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Parametric versus Nonparametric Statistics – When to use them and which is more powerful?

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

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Parametric versus Nonparametric Statistics – When to use them and which is more powerful?

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  1. 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

  2. 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*

  3. Nonparametric Assumptions • Observations are independent • Variable under study has underlying continuity

  4. 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

  5. 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

  6. Differences between independent groups • Two samples – compare mean value for some variable of interest

  7. 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

  8. 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

  9. U = n1n2 + n1(n1+1) – R1 • 2 • U=(7)(5) + (7)(8) – 30 • 2 • 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

  10. Differences between independent groups • Multiple groups

  11. Differences between dependent groups • Compare two variables measured in the same sample • If more than two variables are measured in same sample

  12. Relationships between variables • Two variables of interest are categorical

  13. Level of Measurement Sample Characteristics Correlation 1 Sample 2 Sample K Sample (i.e., >2) Independent Dependent Independent Dependent Categorical or Nominal Χ2 or bi-nomial Χ2 Macnarmar’s Χ2 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 Summary Table of Statistical Tests (Plonskey, 2001)

  14. 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

  15. 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

  16. 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

  17. 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)

  18. Questions?

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