Download presentation
Presentation is loading. Please wait.
Published byJonathan Morgan Modified over 9 years ago
1
DATA ANALYSIS Module Code: CA660 Lecture Block 6: Alternative estimation methods and their implementation
2
2 MAXIMUM LIKELIHOOD ESTIMATION Recall general points: Estimation, definition of Likelihood function for a vector of parameters and set of values x. Find most likely value of = maximise the Likelihood fn. Also defined Log-likelihood (Support fn. S( ) ) and its derivative, the Score, together with Information content per observation, which for single parameter likelihood is given by Why MLE? (Need to know underlying distribution). Properties: Consistency; sufficiency; asymptotic efficiency (linked to variance); unique maximum; invariance and, hence most convenient parameterisation; usually MVUE; amenable to conventional optimisation methods.
3
3 VARIANCE, BIAS & CONFIDENCE Variance of an Estimator - usual form or for k independent estimates For a large sample, variance of MLE can be approximated by can also estimate empirically, using re-sampling* techniques. Variance of a linear function (of several estimates) – (common need in genomics analysis, e.g. heritability), in risk analysis Recall Bias of the Estimator then the Mean Square Error is defined to be: expands to so we have the basis for C.I. and tests of hypothesis.
4
4 COMMONLY-USED METHODS of obtaining MLE Analytical - solving or when simple solutions exist Grid search or likelihood profile approach Newton-Raphson iteration methods EM (expectation and maximisation) algorithm N.B. Log.-likelihood, because max. same value as Likelihood Easier to compute Close relationship between statistical properties of MLE and Log-likelihood
5
5 MLE Methods in outline Analytical : - recall Binomial example earlier Example : For Normal, MLE’s of mean and variance, (taking derivatives w.r.t mean and variance separately), and equivalent to sample mean and actual variance (i.e. /N), - unbiased if mean known, biased if not. Invariance : One-to-one relationships preserved Used: when MLE has a simple solution
6
6 MLE Methods in outline contd. Grid Search – Computational Plot likelihood or log-likelihood vs parameter. Various features Relative Likelihood =Likelihood/Max. Likelihood (ML set =1). Peak of R.L. can be visually identified /sought algorithmically. e.g. Plot likelihood and parameter space range - gives 2 peaks, symmetrical around ( likelihood profile for e.g. well-known mixed linkage analysis problem. Or for similar example of populations following known proportion splits). If now constrain MLE solution unique e.g. = R.F. between genes (possible mixed linkage phase).
7
7 MLE Methods in outline contd. Graphic/numerical Implementation - initial estimate of . Direction of search determined by evaluating likelihood to both sides of . Search takes direction giving increase, because looking for max. Initial search increments large, e.g. 0.1, then when likelihood change starts to decrease or become negative, stop and refine increment. Issues: Multiple peaks – can miss global maximum, computationally intensive ; see e.g. http://statgen.iop.kcl.ac.uk/bgim/mle/sslike_1.html http://statgen.iop.kcl.ac.uk/bgim/mle/sslike_1.html Multiple Parameters - grid search. Interpretation of Likelihood profiles can be difficult, e.g. http://blogs.sas.com/content/iml/2011/10/12/maximum- likelihood-estimation-in-sasiml/ http://blogs.sas.com/content/iml/2011/10/12/maximum- likelihood-estimation-in-sasiml/
8
8 Example in outline Data e.g used to show a linkage relationship (non-independence) between e.g. marker and a given disease gene, or (e.g. between sex and purchase) of computer games. Escapes = individuals who are susceptible, but show no disease phenotype under experimental conditions: (express interest but no purchase record). So define as proportion of escapes and R.F. respectively. is penetrance for disease trait or of purchasing, i.e. P{ that individual with susceptible genotype has disease phenotype}. P{individual of given sex and interested who actually buys} Purpose of expt.-typically to estimate R.F. between marker and gene or proportion of a sex that purchases Use: Support function = Log-Likelihood. Often quite complex, e.g. for above example, might have
9
9 Example contd. Setting 1st derivatives (Scores) w.r.t and w.r.t. Expected value of Score (w.r.t. is zero, (see analogies in classical sampling/hypothesis testing). Similarly for . Here, however, No simple analytical solution, so can not solve directly for either. Using grid search, likelihood reaches maximum at e.g. In general, this type of experiment tests H 0 : Independence between the factors (marker and gene), (sex and purchase) and H 0 : no escapes Uses Likelihood Ratio Test statistics. (M.L.E. 2 equivalent)
10
10 MLE Methods in outline contd. Newton-Raphson Iteration Have Score ( ) = 0 from previously. N-R consists of replacing Score by linear terms of its Taylor expansion, so if ´´ a solution, ´=1st guess Repeat with ´´ replacing ´ Each iteration - fits a parabola to Likelihood Fn. Problems - Multiple peaks, zero Information, extreme estimates Multiple parameters – need matrix notation, where S matrix e.g. has elements = derivatives of S( , ) w.r.t. and respectively. Similarly, Information matrix has terms of form Estimates are L.F. 2 nd 1st Variance of Log-L i.e.S( )
11
11 MLE Methods in outline contd. Expectation-Maximisation Algorithm - Iterative. Incomplete data (Much genomic, financial and other data fit this situation e.g. linkage analysis with marker genotypes of F2 progeny. Usually 9 categories observed for 2- locus, 2-allele model, but 16 = complete info., while 14 give info. on linkage. Some hidden, but if linkage parameter known, expected frequencies can be predicted and the complete data restored using expectation). Steps: (1) Expectation estimates statistics of complete data, given observed incomplete data. -(2) Maximisation uses estimated complete data to give MLE. Iterate till converges (no further change)
12
12 E-M contd. Implementation Initial guess, ´, chosen (e.g. =0.25 say = R.F.). Taking this as “true”, complete data is estimated, by distributional statements e.g. P(individual is recombinant, given observed genotype) for R.F. estimation. MLE estimate ´´ computed. This, for R.F. sum of recombinants/N. Thus MLE, for f i observed count, Convergence ´´ = ´ or
13
13 LIKELIHOOD : C.I. and H.T. Likelihood Ratio Tests – c.f. with 2. Principal Advantage of G is Power, as unknown parameters involved in hypothesis test. Have : Likelihood of taking a value A which maximises it, i.e. its MLE and likelihood under H 0 : N, (e.g. N = 0.5) Form of L.R. Test Statistic or, conventionally - choose; easier to interpret. Distribution of G ~ approx. 2 (d.o.f. = difference in dimension of parameter spaces for L( A ), L( N ) ) Goodness of Fit : notation as for 2, G ~ 2 n-1 : Independence: notation again as for 2
14
14 Likelihood C. I.’s – graphical method Example: Consider the following Likelihood function is the unknown parameter ; a, b observed counts For 4 data sets observed, A: (a,b) = (8,2), B: (a,b)=(16,4) C: (a,b)=(80, 20) D: (a,b) = (400, 100) Likelihood estimates can be plotted vs possible parameter values, with MLE = peak value. e.g. MLE = 0.2, L max =0.0067 for A, and L max =0.0045 for B etc. Set A: Log L max - Log L=Log(0.0067) - Log(0.00091)= 2 gives 95% C.I. so =(0.035,0.496) corresponding to L=0.00091, 95% C.I. for A. Similarly, manipulating this expression, Likelihood value corresponding to 95% confidence interval given as L = (7.389) -1 L max Note: Usually plot Log-likelihood vs parameter, rather than Likelihood. As sample size increases, C.I. narrower and symmetric
15
15 Maximum Likelihood Benefits Strong estimator properties – sufficiency, efficiency, consistency, non-bias etc. as before Good Confidence Intervals Coverage probability realised and intervals meaningful MLE Good estimator of a CI MSE consistent Absence of Bias - does not “stand-alone” – minimum variance important Asymptotically Normal Precise – large sample Inferences valid, ranges realistic
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.