Using Calibration Weighting to Adjust for Nonresponse and Coverage Errors

Using Calibration Weighting to Adjust for Nonresponse and Coverage Errors was written by Phillip S. Kott in 2006. It was published in Survey Methodology (vol. 32, no. 2).

The authors build on Deville and Särndal (1992) and Folsom and Singh (2000). They introduce a novel calibration procedure and examine its properties under nonresponse and coverage error.

Establishing some notation:

• individuals are indexed by k

• the universe of individuals is U with size N

• the sample of individuals is S

• π_k is the probability of selection

• true population parameter is T_y, and the corresponding survey response item is y_k

• the indicator function I_k takes the value 1 if an individual is sampled, i.e. k ∈ S, and 0 otherwise

The true population parameter would be calculated as T_y = Σ_U y_k. An unbiased 'expansion' (design weighted) estimator is t_{y_E} = Σ_U y_kI_k/π_k = Σ_S y_k/π_k. (It is unbiased because sample selection is sufficiently random.) We can also define a_k = I_k/π_k so that t_{y_E} = Σ_S a_ky_k.

Some further notation:

• P auxiliary variables are known for the universe

• individual items are expressed as a row vector: x_k = <x_1,k ... x_P,k>

  • Emphasis here! This is implicitly a row vector!

• true population parameters are also a vector: T_x

Weights w_k are calculated such that T_x = Σ_U x_k = Σ_S w_kx_k (the 'calibration equation'). There is a continuum of such weights, so they should be selected with respect to some loss function and distance from the design weights. It follows that the calibration estimator is t_{y_CAL} = Σ_S w_ky_k. This is also unbiased in the sense that the expected value of random errors is 0.

Some further notation:

• c_k is a constant which may or may not be based on x_k

• q is a vector calculated as

• Again, recall that x_k is implicitly a row vector!

The GREG estimator is t_{y_GREG} = t_{y_E} + q c_ka_kx'_k. We require that Λ = lim_{N -> ∞} c_kx'_kx_k/N is a positive definite matrix such that it is invertible. We can also define w_k = a_k + q c_ka_kx'_k = a_k(1 + c_kx'_kq) so that t_{y_GREG} = Σ_S w_ky_k. This is essentially a 'calibration form' of the GREG estimator, demonstrating that they are equivalent. (Or rather, that GREG estimators are an optimized calibration routine.) It can be proven that this estimator is randomization consistent.

By releasing the requirement that calibrated weights should minimize distance from design weights, weights become defined by w_k = a_k(1 + h_kq) where h_k is another (implicitly row) vector with the same dimension as x_k. (We still have a parallel requirement that the matrix calculated as Σ_S a_kh'_kx_k be invertible.) Some further notes:

• basically h_k replaces c_kx_k

• any components of h_k that are not linear combinations of x_k are instrumental variables

• it follows that q is now calculated as

• for all the same reasons as the GREG estimator, this calibration estimator is randomization consistent

The author further generalizes the calibration estimator using w_{k_GEN} = a_k f(h_kq*) where:

• f is a function with the following properties:

  • monotonic

  • twice-differentiable

  • f(0) = 1

  • f'(0) = 1

• q* is chosen to satisfy the calibration equation

  • can usually be derived iteratives, i.e. start with a zero vector, determine the error compared to known population parameters, add some portion of that error to the working vector, loop until converges

This is generalized because f can be nonlinear. Consider f(h_kq) = exp(x_kq) when all x_k are binary (0 or 1).

Now the author explores the use of this framework for nonresponse weighting. If the possibility of non-sampling error is considered, then f(0) and f'(0) are not constrained to be 1. The logistic function is an option. Each individual has an independent probability of responding as p_k = 1/f(h_kφ).

Calibration can also correct under-coverage errors. This is an extension of simple post-stratification.

Reading Notes

I get lost in section 5, which discusses asymptotic properties to estimate variance. The parts of section 6 which discuss quasi-randomization I also think I'm not 100% following.

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