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EXPERIMENTAL DESIGN AND COMPUTATIONS MIXED MODEL Presented by Arun N
Mixed model <ul><li>Mixed models were developed first to deal with correlated (Gaussian) model. </li></ul><ul><li>Applicat...
What is Mixed Model <ul><li>Mixed models contain both fixed and random effect. </li></ul><ul><li>Fixed effects: </li></ul>...
Examples of fixed and random effect <ul><li>Fixed effect: </li></ul><ul><ul><li>Both male and female genders are included ...
The Mixed model <ul><li>It uses long data format, which include both random and fixed effect. </li></ul><ul><li>It can be ...
The Mixed model  (cont’d) <ul><li>Y = X  β  + Z  γ  +  ε </li></ul><ul><ul><li>Here, </li></ul></ul><ul><ul><ul><li>β = Fi...
Mixed model theory <ul><li>u  and  e  are uncorrelated random variables with 0 mean and covariance, G and R respectively. ...
Procedure for fitting the mixed model <ul><li>One can use the LR test or the lesser of the information criteria.  The smal...
LR Test <ul><li>To test whether one model is significantly better than the other. </li></ul><ul><li>To test random effect ...
Applying the LR test <ul><li>We obtain the -2LL from the unrestricted model.  </li></ul><ul><li>We obtain the -2LL from th...
Advantage of Mixed model <ul><li>It can allow random effects to be properly specified and computed,  unlike the GLM. </li>...
GLM vs Mixed Means  – Average,  LS  – Least square means,  SS  - Sum of squares for ANOVA effect,  OLS  - Ordinary least s...
References <ul><li>Thank you   </li></ul><ul><li>Dallas E. Johnson, Kansas State University, Manhattan, KS. An Introducti...
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Mixed models

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  1. 1. EXPERIMENTAL DESIGN AND COMPUTATIONS MIXED MODEL Presented by Arun N
  2. 2. Mixed model <ul><li>Mixed models were developed first to deal with correlated (Gaussian) model. </li></ul><ul><li>Application of Mixed model has become an attractive tools to evaluate plants in actual breeding programs of breeding organization. </li></ul><ul><li>It is important to properly identify which variables should be modeled as random and which as fixed. </li></ul>
  3. 3. What is Mixed Model <ul><li>Mixed models contain both fixed and random effect. </li></ul><ul><li>Fixed effects: </li></ul><ul><ul><li>Factors for which the only level under consideration are continued in the coding of those effects. </li></ul></ul><ul><li>Random effects: </li></ul><ul><ul><li>Factors for which the level contained in the coding of those factors are a random sample of the total number of levels in the population for that factor. </li></ul></ul>
  4. 4. Examples of fixed and random effect <ul><li>Fixed effect: </li></ul><ul><ul><li>Both male and female genders are included in the factor sex. </li></ul></ul><ul><ul><li>Adult and minor are both included in the factor age group. </li></ul></ul><ul><li>Random effect: </li></ul><ul><ul><li>Sample is a random sample of the target population. </li></ul></ul><ul><ul><li>Repeated measures of heart rate variability in an elderly panel. </li></ul></ul>
  5. 5. The Mixed model <ul><li>It uses long data format, which include both random and fixed effect. </li></ul><ul><li>It can be used to model merely fixed or random effects, by zeroing out the other parameters. </li></ul><ul><li>The F tests for Fixed, Random and Mixed model are different. </li></ul><ul><ul><li>Mixed model has the parameter vector for both of these and can estimate the error covariance matrix for each, it can provide the correct standard errors. </li></ul></ul>
  6. 6. The Mixed model (cont’d) <ul><li>Y = X β + Z γ + ε </li></ul><ul><ul><li>Here, </li></ul></ul><ul><ul><ul><li>β = Fixed effect parameter estimates, </li></ul></ul></ul><ul><ul><ul><li>X = Fixed effect, </li></ul></ul></ul><ul><ul><ul><li>Z = Random effect, </li></ul></ul></ul><ul><ul><ul><li>γ = Random effect parameter estimates, </li></ul></ul></ul><ul><ul><ul><li>ε = Error </li></ul></ul></ul><ul><ul><li>Variance of Y = V = ZGZ’ + r </li></ul></ul><ul><ul><ul><li>G and R requires covariance structure fitting. </li></ul></ul></ul>
  7. 7. Mixed model theory <ul><li>u and e are uncorrelated random variables with 0 mean and covariance, G and R respectively. </li></ul><ul><ul><li>Because the covariance matrix V=ZGZ’ + R </li></ul></ul><ul><ul><li>V is generalized inverse, because V is usually singular & non invertible; AVA = A is an augmented matrix that is invertible. </li></ul></ul><ul><ul><li>It can later be transformed back to V. </li></ul></ul><ul><ul><li>The G and R matrices must be positive definite. </li></ul></ul><ul><ul><li>In the Mixed model, the covariance type of the random (generalized) effects defines the structure of G and a repeated covariance type defines structure of R. </li></ul></ul>
  8. 8. Procedure for fitting the mixed model <ul><li>One can use the LR test or the lesser of the information criteria. The smaller the information criterion, the better the model happens to be. </li></ul><ul><li>We try to go from a larger to a smaller information criterion when we fit the model. </li></ul>
  9. 9. LR Test <ul><li>To test whether one model is significantly better than the other. </li></ul><ul><li>To test random effect for statistical significance. </li></ul><ul><li>To test covariance structure improvement. </li></ul>LR Test – Likelihood ratio
  10. 10. Applying the LR test <ul><li>We obtain the -2LL from the unrestricted model. </li></ul><ul><li>We obtain the -2LL from the restricted model. </li></ul><ul><li>We subtract the latter from the larger former. </li></ul><ul><li>That is a chi-square with df= the difference in the number of parameters. </li></ul><ul><li>We can look this up and determine whether or not it is statistically significant. </li></ul>LL – Log Lambda, df – Degree of freedom
  11. 11. Advantage of Mixed model <ul><li>It can allow random effects to be properly specified and computed, unlike the GLM. </li></ul><ul><li>It can allow correlation of errors, unlike the GLM. It therefore has more flexibility in modeling the error covariance structure. </li></ul><ul><li>It can allow the error terms to exhibit non-constant variability, unlike the GLM, allowing more flexibility in modeling the dependent variable and It can handle missing data. </li></ul>
  12. 12. GLM vs Mixed Means – Average, LS – Least square means, SS - Sum of squares for ANOVA effect, OLS - Ordinary least square, WLS - Weighted least square, F Test – More often used when comparing statistical models GLM Mixed <ul><li>MEANS Statement </li></ul><ul><li>SS type 1,2,3,4 </li></ul><ul><li>Estimate using OLS (or) WLS </li></ul><ul><li>One has to program the correct F test for random effect. </li></ul><ul><li>Losses causes with missing data. </li></ul><ul><li>LSMEANS statement </li></ul><ul><li>SS type 1 and 3 </li></ul><ul><li>Estimated using maximum likelihood, general methods of moments. </li></ul><ul><li>Gives correct standard errors and confidence interval for random effect. </li></ul><ul><li>Can handle missing values. </li></ul>
  13. 13. References <ul><li>Thank you  </li></ul><ul><li>Dallas E. Johnson, Kansas State University, Manhattan, KS. An Introduction to the Analysis of Mixed Models, Statistics and Data Analysis. 28:253 </li></ul><ul><li>Raudenbush, S. W. & A. S. Bryk. 2002. Hierarchical Linear Models: Applications and Data Analysis Methods. 2nd ed. Thousand Oaks CA: Sage. </li></ul><ul><li>Hedeker, D. (1999). MIXNO: a computer program for mixed-effects nominal logistic regression, Journal of Statistical Software 4(5), 1–92. </li></ul><ul><li>Brenton R. Clarke (2008), Linear Models : The theory and application of analysis of variance. Wiley Series in probability and statistics. </li></ul>
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