Statistical Inference II: Interval Estimation, Hypothesis Testing,
2.5 Nonparametric Methods
2.5.1 The Sign Test
The sign test can be used in a large number of situations, including cases that test for central location (mean, median) differences in ordinal data or for correlation in ratio data. But, its most common application is to identify the most preferred alternative, among a set of alternatives, when a sample of n individuals (questionnaires, for example) is used. In this case the data are nominal because the expressions of interest for the n individuals simply indicate a preference. In essence, the objective of this test is to determine whether a difference in preference exists between the alternatives compared. Consider, for example, the case where drivers are asked to indicate their preference for receiving pre-trip travel time information via the Internet vs. receiving the information on their mobile phones. The purpose of the study is to determine whether drivers prefer one method over the other. Letting p indicate the proportion of the population of drivers favoring the Internet, the following hypotheses are to be tested
.
If cannot be rejected, there is no evidence indicating a difference in preference for the two methods of delivering pre-trip information. How- ever, if can be rejected, it can be concluded that driver preferences are different for the two methods. In that case, the method selected by the greater number of drivers can be considered the most preferred method. Interestingly, the logic of the sign test is quite simple. For each individual, a plus sign is used if a preference is expressed for the Internet and a minus sign if the individual expresses an interest for the mobile phone. Because the data are recorded in terms of plus or minus signs, this nonparametric test is called the sign test.
H p H pa 0 0 50 0 50 : . : . ! { H0 H0
TABLE 2.3
A Survey of Nonparametric Testing Methodologies
Test Ref.
Location Parameter Hypotheses
Data on One Sample or Two Related Samples (Paired Samples)
Sign test Arbuthnott (1910)
Wilcoxon signed rank test Wilcoxon (1945, 1947, 1949); Wilcoxon, Katti, and Wilcox (1972)
Data on Two Mutually Independent Samples
Mann–Whitney U test Mann and Whitney (1947)
Wilcoxon rank sum test Wilcoxon (1945, 1947, 1949); Wilcoxon, Katti, and Wilcox (1972)
Median test Brown and Mood (1951); Westenberg (1948)
Tukey’s quick test Tukey (1959)
Normal scores tests Terry (1952); Hoeffding (1951); van der Waerden (1952, 1953); van der Waerden and Nievergelt (1956)
Percentile modified rank tests Gastwirth (1965); Gibbons and Gastwirth (1970)
Wald–Wolfowitz runs tests Wald and Wolfowitz (1940)
Data on k Independent Samples (k u 3)
Kruskal–Wallis One-Way Analysis of Variance Test
Kruskal and Wallis (1952); Iman, Quade, and Alexander (1975)
Steel tests for comparison with a control Steel (1959a,b, 1960, 1961); Rhyne and Steel (1965)
Data on k Related Samples (k u 3)
Friedman two-way analysis of variance test Friedman (1937) Durbin test for balanced incomplete block
designs
Durbin (1951)
Scale or Dispersion Parameter Hypotheses
Data on Two Mutually Independent Samples
Siegel–Tukey test Siegel and Tukey (1960)
Mood test Mood (1954); Laubscher, Steffens, and
deLange (1968)
Freund–Ansari test Freund and Ansari (1957); Ansari and Bradley (1960)
Barton–David test David and Barton (1958) Normal-scores test Klotz (1962); Capon (1961)
Sukhatme test Sukhatme (1957); Laubcher and Odeh (1976)
Rosenbaum test Rosenbaum (1953)
Kamat test Kamat (1956)
Percentile modified rank tests Gastwirth (1965); Gibbons and Gastwirth (1970)
Tests of Independence
Data on Two Related Samples
Spearman rank correlation parameter Spearman (1904); Kendall (1962); Glasser and Winter (1961)
KendallXparameter Kendall (1962); Kaarsemaker and van Wijngaarden (1953)
Data on k Related Samples (k u 3)
Kendall parameter of concordance for complete rankings
Kendall (1962) Kendall parameter of concordance for
balanced incomplete rankings
Durbin (1951)
Partial correlation Moran (1951); Maghsoodloo (1975); Maghsoodloo and Pallos (1981)
Contingency Table Data
Chi square test of independence
Tests of Randomness with General Alternatives
Data on One Sample
Number of runs test Swed and Eisenhart (1943) Runs above and below the median
Runs up and down test Olmstead (1946); Edgington (1961) Rank von Neumann runs test Bartels (1982)
Tests of Randomness with Trend Alternatives
Time Series Data
Daniels test based on rank correlation Daniels (1950) Mann test based on Kendall X parameter Mann (1945)
Cox–Stuart test Cox and Stuart (1955)
Slope Tests in Linear Regression Models
Data on Two Related Samples
Theil test based on Kendall’s X Theil (1950)
Data on Two Independent Samples
Hollander test for parallelism Hollander (1970)
Tests of Equality of k Distributions (k u 2) Data on Two Independent Samples
Chi square test
TABLE 2.3 (CONTINUED)
A Survey of Nonparametric Testing Methodologies
Under the assumption that is true (p = 0.50), the number of signs follows a binomial distribution with p = 0.50, and one would have to refer to the binomial distribution tables to find the critical values for this test. But, for large samples (n > 20), the number of plus signs (denoted by x) can be approximated by a normal probability distribution with mean and standard deviation given by
, (2.25)
and the large sample test-statistic is given by
. (2.26)
Kolmogorov–Smirnov test Kim and Jennrich (1973) Wald–Wolfowitz test Wald and Wolfowitz (1940)
Data on k Independent Samples (k u 3)
Chi square test
Kolmogorov–Smirnov test Birnbaum and Hall (1960)
Tests of Equality of Proportions
Data on One Sample
Binomial test
Data on Two Related Samples
McNemar test McNemar (1962)
Data on Two Independent Samples
Fisher’s exact test Chi square test
Data on k Independent Samples
Chi square test
Data on k Related Samples (k u 3)
Cochran Q test Cochran (1950)
Tests of Goodness of Fit
Chi square test
TABLE 2.3 (CONTINUED)
A Survey of Nonparametric Testing Methodologies
Test Ref. H0 E n n
! ! 0 50 0 25 . . W Z*! x E
W
Example 2.11
Consider the example previously mentioned, in which 200 drivers are asked to indicate their preference for receiving pre-trip travel time infor- mation via the Internet or via their mobile phones. Results show that 72 drivers preferred receiving the information via the Internet, 103 via their mobile phones, and 25 indicated no difference between the two methods. Do the responses to the questionnaire indicate a significant difference between the two methods in terms of how pre-trip travel time informa- tion should be delivered?
Using the sign test, we see that n = 200 – 25 = 175 individuals were able to indicate their preferred method. Using Equation 2.25, we find that the sampling distribution of the number of plus signs has the following properties:
.
In addition, with n = 175 it can be assumed that the sampling distribution is approximately normal. By using the number of times the Internet was selected as the preferred alternative (x = 72), the following value of the test statistic (Equation 2.25) is obtained:
.
From the test result it is obvious that the null hypothesis of no difference in how pre-trip information is provided should be rejected at the 0.05
level of significance (since ). It should be noted that although
the number of plus signs was used to determine whether to reject the null hypothesis that p = 0.50, one could use the number of minus signs; the test results would be the same.