nonparametric lecture.ppt
TRANSCRIPT
Parametric Statistics 1
Assume data are drawn from samples with a certain distribution (usually normal)
Compute the likelihood that groups are related/unrelated or same/different given that underlying model
t-test, Pearson’s correlation, ANOVA…
Parametric Statistics 2
Assumptions of Parametric statistics1. Observations are independent
2. Your data are normally distributed
3. Variances are equal across groups• Can be modified to cope with unequal ∂2
Non-parametric Statistics?
Non-parametric statistics do not assume any underlying distribution
They estimate the distribution AND compute the probability that your groups are the related/the same or unrelated/different
Nonparametric ≠ No parameters
Model structure is not specified a priori but is instead determined from data.
The data are parameterised by the analysis
AKA: “distribution free”
Non-parametric Statistics
Assumptions of non-parametric statistics1. Observations are independent
Non-parametric Statistics?
Non-parametric statistics do not assume any underlying distribution
Estimating or modeling this distribution reduces their power to detect effects…
So never use them unless you have to
Why use a Non-parametric Statistic?
Very small samples (<20 replicates) High probability of violating the assumption of
normality Leads to spurious Type-1 (false alarm) errors
Why use a Non-parametric Statistic?
Outliers more often lead to spurious Type-1 (false alarm) errors in parametric statistics.
Nonparametric statistics reduce data to an ordinal rank, which reduces the impact or leverage of outliers.
Error Type-I error: False Alarm for a bogus effect
reject the null hypothesis when it is really true
Type-II error: Miss a real effect fail to reject our null hypothesis when it is really false
Type-III error: :-) lazy, incompetent, or willful ignorance of the truth
Power
1-alpha
Non-parametric ChoicesData type?
χ2
discrete
Question?
continuous
Number of groups?
Spearman’s Rank
association Different central value
Mann-Whitney UWilcoxon’s Rank Sums
Kruskal-Wallis test
two-groups more than 2
Brown-Forsythe
Difference in ∂2
Non-parametric ChoicesData type?
χ2
discrete
Question?
continuous
Number of groups?
Spearman’s Rank
Like a Pearson’s R
Mann-Whitney UWilcoxon’s Rank Sums
Kruskal-Wallis test
two-groups more than 2Like ANOVA
Like Student’s t
No alternative
Different central value
Brown-Forsythe
Difference in ∂2
Like F-test
association
Chi-Squared (Χ2) χ2 tests the null hypothesis that observed
events occur with an expected frequency in large samples frequencies are distributed as Χ2
e.g. Ho: “This six-sided dice is fair ” Expect all 6 outcomes to occur equally often
Assumptions Observations are independent Outcomes mutually exclusive Sample is not small
Small samples require exact test:, i.e., binomial test
Chi-Squared Χ2 formula
Χ2 = the sum of each squared difference between the observed and expected frequencies divided its expected frequency
Χ2 and contingency tables
Χ2 essentially tests if each cell in a contingency table has its expected value
In a 2-way table, this expectation will be the value of an adjacent cell
Example: coin toss
Random sample of 100 coin tosses, of a coin believed to be fair
We observed number of 45 heads, and and 55 tails
Is the coin fair?
Coin toss
If ho is true, our test statistic is drawn from a Χ2
distribution with df = 1
(45-50)2 + (55-50)2 = 0.5 + 0.5 = 1
50 50
Χ2(1) = 1, p > 0.3
Coin toss Χ2 in R
chisq.test(c(45,55), p=c(.5,.5))
Chi-squared test for given probabilities Χ2 = 1, df = 1, p = 0.3173
Spearman Rank test (ρ (rho)) Named after Charles Spearman,
Non-parametric measure of correlation Assesses how well an arbitrary monotonic
function describes the relationship between two variables,
Does not require the relationship be linear Does not require interval measurement
Spearman Rank test (ρ (rho)) Mathematically, it is simply a Pearson’s r
computed on ranked data d = difference in rank of a given pair n = number of pairs
Alternative test = Kendall's Tau (Kendall's τ)
Mann-Whitney U
AKA: “Wilcoxon rank-sum test Mann & Whitney, 1947; Wilcoxon, 1945
Non-parametric test for difference in the medians of two independent samples Assumptions:
• Samples are independent• Observations can be ranked (ordinal or better)
Mann-Whitney U
U tests the difference in the medians of two independent samples
n1 = number of obs in sample 1
n2 = number of obs in sample 2 R = sum of ranks of the lower-ranked
sample
Mann-Whitney U or t-test? Should you use it over the t-test?
Yes if you have a very small sample (<20)• (central limit assumptions not met)
Possibly if your data are inherently ordinal Otherwise, probably not.
It is less prone to type-I error (spurious significance) due to outliers.
But does not in fact handle comparisons of samples whose variances differ very well (Use unequal variance t-test with rank data)
Aesop: Mann-Whitney U Example
Suppose that Aesop is dissatisfied with his classic experiment in which one tortoise was found to beat one hare in a race.
He decides to carry out a significance test to discover whether the results could be extended to tortoises and hares in general…
Aesop 2: Mann-Whitney U He collects a sample of 6 tortoises and 6 hares,
and makes them all run his race. The order in which they reach the finishing post (their rank order) is as follows:
tort = c(1, 7, 8, 9, 10,11) hare = c(2, 3, 4, 5, 6, 12)
Original tortoise still goes at warp speed, original hare is still lazy, but the others run truer to stereotype.
Aesop 3: Mann-Whitney U
wilcox.test(tort, hare) Wilcoxon = W = 25, p-value = 0.31
Tortoises are not faster (but neither are hares)
tort = c(1, 7, 8, 9, 10,11) (n2 = 6)
hare = c(2, 3, 4, 5, 6, 12) (n1 = 6, R1 =32)
Aesop 4: Mann-Whitney U Wilcoxon = W = 25, p-value = 0.31
Tortoises are not faster (but neither are hares). Welch Two Sample t-test
t = 1.1355, df = 10, p-value = 0.28 Alternative hypothesis: true difference in means is
not equal to 0 95 percent confidence interval:
-2.25 ~ 6.91 sample estimates:
• mean of x = 7.6 mean of y = 5.3
Power comparison with continuous normal data
tort = 1 74 79 81 100 121 hare = 4 9 16 17 18 144 Wilcoxon
W = 25, p = 0.31 t.test
t.test(tort, hare, var.equal = TRUE) t(10) = 1.5, p = 0.16
Wilcoxon signed-rank test (related samples)
Same idea as MW U, generalized to matched samples
Equivalent to non-independent sample t-test
Kruskall-Wallis Non-parametric one-way analysis of variance
by ranks (named after William Kruskal and W. Allen Wallis)
tests equality of medians across groups. It is an extension of the Mann-Whitney U test to
3 or more groups. Does not assume a normal population, Assumes population variances among groups
are equal.