Parametric versus Nonparametric Statistics – When to use them and which is more powerful? Angela Hebel Department of Natural Sciences University of Maryland Eastern Shore April 5, 2002

Parametric Assumptions The observations must be independent The observations must be drawn from normally distributed populations These populations must have the same variances The means of these normal and homoscedastic populations must be linear combinations of effects due to columns and/or rows*

Nonparametric Assumptions Observations are independent Variable under study has underlying continuity

Measurement What are the 4 levels of measurement discussed in Siegel’s chapter? 1. Nominal or Classificatory Scale Gender, ethnic background 2. Ordinal or Ranking Scale Hardness of rocks, beauty, military ranks 3. Interval Scale Celsius or Fahrenheit 4. Ratio Scale Kelvin temperature, speed, height, mass or weight

Nonparametric Methods There is at least one nonparametric test equivalent to a parametric test These tests fall into several categories Tests of differences between groups (independent samples) Tests of differences between variables (dependent samples) Tests of relationships between variables

Differences between independent groups Two samples – compare mean value for some variable of interest Parametric Nonparametric t-test for independent samples Wald-Wolfowitz runs test Mann-Whitney U test Kolmogorov-Smirnov two sample test

Mann-Whitney U Test Nonparametric alternative to two-sample t-test Actual measurements not used – ranks of the measurements used Data can be ranked from highest to lowest or lowest to highest values Calculate Mann-Whitney U statistic U = n1n2 + n1(n1+1) – R1 2

Example of Mann-Whitney U test Two tailed null hypothesis that there is no difference between the heights of male and female students Ho: Male and female students are the same height HA: Male and female students are not the same height

U = n1n2 + n1(n1+1) – R1 2 U=(7)(5) + (7)(8) – 30 U = 35 + 28 – 30 Heights of males (cm) Heights of females (cm) Ranks of male heights Ranks of female heights 193 175 1 7 188 173 2 8 185 168 3 10 183 165 4 11 180 163 5 12 178 6 170 9 n1 = 7 n2 = 5 R1 = 30 R2 = 48 U = n1n2 + n1(n1+1) – R1 2 U=(7)(5) + (7)(8) – 30 U = 35 + 28 – 30 U = 33 U’ = n1n2 – U U’ = (7)(5) – 33 U’ = 2 U 0.05(2),7,5 = U 0.05(2),5,7 = 30 As 33 > 30, Ho is rejected Zar, 1996

Differences between independent groups Multiple groups Parametric Nonparametric Analysis of variance (ANOVA/ MANOVA) Kruskal-Wallis analysis of ranks Median test

Differences between dependent groups Compare two variables measured in the same sample If more than two variables are measured in same sample Parametric Nonparametric t-test for dependent samples Sign test Wilcoxon’s matched pairs test Repeated measures ANOVA Friedman’s two way analysis of variance Cochran Q

Relationships between variables Two variables of interest are categorical Parametric Nonparametric Correlation coefficient Spearman R Kendall Tau Coefficient Gamma Chi square Phi coefficient Fisher exact test Kendall coefficient of concordance

Summary Table of Statistical Tests Level of Measurement Sample Characteristics Correlation 1 Sample 2 Sample K Sample (i.e., >2) Independent Dependent Categorical or Nominal Χ2 or bi-nomial Χ2 Macnarmar’s Χ2 Cochran’s Q   Rank or Ordinal Mann Whitney U Wilcoxin Matched Pairs Signed Ranks Kruskal Wallis H Friendman’s ANOVA Spearman’s rho Parametric (Interval & Ratio) z test or t test t test between groups t test within groups 1 way ANOVA between groups 1 way ANOVA (within or repeated measure) Pearson’s r Factorial (2 way) ANOVA   (Plonskey, 2001)

Advantages of Nonparametric Tests Probability statements obtained from most nonparametric statistics are exact probabilities, regardless of the shape of the population distribution from which the random sample was drawn If sample sizes as small as N=6 are used, there is no alternative to using a nonparametric test Siegel, 1956

Advantages of Nonparametric Tests Treat samples made up of observations from several different populations. Can treat data which are inherently in ranks as well as data whose seemingly numerical scores have the strength in ranks They are available to treat data which are classificatory Easier to learn and apply than parametric tests Siegel, 1956

Criticisms of Nonparametric Procedures Losing precision/wasteful of data Low power False sense of security Lack of software Testing distributions only Higher-ordered interactions not dealt with

Power of a Test Statistical power – probability of rejecting the null hypothesis when it is in fact false and should be rejected Power of parametric tests – calculated from formula, tables, and graphs based on their underlying distribution Power of nonparametric tests – less straightforward; calculated using Monte Carlo simulation methods (Mumby, 2002)

Questions?