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Course: EPIB 677, Fall 2009
School: McGill
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WinBUGS Simple Programs for Common Situations This section of the course will presents seven WinBUGS programs. Each program le contains three parts, the program listing, the data, and sample output using the program on the data. Most of the programs also contain an initial values section. Seven dierent types of simple analyses are presented: 1. Single binomial proportion 2. Single normal nmean, variance known 3....

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WinBUGS Simple Programs for Common Situations This section of the course will presents seven WinBUGS programs. Each program le contains three parts, the program listing, the data, and sample output using the program on the data. Most of the programs also contain an initial values section. Seven dierent types of simple analyses are presented: 1. Single binomial proportion 2. Single normal nmean, variance known 3. Single normal mean, variance unknown 4. Dierence of two binomial proportions 5. Linear regression 6. Logistic regression 7. Hierarchical binomial proportion modelling The online WinBUGS examples manuals (Volumes 1 and 2) contain 36 further types of analyses, almost all of them common to medicine. There is a very good chance that there will be a program already written for most analyses you will want to do (or something very close). 1. Single binomial proportion Simple binomial model with x successes in n trials as the data, and a beta(1,1) prior density. Set up as a bernoulli model with success parameter , but also could have been set up as a binomial (see example 7). Model model { for (i in 1:n) { x[i] ~ dbern(theta); } theta ~ dbeta(1,1); # } Prior density for theta Data list(x=c(0, 0, 0, 0 ,0, 0, 0 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ,1, 1, 1, 1, 1, 1), n=20) Results node mean sd MC error 2.5% theta 0.3177 0.0969 0.001411 0.1474 median 0.3116 97.5% start sample 0.5208 1001 5000 2. Single normal mean, variance known Simple normal mean () problem. The standard deviation of the data is sigma, which is obtained from the precision tau (value = 1). Prior are put on mu, note tiny prior precision, which implies huge variance, so close to a non-informative analysis. Data were randomly generated from a Normal(0,1) distribution. Note use of mean function. Model model { for (i in 1:n) { x[i] ~ dnorm(mu,tau); } mu ~ dnorm(0,0.0001); tau <- 1; sigma <- 1/sqrt(tau); xbar <mean(x[]); } Data list(x=c(-1.10635822, 0.56352639, -1.62101846, 0.06205707, 0.50183464, 0.45905694, -1.00045360, -0.58795638, 1.01602187, -0.26987089, 0.18354493 , 1.64605637, -0.96384666, 0.53842310, -1.11685831, 0.75908479 , 1.10442473 , -1.71124673, -0.42677894 , 0.68031412), n=20) Results node mean mu -0.06355 sd MC error 0.2235 0.002986 2.5% -0.5034 median -0.06203 97.5% start sample 0.3738 1001 5000 3. Single normal mean, variance unknown Identical problem and data to the previous problem, but now variance is also considered unknown. A gamma(0.001, 0.001) prior is put on the precision, very widely dispersed (mean=1, variance = 1000), so close to noninformative. Note the use of an initialization le, nmeanvar.in. This is to avoid potential problems with bad starting values, which are likely in this problem due to very wide prior on the variance (through the precision). Model model { for (i in 1:n) { x[i] ~ dnorm(mu,tau); } mu ~ dnorm(0,0.0001); tau ~ dgamma(0.001,0.001); sigma<-1/sqrt(tau); xbar <- mean(x[]); } Data list(x=c( -1.10635822, 0.56352639, -1.62101846, 0.06205707, 0.50183464, 0.45905694, -1.00045360, -0.58795638, 1.01602187, -0.26987089 , 0.18354493 , 1.64605637, -0.96384666, 0.53842310, -1.11685831, 0.75908479 , 1.10442473 , -1.71124673, -0.42677894 , 0.68031412), n=20) Initial Values list(mu=1, tau=.33) Results node mean sd MC error 2.5% median 97.5% start sample mu -0.06449 0.219 0.002937 -0.4952 -0.06569 0.3707 1001 5000 tau 1.088 0.3466 0.005788 0.5318 1.045 1.858 1001 5000 sigma 0.9971 0.1673 0.002776 0.7338 0.9781 1.372 1001 5000 Try changing some of the data points to be missing by changing the number to NA. For example, change the rst data line -1.10635822, 0.56352639, -1.62101846, 0.06205707, 0.50183464, to -1.10635822, 0.56352639, NA, 0.06205707, NA, You will notice that WinBUGS has automatically detected the missing data, and performed a multiple imputation on its value. Note also that the standard deviation in the estimation of mu slightly increases, since only 18 values truly contribute independent information, rather than 20. Automatic imputation will work for any missing data that are on the left hand side of a WinBUGS statement. To impute other missing items, create a new line that describes its distribution. Note that this allows all data to be used in linear regressions, not only those cases with complete data. node mean sd MC error 2.5% median 97.5% start sample mu -0.00799 0.2329 0.002951 -0.4533 -0.01144 0.4728 1001 5000 sigma 0.9778 0.1817 0.002724 0.7053 0.9526 1.399 1001 5000 tau 1.148 0.395 0.005871 0.5108 1.102 2.011 1001 5000 x[3] 0.01365 1.018 0.01355 -2.018 0.003193 2.012 1001 5000 x[5] -0.02405 1.034 0.01561 -2.092 -0.01312 1.927 1001 5000 4. Dierence of two binomial proportions Program to calculate the posterior distribution related to quantities comparing two independent binomial distributions. Posterior densities for the dierence in proportions, risk ratio, and odds ratio are all calculated. Sample sizes in two groups need not be the same. Model model { for (i in 1:n1) { x[i] ~ dbern(theta1); } theta1 ~ dbeta(1,1); for (i in 1:n2) { y[i] ~ dbern(theta2); } theta2 ~ dbeta(1,1); propdiff <- theta1-theta2 rr <- theta1/theta2 or<- theta1*(1-theta2)/((1-theta1)*theta2) } Data list(x = c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1), n1 = 20, y = c(0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1), n2 = 25) Results node mean sd MC error 2.5% median 97.5% start sample theta1 0.3176 0.09681 0.001334 0.148 0.3118 0.5211 1001 5000 theta2 0.7774 0.07969 0.001067 0.6057 0.7848 0.9103 1001 5000 propdi -0.4598 0.1254 0.001562 -0.6886 -0.4662 -0.1937 1001 5000 or 0.1505 0.112 0.001403 0.03093 0.122 0.4428 1001 5000 rr 0.4134 0.1358 0.001762 0.1862 0.4025 0.7155 1001 5000 5. Linear regression Simple linear regression of dependent variable blood pressure (bp) on age and sex. Data were simulated from: bp = -25 + 2 * sex + 3 * age, so alpha = -25, beta1 = 2, beta = 3. Sigma was set to 5. Posterior means should be somewhat near these values, but with small sample size, do not expect high accuracy. Model model { for (i in 1:n) { mu[i] <- alpha + b.sex*sex[i] + b.age*age[i]; bp[i] ~ dnorm(mu[i],tau); } alpha ~ dnorm(0.0,1.0E-4); b.sex ~ dnorm(0.0,1.0E-4); b.age ~ dnorm(0.0,1.0E-4); tau ~ dgamma(1.0E-3,1.0E-3); sigma <- 1.0/sqrt(tau); } Data list( sex = c(0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1), age = c(59, 52, 37, 40, 67, 43, 61, 34, 51, 58, 54, 31, 49, 45, 66, 48, 41, 47, 53, 62, 60, 33, 44, 70, 56, 69, 35, 36, 68, 38), bp = c(143, 132, 88, 98, 177, 102, 154, 83, 131, 150, 131, 69, 111, 114, 170, 117, 96, 116, 131, 158, 156, 75, 111, 184, 141, 182, 74, 87, 183, 89), n=30) Initial Values list(alpha=50, b.sex=1, b.age=4, tau=1) Results node mean sd MC 2.5% error median 97.5% start sample alpha -23.0 3.615 0.2896 -30.05 -23.15 -15.23 1001 5000 b.age 2.934 0.06587 0.005214 2.792 2.937 3.065 1001 5000 b.sex 1.63 1.52 0.05885 -1.384 1.602 4.533 1001 5000 tau 0.06592 0.01799 3.943E-4 0.03531 0.06459 0.1043 1001 5000 sigma 4.01 0.5809 0.01299 3.097 3.935 5.322 1001 5000 mu[1] 150.1 1.032 0.02828 148.1 150.1 152.2 1001 5000 mu[2] 131.2 1.165 0.03362 128.9 131.2 133.5 1001 5000 mu[3] 85.57 1.418 0.09821 82.89 85.53 88.44 1001 5000 (etc...) 6. Logistic regression Logistic regression model of a bone fracture with independent variables age and sex. The true model had: alpha = -25, b.sex = 0.5, b.age = 0.4. With so few data points and three parameters to estimate, do not expect posterior means/medians to equal the correct values exactly, but all would most likely be in the 95% intervals. Model model { for (i in 1:n) { logit(p[i]) <- alpha + b.sex*sex[i] + b.age*age[i]; frac[i] ~ dbern(p[i]); } alpha ~ dnorm(0.0,1.0E-4); b.sex ~ dnorm(0.0,1.0E-4); b.age ~ dnorm(0.0,1.0E-4); } Data list(sex=c(1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1), age= c(69, 57, 61, 60, 69, 74, 63, 68, 64, 53, 60, 58, 79, 56, 53, 74, 56, 76, 72, 56, 66, 52, 77, 70, 69, 76, 72, 53, 69, 59, 73, 77, 55, 77, 68, 62, 56, 68, 70, 60, 65, 55, 64, 75, 60, 67, 61, 69, 75, 68, 72, 71, 54, 52, 54, 50, 75, 59, 65, 60, 60, 57, 51, 51, 63, 57, 80, 52, 65, 72, 80, 73, 76, 79, 66, 51, 76, 75, 66, 75, 78, 70, 67, 51, 70, 71, 71, 74, 74, 60, 58, 55, 61, 65, 52, 68, 75, 52, 53, 70), frac=c(1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1), n=100) Initial Values list(alpha=0, b.sex=1, b.age=1) Results node alpha b.age b.sex p[1] p[2] p[3] p[4] (etc...) mean sd MC error 2.5% median 97.5% start sample -22.55 5.013 0.6062 -34.33 -21.64 -14.29 1001 4000 0.3559 0.07771 0.009395 0.227 0.3418 0.5338 1001 4000 1.405 0.7719 0.05094 -0.0387 1.374 3.031 1001 4000 0.9575 0.03153 0.002943 0.879 0.9647 0.9952 1001 4000 0.307 0.09828 0.004853 0.13 0.3012 0.5082 1001 4000 0.6308 0.1041 0.003344 0.4166 0.6356 0.8178 1001 4000 0.2477 0.103 0.007281 0.07738 0.2379 0.4728 1001 4000 7. Hierarchical binomial proportion modelling Real data from the MSc thesis of Dr. Michael Schull. Nine MDs made decisions on 133 patients. The decisions involved whether to give the patient thrombolysis or not in the emergency room. There are Canadian guidelines as to whether to thrombolyse such patients or not, so each physician was evaluated as th whether they followed the guidelines for each subject they treated or not. For example, the rst physician followed the guidelines in 19 out of 20 patients she/he treated. While each physician has their own rate of success (following the guidelines), it may be that overall, these rates may themselves have a distribution. The idea is to estimate both each physicians rate, as well as the rate of the next physician. The latter is accomplished by looking at the posterior distribution of the extra variable y. Note the hierarchical structure, the rates follow a distribution. Model model { for (i in 1:nmd) { x[i] ~ dbin(p[i],n[i]); logit(p[i]) <- z[i]; z[i] ~ dnorm(mu,tau); } mu tau y sigma w } ~ dnorm(0,0.001); # Prior distribution for mu ~ dgamma(0.001,0.001); # Prior distribution for tau ~ dnorm(mu, tau) ; # Predictive distribution for rate <- 1/sqrt(tau); # SD on the logit scale <- exp(y)/(1+exp(y)); # Predictive dist back on p-scale Data list(n=c( 20, 6, 24, 13, 12, 4, 24, 12, 18), x=c( 19, 5, 22, 12, 11, 4, 23, 12, 16), nmd=9) Initial Values list(mu=0, tau=1) Resul...

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MatSE 490DJ: Homework 1 Due: 5 February, 2003 1. For a trimer (triangular, planar) molecule made from s- (or pz-) atomic orbitals, or basis functions fi (i=1,3), a. Work out the one-electron (approximate) energy levels within the Hckel approximation
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556: M ATHEMATICAL S TATISTICS ID ISTRIBUTIONS DERIVED FROM N ORMAL RANDOM SAMPLES Suppose that X1 , . . . , Xn N (, 2 ) form a random sample. (a) By the univariate transformation result Z= (b) By the multivariate transformation result T = X St
Allan Hancock College - CLT - 112506
Liv e Sc enarioTeacher activity informationCreating a &quot;live&quot; scenarioThis is a different way of developing your students' observation skills.FSP14ScriptAn example, or create your own. 3 people participate in the action. Person A rushes into t
McGill - MATH - 556
556: M ATHEMATICAL S TATISTICS I D EFINITIONS A ND N OTATION F ROM R EAL A NALYSISDenition: Limits of sequences of reals Sequence {an } has limit a as n , writtennlim an = a for all n &gt; N. We say that {an } is aif, for every &gt; 0, there exists
University of Illinois, Urbana Champaign - MSE - 490
Deposition on &quot;Non-reconstructing&quot; Metallic SurfacesAg/Cu on Ru(0001) substratesStevens and Hwang PRL 74, 2078 (1995) Ag, Cu, and Ru are immiscible in bulk. Lattices: aCu(111) &lt; aRu(0001) &lt; aAg(111) . Large misfit strain on Ru(0001) surface.R
Allan Hancock College - CLT - 112506
Anthropo m etryThe Student Handouts that relate to these activities are found in FSP25 Personal Dossier under the section Anthropometry Table. What are anthropometric measurements? They are simply body dimensions. The main thing about making anthro
McGill - MATH - 557
MATH 557 - A SSIGNMENT 1 S OLUTIONS1 We construct the joint pmf/pdf in each case, and inspect the required conditional pdf. Note that 1-1 transformations of the statistics are also sufficient. (a) For the fX |, (x|, ) = suggesting the sufficient sta
Allan Hancock College - CLT - 112506
O bservation skillsPlayground crime scene activityThis is another way to further develop your students observation skills. It is based on FSP31 an online playground scene. Worksheets for this activity are available from the Personal Dossier FSP25
McGill - MATH - 556
MATH 556 - ASSIGNMENT 1: SOLUTIONS1 The distribution of any discrete random variable, X, can be written as a linear combination of a countable number of point mass measuresPX (B) =i=1pi xi (B)where x1 , x2 , . . . are the countable set of v
Allan Hancock College - CLT - 112508
FSB03blo od spatte rWhat is Forensic Science?Teacher background informationWhat is forensic science?Definition Forensics is the term given to an investigation of a crime using scientific means. It is also used as the name of the application of
Allan Hancock College - CLT - 112507
FSE04forensic ento mologyInsect StructureTeacher background informationClassification is essentially a hierarchy that progressively becomes more specific. For example, a Phylum groups organisms based on similarities in basic body plan and organi
McGill - MATH - 204
Memory.task Memory1 92 73 114 125 101 82 93 134 115 191 62 63 84 165 141 82 63 64 115 51 102 63 144 95 101 42 113 114 235 111 62 63 134 125 141 52 33 134 105 151 72 83 104 195 111 72 73 114 115 11
McGill - MATH - 204
&quot;Batch&quot; &quot;BacLevel&quot;&quot;1&quot; 24&quot;2&quot; 14&quot;3&quot; 11&quot;4&quot; 7&quot;5&quot; 19&quot;1&quot; 15&quot;2&quot; 7&quot;3&quot; 9&quot;4&quot; 7&quot;5&quot; 24&quot;1&quot; 21&quot;2&quot; 12&quot;3&quot; 7&quot;4&quot; 4&quot;5&quot; 19&quot;1&quot; 27&quot;2&quot; 17&quot;3&quot; 13&quot;4&quot; 7&quot;5&quot; 15&quot;1&quot; 33&quot;2&quot; 14&quot;3&quot; 12&quot;4&quot; 12&quot;5&quot; 10&quot;1&quot; 23&quot;2&quot; 16&quot;3&quot; 18&quot;4&quot; 18&quot;5&quot; 20
McGill - MATH - 204
&quot;Dose&quot; &quot;Pull&quot;&quot;A&quot; 27&quot;A&quot; 26.2&quot;A&quot; 28.8&quot;A&quot; 33.5&quot;A&quot; 28.8&quot;B&quot; 22.8&quot;B&quot; 23.1&quot;B&quot; 27.7&quot;B&quot; 27.6&quot;B&quot; 24&quot;C&quot; 21.9&quot;C&quot; 23.4&quot;C&quot; 20.1&quot;C&quot; 27.8&quot;C&quot; 19.3&quot;D&quot; 23.5&quot;D&quot; 19.6&quot;D&quot; 23.7&quot;D&quot; 20.8&quot;D&quot; 23.9
McGill - CS - 360
Algorithm Design TechniquesAssignment 6: Solutions(1) Hitting Set Problem. The Hitting Set problem is in NP. To check a solution a , . . . , a we just see 1 k if every set Bj is hit by at least one of the a . i We use a reduction from Set Cover to
University of Illinois, Urbana Champaign - BIOP - 490
BIOPHYS490M-final paper Yamini Purohit 1Exploring the Structural Basis of Ligand-specific Activation and Antagonism of an AMPA ReceptorIntroduction The ionotropic, glutamate-gated receptors (iGluRs) mediate excitatory fast synaptic transmission in
McGill - MATH - 242
MATH 242 Assignment 4 Solutions1. Let abs(x) = |x|, we use the inequality |a| |b| |a b| for all a, b R from the text, Corollary 2.2.4. Given &gt; 0 we take = &gt; 0 to nd that |a b| &lt; implies that |a| |b| &lt; . Hence abs is uniformly continuous on