Internals and extending the @formula

Internals and extending the formula DSL

This section is intended to help package developers understand the internals of how a @formula becomes a numerical matrix, in order to use, manipulate, and even extend the DSL. The Julia @formula is designed to be as extensible as possible through the normal Julian mechanisms of multiple dispatch.

The lifecycle of a @formula

A formula goes through a number of stages, starting as an expression that's passed to the @formula macro and ending up generating a numeric matrix when ultimately combined with a tabular data source:

  1. "Syntax time" when only the surface syntax is available, when the @formula macro is invoked.
  2. "Schema time" incorporates information about data invariants (types of each variable, levels of categorical variables, summary statistics for continuous variables) and the overall structure of the data, during the invocation of schema.
  3. "Semantics time" incorporates information about the model type (context), and custom terms, during the call to apply_schema.
  4. "Data time" when the actual data values themselves are available.

For in-memory (columnar) tables, there is not much difference between "data time" and "schema time" in practice, but in principle it's important to distinguish between these when dealing with truly streaming data, or large data stores where calculating invariants of the data may be expensive.

Syntax time (@formula)

The @formula macro does syntactic transformations of the formula expression. At this point, only the expression itself is available, and there's no way to know whether a term corresponds to a continuous or categorical variable.

For standard formulae, this amounts to applying the syntactic rules for the DSL operators (expanding * and applying the distributive and associative rules), and wrapping each symbol in a Term constructor:

julia> @macroexpand @formula(y ~ 1 + a*b)
:(Term(:y) ~ ConstantTerm(1) + Term(:a) + Term(:b) + Term(:a) & Term(:b))

Note that much of the action happens outside the @formula macro, when the expression returned by the @formula macro is evaluated. At this point, the Terms are combined to create higher-order terms via overloaded methods for ~, +, and &:

julia> using StatsModels;

julia> dump(Term(:a) & Term(:b))
InteractionTerm{Tuple{Term,Term}}
  terms: Tuple{Term,Term}
    1: Term
      sym: Symbol a
    2: Term
      sym: Symbol b

julia> dump(Term(:a) + Term(:b))
Tuple{Term,Term}
  1: Term
    sym: Symbol a
  2: Term
    sym: Symbol b

julia> dump(Term(:y) ~ Term(:a))
FormulaTerm{Term,Term}
  lhs: Term
    sym: Symbol y
  rhs: Term
    sym: Symbol a
Note

As always, you can introspect which method is called with

julia> @which Term(:a) & Term(:b)
&(terms::AbstractTerm...) in StatsModels at /home/dave/.julia/dev/StatsModels/src/terms.jl:399

The reason that the actual construction of higher-order terms is done after the macro is expanded is that it makes it much easier to create a formula programmatically:

julia> f = Term(:y) ~ sum(term.([1, :a, :b, :c]))
FormulaTerm
Response:
  y(unknown)
Predictors:
  1
  a(unknown)
  b(unknown)
  c(unknown)

julia> f == @formula(y ~ 1 + a + b + c)
true

The major exception to this is that non-DSL calls must be specified using the @formula macro. The reason for this is that non-DSL calls are "captured" and turned into anonymous functions that can be evaluated elementwise, which has to happen at compile time. For instance, the call to log in @formula(y ~ log(a+b)) is converted into the anonymous function (a,b) -> log(a+b).

Internally a lot of the work at syntax time is done by the parse! function.

Schema time (schema)

The next phase of life for a formula requires some information about the data it will be used with. This is represented by a schema, a mapping from placeholder Terms to concrete terms—like ContinuousTerm CategoricalTerm—which represent all the summary information about a data column necessary to create a model matrix from that column.

There are a number of ways to construct a schema, ranging from fully automatic to fully manual.

Fully automatic: schema

The most convenient way to automatically compute a schema is with the schema function. By default, it will create a schema for every column in the data:

julia> using DataFrames    # for pretty printing---any Table will do

julia> df = DataFrame(y = rand(9), a = 1:9, b = rand(9), c = repeat(["a","b","c"], 3))
9×4 DataFrames.DataFrame
│ Row │ y          │ a     │ b         │ c      │
│     │ Float64    │ Int64 │ Float64   │ String │
├─────┼────────────┼───────┼───────────┼────────┤
│ 1   │ 0.236033   │ 1     │ 0.986666  │ a      │
│ 2   │ 0.346517   │ 2     │ 0.555751  │ b      │
│ 3   │ 0.312707   │ 3     │ 0.437108  │ c      │
│ 4   │ 0.00790928 │ 4     │ 0.424718  │ a      │
│ 5   │ 0.488613   │ 5     │ 0.773223  │ b      │
│ 6   │ 0.210968   │ 6     │ 0.28119   │ c      │
│ 7   │ 0.951916   │ 7     │ 0.209472  │ a      │
│ 8   │ 0.999905   │ 8     │ 0.251379  │ b      │
│ 9   │ 0.251662   │ 9     │ 0.0203749 │ c      │

julia> schema(df)
StatsModels.Schema with 4 entries:
  y => y
  a => a
  b => b
  c => c

However, if a term (including a FormulaTerm) is provided, the schema will be computed based only on the necessary variables:

julia> schema(@formula(y ~ 1 + a), df)
StatsModels.Schema with 2 entries:
  y => y
  a => a

julia> schema(Term(:a) + Term(:b), df)
StatsModels.Schema with 2 entries:
  a => a
  b => b

Fully manual: term constructors

While schema is a convenient way to generate a schema automatically from a data source, in some cases it may be preferable to create a schema manually. In particular, schema peforms a complete sweep through the data, and if your dataset is very large or truly streaming (online), then this may be undesirable. In such cases, you can construct a schema from instances of the relevant concrete terms (ContinuousTerm or CategoricalTerm), in a number of ways.

The constructors for concrete terms provide the maximum level of control. A ContinuousTerm stores values for the mean, standard deviation, minimum, and maximum, while a CategoricalTerm stores the StatsModels.ContrastsMatrix that defines the mapping from levels to predictors, and these need to be manually supplied to the constructors:

Warning

The format of the invariants stored in a term are implementation details and subject to change.

julia> cont_a = ContinuousTerm(:a, 0., 1., -1., 1.)
a(continuous)

julia> cat_b = CategoricalTerm(:b, StatsModels.ContrastsMatrix(DummyCoding(), [:a, :b, :c]))
b(DummyCoding:3→2)

The Term-concrete term pairs can then be passed to the StatsModels.Schema constructor (a wrapper for the underlying Dict{Term,AbstractTerm}):

julia> sch1 = StatsModels.Schema(term(:a) => cont_a, term(:b) => cat_b)
StatsModels.Schema with 2 entries:
  a => a
  b => b

Semi-automatic: data subsets

A slightly more convenient method for generating a schema is provided by the concrete_term internal function, which extracts invariants from a data column and returns a concrete type. This can be used to generate concrete terms from data vectors constructed to have the same invariants that you care about in your actual data (e.g., the same unique values for categorical data, and the same minimum/maximum values or the same mean/variance for continuous):

julia> cont_a2 = concrete_term(term(:a), [-1., 1.])
a(continuous)

julia> cat_b2 = concrete_term(term(:b), [:a, :b, :c])
b(DummyCoding:3→2)

julia> sch2 = StatsModels.Schema(term(:a) => cont_a2, term(:b) => cat_b2)
StatsModels.Schema with 2 entries:
  a => a
  b => b

Finally, you could also call schema on a NamedTuple of vectors (e.g., a Tables.ColumnTable) with the necessary invariants:

julia> sch3 = schema((a=[-1., 1], b=[:a, :b, :c]))
StatsModels.Schema with 2 entries:
  a => a
  b => b

Semantics time (apply_schema)

The next stage of life for a formula happens when semantic information is available, which includes the schema of the data to be transformed as well as the context, or the type of model that will be fit. This stage is implemented by apply_schema. Among other things, this instantiates placeholder terms:

julia> f = @formula(y ~ 1 + a + b * c)
FormulaTerm
Response:
  y(unknown)
Predictors:
  1
  a(unknown)
  b(unknown)
  c(unknown)
  b(unknown) & c(unknown)

julia> typeof(f)
FormulaTerm{Term,Tuple{ConstantTerm{Int64},Term,Term,Term,InteractionTerm{Tuple{Term,Term}}}}

julia> f = apply_schema(f, schema(f, df))
FormulaTerm
Response:
  y(continuous)
Predictors:
  1
  a(continuous)
  b(continuous)
  c(DummyCoding:3→2)
  b(continuous) & c(DummyCoding:3→2)

julia> typeof(f)
FormulaTerm{ContinuousTerm{Float64},MatrixTerm{Tuple{InterceptTerm{true},ContinuousTerm{Float64},ContinuousTerm{Float64},CategoricalTerm{DummyCoding,String,2},InteractionTerm{Tuple{ContinuousTerm{Float64},CategoricalTerm{DummyCoding,String,2}}}}}}

This transformation is done by calling apply_schema(term, schema, modeltype) recursively on each term (the modeltype defaults to StatisticalModel when fitting a statistical model, and Nothing if apply_schema is called with only two arguments). Because apply_schema dispatches on the term, schema, and model type, this stage allows generic context-aware transformations, based on both the source (schema) and the destination (model type). This is the primary mechanisms by which the formula DSL can be extended (see below for more details)

Data time (modelcols)

At the end of "schema time", a formula encapsulates all the information needed to convert a table into a numeric model matrix. That is, it is ready for "data time". The main API method is modelcols, which when applied to a FormulaTerm returns a tuple of the numeric forms for the left- (response) and right-hand (predictor) sides.

julia> resp, pred = modelcols(f, df);

julia> resp
9-element Array{Float64,1}:
 0.23603334566204692
 0.34651701419196046
 0.3127069683360675
 0.00790928339056074
 0.4886128300795012
 0.21096820215853596
 0.951916339835734
 0.9999046588986136
 0.25166218303197185

julia> pred
9×7 Array{Float64,2}:
 1.0  1.0  0.986666   0.0  0.0  0.0       0.0
 1.0  2.0  0.555751   1.0  0.0  0.555751  0.0
 1.0  3.0  0.437108   0.0  1.0  0.0       0.437108
 1.0  4.0  0.424718   0.0  0.0  0.0       0.0
 1.0  5.0  0.773223   1.0  0.0  0.773223  0.0
 1.0  6.0  0.28119    0.0  1.0  0.0       0.28119
 1.0  7.0  0.209472   0.0  0.0  0.0       0.0
 1.0  8.0  0.251379   1.0  0.0  0.251379  0.0
 1.0  9.0  0.0203749  0.0  1.0  0.0       0.0203749

modelcols can also take a single row from a table, as a NamedTuple:

julia> using Tables

julia> modelcols(f, first(Tables.rowtable(df)))
(0.23603334566204692, [1.0, 1.0, 0.9866663668987996, 0.0, 0.0, 0.0, 0.0])

Any AbstractTerm can be passed to modelcols with a table, which returns one or more numeric arrays:

julia> t = f.rhs.terms[end]
b(continuous) & c(DummyCoding:3→2)

julia> modelcols(t, df)
9×2 Array{Float64,2}:
 0.0       0.0
 0.555751  0.0
 0.0       0.437108
 0.0       0.0
 0.773223  0.0
 0.0       0.28119
 0.0       0.0
 0.251379  0.0
 0.0       0.0203749

Extending @formula syntax

Package authors may want to create additional syntax to the @formula DSL so their users can conveniently specify particular kinds of models. StatsModels.jl provides mechanisms for such extensions that do not rely on compile time "macro magic", but on standard julian mechanisms of multiple dispatch.

Extensions have three components:

  1. Syntax: the Julia function which is given special meaning inside a formula.
  2. Context: the model type(s) where this extension applies
  3. Behavior: how tabular data is transformed under this extension

These correspond to the stages summarized above (syntax time, schema time, and data time)

An example of custom syntax: poly

As an example, we'll add syntax for specifying a polynomial regression model, which fits a regression using polynomial basis functions of a continuous predictor.

The first step is to specify the syntax we're going to use. While it's possible to use an existing function, the best practice is to define a new function to make dispatch less ambiguous.

using StatsBase
# syntax: best practice to define a _new_ function
poly(x, n) = x^n

# type of model where syntax applies: here this applies to any model type
const POLY_CONTEXT = Any

# struct for behavior
struct PolyTerm{T,D} <: AbstractTerm
    term::T
    deg::D
end

Base.show(io::IO, p::PolyTerm) = print(io, "poly($(p.term), $(p.deg))")

# for `poly` use at run-time (outside @formula), return a schema-less PolyTerm
poly(t::Symbol, d::Int) = PolyTerm(term(t), term(d))

# for `poly` use inside @formula: create a schemaless PolyTerm and apply_schema
function StatsModels.apply_schema(t::FunctionTerm{typeof(poly)},
                                  sch::StatsModels.Schema,
                                  Mod::Type{<:POLY_CONTEXT})
    apply_schema(PolyTerm(t.args_parsed...), sch, Mod)
end

# apply_schema to internal Terms and check for proper types
function StatsModels.apply_schema(t::PolyTerm,
                                  sch::StatsModels.Schema,
                                  Mod::Type{<:POLY_CONTEXT})
    term = apply_schema(t.term, sch, Mod)
    isa(term, ContinuousTerm) ||
        throw(ArgumentError("PolyTerm only works with continuous terms (got $term)"))
    isa(t.deg, ConstantTerm) ||
        throw(ArgumentError("PolyTerm degree must be a number (got $t.deg)"))
    PolyTerm(term, t.deg.n)
end

function StatsModels.modelcols(p::PolyTerm, d::NamedTuple)
    col = modelcols(p.term, d)
    reduce(hcat, [col.^n for n in 1:p.deg])
end

# the basic terms contained within a PolyTerm (for schema extraction)
StatsModels.terms(p::PolyTerm) = terms(p.term)
# names variables from the data that a PolyTerm relies on
StatsModels.termvars(p::PolyTerm) = StatsModels.termvars(p.term)
# number of columns in the matrix this term produces
StatsModels.width(p::PolyTerm) = p.deg

StatsBase.coefnames(p::PolyTerm) = coefnames(p.term) .* "^" .* string.(1:p.deg)

# output

Now, we can use poly in a formula:

julia> data = DataFrame(y = rand(4), a = rand(4), b = [1:4;])
4×3 DataFrames.DataFrame
│ Row │ y          │ a        │ b     │
│     │ Float64    │ Float64  │ Int64 │
├─────┼────────────┼──────────┼───────┤
│ 1   │ 0.236033   │ 0.488613 │ 1     │
│ 2   │ 0.346517   │ 0.210968 │ 2     │
│ 3   │ 0.312707   │ 0.951916 │ 3     │
│ 4   │ 0.00790928 │ 0.999905 │ 4     │

julia> f = @formula(y ~ 1 + poly(b, 2) * a)
FormulaTerm
Response:
  y(unknown)
Predictors:
  1
  (b)->poly(b, 2)
  a(unknown)
  (b)->poly(b, 2) & a(unknown)

julia> f = apply_schema(f, schema(data))
FormulaTerm
Response:
  y(continuous)
Predictors:
  1
  poly(b, 2)
  a(continuous)
  poly(b, 2) & a(continuous)

julia> modelcols(f.rhs, data)
4×6 Array{Float64,2}:
 1.0  1.0   1.0  0.488613  0.488613   0.488613
 1.0  2.0   4.0  0.210968  0.421936   0.843873
 1.0  3.0   9.0  0.951916  2.85575    8.56725
 1.0  4.0  16.0  0.999905  3.99962   15.9985

julia> coefnames(f.rhs)
6-element Array{String,1}:
 "(Intercept)"
 "b^1"
 "b^2"
 "a"
 "b^1 & a"
 "b^2 & a"

And in a linear regression, with simulated data where there is an effect of a^1 and of b^2 (but not a^2 or b^1):

julia> using GLM

julia> sim_dat = DataFrame(a=rand(100).-0.5, b=randn(100).-0.5);

julia> sim_dat.y = randn(100) .+ 1 .+ 2*sim_dat.a .+ 3*sim_dat.b.^2;

julia> fit(LinearModel, @formula(y ~ 1 + poly(a,2) + poly(b,2)), sim_dat)
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Array{Float64,1}},GLM.DensePredChol{Float64,LinearAlgebra.Cholesky{Float64,Array{Float64,2}}}},Array{Float64,2}}

y ~ 1 + poly(a, 2) + poly(b, 2)

Coefficients:
────────────────────────────────────────────────────────────────────────────
              Estimate  Std. Error   t value  Pr(>|t|)  Lower 95%  Upper 95%
────────────────────────────────────────────────────────────────────────────
(Intercept)   0.693521   0.159469    4.34893    <1e-4    0.376934    1.01011
a^1           2.1042     0.383115    5.49235    <1e-6    1.34362     2.86478
a^2           2.34395    1.39244     1.68334    0.0956  -0.420386    5.10829
b^1          -0.180113   0.148698   -1.21127    0.2288  -0.475317    0.11509
b^2           2.89786    0.0794572  36.4707     <1e-56   2.74012     3.05561
────────────────────────────────────────────────────────────────────────────

Making special syntax "runtime friendly"

When used from the @formula macro, special syntax relies on dispatching on the FunctionTerm{MyFunction} type. But when creating a formula at runtime without the @formula macro, FunctionTerms aren't available, and so care must be taken to make sure you provide a runtime replacement. The example for poly above shows how to do this, but we spell it out here in more detail.

The first step is to make sure you can create a schema-less instance of the AbstractTerm that implements your special syntax behavior. For the poly example, that means we need to be able to create a PolyTerm(term(column_name), term(poly_degree)). In order to do this, the types of the term and deg fields aren't specified but are parameters of the PolyTerm type.

The second step is to provide a runtime method for the special syntax function (poly), which accepts arguments in form that's convenient at runtime. For this example, we've defined poly(s::Symbol, i::Int) = PolyTerm(term(s), term(i)):

julia> pt = poly(:a, 3)
poly(a, 3)

julia> typeof(pt) # contains schema-less `Term`
PolyTerm{Term,ConstantTerm{Int64}}
Note

The functions like poly should be exported by the package that provides the special syntax for two reasons. First, it makes run-time term construction more convenient. Second, because of how the @formula macro generates code, the function that represents special syntax must be available in the namespace where @formula is called. This is because calls to arbitrary functions f are lowered to FunctionTerm{tyepof(f)}.

Now we can programmatically construct PolyTerms at run-time:

julia> my_col = :a; my_degree = 3;

julia> poly(my_col, my_degree)
poly(a, 3)

julia> poly.([:a, :b], my_degree)
2-element Array{PolyTerm{Term,ConstantTerm{Int64}},1}:
 poly(a, 3)
 poly(b, 3)

These run-time PolyTerms are "schema-less" though, and to be able to construct a model matrix from them we need to have a way to apply a schema. Thus, the third and final step is to provide an apply_schema method that upgrades a schema-less instance to one with a schema (i.e., one that can be used with modelcols). For example, we've specified apply_schema(pt::PolyTerm, ...) which calls apply_schema on the wrapped pt.term, returning a new PolyTerm with the instantiated result:

julia> pt = apply_schema(PolyTerm(term(:b), term(2)),
                         schema(data),
                         StatisticalModel)
poly(b, 2)

julia> typeof(pt) # now holds a `ContinuousTerm`
PolyTerm{ContinuousTerm{Float64},Int64}

julia> modelcols(pt, data)
4×2 Array{Int64,2}:
 1   1
 2   4
 3   9
 4  16

Now with these methods in place, we can run exactly the same polynomial regression as above (which used @formula(y ~ 1 + poly(a, 2) + poly(b, 2)), but with the predictor names and the polynomial degree stored in variables:

julia> poly_vars = (:a, :b); poly_deg = 2;

julia> poly_formula = term(:y) ~ term(1) + poly.(poly_vars, poly_deg)
FormulaTerm
Response:
  y(unknown)
Predictors:
  1
  poly(a, 2)
  poly(b, 2)

julia> fit(LinearModel, poly_formula, sim_dat)
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Array{Float64,1}},GLM.DensePredChol{Float64,LinearAlgebra.Cholesky{Float64,Array{Float64,2}}}},Array{Float64,2}}

y ~ 1 + poly(a, 2) + poly(b, 2)

Coefficients:
────────────────────────────────────────────────────────────────────────────
              Estimate  Std. Error   t value  Pr(>|t|)  Lower 95%  Upper 95%
────────────────────────────────────────────────────────────────────────────
(Intercept)   0.693521   0.159469    4.34893    <1e-4    0.376934    1.01011
a^1           2.1042     0.383115    5.49235    <1e-6    1.34362     2.86478
a^2           2.34395    1.39244     1.68334    0.0956  -0.420386    5.10829
b^1          -0.180113   0.148698   -1.21127    0.2288  -0.475317    0.11509
b^2           2.89786    0.0794572  36.4707     <1e-56   2.74012     3.05561
────────────────────────────────────────────────────────────────────────────

Defining the context where special syntax applies

The third argument to apply_schema determines the contexts in which the special poly syntax applies.

For instance, it's possible to block interpretation of the poly syntax as special in certain contexts by adding additional (more specific) methods. If for some reason we wanted to block PolyTerms being generated for GLM.LinearModel, then we just need to add the appropriate method:

julia> StatsModels.apply_schema(t::FunctionTerm{typeof(poly)},
                                sch::StatsModels.Schema,
                                Mod::Type{GLM.LinearModel}) = t

Now in the context of a LinearModel, the poly is interpreted as a call to the "vanilla" function defined first, which just raises its first argument to the designated power:

julia> f = apply_schema(@formula(y ~ 1 + poly(b,2) * a),
                        schema(data),
                        GLM.LinearModel)
FormulaTerm
Response:
  y(continuous)
Predictors:
  1
  (b)->poly(b, 2)
  a(continuous)
  (b)->poly(b, 2) & a(continuous)

julia> modelcols(f.rhs, data)
4×4 Array{Float64,2}:
 1.0   1.0  0.488613   0.488613
 1.0   4.0  0.210968   0.843873
 1.0   9.0  0.951916   8.56725
 1.0  16.0  0.999905  15.9985

julia> coefnames(f.rhs)
4-element Array{String,1}:
 "(Intercept)"
 "poly(b, 2)"
 "a"
 "poly(b, 2) & a"

But by using a different context (e.g., the related but more general GLM.GeneralizedLinearModel) we get the custom interpretation:

julia> f2 = apply_schema(@formula(y ~ 1 + poly(b,2) * a),
                         schema(data),
                         GLM.GeneralizedLinearModel)
FormulaTerm
Response:
  y(continuous)
Predictors:
  1
  poly(b, 2)
  a(continuous)
  poly(b, 2) & a(continuous)

julia> modelcols(f2.rhs, data)
4×6 Array{Float64,2}:
 1.0  1.0   1.0  0.488613  0.488613   0.488613
 1.0  2.0   4.0  0.210968  0.421936   0.843873
 1.0  3.0   9.0  0.951916  2.85575    8.56725
 1.0  4.0  16.0  0.999905  3.99962   15.9985

julia> coefnames(f2.rhs)
6-element Array{String,1}:
 "(Intercept)"
 "b^1"
 "b^2"
 "a"
 "b^1 & a"
 "b^2 & a"

The definitions of these methods control how models of each type are fit from a formula with a call to poly:

julia> sim_dat = DataFrame(b=randn(100));

julia> sim_dat.y = randn(100) .+ 1 .+ 2*sim_dat.b .+ 3*sim_dat.b.^2;

julia> fit(LinearModel, @formula(y ~ 1 + poly(b,2)), sim_dat)
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Array{Float64,1}},GLM.DensePredChol{Float64,LinearAlgebra.Cholesky{Float64,Array{Float64,2}}}},Array{Float64,2}}

y ~ 1 + :(poly(b, 2))

Coefficients:
───────────────────────────────────────────────────────────────────────────
             Estimate  Std. Error   t value  Pr(>|t|)  Lower 95%  Upper 95%
───────────────────────────────────────────────────────────────────────────
(Intercept)  0.911363    0.310486   2.93528    0.0042   0.295214    1.52751
poly(b, 2)   2.94442     0.191024  15.4139     <1e-27   2.56534     3.3235
───────────────────────────────────────────────────────────────────────────

julia> fit(GeneralizedLinearModel, @formula(y ~ 1 + poly(b,2)), sim_dat, Normal())
StatsModels.TableRegressionModel{GeneralizedLinearModel{GLM.GlmResp{Array{Float64,1},Normal{Float64},IdentityLink},GLM.DensePredChol{Float64,LinearAlgebra.Cholesky{Float64,Array{Float64,2}}}},Array{Float64,2}}

y ~ 1 + poly(b, 2)

Coefficients:
──────────────────────────────────────────────────────────────────────────
             Estimate  Std. Error  z value  Pr(>|z|)  Lower 95%  Upper 95%
──────────────────────────────────────────────────────────────────────────
(Intercept)  0.829374   0.131582    6.3031    <1e-9    0.571478    1.08727
b^1          2.13096    0.100552   21.1926    <1e-98   1.93388     2.32804
b^2          3.1132     0.0813107  38.2877    <1e-99   2.95384     3.27257
──────────────────────────────────────────────────────────────────────────

(a GeneralizeLinearModel with a Normal distribution is equivalent to a LinearModel)

Summary

"Custom syntax" means that calls to a particular function in a formula are not interpreted as normal Julia code, but rather as a particular (possibly special) kind of term.

Custom syntax is a combination of syntax (Julia function) and term (subtype of AbstractTerm). This syntax applies in a particular context (schema plus model type, designated via a method of apply_schema), transforming a FunctionTerm{syntax} into another (often custom) term type. This custom term type then specifies special behavior at data time (via a method for modelcols).

Finally, note that it's easy for a package to intercept the formula terms and manipulate them directly as well, before calling apply_schema or modelcols. This gives packages great flexibility in how they interpret formula terms.