A ANOVA B Non-parametric tests: Wilcoxon Rank Sum Test

Examining the reading skills of children in the U.S., three methods of education were compared. Several variables were measured before the lessons started. One of the goals of the pretest was to see whether the three groups of children had similar cognitive capacities. One of its variables gave an indication of the "ability of reading garbled sentences", which measures a certain kind of text comprehension. The data for the 22 subjects are given below. The three types of education are called (B)asal, (D)irected Reading as Thinking Activity en (S)trategies. (Source: research done by Jim Baumann and Leah Jones from the School of Education of Purdue University; slightly altered!)

Group B D S

4 7 11 14 7 7 9 12 4 12 10 7 16 16 7 15 15 6 14 9 11 12 8 14 16 13 13 8 12 9 13 7 12 9 6 13 12 8 4 12 9 13 12 9 6 10 8 12 8 9 6 12 13 11 11 10 14 8 8 8 17 8 5 9 10 8

You can read the data from this file (save it to your disk, before importing it to SPSS, as usual). This time, this is a csv-file ("comma separated value"): the delimiter character between cells in a row is not space but a comma.

Most probably, you will import this file in a way so that you get three columns, as the table above. Yet, this is not what you need for further processing. Indeed, each value represents a separate case, so there are 66 cases in total. That is, you want 66 rows, as a row represents a case in SPSS. You will, therefore, cut-and-paste the columns under each other, to get a single column for variable ARGS ("ability of reading garbled sentences"). In case you use an abbreviation for the name of the variable, do not forget to add an explanation ("label" in "Variable View"). Moreover, after cut-and-paste, remove the variables (columns) that have just been emptied ("Data View", right-click on the top of the column, and then "clear").

Yet, you need a second variable to distinguish between cases belonging to the three methods. So introduce a new variable, called METHOD, which has three values: use numerical values, and add labels to the values (1=basal,2=directed, 3=strategies). You will use this second variable to define groups of cases, as your goal is exactly to compare those groups. Probably the simplest way is to enter the values 1, 2 and 3 of METHOD by hand (depending on the way you have cut-and-pasted the columns earlier, but probably cases 1-22 represent method B, 23-44 represent D and 45-66 represent S).

An ANOVA test (M&M ch. 12) compares averages, boxplots show medians. If the two distributions are nearly symmetric, both central measures will display nearly the same values. However, if the number of observations is low, and the variable values are not very diverse, boxplots are not very accurate.

***1. Draw for each group a boxplot (simple boxplot, summaries for
groups of cases).
**

***2. Give for each group (B, D, and S) the mean and the standard
deviation. What is the ratio between the biggest and the smallest
standard deviation? Can we employ an ANOVA test to get reliable information?
**

Hint: You can use "select case" in order to obtain the mean and the standard
deviation of 22 cases only at a time.

***3. Formulate H_0 and H_a. Run ANOVA ("Analyze", "Compare Means"),
and copy the one-way ANOVA table to your
report. What is your conclusion? Formulate the "magic sentence"
with the statistical details in parenthesis.**

Hint: the variable whose mean you want to compare is called "dependent list"
and the variable used to form the groups is called "factor". Namely, the
question is whether the quantitative variable depends on factors
such as the method. Let SPSS also plot a "means plot" and give you data
on descriptive statistics (within 'Options' of the ANOVA window).

After having performed an ANOVA, we proceed by searching for why the null hypothesis has been rejected: which of the three populations differ from the others? We can either employ contrasts, or run posthoc pairwise comparisons of the samples. In SPSS, you find two buttons in the ANOVA window that bring you to these further procedures.

***4. Analyse the contrasts '(D and S) vs. B' (contrast1) and 'D vs. S'
(contrast2). Check with contrast1 whether (D and S) have higher
average scores than B. Check with contrast2 whether the average
scores of D are not equal to the average scores of S. Give for both
contrasts the null hypothesis and the alternative
hypothesis, the t-value, the p-value, and the conclusion.**

Help: open the "contrast" window from the ANOVA window. Add the three
coefficients for the three groups one under the other (enter first
value, click on "add", etc.; the order corresponds to the values in variable
METHOD). Make sure the sum of the coefficients is 0, which can be also checked
in the window. Then, click on "next" to enter the coefficients of a second
contrast.

***5. Perform a Bonferroni-test with alfa=0.05. In which pairwise comparison(s)
are the two groups significantly different? (Look at the stars...)
**

**Optional question: a friend of yours argues that ANOVA is worthless, because
you get a situation in which a=b and b=c, but a and c are different. Such a
situation is impossible. What is your answer?
**

Has the number of female Nobel laureates increase as a result of women's emancipation in recent decades? As of 2008 there, have been 35 female laureates (http://en.wikipedia.org/wiki/List_of_female_Nobel_laureates, http://nobelprize.org/nobel_prizes/lists/women.html), whereas the Nobel prize has been awarded to a man 759 times. (The four people who received the Nobel prize twice has been counted twice; by the way, one of them was a woman.)

It is clear that much more men have received the Nobel prize than women. However, women were awarded the Nobel prize since its earliest years, and at least one woman received the Nobel prize in each decade, with the exception of the 1950's. The number of women increases since the sixties, but the same is true of men, due to several factors: establishing the Nobel prize in economics, and the practice of sharing the prize between three people becoming most frequent (whereas the Nobel prize was not awarded quite often earlier).

I have compiled two files: a list for men and a list of for women. These lists contain only the years, so that we can see the distribution of Nobel prizes per year. I have omitted the 22 cases when an organization was awarded the Nobel prize for peace.

> Import the data from both files to SPSS.
Use a variable called YEAR. Introduce a second variable called GENDER,
with some numerical encoding as you did last week
(e.g., 1 = man, 2 = woman). Do not forget setting the values and the
decimals in the "Variable View".

Hints: it is easiest to first read the two files to two separate
spreadsheets. Using "Compute Variable" (within "Transform"), create
the second variable GENDER (target variable: gender = numeric expression: 1,
for one file, and gender = 2 for the other file). Then, copy-paste
the two columns from the shorter spreadsheet to the end of the longer one.
Finally, do not forget to save what you get in a .sav format.

**>Draw for each group a boxplot (simple boxplot, summaries for
groups of cases; variable: year, category axis: gender): what can you
see (spread, median)?**

**
> Create a histogram showing the distribution of man, and another one
showing the distribution of women.**

You can do it before copy-pasting. Alternatively, use the "select case"
function (condition gender=2, refer to
lab 5). Do not forget to
unselect the case afterward.

***6. Copy the two histograms to your report. Add captions. Compare
the two distributions: Are they similar? What shape(s) do these
distributions follow?**

(NB: Please, do not argue that they are close to a Normal distribution!
Do you expect to decrease the number of Nobel prizes in the future?)

Here are a few ways to pose the same question (make sure you understand each of them, and why they are related):

- Has the proportion of women being awarded a Nobel prize increased in the second half of the history of the prize, with respect to the same proportion in the first half of its history? (Cf. tests on proportions.)
- Is there a correlation between the variables YEAR and GENDER?
(Cf. scatterplot and Pearson's correlation coefficient
*r*.) - Are variables YEAR and GENDER independent from each other? (Cf. chi-square test.)

**
*7. Choose any of these methods (preferably one that you haven't done yet).
Report your results, and explain them.
**

Here are some further ways to formulate our question:

- Has the number of women laureates increase more quickly recently than the number of men laureate?
- If we look at percentages, and not absolute numbers, are the two distributions the same, or different?
- Is the
*cumulative proportion*of the two distributions different? - If we list all laureates, are the few women distributed equally among the men?
- Is the median of the two populations different?

These last four questions bring us to nonparametric tests, which you can find in SPSS under "Analyze, Nonparametric Tests". We obviously have two independent samples, and we focus on Mann-Whitney U, a variant of Wilcoxon Rank Sum Test (cf. M&M, p. 15-8).

***8. Perform the test, report the results and draw a conclusion.**

***9. M&M (15.1) proposes two alternative interpretation of what is tested
by the Wilcoxon rank sum test: either the identity of the two distributions
(no parameter involved at all; hence the name "nonparametric test"),
or the equality of medians (an unusual parameter involved).
Explain which interpretation makes sense in the present case?**

***10. Why do we need to "fall back" to a nonparametric test in the
present case? Give at least two reasons.**

Hints: Did you get anything useful at question 7? Could you employ a
chi-square test (cf. criteria of its use and footnote by SPSS below the table)?
Does the shape of the distribution suggest using a traditional ("Normal")
test? Is the variable being discussed nominal, ordinal or really numerical?

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