Hlm-Centering
Centering
HLM offers the options to use predictors as they are, or to use them after grand- or group-mean centering them. The choice of centering method is dictated by the question studied, and great care should be taken to select a form of centering appropriate to the model considered, as the interpretation of coefficients in the model is dependent on the type of centering used. We start with an example where a simple linear transformation of a predictor is used to ensure that the interpretation of the average estimated outcome is useful, after which some examples of the different types of centering are given for the same data set. An intercept term is generally interpreted as the expected average of the dependent variable given that all predictors in a model are equal to zero. This will be the case, for example, in a model where the test score of students nested within schools are studied. A single level-1 predictor IQ is included in the model in an attempt to explain the variability in the test scores of the students, and only the intercept is allowed to vary randomly over the schools (level-2 units). Model 1:
The meaning of the intercept in the Level-1 model depends upon the location of the level-1 predictors included in the model. From the mixed model formulation given below, we see that can be interpreted as the expected test score for a student from group j with intelligence quotient equal to zero. Although a valid numerical interpretation, a more appropriate choice of the location of IQ will render the results more easily interpretable.
In the next example, data of a financial nature are studied. Seven consecutive annual measurements from a number of firms on both their annual return on assets (represented here by the variable ROA) and the natural logarithm of their total assets in the same year (represented by the predictor LNASTS) are available. Model 2:
The model can be rewritten as
39
hlm6
In this example, can be interpreted as the
HLM offers the options to use predictors as they are, or to use them after grand- or group-mean centering them. The choice of centering method is dictated by the question studied, and great care should be taken to select a form of centering appropriate to the model considered, as the interpretation of coefficients in the model is dependent on the type of centering used. We start with an example where a simple linear transformation of a predictor is used to ensure that the interpretation of the average estimated outcome is useful, after which some examples of the different types of centering are given for the same data set. An intercept term is generally interpreted as the expected average of the dependent variable given that all predictors in a model are equal to zero. This will be the case, for example, in a model where the test score of students nested within schools are studied. A single level-1 predictor IQ is included in the model in an attempt to explain the variability in the test scores of the students, and only the intercept is allowed to vary randomly over the schools (level-2 units). Model 1:
The meaning of the intercept in the Level-1 model depends upon the location of the level-1 predictors included in the model. From the mixed model formulation given below, we see that can be interpreted as the expected test score for a student from group j with intelligence quotient equal to zero. Although a valid numerical interpretation, a more appropriate choice of the location of IQ will render the results more easily interpretable.
In the next example, data of a financial nature are studied. Seven consecutive annual measurements from a number of firms on both their annual return on assets (represented here by the variable ROA) and the natural logarithm of their total assets in the same year (represented by the predictor LNASTS) are available. Model 2:
The model can be rewritten as
39
hlm6
In this example, can be interpreted as the