Sensitivity of a mean to nonignorable nonresponse or selection

A diagnostic step (it does not change any weight) that gauges how much the weighted mean of a study variable could move if response, or participation in a non-probability sample, depended on the outcome itself beyond the observed auxiliaries. It implements the proxy pattern-mixture model of Andridge and Little (2011): the auxiliaries are reduced to a single proxy (the respondent regression prediction of y), and a single sensitivity parameter phi in [0, 1] moves the mechanism from ignorable given the proxy (phi = 0, MAR) to depending only on the outcome (phi = 1). Evaluated over a grid of phi, the adjusted means form an ignorance interval to read alongside the sampling confidence interval; see nr_sensitivity() and the report block.

Usage

step_nr_sensitivity(
  spec,
  y,
  formula,
  respondent = NULL,
  eligible = NULL,
  phi = c(0, 0.25, 0.5, 0.75, 1),
  id = NULL
)

Arguments

spec: a weighting_spec.
y: the study variable (bare column name), observed for respondents and NA for nonrespondents.
formula: one-sided formula of the auxiliaries for the proxy, observed for all units, e.g. ~ region + sex + age.
respondent: optional response/participation indicator (bare column or condition). Defaults to !is.na(y).
eligible: optional in-scope indicator (bare column or condition), the mirror of the argument in step_nonresponse(). Out-of-scope (ineligible) units are neither respondents nor nonrespondents and must be excluded, or they would be counted as nonrespondents and pull the estimate toward their proxy mean. Give it in any household survey that has ineligible units. Default NULL treats every active unit as in scope.
phi: the sensitivity grid, values in [0, 1]; 0 (MAR) is always added. Little et al. (2020) suggest 0.5 as a central value; above 0.5 the implied mechanism is often unrealistically strong.
id: optional stable step id.

Value

the input weighting_spec with this diagnostic step appended.

Details

The adjusted mean at sensitivity $\phi$ is $$\mu(\phi) = \bar{y}_r + (1 - \pi)\,\frac{s_{yr}}{s_{xr}}\,m(\phi)\,(\bar{x}_{nr} - \bar{x}_r),$$ with slope $m(\phi) = \frac{(1-\phi)\rho + \phi}{(1-\phi) + \phi\rho}$, so that $m(0) = \rho$ (ignorable given the proxy) and $m(1) = 1/\rho$.

The proxy correlation rho (the multiple correlation of y on the auxiliaries among respondents) sets how informative the auxiliaries are: a weak proxy widens the ignorance interval (at phi = 1 the slope is 1/rho). The step reads the base design weights, so place it anywhere in the recipe; it needs the nonrespondents still present (a study variable that is NA for them, or an explicit respondent indicator).

References

Andridge, R. R. and Little, R. J. A. (2011). Proxy pattern-mixture analysis for survey nonresponse. Journal of Official Statistics 27(2), 153-180.