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.
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.
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"))
A simple container for a matrix used for coding categorical factors.
Attributes:
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 nbyn for a full rank coding or nby(n1) for a reduced rank coding, though other options are possible.
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]".
Treatment coding (also known as dummy coding).
This is the default coding.
For reducedrank 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 fullrank 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 [1]: dmatrix("C(a, Treatment)", balanced(a=3))
Out[1]:
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 [2]: dmatrix("0 + C(a, Treatment)", balanced(a=3))
Out[2]:
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 [3]: dmatrix("C(a, Treatment(1))", balanced(a=3))
Out[3]:
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 [4]: dmatrix("C(a, Treatment('a2'))", balanced(a=3))
Out[4]:
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”.
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 Taylordecomposed into the sum of a linear, quadratic, etc. components.
For reducedrank coding, you get a linear column, a quadratic column, etc., up to the number of levels provided.
For fullrank coding, the same scheme is used, except that the zeroorder 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 [1]: dmatrix("C(a, Poly)", balanced(a=4))
Out[1]:
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 [2]: dmatrix("0 + C(a, Poly)", balanced(a=3))
Out[2]:
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 [3]: dmatrix("C(a, Poly([1, 2, 10]))", balanced(a=3))
Out[3]:
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 dummycoded, regardless of what contrast you have set.)
Deviation coding (also known as sumtozero coding).
Compares the mean of each level to the meanofmeans. (In a balanced design, compares the mean of each level to the overall mean.)
For fullrank 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 [1]: dmatrix("C(a, Sum)", balanced(a=4))
Out[1]:
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 [2]: dmatrix("0 + C(a, Sum)", balanced(a=4))
Out[2]:
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 [3]: dmatrix("C(a, Sum(1))", balanced(a=3))
Out[3]:
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 [4]: dmatrix("C(a, Sum('a1'))", balanced(a=3))
Out[4]:
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.
Helmert contrasts.
Compares the second level with the first, the third with the average of the first two, and so on.
For fullrank 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 [1]: dmatrix("C(a, Helmert)", balanced(a=4))
Out[1]:
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 [2]: dmatrix("0 + C(a, Helmert)", balanced(a=4))
Out[2]:
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.
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 fullrank 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)
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 stateful transform that centers input data, i.e., subtracts the mean.
If input has multiple columns, centers each column separately.
Equivalent to standardize(x, rescale=False)
A stateful transform that standardizes input data, i.e. it subtracts the mean and divides by the sample standard deviation.
Either centering or rescaling or both can be disabled by use of keyword arguments. The ddof argument controls the delta degrees of freedom when computing the standard deviation (cf. numpy.std()). The default of ddof=0 produces the maximum likelihood estimate; use ddof=1 if you prefer the square root of the unbiased estimate of the variance.
If input has multiple columns, standardizes each column separately.
Note
This function computes the mean and standard deviation using a memoryefficient online algorithm, making it suitable for use with large incrementally processed datasets.
standardize(x, center=True, rescale=True, ddof=0)
A stateful transform that standardizes input data, i.e. it subtracts the mean and divides by the sample standard deviation.
Either centering or rescaling or both can be disabled by use of keyword arguments. The ddof argument controls the delta degrees of freedom when computing the standard deviation (cf. numpy.std()). The default of ddof=0 produces the maximum likelihood estimate; use ddof=1 if you prefer the square root of the unbiased estimate of the variance.
If input has multiple columns, standardizes each column separately.
Note
This function computes the mean and standard deviation using a memoryefficient online algorithm, making it suitable for use with large incrementally processed datasets.
Generates a Bspline basis for x, allowing nonlinear fits. The usual usage is something like:
y ~ 1 + bs(x, 4)
to fit y as a smooth function of x, with 4 degrees of freedom given to the smooth.
Parameters: 


A spline with degree=0 is piecewise constant with breakpoints at each knot, and the default knot positions are quantiles of the input. So if you find yourself in the situation of wanting to quantize a continuous variable into equalsized bins with a constant effect across each bin, you can use bs(x, num_bins, degree=0).
Similarly, a spline with degree=1 is piecewise linear with breakpoints at each knot.
The default is degree=3, which gives a cubic bspline.
This is a stateful transform (for details see Stateful transforms). If knots, lower_bound, or upper_bound are not specified, they will be calculated from the data and then the chosen values will be remembered and reused for prediction from the fitted model.
Using this function requires scipy be installed.
Note
This function is very similar to the R function of the same name. In cases where both return output at all (e.g., R’s bs will raise an error if degree=0, while patsy’s will not), they should produce identical output given identical input and parameter settings.
Warning
I’m not sure on what the proper handling of points outside the lower/upper bounds is, so for now attempting to evaluate a spline basis at such points produces an error. Patches gratefully accepted.
New in version 0.2.0.