1Foundation of Mathematical Biology / Foundation of Mathematical Biology

UCSF		Conclusions

The most important thing to understand is the relationship between statistics and probability distributions

Many statistics have been developed that have known distributions, and of which we can make use for tests of hypotheses

If you need to make use of serious parametric statistics, you should learn a package like R or S-plus

Statistics books are notoriously opaque in terms of just saying what the formula is!

If you are actually concerned with knowing whether you have a real result, use non-parametric tests with resampling methods to decide significance (next lectures).

UCSF

BP-203: Foundations of Mathematical Biology

Statistics Lecture II: October 23, 2001, 2pm

Instructors: Ajay N. Jain, PhD Jane Fridlyand, PhD

Email: [e-mail]

Copyright © 2001

All Rights Reserved

UCSF		Non-parametric statistics

Parametric statistics

♦Require an assumption about the underlying distributions of data

♦With those assumptions, they often provide sensitive tests of significance

♦However, the assumptions are often not reasonable, and it can be difficult to establish reasonableness

Non-parametric statistics

♦Make no assumption about distributional characteristics

♦Lacking those assumptions, they may not be as sensitive as appropriately applied parametric tests

♦However, one avoids the issue of whether a set of distributional assumptions is correct

♦Note: if your data are so close to the edge of nominal significance that you need to play games with different statistical tests, you have bigger problems to worry about.

UCSF		Resampling and permutation-based methods

Non-parametric statistics reduce reliance on distributional assumptions about your data

However, the distributions of the statistics themselves are often derived based on approximations that involve other assumptions

Resampling and permutation-based methods move toward deriving everything from the data observed

UCSF		Lecture II

Non-parametric statistical tests

♦Unpaired data

•Rank sum test

•Comparisons of distributions

·Kolmogorov-Smirnov test

·Receiver-operator characteristic curves: measure separation of distributions

♦Paired data

•Signed rank test

•Spearman’s rank correlation

•Kendall’s Tau

Resampling methods: Jane Fridlyand, PhD

♦The bootstrap (Efron), Jacknife

♦Using resampling methods for confidence intervals and hypothesis testing

UCSF		Lecture III (Tuesday)

General procedure for applying permutation-based methods to derive significance

Application of resampling and permutation methods to array-based data

General strategy for designing experiments involving large numbers of measurements

Homework will be assigned (it will not require programming)

UCSF		A test of location: The rank-sum test

The rank sum test is used to test the null hypothesis that two population distribution functions corresponding to two random samples are identical against the alternative hypothesis that they differ by location

Also called the Wilcoxon or Mann-Whitney rank sum test (Wilcoxon invented it)

UCSF		A test of location: The rank-sum test

Two samples with n1 and n2 observations

♦Order all observations from low to high

♦Assign to each observation its rank

♦For ties, assign each observation the average rank (e.g. if 3 tied observations occupy ranks 9, 10, 11, we assign each 10)

♦Sum the ranks for the set 1

♦Sum the ranks for the set 2

♦If n1 = n2, take the smaller sum: this is T

♦If not:

T1 = sum _ of _ smaller _ n

T2 = n1 (n1 + n2 +1) −T1

T = min(T1,T2 )

UCSF		Example: Rank sum test

So, T = 60

For small n, we can look up the numbers in a table

UCSF		Rank sum significance

For larger n, we must use an approximation based on the normal distribution:

Z = ( µ −T − 0.5) / σ

µ = n1 (n1 + n2 +1) / 2

σ = n2 µ / 6

If Z > 1.96 we have significance at p = 0.05