666.07 - Maximum Likelihood Estimation for Allele...

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Unformatted text preview: Maximum Likelihood Estimation for Allele Frequencies Biostatistics 666 Lecture 7 Last Three Lectures: Introduction to Coalescent Models z Computationally efficient framework • Alternative to forward simulations z Predictions about sequence variation • Number of polymorphisms • Frequency of polymorphisms Coalescent Models: Key Ideas z Proceed backwards in time z Genealogies shaped by • Population size • Population structure • Recombination rates z Given a particular genealogy ... • Mutation rate predicts variation Next Series of Lectures z Estimating allele and haplotype frequencies from genotype data • Maximum likelihood approach • Application of an E-M algorithm z Challenges • Using information from related individuals • Allowing for non-codominant genotypes • Allowing for ambiguity in haplotype assignments Objective: Parameter Estimation z Learn about population characteristics • E.g. allele frequencies, population size z Using a specific sample • E.g. a set sequences, unrelated individuals, or even families Maximum Likelihood z A general framework for estimating model parameters z Find the set of parameter values that maximize the probability of the observed data z Applicable to many different problems Example: Allele Frequencies z Consider… • A sample of n chromosomes • X of these are of type “a” • Parameter of interest is allele frequency… X n X p p X n X n p L − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ) 1 ( ) , | ( Evaluate for various parameters 0.000 0.0 1.0 0.006 0.2 0.8 0.111 0.4 0.6 0.251 0.6 0.4 0.088 0.8 0.2 0.000 1.0 0.0 L 1-p p Likelihood Plot 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0 Allele Frequency Likelihood For n = 10 and X = 4 In this case z The likelihood tells us the data is most probable if p = 0.4 z The likelihood curve allows us to evaluate alternatives… • Is p = 0.8 a possibility? • Is p = 0.2 a possibility? Example: Estimating 4N µ z Consider S polymorphisms in sample of n sequences… z Where P n is calculated using the Q n and P 2 functions defined previously ) | ( ) , | ( θ θ S P S n L n = Likelihood Plot 4N µ Likelihood With n = 5, S = 10 MLE Maximum Likelihood Estimation z Two basic steps… z In principle, applicable to any problem where a likelihood function exists ) | ( maximizes that ˆ of value Find b) ) | ( ) | ( function likelihood down Write a) x L x f x L θ θ θ θ ∝ MLEs z Parameter values that maximize likelihood • θ where observations have maximum probability z Finding MLEs is an optimization problem z How do MLEs compare to other estimators? Comparing Estimators z How do MLEs rate in terms of … • Unbiasedness • Consistency • Efficiency z For a review, see Garthwaite, Jolliffe, Jones (1995) Statistical Inference , Prentice Hall Analytical Solutions z Write out log-likelihood …...
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This note was uploaded on 12/26/2011 for the course BIO 666 taught by Professor Staff during the Fall '06 term at University of Michigan.

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666.07 - Maximum Likelihood Estimation for Allele...

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