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### 13-AnalysisOfVariance

Course: STAT 511, Spring 2011
School: Iowa State
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Word Count: 1705

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Of ANalysis VAriance (ANOVA) for a sequence of models Model comparison can be generalized to a sequence of models (not just one full and one reduced model) Context: usual nGM model: y = X + , Let X1 = 1 and Xm = X. But now, we have a sequence of models &quot;in between&quot; 1 and X Suppose X2 , . . . , Xm-1 are design matrices satisfying C(X1 ) &lt; C(X2 ) &lt; . . . &lt; C(Xm-1 ) &lt;...

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Iowa State - STAT - 511
THE AITKEN MODELAnalysis of averagesExamples - 1y = X + , (0, 2 V)Identical to the Gauss-Markov Linear Model except that Var ( ) = 2 V instead of 2 I.V is assumed to be a known nonsingular Variance matrix.The Normal Theory Aitken Model adds an assu
Iowa State - STAT - 511
THE AITKEN MODELy = X + , (0, 2 V)Identical to the Gauss-Markov Linear Model except that Var ( ) = 2 V instead of 2 I. V is assumed to be a known nonsingular Variance matrix. The Normal Theory Aitken Model adds an assumption of normality: N(0, 2 V) Obs
Iowa State - STAT - 511
the bootstrapProcess optimization Many physical processes can be described (at least approximately) as quadratic functions of input variable(s)ExamplesWe've seen a lot about inference on C in a nGM (or nAitken) modelA huge number of questions can be a
Iowa State - STAT - 511
the bootstrapWe've seen a lot about inference on C in a nGM (or nAitken) model A huge number of questions can be answered by appropriate choice of C key point is that C is a linear function of y What if quantity of interest is not a linear function of y?
Iowa State - STAT - 511
Randomization/permutation testsBootstrapping preserves the fixed effectsRandomization / Permutation testsResampling from a single pool of observations tests HoR : F1 (x) = F2 (x)Notice a subtle point: HoR is slightly more general than Ho:1 = 2H0R is
Iowa State - STAT - 511
Randomization / Permutation testsBootstrapping preserves the fixed effectsDifference of two means: resample Y1i and resample Y2i bootstrap estimates, 1B - 2B , are centered on/near Y1 - Y2 , ^ ^ which estimates 1 - 2 Regression bootstrap: ^ resample ^i
Iowa State - STAT - 511
LINEAR MIXED-EFFECT MODELSSeedling weight in 2 genotype study from Aitken model section. Seedling weight measured on each seedling. Two (potential) sources of variation: among flats and among seedlings within a flat. Yijk = + i + Tij + Tij ijk ijkExamp
Iowa State - STAT - 511
LINEAR MIXED-EFFECT MODELSStudies / data / models seen previously in 511 assumed a single source of &quot;error&quot; variation y = X + . are fixed constants (in the frequentist approach to inference) is the only random effect What if there are multiple sources of
Iowa State - STAT - 511
Experimental Designs and LME'sOne example:LME models provide one way to model correlations among observationsVery useful for experimental designs where there is more than one size of experimental unitOr designs where the observation unit is not the sa
Iowa State - STAT - 511
Experimental Designs and LME'sLME models provide one way to model correlations among observations Very useful for experimental designs where there is more than one size of experimental unit Or designs where the observation unit is not the same as the exp
Iowa State - STAT - 511
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELSThis is the commonly-used model for a CRD with t treatments, n experimental units per treatment, and m observations per experimental unit. We can write the model as y = X + Zu + , where X=[
Iowa State - STAT - 511
THE ANOVA APPROACH TO THE ANALYSIS OF LINEAR MIXED EFFECTS MODELSA model for expt. data with subsampling yijk = + i + uij + eijk , (i = 1, ., t; j = 1, ., n; k = 1, ., m) = (, i , ., t ) , u = (u11 , u12 , ., utn ) , = (e111 , e112 , ., etnm ) , IRt+1 ,
Iowa State - STAT - 511
Two approaches for E MSRCBD with random blocks and multiple obs. per blockijkYijk = + i + j + ij +where i cfw_1, . . . , B, j cfw_1, . . . , T, k cfw_1, . . . , N.with ANOVA table:Expected Mean Squares from two different sources Source 1: Searle (19
Iowa State - STAT - 511
Two approaches for E MSRCBD with random blocks and multiple obs. per block Yijk = + i + j + ij +ijkwhere i cfw_1, . . . , B, j cfw_1, . . . , T, k cfw_1, . . . , N. with ANOVA table: Source Blocks Treatments BlockTrt Error C. total df B-1 T-1 (B-1)(T-1
Iowa State - STAT - 511
ANOVA ANALYSIS OF A BALANCED SPLIT-PLOT EXPERIMENTFieldBlock 1 0 100 150 50 150 100 50 100 150 0 50Plot Genotype B0Genotype CGenotype AExample: the corn genotype and fertilization response studyBlock 2 150 100 Block 3 100 50 0 0 150 50 0 0Main pl
Iowa State - STAT - 511
ANOVA ANALYSIS OF A BALANCED SPLIT-PLOT EXPERIMENTExample: the corn genotype and fertilization response study Main plots: genotypes, in blocks Split plots: fertilization 2 way factorial treatment structure split plot variability nested in main plot varia
Iowa State - STAT - 511
IDENTIFYING AN APPROPRIATE MODELGiven a description of a study, how do you construct an appropriate model?Context: more than one size of e.u.A made-up example, intended to be complicated (but far from being the most complicated I've seen)A study of th
Iowa State - STAT - 511
IDENTIFYING AN APPROPRIATE MODELGiven a description of a study, how do you construct an appropriate model? Context: more than one size of e.u. A made-up example, intended to be complicated (but far from being the most complicated I've seen) A study of th
Iowa State - STAT - 511
MAXIMUM LIKELIHOOD and REML ESTIMATION IN THE GENERAL LINEAR MODELGiven a value of the parameter vector , f (w|) is a real-valued function of w.Suppose f (w|) is the probability density function (pdf ) or probability mass function (pmf ) of a random vec
Iowa State - STAT - 511
MAXIMUM LIKELIHOOD and REML ESTIMATION IN THE GENERAL LINEAR MODELc 2011 Dept. Statistics (Iowa State University)Stat 511 section 211 / 23Suppose f (w|) is the probability density function (pdf ) or probability mass function (pmf ) of a random vector
Iowa State - STAT - 511
Prediction of random variablesKey distinction between fixed and random effects:Estimate means of fixed effects Estimate variance of random effectsBut in some instances, want to predict FUTURE values of a random effectExample (from Efron and Morris, 19
Iowa State - STAT - 511
Prediction of random variablesKey distinction between fixed and random effects:Estimate means of fixed effects Estimate variance of random effectsBut in some instances, want to predict FUTURE values of a random effect Example (from Efron and Morris, 19
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