16 Nonparametric Tests Why Use Nonparametric Tests? Chapter Why Use Nonparametric Tests? One-Sample Runs Test Wilcox on Signed-Rank Test Mann-Whitney Test Kruskal-Wallis Test for Independent Samples Friedman Test for Related Samples Spearman Rank Correlation Test McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

Why Use Nonparametric Tests? Parametric hypothesis tests require the estimation of one or more unknown parameters (e.g., population mean or variance). Often, unrealistic assumptions are made about the normality of the underlying population. Large sample sizes are often required to invoke the Central Limit Theorem.

Why Use Nonparametric Tests? Nonparametric or distribution-free tests - usually focus on the sign or rank of the data rather than the exact numerical value. - do not specify the shape of the parent population. - can often be used in smaller samples. - can be used for ordinal data.

Why Use Nonparametric Tests? Advantages and Disadvantages of Nonparametric Tests Table 16.1

Why Use Nonparametric Tests? Some Common Nonparametric Tests Figure 16.1

One-Sample Runs Test Wald-Wolfowitz Runs Test The one-sample runs test (Wald-Wolfowitz test) detects non-randomness. Ask – Is each observation in a sequence of binary events independent of its predecessor? A nonrandom pattern suggests that the observations are not independent. The hypotheses are H0: Events follow a random pattern H1: Events do not follow a random pattern 16-6

One-Sample Runs Test Wald-Wolfowitz Runs Test To test the hypothesis, first count the number of outcomes of each type. n1 = number of outcomes of the first type n2 = number of outcomes of the second type n = total sample size = n1 + n2 A run is a series of consecutive outcomes of the same type, surrounded by a sequence of outcomes of the other type.

DAAAAAAADDDDAAAAAAAADDAAAAAAAADDDDAAAAAAAAAA One-Sample Runs Test Wald-Wolfowitz Runs Test For example, consider the following series representing 44 defective (D) or acceptable (A) computer chips: DAAAAAAADDDDAAAAAAAADDAAAAAAAADDDDAAAAAAAAAA The grouped sequences are: A run can be a single outcome if it is preceded and followed by outcomes of the other type.

One-Sample Runs Test Wald-Wolfowitz Runs Test There are 8 runs (R = 8). n1 = number of defective chips (D) = 11 n2 = number of acceptable chips (A) = 33 n = total sample size = n1 + n2 = 11 + 33 = 44 The hypotheses are: H0: Defects follow a random sequence H1: Defects follow a nonrandom sequence

One-Sample Runs Test Wald-Wolfowitz Runs Test When n1 > 10 and n2 > 10, then the number of runs R may be assumed to be normally distributed with mean mR and standard deviation sR. calc

One-Sample Runs Test Wald-Wolfowitz Runs Test The test statistic is: For a given level of significance a, find the critical value za for a two-tailed test. Reject the hypothesis of a random pattern if z < -za or if z > +za . calc

One-Sample Runs Test Wald-Wolfowitz Runs Test Decision rule for large-sample runs tests: Figure 16.2

Wilcox on Signed-Rank Test The Wilcox on signed-rank test compares a single sample median with a benchmark using only ranks of the data instead of the original observations. It is used to compare paired observations. Advantages are - freedom from the normality assumption, - robustness to outliers - applicability to ordinal data. The population should be roughly symmetric.

Wilcox on Signed-Rank Test To compare the sample median (M) with a benchmark median (M0), the hypotheses are: When evaluating the difference between paired observations, use the median difference (Md) and zero as the benchmark.

Wilcox on Signed-Rank Test Calculate the difference between the paired observations. Rank the differences from smallest to largest by absolute value. Add the ranks of the positive differences to obtain the rank sum W.

Wilcox on Signed-Rank Test For small samples, a special table is required to obtain critical values. For large samples (n > 20), the test statistic is approximately normal. Use Excel or Appendix C to get a p-value. Reject H0 if p-value < a. calc

Mann-Whitney Test The Mann-Whitney test is a nonparametric test that compares two populations. It does not assume normality. It is a test for the equality of medians, assuming - the populations differ only in centrality, - equal variances The hypotheses are H0: M1 = M2 (no difference in medians) H1: M1 ≠ M2 (medians differ)

Mann-Whitney Test Performing the Test Step 1: Sort the combined samples from lowest to highest. Step 2: Assign a rank to each value. If values are tied, the average of the ranks is assigned to each. Step 3: The ranks are summed for each column (e.g., T1, T2). Step 4: The sum of the ranks T1 + T2 must be equal to n(n + 1)/2, where n = n1 + n2.

Mann-Whitney Test Performing the Test Step 5: Calculate the mean rank sums T1 and T2. Step 6: For large samples (n1 < 10, n2 > 10), use a z test. calc Step 7: For a given a, reject H0 if z < -za or z > +za

Kruskal-Wallis Test for Independent Samples The Kruskal-Wallis (K-W) test compares c independent medians, assuming the populations differ only in centrality. The K-W test is a generalization of the Mann-Whitney test and is analogous to a one-factor ANOVA (completely randomized model). Groups can be of different sizes if each group has 5 or more observations. Populations must be of similar shape but normality is not a requirement.

Kruskal-Wallis Test for Independent Samples Performing the Test First, combine the samples and assign a rank to each observation in each group. For example: When a tie occurs, each observation is assigned the average of the ranks. Table 16.7

Kruskal-Wallis Test for Independent Samples Performing the Test Next, arrange the data by groups and sum the ranks to obtain the Tj’s. Remember, STj = n(n+1)/2. Table 16.8

Kruskal-Wallis Test for Independent Samples Performing the Test The hypotheses to be tested are: H0: All c population medians are the same H1: Not all the population medians are the same For a completely randomized design with c groups, the tests statistic is where n = n1 + n2 + … + nc nj = number of observations in group j Tj = sum of ranks for group j calc

Kruskal-Wallis Test for Independent Samples Performing the Test The H test statistic follows a chi-square distribution with n = c – 1 degrees of freedom. This is a right-tailed test, so reject H0 if H > c2a or if p-value < a.

Friedman Test for Related Samples The Friedman test determines if c treatments have the same central tendency (medians) when there is a second factor with r levels and the populations are assumed to be the same except for centrality. This test is analogous to a two-factor ANOVA without replication (randomized block design) with one observation per cell. The groups must be of the same size. Treatments should be randomly assigned within blocks. Data should be at least interval scale.

Friedman Test for Related Samples In addition to the c treatment levels that define the columns, the Friedman test also specifies r block factor levels to define each row of the observation matrix. The hypotheses to be tested are: H0: All c populations have the same median H1: Not all the populations have the same median Unlike the Kruskal-Wallis test, the Friedman ranks are computed within each block rather than within a pooled sample.

Friedman Test for Related Samples Performing the Test First, assign a rank to each observation within each row. For example, within each Trial: When a tie occurs, each observation is assigned the average of the ranks.

Friedman Test for Related Samples Performing the Test Compute the test statistic: where r = the number of blocks (rows) c = the number of treatments (columns) Tj = the sum of ranks for treatment j calc

Friedman Test for Related Samples Performing the Test The Friedman test statistic F, follows a chi-square distribution with n = c – 1 degrees of freedom. Reject H0 if F > c2a or if p-value < a.

Spearman Rank Correlation Test Spearman’s rank correlation coefficient (Spearman’s rho) is an overall nonparametric test that measures the strength of the association (if any) between two variables. This method does not assume interval measurement. The sample rank correlation coefficient rs ranges from -1 < rs < +1.

Spearman Rank Correlation Test The sign of rs indicates whether the relationship is direct – ranks tend to vary in the same direction, or inverse – ranks tend to vary in opposite directions The magnitude of rs indicated the degree of relationship. If rs is near 0 – there is little or no agreement between rankings rs is near +1 – there is strong direct agreement rs is near -1 – there is strong inverse agreement

Spearman Rank Correlation Test Performing the Test First, rank each variable. For example, If more than one value is the same, assign the average of the ranks. Table 16.11

Spearman Rank Correlation Test Performing the Test The sums of ranks within each column must always be n(n+1)/2. Next, compute the difference in ranks di for each observation. The rank differences should sum to zero.

Spearman Rank Correlation Test Performing the Test Calculate the sample rank correlation coefficient rs. where di = difference in ranks for case i n = sample size For a right-tailed test, the hypotheses to be tested are H0: True rank correlation is zero (rs < 0) H1: True rank correlation is positive (rs > 0)

Spearman Rank Correlation Test Performing the Test If n is large (at least 20 observations), then rs may be assumed to follow the Student’s t distribution with degrees of freedom n = n - 1 Reject H0 if t > ta or if p-value < a. calc

Correlation versus Causation Caution: correlation does not prove causation. Correlations may prove to be “significant” even when there is no causal relation between the two variables. However, causation is not ruled out. Multiple causes may be present.

Applied Statistics in Business & Economics End of Chapter 16 16-37