Master Linear Mixed Models

Linear Mixed Model for Autism Study

• The linear mixed model assumes that there are random effects and random errors.

• The random effects are child-specific and assumed to follow a multivariate normal distribution.

• The random errors are associated with social development scores and assumed to follow a multivariate normal distribution.

• The model assumes that the random effect vectors are independent between children and that the random error vectors between children are independent of each other.

• The model is specified with a mean vector of 0 and an unstructured covariance matrix for the random effects and a diagonal covariance matrix for the random errors.

• Different structures for the covariance matrix of the random errors can be compared to see which fits the data appropriately.

• Hypothesis tests for the random effects are done using REML-based likelihood ratio tests.

• Hypothesis tests for the fixed effects are done using ML-based likelihood ratio tests.

• The quadratic effect of age on social development was found to differ between children and was included in the model.

• The interaction effect of age squared and initial language development was not found to be significant and was dropped from the model.

• The interaction effect of age and initial language development was found to be significant and was retained in the model.

• The quadratic effect of age was found to be in doubt and was tested, but the results were inconclusive.Analysis of Language Development in Children: Model Selection and Diagnostics

• The study examines the relationship between language development in children and various demographic factors.

• Model (3) is selected as the final model, which includes quadratic effects of Age and interaction terms with initial language development level.

• The marginal value of Vsaeti is the expected value of Vsaeti at a given age, and is the same for children with the same initial language development level.

• The linear effect of Age on Vsae for children with a Low level of initial language development is represented by β1.

• The model also allows for conditional values of Vsaeti, which are specific to each child and include random linear and quadratic effects of Age.

• Predicted marginal and conditional values of Vsae are calculated using GLS estimates of fixed and random effects.

• Residuals are the difference between observed and predicted values of Vsae.

• Model diagnostics include checking the agreement between predicted and observed values of Vsae, assumptions of the LMM, and examining the distribution of random effects and errors.

• There are outliers associated with child 49, 180, and 124 in the data.

• The constant variance and normality assumption for εij as a function of Age are not severely violated.

• The model is refit without the outliers and produces similar results.

• The study highlights the importance of model selection and diagnostics in LMMs for accurately analyzing data.Analysis of Social Development Trajectories for Autistic Children

• The social development for autistic children improves linearly over time on average.

• The level of language development at age 2 (initial language development) influences the social development trajectory over time for autistic children.

• Children with a High level of initial language development have greater mean social development at age 2 than those with a Low level.

• The mean social development for children with a High level of initial language development improves faster over time than those with a Low level.

• There are varying social development trajectories over time for children at all levels of initial language development.

• Some children show no improvement over time, while others improve linearly or accelerate over time.

• The observations of all 158 children were used to fit model (3).

• The estimate of the variance-covariance matrix, D, of model (3) is ô.

• The estimate of the correlation matrix of the random effect vector, µi, of model (3) is ñ.

• The estimate of the variance-covariance matrix, R, of model (3) is 38.790.

• Estimates of the fixed effects in model (3) are presented in Table 6.

• The estimates of the fixed effects suggest that the mean Vsae score for children with a High level of initial language development accelerates faster over time than those with a Low level.