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Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Statistics and Data Analysisfor Nursing Research
Second Edition
CHAPTER
Chi-Square and Nonparametric Tests
8
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Nonparametric Tests
• Nonparametric tests are used primarily when: – (1) Outcomes are not measured on an interval
or ratio scale and/or – (2) Assumptions for parametric test are
severely violated Especially when sample sizes are small
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Which Nonparametric Test?
• Selection of a nonparametric test depends mostly on:– Number of groups being compared
Two versus three or more groups
– Type of comparison being made Independent groups (between-subjects
design) Dependent groups (within-subjects/
repeated measures, correlated groups designs)
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Tests for Independent Groups
Number of Groups
Level of Measurement (Outcome Variable)
Nominal Measures
Ordinal Measures
Two Groups
Chi-square test Mann-Whitney U test
Three or More Groups
Chi-square test Kruskal-Wallis test
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Chi-Square Test of Independence
• Tests relationships between two categorical variables in a crosstabs table– Put differently, it is a test of differences in
proportions between groups
• Also called Pearson’s chi-square test
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Chi-Square Test Hypotheses
• Null hypothesis: The two categorical variables are independent (unrelated)– I.e., proportions across groups are equal
• Alternative hypothesis: The two variables are not independent—they are related– I.e., proportions across groups are not equal
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Assumptions for Chi-Square Test
• Random sampling of observations from the population
• Each observation is independent (i.e., not appropriate for correlated groups or repeated measurements)
• Each cell in the contingency table must have an expected frequency greater than 0
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
General Logic for Chi-Square Test
• If null hypothesis is true, there should be no differences in proportions (relative frequencies) for groups being compared
• So, the test contrasts observed frequencies in each cell of a crosstabs table with expected frequencies—that is, the frequencies that would be expected if the null hypothesis were true
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Contingency Table Example
Experiment-al Group (E)
Control Group (C)
Total
Incontinent 1020.0%
2040.0%
30
30.0%
Not Incontinent
4080.0%
3060.0%
7070.0%
Total 50100.0%
50100.0%
100100.0%
• A higher proportion of Cs than Es were incontinent—but is this just random fluctuation?
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Observed Versus Expected Frequencies
E Group C Group TotalIncontinent 10 (20.0%)
(CELL A)20 (40.0%) 30 (30.0%)
Not Incontinent 40 (80.0%) 30 (60.0%) 70 (70.0%)
Total 50 (100.0%) 50 (100.0%) 100 (100.0%)
• If null true, both Es and Cs would have 30% incontinent (see Total Row %): 15 each
• Cell A, Observed (OA) = 10 Expected (EA) = 15
• EA = Row TotA (30) × Col TotA (50) ÷ N (100)
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
• If null hypothesis is true, all observed values (O) would equal expected values (E)
• O – E differences would be zero• Population value of the chi-square statistic
(χ2 ) if null is true = 0.0• Sampling distributions of the statistic are
asymmetric around the value of 0.0
Sampling Distribution
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
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Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
• For each cell, compute the following:
(O – E)2 ÷ E
• Cell A in our example: (10 - 15)2 ÷ 15 = 1.67
• Then add all the cell components together to obtain χ2
• In our example with four cells: χ2 = 1.67 + 1.67 + .71 + .71 = 4.46
Computation of Chi-Square
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
• Table of critical values requires knowing (1) df and (2) significance criterion (e.g., .05)
• In χ2 , df =(Rows – 1) × (Columns – 1)– Here: df = (2 – 1) × (2 – 1) = 1
• If calculated χ2 > tabled value, results are significant– Critical value for df = 1 and α = .05: 3.84
χ2 = 4.46, so null hypothesis is rejected
Testing Significance of Chi-Square
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
• If expected value for multiple cells is < 5, use Fisher’s exact test
• Sometimes for 2 × 2 tables, a correction factor is applied: Yates continuity correction– Reduces value of chi-square– Avoid if expected frequencies are large, as it
can lead to Type II errors
Related Issues for Chi-Square
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
• Index summarizing strength of relationship in a 2 × 2 design: phi (φ)– Phi varies from 0 to 1, and can be interpreted
as Pearson’s r
• Index summarizing strength of relationship in larger tables: Cramér’s V – V also varies from 0 to 1– V = φ in a 2 × 2 design
Magnitude of Effects
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
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Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
• Magnitude of effects in a 2 × 2 design most often expressed through risk indexes previously discussed:– Odds ratio (OR)– Relative risk (RR)
• These indexes especially likely to be used in meta-analyses
Alternative Effect Size: Risk Indexes
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
• Cramér’s V can be used to estimate needed sample size for 2 × 3 tables or larger– Need estimate of V, desired power (usually .
80) and alpha (usually .05); – Then, consult a table to get estimate of
needed N, for contingency tables of specified dimensions
– E.g., if V = .20 (2 × 3) N = 241
Power Analysis
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Power Analysis (cont’d)
• For a 2 × 2 design, sample size needs most often obtained using estimates of proportions in the two groups
• Effect size larger (and sample size needs smaller) for estimated differences at the extremes:– Group 1 = .10, Group 2 = .20, n = 219– Group 1 = .40, Group 2 = .50, n = 407– Group 1 = .80, Group 2 = .90, n = 219
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Other Tests for Independent Groups
• For ordinal-level data—or for higher-level data when parametric tests cannot be used because of a violated assumption:– Two groups—Mann-Whitney U test– Three or more groups—Kruskal-Wallis test
• Both are rank tests that examine differences between groups in location
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Mann-Whitney U Test
• Tests the null hypothesis that two population distributions are identical– The nonparametric analog of the
independent groups t-test
• For ns > 20, normal distribution (z values) can be used
• When displaying outcomes in a table, median values of the dependent variable for the two groups are often shown
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Mann-Whitney U Test: SPSS Output
• One panel shows mean ranks and sum of ranks for the two groups
• Main panel presents test statistics (key values highlighted)
• Assume both ns = 9
Apgar Score
Mann-Whitney U 16.000
Wilcoxon W 46.000
Z 1.94
Asympt. Sig. .043
Exact Sig. .049
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Kruskal-Wallis Test
• Tests the null hypothesis that three or more population distributions are identical:– The nonparametric analog of one-way
ANOVA
• Compares the ranks of the values for the groups
• Test statistic is H, which follows chi-square distribution
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Kruskal-Wallis Test: Nature of Effects
• Significant result only indicates that there is a difference among the groups—does not indicate which ones
• To isolate groups that are significantly different: Use the Dunn procedure– This involves using Mann-Whitney U for
all possible pairs
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Bonferroni Correction
• When the Dunn procedure is used, the risk of a Type I error increases—there are more opportunities for “chance fluctuations” to appear significant
• Need to use a Bonferroni correction to adjust the risk of a Type I error with multiple tests
• Correction involves dividing desired alpha by number of pairs for which Mann-Whitney U tests are run
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Bonferroni Correction (cont’d)
• Table indicates corrected alpha for different number of groups in the Kruskal-Wallis test
• Assumes desired α = .05
Groups Pairs Corrected α
3 3 .017
4 6 .008
5 10 .005
6 15 .003
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Tests for Dependent Groups
Number of Groups (or
Measurement Periods)
Level of Measurement (Outcome Variable)
Nominal Measures
Ordinal Measures
Two McNemar Test Wilcoxon Signed-Ranks Test
Three or More Cochran’s Q Test
Friedman Test
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
McNemar Test
• Tests differences in proportions for the same people measured twice (or for paired groups, like mothers/ daughters)
• Yields a statistic distributed as a chi-square, with df = 1
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Wilcoxon Signed-Ranks Test
• Tests differences in ordinal-level measures for the same people measured twice (or for paired groups, like Sibling A/Sibling B)– The nonparametric analog of a paired t-test
• Another example of a rank test• For n > 10, it follows a normal distribution,
so the test statistic is z
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Cochran’s Q Test
• Tests differences in proportions for the same people measured three or more times (or correlated groups)
• Yields a statistic distributed as a chi-square, with df = 1
• Not many applications in the nursing literature
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Friedman Test
• Tests differences in ordinal-level measures for the same people measured three or more times (or for correlated groups)– The nonparametric analog of an RM-ANOVA
• Another example of a rank test• Test statistic is a chi-square with (k – 1)
degrees of freedom (k = number of measurements)
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Parametric-Nonparametric Analogs
Parametric Test Nonparametric Test
Independent groups t-test
Mann-Whitney U test
Dependent groups t-test
Wilcoxon signed-ranks test
One-way ANOVA Kruskal-Wallis test
RM-ANOVA Friedman test
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Nurse Researchers’ Use of Nonparametrics
• Very crude approximation, using counts in PubMed for 2006-2007 for nursing subset—number of “hits”– Chi-square: 273 WINNER!!!– Mann-Whitney: 38– Kruskal-Wallis: 15– McNemar test: 6– Wilcoxon signed-ranks: 5– Friedman test: 2– Cochran’s Q test: 0
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
SPSS and Chi-Square Tests
• Analyze Descriptives Crosstabs
• Click Statistics to get statistical tests
• Click Cells to get expected frequencies
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
SPSS and Chi-Square Tests (cont’d)
• Crosstabs: Statistics includes:
• The chi-square statistic
• Phi and Cramer’s V• The McNemar test
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
SPSS and Other Nonparametric Tests
• Other nonparametric tests discussed in this chapter are available through Analyze Nonparametric Tests
• Options within this class include:– 2 Independent samples (Mann-Whitney)– K Independent samples (Kruskal-Wallis)– 2 Related samples (Wilcoxon signed-ranks)– K related samples (Cochran’s Q, Freidman)
Copyright ©2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
SPSS and Other Nonparametric Tests (cont’d)
• E.g., 2 Independent samples
• Move DV into Test Variable List
• Move IV into Grouping Variable list (& provide codes for the 2 groups)
• Click Mann-Whitney