Section 13.6-1Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Lecture Slides

Elementary Statistics Twelfth Edition

and the Triola Statistics Series

by Mario F. Triola

Section 13.6-2Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Chapter 13Nonparametric Statistics

13-1 Review and Preview

13-2 Sign Test

13-3 Wilcoxon Signed-Ranks Test for Matched Pairs

13-4 Wilcoxon Rank-Sum Test for Two Independent Samples

13-5 Kruskal-Wallis Test

13-6 Rank Correction

13-7 Runs Test for Randomness

Section 13.6-3Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Key Concept

This section describes the nonparametric method of the rank correlation test, which uses paired data to test for an association between two variables.

In Chapter 10 we used paired sample data to compute values for the linear correlation coefficient r, but in this section we use ranks as a the basis for computing the rank correlation coefficient rs.

Section 13.6-4Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Definition

The rank correlation test (or Spearman’s rank correlation test) is a non-parametric test that uses ranks of sample data consisting of matched pairs.

It is used to test for an association between two variables.

Section 13.6-5Copyright © 2014, 2012, 2010 Pearson Education, Inc.

1. With rank correlation, we can analyze paired data that are ranks or can be converted to ranks. This method does not require a normal distribution for any population.

2. Rank correlation can be used to detect some (not all) relationships that are not linear.

Advantages

Section 13.6-6Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Objective

Compute the rank correlation coefficient rs and use it to test for an association between two variables.

Then we can test the following:

Section 13.6-7Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Notation

rs = rank correlation coefficient for sample paired data (rs is a sample statistic)

ρs = rank correlation coefficient for all the population data (ρsis a population parameter)

n = number of pairs of sample data

d = difference between ranks for the two values within an individual pair

Section 13.6-8Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Requirements

The paired data are a simple random sample and the data are ranks or can be converted to ranks.

Note: Unlike the parametric methods of Section 10-2, there is no requirement that the sample pairs of data have a bivariate normal distribution. There is no requirement of a normal distribution for any population.

Section 13.6-9Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Test Statistic

First convert the data to ranks. Then calculate:

No ties: After converting the data in each sample to ranks, if there are no ties among ranks for either variable, the exact value of the test statistic can be calculated using this formula:

Section 13.6-10Copyright © 2014, 2012, 2010 Pearson Education, Inc.

P-values: Sometimes found using technology, provided the technology used has this function.

Critical values:

If n ≤ 30, critical values are found in Table A-9.

If n > 30, use this formula, where the value of z corresponds to the significance level (for example α = 0.05, z = 1.96).

P-Values and Critical Values

Section 13.6-11Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Disadvantage

A not very serious disadvantage of rank correlation is its efficiency rating of 0.91.

This efficiency rating shows that with all other circumstances being equal, the nonparametric approach of rank correlation requires 100 pairs of sample data to achieve the same results as only 91 pairs of sample observations analyzed through parametric methods, assuming that the stricter requirements of the parametric approach are met.

Section 13.6-12Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Rank Correlation Testing

Section 13.6-13Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Rank Correlation Testing

Section 13.6-14Copyright © 2014, 2012, 2010 Pearson Education, Inc.

ExampleTable 13-7 lists quality rankings and prices of 37-inch LCD televisions. Find the value of the rank correlation coefficient and use it to determine whether or not there is a correlation between quality and price.

Use a 0.05 significance level. Based on the result, does it appear that you can get better quality by spending more?

Section 13.6-15Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - ContinuedRequirement Check: We assume we have a simple random sample.

The hypotheses are:

On the following slide, we convert to ranks.

Section 13.6-16Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example

We have no ties, so the exact value of the test statistic can be calculated as shown on the next slide:

Section 13.6-17Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - ContinuedTest Statistic:

We refer to Table A-9 to find the critical values of ±0.786.

Because the test statistic rs = 0.429 is between the critical values, we fail to reject the null hypothesis.

Section 13.6-18Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example - Continued

There is not sufficient evidence to support a claim of a correlation between quality and price.

Based on the given sample data, it appears you don’t necessarily get better quality by paying more.

Section 13.6-19Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Rank correlation methods sometimes allow us to detect relationships that we cannot detect with the methods of Chapter 10.

Below is a scatterplot that shows an S-shaped pattern:

Example

Section 13.6-20Copyright © 2014, 2012, 2010 Pearson Education, Inc.

The methods of Chapter 10 give r = 0.590 and critical values of ±0.632, suggesting that there is not a linear relationship between x and y.

If we use the methods of this section, we get rs = 1 and critical values of ±0.648, suggesting that there is a correlation between x and y.

With rank correlations, we can sometimes detect relationships that are not linear.

Example - Continued