Statistics in Applied Science and Technology Chapter14. Nonparametric Methods

Key Concepts in this Chapter Nonparametric methods Nonparametric methods Distribution-free methods Distribution-free methods Ranks of observation Ranks of observation Wilcoxon Rank-Sum Test Wilcoxon Rank-Sum Test Kruskal-Wallis One-Way ANOVA By Ranks Kruskal-Wallis One-Way ANOVA By Ranks Spearman Rank-Order Correlation Coefficient (r s ) Spearman Rank-Order Correlation Coefficient (r s )

Rationale for Nonparametric Methods Nonparametric methods, often referred to as distribution-free methods, do not require any assumption about the shape of the underlying population distribution or sample size. Nonparametric methods, often referred to as distribution-free methods, do not require any assumption about the shape of the underlying population distribution or sample size. Nonparametric methods are appropriate when dealing with data that are measured on a nominal or ordinal scale. Nonparametric methods are appropriate when dealing with data that are measured on a nominal or ordinal scale.

Advantages and Disadvantages Advantages: Advantages:  No restrictive assumptions such as normality of the observations and large sample size.  Easy and speedy computation  Good for nominal or ordinal data Disadvantages: Disadvantages:  Less efficient (require larger sample size to reject a false H 0 )  Less specific  Minimal utilization of distribution

Inherent Characteristic of Nonparametric Methods Nonparametric methods deal with ranks rather than values of the observations Nonparametric methods deal with ranks rather than values of the observations Computation is simple Computation is simple

Wilcoxon Rank-Sum Test (I) Wilcoxon Rank-Sum Test is used to test if there is difference in the two population distributions Wilcoxon Rank-Sum Test is used to test if there is difference in the two population distributions Corresponds to the t test for two independent sample means Corresponds to the t test for two independent sample means No assumptions are necessary No assumptions are necessary

Wilcoxon Rank-Sum Test (II) H 0 : No difference in two population distribution H 0 : No difference in two population distribution H 1 : There is a difference in two population distribution H 1 : There is a difference in two population distribution Test Statistics: Z test using sum of the ranks Test Statistics: Z test using sum of the ranks

Wilcoxon Rank-Sum Test (III) Test Statistics Z can be calculated by: Where: W 1 is the sum of ranks of the sample W e is the expected sum of the ranks assuming H 0 is true.  w is the standard error W e can be found using the following equation:  W can be found using the following equation: Where: n 1 and n 2 is the number of observations in two samples, respectively. Where: n 1 and n 2 are defined as above.

Wilcoxon Rank-Sum Test (IV) Decision Rule: At  of 0.05, reject H 0 if Z is above 1.96 or below –1.96. At  of 0.01, reject H 0 if Z is above 2.56 or below –2.56. Decision Rule: At  of 0.05, reject H 0 if Z is above 1.96 or below –1.96. At  of 0.01, reject H 0 if Z is above 2.56 or below –2.56.

Kruskal-Wallis One-Way ANOVA By Ranks (I) Nonparametric equivalent of the one-way ANOVA (the one we discussed in chapter 10). Nonparametric equivalent of the one-way ANOVA (the one we discussed in chapter 10). Appropriate when underlying population is not normally distributed or the samples do not have equal variances. Appropriate when underlying population is not normally distributed or the samples do not have equal variances. Appropriate when data is ordinal Appropriate when data is ordinal

Kruskal-Wallis One-Way ANOVA By Ranks (II) H 0 : No differences among more than two population distributions (K groups) H 0 : No differences among more than two population distributions (K groups) H 1 : There is at least one group has a different population distribution than others H 1 : There is at least one group has a different population distribution than others Test Statistics: H test using sum of the ranks Test Statistics: H test using sum of the ranks

Kruskal-Wallis One-Way ANOVA By Ranks (III) Test statistics H can be calculated by the following: Where: k = the number of groups n j = the number of observations in the jth group N = total number of observations in all groups R j = the sum of ranks in the jth group

Kruskal-Wallis One-Way ANOVA By Ranks (IV) Decision Rule: Reject H 0 when calculated H is more than critical H which can be found in Appendix F (textbook pg. 298) Decision Rule: Reject H 0 when calculated H is more than critical H which can be found in Appendix F (textbook pg. 298) Tied Observations will somewhat influence H, a term introduced in the denominator can correct this effect. (pg.230) Tied Observations will somewhat influence H, a term introduced in the denominator can correct this effect. (pg.230)

Spearman Rank-Order Correlation Coefficient (r s ) Appropriate when two interval-ratio variables deviate away from normal distribution Appropriate when two interval-ratio variables deviate away from normal distribution Appropriate when we deal with two ordinal variables that have a broad range of many different categories since using Gamma because somewhat inconvenient. Appropriate when we deal with two ordinal variables that have a broad range of many different categories since using Gamma because somewhat inconvenient.

Spearman Rank-Order Correlation Coefficient (r s ) r s may take on values from –1 to +1. Values close to  1 indicate a strong correlation; values close to zero indicate a weak association. The sign of rs indicates the direction of association. r s may take on values from –1 to +1. Values close to  1 indicate a strong correlation; values close to zero indicate a weak association. The sign of rs indicates the direction of association. r s 2 represents the proportional reduction in errors of prediction when predicting rank on one variable from rank on the other variable, as compared to predicting rank while ignoring the other variable. r s 2 represents the proportional reduction in errors of prediction when predicting rank on one variable from rank on the other variable, as compared to predicting rank while ignoring the other variable.

Calculation of r s Where: d i is the difference between the paired ranks n is the number of pairs

Is r s “statistically significant”? If sample size is at least 10; x and y represent randomly selected and independent pairs of ranks If sample size is at least 10; x and y represent randomly selected and independent pairs of ranks We can use t test to test hypothesis: We can use t test to test hypothesis:  H 0 :  s = 0  H 1 :  s  0

Is r s “statistically significant”? t test calculation: t test calculation: With n-2 df