Generalized Linear Mixed Models

General Linear Model

The general linear model is useful in answering research questions about the impact of one or multiple predictor variables on an outcome variable. A predictor variable is a variable that is being manipulated (ideally) in an experiment to observe its impact on the outcome variable, whereas the outcome variable is a variable that is hypothesized to be changed by the predictor variable(s). Alternative names for the predictor variable are explanatory variable or independent variable. Likewise, the outcome variables are sometimes referred to as response variables or dependent variables.

Statistically, a general linear model represents conventional linear regression models with a continuous outcome variable predicted by one or more continuous and/or categorical variables. A general linear model includes a simple linear model, a multiple linear model, as well as the analysis of variance and the analysis of covariance. In a general linear model, the model can be expressed in the following equation:

$$y_i={\beta }_0+{\beta }_1{x}_{1i}+\dots +{\beta }_j{x}_{ji}+e_i.$$

The outcome variable yi is modeled by a linear function of the predictor variables xji, plus an error term (ei). The subscript i indicates the ith observation, whereas the subscript j indicates the jth predictor variable.

The word linear in a general linear model implies that a combination of parameters β0 … βj could predict the observed values of the outcome variable. The word general refers to the fact that the outcome variable is dependent on potentially more than one predictor variable and it is normally distributed.

[Page 731]The previous equation can be rewritten with a shortcut as:

$$y=\text{X}\beta +\text{e}.$$

Each bold letter is a matrix. The matrix y represents the values of the outcome variable for all N observations, with a size of N × 1. The matrix X stores the values of all j predictor variables for all N observations, with a size of N × j. The matrix β indicates the estimated values of the general linear model parameters, with a size of j × 1. The matrix e shows the difference between the predicted and the observed value of the outcome variable (i.e., residual/error) for all N observations, with a size of N × 1.

A general linear model is the most widely used statistical model, while it has to meet certain assumptions: (a) linearity, (b) data independency, and (c) the residuals are independent of each other and normally distributed, ei ∼ N (0, σ2). The linearity assumption means there is a linear relationship between the predictor variables and the outcome variable. In other words, a one-unit change in the predictor variable is expected to bring about the same amount of change in the outcome variable for all observations. The data independency assumption means each observation is independent of another.

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