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15 Pages

### 22BLUP

Course: STAT 511, Spring 2011
School: Iowa State
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Word Count: 1115

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of Prediction random variables Key distinction between fixed and random effects: Estimate means of fixed effects Estimate variance of random effects But in some instances, want to predict FUTURE values of a random effect Example (from Efron and Morris, 1975, JASA 70:311-319): Baseball players. Given a player's performance in the beginning of the season, predict performance in rest of season. c 2011 Dept....

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of Prediction random variables Key distinction between fixed and random effects: Estimate means of fixed effects Estimate variance of random effects But in some instances, want to predict FUTURE values of a random effect Example (from Efron and Morris, 1975, JASA 70:311-319): Baseball players. Given a player's performance in the beginning of the season, predict performance in rest of season. c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 1 / 15 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 2 / 15 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 3 / 15 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 4 / 15 Best predictor is found by "Shrinking" obs. performance towards overall mean. So, how much shrinkage is needed? How do we compute optimal predictor? General answer using a linear mixed effects model y = X + Zu + , where u N 0 0 , G 0 0 R Given data y, what is our best guess for values in the unobserved vector u ? c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 5 / 15 Because u is a random vector rather than a fixed parameter, we talk about predicting u rather than estimating u . We seek a Best Linear Unbiased Predictor (BLUP) for u, which we ^ will denote by u To be a BLUP, we require 1. 2. 3. ^ u is a linear function of y. ^ u is unbiased for u so that E(^ - u) = 0. u Var(^ - u) is no "larger" than Var(v - u), where v is any other linear u and unbiased predictor. It turns out that the BLUP for u is the BLUE of E(u|y). c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 6 / 15 What does E(u|y) look like? We will use the following result about conditional distributions for multivariate normal vectors. Suppose w1 1 11 12 N , w2 2 21 22 where 11 12 21 22 is a positive definite variance matrix. Then the conditional distribution of w2 given w1 is as follows (w2 |w1 ) N(2 + 21 -1 (w1 - 1 ), 22 - 21 -1 12 ) 11 11 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 7 / 15 Now note that y u Thus, y u N d = X 0 + Z I I 0 u X 0 X 0 , Z I I 0 , G 0 0 R Z I I 0 = N ZGZ + R ZG GZ G c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 8 / 15 Thus, E(u|y) = 0 + GZ (ZGZ + R)-1 (y - X) = GZ -1 (y - X) Thus, the BLUP of u is ^ GZ -1 (y - X g ) = GZ -1 (y - X(X -1 X)- X -1 y) = GZ -1 (I - X(X -1 X)- X -1 )y For the usual case in which G and = ZGZ + R are unknown, we replace the matrices by estimates and approximate the BLUP of u ^^ ^ ^ -1 by GZ (y - X ) c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 9 / 15 Often we wish to make predictions of quantities like C + du for some estimable C. ^ The BLUP of such a quantity is C + d^, the BLUE of C plus d u g times the BLUP of u Baseball players example is slightly complicated because quantities of interest are proportional to Binomial random variables. Simpler example, using Normal random variables: An old problem, a variation of a homework problem (# 23 on p. 164) of Mood, A.M. (1950) Introduction to the Theory of Statistics. New York: McGraw-Hill. c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 10 / 15 Suppose intelligence quotients (IQs) for a population students of 2 are normally distributed with a mean and variance u An IQ test was given to an i.i.d. sample of such students. Given the IQ of a student, the test score for that student is normally distributed with a mean equal to the student's IQ and a 2 variance e and is independent of the test score of any other students. 2 2 Suppose it is known that u /e = 9 If the sample mean of the students' test scores was 100, what is the best prediction of the IQ of a student who scored 130 on the test? c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 11 / 15 2 Suppose u1 , . . . , un , N(0, u ) independent of 2 e1 , . . . , en , N(0, e ). i.i.d. i.i.d. If we let + ui denote the IQ of student i(i = 1, ..., n), then the IQs 2 of the students are N(, u ) as in the statement of the problem. If we let yi = + ui + ei denote the test score of student 2 i(i = 1, . . . , n), then (yi | + ui ) N( + ui , e ) as in the problem statement. We have y = X + Zu + , where 2 2 2 2 X = 1, = , Z = I, G = u I, R = e I, = ZGZ + R = (u + e ) I. ^ Thus = (X -1 X)- X -1 y = (1 1)- 1 y = . and y GZ g -1 = 2 u 2 2 u +e I 2 u 2 2 (y u +e ^ ^ Thus, the BLUP for u is u = GZ -1 (y - X g ) = The ith element of this vector is ^i = u c 2011 Dept. Statistics (Iowa State University) 2 u 2 + 2 u e - 1. ) y (yi - . ). y Stat 511 section 22 12 / 15 Thus, the BLUP for the IQ of student i, + ui , is 2 2 2 + ^i = . + 2u 2 (yi - . ) = 2u 2 yi + 2e 2 . ^ u y y y + + + u e u e u e Note that the BLUP is a convex combination of the individual score and the overall mean score 2 2 u y + 2e 2 . y 2 + 2 i + u e u e Because 2 u 2 2 u +e = 2 u 2 e 2 u 2 +1 e 2 u 2 e is assumed to be 9, the weights are = 9 9+1 = 0.9 and 2 e 2 2 u +e = 0.1. We predict 0.9(130) + 0.1(100) = 127 to be the IQ of a student who scored 130 on the test. US College test results (e.g. SAT) now include information about "If you take this test again, your score is predicted to be ..." If above average, predicted to drop from current score. I suspect these predictions are BLUPs. c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 13 / 15 An extension that illustrates an important property of BLUP's IQ problem, except now, yi is the average of ni test scores for a student Some student's scores based on ni = 1 test, others on ni = 5 tests 2 Now, (yi | + ui ) N( + ui , e /ni ) ^ Now, is a weighted average of student scores g The BLUP for ui is ^i = u 2 u ^ (yi - g ) 2 2 u + e /ni ^ Again a convex combination of yi and overall average, g . But now weights are 2 u 2 + 2 /n u i e and 2 e /ni 2 + 2 /n u i e c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 14 / 15 Numerical illustration of weights 2 2 2 2 u /e = 9 u /e = 1 ^ ^ Number of tests yi g yi g 1 0.9 0.1 0.5 0.5 2 0.947 0.053 0.667 0.333 3 0.964 0.036 0.75 0.25 4 0.973 0.027 0.80 0.2 10 0.989 0.011 0.909 0.0909 More tests = more precise information about an individual: BLUP is closer to the data value, yi Fewer tests = less precise information about an individual: ^ BLUP is closer to the estimated population mean, g More variability between individuals: BLUP is closer to the data value, yi c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 15 / 15
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Iowa State - STAT - 511
A collection of potentially useful modelsWe've already seen two very common mixed models:for subsampling for designed experiments with multiple experimental unitsHere are three more general classes of modelsRandom coefficient models, aka multi-level m
Iowa State - STAT - 511
A collection of potentially useful modelsA regression where all coefficients vary between groups Example: Strength of parachute lines.Random coefficient modelsWe've already seen two very common mixed models:for subsampling for designed experiments wit
Iowa State - STAT - 511
Choosing among possible random effects structuresGoal is a model that:Fits the data reasonably well Is not too complicatedINFORMATION CRITERIA: AIC and BICSometimes random effects structure specified by the experimental designe.g. for experimental st
Iowa State - STAT - 511
Choosing among possible random effects structuresSometimes random effects structure specified by the experimental designe.g. for experimental study, need a random effect for each e.u.Sometimes subject matter information informs the choicee.g. expect a
Iowa State - STAT - 511
NONLINEAR MODELSSo far the models we have studied this semester have been linear in the sense that our model for the mean has been a linear function of the parameters. We have assumed E(y) = X f (Xi , ) = Xi is said to be linear in the parameters of beca
Iowa State - STAT - 511
NONLINEAR MODELSFor example, if Xi1 = 1 Xi2 = Amount of fertilizer applied to plot i Xi3 = (Amount of fetrtilizer applied to plot i)2 Xi4 = log(Concentration of fungicide on plot i) f (Xi , ) = Xi = Xi1 1 + Xi2 2 + Xi3 3 + Xi4 4 = 1 + ferti 2 + fert2 3 +
Iowa State - STAT - 511
GENERALIZED LINEAR MODELSConsider the normal theory Gauss-Markov linear model y = X + , N(0, 2 I). Does not have to be written as function + error Could specify distribution and model(s) for its parameters i.e., yi N(i , 2 ), where i = Xi for all i = 1,
Iowa State - STAT - 511
GENERALIZED LINEAR MODELSConsider the normal theory Gauss-Markov linear model y = X + , N(0, 2 I).Does not have to be written as function + errorCould specify distribution and model(s) for its parametersIn each example, all responses are independent a
Iowa State - STAT - 511
Logistic Regr. Model for Binomial Count DataBernoulli model appropriate for 0/1 response on an individual What if data are # events out of # trials per subject? Example: Toxicology study of the carcenogenicity of aflatoxicol.(from Ramsey and Schaefer, T
Iowa State - STAT - 511
Logistic Regr. Model for Binomial Count Data Bernoulli model appropriate for 0/1 response on an individual0.8 What if data are # events out of # trials per subject? Example: Toxicology study of the carcenogenicity of aflatoxicol.0.6 0.4But, all f
Iowa State - STAT - 511
Generalized Linear Mixed ModelsGLM + Mixed effects Goal: Add random effects or correlations among observations to a model where observations arise from a distribution in the exponential-scale family (other than the normal) Why:More than one source of va
Iowa State - STAT - 511
Generalized Linear Mixed ModelsAnother look at the canonical LME: Y = X + Zu + Consider each level of variation separately. A hierarchical or multi-level model = X + Zu N(X, ZGZ ) Y| = + N(, ) Y|u = X + Zu + N(X + Zu, ) Above specifies the conditional di
Iowa State - STAT - 511
Methods for large P, small N problemsRegression has (at least) three major purposes:1. Estimate coefficients in a pre-specified model 2. Discover an appropriate model 3. Predict values for new observationsRegression includes classification because clas
Iowa State - STAT - 511
Nonparametric regression using smoothing splinesSmoothing is fitting a smooth curve to data in a scatterplot Will focus on two variables: Y and one X Our model: yi = f (xi ) + i , where 1 , 1 , . . . n are independent with mean 0 f is some unknown smooth
Iowa State - STAT - 511
Nonparametric regression using smoothing splinesWhy estimate f ?Smoothing is fitting a smooth curve to data in a scatterplotWill focus on two variables: Y and one X yi = f (xi ) + i ,Our model:can see features of the relationship between X and Y that
Iowa State - STAT - 511
Smoothing - part 2Next page: fitted penalized regression splines for 3 smoothing parameters: 0, 100, and 5.7 5.7 is the &quot;optimal&quot; choice, to be discussed shortly &quot;optimal&quot; curve is a sequence of straight lines continuous, but 1st derivative is not contin
Iowa State - STAT - 511
Smoothing - part 26.5~0 100 5.7 Next page: fitted penalized regression splines for 3 smoothing parameters: 0, 100, and 5.76.0 5.55.7 is the &quot;optimal&quot; choice, to be discussed shortly&quot;optimal&quot; curve is a sequence of straight lines5.0 continuous, b
Iowa State - STAT - 511
Smoothing - part 3Penalized splines is not the only way to estimate f (x) when y = f (x) + Two others are kernel smoothing and the Lowess (Loess) smoother I'll only talk about Lowess Penalized splines and Lowess have same goal. Lowess is more ad-hoc. Onl
Iowa State - STAT - 511
A simple algorithm that doesn't work well: Penalized splines is not the only way to estimate f (x) when y = f (x) + Two others are kernel smoothing and the Lowess (Loess) smoother I'll only talk about Lowess Penalized splines and Lowess have same goal.Sm
Iowa State - STAT - 511
Classification and Regression TreesWhat if you have many X variables? Could imagine estimating f (X1 , X2 , . . . , Xk ) But increasingly difficult beyond k = 2 or k = 3 &quot;Curse of dimensionality&quot; In high dimensions, every point is isolated (see next slid
Iowa State - STAT - 511
EXAMPLE ANALYSIS OF AN UNBALANCED TWO-FACTOR EXPERIMENTAn experiment was conducted to study the effect of storage time and storage temperature on the amount of active ingredient present in a drug at the end of storage. Sixteen vials of the drug, each con
Iowa State - STAT - 511
EXAMPLE ANALYSIS OF AN UNBALANCED TWO-FACTOR EXPERIMENTStorage Time 3 months 6 months 6 6 7 7 16 2 5 9 12 15 Storage Temperature 30 C 20 CAn experiment was conducted to study the effect of storage time and storage temperature on the amount of active ing
Iowa State - STAT - 511
3000qresid(bacteria.lm)10002000q q q q q q q q q q q q q0-2000-1000qq-3000q010002000300040005000fitted(bacteria.lm)
Iowa State - STAT - 511
Percentile bootstrap con.dence intervalsSuppose that a quantity = (F ) is of interest and that Tn = (the empirical distribution of Y1 ; Y2 ; : : : ; Yn ) Based on B bootstrapped values Tn1 ; Tn2 ; : : : ; TnB , de.ne ordered values Tn(1) Tn(2) Tn(B)Adop
Iowa State - STAT - 511
Bootstrap resamplingPhilip M. Dixon Volume 1, pp 212220 in Encyclopedia of Environmetrics (ISBN 0471 899976) Edited by Abdel H. El-Shaarawi and Walter W. Piegorsch John Wiley &amp; Sons, Ltd, Chichester, 2002Bootstrap resamplingThe bootstrap is a resampli
Iowa State - STAT - 511
Stat 511Homework 1 - correctedSpring 2011Due: 11am, Wednesday Jan 19 (because no class Monday Jan 17) Please review Ken Koehlers notes on vectors and matrices, available at http:/www.public.iastate.edu/kkoehler/stat511/sect2.4page.pdf You may skip/skim
Iowa State - STAT - 511
Stat 511Homework 1 solutions( Total pts: 20) 0 1 -1 5 , and C = -1 1 . 2 -1 2 1 Spring 2011Let A =1 2 3 ,B= 0 1 01. C + A = 2. BA =1 1 5 . 1 2 1-1 3 -3 . 2 3 63. AB is not well defined. 4. tr(B)= -1+ -1=-2. 5. BAC = -9 -1 9 11 2 3 6 CBA = 3 0 9 AC
Iowa State - STAT - 511
Stat 511 Due: 11am, Monday, 24 Jan 2011Homework 2 - correctedSpring 20111. Consider a factor effects model for a study with a balanced two-way factorial treatment design: Yijk = + i + j + ij + ijk , for i = 1, 2, j = 1, 2, 3, and k = 1, 2. The &quot;LSMEANS
Iowa State - STAT - 511
Stat 511Homework 2 Solution (pts:20)Spring 2011In the solution, notice that you do have different choices of A and doesn't affect the estimation. 1. Consider a factor effects model for a study with a balanced two-way factorial treatment design: Yijk =
Iowa State - STAT - 511
Stat 511 Due: 11am, Monday Jan 31Homework 3Spring 20111. Consider a factor effects model for a 2-way ANOVA with 2 levels (a and b) of temperature and 2 levels of pressure (A and B). The table of cell means is: Pressure Temp. A B a aA aB b bA bB Please
Iowa State - STAT - 511
Stat 511Homework 3 Solution (pts:20)Spring 20111. Consider a factor effects model for a 2-way ANOVA with 2 levels (a and b) of temperature and 2 levels of pressure (A and B). The table of cell means is: Pressure Temp. A B a aA aB b bA bB Please answer
Iowa State - STAT - 511
Stat 511 Due: 11am, Monday Feb 7Homework 4Spring 2011The week after this we will use the eigen decomposition of a variance-covariance matrix, specifically the concept of an inverse square root matrix. Please look at pages 78 - 107 Ken Koehler's notes.
Iowa State - STAT - 511
Stat 511Homework 4 SolutionSpring 2011 up experiment. They were: 1. Last HW had the vector and X matrix for a very messed 1 2 3 4 5 6 1 1 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0
Iowa State - STAT - 511
Stat 511 Due: 11am, Monday Feb 14Homework 5Spring 20111. This problem was set to reinforce a point made in lecture about power of various tests in an ANOVA. Consider a study with 4 treatments in a 2 x 2 factorial. The investigators tell you that 1.2 un
Iowa State - STAT - 511
Stat 511 Due: 11am, Monday Feb 14Homework 5 SolutionSpring 20111. This problem was set to reinforce a point made in lecture about power of various tests in an ANOVA. Consider a study with 4 treatments in a 2 x 2 factorial. The investigators tell you th
Iowa State - STAT - 511
Stat 511 Due: 11am, Monday Feb 21Homework 6Spring 20111. The data in bacteria.txt are from a study of bacterial growth as a function of sugar concentration in the growth medium. This is a completely randomized design with five replicates of four sugar
Iowa State - STAT - 511
Stat 511Homework 6 PMD solutionSpring 2011Note: These are sketchy because I'm short of time and Yun has more important things to worry about. They give you a sense of my answers. I will return HW before the midterm if I get it from Yun.1. The data in
Iowa State - STAT - 511
Stat 511 Due: 11am, Monday Mar 7Homework 7Spring 20111. Hazardous waste sites often contain multiple nasty chemicals. At one site in New Jersey, the state regulators needed to assess the correlation between the concentration of hexavalent Chromium, Cr+
Iowa State - STAT - 511
Stat 511 Due: 11am, Monday Mar 7Homework 7 SolutionSpring 20111. Hazardous waste sites often contain multiple nasty chemicals. At one site in New Jersey, the state regulators needed to assess the correlation between the concentration of hexavalent Chro
Iowa State - STAT - 511
Stat 511 Due: 11am, Monday Mar 21Homework 8Spring 20111. The data in range.txt come from a field study of plant response to fertilization. There are five treatments (a control and four different fertilization regimes) randomly assigned to plots arrange
Iowa State - STAT - 511
Stat 511Homework 8 solutionSpring 20111. The data in range.txt come from a field study of plant response to fertilization. There are five treatments (a control and four different fertilization regimes) randomly assigned to plots arranged into blocks. T
Iowa State - STAT - 511
Stat 511 Due: 5 pm, Tuesday Mar 29Homework 9Spring 20111. Last homework, I described the New Zealand pasture fertilization study. The data are in ryegrass.txt. This week, we analyze the data. (a) Last week, you derived the E MS for variety. Some of the
Iowa State - STAT - 511
Stat 511Homework 9 SolutionSpring 20111. Last homework, I described the New Zealand pasture fertilization study. The data are in ryegrass.txt. This week, we analyze the data. (a) Last week, you derived the E MS for variety. Some of the other E MS are:
Iowa State - STAT - 511
Stat 511 Due: 5pm, Tuesday Apr 5Homework 10Spring 2011The first two problems present you with a description of a study. Each has two (or more) sizes of experimental unit. Some of the &quot;experimental&quot; units are not randomly assigned to treatments. Please
Iowa State - STAT - 511
Stat 511 Due: 5pm, Tuesday Apr 5Homework 10 solutionsSpring 2011The first two problems present you with a description of a study. Each has two (or more) sizes of experimental unit. Some of the &quot;experimental&quot; units are not randomly assigned to treatment
Iowa State - STAT - 511
Stat 511 Due: 5pm, Tuesday Apr 19Homework 11 - correctedSpring 20111. The data in school.txt come from an educational study. The full study has 3 treatments, but we will only look at data from one treatment. Treatments were randomly assigned to 8th gra
Iowa State - STAT - 511
Stat 511 Due: 5pm, Tuesday Apr 19Homework 11 solutionSpring 20111. The data in school.txt come from an educational study. The full study has 3 treatments, but we will only look at data from one treatment. Treatments were randomly assigned to 8th grade
Iowa State - STAT - 511
Stat 511 Due: 5pm, Tuesday Apr 26Homework 12 SolutionSpring 20111. If y P oisson(), the pmf of y is f (y) =y e- y!Equation (1) on slide 2 of section 26 gives the exponentional-scale family form of a distribution. Write the pmf of y in exponential-sca
Iowa State - STAT - 511
Stat 511Homework 13 - Corrected - FYI only, not to be turned inSpring 2011Due: 5pm, Tuesday Apr 261. The data in elnino.txt are records of sea surface temperature in four regions of the Pacific ocean from 1950 to current. The data labelled NINO1+2 is
Iowa State - STAT - 511
Stat 511 Due: 5pm, Tuesday Apr 26Homework 13 - SolutionsSpring 20111. The data in elnino.txt are records of sea surface temperature in four regions of the Pacific ocean from 1950 to current. The data labelled NINO1+2 is the average sea surface temperat
Iowa State - STAT - 511
Stat 511 1. Echinaceae variance componentsMidterm II - Answers7 April 2011(a) Source df Plant 24 Tissue 2 Plant*Tissue 48 Extract(P*T) 75 Clarify(E*P*T) 150 error 300 Notes: Tissues and plants are crossed. Each combination is a biological sample. Extra
Iowa State - STAT - 511
Stat 511Midterm II7 April 20111. A large group of ISU chemists and biologists is studying a chemical produced by Echinacea purpurea plants. This chemical is one of the main &quot;active&quot; chemicals in Echinacea herbal medicines. Measuring the concentraion of
Iowa State - STAT - 511
Stat 511Midterm I24 February 20111. A psychological study of memory randomly assigned six subjects to one of three combinations of study time (1 minute or 5 minutes) and refresher (present or absent). The three treatments had the following structure: R
Iowa State - STAT - 511
Stat 511 1. (a)Midterm I solutions24 February 2011 X= 1 1 1 1 1 11 1 1 1 0 00 0 0 0 1 11 1 0 0 1 10 0 1 1 0 0 If you only wrote out three rows, you're describing a study with 3 subjects, not this study with 6 subjects. (b) In terms of model para
Iowa State - STAT - 511
46qqqqresid(sdl.lme)2q q q q q q q q q q q q q q q q qqqqq0qqqqqqq-2qqq qq-4q101214 fitted(sdl.lme)1618
Iowa State - STAT - 511
Stat 511 Final ExamMay 4, 2009 Prof. Vardeman(This exam will scored on a 160 point basis.)I have neither given nor received unauthorized assistance on this exam._ Name_ Name Printed11. A marketing study used as an example in Neter et al. concerned
Iowa State - STAT - 511
Stat 511 Final ExamMay 4, 2009 Prof. Vardeman(This exam will scored on a 160 point basis.)I have neither given nor received unauthorized assistance on this exam._ Name_ Name Printed11. A marketing study used as an example in Neter et al. concerned
Iowa State - STAT - 511
Stat 511 HW#6 Spring 2009 This assignment consists of problems on mixed linear models. Most are repeats from HW #10 of 2003 and HW#8 of 2004. All of these problems requiring computing should be done using both the lme()function in the nlme package in R (u
Iowa State - STAT - 511
STAT 511 HW#6 SPRING 2009 PROBLEM 1:y&lt;-c(6.0,6.1,8.6,7.1,6.5,7.4,9.4,9.9,9.5,7.5,6.4,9.1,8.7) A&lt;-c(1,1,1,1,1,1,2,2,2,2,2,2,2) B&lt;-c(1,1,2,2,2,2,1,1,2,2,3,3,3) options(contrasts=c(&quot;contr.sum&quot;,&quot;contr.sum&quot;) a) The pooled variance from the 5 samples (of sizes
Iowa State - STAT - 511
STAT 511 HW#7 SPRING 2009PROBLEM 1:bootstrap&lt;-function(x,nboot,theta) cfw_ data&lt;-matrix(sample(x,size=length(x)*nboot,replace=T),nrow=nboot) return(apply(data,1,theta) library(MASS) compound1&lt;-c(3.03,5.53,5.60,9.30,9.92,12.51,12.95,15.21,16.04,16.84) c
Iowa State - PHYSICS - 198
Physics 198Exam IFall 20111. The slope of a graph of position versus time is equal to what quantity? A) B) C) D) E) Velocity Acceleration Force Mass Kinetic energy x . tSolution: The slope of a graph of position versus time is velocity: v =2. What is