Overview of non parametric methods

Parametric methods. Normality

Parametric methods. Normality

Parametric methods. Sample size.

Qualitative or Quantitative Parametric methods. Problems in measurement (scale of measurement) Stanley Smith Stevens' typology #   Scale type   Logical/math operations   allowed   Examples:      Variable name                            (data values) Measure of central tendency Qualitative or Quantitative 1 Nominal =/≠ Dichotomous:     Gender                              (male vs. female) Non-dichotomous: Nationality                              (American/Chinese/etc) Mode Qualitative 2 Ordinal =/≠ ; </> Dichotomous:     Health (healthy vs. sick),                          Truth (true vs. false),                          Beauty (beautiful vs. ugly) Non-dichotomous: Opinion ('completely agree'/ 'mostly agree'/ 'mostly disagree'/ 'completely disagree') Median 3 Interval =/≠ ; </> ; +/−   Date (from 9999 BC to 2013 AD)   Latitude (from +90° to −90°) Arithmetic Mean Quantitative 4 Ratio =/≠ ; </> ; +/− ; ×/÷   Age (from 0 to 99 years) Geometric Mean

«Loщ-quality» data Non-normal distribution Small sample size Low level of measurement

nonparametric methods and nonparametric methods (parameter-free methods or distribution-free methods.)

Nonparametric methods Descriptive statistics; Tests of differences between groups (independent samples); Tests of differences between variables (dependent samples); Tests of relationships between variables.

Nonparametric methods Descriptive statistics G = (x1 *x2 *...*xn )1/n - geometric mean log(G) = {Σ[log(xi)]}/n

Nonparametric methods Differences between dependent groups. 2 variables Sign test Wilcoxon's matched pairs test. More than 2 vars Cochran's Q test

Nonparametric methods Sign test Peanut Butter Taste Test As part of a market research study, a sample of 36 consumers were asked to taste two brands of peanut butter and indicate a preference. Do the data shown below indicate a significant difference in the consumer preferences for the two brands? 18 preferred Hoppy Peanut Butter 12 preferred Pokey Peanut Butter 6 had no preference The analysis is based on a sample size of 18 + 12 = 30.

Nonparametric methods Sign test Peanut Butter Taste Test p = Pr(X > Y) H0: p = 0.50 W be the number of pairs for which yi − xi > 0 Reject H0 if z < -1.96 or z > 1.96 z = (18 - 15)/2.74 = 3/2.74 = 1.095

Nonparametric methods Wilcoxon Signed-Rank Test District Office Overnight NiteFlite Seattle 32 hrs. 25 hrs. Los Angeles 30 24 Boston 19 15 Cleveland 16 15 New York 15 13 Houston 18 15 Atlanta 14 15 St. Louis 10 8 Milwaukee 7 9 Denver 16 11

Nonparametric methods Wilcoxon Signed-Rank Test District Office Differ. Diff. Rank Sign. Rank Seattle 7 10 +10 Los Angeles 6 9 +9 Boston 4 7 +7 Cleveland 1 1.5 +1.5 New York 2 4 +4 Houston 3 6 +6 Atlanta -1 1.5 -1.5 St. Louis 2 4 +4 Milwaukee -2 4 -4 Denver 5 8 +8 +44

Nonparametric methods . Nonparametric methods Wilcoxon Signed-Rank Test Compute the differences between the paired observations. Discard any differences of zero. Rank the absolute value of the differences from lowest to highest. Tied differences are assigned the average ranking of their positions. Give the ranks the sign of the original difference in the data. Sum the signed ranks. . . . next we will determine whether the sum is significantly different from zero.

Nonparametric methods Differences between independent groups. 2 variables Mann-Whitney U test Wald-Wolfowitz runs test More than 2 vars Kruskal-Wallis analysis of ranks

Nonparametric methods . Nonparametric methods Mann-Whitney U test First, rank the combined data from the lowest to the highest values, with tied values being assigned the average of the tied rankings. Then, compute U, the sum of the ranks for the each sample. The smaller value of U1 and U2 is the one used when consulting significance tables. If U < = Table value H1, else H0 U is just number of wins out of all pairwise contests

Nonparametric methods Mann-Whitney U test, another way T H H H H H T T T T T H

Nonparametric methods Mann-Whitney U test

Nonparametric methods Tests of relationships between variables. Spearman, Kendall tau, Gamma correlation coefficients 2x2 Tables

Nonparametric methods Spearman correllation ui = rank of item i with respect to one variable vi = rank of item i with respect to a second variable di = ui - vi

Nonparametric methods Spearman correllation Car Age (months) Xi Minimal Stopping dist at 40 kph (metres) Yi Age Rank (ui) Stopping Rank (vi) Differences of the Ranks (di = ui-vi) A 9 28.4 1 B 15 29.3 2 C 24 37.6 3 7 -4 D 30 36.2 4 4.5 -0.5 E 38 36.5 5 6 -1 F 46 35.3 G 53 2.5 H 60 44.1 8 I 64 44.8 J 76 47.2 10 d2=32.5

Nonparametric methods Classification and clustering

Nonparametric methods 2x2 Tables Suppose that you are considering whether to introduce a new formula for a successful soft drink. Before finally deciding on the new formula, you conduct a survey in which you ask male and female respondents to express their preference for either the old or new soft drink. Assume that out of 50 males, 41 prefer the new formula over the old formula; out of 50 females, only 27 prefer the new formula. 41 9 27 23 Chi-square (df=1) 9.01 p= .0027 V-square (df=1) 8.92 p= .0028 Yates corrected Chi-square 7.77 p= .0053 Phi-square .09007 Fisher exact p, one-tailed p= .0025 two-tailed p= .0049 McNemar Chi-square (A/D) 4.52 p= .0336 Chi-square (B/C) 8.03 p= .0046