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 1. Tests of differences between groups (independent samples) 2. Tests of differences between variables (dependent samples) 3. 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 KolmogorovSmirnov 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 2 U = 35 + 28 – 30 U = 33 U’ = n1n2 – U 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 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 R2 = 48 Differences between independent groups  Multiple groups Parametric Nonparametric Analysis of Kruskal-Wallis variance analysis of (ANOVA/ ranks MANOVA) Median test Differences between dependent groups   Compare two variables measured in the same sample Parametric t-test for dependent samples If more than two variables are measured in Repeated same sample measures ANOVA Nonparametric Sign test Wilcoxon’s matched pairs test Friedman’s two way analysis of variance Cochran Q Relationships between variables Parametric Nonparametric Correlation coefficient Spearman R Kendall Tau Coefficient Gamma  Two variables of interest are categorical Chi square Phi coefficient Fisher exact test Kendall coefficient of concordance Summary Table of Statistical Tests Level of Measurement Sample Characteristics 1 Sample Categorical or Nominal Χ2 or binomial Rank or Ordinal Parametric (Interval & Ratio) z test or t test 2 Sample Correlation K Sample (i.e., >2) Independent Dependent Independent Dependent Χ2 Macnarmar’ s Χ2 Χ2 Cochran’s Q Mann Whitney U Wilcoxin Matched Pairs Signed Ranks Kruskal Wallis H Friendman’s ANOVA Spearman’s rho 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?