Lab 10 Non-Parametric Hypothesis Testing
10.1 Introduction
Researchers often begin a project with a hypothesis and then gather data to see if the hypothesis supports an underlying theory in a process commonly called the scientific method. Categorical data gathered as part of the research project are analyzed using nonparametric techniques and two of the most commonly-used tests are described in this lab. (Note: the concept of “hypothesis” is discussed in Parametric Hypothesis Testing.)
10.2 Kruskal-Wallis H
This test is used to determine if there are any significant differences in three or more groups of data that are not normally-distributed, often categorical. Imagine that a researcher wanted to determine if there was a difference in smoking habit by age group. The subjects were interviewed and asked about how many packs they smoked per week. This data were skewed to the right since a few subjects smoked heavily but most were non-smokers or only smoked a few packs per week. The subjects were also divided into age groups: <20, 20-29, 30-39, 40-49, >49. The researcher would then use a Kruskal-Wallis H test to see if there was a significant difference in smoking habit by age group since the dependent variable (packs smoked) was not normally distributed.
10.2.1 Demonstration: Kruskal-Wallis H
The R kruskal.test function requires the two variables being compared to be input in the form of y ~ x, where y is the dependent variable (measured outcomes) and x is the independent variable (the groups used to divide the measured outcomes). Also the data source is specified with a data = parameter. In the case of the smoking example mentioned in the previous paragraph, the number of packs smoked would be the dependent variable, the measured outcome, while the age group would be the independent variable.
The following one-line script generates a kruskal.test from the airquality data frame. The amount of Ozone in the air was measured every day for several months. To see if there is any significant difference between the months a Kruskal-Wallis H would be calculated where the Ozone is the measured outcome and the Month is the grouping variable.
The kruskal.test function returns a lot of information, most of which is beyond the scope of this lab:
Kruskal-Wallis rank sum test
data: Ozone by Month
Kruskal-Wallis chi-squared = 29.267, df = 4, p-value = 6.901e-06
While a kruskal.test function returns information that would be useful in a more thorough statistical analysis, this lab is only concerned with the p-value, 6.901e-06, which is found at the end of Line 3. Because this is less than 0.05 (5%) it would be considered a significant result. Thus, the null hypothesis would be rejected (that there was no difference in Ozone by Month). Notice that this test does not indicate which month had the greatest ozone reading or if all months had some sort of significant variance from the mean, just that there is a significant difference between the months.
10.2.2 Guided Practice: Kruskal-Wallis H
Using the ChickWeight data frame, calculate a Kruskal-Wallis H for the weight output when grouped by Diet.
10.2.3 Activity: Kruskal-Wallis H
Using the cafe data, conduct a Kruskal-Wallis H and report the p-value for the following variables. Note: in the document submitted for this lab, the Activity should have a simple listing, something like illustrated here. (Notes: these are not the correct answers to the listed tests. To indicate tiny values use scientific notation in a form like 1.6e-05 since that is easier to type.) Record the p-values in the deliverable document for this lab.
- 6.352e-09
- 0.0059
- 4.028e-06
- 0.3275
Here are the variables to test:
| Num | y | x |
|---|---|---|
| 1 | Tip | Meal |
| 2 | Bill | Day |
| 3 | Party Size (ptysize) | Recommend (recmd) |
| 4 | Length | Sex |
10.3 Mann-Whitney U
This test is used to determine if there are any significant differences in two groups of data that are not normally-distributed, often categorical. Imagine that a movie producer wanted to know if there was a difference in the way the audience in two different cities responded to a movie. The null hypothesis (H0) is “There is no difference in movie-goers’ opinions between these two cities.” The alternate hypothesis (Ha) is “Movie-goers’ opinions are significantly different by city.” As the audience members left the theater they would be asked to rate the movie on a scale of one to five stars. The ratings for the two cities would be collected and then a Mann-Whitney test would be used to determine if the difference in ratings between the cities was significant.
10.3.1 Demonstration: Mann-Whitney U
R uses the wilcox.test function for several different types of nonparametric tests. It will automatically compute a Mann-Whitney U test when the dependent variable is numeric and the independent variable is binary (that is, only two levels). The wilcox.test function requires the two variables being compared to be input in the form of y ~ x, where y is the dependent variable (measured outcomes) and x is the independent variable (the two groups used to divide the measured outcomes). Also the data source is specified with a data = parameter. In the case of the movie analysis example mentioned in the previous paragraph, the number of movie rating would be the dependent variable, the measured outcome, while the city would be the independent, or grouping, variable.
The following one-line script generates a Mann-Whitney U from the CO2 data frame. The amount of CO2 uptake of a group of plants was measured while chilled or not chilled. To see if there is any significant difference between the cold tolerance of those plants, a Mann-Whitney U test would be calculated where the uptake is the measured outcome and the Treatment is the grouping variable.
The wilcox.test function returns a lot of information, most of which is beyond the scope of this lab:
Warning message: cannot compute exact p-value with ties
Wilcoxon rank sum test with continuity correction
data: uptake by Treatment
W = 1187.5, p-value = 0.006358
alternative hypothesis: true location shift is not equal to 0
While a wilcox.test function returns information that would be useful in a more thorough statistical analysis, this lab is only concerned with the p-value, 0.006358, which is found at the end of Line 4. Because this is less than 0.05 (5%) it would be considered a significant result.
10.3.2 Guided Practice: Mann-Whitney U
Using the mtcars data frame, calculate a Mann-Whitney U for the disp output when grouped by am.
10.3.3 Activity: Mann-Whitney U
Using the cafe data, conduct a Mann-Whitney U and report the p-value for the following variables. Note: in the document submitted for this lab, the Activity should have a simple listing, something like illustrated here. (Notes: these are not the correct answers to the listed tests. To indicate tiny values use scientific notation in a form like 1.6e-05 since that is easier to type.) Record the p-values in the deliverable document for this lab.
- 6.352e-09
- 0.0059
Here are the variables to test:
| Num | y | x |
|---|---|---|
| 1 | Food | Recommend (recmd) |
| 2 | Service (svc) | Preference (pref) |
10.4 Deliverable
Complete the activities in this lab and consolidate the responses into a single document. Name the document with your name and “Lab 10,” like “George Self Lab 10” and submit that document for grade.