Nonparametric tests and ANOVAs: What you need to know

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Nonparametric testsand ANOVAs What you need
to know
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Nonparametric tests

• Nonparametric tests are usually based on ranks
• There are nonparametric versions of most
  parametric tests

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Parametric
Nonparametric
One-sample and Paired t-test
Sign test
Mann-Whitney U-test
Two-sample t-test
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Quick Reference Summary Sign Test

• What is it for? A non-parametric test to compare
  the medians of a group to some constant
• What does it assume? Random samples
• Formula Identical to a binomial test with po
  0.5. Uses the number of subjects with values
  greater than and less than a hypothesized median
  as the test statistic.

P 2 Prx?X
P(x) probability of a total of x successes p
probability of success in each trial n total
number of trials
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Sign test
Null hypothesis Median mo
Sample
Test statistic x number of values greater than
mo
Null distribution Binomial n, 0.5
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Quick Reference Summary Mann-Whitney U Test

• What is it for? A non-parametric test to compare
  the central tendencies of two groups
• What does it assume? Random samples
• Test statistic U
• Distribution under Ho U distribution, with
  sample sizes n1 and n2
• Formulae

n1 sample size of group 1 n2 sample size of
group 2 R1 sum of ranks of group 1
Use the larger of U1 or U2 for a two-tailed test
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Mann-Whitney U test
Null hypothesis The two groups Have the same
median
Sample
Test statistic U1 or U2 (use the largest)
Null distribution U with n1, n2
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Mann-Whitney U test

• Large-sample approximation

Use this when n1 n2 are both gt 10 Compare to the
standard normal distribution
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Mann-Whitney U Test

• If you have ties
• Rank them anyway, pretending they were slightly
  different
• Find the average of the ranks for the identical
  values, and give them all that rank
• Carry on as if all the whole-number ranks have
  been used up

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Example
Data
14 2 5 4 2 14 18 14
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Example
Sorted Data
Data
22 4 5 14 14 14 18
14 2 5 4 2 14 18 14
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Example
Sorted Data
Data
22 4 5 14 14 14 18
14 2 5 4 2 14 18 14
TIES
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Example
Sorted Data
Data
Rank them anyway, pretending they were slightly
different
22 4 5 14 14 14 18
14 2 5 4 2 14 18 14
TIES
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Example
Sorted Data
Rank A
Data
22 4 5 14 14 14 18
12 3 4 5 6 7 8
14 2 5 4 2 14 18 14
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Example
Sorted Data
Rank A
Data
Find the average of the ranks for the identical
values, and give them all that rank
22 4 5 14 14 14 18
12 3 4 5 6 7 8
14 2 5 4 2 14 18 14
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Example
Sorted Data
Rank A
Data
Average 1.5
22 4 5 14 14 14 18
12 3 4 5 6 7 8
14 2 5 4 2 14 18 14
Average 6
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Example
Sorted Data
Rank A
Data
Rank
22 4 5 14 14 14 18
12 3 4 5 6 7 8
14 2 5 4 2 14 18 14
1.51.5 3 4 6 6 6 8
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Example
Sorted Data
Rank A
Data
Rank
22 4 5 14 14 14 18
12 3 4 5 6 7 8
14 2 5 4 2 14 18 14
1.51.5 3 4 6 6 6 8
These can now be used for the Mann-Whitney U test
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Benefits and Costs of Nonparametric Tests

• Main benefit
• Make fewer assumptions about your data
• E.g. only assume random sample
• Main cost
• Reduce statistical power
• Increased chance of Type II error

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When Should I Use Nonparametric Tests?

• When you have reason to suspect the assumptions
  of your test are violated
• Non-normal distribution
• No transformation makes the distribution normal
• Different variances for two groups

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Quick Reference Summary ANOVA (analysis of
variance)

• What is it for? Testing the difference among k
  means simultaneously
• What does it assume? The variable is normally
  distributed with equal standard deviations (and
  variances) in all k populations each sample is a
  random sample
• Test statistic F
• Distribution under Ho F distribution with k-1
  and N-k degrees of freedom

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Quick Reference Summary ANOVA (analysis of
variance)

• Formulae

mean of group i overall mean
ni size of sample i N total sample size
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ANOVA
Null hypothesis All groups have the same mean
k Samples
Test statistic
Null distribution F with k-1, N-k df
compare
How unusual is this test statistic?
P gt 0.05
P lt 0.05
Reject Ho
Fail to reject Ho
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Quick Reference Summary ANOVA (analysis of
variance)

• Formulae

There are a LOT of equations here, and this is
the simplest possible ANOVA
mean of group i overall mean
ni size of sample i N total sample size
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(No Transcript)
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dfgroup k-1 dferror N-k
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Sum of Squares
Mean Squares
F-ratio
df
dfgroup k-1 dferror N-k
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ANOVA Tables
Source of variation Sum of squares df Mean Squares F ratio P
Treatment
Error
Total
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ANOVA Tables
Source of variation Sum of squares df Mean Squares F ratio P
Treatment
Error
Total
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ANOVA Tables
Source of variation Sum of squares df Mean Squares F ratio P
Treatment k-1
Error N-k
Total N-1
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ANOVA Tables
Source of variation Sum of squares df Mean Squares F ratio P
Treatment k-1
Error N-k
Total N-1
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ANOVA Tables
Source of variation Sum of squares df Mean Squares F ratio P
Treatment k-1
Error N-k
Total N-1
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ANOVA Tables
Source of variation Sum of squares df Mean Squares F ratio P
Treatment k-1
Error N-k
Total N-1

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ANOVA Table Example
Source of variation Sum of squares df Mean Squares F ratio P
Treatment 7.22 2
Error 9.41 19
Total
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ANOVA Table Example
Source of variation Sum of squares df Mean Squares F ratio P
Treatment 7.22 2 3.61 7.29 0.004
Error 9.42 19 0.50
Total 16.64 21
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Additions to ANOVA

• R2 value how much variance is explained?
• Comparisons of groups planned and unplanned
• Fixed vs. random effects
• Repeatability

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Two-Factor ANOVA

• Often we manipulate more than one thing at a time
• Multiple categorical explanitory variables
• Example sex and nationality

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Two-factor ANOVA

• Dont worry about the equations for this
• Use an ANOVA table

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Two-factor ANOVA

• Testing three things
• Means dont differ among treatment 1
• Means dont differ among treatment 2
• There is no interaction between the two treatments

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Two-factor ANOVA Table
Source of variation Sum of Squares df Mean Square F ratio P
Treatment 1 SS1 k1 - 1 SS1 k1 - 1 MS1 MSE
Treatment 2 SS2 k2 - 1 SS2 k2 - 1 MS2 MSE
Treatment 1 Treatment 2 SS12 (k1 - 1)(k2 - 1) SS12 (k1 - 1)(k2 - 1) MS12 MSE
Error SSerror XXX SSerror XXX
Total SStotal N-1