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Approximate random distribution of coefficients of correlation for two random variates

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Lecture 2 Randomization techniques. Approximate random distribution of coefficients of correlation for two random variates. The standard normal distribution. g = 0.03. Under a normal approximation we can use Z-transformed score for statistical infering .

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Lecture 2

Randomizationtechniques

Approximate random distribution of coefficients of correlation for two random variates

The standard normaldistribution

g= 0.03

Under a normalapproximation we canuseZ-transformedscore for statisticalinfering.

TheFisheriansignificancelevels

P(m - s < X < m + s) = 68%

P(m - 1.65s < X < m + 1.65s) = 90%

P(m - 1.96s < X < m + 1.96s) = 95%

P(m - 2.58s < X < m + 2.58s) = 99%

P(m - 3.29s < X < m + 3.29s) = 99.9%

Z is standard normallydistributed

AveragetemperaturedifferenceinEuropeancountries/islands

Reshuffling

Consider the coefficient of correlation. Statistical significance of r > 0 (H1) is tested against the null hypothesis H0 of r = 0. Most statistics programs do this using Fisher’s Z-transformation

Parameters and standard errors

Probabilitylevel

Permutation test probability

Bootstrapprobability

Permutationtesting

We reorder one of thevariablesat random (at least 1000 times)

We calculatethemean, standard deviation, and theupper and lowerconfidenceintervals.

Thisgivesus an estimate of howprobableistheobservedcorrelation.

Thedistribution of randomizedcorrelationcoefficients

Lower two-sided 1% confidence limit

Upper two-sided 1% confidence limit

Observedvalue

Probabilitylevel for r = 0.457: P = 0.0006

Thedistributionis not symmetric.

We can’tuseZ-transformedvalues (thenormalapproximation)

We can’tuse a t-test.

We have to usetheupper and lowerprobabilitylevels. We getthemdirectlyfromthe random distribution

Jackknifing

Thejackknifed standard error of thecoefficient of variation

Bootstrapping

Take theoriginalvalues and calculatetheparameteryouneed

Take 1000 random samples of differentsize

Calculate 1000 parametersfromthebootstrapsamples

Comparetheobservedvaluewiththeparametersdistribution and calulatetheconfidencelimits for theobservedvalue

We useatleast 1000 random samples and calculate for eachsample CV. The standard deviation of thses CV valuesis an estimate of the standard error of theoriginal CV.

The standard error of a distribution is identical to the standard deviation of the sample.

Bootstrapdistribution

Themean CV valuesarebased on samples of differentsize.

Thescoresaretherefore of differentvalue.

We have to useweighedaverages

Nullmodels

Do the beak length of Darwin finches as a measure of resource usage differ more or less than expected just by chance?

Darwin finch

Photo:Guardian Unlimited

Theclassicalmethod to answerthisquestionis to comparetheobservedvarianceinbeaklengthdifferenceswiththoseobtainedfrom a random draw of beaklengthinsidetheobservedrange (smallest and largestbeaksizebeingfixed).

Thisis a null model approach

We test whether this null model approach is reliable

We haverandomlyassignedbeaklength of 20 speciesmeasuredin mm

Randomizedvariances

Observedvariance

Thenulldistributiongivesusdirectlythe H0probability.

P (H0) = 21/1000 = 0.021

Meningitisin Europe

Distribution of forestsin Europe

Istheprobability of Meningitisinfectioncorrelated to thedistribution of forestsin Europe?

We usethecorefficient of correlationbetweentheentries of bothgrids

R = 0.06; P(R=0) > 0.1.

We use a gridaproach

Thedistancebetweenthesitesmight be of importance.

Meningitisin Europe

Distribution of forestsin Europe

We reshufflerows and columnsonly to getthenull model distribution.

P (H0) = 26/1000 = 0.026

Mantel test

TheMantel test is a test for thecorrelationbetweentwodistancematrices.

Ittestswhetherdistancesarecorrelated.

Coefficient of correlationbetweenmatrixentries

For convenience we useZ-transformed data