Choosing the Right Test for Non-normal Distributions in Behavioral Sciences, Slides of Behavioural Science

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Statistics for the Behavioral

Sciences

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.

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.

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 U 1 and U 2.

 Look up U in the U table.

Observations Ranks TV Favorable TV Unfavorable TV Favorable TV Unfavorable

0 1. 0 1. 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 R 1 = 72 R 2 = 48

Testing U

 H 0 : Population distribution 1 = population distribution 2 H 1 : 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

 Reject the null (equal groups) when U is less than the critical U in the table.

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:  H 0 : Pop A = Pop B = Pop C  H 1 : H 0 is false.

 Convert data to ranks and then use the H formula.

 With n > 4, look up in χ^2 table.

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, χ 2 , 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 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.