STANDARD DISTRIBUTIONS ExamplesExtensions

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STANDARD DISTRIBUTIONS -Examples/Extensions

• GENETIC LINKAGE and MAPPING
• Linkage Phase - chromatid associations of
  alleles of linked loci
• - same
  chromosome coupled, different repulsion
• Genetic Recombination - define R.F. (gametes or
  phenotypes) homologous case - greater the
  distance, greater chance of recombining.High
  interference problem for multiple locus models.
  R.F. between loci not additive. Need Mapping
  Function
• Haldanes Mapping Function
• Assume crossovers occur randomly along
  chromosome length and average number ?, model
  as Poisson
• PNo crossover e - ? and
  PCrossover 1- e - ?

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Examples - continued

• Precombinant 0.5?P(Crossover (each pair of
  homologs, with one crossover results in one-half
  recombinant gametes)
• Define Expected No. recombinants as mapping
  function (m 0.5 ?)
• R.F. r 0.5(1-e
  -2m) (form of Haldanes M.F.)
• with inverse m - 0.5 ln (1-2r)
• converting an estimated R.F. to Haldanes
  map distance
• Thus, for locus order ABC
• mAC mAB
  mBC (since mAB - 0.5ln(1-2rAB) )etc.
• Substituting for each of these gives us the
  usual relationship between R.F.s (no
  interference situation)
• Net Effect - transform to straight line mAC vs
  mAB or mBC
• In practice - too simple only applies to
  specific conditions may not relate directly to
  physical distance -(common M.F. problem).

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Example expanded- LINKAGE/MAPPING

• Genetic Map -Models linear arrangement of group
  of genes / markers (easily identified genetic
  features - e.g. change in known gene, piece of
  DNA -no known function). Map based on homologous
  recombination during meiosis. If two or more
  markers located close together on chromosome,
  alleles usually inherited through meiosis
• 4 steps after marker data obtained. Pairwise
  linkage - all 2-locus combinations (based on
  observed and expected frequencies of genotypic
  classes). Grouping markers into Linkage Groups
  (based on R.F.s, significance level etc.). No.
  of linkage groups should ? haploid no.
  chromosomes for organism. Ordering of markers
  (key step, computationally demanding, precision
  important). Estimation multipoint R.F. (physical
  distance - no. of DNA base pairs between two
  genes vs map distance transformation of R.F.).
  Ultimate physical map DNA sequence (restriction
  map also common)

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Example

• RECOMBINANTS and MULTINOMIAL
• Binomial No. of recombinant gametes, produced by
  a heterozygous parent for a 2-locus model, with ?
  Pgamete recombinant (R.F.)
• So for r recombinants in sample of n
• Multinomial 3-locus model (A,B,C) - 4 possible
  classes of gametes
• (non-recombinants, AB recombinants, BC
  recombinants and double recombinants loci ABC).
  Joint probability distribution for r.v.s
  counting number in each class
• where abcdn and P1, P2, P3, P4 are
  probabilities of observing a member of each of 4
  classes respectively

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Sampling and Sampling Distributions

• Random Sampling. If the same element can not be
  selected more than once, we say that the sample
  is drawn without replacement otherwise, the
  sample is said to be drawn with replacement.The
  usual convention in sampling is that lower case
  letters are used to designate the sample
  characteristics. Thus if the sample size is n,
  its elements are designated, x1, x2, , xn , its
  mean is and its modified variance is
• The corresponding parent population
  characteristics are N (or infinity), X and S2.

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Central Limit Theorem

• Suppose that we repeatedly draw random
  samples of size n (with replacement) from a
  distribution with mean m and variance s2. Let
• be the
  collection of sample averages and let
• where the collection is
  called the sampling distribution of means.
• Central Limit Theorem.If X1, X2, Xn are a
  random sample of r.v. X, (mean ?, variance ?2),
  then, in the limit, as n ??, the sampling
  distribution of means has a Standard Normal
  distribution, N(0,1)

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Probabilities for sampling distribution

• Statements
  for large n
• U standardized Normal deviate
• and, in particular
• In general, closer r.v. X to Normal, the faster
  the approximation approaches U. Generally n? 30
  ? Large sample theory

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Attribute and Proportionate Sampling

• If the sample elements are a measurement of some
  characteristic, we are said to have attribute
  sampling.
• If all the sample elements are 1 or 0
  (success/failure, agree/ do-not-agree), we have
  proportionate sampling.
• For proportionate sampling, the sample average
  and the sample proportion are synonymous,
  just as are the mean m and proportion p (or P)
  for the parent population. From our results on
  the binomial distribution, the sample variance is
  p (1 - p) and the variance of the parent
  distribution is P (1 - P).

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Probability Statements

• If X and Y independent Binomially distributed
  r.v.s parameters n, p and m, p respectively,
  then XY B(nm, p) - (show e.g. by m.g.f.s)
• So, YX1 X2. Xn B(n, p) for the IID
  XB(1, p).
• Since we know ?Y np, ?Y?(npq) and, clearly
  then
• and, further
  sampling distribution of a proportion

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Probability Statements- C.L.T. and Approximation
summary

• General form of theorem - an infinite sequence of
  independent r.v.s, with means, variances as
  before, then approximation ? U for n large
  enough. Note No condition on form of
  distribution of Xs (raw data)
• Strictly - for approximations of discrete
  distributions, can improve by considering
  correction for continuity
• e.g.

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Generalising Sampling Distn. concept

• For the sampling distribution of any statistic.
  We say that a sample characteristic is an
  unbiased estimator of the parent population
  characteristic, if the mean of the corresponding
  sampling distribution is equal to the parent
  characteristic.
• Lemma. The sample average (proportion ) is an
  unbiased estimator of the parent average
  (proportion) Recall Sampling without
  replacement from finite population -
  Hypergeometric. The quantity Ö ( N - n) / ( N -
  1) is called the finite population correction
  (fpc). If the parent population is infinite or if
  sampling with replacement the fpc 1.
• Lemma. E s S ? fpc.

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NON-PARAMETRICS/DISTRIBUTION FREE

• Standard Probability Distributions -do not apply
  to data or sampling distributions or test
  statistics - uncertainty because small or
  unreliable data sets, non-independence etc.
  Parameter estimation - not key issue.
• Example or Empirical-basis. Weaker assumptions.
  Less information.
• e.g. median, not mean. Simple hypothesis
  testing rather than estimation.
• Counts - nominal, ordinal (natural
  non-parametric).
• Power and efficiency
• Nonparametric Hypothesis Tests -main focus of
  interest here (standard parallels to the
  parametric case).
• e.g. H.T. of locus orders have complex test
  statistic distributions, for which need to
  construct empirical probability distributions.
  Usually obtained, by assuming the null hypothesis
  using re-sampling techniques, e.g. permutation
  tests, bootstrap, jacknife

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LIKELIHOOD - DEFINITIONS

• Suppose X can take a set of values x1,x2,with
• where is a vector of parameters
  affecting observed xs
• e.g. . So can say
  something about PX if we
• know say
• But not usually case, i.e. observe xs, knowing
  nothing of
• Assuming xs a random sample size n from a known
  distribution, then likelihood for
• Finding most likely for given data is
  equivalent to Maximising the Likelihood function.
  
• where M.L.E. is

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LIKELIHOOD SCORE and INFO. CONTENT

• The Log-likelihood is a support function S(?)
  evaluated at point,
• ? say
• Support function for any other point, say ?
  can be obtained approx., using the Taylor
  expansion
• and this is the basis of the Newton-Raphson
  iteration for the M.L.E.
• SCORE first derivative of support function
  w.r.t. the parameter
• 
  or, numerically,
• INFORMATION CONTENT evaluated at (i) arbitrary
  point Observed Info. (ii)support function
  maximum Expected Info.
• 
  

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Example -Binomial variable(e.g. use of Score,
Expected Info. Content to determine type of
mapping population and sample size for genomics
experiments)

• Likelihood function
• Log-likelihood
• Assuming n constant , then first term invariant
  w.r.t. p S(? ) at point p
• Maximising w.r.t. p gives M.L.E.
  with SCORE

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Bayesian Estimation- in context

• Parametric Estimation - in the classical
  approach f(x,?) for a r.v. X of density f(x) ,
  with ? the unknown parameter indicates the
  dependency of the distribution on the parameter
  to be estimated.
• Bayesian Estimation- ? is a random variable, so
  appropriate to consider the density as
  conditional and write f(x/ ? )
• Given a random sample X1, X2, Xn the
  sample random variables can be considered jointly
  distributed with parameter r.v. ?. So, joint pdf
• Objective - to form an estimator that gives a
  value of ? dependent on observations of the
  sample random variables. Thus conditional density
  of ? given X1, X2, Xn also plays a role. This is
  the posterior density

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Bayes - contd.

• Posterior Density
• Relationship - prior and posterior
• where ?(?) prior density of ?
• Value Close to MLE for large n, or for small n
  if sample values compatible with the prior
  distribution, strong sample basis, simpler to
  calculate.