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.

Linear Mixed Model for Autism Study

• The linear mixed model assumes that the response variable follows a normal distribution.

• The model includes both fixed effects (predictors) and random effects (individual-specific effects).

• The random effect vectors are assumed to be independent between children.

• The model assumes that the random effects and error terms are normally distributed with zero means and specific covariance matrices.

• The variance-covariance parameter vector of the random effects and error terms can be estimated using maximum likelihood or restricted maximum likelihood.

• The covariance matrix for the error terms is assumed to be diagonal, but different structures can be tested to see which fits the data better.

• The hypothesis tests for random effects are based on the likelihood ratio test using restricted maximum likelihood.

• The hypothesis tests for fixed effects are based on the likelihood ratio test using maximum likelihood.

• The quadratic effect of Age2 on Vsae was found to differ between children, and was included in the final model.

• The interaction effect of Age2Sq × Sicdegp was tested, and found to be insignificant and removed from the final model.

• The interaction effect of Age2 × Sicdegp was found to be significant and included in the final model.

• There was some doubt over whether the quadratic effect of Age2 should be included in the final model, but it was found to be significant and included.Analysis of Language Development Data using Linear Mixed Effects Models

• The quadratic effect of Age2 is found to be significant in the language development model, indicating its retention in the final model.

• The marginal value of Vsaeti is the same for children who belong to the same initial language development level.

• The linear effect of Age2 on Vsae for children who have a Low level of initial language development is represented by β1.

• The model used for examining research questions is model (3), which includes various fixed and random effects.

• The predicted marginal value of Vsaeti is given by the GLS estimates of the fixed effects.

• Marginal residuals are usually analyzed in marginal models rather than in models that include random effects.

• Each child has their own conditional value of Vsaeti, which is the expected value of Vsaeti in model (3) at a given age, conditional on the child-specific random linear and quadratic effects of Age2.

• Cook’s distance is used to identify influential observations when regressing the observed values of Vsae on the predicted conditional values of Vsae.

• Histograms and scatter plots are used to examine the assumptions associated with the random effects and errors.

• Outliers are identified in the data, associated with child 49, 180, and 124.

• The data fits the marginal and conditional parts of the model quite well, with no severe violation of constant variance and normality assumption for εij as a function of Age.

• Refitting the model without the outliers did not significantly change the results.Analysis of Social Development Trajectories in Autistic Children

• The study analyzed social development trajectories in autistic children over time.

• The model used included fixed effects for initial language development level and age, and random effects for individual variability.

• The estimate of the variance-covariance matrix for the random effects suggests that the quadratic effect of age decreases as the linear effect increases.

• The confidence interval for the correlation between the random effects does not contain 0, indicating that the effects are not similar.

• The estimate of the variance-covariance matrix of the model shows that the variability of Vsae scores for a child increases with age.

• Estimates of the fixed effects show that initial language development level significantly influences social development trajectories over time.

• Children with high levels of initial language development have greater mean social development and improve faster over time than those with low levels.

• The mean social development of some children does not improve over time, while others improve linearly or accelerate.

• Confidence intervals for the fixed effects show significant differences between the mean social development of children with high and low levels of initial language development.

• The linear effect of age on social development is greater for children with high levels of initial language development than those with low levels.

• The study used linear mixed models and likelihood ratio testing to analyze the data.

• References include resources on linear mixed models and longitudinal data analysis.

Linear Mixed Model for Autism Study

• The linear mixed model assumes that the response variable follows a normal distribution.

• The model includes both fixed effects (predictors) and random effects (individual-specific effects).

• The random effect vectors are assumed to be independent between children.

• The model assumes that the random effects and error terms are normally distributed with zero means and specific covariance matrices.

• The variance-covariance parameter vector of the random effects and error terms can be estimated using maximum likelihood or restricted maximum likelihood.

• The covariance matrix for the error terms is assumed to be diagonal, but different structures can be tested to see which fits the data better.

• The hypothesis tests for random effects are based on the likelihood ratio test using restricted maximum likelihood.

• The hypothesis tests for fixed effects are based on the likelihood ratio test using maximum likelihood.

• The quadratic effect of Age2 on Vsae was found to differ between children, and was included in the final model.

• The interaction effect of Age2Sq × Sicdegp was tested, and found to be insignificant and removed from the final model.

• The interaction effect of Age2 × Sicdegp was found to be significant and included in the final model.

• There was some doubt over whether the quadratic effect of Age2 should be included in the final model, but it was found to be significant and included.Analysis of Language Development Data using Linear Mixed Effects Models

• The quadratic effect of Age2 is found to be significant in the language development model, indicating its retention in the final model.

• The marginal value of Vsaeti is the same for children who belong to the same initial language development level.

• The linear effect of Age2 on Vsae for children who have a Low level of initial language development is represented by β1.

• The model used for examining research questions is model (3), which includes various fixed and random effects.

• The predicted marginal value of Vsaeti is given by the GLS estimates of the fixed effects.

• Marginal residuals are usually analyzed in marginal models rather than in models that include random effects.

• Each child has their own conditional value of Vsaeti, which is the expected value of Vsaeti in model (3) at a given age, conditional on the child-specific random linear and quadratic effects of Age2.

• Cook’s distance is used to identify influential observations when regressing the observed values of Vsae on the predicted conditional values of Vsae.

• Histograms and scatter plots are used to examine the assumptions associated with the random effects and errors.

• Outliers are identified in the data, associated with child 49, 180, and 124.

• The data fits the marginal and conditional parts of the model quite well, with no severe violation of constant variance and normality assumption for εij as a function of Age.

• Refitting the model without the outliers did not significantly change the results.Analysis of Social Development Trajectories in Autistic Children

• The study analyzed social development trajectories in autistic children over time.

• The model used included fixed effects for initial language development level and age, and random effects for individual variability.

• The estimate of the variance-covariance matrix for the random effects suggests that the quadratic effect of age decreases as the linear effect increases.

• The confidence interval for the correlation between the random effects does not contain 0, indicating that the effects are not similar.

• The estimate of the variance-covariance matrix of the model shows that the variability of Vsae scores for a child increases with age.

• Estimates of the fixed effects show that initial language development level significantly influences social development trajectories over time.

• Children with high levels of initial language development have greater mean social development and improve faster over time than those with low levels.

• The mean social development of some children does not improve over time, while others improve linearly or accelerate.

• Confidence intervals for the fixed effects show significant differences between the mean social development of children with high and low levels of initial language development.

• The linear effect of age on social development is greater for children with high levels of initial language development than those with low levels.

• The study used linear mixed models and likelihood ratio testing to analyze the data.

• References include resources on linear mixed models and longitudinal data analysis.