*Introduction to Analysis-of-Variance Procedures* |

## General Linear Models

An analysis-of-variance model can be written as a linear model, which is
an equation that predicts the response as a linear function of
parameters and design
variables. In general,

where *y*_{i} is the response for the *i*th observation,
are unknown parameters to be estimated, and
*x*_{ij} are design variables.
Design variables for analysis of variance are indicator
variables; that is, they are always either 0 or 1.
The simplest model is to fit a single mean to all observations.
In this case there is only one parameter, , and one
design variable, *x*_{0i}, which always has the value of 1:

The least-squares estimator of is the mean of the *y*_{i}.
This simple model underlies all more complex models, and
all larger models are compared to this simple mean model.
In writing the parameterization of a linear model,
is usually referred to as the *intercept*.
A one-way model is written by introducing an indicator
variable for each level of the classification variable.
Suppose that a variable A has four levels,
with two observations per level.
The indicator variables are created as follows:

| | Intercept | | A1 | | A2 | | A3 | | A4 |

| | 1 | | 1 | | 0 | | 0 | | 0 |

| | 1 | | 1 | | 0 | | 0 | | 0 |

| | 1 | | 0 | | 1 | | 0 | | 0 |

| | 1 | | 0 | | 1 | | 0 | | 0 |

| | 1 | | 0 | | 0 | | 1 | | 0 |

| | 1 | | 0 | | 0 | | 1 | | 0 |

| | 1 | | 0 | | 0 | | 0 | | 1 |

| | 1 | | 0 | | 0 | | 0 | | 1 |

The linear model for this example is

To construct crossed and nested effects, you can simply
multiply out all combinations of the main-effect columns.
This is described in detail in "Specification of Effects" in Chapter 30, "The GLM Procedure."

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