Non-parametric stats
- Created by: Sam_dearnx
- Created on: 24-01-17 21:11
Rank transformation
- Generic way to deal with heavily skewed or otherwise difficult data
- Assign 1 to lowest value, 2 to next lowest etc
- Average score of group for same values (ties)
Effects of rank transformation
- Will flatten any distribution and remove gaps between outcome value ranges.
- Moves outliers closer to the centre of the data
- Hence deals with the most serious issues in non-normal data: modality, skew, outliers
- E.g. identical rank scores for log-normal and transformed normal distributions
Non-parametric correlation methods
- Spearman’s rank
o For non-normal but continuous data (interval and ratio)
o Can also be used for ordinal data with many levels.
o Works on rank transformations
- Kendall’s tau
o Specifically designed for discrete data (i.e whole numbers only) and ordinal data
o E.g. rating on Likert scales
Spearman's rank correlation
- Step 1: rank transformation of the x and y variables
- Step 2: Pearson’s r correlation on the transformed variables
- Stats software will perform step one implicitly, you don’t need to do the rank transformation first.
- Correlation coefficient denoted by Greek letter p (rho) or rs
- Rho has the same range (12, 1) and interpretation as Pearson’s R
- Where rank(x) and rank(y) denote the rank-transformed variables x and y, respectively.
Spearman's rho vs Pearson R
- Pearson’s r assesses the linear relationship
- Spearman’s assesses only a monotonic relationship
- Similar results if x and y are near-normal – use Pearson
- Different if not normal (outliers) – use Spearman
Kendall's tau rank correlation
- Step 1: rank transformation of the x and y variables
- Step 2: count concordant and discordant pairs
- Concordant pair: ranks for both observations (x-y pairs i and j) agree (xi > xj and yi > yj or xi < xj and yi < yj)
- Discordant pair: where ranks for both observations (x-y pairs i and j) differ (xi > xj and yi < yj or xi > xj and yi < yj)
- correlation coefficient denoted by Greek letter (tau)
- Total number of pairs is n(n-1)/2, used as scaling factor
- Tau has range (-1,1), same interpretation as r and rho
Tau = number of concordant pairs - number of discondant pairs / n (n-1)/2
Kendall's tau rank correlation
- Step 1: rank transformation of the x and y variables
- Step 2: count concordant and discordant pairs
- Concordant – both the x values are larger than the y values
- Discordant – x value is larger and a y value is smaller
- correlation coefficient denoted by Greek letter (tau)
- Total number of pairs is n(n-1)/2, used as scaling factor
- Tau has range (-1,1), same interpretation as r and rho
Tau = number of concordant pairs - number of discondant pairs / n (n-1)/2
Accounting for ties
- Ties: Pairs which are neither concordant nor discordant. i.e. where ranks for x or y in a pair do not differ.
- Tau-A: No adjustment for ties. This can keep true range for tau smaller than (-1, 1)
- Tau-B: Adjustment for ties to keep range of tau in (-1, 1). Easier to interpret. Standard method used in most stats packages
- Tau-C: Another way of adjusting for ties. Not frequently used.
Kendall's tau vs Spearman's rho
- Both typically yield similar results
- Tau is more robust for significance testing in smaller samples
- Tau is particularly suitable for discrete and ordinal data
Non-parametric tests for group differences
- Wilcoxon signed-rank test
o For paired samples
o Alternative to paired t test
- Mann-Whitney U
o For independent samples
o Alternative to independent samples t test
- Binomial test
o For binomial data
Wilcoxon signed rank test
- Frank Wilcoxon (1945): signed-rank method
o You have n paired observations x1 and x2
o For each pair, calculate:
§ absolute value difference (always positive): |1x1i - 2i| or abs(x1i-x2i)
§ sign of difference (+ or -): (x1i - x2i)
o exclude pairs with 1x1i - x2i
o reduce sample size by excluded pairs to get nr
o rank pairs by absolute difference and multiply rank with sign
o Sum of signed ranks gives W (Wilcoxon test statistic)
wilcoxon test statistic
- Similar to t test, W follows a known distribution that approximates normal as n increases
- W and degrees of freedom (n-1) can then be used to derive Z statistic and significance (p)
- Stats program will do all this for you and give you W, Z and p
The Mann-Whitney U test
- Developed by Mann and Whitney in 1947 and Wilcoxon (1945), also known as Wilcoxon ranked-sum test
- For independent samples
- Ranked-sum method:
o you have two samples with sample sizes n1, n2
o Combine both samples and rank all values
o Add up the ranks which came from each sample to find sum of ranks r1, r2
o Use smaller U as the U test statistic
Mann-whitney I test interpretation
- U can range from 0 (complete separation between the groups, most likely to reject null hypothesis) to n1 * n2 (no separation between the groups, accept H0)
- U and df used to derive Z
- Z is normally distributed for larger samples
- Z used to derive significance (p), all usually included in stats program output.
- Having numerous tied ranks leads to problems
o Complex correction to standard deviation of U in stats software
o Not an issue if only few ties are present
Advantages of ranked group difference tests
- They do not require normally distributed data
- Ranking method provides robustness against outliers
- Also applicable to ordinal data, unlike t tests
- But if you have normal data use t test, more power
- Non-parametric tests that do not use rankings:
o For 2 sets of binomial data use binomial test
o Chi square or McNemar for nominal data
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