Parametric Test (Normal Distributions)  Nonparametric Test (Nonnormal Distributions)  Purpose of Test 


Rank sum test  Compares means of two independent samples 
Paired 
Sign test  Examines a set of differences of means 
Pearson correlation coefficient  Rank correlation test  Assesses the linear association between two variables. 
Oneway analysis of variance ( 
KruskalWallis test  Compares three or more groups 
Twoway analysis of variance  Runs test  Compares groups classified by two different factors 
The Sign Test
One of the simplest nonparametric tests is the sign test. The sign test examines the difference in the medians of matched data sets. It is important to note that we use the sign test only when testing if there is a difference between the matched pairs of observations. This test does not measure the magnitude of the relationship
For example, we would use the sign test when assessing if a certain drug or treatment had an impact on a population or if a certain program made a difference in behavior. We first determine whether there is a positive or negative difference between each of the matched pairs. To determine this, we arrange the data in such a way that it is easy to identify what type of difference that we have. Let’s take a look at an example to help clarify this concept.
Example: Suppose we have a school psychologist who is interested in whether or not a behavior intervention program is working. He examines 8 middle school classrooms and records the number of referrals written per month both before and after the intervention program. Below are his observations:
Observation Number  Referrals Before Program  Referrals After Program 

1  8  5 
2  10  8 
3  2  3 
4  4  1 
5  6  4 
6  4  1 
7  5  7 
8  9  6 
Since we need to determine the number of observations where there is a positive difference and the number of observations where there is a negative difference, it is helpful to add an additional column to the table to classify each observation as such (see below). We ignore all zero or equal observations.
Observation Number  Referrals Before Program  Referrals After Program  Change 

1  8  5 

2  10  8 

3  2  3 

4  4  1 

5  6  4 

6  4  1 

7  5  7 

8  9  6 

The test statistic we use is
If the sample has fewer than 30 observations, we use the
Our example has only 8 observations, so we calculate our
Similar to other hypothesis tests using standard scores, we establish null and alternative hypotheses about the population and use the test statistic to assess these hypotheses. As mentioned, this test is used with paired data and examines whether the medians of the two data sets are equal. When we conduct a pretest and a posttest using matched data, our null hypothesis is that the difference between the data sets will be zero. In other words, under our null hypothesis, we would expect there to be some fluctuations between the pretest and posttest, but nothing of significance. Therefore, our null and alternative hypotheses would be as follows:
With the sign test, we set criterion for rejecting the null hypothesis in the same way as we did when we were testing hypotheses using parametric tests. For the example above, if we set
When we use the sign test to evaluate a hypothesis about the median of a population, we are estimating the likelihood, or the probability, that the number of successes would occur by chance if there was no difference between pretest and posttest data. When working with small samples, the sign test is actually the binomial test, with the null hypothesis being that the proportion of successes will equal 0.5.
Example: Suppose a physical education teacher is interested in the effect of a certain weighttraining program on students’ strength. She measures the number of times students are able to lift a dumbbell of a certain weight before the program and then again after the program. Below are her results:
Before Program  After Program  Change 

12  21 

9  16 

11  14 

21  36 

17  28 

22  20 

18  29 

11  22 

If the program had no effect, then the proportion of students with increased strength would equal 0.5. Looking at the data above, we see that 7 of the 8 students had increased strength after the program. But is this statistically significant? To answer this question, we use the binomial formula, which is as follows:
Using this formula, we need to determine the probability of having either 7 or 8 successes as shown below:
To determine the probability of having either 7 or 8 successes, we add the two probabilities together and get
Using the Sign Test to Examine Categorical Data
We can also use the sign test to examine differences and evaluate hypotheses with categorical data sets. Recall that we typically use the chisquare distribution to assess categorical data. We could use the sign test when determining if one categorical variable is really more than another. For example, we could use this test if we were interested in determining if there were equal numbers of students with brown eyes and blue eyes. In addition, we could use this test to determine if equal numbers of males and females get accepted to a fouryear college.
When using the sign test to examine a categorical data set and evaluate a hypothesis, we use the same formulas and methods as if we were using nominal data. The only major difference is that instead of labeling the observations as positives or negatives, we would label the observations with whatever dichotomy we want to use (male/female, brown/blue, etc.) and calculate the test statistic, or probability, accordingly. Again, we would not count zero or equal observations.
Example: The UC admissions committee is interested in determining if the numbers of males and females who are accepted into fouryear colleges differ significantly. They take a random sample of 200 graduating high school seniors who have been accepted to fouryear colleges. Out of these 200 students, they find that there are 134 females and 66 males. Do the numbers of males and females accepted into colleges differ significantly? Since we have a large sample, calculate the
To answer this question using the sign test, we would first establish our null and alternative hypotheses:
This null hypothesis states that the median numbers of males and females accepted into UC schools are equal.
Next, we use
To calculate our test statistic, we use the following formula:
However, instead of the numbers of positive and negative observations, we substitute the number of females and the number of males. Because we are calculating the absolute value of the difference, the order of the variables does not matter. Therefore, our
With a calculated test statistic of 4.74, we can reject the null hypothesis and conclude that there is a difference between the number of graduating males and the number of graduating females accepted into the UC schools.
The Benefit of Using the Sign Rank Test
As previously mentioned, the sign test is a quick and easy way to test if there is a difference between pretest and posttest matched data. When we use the sign test, we simply analyze the number of observations in which there is a difference. However, the sign test does not assess the magnitude of these differences.
A more useful test that assesses the difference in size between the observations in a matched pair is the sign rank test. The sign rank test (also known as the Wilcoxon sign rank test) resembles the sign test, but it is much more sensitive. Similar to the sign test, the sign rank test is also a nonparametric alternative to the paired Student’s
The main difference with the sign rank test is that under this test, the hypothesis states that the difference between observations in each data pair (pretest and posttest) is equal to zero. Essentially, the null hypothesis states that the two variables have identical distributions. The sign rank test is much more sensitive than the sign test, since it measures the difference between matched data sets. Therefore, it is important to note that the results from the sign and the sign rank test could be different for the same data set.
To conduct the sign rank test, we first rank the differences between the observations in each matched pair, without regard to the sign of the difference. After this initial ranking, we affix the original sign to the rank numbers. All equal observations get the same rank and are ranked with the mean of the rank numbers that would have been assigned if they had varied. After this ranking, we sum the ranks in each sample and then determine the total number of observations. Finally, the one sample
It is important to remember that the sign rank test is more precise and sensitive than the sign test. However, since we are ranking the nominal differences between variables, we are not able to use the sign rank test to examine the differences between categorical variables. In addition, this test can be a bit more time consuming to conduct, since the figures cannot be calculated directly in Excel or with a calculator.
Lesson Summary
We use nonparametric tests when the assumptions of normality and homogeneity of variance are not met.
There are several different nonparametric tests that we can use in lieu of their parametric counterparts. These tests include the sign test, the sign rank test, the ranksum test, the KruskalWallis test, and the runs test.
The sign test examines the difference in the medians of matched data sets. When testing hypotheses using the sign test, we can calculate the standard
We can also use the sign test to examine differences and evaluate hypotheses with categorical data sets.
A more precise test that assesses the difference in size between the observations in a matched pair is the sign rank test.
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