4/6/16. Non-parametric Test. Overview. Stephen Opiyo. Distinguish Parametric and Nonparametric Test Procedures

Transcription

1 Non-parametric Test Stephen Opiyo Overview Distinguish Parametric and Nonparametric Test Procedures Explain commonly used Nonparametric Test Procedures Perform Hypothesis Tests Using Nonparametric Procedures Hypothesis Testing Parametric ØTTest ØANOVA Non-Parametric ØU-Test ØKruskal-Wallis Overview of Hypothesis testing

2 Parametric Test Procedures Involve population parameters (Mean). Have stringent assumptions (Normality). Examples: TTest and ANOVA. Parametric Assumptions The observations must be independent The observations must be drawn from normally distributed populations Nonparametric Test Procedures Data not normally distributed Data measured on any scale (ratio or interval, ordinal or nominal). Example: Mann-Whitney U test, Kruskal-Wallis etc.

3 Mann-Whitney U Test Nonparametric alternative to two-sample TTest. Actual measurements not used ranks of the measurements are used. Data can be ranked from highest to lowest or lowest to highest values. Mann-Whitney U statistic equation. Calculate U and U. U = n n + n (n +) - R U = n n -U Mann-Whitne y U Test: Sample Size Consideration Size of sample : n Size of sample : n If both n and n are 0, the small sample procedure is appropriate. If either n or n is greater than 0, the large sample procedure is appropriate. Example of Mann-Whitney U test Two tailed null hypothesis that there is no difference between the of male and female students H o: Male and female students are the same height H a: Male and female students are not the same height 3

4 Example of Mann-Whitney U test males females n = 7 n = 5 Rank the of males and females males females males females male female n = 7 n = 5 R = 30 R = n = 7 n = 5 U = nn + n(n+) R males females male female U=(7)(5) + (7)(8) 30 U = U = 33 U = nn U n = 7 n = 5 R = 30 R = 48 U = (7)(5) 33 U = The smaller value of U and U is the one used when consulting significance tables 4

5 U = nn + n(n+) R males females male female U=(7)(5) + (7)(8) 30 U = U = 33 U = nn U U = (7)(5) n = 7 n = 5 R = 30 R = 48 U = To be statistically significant, the obtained U has to be equal to or less than this critical value. U 0.05(,7,5) = U 0.05(5,7) = 5 As < 5, Ho is rejected Mann-Whitney U Test: Formulas for Large Sample Case If either n or n is > 0, the sampling distribution of U is approximately normal. ( ) U = n n + n n + W where : n = number in group = number in group n W = sum or the ranks of values in group µ = n n U σ = n n n + n + U Z = U µ σ U U ( ) 5

6 Comparing Three or More Populations: Kruskal-Wallis H-Test Tests the equality of more than two (p) population probability distributions Corresponds to ANOVA. Uses c distribution with p df Kruskal-Wallis H-Test for Comparing k Probability Distributions H 0 : The k probability distributions are identical H a : At least two of the k probability distributions differ in location. Squared total of each group Test statistic:! R " j H = 3 + $ n( n ) + n % & j ' ( n ) 6

7 Kruskal-Wallis H-Test for Comparing k Probability Distributions where n j = Number of measurements in sample j R j = Rank sum for sample j, where the rank of each measurement is computed according to its relative magnitude in the totality of data for the k samples n = Total sample size = n + n n k Kruskal-Wallis H-Test for Comparing k Probability Distributions Rejection region: H > χ α with (k ) degrees of freedom Ties: Assign tied measurements the average of the ranks they would receive if they were unequal but occurred in successive order. For example, if the third-ranked and fourth-ranked measurements are tied, assign each a rank of (3 + 4)/ = 3.5. The number should be small relative to the total number of observations. Conditions Required for the Validity of the Kruskal-Wallis H-Test. The k samples are random and independent.. The k probability distributions from which the samples are drawn are continuous 7

8 Kruskal-Wallis H-Test Procedure. Assign ranks, R i, to the n combined observations Smallest value = ; largest value = n Average ties. Sum ranks for each group 3. Compute test statistic! R " j H = 3 + $ n( n ) + n % & j ' Squared total of each group ( n ) Kruskal-Wallis H-Test Example A production manager wants to see if three filling machines have different filling times. He assigns 5 similarly trained and experienced workers, 5 per machine, to the machines. At the.05 level of significance, is there a difference in the distribution of filling times? Mach Mach Mach H 0: Identical Distrib. H a: At Least Differ a =.05 df = p = 3 = Critical Value(s): a = c 8

9 Mach Mach Mach Mach Mach Mach3 Mach Mach Mach Mach Mach Mach3 Mach Mach Mach Mach Mach Mach3 9

10 Mach Mach Mach Mach Mach Mach3 3 Mach Mach Mach Mach Mach Mach Mach Mach Mach Tot al Mach Mach Mach

11 ! R " j H = 3 + $ n( n ) + n % & j ' ( n ) ( 65) ( 38) ( 7)! "! " = $ + + % 3( 6) $ ( 5)( 6) $ %% & & ''! " = $ %( 9.6) 48 & 40 ' H = n =.58 n n + j R j R ( ) ( ) H 0: Identical Distrib. H a: At Least Differ a =.05 df = p = 3 = Critical Value(s): a =.05 c Tes t St at is t ic : H =.58 Decision: Reject at a =.05 Conclusion: There is evidence population distrib. are different Post hoc after Kruskal-Wallis Test post-hoc Nemenyi Test