Nonparametric Approaches

1
Non-parametric Approaches

• The Bootstrap

2
Non-parametric?

• Non-parametric or distribution-free tests have
  more lax and/or different assumptions
• Properties
• No assumption about the underlying distribution
  being normal
• More sensitive to medians than means (which is
  good if youre interested in the median)
• Some may not be very affected by outliers
• Rank tests

3
Parametric vs. Non-parametric

• Parametric tests will typically be used when
  assumptions are met as they will usually have
  more power
• Though the non-parametric tests might be able to
  match that power under certain conditions
• Non-parametric tests are often used with small
  samples or violations of assumptions

4
Some commonly used nonparametric analyses

• Chi-Square
• Chi-Square analysis involves categorical
  variables/frequency data only
• Example
• Party Republican Democrat
• Vote Yes No
• In this case, we cannot meet the assumptions
  common to our typical tests, but the goal would
  still be to understand the relationship between
  the variables involved
• Chi-square analysis examines such relationships
  regarding frequencies of cells, and we can still
  get measures of the strength of the association
• See effect size handout

5
Common Rank tests

• Wilcoxon t for independent and dependent samples,
  Mann-Whitney U
• Kruskal-Wallis, Friedman for more than 2 groups
• Basic procedure
• Rank the DV and get sums of the ranks for the
  groups
• Construct a test statistic based on the ranked
  data
• Advantage
• Normality not necessary
• Insensitive to outliers
• Disadvantage
• Ranked data is not in original units and so
  therefore may be less interpretable
• May lack power, particularly when parametric
  assumptions hold

6
Transformation of data

• So if I dont like my data I just change it?
• Think about what youre studying
• Is depression a function of Likert scale
  questions?
• Is reaction time inherently related to learning?
• Tukey reexpressions
• Our original numbers are already useful fictions,
  and if we think of them as such, transforming
  them into something else may not seem so
  far-fetched

7
Some common transformations

• Logarithmic Positively skewed
• Square root Count data
• e.g.
• Reciprocal (1/x) When there are very extreme
  outliers
• Arcsine Proportional data
• e.g.
• Other measures of location Heavy tailed data
• e.g. Trimmed mean

8
When to transform?

• Not something to think about doing straight away
  at any little sign of trouble
• Even if your groups are skewed in a similar
  manner parametric tests may hold
• Shop around
• Try different transformations to see if one works
  better for your problem regarding the
  distribution of values (but not just to get a sig
  p-value)

9
Note

• Transformations will not necessarily solve
  problems with outliers
• Also, if inferences are based on e.g. the mean of
  the transformed data, we cannot simply transform
  the values back to the original and act as though
  the inferences still hold (e.g. for µ)
• In the end, wed rather keep our data in original
  units and those transformations should be a last
  resort

10
More recent developments

• The Bootstrap
• The basic idea involves sampling with replacement
  from the sample data to produce random samples of
  size n
• Each of these samples provides an estimate of the
  parameter of interest
• Repeating the sampling a large number of times
  provides information on the variability of the
  estimate i.e. its standard error
• Necessary for any inferential test

11
TV Example
How many hours of TV watched yesterday
12
Bootstrap

• 1000 samples
• Distribution of Means of each sample ?
• Mean 3.951

13
How?

• Two Examples

basic R function boot(11000) for (i in boot)
bootimean(sample(TVdata, replaceT)) mean(boot)
hist(boot, col"blue", border"red")
uses the bootstrap package library(bootstrap) boo
tmeanbootstrap(TVdata, thetamean,
nboot1000) mean(bootmeanthetastar) hist(bootmean
thetastar)
14
Bootstrap

• Hypothetical situation
• If we cannot assume normality, how would we go
  about getting a confidence interval for a
  particular statistic?
• How would you get a confidence interval for
  robust measures and other statistics?
• Solution
• Resample (with replacement) from our own data
  based on its distribution
• Treat our sample distribution as a population
  distribution and take random samples from it
• So what we have done is, instead of assuming some
  sampling distribution of a particular shape and
  size, weve created it ourselves and derived our
  interval estimate from it
• From this we can create confidence intervals and
  perform other inferential procedures

15
Hypothesis Testing

• Comparing independent groups
• Step 1 compute the bootstrap mean and bootstrap
  sd as before, but for each group
• Each time you do so, calculate T
• This creates your own t distribution.

16
Hypothesis Testing

• Use the quantile points corresponding to your
  confidence level from it in computing your
  confidence interval on the difference betweens,
  rather than the tcv from typical distributions
• Note however that your T will not be the same
  for the upper and lower bounds
• Unless your bootstrap distribution was perfectly
  symmetrical
• Not likely to happen

17
So why use?

• Accuracy and control of type I error rate
• Most of the problems associated with both
  accuracy and maintenance of type I error rate are
  reduced using bootstrap methods compared to
  Students t
• Wilcox goes further to suggest that there may be
  in fact very few situations, if any, in which the
  traditional approach offers any advantage over
  the bootstrap approach
• The problem of outliers and the basic statistical
  properties of means and variances as remain
  however