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NON-PARAMETRIC DATA

NON-PARAMETRIC DATA. Parametric Tests. Parametric tests assume that data fits a Normal distribution When you plot the data as a frequency histogram, it should look like a bell-shaped curve. Non-parametric Tests. However, biological variables are often not normally distributed

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NON-PARAMETRIC DATA

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  1. NON-PARAMETRIC DATA

  2. Parametric Tests • Parametric tests assume that data fits a Normal distribution • When you plot the data as a frequency histogram, it should look like a bell-shaped curve

  3. Non-parametric Tests • However, biological variables are often not normally distributed • Cannot use parametric tests to analyse data as assumptions are not met • Use non-parametric tests instead

  4. How do we know if data are normally distributed? • Tests of normality: • Kolmogorov-Smirnov (Liliefors) Test (extremely large data sets) • Shapiro-Wilk Test (small sample sizes) • Test null hypothesis that data do not deviate from a normal distribution

  5. Skewness & Kurtosis • Give an indication of deviation from normality • Skewness: • Normal distribution = 0 • Positive: long right tail • Negative: long left tail • Kurtosis: • Normal distribution = 0 • Positive: cluster more (leptokurtic) • Negative: cluster less (platykurtic)

  6. Transformations • Sometimes data can be normalised with an appropriate transformation • Perform parametric statistical tests on the transformed data

  7. Types of Transformations • Log: • Take log of data • Can use either natural log or log to base 10 Histograms of number of Eastern Mud Minnows per 75 m section of stream. Untransformed data on left, log-transformed data on right.

  8. Types of Transformations • Square root: • Take square root of data • Used when variable is a count • Arcsine: • Take arcsine of the square root of a number • Used when response variable is a proportion • Data must range between -1 and 1 • Reciprocal: • Time to occurrence of an event • Replace y with 1/y

  9. Example • Data on number of mud minnows per section of stream • Does this data follow a Normal distribution? • Use this data to perform a: • Square root transformation • Log transformation • Reciprocal transformation

  10. Tests of Normality – SPSS From the menus select: Analyse Descriptive statistics Explore Click on Plots and select Normality plots with tests ClickContinueand OK Tests of Normality – look at sig. values Normal Q-Q plots – data should be in a diagonal line, not curved

  11. Transformations in SPSS From the menus select: Transform Compute variable Type in a new variable name under Target Variable. The transformed data will appear as this variable in your spreadsheet. Specify the desired formula to calculate the transformed data in the Numeric Expression box

  12. MANN-WHITNEY U TEST

  13. Mann-Whitney U Test • Non-parametric equivalent of 2-sample t-test • Tests central tendency between two samples, does NOT test means or medians • Use sample sizes instead of degrees of freedom to look up critical values • Also known as Wilcox rank test, or Kendall’s S test

  14. Treatment A nA= 11 Treatment B nB = 10 Example HO: No difference between treatments A and B HA: Treatment A is more effective Treatments are supposed to prevent fungal growth 21 plants Treatment applied over 15 weeks

  15. Example Fungal growth ratings: Lower values indicate less fungal growth

  16. Example Data is NOT normally distributed:

  17. Example • Assemble all measures into a single set • Rank-order from lowest to highest

  18. Example • Substitute rank values for raw measures in table • Sum up the ranks of each group

  19. Ui = n1n2+ ni(ni+1) - Ri 2 Formula for U niis the sample size in group i Ri is the sum of ranks in group i

  20. U U' = n1n2+ = n1n2+ n1(n1+1) n2(n2+1) - R1 - R2 2 2 Example UA = 110 + 66 – 96.5 = 79.5 UB = 110 + 55 – 134.5 = 30.5

  21. Example • Sample sizes: n1 = 10 and n2 = 11 • U = 79.5 (use the larger of the two) • Alpha = 0.05 • Two-tailed Ucrit = 84 (Do not reject HO) • One-tailed Ucrit = 79 (Reject HO)

  22. SPSS Non-Parametric Tests

  23. Non-Parametric Tests • Non-parametric equivalents of t-tests and ANOVAs can be performed under this option • Usage is similar to the parametric tests

  24. Mann-Whitney U Test • From the menus select: Analyze Nonparametric Tests Independent Samples • Under the Objectivestab select “Automatically compare distributions across groups” • Under the Fields tab select one or more dependent variables and one grouping variable • Specify additional criteria under the Settings tab

  25. Mann-Whitney U Test • Example: • Use the same example of fungal growth on plants to perform a Mann-Whitney U test in SPSS

  26. Kruskal-Wallis Test • Non-parametric equivalent of the one-way ANOVA • Less powerful than ANOVA • Performed on ranked data • Mathematically identical to the Mann-Whitney U test • Uses H as a test statistic

  27. Kruskal-Wallis Test • Example: • Test whether there is a difference between the 3 groups

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