# StatsModels.jl API

## Formulae and terms

StatsModels.@formulaMacro
@formula(ex)

Capture and parse a formula expression as a Formula struct.

A formula is an abstract specification of a dependence between left-hand and right-hand side variables as in, e.g., a regression model. Each side specifies at a high level how tabular data is to be converted to a numerical matrix suitable for modeling. This specification looks something like Julia code, is represented as a Julia Expr, but uses special syntax. The @formula macro takes an expression like y ~ 1 + a*b, transforms it according to the formula syntax rules into a lowered form (like y ~ 1 + a + b + a&b), and constructs a Formula struct which captures the original expression, the lowered expression, and the left- and right-hand-side.

Operators that have special interpretations in this syntax are

• ~ is the formula separator, where it is a binary operator (the first argument is the left-hand side, and the second is the right-hand side.
• + concatenates variables as columns when generating a model matrix.
• & represents an interaction between two or more variables, which corresponds to a row-wise kronecker product of the individual terms (or element-wise product if all terms involved are continuous/scalar).
• * expands to all main effects and interactions: a*b is equivalent to a+b+a&b, a*b*c to a+b+c+a&b+a&c+b&c+a&b&c, etc.
• 1, 0, and -1 indicate the presence (for 1) or absence (for 0 and -1) of an intercept column.

The rules that are applied are

• The associative rule (un-nests nested calls to +, &, and *).
• The distributive rule (interactions & distribute over concatenation +).
• The * rule expands a*b to a+b+a&b (recursively).
• Subtraction is converted to addition and negation, so x-1 becomes x + -1 (applies only to subtraction of literal 1).
• Single-argument & calls are stripped, so &(x) becomes the main effect x.
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StatsModels.termFunction
term(x)

Wrap argument in an appropriate AbstractTerm type: Symbols and AbstractStrings become Terms, and Numbers become ConstantTerms. Any AbstractTerms are unchanged. AbstractStrings are converted to symbols before wrapping.

Example

julia> ts = term.((1, :a, "b"))
1
a(unknown)
b(unknown)

julia> typeof(ts)
Tuple{ConstantTerm{Int64}, Term, Term}
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StatsModels.modelcolsFunction
modelcols(t::AbstractTerm, data)

Create a numerical "model columns" representation of data based on an AbstractTerm. data can either be a whole table (a property-accessible collection of iterable columns or iterable collection of property-accessible rows, as defined by Tables.jl or a single row (in the form of a NamedTuple of scalar values). Tables will be converted to a NamedTuple of Vectors (e.g., a Tables.ColumnTable).

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modelcols(ts::NTuple{N, AbstractTerm}, data) where N

When a tuple of terms is provided, modelcols broadcasts over the individual terms. To create a single matrix, wrap the tuple in a MatrixTerm.

Example

julia> using StableRNGs; rng = StableRNG(1);

julia> d = (a = [1:9;], b = rand(rng, 9), c = repeat(["d","e","f"], 3));

julia> ts = apply_schema(term.((:a, :b, :c)), schema(d))
a(continuous)
b(continuous)
c(DummyCoding:3→2)

julia> cols = modelcols(ts, d)
([1, 2, 3, 4, 5, 6, 7, 8, 9], [0.5851946422124186, 0.07733793456911231, 0.7166282400543453, 0.3203570514066232, 0.6530930076222579, 0.2366391513734556, 0.7096838914472361, 0.5577872440804086, 0.05079002172175784], [0.0 0.0; 1.0 0.0; … ; 1.0 0.0; 0.0 1.0])

julia> reduce(hcat, cols)
9×4 Matrix{Float64}:
1.0  0.585195   0.0  0.0
2.0  0.0773379  1.0  0.0
3.0  0.716628   0.0  1.0
4.0  0.320357   0.0  0.0
5.0  0.653093   1.0  0.0
6.0  0.236639   0.0  1.0
7.0  0.709684   0.0  0.0
8.0  0.557787   1.0  0.0
9.0  0.05079    0.0  1.0

julia> modelcols(MatrixTerm(ts), d)
9×4 Matrix{Float64}:
1.0  0.585195   0.0  0.0
2.0  0.0773379  1.0  0.0
3.0  0.716628   0.0  1.0
4.0  0.320357   0.0  0.0
5.0  0.653093   1.0  0.0
6.0  0.236639   0.0  1.0
7.0  0.709684   0.0  0.0
8.0  0.557787   1.0  0.0
9.0  0.05079    0.0  1.0
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StatsModels.termnamesFunction
termnames(model::StatisticalModel)

Return the names of terms used in the formula of model.

This is a convenience method for termnames(formula(model)), which returns a two-tuple of termnames applied to the left and right hand sides of the formula.

For RegressionModels with only continuous predictors, this is the same as (responsename(model), coefnames(model)) and coefnames(formula(model)).

For models with categorical predictors, the returned names reflect the variable name and not the coefficients resulting from the choice of contrast coding.

See also coefnames.

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termnames(t::FormulaTerm)

Return a two-tuple of termnames applied to the left and right hand sides of the formula.

Note

Until apply_schema has been called, literal 1 in formulae is interpreted as ConstantTerm(1) and will thus appear as "1" in the returned term names.

julia> termnames(@formula(y ~ 1 + x * y + (1+x|g)))
("y", ["1", "x", "y", "x & y", "(1 + x) | g"])

Similarly, formulae with an implicit intercept will not have a "1" in their variable names, because the implicit intercept does not exist until apply_schema is called (and may not exist for certain model contexts).

julia> termnames(@formula(y ~ x * y + (1+x|g)))
("y", ["x", "y", "x & y", "(1 + x) | g"])
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termnames(term::AbstractTerm)

Return the name of the statistical variable associated with a term.

Return value is either a String, an iterable of Strings or the empty vector if there is no associated variable (e.g. termnames(InterceptTerm{false}())).

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### Higher-order terms

StatsModels.FormulaTermType
FormulaTerm{L,R} <: AbstractTerm

Represents an entire formula, with a left- and right-hand side. These can be of any type (captured by the type parameters).

Fields

• lhs::L: The left-hand side (e.g., response)
• rhs::R: The right-hand side (e.g., predictors)
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StatsModels.InteractionTermType
InteractionTerm{Ts} <: AbstractTerm

Represents an interaction between two or more individual terms.

Generated by combining multiple AbstractTerms with & (which is what calls to & in a @formula lower to)

Fields

• terms::Ts: the terms that participate in the interaction.

Example

julia> using StableRNGs; rng = StableRNG(1);

julia> d = (y = rand(rng, 9), a = 1:9, b = rand(rng, 9), c = repeat(["d","e","f"], 3));

julia> t = InteractionTerm(term.((:a, :b, :c)))
a(unknown) & b(unknown) & c(unknown)

julia> t == term(:a) & term(:b) & term(:c)
true

julia> t = apply_schema(t, schema(d))
a(continuous) & b(continuous) & c(DummyCoding:3→2)

julia> modelcols(t, d)
9×2 Matrix{Float64}:
0.0       0.0
1.88748   0.0
0.0       1.33701
0.0       0.0
0.725357  0.0
0.0       0.126744
0.0       0.0
4.93994   0.0
0.0       4.33378

julia> modelcols(t.terms, d)
([1, 2, 3, 4, 5, 6, 7, 8, 9], [0.236781883208121, 0.9437409715735081, 0.4456708824294644, 0.7636794266904741, 0.14507148958283067, 0.021124039581375875, 0.15254507694061115, 0.617492416565387, 0.48153065407402607], [0.0 0.0; 1.0 0.0; … ; 1.0 0.0; 0.0 1.0])
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StatsModels.FunctionTermType
FunctionTerm{F,Args} <: AbstractTerm

Represents a call to a Julia function. The first type parameter is the type of the captured function (e.g., typeof(log)), and the second is the type of the captured arguments (e.g., a Vector of AbstractTerms).

Nested function calls are captured as further FunctionTerms.

Fields

• f::F: the captured function (e.g., log)
• args::Args: the arguments of the call passed to @formula, each captured as an AbstractTerm. Usually this is a Vector{<:AbstractTerm}.
• exorig::Expr: the original expression passed to @formula

Type parameters

• F: the type of the captured function (e.g., typeof(log))
• Args: the type of container of captured arguments.

Example

julia> f = @formula(y ~ log(1 + a + b))
FormulaTerm
Response:
y(unknown)
Predictors:
(a,b)->log(1 + a + b)

julia> typeof(f.rhs)
FunctionTerm{typeof(log), Vector{FunctionTerm{typeof(+), Vector{AbstractTerm}}}}

julia> typeof(only(f.rhs.args))
FunctionTerm{typeof(+), Vector{AbstractTerm}}

julia> only(f.rhs.args).args
3-element Vector{AbstractTerm}:
1
a(unknown)
b(unknown)

julia> f.rhs.f(1 + 3 + 4)
2.0794415416798357

julia> modelcols(f.rhs, (a=3, b=4))
2.0794415416798357

julia> modelcols(f.rhs, (a=[3, 4], b=[4, 5]))
2-element Vector{Float64}:
2.0794415416798357
2.302585092994046
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### Concrete terms

These are all generated by apply_schema.

StatsModels.ContinuousTermType
ContinuousTerm <: AbstractTerm

Represents a continuous variable, with a name and summary statistics.

Fields

• sym::Symbol: The name of the variable
• mean::T: Mean
• var::T: Variance
• min::T: Minimum value
• max::T: Maximum value
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StatsModels.CategoricalTermType
CategoricalTerm{C,T,N} <: AbstractTerm

Represents a categorical term, with a name and ContrastsMatrix

Fields

• sym::Symbol: The name of the variable
• contrasts::ContrastsMatrix: A contrasts matrix that captures the unique values this variable takes on and how they are mapped onto numerical predictors.
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StatsModels.InterceptTermType
InterceptTerm{HasIntercept} <: AbstractTerm

Represents the presence or (explicit) absence of an "intercept" term in a regression model. These terms are generated from ConstantTerms in a formula by apply_schema(::ConstantTerm, schema, ::Type{<:StatisticalModel}). A 1 yields InterceptTerm{true}, and 0 or -1 yield InterceptTerm{false} (which explicitly omits an intercept for models which implicitly includes one via the implicit_intercept trait).

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ShiftedArrays.leadFunction
    lead(term, nsteps::Integer)

This @formula term is used to introduce lead variables.
For example lead(x,1) effectively adds a new column containing
the value of the x column from the next row.
If there is no such row (e.g. because this is the last row),
then the lead column will contain missing for that entry.

Note: this is only a basic row-wise lead operation.
It is up to the user to ensure that data is sorted by the temporal variable,
and that observations are spaced with regular time-steps.
(Which may require adding extra-rows filled with missing values.)
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ShiftedArrays.lagFunction
    lag(term, nsteps::Integer)

This @formula term is used to introduce lagged variables.
For example lag(x,1) effectively adds a new column containing
the value of the x column from the previous row.
If there is no such row (e.g. because this is the first row),
then the lagged column will contain missing for that entry.

Note: this is only a basic row-wise lag operation.
It is up to the user to ensure that data is sorted by the temporal variable,
and that observations are spaced with regular time-steps.
(Which may require adding extra-rows filled with missing values.)
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StatsModels.collect_matrix_termsFunction
collect_matrix_terms(ts::TupleTerm)
collect_matrix_terms(t::AbstractTerm) = collect_matrix_term((t, ))

Depending on whether the component terms are matrix terms (meaning they have is_matrix_term(T) == true), collect_matrix_terms will return

1. A single MatrixTerm (if all components are matrix terms)
2. A tuple of the components (if none of them are matrix terms)
3. A tuple of terms, with all matrix terms collected into a single MatrixTerm in the first element of the tuple, and the remaining non-matrix terms passed through unchanged.

By default all terms are matrix terms (that is, is_matrix_term(::Type{<:AbstractTerm}) = true), the first case is by far the most common. The others are provided only for convenience when dealing with specialized terms that can't be concatenated into a single model matrix, like random effects terms in MixedModels.jl.

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StatsModels.is_matrix_termFunction
is_matrix_term(::Type{<:AbstractTerm})

Does this type of term get concatenated with other matrix terms into a single model matrix? This controls the behavior of the collect_matrix_terms, which collects all of its arguments for which is_matrix_term returns true into a MatrixTerm, and returns the rest unchanged.

Since all "normal" terms which describe one or more model matrix columns are matrix terms, this defaults to true for any AbstractTerm.

An example of a non-matrix term is a random effect term in MixedModels.jl.

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### Protection

For more fine-grained control over whether function calls are treated as normal Julia calls ("protected" and captured as FunctionTerms) or as @formula syntax ("unprotected").

StatsModels.protectFunction
protect(term::T)

Create a Protected context for interpreting term (and descendents) during apply_schema.

Outside a @formula, acts as a constructor for the singleton Protected{T}.

Example

julia> d = (y=rand(4), a=[1:4;], b=rand(4));

julia> f = @formula(y ~ 1 + protect(a+b));

julia> modelmatrix(f.rhs, d)
4×2 Matrix{Float64}:
1.0  1.91493
1.0  2.19281
1.0  3.77018
1.0  4.78052

julia> d.a .+ d.b
4-element Vector{Float64}:
1.9149290036628313
2.1928081162458755
3.7701803478856664
4.7805192636751865
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StatsModels.unprotectFunction
unprotect(term)
unprotect(::Protected{T})

Inside a [@formula], removes Protected status for the argument term(s). This allows the @formula-specific interpretation of calls to +, &, *, and ~ to be restored inside an otherwise Protected context.

When called outside a @formula, unwraps Protected{T} to T.

Example

julia> d = (y=rand(4), a=[1.:4;], b=rand(4));

julia> f = @formula(y ~ 1 - unprotect(a&b));

julia> modelmatrix(f, d)
4×1 Matrix{Float64}:
0.08507099633716864
0.6143837675082491
-1.310541043656999
-2.1220770547007453

julia> 1 .- d.a .* d.b
4-element Vector{Float64}:
0.08507099633716864
0.6143837675082491
-1.310541043656999
-2.1220770547007453
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StatsModels.@support_unprotectMacro
StatsModels.@support_unprotect f sch_types...

Generate methods necessary for function f to support unprotect inside of a @formula with a schema of types sch_types. If not specified, sch_types defaults to Schema, FullRank (the two schema types defined in StatsModels itself).

Any function call that occurs as a child of a protected call is also protected by default. In order to support unprotecting functions/operators that work directly on Terms (like the built-in "special" operators +, &, *, and ~), we need to add methods for apply_schema(::FunctionTerm{typeof(f)}, ...) that calls f on the captured arguments before further schema application.

This macro generates the necessary method for f. For this to do something reasonable, a few conditions must be met:

1. Methods must exist for f(args::AbstractTerm...) matching the specific signatures that users provide when calling f in @formula (and usually, returns an AbstractTerm of some kind).

2. The custom term type returned by new_term = f(args::AbstractTerm...) needs to do something reasonable when modelcols is called on it.

3. The thing returned by modelcols(new_term, data) needs to be something that can be consumed as input to whatever the parent call was for f in the original formula expression.

To take a concrete example, if we have a function g that can do something meaningful with the output of modelcols(::InteractionTerm, ...), then when a user provides something like

@formula(g(unprotect(a & b)))

that gets lowered to

FunctionTerm(g, [FuntionTerm(&, [Term(:a), Term(:b)], ...)], ...)

and we need to convert it to something like

FuntionTerm(g, [Term(:a) & Term(:b)], ...)

during schema application, which is what the method generated by @support_unprotect & does.

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StatsModels.ProtectedType
struct Protected{Ctx}

Represent a context in which @formula DSL syntax (e.g. & to construct InteractionTerm rather than bitwise-and) and apply_schema transformations should not apply. This is automatically applied to the arguments of a FunctionTerm, meaning that by default calls to +, &, or ~ inside a FunctionTerm will be interpreted as calls to the normal Julia functions, rather than term union, interaction, or formula separation.

The only special behavior with apply_schema inside a Protected context is when a call to unprotect is encountered. At that point, everything below the call to unprotect is treated as formula-specific syntax.

A Protected context is created inside a FunctionTerm automatically, but can be manually created with a call to protect. 

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## Schema

StatsModels.SchemaType
StatsModels.Schema

Struct that wraps a Dict mapping Terms to their concrete forms. This exists mainly for dispatch purposes and to support possibly more sophisticated behavior in the future.

A Schema behaves for all intents and purposes like an immutable Dict, and delegates the constructor as well as getindex, get, merge!, merge, keys, and haskey to the wrapped Dict.

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StatsModels.schemaFunction
schema([terms::AbstractVector{<:AbstractTerm}, ]data, hints::Dict{Symbol})
schema(term::AbstractTerm, data, hints::Dict{Symbol})

Compute all the invariants necessary to fit a model with terms. A schema is a dict that maps Terms to their concrete instantiations (either CategoricalTerms or ContinuousTerms. "Hints" may optionally be supplied in the form of a Dict mapping term names (as Symbols) to term or contrast types. If a hint is not provided for a variable, the appropriate term type will be guessed based on the data type from the data column: any numeric data is assumed to be continuous, and any non-numeric data is assumed to be categorical.

Returns a StatsModels.Schema, which is a wrapper around a Dict mapping Terms to their concrete instantiations (ContinuousTerm or CategoricalTerm).

Example

julia> using StableRNGs; rng = StableRNG(1);

julia> d = (x=sample(rng, [:a, :b, :c], 10), y=rand(rng, 10));

julia> ts = [Term(:x), Term(:y)];

julia> schema(ts, d)
StatsModels.Schema with 2 entries:
x => x
y => y

julia> schema(ts, d, Dict(:x => HelmertCoding()))
StatsModels.Schema with 2 entries:
x => x
y => y

julia> schema(term(:y), d, Dict(:y => CategoricalTerm))
StatsModels.Schema with 1 entry:
y => y

Note that concrete ContinuousTerm and CategoricalTerm and un-typed Terms print the same in a container, but when printed alone are different:

julia> sch = schema(ts, d)
StatsModels.Schema with 2 entries:
x => x
y => y

julia> term(:x)
x(unknown)

julia> sch[term(:x)]
x(DummyCoding:3→2)

julia> sch[term(:y)]
y(continuous)
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StatsModels.concrete_termFunction
concrete_term(t::Term, data[, hint])

Create concrete term from the placeholder t based on a data source and optional hint. If data is a table, the getproperty is used to extract the appropriate column.

The hint can be a Dict{Symbol} of hints, or a specific hint, a concrete term type (ContinuousTerm or CategoricalTerm), or an instance of some <:AbstractContrasts, in which case a CategoricalTerm will be created using those contrasts.

If no hint is provided (or hint==nothing), the eltype of the data is used: Numbers are assumed to be continuous, and all others are assumed to be categorical.

Example

julia> concrete_term(term(:a), [1, 2, 3])
a(continuous)

julia> concrete_term(term(:a), [1, 2, 3], nothing)
a(continuous)

julia> concrete_term(term(:a), [1, 2, 3], CategoricalTerm)
a(DummyCoding:3→2)

julia> concrete_term(term(:a), [1, 2, 3], EffectsCoding())
a(EffectsCoding:3→2)

julia> concrete_term(term(:a), [1, 2, 3], Dict(:a=>EffectsCoding()))
a(EffectsCoding:3→2)

julia> concrete_term(term(:a), (a = [1, 2, 3], b = [0.0, 0.5, 1.0]))
a(continuous)
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StatsModels.apply_schemaFunction
apply_schema(t, schema::StatsModels.Schema[, Mod::Type = Nothing])

Return a new term that is the result of applying schema to term t with destination model (type) Mod. If Mod is omitted, Nothing will be used.

When t is a ContinuousTerm or CategoricalTerm already, the term will be returned unchanged unless a matching term is found in the schema. This allows selective re-setting of a schema to change the contrast coding or levels of a categorical term, or to change a continuous term to categorical or vice versa.

When defining behavior for custom term types, it's best to dispatch on StatsModels.Schema for the second argument. Leaving it as ::Any will work in most cases, but cause method ambiguity in some.

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apply_schema(t::AbstractTerm, schema::StatsModels.FullRank, Mod::Type)

Apply a schema, under the assumption that when a less-than-full rank model matrix would be produced, categorical terms should be "promoted" to full rank (where a categorical variable with $k$ levels would produce $k$ columns, instead of $k-1$ in the standard contrast coding schemes). This step is applied automatically when Mod <: StatisticalModel, but other types of models can opt-in by adding a method like

StatsModels.apply_schema(t::FormulaTerm, schema::StatsModels.Schema, Mod::Type{<:MyModelType}) =
apply_schema(t, StatsModels.FullRank(schema), mod)

See the section on Modeling categorical data in the docs for more information on how promotion of categorical variables works.

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## Modeling

StatsAPI.fitFunction

Fit a statistical model.

fit(Mod::Type{<:StatisticalModel}, f::FormulaTerm, data, args...;
contrasts::Dict{Symbol}, kwargs...)

Convert tabular data into a numeric response vector and predictor matrix using the formula f, and then fit the specified model type, wrapping the result in a TableRegressionModel or TableStatisticalModel (as appropriate).

This is intended as a backstop for modeling packages that implement model types that are subtypes of StatsAPI.StatisticalModel but do not explicitly support the full StatsModels terms-based interface. Currently this works by creating a ModelFrame from the formula and data, and then converting this to a ModelMatrix, but this is an internal implementation detail which may change in the near future.

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StatsAPI.gvifFunction
gvif(m::RegressionModel; scale=false)

Compute the generalized variance inflation factor (GVIF).

If scale=true, then each GVIF is scaled by the degrees of freedom for (number of coefficients associated with) the predictor: $GVIF^(1 / (2*df))$.

The GVIF measures the increase in the variance of a (group of) parameter's estimate in a model with multiple parameters relative to the variance of a parameter's estimate in a model containing only that parameter. For continuous, numerical predictors, the GVIF is the same as the VIF, but for categorical predictors, the GVIF provides a single number for the entire group of contrast-coded coefficients associated with a categorical predictor.

See also vif.

References

Fox, J., & Monette, G. (1992). Generalized Collinearity Diagnostics. Journal of the American Statistical Association, 87(417), 178. doi:10.2307/2290467

StatsModels.lrtestFunction
lrtest(mods::StatisticalModel...; atol::Real=0.0)

For each sequential pair of statistical models in mods..., perform a likelihood ratio test to determine if the first one fits significantly better than the next.

A table is returned containing degrees of freedom (DOF), difference in DOF from the preceding model, log-likelihood, deviance, chi-squared statistic (i.e. absolute value of twice the difference in log-likelihood) and p-value for the comparison between the two models.

Optional keyword argument atol controls the numerical tolerance when testing whether the models are nested.

Examples

Suppose we want to compare the effects of two or more treatments on some result. Our null hypothesis is that Result ~ 1 fits the data as well as Result ~ 1 + Treatment.

julia> using DataFrames, GLM

julia> dat = DataFrame(Result=[1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1],
Treatment=[1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2],
Other=string.([1, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1]));

julia> nullmodel = glm(@formula(Result ~ 1), dat, Binomial(), LogitLink());

julia> model = glm(@formula(Result ~ 1 + Treatment), dat, Binomial(), LogitLink());

julia> bigmodel = glm(@formula(Result ~ 1 + Treatment + Other), dat, Binomial(), LogitLink());

julia> lrtest(nullmodel, model, bigmodel)
Likelihood-ratio test: 3 models fitted on 12 observations
────────────────────────────────────────────────────
DOF  ΔDOF   LogLik  Deviance   Chisq  p(>Chisq)
────────────────────────────────────────────────────
[1]    1        -8.1503   16.3006
[2]    2     1  -7.9780   15.9559  0.3447     0.5571
[3]    4     2  -7.0286   14.0571  1.8988     0.3870
────────────────────────────────────────────────────

julia> lrtest(bigmodel, model, nullmodel)
Likelihood-ratio test: 3 models fitted on 12 observations
────────────────────────────────────────────────────
DOF  ΔDOF   LogLik  Deviance   Chisq  p(>Chisq)
────────────────────────────────────────────────────
[1]    4        -7.0286   14.0571
[2]    2    -2  -7.9780   15.9559  1.8988     0.3870
[3]    1    -1  -8.1503   16.3006  0.3447     0.5571
────────────────────────────────────────────────────
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StatsAPI.responseFunction
response(model::RegressionModel)

Return the model response (a.k.a. the dependent variable).

StatsAPI.vifFunction
vif(m::RegressionModel)

Compute the variance inflation factor (VIF).

The VIF measures the increase in the variance of a parameter's estimate in a model with multiple parameters relative to the variance of a parameter's estimate in a model containing only that parameter.

See also gvif.

Warning

This method will fail if there is (numerically) perfect multicollinearity, i.e. rank deficiency. In that case though, the VIF is not particularly informative anyway.

### Traits

StatsModels.implicit_interceptFunction
implicit_intercept(T::Type)
implicit_intercept(x::T) = implicit_intercept(T)

Return true if models of type T should include an implicit intercept even if none is specified in the formula. Is true by default for all T<:StatisticalModel, and false for others. To specify that a model type T includes an intercept even if one is not specified explicitly in the formula, overload this function for the corresponding type: implicit_intercept(::Type{<:T}) = true

If a model has an implicit intercept, it can be explicitly excluded by using 0 in the formula, which generates InterceptTerm{false} with apply_schema.

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StatsModels.drop_interceptFunction
drop_intercept(T::Type)
drop_intercept(x::T) = drop_intercept(T)

Define whether a given model automatically drops the intercept. Return false by default. To specify that a model type T drops the intercept, overload this function for the corresponding type: drop_intercept(::Type{<:T}) = true

Models that drop the intercept will be fitted without one: the intercept term will be removed even if explicitly provided by the user. Categorical variables will be expanded in the rank-reduced form (contrasts for n levels will only produce n-1 columns).

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### Wrappers

Warning

These are internal implementation details that are likely to change in the near future. In particular, the ModelFrame and ModelMatrix wrappers are dispreferred in favor of using terms directly, and can in most cases be replaced by something like

# instead of ModelMatrix(ModelFrame(f::FormulaTerm, data, model=MyModel))
sch = schema(f, data)
f = apply_schema(f, sch, MyModel)
response, predictors = modelcols(f, data)
StatsModels.ModelFrameType
ModelFrame(formula, data; model=StatisticalModel, contrasts=Dict())

Wrapper that encapsulates a FormulaTerm, schema, data table, and model type.

This wrapper encapsulates all the information that's required to transform data of the same structure as the wrapped data frame into a model matrix (the FormulaTerm), as well as the information about how that formula term was instantiated (the schema and model type)

Creating a model frame involves first extracting the schema for the data (using any contrasts provided as hints), and then applying that schema with apply_schema to the formula in the context of the provided model type.

Constructors

ModelFrame(f::FormulaTerm, data; model::Type{M} = StatisticalModel, contrasts::Dict = Dict())

Fields

• f::FormulaTerm: Formula whose left hand side is the response and right hand side are the predictors.
• schema::Any: The schema that was applied to generate f.
• data::D: The data table being modeled. The only restriction is that data is a table (Tables.istable(data) == true)
• model::Type{M}: The type of the model that will be fit from this model frame.

Examples

julia> df = (x = 1:4, y = 5:8)
julia> mf = ModelFrame(@formula(y ~ 1 + x), df)
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StatsModels.ModelMatrixType
ModelMatrix(mf::ModelFrame)

Convert a ModelFrame into a numeric matrix suitable for modeling

Fields

• m::AbstractMatrix{<:AbstractFloat}: the generated numeric matrix
• assign::Vector{Int} the index of the term corresponding to each column of m.

Constructors

ModelMatrix(mf::ModelFrame)
# Specify the type of the resulting matrix (default Matrix{Float64})
ModelMatrix{T <: AbstractMatrix{<:AbstractFloat}}(mf::ModelFrame)
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StatsModels.TableStatisticalModelType

Wrapper for a StatisticalModel that has been fit from a @formula and tabular data.

Most functions from the StatsBase API are simply delegated to the wrapped model, with the exception of functions like fit, predict, and coefnames where the tabular nature of the data means that additional processing is required or information provided by the formula.

Fields

• model::M the wrapped StatisticalModel.
• mf::ModelFrame encapsulates the formula, schema, and model type.
• mm::ModelMatrix{T} the model matrix that the model was fit from.
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StatsModels.TableRegressionModelType

Wrapper for a RegressionModel that has been fit from a @formula and tabular data.

Most functions from the StatsBase API are simply delegated to the wrapped model, with the exception of functions like fit, predict, and coefnames where the tabular nature of the data means that additional processing is required or information provided by the formula.

Fields

• model::M the wrapped RegressioModel.
• mf::ModelFrame encapsulates the formula, schema, and model type.
• mm::ModelMatrix{T}` the model matrix that the model was fit from.
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