Introduction to LikelihoodPowerPoint PresentationSlide 3Slide 4Cumulative Probability & P ValuesSlide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Which hypotheses do we reject?Slide 14Which hypotheses do we NOT reject: CONFIDENCE INTERVALLet’s take another approachSlide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Defining LikelihoodDeriving the Maximum Likelihood EstimateSlide 28Slide 29Slide 30Slide 31Building Confidence Intervals Likelihood Ratio TestSlide 33Slide 34Slide 35Slide 36Comparing Confidence IntervalsAdvantages of LikelihoodThis presentation is made available through a Creative Commons Attribution-Noncommercial license. Details of the license and permitted uses are available at http://creativecommons.org/licenses/by-nc/3.0/ © 2010 Steve Bellan and the Meaningful Modeling of Epidemiological Data ClinicIntroduction to LikelihoodMeaningful Modeling of Epidemiologic Data, 2012AIMS, Muizenberg, South AfricaSteve Bellan, MPH, PhDDepartment of Environmental Science, Policy & ManagementUniversity of California at Berkeleybarplot(dbinom(x = 0:100, size = 100, prob = .3), names.arg = 0:size)In a population of 1,000,000 people with a true prevalence of 30%, the probability distribution of number of positive individuals if 100 are sampled: f (x) =100xæ è ç ö ø ÷ (0.3)x(0.7)100- xIn a population of 1,000,000 people with a true prevalence of 30%, the probability distribution of number of positive individuals if 100 are sampled: f (x) =100xæ è ç ö ø ÷ (0.3)x(0.7)100- x> rbinom(n = 1, size = 100, prob = .3)[1] 28We sample 100 people once and 28 are positive:> rbinom(n = 1, size = 100, prob = .3)[1] 28We sample 100 people once and 28 are positive:We don’t know the true prevalence!But we can calculate the probability of 28 or a more extreme value occurring for a given prevalence.0 2 4 6 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98number HIV+probability0.00 0.02 0.04 0.06 0.08We sample 100 people once and 28 are positive.p-value = 0.74> 2*pbinom(28,100,.3))[1] 0.7535564Cumulative Probability & P Valuesfor 30% prevalence:p(28 or a more extreme value occurring) =0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99If true prevalence were 15%, then p(28 or more extreme) isnumber HIV+probability0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.142*pbinom(28−1, 100, 0.15, lower.tail = FALSE)p = 0.00123x20 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99If true prevalence were 20%, then p(28 or more extreme) isnumber HIV+probability0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.142*pbinom(28−1, 100, 0.2, lower.tail = FALSE)p = 0.0683x20 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99If true prevalence were 25%, then p(28 or more extreme) isnumber HIV+probability0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.142*pbinom(28−1, 100, 0.25, lower.tail = FALSE)p = 0.555x20 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99If true prevalence were 30%, then p(28 or more extreme) isnumber HIV+probability0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.142*pbinom(28, 100, 0.3, lower.tail = TRUE)p = 0.754x20 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99If true prevalence were 35%, then p(28 or more extreme) isnumber HIV+probability0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.142*pbinom(28, 100, 0.35, lower.tail = TRUE)p = 0.17x20 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99If true prevalence were 40%, then p(28 or more extreme) isnumber HIV+probability0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.142*pbinom(28, 100, 0.4, lower.tail = TRUE)p = 0.0169x20 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 15 %number HIV+probability0.00 0.04 0.08 0.12p = 0.001230 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 20 %number HIV+probability0.00 0.04 0.08 0.12p = 0.06830 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 25 %number HIV+probability0.00 0.04 0.08 0.12p = 0.5550 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 30 %number HIV+probability0.00 0.04 0.08 0.12p = 0.7540 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 35 %number HIV+probability0.00 0.04 0.08 0.12p = 0.170 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 40 %number HIV+probability0.00 0.04 0.08 0.12p = 0.0169x2x2x2x2x2x2Which hypotheses do we reject?IF GIVEN THE HYPOTHESIS p value < cutoff THEN REJECT HYPOTHESISCutoff usually chosen as α = 0.050 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 15 %number HIV+probability0.00 0.04 0.08 0.12p = 0.001230 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 20 %number HIV+probability0.00 0.04 0.08 0.12p = 0.06830 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 25 %number HIV+probability0.00 0.04 0.08 0.12p = 0.5550 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 30 %number HIV+probability0.00 0.04 0.08 0.12p = 0.7540 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 35 %number HIV+probability0.00 0.04 0.08 0.12p = 0.170 5 11 18 25 32 39 46 53 60 67 74 81 88 95hypothetical prevalence: 40 %number HIV+probability0.00 0.04 0.08 0.12p = 0.0169Which hypotheses do we reject?0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0hypothetical prevalencep−value0.195 0.37895% CI includes HIV prevalences of 19.5% to 37.8%Which hypotheses do we NOT reject: CONFIDENCE INTERVALWe don’t know the true prevalence, but the probability that we had exactly 28/100 with 30% prevalence is:> dbinom(x = 28, size = 100, prob = .3)[1] 0.08041202> rbinom(n = 1, size = 100, prob = .3)[1] 28We sample 100 people once and 28 are positive:Let’s take another approachWhich prevalence gives the greatest probability of observing exactly 28/100?Which of these prevalence values is most likely given our data?Maximum Likelihood Estimate parameter value giving greatest probability of the data having occurred.MLE = 28/100 = 0.28What do you think is the MLE here?true unknown value = 0.30different null hypothesesDefining Likelihood• L(parameter | data) = p(data | parameter)•Not a probability distribution.•Probabilities taken from many different distributions. f (x | p) =nxæ è ç ö ø ÷ px(1- p)n- xfunction of xPDF: L( p| x) =nxæ è