In a standard regression model - the parameter (eg the slope or intercept)  is fixed to a single value - in a random coefficient model it is allowed to vary according to a distribution. For example if modelling the relation between attainment score and a pre-score for pupils,the relation could be allowed to vary (ie random) between different schools. In this version the RC model is called a multilevel model- the key difference is that you are modelling variances and not just means ( that is he overall or fixed intercept and slope) as in standard regression model.

Here is an online course

http://www.bristol.ac.uk/cmm/learning/online-course/index.html

and here are some free book-length materials on Rgate

https://www.researchgate.net/publication/260771330_Developing_multilevel_models_for_analysing_contextuality_heterogeneity_and_change_using_MLwiN_Volume_1_%28updated_June_2014%29

You can also include variables in your model to try and account for this parameter heterogeneity;

and sometimes the variance is itself represents a research question -

see

https://www.researchgate.net/publication/225303320_Multilevel_Modeling_of_Social_Segregation

and sometimes the estimates can be technically helpful as they are automatically precision-weighted so that (say) school differences based on few pupils will be downweighted in the analysis

see

https://www.researchgate.net/publication/266383638_A_model-_based_approach_to_the_analysis_of_a_large_table_of_counts_occupational_class_patterns_in_among_Australians_by_ancestry_generation_and_age_group

Book Developing multilevel models for analysing contextuality, he...

Article Multilevel Modeling of Social Segregation

Article A model- based approach to the analysis of a large table of ...

In a standard regression model - the parameter (eg the slope or intercept)  is fixed to a single value - in a random coefficient model it is allowed to vary according to a distribution. For example if modelling the relation between attainment score and a pre-score for pupils,the relation could be allowed to vary (ie random) between different schools. In this version the RC model is called a multilevel model- the key difference is that you are modelling variances and not just means ( that is he overall or fixed intercept and slope) as in standard regression model.

Here is an online course

http://www.bristol.ac.uk/cmm/learning/online-course/index.html

and here are some free book-length materials on Rgate

https://www.researchgate.net/publication/260771330_Developing_multilevel_models_for_analysing_contextuality_heterogeneity_and_change_using_MLwiN_Volume_1_%28updated_June_2014%29

You can also include variables in your model to try and account for this parameter heterogeneity;

and sometimes the variance is itself represents a research question -

see

https://www.researchgate.net/publication/225303320_Multilevel_Modeling_of_Social_Segregation

and sometimes the estimates can be technically helpful as they are automatically precision-weighted so that (say) school differences based on few pupils will be downweighted in the analysis

see

https://www.researchgate.net/publication/266383638_A_model-_based_approach_to_the_analysis_of_a_large_table_of_counts_occupational_class_patterns_in_among_Australians_by_ancestry_generation_and_age_group

Book Developing multilevel models for analysing contextuality, he...

Article Multilevel Modeling of Social Segregation

Article A model- based approach to the analysis of a large table of ...

Hi Stephen, one other thing to consider about random effects models are the application to analysis of variance. In ANOVA, a random effects model is also called a variance components analysis (Type II ANOVA) - because, much like Kelvyn explained above, the analysis of random effects means that you are selecting from some definable population of sources of systematic variability and rather than assessing group means, you are focusing on group variances.

The application of a random effects model is dependent upon the features of the research design. If you cannot justify that you have a fixed-factor (typically manipulated) research design, you would most likely want to apply a random effects model.

There is so much that could be detailed here, I would recommend looking up a good multivariate book like the Tabachnick & Fidell text "Using Multivariate Statistics" for a thorough explanation.

Thank you Kelvyn and Brendan. Your explanations have helped a lot to clarify this question for me. I will definitely read more from the resources you have suggested.

Stephen - 

Attached is a link to a short paper with regard to this. I like the way it goes smoothly from equation 1there to eq2 to eq3.  (I'm not wild about ignoring heteroscedasticity in the error term in those equations though. However, it does indicate further down that "z" can be used to 'model variance heterogeneity' here.)  

Jim 

http://statprob.com/encyclopedia/LinearMixedModels.html

Dear All

Can anyone explain to me what are the differences between the heterogeneous coefficient and random coefficient models?. In Bayesian approach, I found the literature used non-hierarchical model for heterogeneous coefficient (LeSage, 2017) and hierarchical model for the random coefficient model (Koop, 2003). Thank you

Can anyone help me to get real life application of Response Surface Methodology with random coefficients.