Double click on the My
Computer icon on the desktop. Then double click on the campus_share on 'curtis' (U:) drive and then the Class_Share
folder. Finally, double click on the Math
folder and then the Math217 folder. Make copies of the AlcoholIntake.MPJ, ReactionTime2.MPJ, and WeightLoss.MPJ files and put them in your account. Then
open the WeightLoss.MPJ file from your account
Description of WeightLoss.MPJ
These
are data we just talked about as a class when I introduced the sign test. They
are the weight losses of 10 people (let’s assume a random or representative
sample) on a special diet.
Suppose you have a single
sample of data (or you have a set of paired data that reduces to a single
sample of differences). The Wilcoxon signed tank test
has a null hypothesis that the median of the population is a certain number
(when dealing with paired data, the null hypothesis is that the median is 0,
but only if the two paired distributions have the same shape). The test
statistic takes into consideration both the sign and the rank of the sample
data differences (this is an improvement over the simple sign test). The null
distribution of the test statistic is more complicated than with the simple
sign test. Hence, you’ll leave Minitab to do the calculations. This test is the nonparametric equivalent to
the one-sample or paired t-test (although it tests different hypotheses).
Create a dotplot and a boxplot of the
weight loss values. Clearly, there is a strong outlier, which can affect the
test results, and brings into question the condition of normality of the
population. So we should be hesitant to use the one-sample t-test. One way to handle this situation is to perform both the t-test and the Wilcoxon
signed rank test, and see if they agree.
First perform the t-test. From the Stat menu select Basic
Statistics>1-Sample t. Select Weight
Loss as the “Sample in columns:” variable. Then type in 0 as the
hypothesized mean (and from the Options
button select a one-sided alternative). Now perform the Wilcoxon
signed rank test. From the Stat menu
select Nonparametrics>1-Sample Wilcoxon.
Choose Weight Loss as your variable
and 0 as your test median (and greater than as your alternative).
Now compare the two
results. Note the p-values are quite
close, and they both indicate significant results. (That is, we have strong
evidence that the average weight loss for everyone on the special diet is greater
than 0 pounds.)
Description of AlcoholIntake.MPJ
In
1986, a group of researchers studied a social skills training program for
alcoholics. Twenty-three “alcohol-dependent” male inpatients at an alcohol
treatment center were randomly assigned to two groups. The control group
patients were given a traditional treatment program. The treatment group
patients were given the traditional treatment program plus a class in social
skills training. After being discharged from the program, each patient reported
the quantity of alcohol consumed over the next year (reports were verified by
other sources). The data are shown in the Minitab worksheet.
Suppose you have
independent samples from two different populations. The Mann-Whitney test
allows you to test for equality of medians (not means) without the added condition
of normality (although it does require a condition of similar shapes for the
two distributions—if the condition of similar shapes is not met, then the null
hypothesis of the Mann-Whitney test is that the variable values for the two
different treatments have the same distribution). The test consists of ranking
the entire set of observations (from both populations), and then summing the
ranks for the first sample. Again, you will have Minitab do the calculations
for you. This test is the nonparametric
equivalent to the two-sample t-test (although it tests different hypotheses).
Of interest is whether
the average alcohol intakes are different for the two groups. Create side-by-side
histograms of the two samples of data. Neither of the sample distributions
looks extremely non-normal, but you may be concerned about the low value for
the treatment group and the high peak for the control group. If so, you can
perform both the two-sample t-test
and the Mann-Whitney test.
First do the t-test. From the Stat menu select Basic
Statistics>2-Sample t. Choose “Samples in different columns:” and enter control alcohol intake (ozs)
and treatment alchohol
intake (ozs) as the two variables, and then click
on OK. What does the Minitab output
tell you?
Now do the Mann-Whitney
test. From the Stat menu select Nonparametrics>Mann-Whitney. Enter the two
variables as the two samples. Then click on OK.
Note the p-value for the Mann-Whitney
test is very close to the p-value for
the two-sample t-test, and they both
indicate significant results. (The average alcohol consumptions are
significantly different. Furthermore, based on the data we can see the
treatment group drank significantly less on average.)
Description of
ReactionTime2.MPJ
Suppose an experiment is
done where the response variable is the time (in minutes) for a certain
chemical reaction to occur and the one factor is temperature (60, 80, or 100
degrees Fahrenheit). The data in this Minitab project are results from this
hypothetical experiment.
Kruskal-Wallis Test
We’d like to know if
there is a difference in reaction times based on temperature. Look at boxplots and numerical summaries of the data (reaction time
separately for each temperature). Do the ANOVA conditions appear to be met? The
two outliers make both the equal-variance and normality conditions very
questionable. Hence, we shouldn’t use the ANOVA analysis.
The Kruskal-Wallis test
is a non-parametric analog to one-way ANOVA (although it tests different
hypotheses). The Kruskal-Wallis test ranks all the responses from all groups
together and then applies one-way ANOVA to the ranks rather than to the
original observations. The null hypothesis is that the reaction times have the
same distribution in all groups (versus the reaction times are systematically
higher in some groups than in others). Additionally, if all the population
distributions of reaction times follow the same shape (not necessarily normal),
then the null hypothesis is that the median reactions times are all the same.
To
perform the Kruskal-Wallis test, select Nonparametrics>Kruskal-Wallis
from the Stat menu. Select Reaction
Time as the response and Temperature
as the factor. The small p-value indicates that the distribution
of reaction times are systematically higher in some groups than in
others.
Just for kicks, also
perform the one-way ANOVA analysis (Stat>ANOVA>One-Way).
Notice that the ANOVA analysis does not show a significance difference between
groups—most likely because the test is on averages (and requires conditions),
which are heavily impacted by outliers.
Bottom Line
Suppose
you have a one- or two-sample (or ANOVA) problem where you want to do inference
on the mean(s). If the normality
assumption seems to be met, then use the t-test
(when the normality assumption is met, the corresponding nonparametric test is
much less powerful than the t-test).
If the normality assumption is not met, then perform both the t-test and the corresponding
nonparametric test. If the results agree, then you can feel better about using
the t-test (and you can report the t-test results, as laypeople will be
much more familiar with t-tests than
nonparametric tests). If the results do not agree, then it’s probably best to
stick with the nonparametric test results.