1 / 38

The Method of Likelihood

The Method of Likelihood. Hal Whitehead BIOL4062/5062. What is likelihood Maximum likelihood Maximum likelihood estimation Likelihood ratio tests Likelihood profile confidence intervals Model selection: Likelihood ratio tests Akaike Information Criterion (AIC)

iain
Download Presentation

The Method of Likelihood

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. The Method of Likelihood Hal Whitehead BIOL4062/5062

  2. What is likelihood • Maximum likelihood • Maximum likelihood estimation • Likelihood ratio tests • Likelihood profile confidence intervals • Model selection: • Likelihood ratio tests • Akaike Information Criterion (AIC) • Likelihood and least-squares • Calculating likelihood

  3. The Method of Likelihood Observations: Y = {y1,y2,y3,...} e.g. Weights of 30 crabs of known age and sex Model specified by: μ1, μ2, μ3,… e.g. y = μ1 + μ2·√Age + μ3·Sex(0:1) + μ4·e where e ~ N(0, 1) The LIKELIHOOD of Y is: L = Probability (Y| Model & μ1, μ2, μ3,... )

  4. Likelihood The LIKELIHOOD of Y is: L = Probability (Y| Model & μ1, μ2, μ3,... ) The LIKELIHOOD that Z became a criminal: Probability Z became a criminal given what we what we know of Z’s characteristics and how those characteristics translate into the probability of being a criminal

  5. The LIKELIHOOD of Y is: L = Probability (Y| Model & μ1, μ2, μ3,…) We can work this outif we know μ1, μ2, μ3,… Weights of 30 crabs of known age and sex y = μ1 + μ2·√Age + μ3·Sex(0:1) + μ4·e e.g Prob. of these 30 weights is 0.04 if: female wt at age 0, μ1 = 30.0 growth parameter, μ2 = 0.7 excess male weight, μ3 = 5.0 residual s.d., μ4 = 6.3 L(μ1=30,μ2=0.7,μ3=5.0, μ4=6.3)=0.04

  6. Maximum Likelihood Estimators If we do not know μ1, μ2, μ3,... MAXIMUM LIKELIHOOD of Y is: L(μ1,μ2,μ3,...) = Max{Prob.(Y| μ1, μ2, μ3,... )} μ1,μ2,… e.g Max prob. of 30 weights is 0.12 when: female wt at age 0, μ1 = 28.4 growth parameter, μ2 = 0.31 excess male weight, μ3 = 1.7 residual s.d., μ4 = 3.9

  7. Maximum likelihood Maximum likelihood estimator of μ1 Maximum Likelihood Likelihood μ1

  8. Imprecise estimate Maximum Likelihood Precise estimate Likelihood μ1

  9. Likelihood Ratio Tests If: μ1,μ2,μ3,…,μt is true model μ1,μ2,μ3,…,μt,...,μg is more general model then: G = 2∙Log[L(μ1,μ2,μ3,…,μg)/L(μ1,μ2,μ3,…,μt)] (twice the log of the ratios of the maximum likelihoods) is distributed as χ² with g-t degrees of freedom for large sample sizes (asymptotically) If G is unexpectedly large then data are unlikely to be from model μ1,μ2,μ3,…,μt

  10. Likelihood Ratio Tests G = 2·Log[L(μ1,μ2,μ3,…,μg)/L(μ1,μ2,μ3,…,μt)] This is the "G-test for goodness-of-fit": null hypothesis: μ1,μ2,μ3,…,μt alternative hypothesis: μ1,μ2,μ3,…,μt,...,μg

  11. Likelihood: an example Expect Find Wild Type 75% 80 Mutants 25% 10 Total 100% 90

  12. Null hypothesis: Binomial Distribution with q = 0.75 Expect Find Wild Type 75% 80 Mutants 25% 10 Total 100% 90 Likelihood(q=0.75) = 90C10 ·0.7580 ·0.2510 = .000551

  13. Maximum Likelihood Estimator Alternative hypothesis: Binomial Distribution with q = ? Expect Find Wild Type 75% 80 Mutants 25% 10 Total 100% 90 Likelihood(q) = 90C10 ·q80 ·(1-q)10 This has a maximum value when q = 80/90 = 0.89 Max Likelihood(q) = 90C10 ·(0.89)80 ·(1-0.89)10 = 0.1236

  14. Likelihood Ratio Test Expect Find Wild Type 75% 80 Mutants 25% 10 Total 100% 90 • G = 2 ·Log { Max Likelihood (q) } • Likelihood (q = 0.75) • = 2 · Log(0.1236/ 0.000551) = 10.96 • is distributed as χ² with 1 d.f. if q=0.75 • significantly large (P<0.01) in χ²(1) • so: reject null hypothesis.

  15. Profile LikelihoodConfidence Intervals Likelihood μ1

  16. Maximum likelihood 2 95% c.i. Maximum likelihood estimator of μ1 Profile LikelihoodConfidence Intervals Log- Likelihood μ1

  17. MLE(0) μ2 -2 μ1 Profile LikelihoodConfidence Intervals Log-Likelihood Contours(relative to maximum likelihood) 95% Confidence region

  18. Model SelectionUsing Likelihood-Ratio Tests Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4 ·e M(1): y = μ1 + μ2 ·√Age + μ4 ·e M(2): y = μ1 + μ2 ·√Age + μ3 ·Sex(0:1) + μ4 ·e

  19. Model SelectionUsing Likelihood-Ratio Tests Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4·e Log(L)= -23.04 M(1): y = μ1 + μ2 ·√Age + μ4 ·e Log(L)= -20.34 M(2): y = μ1 + μ2 ·√Age + μ3 ·Sex(0:1) + μ4 ·e Log(L)= -19.84

  20. Model SelectionUsing Likelihood-Ratio Tests Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4 ·e Log(L)= -23.04 M(1): y = μ1 + μ2 ·√Age + μ4 ·e Log(L)= -20.34 M(2): y = μ1 + μ2 · √Age + μ3 ·Sex(0:1) + μ4 ·e Log(L)= -19.84 G(M(0)vs.M(1)) = 2x(-20.34 - (-23.04)) = 5.40 G(M(1)vs.M(2)) = 2x(-19.84 - (-20.34)) = 1.00 G(M(0)vs.M(2)) = 2x(-19.84 - (-23.04)) = 6.40

  21. Model SelectionUsing Likelihood-Ratio Tests Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4 ·e Log(L)= -23.04 M(1): y = μ1 + μ2 · √Age + μ4 ·e Log(L)= -20.34 M(2): y = μ1 + μ2 · √Age + μ3 ·Sex(0:1) + μ4 ·e Log(L)= -19.84 G(M(0)vs.M(1)) = 2x(-20.34 - (-23.04)) = 5.40 P(χ²(1))<0.05 G(M(1)vs.M(2)) = 2x(-19.84 - (-20.34)) = 1.00 P(χ²(1))>0.10 G(M(0)vs.M(2)) = 2x(-19.84 - (-23.04)) = 6.40 P(χ²(2))<0.05

  22. Model SelectionUsing Likelihood-Ratio Tests Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4 ·e Log(L)= -23.04 M(1): y= μ1 + μ2 ·√Age + μ4 ·eLog(L)= -20.34 M(2): y = μ1 + μ2 · √Age + μ3 · Sex(0:1) + μ4 ·e Log(L)= -19.84 G(M(0)vs.M(1)) = 2x(-20.34 - (-23.04)) = 5.40 P(χ²(1))<0.05 G(M(1)vs.M(2)) = 2x(-19.84 - (-20.34)) = 1.00 P(χ²(1))>0.10 G(M(0)vs.M(2)) = 2x(-19.84 - (-23.04)) = 6.40 P(χ²(2))<0.05

  23. Model SelectionUsing Likelihood-Ratio Tests Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4 ·e Log(L)= -23.04 M(1): y = μ1 + μ2 · √Age + μ4 ·e Log(L)= -20.34 M(2): y = μ1 + μ2 · √Age + μ3 · Sex(0:1) + μ4 ·e Log(L)= -19.84 G(M(0)vs.M(1)) = 2x(-20.34 - (-23.04)) = 5.40 P(χ²(1))<0.05 G(M(1)vs.M(2)) = 2x(-19.84 - (-20.34)) = 1.00 P(χ²(1))>0.10 G(M(0)vs.M(2)) = 2x(-19.84 - (-23.04)) = 6.40 P(χ²(2))<0.05 But: What is critical p-value?

  24. Model SelectionUsing Likelihood-Ratio Tests Weights of 30 crabs of known age and sex: M(1): y = μ1 + μ2 ·√Age + μ4 ·e M(3): y = μ1 + μ3 ·Sex(0:1) + μ4 ·e But: Cannot compare M(1) and M(3) using likelihood-ratio tests

  25. Model SelectionUsing Likelihood-Ratio Tests • What is critical p-value? • Cannot compare models which are not subsets of one another using likelihood-ratio tests So: Akaike Information Criteria (AIC)

  26. Akaike Information Criteria (AIC) • Kullback-Leibler Information (KLI): • “information lost when model M(0) is used to approximate model M(1)” • “distance from M(0) to M(1)” • AIC(M) = - 2xLog(Likelihood(M)) + 2xK(M) • K(M) is number of estimable parameters of model M • AIC is an estimate of the expected relative distance (KLI) between a fitted model, M, and the unknown true mechanism that generated the data

  27. Akaike Information Criteria (AIC) • AIC(M) = - 2xLog(Likelihood(M)) + 2xK(M) • K(M) is number of estimable parameters • In model selection: choose model with smallest AIC • least expected relative distance between M, and the unknown true mechanism that generated the data

  28. Model SelectionUsing AIC Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4 · e M(1): y = μ1 + μ2 · √Age + μ4 · e M(2): y = μ1 + μ2 · √Age + μ3 · Sex(0:1) + μ4 · e M(3): y = μ1 + μ3 · Sex(0:1) + μ4 · e

  29. Model SelectionUsing AIC Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4 · e AIC=50.08 M(1): y = μ1 + μ2 · √Age + μ4 · e AIC=46.68 M(2): y = μ1 + μ2 · √Age + μ3 · Sex(0:1) + μ4 · e AIC=47.68 M(3): y = μ1 + μ2 · Sex(0:1) + μ4 · e AIC=49.95

  30. Model SelectionUsing AIC Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4 · e AIC=50.08 M(1): y= μ1 + μ2 · √Age + μ4 · eAIC=46.68 M(2): y = μ1 + μ2 · √Age + μ3 · Sex(0:1) + μ4 · e AIC=47.68 M(3): y = μ1 + μ3 · Sex(0:1) + μ4 · e AIC=49.95

  31. Model SelectionUsing AIC • Differences in AIC between models: ΔAIC • Support for less favoured model • ΔAIC: 0-2 Substantial • ΔAIC: 4-7 Considerably less • ΔAIC: >10 Essentially none

  32. Model SelectionUsing AIC Weights of 30 crabs of known age and sex: M(0): y = μ1 + μ4 · e AIC=50.08 Unlikely M(1): y= μ1 + μ2 · √Age + μ4 · eAIC=46.68 BEST M(2): y = μ1 + μ2·√Age + μ3·Sex(0:1) + μ4·e AIC=47.68 Good M(3): y = μ1 + μ3 · Sex(0:1) + μ4 · e AIC=49.95 Unlikely

  33. Modifications to AIC AIC for small sample sizes: AICC= - 2x(Log-Likelihood) + 2xKxn/(n-K-1) n is sample size AIC for overdispersed count data: QAIC= - 2xLog-Likelihood/c + 2xK c is “variance inflation factor” (c=χ²/df)

  34. Burnham, K. P., and D. R. Anderson2002Model selection and multimodel inference: a practical information-theoretic approach, 2nd ed. New York: Springer-Verlag

  35. Likelihood and Least-Squares • If errors are normally distributed • least squares and maximum-likelihood estimates of parameters are the same • but not σ2 estimators • Likelihood is a more powerful and theoretically-based technique

  36. AIC and Least-Squares • If all models assume normal errors with constant variance: • AIC = n.Log(σ2) + 2.K • σ2 = Σei2/n (the MLE of σ2) • K is total no of estimated regression parameters, including the intercept and σ2

  37. Calculating Likelihoods • Analytical formulae • Compute by multiplying probabilities • Estimate by simulation • number of times data are obtained in 1,000 simulations given model and parameters

  38. The Method of Likelihood • Probability of data given model • Estimate parameters using maximum likelihood • Estimate confidence intervals using likelihood profiles • Compare models using • likelihood ratio tests • Akaike Information Criterion (AIC)

More Related