delicatessen.estimating_equations.regression.ee_meta_regression

ee_meta_regression(theta, point_est, var_est, X=None)

Estimating equation for random-effects meta-regression using the Paule-Mandel method. This estimating equation allows for one to regress multiple point estimates with the inverse-variance weighted average. Importantly, this estimating equation also incorporates heterogeneity between studies via a random effect. The corresponding estimating equations are

\[\begin{split}\sum_{j=1}^n \begin{bmatrix} \frac{1}{V_j + \tau^2} \times (E_j - X_j \beta^T) \\ \frac{(E_j - \mu)^2}{V_j + \tau^2} - \frac{k-1}{k} \end{bmatrix} = 0\end{split}\]

where \(\theta = (\beta, \tau^2)\), \(\beta\) is the weighted meta-regressionmean cofficients, \(\tau^2\) is the between-study heterogeneity term, \(E_j\) is the point estimate from study \(j\), \(X_j\) is the design matrix consisting of the study covariates, and \(V_j\) is the variance estimate.

Here, delicatessen solves for the log-transformed \(\tau^2\) rather than \(\tau^2\) directly. This is due to \(\tau^2 > 0\) and the log constraint enforces this to be non-negative. Note that issues will arise if \(\tau^2 = 0\), but a random-effect model would not be appropriate in that setting.

Note

Due to some of the underlying computations (an inversion of a matrix), deriv_method='exact' cannot be used with meta-regression.

Parameters

  • theta (ndarray, list, vector) – Theta consists of p+1 values.

  • point_est (ndarray, list, vector) – 1-dimensional vector of s point estimates. Note that these should be on a linear scale (e.g., risk or mean differences, log risk ratios)

  • var_est (ndarray, list, vector) – 1-dimensional vector of s corresponding variance estimates. Note that these should be on the same scale as those provided in point_est.

  • X (ndarray, list, vector, None) – 2-dimensional vector of p covariates and s corresponding observations. Default is None, which uses an intercept-only meta-regression model.

Returns

Returns (p`+1)-by-`n NumPy array evaluated for the input theta.

Return type

array

Examples

Construction of estimating equations with ee_meta_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_meta_regression

Some generic data to estimate a random-effects meta-regression model with.

>>> d = pd.DataFrame()
>>> d['X'] = [0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 0]
>>> d['P'] = [-0.89, -1.58, -1.35, -1.44, -0.22, -0.78, -1.62, 0.02, -0.46, -1.37, -0.33, 0.44, -0.02]
>>> d['V'] = [0.32, 0.19, 0.41, 0.02, 0.05, 0.01, 0.22, 0.01, 0.06, 0.07, 0.02, 0.53, 0.07]
>>> d['I'] = 1

Defining psi, or the stacked estimating equations

>>> def psi(theta):
>>>     return ee_meta_regression(theta, point_est=d['P'], var_est=d['V'], X=d[['I', 'X']])

Calling the M-estimation procedure (init requires 3 values, since there is 2 coefficients in the model).

>>> estr = MEstimator(psi, init=[np.mean(d['P']), 0., 0.])
>>> estr.estimate(solver='lm')

When considering ratio measures, the meta-regression model should be fit using the log-transformed point estimates (and the variance estimate for the log-ratio). delicatessen does not implement these transformations.

References

Paule RC & Mandel J. (1982). Consensus values and weighting factors. Journal of research of the National Bureau of Standards, 87(5), 377.