By Hui Bian

Office for Faculty Excellence

A parametric statistical test is a test whose model specifies certain conditions about the parameters of the population from which the research sample was drawn. Conditions include:

The observations must be independent.

The observations must be drawn from normally distributed populations

These populations must have the same variances

Variables involved must have been measured in at least an interval scale.

A non-parametric statistical test is a test whose model does NOT specify conditions about the parameters of the population from which the sample was drawn.

Do not require measurement so strong as that required for the parametric tests.

Most non-parametric tests apply to data in an ordinal scale, and some apply to data in nominal scale.

Why non-parametric tests? They do not make numerous or stringent assumptions about parameters.

Or call those tests distribution-free

Statistical reference is concerned two types of problem:

Estimation of population parameters

Tests of hypotheses

When the data under analysis are met those assumptions for parametric tests, we should choose parametric tests because they are more powerful than non-parametric tests.

Non-parametric tests focus on order or ranking Data are changed from scores to ranks or signs

A parametric test focuses on the mean difference, and equivalent non-parametric test focuses on the difference between medians.

Three types of tests A one-sample test analyzes one field.

A test for related samples compares two or more fields for the same set of cases.

An independent-samples test analyzes one field that is grouped by categories of another field.

What is your objective? Automatically compare observed data to hypothesized. Binomial test: categorical fields with only two categories; Chi-Square test: all other categorical fields; and Kolmogorov-Smirnov test: continuous fields.

Test sequence for randomness. Runs test: test the observed sequence of data values for randomness.

Custom analysis. Manually choose your test (Click Settings tab). This setting is automatically selected if you subsequently make changes to options on the Settings tab that are incompatible with the currently selected objective.

How to run one-sample nonparametric tests Go to Analyze > Nonparametric Tests > One Sample

Example 1: Binomial test (variable with only two categories)

We want to know if there is no difference between the proportion of males and females (H0).

Use Q2 (gender) as Field variable

Binomial test: Click Field tab, then click Run

SPSS output of binomial test

the proportions of females and males in this sample significantly differ.

Double click the table in the output, we can get a new window called Model Viewer

Model Viewer

Another way to get Binomial test Go to Analyze > Nonparametric Tests > Legacy Dialogue > Binomial Test

SPSS Output

Chi-square goodness of fit: it allows us to test whether the observed proportions for a categorical variable differ from hypothesized proportions.

Example: we want to know whether the four grade levels have equal frequencies.

Use Q3r (has four grade levels: 9th, 10th, 11th, and 12th)

We can let SPSS automatically choose expected values for us (the probability should be 25% for each grade level)

We can also customize our analysis

Objective: customize analysis

Fields: Q3r

Settings: Choose Customize tests, check the second box

Click Options: we want equal probability

In the Category column, specify category values. In the Relative Frequency column, specify a value greater than 0 for each category. Custom frequencies are treated as ratios so that, for example, specifying frequencies 1, 2, and 3 is equivalent to specifying frequencies 10, 20, and 30, and both specify that 1/6 of the records are expected to fall into the first category, 1/3 into the second, and 1/2 into the third.

SPSS Output

Kolmogorov-Smirnov test: it is applied to continuous fields. This produces a one-sample test of whether the sample cumulative distribution function for a field is homogenous with a uniform, normal, Poisson, or exponential distribution.

Example: we want to know If Q30 has a normal distribution.

We can let SPSS automatically choose test for us or customize analysis

Q30 is a continuous variable.

Click Options: check Normal

SPSS outputs

One sample median test: Wilcoxon signed rank test that allows us to test whether a sample median differs significantly from a hypothesized value.

Example: we use Q80 as a field. We want to know whether the median of Q80 differs from 3.

Click settings

SPSS Outputs

Independent sample nonparametric tests identify differences between two or more groups using one or more nonparametric tests. Nonparametric tests do not assume your data follow the normal distribution.

2 Independent samples: grouping variable has two categories.

K Independent samples: grouping variable has more than two (k) categories

2 Independent samples

Example: we want to know whether there are gender difference on Q30 and Q80

Go to analyze > Nonparametric Test > Legacy Dialogue > 2 Independent Samples

Another way to run the analysis

Go to Analyze > Non Parametric Tests > Independent Samples

Choose Automatically compare distributions across groups

Click Settings

We can let SPSS make decision for us.

We also can customize tests Mann-Whitney U for two groups

Kruskal-Wallis 1-way ANOVA for more than two groups

2 Independent samples

SPSS Outputs

SPSS Outputs

K Independent samples (more than two groups)

Example: we want to know whether there are grade (Q3r) difference on Q30 and Q80

Go to analyze > Nonparametric Test > Legacy Dialogue > K Independent Samples

Another way to analyze data

K Independent samples: SPSS output

SPSS Output

2 Related samples: Wilcoxon signed rank sum test (identify differences between two related fields).

Example: we want to know whether there is a difference between pre and post test scores of drug use (assume Q30 is a pre score and Q41 is a post score)

Go to Analyze > Nonparametric Tests > Legacy Dialogue > 2 Related Samples

2 Related samples

2 Related samples : SPSS Outputs

Another way to analyze data

Go to Analyze > Nonparametric Tests > Related Samples

Click Settings

McNemar’s test (2 samples): test for change in binary data.

Cochran’s Q (k samples) can be applied to categorical fields.

Test change for Multinomial data (2 samples): applied to ordinal fields.

Compare median difference to hypothesized (2 samples): applied to continuous fields.

Estimate confidence interval (2 samples): median difference of related samples

Quantity association (k samples): produces a measure of agreement among judges or raters.

Compare distributions (k samples): applied to continuous fields.

SPSS Output

K related samples

Example: we use Q30, Q41, and Q42 as three related fields to test the change.

From Legacy Dialogue

SPSS output