1 / 20

DATA ANALYSIS

DATA ANALYSIS. Module Code: CA660 Lecture Block 5. ESTIMATION /H.T . Rationale, Other Features & Alternatives. Estimator validity – how good? Basis statistical properties (variance, bias, distributional etc.)

melita
Download Presentation

DATA ANALYSIS

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. DATA ANALYSIS Module Code: CA660 Lecture Block 5

  2. ESTIMATION /H.T. Rationale, OtherFeatures & Alternatives • Estimator validity– how good? • Basisstatistical properties (variance, bias, distributional etc.) • Bias where is the point estimate,  the true parameter. Bias can be positive, negative or zero. • Permits calculation of other properties, e.g. • where this quantity and variance of estimator are the same if estimator is unbiased. • Obtained by both analytical and“bootstrap methods” • Similarly, for continuous variables • or for b bootstrap replications,

  3. Estimation/H.T. Rationale etc. - contd. • For any,estimator , even unbiased,there is a difference between estimator and true parameter = sampling error • Hence the need for probability statements around • C.L. for estimator = (T1 , T2), similarly to before and  the confidence coefficient. If the estimator is unbiased, in other words,  = P{that true parameter falls into the interval}. • In general, confidence intervals can be determined using parametricand non-parametric approaches, where parametric construction needs a pivotal quantity= variable which is a function of parameter and data, but whose distribution does not depend on the parameter.

  4. Related issues in Hypothesis Testing -POWER of the TEST • Probability of False Positiveand False Negativeerrors • e.g.false positive if linkage between two genes declared, when really independent • Hypothesis Test Result • Fact Accept H0Reject H0 • H0 True1- False positive • = Type I error = • H0 False False negative Power of the Test • =Type II error= = 1-  • Power of the TestorStatistical Power=probability of rejecting H0when correct to do so. (Related strictly to alternative hypothesis and )

  5. Example on Type II Error and Power • Suppose have a variable, with known population S.D. = 3.6. From the population, a r.s. size n=100, used to test at =0.05, that • critical values of for a 2-sided test are: • for =0.05 where for , i = upper or lower and 0  under H0 • So substituting our values gives: • But, if H0false,  is not17.5, but some other value e.g. 16.5 say ??

  6. Example contd. • Want new distributionwith mean  = 16.5, i.e. new distribution isshifted w.r.t. the old. • Thus the probability of the Type II error - failingtoreject false H0is the area under the curve in the new distribution which overlaps thenon-rejectionregion specified under H0 • So, this is • Thus, probability of taking the appropriate action(rejectingH0 when this is false) is 0.791 = Power

  7. Shifting the distributions Non-Rejection region Rejection region /2 Rejection region /2 16.79 17.5 18.21 16.5 

  8. Example contd. Power under alternative  for given  • Possible values of  1- • under H1for H0 false • 16.0 0.0143 0.9857 • 16.5 0.2090 0.7910 • 17.0 0.7190 0.2810 • 18.0 0.7190 0.2810 • 18.5 0.2090 0.7910 • 19.0 0.0143 0.9857 • Balancing and :  tends to be largec.f.  unless original hypothesis way off. So decision based on a rejectedH0more conclusive than one based on H0not rejected, as probability of being wrong is larger in the latter case.

  9. SAMPLE SIZE DETERMINATION • Example: Suppose wanted to design a genetic mapping experiment, or comparative product survey. Conventional experimental design - ANOVA), genetic marker type (or product type) and sample sizeconsidered. • Questions might include: • What is the statistical powerto detect linkage for certain progeny size? (or common ‘shared’ consumer preferences, say) • What is the precision of estimated R.F. (or grouped preferences) when sample size is N? • Sample size needed for specificStatistical Power • Sample size needed for specific Confidence Interval

  10. Sample size - calculation based on C.I. For some parameter , Normal approximation approach valid, C.I. are U =standardized normal deviate (S.N.D.) and rangeis from lower to upper limits, i.e. is just a precision measurement for the estimator Given a true parameter  , So manipulation gives:

  11. Sample size - calculation based on Power (firstly, what affects power)? • Suppose  = 0.05,  =3.5, n=100, testing H0: 0=25 when true  =24; assume H1 : 1< 25. Sample mean found = 24.45. • One-tailedtest (U = 1.645) : shift small, lower limit of original distribution virtually coincides with actual sample value • Under H1Power = 0.50+0.39 = 0.89; correct decision 89% of time • Note:Two-sided testat  = 0.05 gives critical values, under H0given by • : equivalently UL= + 0.89, Uu= 4.82 for H1 • In general: substitute for limits & then recalculate for new  = 1 • So, P{do not rejectH0: =25 when true mean =24} = 0.1867 =  (Type II) • Thus, Power= 1 - 0.1867 = 0.8133

  12. Sample Size and Power contd. • Suppose, n=25, other values same. 1-tailed now • Power = 0.4129 • Suppose  = 0.01, critical values 2-tailed • with, equivalently,UL = + 0.29, UU = +5.43 • So, P{do not rejectH0: =25 when true mean =24} = 0.1141 • Power =0.8859 • FACTORS: , n and type of test (1- or 2-sided), true parameter value • where subscripts 0 and 1 refer to null and alternative, and  value taken as ‘generic’ (either all in one tail, 1-sided test/limit or split between two, 2-sided test/limit)

  13. ‘Other’ Estimation/Test MethodsNON-PARAMETRICS/DISTN FREE • Standard Pdfscan not be assumed for data, sampling distributions or test statistics – uncertain due to small or unreliable data sets, non-independence etc. Parameter estimation - not key issue. • Example/ Empirical-basis. Weaker assumptions. Less ‘information’ e.g. median used. Simple hypothesis testing as opposed to estimation. Power and efficiency are issues. • Counts - nominal, ordinal (natural non-parametric data type/ measure). • Nonparametric Hypothesis Tests- (has parallels to parametric case). • e.g. H.T. of locus orders requires complex ‘test statistic’ distribution, so need to construct empirical pdf. Usually, assume the null hypothesis and use re-samplingtechniques, e.g. permutation tests, bootstrap, jacknife.

  14. LIKELIHOOD METHOD - DEFINITIONS • Suppose X can take a set of values x1,x2,…with • whereis a vector of parameters affecting observed x’s • e.g. . So can say something about P{X} if we • know, say, • Butnot usually case, i.e. observe x’s, knowing nothing of • Assuming x’s a random sample size n from aknowndistribution, then • likelihoodfor • Finding most likely or ‘s for given data is equivalent to Maximising the Likelihood function, (where M.L.E. is )

  15. LIKELIHOOD –SCORE and INFO. CONTENT • The Log-likelihoodis a supportfunction[S()] evaluated at point,  ´say • Support function for any other point, say  ´´ can also be obtained – basis for computational iterations for MLE e.g. using Newton-Raphson • SCORE = first derivative of support function w.r.t. the parameter • or, numerically/discretely, • INFORMATION CONTENT evaluated at (i) arbitrary point =Observed Info.(ii)support function maximum = Expected Info.

  16. 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 Assume n constant, so first term invariant w.r.t. p = S( ) at point p Maximisingw.r.t. p i.e. set the derivative of S w.r.t. to 0 so SCOREsoM.L.E. How does it work, why bother?

  17. Numerical Examples See some data sets and test examples: Basics: http://www.unc.edu/~monogan/computing/r/MLE_in_R.pdf Context: http://statgen.iop.kcl.ac.uk/bgim/mle/sslike_1.htmlAll sections useful, but especially examples, sections 1-3 and 6 Also, e.g. for R http://www.montana.edu/rotella/502/binom_like.pdf http://www.unc.edu/~monogan/computing/r/MLE_in_R.pdf forSPSS– see e.g. tutorial for data sets or http://www.spss.ch/upload/1126184451_Linear%20Mixed%20Effects%20Modeling%20in%20SPSS.pdfgeneral for mixed Linear Models For SAS – of possible interest http://blogs.sas.com/content/iml/2011/10/12/maximum-likelihood-estimation-in-sasiml/

  18. Bayesian Estimation- in context • Parametric Estimation- in “classical approach” f(x,) for a r.v. X of density f(x) , with  the unknown parameter  dependency of distribution on parameter to be estimated. • Bayesian Estimation- is a random variable, so can consider the density as conditional and write f(x|  ) • Given a random sampleX1,X2,… Xnthe sample random variables are jointly distributed with parameter r.v. . So, joint pdf • Objective - to form an estimator that gives 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

  19. Bayes - contd. • Posterior Density • Relationship - prior and posterior: • where () prior density of  • Value: Close to MLEforlarge n, or for small n if sample values compatible with the prior distribution. Also, has strong sample basis, simpler to calculate than M.L.E.

  20. Estimator Comparison in brief. • Classical: uses objective probabilities, intuitive estimators, additional assumptions for sampling distributions: good properties for some estimators. • Moments : { less calculation, less efficient. Despite analytical solutions & low bias, not well-used for large-scale data because poorer asymptotic properties; even simple solutions may not be unique.} • Bayesian - subjective prior knowledge, sample info. close to MLE under certain conditions - see earlier. • LSE- if assumptions met,  ’s unbiased + variances obtained, {(XTX)-1} . Few assumptions for response variable distributions, just expectations, variance-covariance structure. (Unlike MLE where need to specify joint prob. distribution of variables). Requires additional assumptions for sampling distns. Close to MLE if these are met. Computation easier.

More Related