# Non-probability samples

Opt-in panels, volunteer surveys and river samples have no design weights: the inclusion probabilities are unknown and, worse, related to what we measure. A weightflow recipe treats such a sample the same way it treats a probability sample – an explicit, auditable sequence of adjustments with recipe-aware variances – but the first steps change: instead of expanding design weights, we estimate participation and adjust for it against a probability reference.

## A biased volunteer sample

We take a volunteer sample from the bundled `population` in which men over- participate, so the sample mean of `income` is biased. A small probability survey serves as the reference, carrying the design weights of a real reference survey.

    set.seed(1)
    N   <- nrow(population)
    p   <- plogis(-2 + 0.9 * (population$sex == "M") - 0.02 * (population$age - 45))
    vol <- population[runif(N) < p, c("region", "sex", "age", "income")]  # volunteers
    ref <- population[sample(N, 1000), c("region", "sex", "age")]         # reference
    ref$d <- N / 1000                                                     # its design weights

    truth <- mean(population$income)
    c(truth = truth, naive_volunteer_mean = mean(vol$income))
    #>                truth naive_volunteer_mean
    #>             19298.11             19827.23

## Declaring a non-probability sample

Start the recipe with `base_weights = NULL` and `nonprob = TRUE`: every unit begins with a base weight of 1, and the object records that the sample is non-probability (the report declares it and adds the methodological caveat).

    ref_sample <- reference_sample(ref, weights = "d")

## Pseudo-weighting

`step_pseudoweight()` fits a participation model that separates the volunteers from the reference and assigns each volunteer the inverse-propensity pseudo-weight `(1 - p) / p` (Elliott and Valliant 2017), so the weighted volunteers reproduce the population. The two samples are stacked internally – no manual pooling.

    spec <- weighting_spec(vol, base_weights = NULL, nonprob = TRUE) |>
      step_pseudoweight(reference = ref_sample, formula = ~ region + sex + age,
                        engine = "logit")
    fit <- prep(spec)

    w <- fit$final_weight
    c(pseudoweighted_mean = weighted.mean(vol$income, w),
      sum_of_weights = sum(w), N = N)     # the (1-p)/p pseudo-weights sum to N
    #> pseudoweighted_mean      sum_of_weights                   N
    #>           19752.506            4480.998            4495.000

The pseudo-weighted mean is closer to the truth than the naive volunteer mean, and the weights sum to the population size (a `1/p` mistake would overshoot). Flexible learners (`engine = "tree" / "forest" / "boost"`, optionally cross-fitted) are available for the participation model, exactly as for response-propensity models.

## Doubly robust: add calibration

Pseudo-weighting is consistent if the participation model is right; calibration is consistent if a linear outcome model is right. Doing both is doubly robust – correct if either holds. Calibrate the pseudo-weighted sample to the reference:

    weighting_spec(vol, base_weights = NULL, nonprob = TRUE) |>
      step_pseudoweight(reference = ref_sample, formula = ~ region + sex + age,
                        engine = "logit") |>
      step_calibrate(method = "raking", formula = ~ region + sex,
                     population = ref_sample)

## How large is the sample, really?

A big opt-in sample can carry a small effective one. `data_defect()` brings the data-defect view of Meng (2018): the effective size is governed by the correlation between the outcome and participation, not by the raw count. Because that correlation is not observable from the sample alone, the effective size is reported across a grid of plausible values – an ignorance range, not a single number.

    dd <- data_defect(fit)
    dd
    #> Data-defect diagnostics (non-probability sample)
    #>   n = 916   N = 4,481   f = 0.2044
    #>   strongest covariate-participation correlation: |r| = 0.177 (sexM)
    #>   effective size by residual data-defect correlation (Meng 2018):
    #>     |rho| = 0.001  ->  n_eff = 256,943
    #>     |rho| = 0.005  ->  n_eff = 10,278
    #>     |rho| = 0.010  ->  n_eff = 2,569
    #>     |rho| = 0.050  ->  n_eff = 103
    #>     |rho| = 0.100  ->  n_eff = 26
    #>   (rho on the target variable is not observable; read as an ignorance range)

## Variance

The recipe-aware bootstrap re-runs the whole non-probability recipe – the participation model included – on each replicate, so the standard error carries the variability of the pseudo-weighting, not just the spread of the final weights.

    boot <- bootstrap_weights(spec, replicates = 200, seed = 1, progress = FALSE)
    boot_mean(boot, "income")
    #>   estimate       se ci_lower ci_upper
    #> 1 19752.51 327.3892 19110.84 20394.18

## Nonignorable selection

Everything above assumes participation is ignorable given the covariates. When it may depend on the outcome itself, `step_nr_sensitivity()` reports how the estimate moves across a single sensitivity parameter (the proxy pattern-mixture model of Andridge and Little 2011), producing an ignorance interval to read next to the sampling confidence interval. See `?step_nr_sensitivity`.

## References

Elliott and Valliant (2017), Statistical Science; Meng (2018), Annals of Applied Statistics; Andridge and Little (2011), JSSAM.