# `patsy.builtins` API reference

This module defines some tools that are automatically made available to code evaluated in formulas. You can also access it directly; use `from patsy.builtins import *` to import the same variables that formula code receives automatically.

patsy.builtins.C(data, contrast=None, levels=None)

  Marks some data as being categorical, and specifies how to interpret it.

  This is used for three reasons:

  - To explicitly mark some data as categorical. For instance, integer data is by default treated as numerical. If you have data that is stored using an integer type, but where you want patsy to treat each different value as a different level of a categorical factor, you can wrap it in a call to C to accomplish this. E.g., compare:

        dmatrix("a", {"a": [1, 2, 3]})
        dmatrix("C(a)", {"a": [1, 2, 3]})

  - To explicitly set the levels or override the default level ordering for categorical data, e.g.:

        dmatrix("C(a, levels=["a2", "a1"])", balanced(a=2))

  - To override the default coding scheme for categorical data. The contrast argument can be any of:

    - A `ContrastMatrix` object

    - A simple 2d ndarray (which is treated the same as a ContrastMatrix object except that you can’t specify column names)

    - An object with methods called code_with_intercept and code_without_intercept, like the built-in contrasts (`Treatment`, `Diff`, `Poly`, etc.). See Coding categorical data for more details.

    - A callable that returns one of the above.

class patsy.builtins.ContrastMatrix(matrix, column_suffixes)

  A simple container for a matrix used for coding categorical factors.

  Attributes:

  matrix

    A 2d ndarray, where each column corresponds to one column of the resulting design matrix, and each row contains the entries for a single categorical variable level. Usually n-by-n for a full rank coding or n-by-(n-1) for a reduced rank coding, though other options are possible.

  column_suffixes

    A list of strings to be appended to the factor name, to produce the final column names. E.g. for treatment coding the entries will look like `"[T.level1]"`.

class patsy.builtins.Diff

  Backward difference coding.

  This coding scheme is useful for ordered factors, and compares the mean of each level with the preceding level. So you get the second level minus the first, the third level minus the second, etc.

  For full-rank coding, a standard intercept term is added (which gives the mean value for the first level).

  Examples:

      # Reduced rank
      In [1]: dmatrix("C(a, Diff)", balanced(a=3))
      Out[1]:
      DesignMatrix with shape (3, 3)
        Intercept  C(a, Diff)[D.a1]  C(a, Diff)[D.a2]
                1          -0.66667          -0.33333
                1           0.33333          -0.33333
                1           0.33333           0.66667
        Terms:
          'Intercept' (column 0)
          'C(a, Diff)' (columns 1:3)

      # Full rank
      In [2]: dmatrix("0 + C(a, Diff)", balanced(a=3))
      Out[2]:
      DesignMatrix with shape (3, 3)
        C(a, Diff)[D.a1]  C(a, Diff)[D.a2]  C(a, Diff)[D.a3]
                       1          -0.66667          -0.33333
                       1           0.33333          -0.33333
                       1           0.33333           0.66667
        Terms:
          'C(a, Diff)' (columns 0:3)

  code_with_intercept(levels)
  code_without_intercept(levels)

class patsy.builtins.Helmert

  Helmert contrasts.

  Compares the second level with the first, the third with the average of the first two, and so on.

  For full-rank coding, a standard intercept term is added.

  Warning

  There are multiple definitions of ‘Helmert coding’ in use. Make sure this is the one you expect before trying to interpret your results!

  Examples:

      # Reduced rank
      In [3]: dmatrix("C(a, Helmert)", balanced(a=4))
      Out[3]:
      DesignMatrix with shape (4, 4)
        Intercept  C(a, Helmert)[H.a2]  C(a, Helmert)[H.a3]  C(a, Helmert)[H.a4]
                1                   -1                   -1                   -1
                1                    1                   -1                   -1
                1                    0                    2                   -1
                1                    0                    0                    3
        Terms:
          'Intercept' (column 0)
          'C(a, Helmert)' (columns 1:4)

      # Full rank
      In [4]: dmatrix("0 + C(a, Helmert)", balanced(a=4))
      Out[4]:
      DesignMatrix with shape (4, 4)
        Columns:
          ['C(a, Helmert)[H.intercept]',
           'C(a, Helmert)[H.a2]',
           'C(a, Helmert)[H.a3]',
           'C(a, Helmert)[H.a4]']
        Terms:
          'C(a, Helmert)' (columns 0:4)
        (to view full data, use np.asarray(this_obj))

  This is equivalent to R’s contr.helmert.

  code_with_intercept(levels)
  code_without_intercept(levels)

patsy.builtins.I(x)

  The identity function. Simply returns its input unchanged.

  Since Patsy’s formula parser ignores anything inside a function call syntax, this is useful to ‘hide’ arithmetic operations from it. For instance:

      y ~ x1 + x2

  has `x1` and `x2` as two separate predictors. But in:

      y ~ I(x1 + x2)

  we instead have a single predictor, defined to be the sum of `x1` and `x2`.

class patsy.builtins.Poly(scores=None)

  Orthogonal polynomial contrast coding.

  This coding scheme treats the levels as ordered samples from an underlying continuous scale, whose effect takes an unknown functional form which is Taylor-decomposed into the sum of a linear, quadratic, etc. components.

  For reduced-rank coding, you get a linear column, a quadratic column, etc., up to the number of levels provided.

  For full-rank coding, the same scheme is used, except that the zero-order constant polynomial is also included. I.e., you get an intercept column included as part of your categorical term.

  By default the levels are treated as equally spaced, but you can override this by providing a value for the scores argument.

  Examples:

      # Reduced rank
      In [5]: dmatrix("C(a, Poly)", balanced(a=4))
      Out[5]:
      DesignMatrix with shape (4, 4)
        Intercept  C(a, Poly).Linear  C(a, Poly).Quadratic  C(a, Poly).Cubic
                1           -0.67082                   0.5          -0.22361
                1           -0.22361                  -0.5           0.67082
                1            0.22361                  -0.5          -0.67082
                1            0.67082                   0.5           0.22361
        Terms:
          'Intercept' (column 0)
          'C(a, Poly)' (columns 1:4)

      # Full rank
      In [6]: dmatrix("0 + C(a, Poly)", balanced(a=3))
      Out[6]:
      DesignMatrix with shape (3, 3)
        C(a, Poly).Constant  C(a, Poly).Linear  C(a, Poly).Quadratic
                          1           -0.70711               0.40825
                          1           -0.00000              -0.81650
                          1            0.70711               0.40825
        Terms:
          'C(a, Poly)' (columns 0:3)

      # Explicit scores
      In [7]: dmatrix("C(a, Poly([1, 2, 10]))", balanced(a=3))
      Out[7]:
      DesignMatrix with shape (3, 3)
        Intercept  C(a, Poly([1, 2, 10])).Linear  C(a, Poly([1, 2, 10])).Quadratic
                1                       -0.47782                           0.66208
                1                       -0.33447                          -0.74485
                1                        0.81229                           0.08276
        Terms:
          'Intercept' (column 0)
          'C(a, Poly([1, 2, 10]))' (columns 1:3)

  This is equivalent to R’s `contr.poly`. (But note that in R, reduced rank encodings are always dummy-coded, regardless of what contrast you have set.)

  code_with_intercept(levels)
  code_without_intercept(levels)

patsy.builtins.Q(name)

  A way to ‘quote’ variable names, especially ones that do not otherwise meet Python’s variable name rules.

  If `x` is a variable, `Q("x")` returns the value of `x`. (Note that `Q` takes the string `"x"`, not the value of `x` itself.) This works even if instead of `x`, we have a variable name that would not otherwise be legal in Python.

  For example, if you have a column of data named `weight.in.kg`, then you can’t write:

      y ~ weight.in.kg

  because Python will try to find a variable named `weight`, that has an attribute named `in`, that has an attribute named `kg`. (And worse yet, `in` is a reserved word, which makes this example doubly broken.) Instead, write:

      y ~ Q("weight.in.kg")

  and all will be well. Note, though, that this requires embedding a Python string inside your formula, which may require some care with your quote marks. Some standard options include:

      my_fit_function("y ~ Q('weight.in.kg')", ...)
      my_fit_function('y ~ Q("weight.in.kg")', ...)
      my_fit_function("y ~ Q(\"weight.in.kg\")", ...)

  Note also that `Q` is an ordinary Python function, which means that you can use it in more complex expressions. For example, this is a legal formula:

      y ~ np.sqrt(Q("weight.in.kg"))

class patsy.builtins.Sum(omit=None)

  Deviation coding (also known as sum-to-zero coding).

  Compares the mean of each level to the mean-of-means. (In a balanced design, compares the mean of each level to the overall mean.)

  For full-rank coding, a standard intercept term is added.

  One level must be omitted to avoid redundancy; by default this is the last level, but this can be adjusted via the omit argument.

  Warning

  There are multiple definitions of ‘deviation coding’ in use. Make sure this is the one you expect before trying to interpret your results!

  Examples:

      # Reduced rank
      In [8]: dmatrix("C(a, Sum)", balanced(a=4))
      Out[8]:
      DesignMatrix with shape (4, 4)
        Intercept  C(a, Sum)[S.a1]  C(a, Sum)[S.a2]  C(a, Sum)[S.a3]
                1                1                0                0
                1                0                1                0
                1                0                0                1
                1               -1               -1               -1
        Terms:
          'Intercept' (column 0)
          'C(a, Sum)' (columns 1:4)

      # Full rank
      In [9]: dmatrix("0 + C(a, Sum)", balanced(a=4))
      Out[9]:
      DesignMatrix with shape (4, 4)
        C(a, Sum)[mean]  C(a, Sum)[S.a1]  C(a, Sum)[S.a2]  C(a, Sum)[S.a3]
                      1                1                0                0
                      1                0                1                0
                      1                0                0                1
                      1               -1               -1               -1
        Terms:
          'C(a, Sum)' (columns 0:4)

      # Omit a different level
      In [10]: dmatrix("C(a, Sum(1))", balanced(a=3))
      Out[10]:
      DesignMatrix with shape (3, 3)
        Intercept  C(a, Sum(1))[S.a1]  C(a, Sum(1))[S.a3]
                1                   1                   0
                1                  -1                  -1
                1                   0                   1
        Terms:
          'Intercept' (column 0)
          'C(a, Sum(1))' (columns 1:3)

      In [11]: dmatrix("C(a, Sum('a1'))", balanced(a=3))
      Out[11]:
      DesignMatrix with shape (3, 3)
        Intercept  C(a, Sum('a1'))[S.a2]  C(a, Sum('a1'))[S.a3]
                1                     -1                     -1
                1                      1                      0
                1                      0                      1
        Terms:
          'Intercept' (column 0)
          "C(a, Sum('a1'))" (columns 1:3)

  This is equivalent to R’s contr.sum.

  code_with_intercept(levels)
  code_without_intercept(levels)

class patsy.builtins.Treatment(reference=None)

  Treatment coding (also known as dummy coding).

  This is the default coding.

  For reduced-rank coding, one level is chosen as the “reference”, and its mean behaviour is represented by the intercept. Each column of the resulting matrix represents the difference between the mean of one level and this reference level.

  For full-rank coding, classic “dummy” coding is used, and each column of the resulting matrix represents the mean of the corresponding level.

  The reference level defaults to the first level, or can be specified explicitly.

      # reduced rank
      In [12]: dmatrix("C(a, Treatment)", balanced(a=3))
      Out[12]:
      DesignMatrix with shape (3, 3)
        Intercept  C(a, Treatment)[T.a2]  C(a, Treatment)[T.a3]
                1                      0                      0
                1                      1                      0
                1                      0                      1
        Terms:
          'Intercept' (column 0)
          'C(a, Treatment)' (columns 1:3)

      # full rank
      In [13]: dmatrix("0 + C(a, Treatment)", balanced(a=3))
      Out[13]:
      DesignMatrix with shape (3, 3)
        C(a, Treatment)[a1]  C(a, Treatment)[a2]  C(a, Treatment)[a3]
                          1                    0                    0
                          0                    1                    0
                          0                    0                    1
        Terms:
          'C(a, Treatment)' (columns 0:3)

      # Setting a reference level
      In [14]: dmatrix("C(a, Treatment(1))", balanced(a=3))
      Out[14]:
      DesignMatrix with shape (3, 3)
        Intercept  C(a, Treatment(1))[T.a1]  C(a, Treatment(1))[T.a3]
                1                         1                         0
                1                         0                         0
                1                         0                         1
        Terms:
          'Intercept' (column 0)
          'C(a, Treatment(1))' (columns 1:3)

      In [15]: dmatrix("C(a, Treatment('a2'))", balanced(a=3))
      Out[15]:
      DesignMatrix with shape (3, 3)
        Intercept  C(a, Treatment('a2'))[T.a1]  C(a, Treatment('a2'))[T.a3]
                1                            1                            0
                1                            0                            0
                1                            0                            1
        Terms:
          'Intercept' (column 0)
          "C(a, Treatment('a2'))" (columns 1:3)

  Equivalent to R `contr.treatment`. The R documentation suggests that using `Treatment(reference=-1)` will produce contrasts that are “equivalent to those produced by many (but not all) SAS procedures”.

  code_with_intercept(levels)
  code_without_intercept(levels)