delicatessen.estimating_equations.regression.ee_mlogit

ee_mlogit(theta, X, y, weights=None, offset=None)

Estimating equation for multinomial logistic regression. This estimating equation functionality supports unranked categorical outcome data, unlike ee_regression and ee_glm.

Unlike the other regression estimating equations, ee_mlogit expects a matrix of indicators for each possible value of y, with the first column being used as the referent category. In other words, the outcome variable is a matrix of dummy variables that includes the reference. The estimating equation for column \(r\) of the indicator variable \(Y_{r}\) of a \(Y\) with \(k\) unique categories is

\[\sum_{i=1}^n \left\{ Y_{r,i} - \frac{\exp(X_i^T \theta_r)}{1 + \sum_{j=2}^{k} \exp(X_i^T \theta_j)} \right\} X_i = 0\]

where \(\theta_r\) are the coefficients correspond to the log odds ratio comparing \(Y_r\) to all other categories of \(Y\). Here, \(\theta\) is a 1-by-(b :math`times` (k-1)) array, where b is the distinct covariates included as part of X. So, the stack of estimating equations consists of (k-1) estimating equations of the dimension \(X_i\). For example, if X is a 3-by-n matrix and \(Y\) has three unique categories, then \(\theta\) will be a 1-by-6 array.

Parameters

  • theta (ndarray, list, vector) – Theta in this case consists of b \(\times\) (k-1) values. Therefore, initial values should consist of the same number as the number of columns present in the design matrix for each category of the outcome matrix besides the reference.

  • X (ndarray, list, vector) – 2-dimensional design matrix of n observed covariates for b variables.

  • y (ndarray, list, vector) – 2-dimensional indicator matrix of n observed outcomes.

  • weights (ndarray, list, vector, None, optional) – 1-dimensional vector of n weights. Default is None, which assigns a weight of 1 to all observations.

  • offset (ndarray, list, vector, None, optional) – A 1-dimensional offset to be included in the model. Default is None, which applies no offset term.

Returns

Returns a (b \(\times\) (k-1))-by-n NumPy array evaluated for the input theta.

Return type

array

Examples

Construction of a estimating equation(s) with ee_regression should be done similar to the following

>>> import numpy as np
>>> import pandas as pd
>>> from delicatessen import MEstimator
>>> from delicatessen.estimating_equations import ee_mlogit

Some generic data to estimate a multinomial logistic regression model

>>> d = pd.DataFrame()
>>> d['W'] = [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
>>> d['Y'] = [1, 1, 1, 1, 2, 2, 3, 3, 3, 1, 2, 2, 3, 3]
>>> d['C'] = 1

First, notice that Y needs to be pre-processed for use with ee_mlogit. To prepare the data, we need to convert d['Y'] into a matrix of indicator variables. We can do this manually by

>>> d['Y1'] = np.where(d['Y'] == 1, 1, 0)
>>> d['Y2'] = np.where(d['Y'] == 2, 1, 0)
>>> d['Y3'] = np.where(d['Y'] == 3, 1, 0)

This can also be accomplished with pd.get_dummies(d['Y'], drop_first=False).

For the reference category, we want to have Y=1 as the reference. Therefore, Y1 will be the first column in y. The pair of matrices are

>>> y = d[['Y1', 'Y2', 'Y3']]
>>> X = d[['C', 'W']]

Defining psi, or the stacked estimating equations

>>> def psi(theta):
>>>     return ee_mlogit(theta, X=X, y=y)

Calling the M-estimator (note that init requires 4 values, since X.shape[1] is 2 and y.shape[1] is 3).

>>> estr = MEstimator(stacked_equations=psi, init=[0., 0., 0., 0.])
>>> estr.estimate()

Inspecting the parameter estimates, variance, and confidence intervals

>>> estr.theta
>>> estr.variance
>>> estr.confidence_intervals()

Here, the first two values of theta correspond to Y2 and the last two values of theta correspond to Y3.

A weighted multinomial logistic regression can be implemented by specifying the weights argument. An offset can be added by specifying the offset argument.

References

Kwak C & Clayton-Matthews A. (2002). Multinomial logistic regression. Nursing Research, 51(6), 404-410.