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): 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

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

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. 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: 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. 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 Ranks

TV Favorable 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 11

20 12

42 13

43 14

49 15

R1 = 72 R2 = 48

Calculating U

20

723656

722

)18(8)7)(8(

2

)1(1

11211

Rnn

nnU

36

482856

482

)17(7)7)(8(

2

)1(2

22212

Rnn

nnU

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: 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

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: 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 2 table.

A Repertoire of Hypothesis Tests

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

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, , U, T, H).

t and z tests can be directional.

Deciding Which Test to Use

Is data qualitative or quantitative? If qualitative use Chi-square.

How many groups are there? 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

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

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

Two samples, independent groups – Mann-Whitney (U). Like an independent sample t-test.

Two samples, paired, matched or repeated measures – Wilcoxon (T). Like a paired sample t-test.

Three or more samples, independent groups – Kruskal-Wallis (H). Like a one-way ANOVA.

Summary of Qualitative Tests

Chi Square (2) – one variable. 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.