PSY 307 – Statistics for the
Behavioral Sciences
Chapter 20 – Tests for Ranked Data,
Choosing Statistical Tests
What To Do with Non-normal
Distributions

Tranformations (pg 382):
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The shape of the distribution can be
changed by applying a math operation
to all observations in the data set.
Square roots, logs, normalization
(standardization).
Rank order tests (pg 387):

Use a nonparametric statistic that has
different assumptions about the shape
of the underlying distribution.
Pros and Cons
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Tranformations must be described
in the Results section of your
manuscript.
Effects of transformations on the
validity of your t or F statistical
tests is unclear.
Nonparametric tests may be
preferable but make probability of
Type II error greater.
Nonparametric Tests


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A parameter is any descriptive
measure of a population, such as a
mean.
Nonparametric tests make no
assumptions about the form of the
underlying distribution.
Nonparametric tests are less
sensitive and thus more susceptible
to Type II error.
When to Use Nonparametric Tests

When the distribution is known to
be non-normal.


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When a small sample (n < 10) contains
extreme values.
When two or more small samples have
unequal variances.
When the original data consists of
ranks instead of values.
Mann-Whitney Test (U Test)


The nonparametric equivalent of the
independent group t-test.
Hypotheses:
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H0: Pop. Dist. 1 = Pop. Dist. 2
H1: Pop. Dist. 1 ≠ Pop. Dist. 2
The nature of the inequality is
unspecified (e.g., central tendency,
variability, shape).
Calculating the U-Test

Convert data in both samples to
ranks.




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With ties, rank all values then give all
equal values the mean rank.
Add the ranks for the two groups.
Substitute into the formula for U.
U is the smaller of U1 and U2.
Look up U in the U table.
Observations
TV Favorable
Ranks
TV Unfavorable
TV Favorable
TV Unfavorable
0
1.5
0
1.5
1
3
2
4
4
5
5
7
5
7
5
7
10
9
12
10
14
20
49
11
12
42
13
43
14
15
R1 = 72
R2 = 48
Calculating U
n1 (n1  1)
 R1
2
8(8  1)
 (8)(7) 
 72
2
 56  36  72
 20
U1  n1n2 
n2 (n2  1)
 R2
2
7(7  1)
 (8)(7) 
 48
2
 56  28  48
 36
U 2  n1n2 
U = whichever is smaller – U1 or U2
= 20
Testing U


H0: Population distribution 1 =
population distribution 2
H1: Population distribution 1 ≠
population distribution 2
Look up critical values in U Table.


Instead of degrees of freedom, use n’s
for the two groups to find the cutoff.
Since 20 is larger than 10, retain
the null (not reject).
Interpretation of U


U represents the number of times
individual ranks in the lower group
exceed those in the higher group.
When all values in one group
exceed those in the other, U will be
0.

Reject the null (equal groups) when U
is less than the critical U in the table.
Directional U-Test


Similar variance is required in order
to do a directional U-test.
The directional hypothesis states
which group will exceed which:


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H0: Pop Dist 1 ≥ Pop Dist 2
H1: Pop Dist 1 < Pop Dist 2
In addition to calculating U, verify
that the differences in mean ranks
are in the predicted direction.
Wilcoxon T Test

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Equivalent to paired-sample t-test
but used with non-normal
distributions and ranked data.
Compute difference scores.
Rank order the difference scores.
Put plus ranks in one group, minus
ranks in the other. Sum the ranks.
Smallest value is T. Look up in T
table. Reject null if < than critical T.
Kruskal-Wallis H Test


Equivalent to one-way ANOVA for
ranked data or non-normal
distributions.
Hypotheses:


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H0: Pop A = Pop B = Pop C
H1: H0 is false.
Convert data to ranks and then use
the H formula.
With n > 4, look up in c2 table.
A Repertoire of Hypothesis Tests
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z-test – for use with normal
distributions when σ is known.
t-test – for use with one or two
groups, when σ is unknown.
F-test (ANOVA) – for comparing
means for multiple groups.
Chi-square test – for use with
qualitative data.
Null and Alternative Hypotheses
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How you write the null and
alternative hypothesis varies with
the design of the study – so does
the type of statistic.
Which table you use to find the
critical value depends on the test
statistic (t, F, c2, U, T, H).
t and z tests can be directional.
Deciding Which Test to Use
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Is data qualitative or quantitative?

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How many groups are there?

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If qualitative use Chi-square.
If two, use t-tests, if more use ANOVA
Is the design within or between
subjects?
How many independent variables
(IVs or factors) are there?
Summary of t-tests
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Single group t-test for one sample
compared to a population mean.
Independent sample t-test – for
comparing two groups in a
between-subject design.
Paired (matched) sample t-test –
for comparing two groups in a
within-subject design.
Summary of ANOVA Tests
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One-way ANOVA – for one IV,
independent samples
Repeated Measures ANOVA – for one or
more IVs where samples are repeated,
matched or paired.
Two-way (factorial) ANOVA – for two or
more IVs, independent samples.
Mixed ANOVA – for two or more IVs,
between and within subjects.
Summary of Nonparametric Tests
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Two samples, independent groups –
Mann-Whitney (U).
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Two samples, paired, matched or
repeated measures – Wilcoxon (T).
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Like an independent sample t-test.
Like a paired sample t-test.
Three or more samples, independent
groups – Kruskal-Wallis (H).

Like a one-way ANOVA.
Summary of Qualitative Tests
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Chi Square (c2) – one variable.

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Tests whether frequencies are equally
distributed across the possible categories.
Two-way Chi Square – two variables.

Tests whether there is an interaction
(relationship) between the two variables.