0 0.05 0.1 0.15 0.2 0.25 0.3 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Frequency Correlation Approximate random distribution of coefficients of correlation for two random variates g= 0.03 Under a normal approximation we can use Z-transformed score for statistical infering. Exp StdDev Exp Obs Z 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% The Fisherian significance levels The standard normal distribution Z is standard normally distributed 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 3 6 9 12 15 18 X f(x n=20 0 0.02 0.04 0.06 0.08 0.1 0.12 0 6 12 18 24 30 36 42 48 X f(x n=50 0 0.05 0.1 0.15 0.2 0.25 0.3 0 2 4 6 8 10 X f(x n=10 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 X f(x + s - s 0.68 +2 s -2 s 0.95 Lecture 2 Randomization techniques
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Approximate random distribution of coefficients of correlation for two random variates
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 . - PowerPoint PPT Presentation
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Approximate random distribution of coefficients of correlation for two random variates
g= 0.03
Under a normal approximation we can use Z-transformed score for
statistical infering.
ExpStdDevExpObsZ
P(m - s < X < m + s) = 68%P(m - 1.65s < X < m + 1.65s) =
Average temperature difference in European countries/islands
Permutation test probability
Bootstrap probability
Probability level
Parameters and standard errors
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
1 1 rZ ln2 1 r
Reshuffling
Permutation testing
Random number ln area ln Delta T r Sim r Average r Average r0.247012838 11.33704 2.833213 0.457176 0.14894 0.08609641 =+ŚREDNIA(H2:H21)0.303300878 12.65321 2.70805 0.014534 StdDev r StdDev r0.725633833 9.917045 2.995732 0.157997 0.16530152 +ODCH.STANDARDOWE(H2:H21)0.258217857 0.667829 1.94591 0.0310330.632451857 7.243513 2.70805 -0.14119 t t0.254528292 7.696213 3.135494 0.268839 10.0393331 (H2-J2)/J4*20^0.50.980671601 13.01692 2.70805 0.117112 P(t) P(t)0.522396276 10.62825 2.995732 0.137361 4.9403E-09 +ROZKŁAD.T(J7,19,2)0.683545674 11.08702 3.044522 0.214470.773648713 7.887209 1.609438 0.159525 Z Z0.359562515 10.3264 2.302585 -0.05251 2.24486312 +(G2-H2)/J40.128137778 12.68838 2.564949 -0.23382 P(Z) P(Z)0.573061911 11.7905 2.564949 0.072888 0.03687629 =ROZKŁAD.T(J12,19,2)0.025421522 12.78555 3.044522 -0.046160.087309492 11.42796 2.484907 0.2224670.20159921 9.132379 2.944439 -0.143290.438208554 12.40519 2.944439 -0.05720.575893524 13.13427 2.772589 0.4491860.931176694 10.1401 2.639057 0.1675530.0309793 10.67112 3.044522 0.234201
We reorder one of the variables at random (at least
1000 times)
We calculate the mean, standard deviation, and the upper and lower confidence intervals.This gives us an estimate of how probable is the observed correlation.
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The distribution of randomized correlation coefficients
Observed value
The distribution is not symmetric.We can’t use Z-transformed values (the normal approximation)We can’t use a t-test.
Lower two-sided 1% confidence
limit
Upper two-sided 1% confidence
limit
We have to use the upper and lower probability levels. We get them directly from the random distribution
We use at least 1000 random samples and calculate for each sample CV. The standard deviation of thses CV values is an estimate of the standard error of the original CV.
The standard error of a distribution is identical to the standard deviation of the sample.
The mean CV values are based on samples of different size. The scores are therefore of different value.
We have to use weighed averages
Monte Carlo simulation.
Null models
Darwin finch
Photo:Guardian Unlimited
Do the beak length of Darwin finches as a measure of resource usage differ more or less than expected just by chance?
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The classical method to answer this question is to compare the observed variance in beak length differences with those obtained from a random draw of beak length inside
the observed range (smallest and largest beak size being fixed).
This is a null model approach
We test whether this null model approach is reliable
We have randomly assigned beak length of 20 species measured in mmOrdiginal
We reshuffle rows and columns only to get the null model distribution.
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P (H0) = 26/1000 = 0.026
Mantel testSequence
Caruabus coriaceus A T T T G C A T G C ACarabus auronitens A G T A A C A G G G ACarabus cancellatus A C G T G C A T C C TCarabus auratus A T A T G C T T G G T