Random Effects

Total Page:16

File Type:pdf, Size:1020Kb

Random Effects Random Effects Lukas Meier, Seminar für Statistik New Philosophy… . Up to now: treatment effects were fixed, unknown parameters that we were trying to estimate. Such models are also called fixed effects models. Now: Consider the situation where treatments are random samples from a large population of potential treatments. Example: Effect of machine operators that were randomly selected from a large pool of operators. In this setup, treatment effects are random variables and therefore called random effects. The corresponding model will be a random effects model. 1 New Philosophy… . Why would we be interested in a random effects situation? . It is a useful way of thinking if we want to make a statement (conclusion) about the population of all treatments. In the operator example we shift the focus away from the individual operators (treatments) to the population of all operators (treatments). Typically, we are interested in the variance of the treatment population. E.g., what is the variation from operator to operator? 2 Examples of Random Effects Randomly select… …from… clinics …all clinics in a country. school classes …all school classes in a region. investigators …a large pool of investigators. series in quality control …all series in a certain time period. … … 3 Carton Experiment One (Oehlert, 2000) . Company with 50 machines that produce cardboard cartons. Ideally, strength of the cartons shouldn’t vary too much. Therefore, we want to have an idea about . “machine-to-machine” variation . “sample-to-sample” variation on the same machine. Perform experiment: . Choose 10 machines at random (out of the 50) . Produce 40 cartons on each machine . Test resulting cartons for strength ( response) 4 Carton Experiment One (Oehlert, 2000) . Model so far: 푌푖푗 = 휇 + 훼푖 + 휖푖푗, where 훼푖 is the (fixed) effect of machine 푖 and 휀푖푗 are the errors with the usual assumptions. However, this model does not reflect the sampling mechanism from above. If we repeat the experiment, the selected machines change and therefore also the meaning of the parameters: they typically correspond to a different machine! . Moreover, we want to learn something about the population of all machines. 5 Carton Experiment One (Oehlert, 2000) . New: Random effects model: Parameter 푌푖푗 = 휇 + 훼푖 + 휖푖푗, Random variable with effect of 2 machine . 훼푖 i. i. d. ∼ 푁 0, 휎훼 2 . 휖푖푗 i. i. d. ∼ 푁 0, 휎 . This looks very similar to the old model, however the 훼푖’s are now random variables! . That small change will have a large impact on the properties of the model and on our way to analyze such kind of data. 6 Carton Experiment One (Oehlert, 2000) . Properties of random effects model: 2 2 . Var 푌푖푗 = 휎훼 + 휎 variance components 0 푖 ≠ 푘 different machines . Cor 푌 , 푌 = 휎2/(휎2 + 휎2) 푖 = 푘, 푗 ≠ 푙 푖푗 푘푙 훼 훼 same machine 1 푖 = 푘, 푗 = 푙 intraclass correlation Reason: Observations from the same machine “share” the same random value 훼푖 and are therefore correlated. Conceptually, we could also put all the correlation structure into the error term and forget about the 훼푖’s, i.e. 푌푖푗 = 휇 + 휖푖푗 where 휖푖푗 has the appropriate correlation structure from above. Sometimes this interpretation is a useful way of thinking. 7 Random vs. Fixed: Overview . Comparison between random and fixed effects models Term Fixed effects model Random effects model fixed, unknown 훼 훼 i. i. d. ∼ 푁(0, 휎2) 푖 constant 푖 훼 Side constraint on 훼푖 needed not needed 퐸[푌푖푗] 휇 + 훼푖 휇, but 퐸 푌푖푗 훼푖 = 휇 + 훼푖 2 2 2 Var(푌푖푗) 휎 휎훼 + 휎 0 푖 ≠ 푘 2 2 2 Corr(푌푖푗, 푌푘푙) = 0 (푗 ≠ 푙) = 휎훼 /(휎훼 + 휎 ) 푖 = 푘, 푗 ≠ 푙 1 푖 = 푘, 푗 = 푙 . A note on the sampling mechanism: . Fixed: Draw new random errors only, everything else is kept constant. Random: Draw new “treatment effects” and new random errors (!) 8 Think of 3 specific Illustration of Correlation Structure machines Fixed case: 3 different fixed treatment levels 훼푖. We (repeatedly) sample 2 observations per treatment level: Think of 2 carton 푌푖푗 = 휇 + 훼푖 + 휖푖푗 samples 훼3 = 3.5 훼2 = 1 훼1 = −4.5 9 Illustration of Correlation Structure Think of 2 carton Random case: samples Whenever we draw 2 observations 푌푖1 and 푌푖2 we first have to draw a new (common) random treatment effect 훼푖. Think of a 8 random machine. 6 4 2 2 i Y 0 -2 -4 -6 -10 -5 0 5 10 10 Yi1 Carton Experiment Two (Oehlert, 2000) . Let us extend the previous experiment. Assume that machine operators also influence the production process. Choose 10 operators at random. Each operator will produce 4 cartons on each machine (hence, operator and machine are crossed factors). All assignments are completely randomized. 11 Carton Experiment Two (Oehlert, 2000) 2 푁 0, 휎2 푁 0, 휎2 2 . Model: 푁 0, 휎훼 훽 훼훽 푁 0, 휎 푌푖푗푘 = 휇 + 훼푖 + 훽푗 + 훼훽 푖푗 + 휖푖푗푘, main effect main effect interaction with machine operator machine×operator . 훼푖, 훽푗, 훼훽 푖푗, 휖푖푗푘 independent and normally distributed. 2 2 2 2 . Var 푌푖푗푘 = 휎훼 + 휎훽 + 휎훼훽 + 휎 (different variance components). Measurements from the same machine and / or operator are again correlated. The more random effects two observations share, the larger the correlation. It is given by sum of shared variance components sum of all variance components . E.g., correlation between two (different) observations from the same operator on different machines is given by 2 휎훽 2 2 2 2 휎훼 + 휎 + 휎 + 휎 훽 훼훽 12 Carton Experiment Two (Oehlert, 2000) . Hierarchy is typically less problematic in random effects models. 1) What part of the variation is due to general machine-to-machine 2 variation? 휎훼 2) What part of the variation is due to operator-specific machine 2 variation? 휎훼훽 Could ask question (1) even if interaction is present (question (2)). Extensions to more than two factors straightforward. 13 ANOVA for Random Effects Models (balanced designs) . Sums of squares, degrees of freedom and mean squares are being calculated as if the model would be a fixed effects model (!) . One-way ANOVA (퐴 random, 푛 observations per cell) Source df SS MS E[MS] 2 2 퐴 푔 − 1 … … 휎 + 푛휎훼 Error 푁 − 푔 … … 휎2 . Two-way ANOVA (퐴, 퐵, 퐴퐵 random, 푛 observations per cell) Source df SS MS E[MS] 2 2 2 퐴 푎 − 1 … … 휎 + 푏 ⋅ 푛 ⋅ 휎훼 + 푛 ⋅ 휎훼훽 2 2 2 퐵 푏 − 1 … … 휎 + 푎 ⋅ 푛 ⋅ 휎훽 + 푛 ⋅ 휎훼훽 2 2 퐴퐵 (푎 − 1)(푏 − 1) … … 휎 + 푛 ⋅ 휎훼훽 Error 푎푏(푛 − 1) … … 휎2 14 One-Way ANOVA with Random Effects . We are now formulating our null-hypothesis with respect 2 to the parameter 휎훼 . 2 2 . To test 퐻0: 휎훼 = 0 vs. 퐻퐴: 휎훼 > 0 we use the ratio 푀푆퐴 퐹 = ∼ 퐹푔−1,푁−푔 under 퐻0 푀푆퐸 Exactly as in the fixed effect case! . Why? Under the old and the new 퐻0 both models are the same! 15 Two-Way ANOVA with Random Effects 2 . To test 퐻0: 휎훼 = 0 we need to find a term which has identical 퐸[푀푆] under 퐻0. 푀푆퐴 . Use 푀푆퐴퐵, i.e. 퐹 = ∼ 퐹푎−1, 푎−1 푏−1 under 퐻0. 푀푆퐴퐵 2 . Similarly for the test 퐻0: 휎훽 = 0. The interaction will be tested against the error, i.e. use 푀푆퐴퐵 퐹 = ∼ 퐹 푎−1 푏−1 , 푎푏 푛−1 푀푆퐸 2 under 퐻0: 휎훼훽 = 0. In the fixed effect case we would test all effects against the error term (i.e., use 푀푆퐸 instead of 푀푆퐴퐵 to build 퐹-ratio)! 16 Didn’t look at this Two-Way ANOVA with Random Effects column when analyzing factorials . Reason: ANOVA table for fixed effects: Shorthand notation for a Source df E[MS] term depending on 훼′푠 퐴 푎 − 1 휎2 + 푏 ⋅ 푛 ⋅ 푄 훼 푖 퐵 푏 − 1 휎2 + 푎 ⋅ 푛 ⋅ 푄(훽) 퐴퐵 (푎 − 1)(푏 − 1) 휎2 + 푛 ⋅ 푄(훼훽) Error 푎푏(푛 − 1) 휎2 . E.g, 푆푆퐴 (푀푆퐴) is being calculated based on column-wise means. In the fixed effects model, the expected mean squares do not “contain” any other component. 17 Two-Way ANOVA with Random Effects . In a random effects model, a column-wise mean is “contaminated” with the average of the corresponding interaction terms. In a fixed effects model, the sum (or mean) of these interaction terms is zero by definition. In the random effects model, this is only true for the expected value, but not for an individual realization! . Hence, we need to check whether the variation from “column to column” is larger than term based on error and interaction term. 18 Point Estimates of Variance Components . We do not only want to test the variance components, we also want to have estimates of them. 2 2 2 2 . I.e., we want to determine 휎 훼 , 휎 훽 , 휎 훼훽, 휎 etc. Easiest approach: ANOVA estimates of variance components. Use columns “MS” and “E[MS]” in ANOVA table, solve the corresponding equations from bottom to top. Example: One-way ANOVA 2 . 휎 = 푀푆퐸 2 푀푆퐴−푀푆퐸 . 휎 = 훼 푛 19 Point Estimates of Variance Components . Advantage: Can be done using standard ANOVA functions (i.e., no special software needed). Disadvantages: . Estimates can be negative (in previous example if 푀푆퐴 < 푀푆퐸). Set them to zero in such cases. Not always as easy as here. This is like a method of moments estimator. More modern and much more flexible: restricted maximum-likelihood estimator (REML). 20 Point Estimates of Variance Components: REML . Think of a modification of maximum likelihood estimation that removes bias in estimation of variance components. Theory complicated (still ongoing research). Software implementation in R-package lme4 (or lmerTest) .
Recommended publications
  • Linear Mixed-Effects Modeling in SPSS: an Introduction to the MIXED Procedure
    Technical report Linear Mixed-Effects Modeling in SPSS: An Introduction to the MIXED Procedure Table of contents Introduction. 1 Data preparation for MIXED . 1 Fitting fixed-effects models . 4 Fitting simple mixed-effects models . 7 Fitting mixed-effects models . 13 Multilevel analysis . 16 Custom hypothesis tests . 18 Covariance structure selection. 19 Random coefficient models . 20 Estimated marginal means. 25 References . 28 About SPSS Inc. 28 SPSS is a registered trademark and the other SPSS products named are trademarks of SPSS Inc. All other names are trademarks of their respective owners. © 2005 SPSS Inc. All rights reserved. LMEMWP-0305 Introduction The linear mixed-effects models (MIXED) procedure in SPSS enables you to fit linear mixed-effects models to data sampled from normal distributions. Recent texts, such as those by McCulloch and Searle (2000) and Verbeke and Molenberghs (2000), comprehensively review mixed-effects models. The MIXED procedure fits models more general than those of the general linear model (GLM) procedure and it encompasses all models in the variance components (VARCOMP) procedure. This report illustrates the types of models that MIXED handles. We begin with an explanation of simple models that can be fitted using GLM and VARCOMP, to show how they are translated into MIXED. We then proceed to fit models that are unique to MIXED. The major capabilities that differentiate MIXED from GLM are that MIXED handles correlated data and unequal variances. Correlated data are very common in such situations as repeated measurements of survey respondents or experimental subjects. MIXED extends repeated measures models in GLM to allow an unequal number of repetitions.
    [Show full text]
  • Application of Random-Effects Probit Regression Models
    Journal of Consulting and Clinical Psychology Copyright 1994 by the American Psychological Association, Inc. 1994, Vol. 62, No. 2, 285-296 0022-006X/94/S3.00 Application of Random-Effects Probit Regression Models Robert D. Gibbons and Donald Hedeker A random-effects probit model is developed for the case in which the outcome of interest is a series of correlated binary responses. These responses can be obtained as the product of a longitudinal response process where an individual is repeatedly classified on a binary outcome variable (e.g., sick or well on occasion t), or in "multilevel" or "clustered" problems in which individuals within groups (e.g., firms, classes, families, or clinics) are considered to share characteristics that produce similar responses. Both examples produce potentially correlated binary responses and modeling these per- son- or cluster-specific effects is required. The general model permits analysis at both the level of the individual and cluster and at the level at which experimental manipulations are applied (e.g., treat- ment group). The model provides maximum likelihood estimates for time-varying and time-invari- ant covariates in the longitudinal case and covariates which vary at the level of the individual and at the cluster level for multilevel problems. A similar number of individuals within clusters or number of measurement occasions within individuals is not required. Empirical Bayesian estimates of per- son-specific trends or cluster-specific effects are provided. Models are illustrated with data from
    [Show full text]
  • 10.1 Topic 10. ANOVA Models for Random and Mixed Effects References
    10.1 Topic 10. ANOVA models for random and mixed effects References: ST&DT: Topic 7.5 p.152-153, Topic 9.9 p. 225-227, Topic 15.5 379-384. There is a good discussion in SAS System for Linear Models, 3rd ed. pages 191-198. Note: do not use the rules for expected MS on ST&D page 381. We provide in the notes below updated rules from Chapter 8 from Montgomery D.C. (1991) “Design and analysis of experiments”. 10. 1. Introduction The experiments discussed in previous chapters have dealt primarily with situations in which the experimenter is concerned with making comparisons among specific factor levels. There are other types of experiments, however, in which one wish to determine the sources of variation in a system rather than make particular comparisons. One may also wish to extend the conclusions beyond the set of specific factor levels included in the experiment. The purpose of this chapter is to introduce analysis of variance models that are appropriate to these different kinds of experimental objectives. 10. 2. Fixed and Random Models in One-way Classification Experiments 10. 2. 1. Fixed-effects model A typical fixed-effect analysis of an experiment involves comparing treatment means to one another in an attempt to detect differences. For example, four different forms of nitrogen fertilizer are compared in a CRD with five replications. The analysis of variance takes the following familiar form: Source df Total 19 Treatment (i.e. among groups) 3 Error (i.e. within groups) 16 And the linear model for this experiment is: Yij = µ + i + ij.
    [Show full text]
  • Chapter 17 Random Effects and Mixed Effects Models
    Chapter 17 Random Effects and Mixed Effects Models Example. Solutions of alcohol are used for calibrating Breathalyzers. The following data show the alcohol concentrations of samples of alcohol solutions taken from six bottles of alcohol solution randomly selected from a large batch. The objective is to determine if all bottles in the batch have the same alcohol concentrations. Bottle Concentration 1 1.4357 1.4348 1.4336 1.4309 2 1.4244 1.4232 1.4213 1.4256 3 1.4153 1.4137 1.4176 1.4164 4 1.4331 1.4325 1.4312 1.4297 5 1.4252 1.4261 1.4293 1.4272 6 1.4179 1.4217 1.4191 1.4204 We are not interested in any difference between the six bottles used in the experiment. We therefore treat the six bottles as a random sample from the population and use a random effects model. If we were interested in the six bottles, we would use a fixed effects model. 17.3. One Random Factor 17.3.1. The Random‐Effects One‐way Model For a completely randomized design, with v treatments of a factor T, the one‐way random effects model is , :0, , :0, , all random variables are independent, i=1, ..., v, t=1, ..., . Note now , and are called variance components. Our interest is to investigate the variation of treatment effects. In a fixed effect model, we can test whether treatment effects are equal. In a random effects model, instead of testing : ..., against : treatment effects not all equal, we use : 0,: 0. If =0, then all random effects are 0.
    [Show full text]
  • Menbreg — Multilevel Mixed-Effects Negative Binomial Regression
    Title stata.com menbreg — Multilevel mixed-effects negative binomial regression Syntax Menu Description Options Remarks and examples Stored results Methods and formulas References Also see Syntax menbreg depvar fe equation || re equation || re equation ::: , options where the syntax of fe equation is indepvars if in , fe options and the syntax of re equation is one of the following: for random coefficients and intercepts levelvar: varlist , re options for random effects among the values of a factor variable levelvar: R.varname levelvar is a variable identifying the group structure for the random effects at that level or is all representing one group comprising all observations. fe options Description Model noconstant suppress the constant term from the fixed-effects equation exposure(varnamee) include ln(varnamee) in model with coefficient constrained to 1 offset(varnameo) include varnameo in model with coefficient constrained to 1 re options Description Model covariance(vartype) variance–covariance structure of the random effects noconstant suppress constant term from the random-effects equation 1 2 menbreg — Multilevel mixed-effects negative binomial regression options Description Model dispersion(dispersion) parameterization of the conditional overdispersion; dispersion may be mean (default) or constant constraints(constraints) apply specified linear constraints collinear keep collinear variables SE/Robust vce(vcetype) vcetype may be oim, robust, or cluster clustvar Reporting level(#) set confidence level; default is level(95)
    [Show full text]
  • A Simple, Linear, Mixed-Effects Model
    Chapter 1 A Simple, Linear, Mixed-effects Model In this book we describe the theory behind a type of statistical model called mixed-effects models and the practice of fitting and analyzing such models using the lme4 package for R. These models are used in many different dis- ciplines. Because the descriptions of the models can vary markedly between disciplines, we begin by describing what mixed-effects models are and by ex- ploring a very simple example of one type of mixed model, the linear mixed model. This simple example allows us to illustrate the use of the lmer function in the lme4 package for fitting such models and for analyzing the fitted model. We also describe methods of assessing the precision of the parameter estimates and of visualizing the conditional distribution of the random effects, given the observed data. 1.1 Mixed-effects Models Mixed-effects models, like many other types of statistical models, describe a relationship between a response variable and some of the covariates that have been measured or observed along with the response. In mixed-effects models at least one of the covariates is a categorical covariate representing experimental or observational “units” in the data set. In the example from the chemical industry that is given in this chapter, the observational unit is the batch of an intermediate product used in production of a dye. In medical and social sciences the observational units are often the human or animal subjects in the study. In agriculture the experimental units may be the plots of land or the specific plants being studied.
    [Show full text]
  • BU-1120-M RANDOM EFFECTS MODELS for BINARY DATA APPLIED to ENVIRONMENTAL/ECOLOGICAL STUDIES by Charles E. Mcculloch Biometrics U
    RANDOM EFFECTS MODELS FOR BINARY DATA APPLIED TO ENVIRONMENTAL/ECOLOGICAL STUDIES by Charles E. McCulloch Biometrics Unit and Statistics Center Cornell University BU-1120-M April1991 ABSTRACT The outcome measure in many environmental and ecological studies is binary, which complicates the use of random effects. The lack of methods for such data is due in a large part to both the difficulty of specifying realistic models, and once specified, to their computational in tractibility. In this paper we consider a number of examples and review some of the approaches that have been proposed to deal with random effects in binary data. We consider models for random effects in binary data and analysis and computation strategies. We focus in particular on probit-normal and logit-normal models and on a data set quantifying the reproductive success in aphids. -2- Random Effects Models for Binary Data Applied to Environmental/Ecological Studies by Charles E. McCulloch 1. INTRODUCTION The outcome measure in many environmental and ecological studies is binary, which complicates the use of random effects since such models are much less well developed than for continuous data. The lack of methods for such data is due in a large part to both the difficulty of specifying realistic models, and once specified, to their computational intractibility. In this paper we consider a number of examples and review some of the approaches that have been proposed to deal with random effects in binary data. The advantages and disadvantages of each are discussed. We focus in particular on a data set quantifying the reproductive success in aphids.
    [Show full text]
  • Lecture 34 Fixed Vs Random Effects
    Lecture 34 Fixed vs Random Effects STAT 512 Spring 2011 Background Reading KNNL: Chapter 25 34-1 Topic Overview • Random vs. Fixed Effects • Using Expected Mean Squares (EMS) to obtain appropriate tests in a Random or Mixed Effects Model 34-2 Fixed vs. Random Effects • So far we have considered only fixed effect models in which the levels of each factor were fixed in advance of the experiment and we were interested in differences in response among those specific levels . • A random effects model considers factors for which the factor levels are meant to be representative of a general population of possible levels. 34-3 Fixed vs. Random Effects (2) • For a random effect, we are interested in whether that factor has a significant effect in explaining the response, but only in a general way. • If we have both fixed and random effects, we call it a “mixed effects model”. • To include random effects in SAS, either use the MIXED procedure, or use the GLM procedure with a RANDOM statement. 34-4 Fixed vs. Random Effects (2) • In some situations it is clear from the experiment whether an effect is fixed or random. However there are also situations in which calling an effect fixed or random depends on your point of view, and on your interpretation and understanding. So sometimes it is a personal choice. This should become more clear with some examples. 34-5 Random Effects Model • This model is also called ANOVA II (or variance components model). • Here is the one-way model: Yij=µ + α i + ε ij αN σ 2 i~() 0, A independent ε~N 0, σ 2 ij () 2 2 Yij~ N (µ , σ A + σ ) 34-6 Random Effects Model (2) Now the cell means µi= µ + α i are random variables with a common mean.
    [Show full text]
  • Panel Data 4: Fixed Effects Vs Random Effects Models Richard Williams, University of Notre Dame, Last Revised March 20, 2018
    Panel Data 4: Fixed Effects vs Random Effects Models Richard Williams, University of Notre Dame, https://www3.nd.edu/~rwilliam/ Last revised March 20, 2018 These notes borrow very heavily, sometimes verbatim, from Paul Allison’s book, Fixed Effects Regression Models for Categorical Data. The Stata XT manual is also a good reference. This handout tends to make lots of assertions; Allison’s book does a much better job of explaining why those assertions are true and what the technical details behind the models are. Overview. With panel/cross sectional time series data, the most commonly estimated models are probably fixed effects and random effects models. Population-Averaged Models and Mixed Effects models are also sometime used. In this handout we will focus on the major differences between fixed effects and random effects models. Several considerations will affect the choice between a fixed effects and a random effects model. 1. What is the nature of the variables that have been omitted from the model? a. If you think there are no omitted variables – or if you believe that the omitted variables are uncorrelated with the explanatory variables that are in the model – then a random effects model is probably best. It will produce unbiased estimates of the coefficients, use all the data available, and produce the smallest standard errors. More likely, however, is that omitted variables will produce at least some bias in the estimates. b. If there are omitted variables, and these variables are correlated with the variables in the model, then fixed effects models may provide a means for controlling for omitted variable bias.
    [Show full text]
  • Fixed Vs. Random Effects
    Statistics 203: Introduction to Regression and Analysis of Variance Fixed vs. Random Effects Jonathan Taylor - p. 1/19 Today’s class ● Today’s class ■ Random effects. ● Two-way ANOVA ● Random vs. fixed effects ■ ● When to use random effects? One-way random effects ANOVA. ● Example: sodium content in ■ beer Two-way mixed & random effects ANOVA. ● One-way random effects model ■ ● Implications for model Sattherwaite’s procedure. ● One-way random ANOVA table ● Inference for µ· ● 2 Estimating σµ ● Example: productivity study ● Two-way random effects model ● ANOVA tables: Two-way (random) ● Mixed effects model ● Two-way mixed effects model ● ANOVA tables: Two-way (mixed) ● Confidence intervals for variances ● Sattherwaite’s procedure - p. 2/19 Two-way ANOVA ● Today’s class ■ Second generalization: more than one grouping variable. ● Two-way ANOVA ● Random vs. fixed effects ■ ● When to use random effects? Two-way ANOVA model: observations: ● Example: sodium content in ≤ ≤ ≤ ≤ ≤ ≤ beer (Yijk); 1 i r; 1 j m; 1 k nij : r groups in first ● One-way random effects model grouping variable, m groups ins second and nij samples in ● Implications for model ● One-way random ANOVA (i; j)-“cell”: table ● Inference for µ· 2 ● 2 Estimating σµ Yijk = µ + αi + βj + (αβ)ij + "ijk; "ijk ∼ N(0; σ ): ● Example: productivity study ● Two-way random effects ■ model Constraints: ● ANOVA tables: Two-way ◆ r (random) αi = 0 ● i=1 Mixed effects model ◆ Pm ● Two-way mixed effects model j=1 βj = 0 ● ANOVA tables: Two-way Pm (mixed) ◆ (αβ) = 0; 1 ≤ i ≤ r ● Confidence intervals for j=1 ij variances ◆ Pr ● Sattherwaite’s procedure (αβ)ij = 0; 1 ≤ j ≤ m: Pi=1 - p.
    [Show full text]
  • 2.11 the Random Effects Model • So Far We Have Assumed the Factor Levels Were Fixed. That Is, the Factor Levels Were Set at Fi
    2.11 The Random Effects Model • So far we have assumed the factor levels were fixed. That is, the factor levels were set at fixed levels by the experimenter. • In many one-factor CRDs, the a factor levels are randomly selected from a population of levels. For this type of experiment, the factor is called a random factor, and the associated effects are called random effects. • In theory, for random effects, we assume the population is infinite. In practice, it is acceptable if the number of randomly selected factor levels (a) is small relative to the number of levels in the population (N). In general, we want a=N < :10 (or, < 10%). • The random effects model for a one-factor CRD is: yij = µ + τi + ij (3) where both τi and ij are random variables. The model assumptions are 2 2 { The ij's are IID N(0; σ ) and the τi's are IID N(0; στ ). { τi and ij are independent for all i; j. Note: IID means `independent and identically distributed'. 2 2 • τi is a random variable with Var(τi) = στ . στ is the variance associated with the distribution or 2 population of all τi's. We assume τi is independent of the random error ij which has variance σ . 2 2 • The variances στ and σ are called variance components. 2 • The hypotheses of interest involve the variance component στ : and 2 • If στ = 0, then the random τi effects are identically 0. In this case, the variability of the τbi estimates (i = 1; 2; : : : ; a) should be close to 0 relative to the MSE.
    [Show full text]
  • One-Way Random Effects Model (Consider Balanced Case First) in the One-Way Layout There Are a Groups of Observations, Where Each
    One-way Random Effects Model (Consider Balanced Case First) In the one-way layout there are a groups of observations, where each group corresponds to a different treatment and contains experimental units that were randomized to that treatment. For this situation, the one-way fixed effect anova model allows for distinct group or treatment effects and esti- mates a population treatment means, which are the quantities of primary interest. Recall that for such a situation, the one-way (fixed effect) anova model takes the form: P ∗ yij = µ + αi + eij; where i αi = 0. ( ) However, a one-way layout is not the only situation where we may en- counter grouped data where we want to allow for distinct group effects. Sometimes the groups correspond to levels of a blocking factor, rather than a treatment factor. In such situations, it is appropriate to use the random effects version of the one-way anova model. Such a model is sim- ilar to (*), but there are important differences in the model assumptions, the interpretation of the fitted model, and the framework and scope of inference. Example | Calcium Measurement in Turnip Greens: (Snedecor & Cochran, 1989, section 13.3) Suppose we want to obtain a precise measurement of calcium con- centration in turnip greens. A single calcium measurement nor even a single turnip leaf is considered sufficient to estimate the popula- tion mean calcium concentration, so 4 measurements from each of 4 leaves were obtained. The resulting data are given below. Leaf Calcium Concentration Sum Mean 1 3.28 3.09 3.03 3.03 12.43 3.11 2 3.52 3.48 3.38 3.38 13.76 3.44 3 2.88 2.80 2.81 2.76 11.25 2.81 4 3.34 3.38 3.23 3.26 13.21 3.30 • Here we have grouped data, but the groups do not correspond to treatments.
    [Show full text]