delicatessen.estimating_equations.basic.ee_meta_random

ee_meta_random(theta, point_est, var_est)

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

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

where \(\theta = (\mu, \tau^2)\), \(\mu\) is the weighted mean across the studies, \(\tau^2\) is the between-study heterogeneity term, \(E_j\) is the point estimate from study \(j\), 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

Estimates and variances are assumed to be on the same scale. Further, the estimates should be on a scale for which a linear combination is valid (e.g., log-transformed for ratio measures).

Parameters

  • theta (ndarray, list, vector) – Theta consists of 2 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.

Returns

Returns 2-by-n NumPy array evaluated for the input theta.

Return type

array

Examples

Construction of estimating equations with ee_meta_random should be done similar to the following

>>> import numpy as np
>>> from delicatessen import MEstimator
>>> from delicatessen.estimating_equations import ee_meta_random

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

>>> p_est = [-0.186, 0.235, 0.037, -0.904, 0.218, -0.135, 0.68, -1.254, -0.154, -1.12]
>>> v_est = [0.183, 0.186, 0.054, 0.133, 0.182, 0.201, 0.154, 0.139, 0.115, 0.125]

Defining psi, or the stacked estimating equations

>>> def psi(theta):
>>>     return ee_meta_random(theta=theta, point_est=p_est, var_est=v_est)

Calling the M-estimation procedure (init requires 2 values). As starting values, the mean of the point estimates for \(\mu\) is reasonable. For \(\tau^2\), a starting value of 0 correspond to \(\exp(0) = 1\).

>>> estr = MEstimator(psi, init=[np.mean(p_est), 0.])
>>> estr.estimate()

Inspecting the parameter estimates

>>> estr.theta[0]  # Estimate for mu          (-0.269)
>>> estr.theta[1]  # Estimate for log(tau^2)  (-1.346)

When considering ratio measures, the meta-analysis model should be fit using the log-transformed point estimates (and the variance estimate for the log-ratio). delicatessen does not implement these transformations and expects the user to provide estimates in a format for which the mean is an appropriate summary.

References

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