Introduction to nonparametric statistics

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Introduction to non-parametric statistics

• Cord Heuer and Peter Davies
• EpiCentre, Massey University

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Ingredients

• Relationship between parametric and
  non-parametric procedures
• Circumstances for using NP procedures
• Sign and rank transformations
• Introduce most common NP procedures
• Case study - NP vs. P tests
• Limitations of NP methods
• Example abortion rates

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What options ?

• Robustness
• Outliers
• Transforming data
• Other distributions
• Non-parametric tests

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Do a 30 second pulse count!
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Types of Data

• Nominal - no numerical value
• Ordinal - order or rank
• Discrete - counts
• Continuous - interval, ratio

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Parametric statistics

• Inference based on parameters (m, s) of
  distributions
• Require measurement on continuous (interval or
  ratio) scale
• Other assumptions (e.g. homogeneity of variance)

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Parametric procedures assume a distribution, e.g.
normal
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Types of data and analysis

• Nominal
• Ordinal
• Discrete
• Continuous

Non-parametric
Parametric
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What to do with non-normal data?

• Ignore and proceed
• Transform data and use parametric methods
• Use non-parametric procedures

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Non-parametric statistics

• Inference does not rely on estimation of
  distribution parameters
• Distribution-free statistics
• Developed for nominal and ordinal data or data of
  unknown distribution
• Can be used with continuous and discrete data
  when assumptions of parametric tests are not met

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Why non-parametric statistics?

• Need to analyse
• Crude data (nominal, ordinal)
• Data derived from small samples
• Data that do not follow a normal distribution
• Data of unknown distribution

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Departures from normality are common
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Transformation of data to meet assumptions
Log transformation of right skewed distribution
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Annual incidence of abortion in 602 dairy herds
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Use non-parametric statistics when

• Nominal data or data converted to counts and
  measurement scale ignored
• Ordinal data or data converted to ranks

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Non-parametric statistics

• All common parametric tests have non-parametric
  counterparts
• Use with continuous data involves loss of
  information and lower power
• Non-parametric procedures can have greater power
  when data not normal
• If assumptions hold, use parametric methods
• NP methods also have one assumption
• independence

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Non-parametric options
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Signs and ranks

• NP methods use relatively simple approaches to
  data
• Signs and ranks
• Higher order data transformed to signs or ranks
  for NP analysis

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Signs and the sign transformation

• Information in the data ignored apart from
  direction that each point differs from a
  reference point
• Better (), worse (-), no change
• Comparison of blood pressure when receiving
  treatment vs. placebo

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Sign transformation - blood pressure

• Paired data
• Difference in BP determined by subtracting value
  on treatment from value on placebo
• Could use actual values (paired t-test)
• Convert each difference to signs ( or -)
  ignoring zero values
• Use minimal information in data, therefore loss
  of power

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Ranks

• Transform data to ordinal form as ranks
• More information retained than with signs
• Rank tests have greater power than sign tests

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5.4 5.6
105.8 103.6
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Ties
OBS Rank 12 1 13 2 14 3 15 4.5
15 4.5 16 6 17 7 18 8 18 8 18 8 19 11
20 12

• Ties occur when two observations return the same
  value
• Ties assigned the mean rank of the tied values
• Software packages detect and adjust for tied
  values

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Case study - response to vaccination against
canine parvovirus

• Paired serum samples collected from adult dogs
  before and 2 weeks after vaccination
• Serological results reported as titres using
  2-fold dilution (140 180 1160.)
• Interest in whether vaccination affects test
  results
• Examine distributions

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Nature of the data?
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Histogram of test results before vaccination
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Histogram of test results after vaccination
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Options for analysis
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Paired t-test
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The Sign test

• Most crude and insensitive test
• Ideal quick and dirty test
• If sign test shows significance but method B does
  not, question method B
• Ignores information other than direction
• Can be used when distribution is asymmetric

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The sign test

• Under null hypothesis, and - values should
  occur with equal probability and mean difference
  should equal zero
• Ignore any zero values and count and - values
• P () P (-) 0.5 (null hypothesis)
• Binomial test of observed data

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Do a 30 second pulse count!
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Sign test of vaccination data
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Sign test of vaccination data
Probability of throwing 0 or 1 heads in 13
tosses of a coin
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Wilcoxon signed ranks test

• Based on ranks - takes magnitude into account
• Higher power than sign test
• more weight to pairs that show large differences
  than to pairs that show small differences
• Use whenever one sample t-test used
• Can also test the hypothesis that 2 variables
  have the same distribution
• BUT data must have a symmetric distribution

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Wilcoxon signed ranks test

• Rank the absolute value (i.e. ignoring sign) of
  differences from smallest to largest, ignoring
  values of zero.
• Sum the ranks assigned to positive values, then
  to negative values.
• The smaller value of the positive or negative
  rank sums is the Wilcoxon signed rank statistic
  (W).
• P probability of this rank sum occurring
  under the null hypothesis

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Wilcoxon signed ranks test

• H0 each observation is from a symmetrical
  distribution with mean zero
• Positive and negative results equally likely
• Total sum of ranks fixed by N (n n-)
• Need only consider one group (smallest)
• If N 10, sum of ranks 55
• If sum of ranks of n- 18, then sum of ranks
  of n must equal 37

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Wilcoxon signed ranks test

• With 13 pairs of which 3 were zero, there are 10
  pairs available for analysis
• 10 pairs could give a total of 55 ranks if all
  were on the same side (10911) 55
• Ho 27/55 expected (if half were above/below 0)
• And 18/55 observed
• 10/55 would correspond to p 0.10
• Not enough evidence to reject Ho

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Wilcoxon signed ranks test

• If N gt 16 use approximation
• Where T smaller rank sum

N 10 (ie. lt16!)
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Wilcoxon signed ranks test - vaccination data
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Log transformed results before vaccination
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Log transformed results after vaccination
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Paired t-test after log transformation
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Summary of vaccine data

• All methods indicate significant difference
• t - test on untransformed data has lowest power
• WSR has greater power than ST
• t-test after transformation has highest power
• Transformation does not affect ST
• Transformation can affect WSR (rank order)
• aKolmogorov-Smirnow test of transformed values
  fits a normal distribution

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Independent groups

• 2 groups
• Wilcoxon-Mann-Whitney test
• gt2 groups
• Kruskal Wallis test
• gt2 groups and categorical covariates
• Friedman test

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Correlation

• Spearman rank test

Spearman rank correlation rs 1 - 6SUM(diff2)
/ (n(n2-1)) Example rs 1 6 8 /
(7(72-1)) 0.857