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# The Formula, ModelFrame and ModelMatrix Types

In regression analysis, we often want to describe the relationship between a response variable and one or more input variables in terms of main effects and interactions. To facilitate the specification of a regression model in terms of the columns of a DataFrame, the DataFrames package provides a Formula type, which is created using the @formula macro in Julia:

fm=@formula(Z~X+Y)

A Formula object can be used to transform a DataFrame into a ModelFrame object:

df=DataFrame(X=randn(10),Y=randn(10),Z=randn(10))mf=ModelFrame(@formula(Z~X+Y),df)

A ModelFrame object is just a simple wrapper around a DataFrame. For modeling purposes, one generally wants to construct a ModelMatrix, which constructs a Matrix{Float64} that can be used directly to fit a statistical model:

mm=ModelMatrix(ModelFrame(@formula(Z~X+Y),df))

Note that mm contains an additional column consisting entirely of 1.0 values. This is used to fit an intercept term in a regression model.

In addition to specifying main effects, it is possible to specify interactions using the & operator inside a Formula:

mm=ModelMatrix(ModelFrame(@formula(Z~X+Y+X&Y),df))

If you would like to specify both main effects and an interaction term at once, use the * operator inside a Formula:

mm=ModelMatrix(ModelFrame(@formula(Z~X*Y),df))

You can control how categorical variables (e.g., PooledDataArray columns) are converted to ModelMatrix columns by specifying contrasts when you construct a ModelFrame:

mm=ModelMatrix(ModelFrame(@formula(Z~X*Y),df,contrasts=Dict(:X=>HelmertCoding())))

Contrasts can also be modified in an existing ModelFrame:

mf=ModelFrame(@formula(Z~X*Y),df)contrasts!(mf,X=HelmertCoding())

The construction of model matrices makes it easy to formulate complex statistical models. These are used to good effect by the GLM Package.