Linear Mixed Models in JMP Pro

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Presentation transcript:

Linear Mixed Models in JMP Pro Jian Cao March 10, 2016 Salt Lake City

It depends Why Mixed Models In general when there exists correlation among responses or an important explanatory variable is missing, failure to account for that leads to biased estimates of the effects of treatment and other factors! Copyright © 2010, SAS Institute Inc. All rights reserved.

Applications of Mixed Models Allow intercept and slope to vary randomly across subjects (Random Coefficients Models) Analysis of repeated measures and correlated response data where within-subject errors are correlated. Randomized block designs. Cross-over designs. Or split-plot designs where hard-to-change and easy-to-change factors result in multiple error terms Control for unobserved individual heterogeneity as a random effect (Panel Data Models) Subjects are hierarchical or multi-level (e.g., students within schools) (HLM or MLM) Spatial Regression—modeling spatially correlated data …… Copyright © 2010, SAS Institute Inc. All rights reserved.

Linear mixed models are generalization of multiple regression models 𝜷 --fixed-effect parameters, X – Fixed effects Fit to a cross sectional random sample Linear Mixed Model is generalization of the standard regression model: 𝒚=𝑿𝜷+𝒁𝜸+ε 𝜸 -- Random effect parameters, Z – Random effects ε -- Random errors 𝐶𝑜𝑣 𝜸,𝜺 = 𝟎 (i.e., no correlation between 𝜸 and ε ) Additional data needed on each individual in order to estimate 𝜸 and covariance structures Copyright © 2010, SAS Institute Inc. All rights reserved.

Assumptions 𝜸~𝑵 𝟎,𝑮 𝜺~𝑵 𝟎,𝑹 𝐂𝐨𝐯 𝜸,𝜺 = 𝟎 𝜸 and 𝜺 are assumed normally distributed 𝜸~𝑵 𝟎,𝑮 𝜺~𝑵 𝟎,𝑹 𝐂𝐨𝐯 𝜸,𝜺 = 𝟎 G and R are the variance-covariance matrices for 𝜸 and 𝜺, respectively. No co-variances between 𝜸 and 𝜺 In JMP, an unstructured covariance structure is assumed for G, and several common structures are available for R. Maximizes the marginal log likelihood (REML) of the data 𝒀 ~ 𝑵(𝑿𝜷,𝑽) where V=𝒁𝑮 𝒁 ′ +𝑹 to jointly estimate both the fixed-effect parameters, 𝜷, and co-variances, 𝑽. REML performs maximum likelihood estimation of a restricted likelihood function that does not depend on the fixed-effect parameters. This yields estimates of the variance components that are then used to obtain estimates of the fixed effects. Estimates of precision are based on estimates of the covariance matrix for the parameters. (less biased) Even when the data are unbalanced, REML provides useful estimates, tests, and confidence intervals. Copyright © 2010, SAS Institute Inc. All rights reserved.

Where is Mixed Model in JMP Pro Analyze => Fit Model and choose Mixed Model personality New in JMP Pro 12: (1) Improved algorithm that enables JMP to run much faster for large data than in JMP Pro 11; (2) Variograms for isotropic spatial models: visualize and compare spatial correlation structures. Copyright © 2010, SAS Institute Inc. All rights reserved.