How would you post-stratify an age-skewed online panel to census shares, and when does bias remain?

## The mechanics You have an online panel whose respondents skew young, and you know the population age distribution from a census. **Post-stratification** adjusts after the fact, using population totals that you did not control at sampling time. For each age cell `c`, let `P_c` be its population share and `p_c` its share of respondents. The weight is `w_c = P_c / p_c` A cell that is 10% of the population but 20% of respondents gets `w = 0.5`; a cell that is 25% of the population but 10% of respondents gets `w = 2.5`. The post-stratified mean is the population-weighted average of the cell means: `estimate = sum over c of (P_c x mean of Y within cell c)` Equivalently, a weighted mean of individual responses using `w_c` as each respondent's weight. Notice that the cell means come from your data, but the mixing proportions come from the census. That is precisely the trade: you accept your respondents' answers within each cell, and you replace your sample's composition with the population's. ## What it assumes The estimate is unbiased when, **within each cell**, respondents are exchangeable with non-respondents on the outcome. In words: the panel may be short of 65-year-olds, but the 65-year-olds it does have must be typical 65-year-olds for the question at hand. This assumption is doing all the work, and it is untestable from the survey alone. If the online panel's older respondents are the technologically comfortable ones, and technological comfort predicts the answer, weighting them up amplifies their unrepresentativeness instead of correcting it. Post-stratification removes only the portion of the bias that runs **through the weighting variables**; whatever remains inside the cells survives untouched. ## Practical conditions - **Trustworthy external totals.** The population shares must come from a source you did not derive from the sample. Weighting to shares estimated from the same respondents is circular and does nothing. - **Adequate cell sizes.** A cell with three respondents produces a wildly unstable cell mean, and multiplying it by a large weight injects that instability straight into the estimate. Collapse thin cells into neighbours, or accept a coarser cell definition. - **Only marginals known?** If the census gives you age totals and region totals but not the joint age-by-region table, use **raking** (iterative proportional fitting): adjust weights to match age marginals, then region marginals, then age again, cycling until both are matched to tolerance. ## The variance cost Weighting is not free. Unequal weights increase the variance of the estimate, which is the same as saying the **effective sample size falls below the nominal one**. A survey with 4,000 respondents and a handful of very large weights can carry the precision of a few hundred. This is why practitioners **trim or cap** weights: capping reduces variance and reintroduces some of the bias you were removing, so the cap level is an explicit bias-variance decision that should be documented, not chosen silently. ## Choosing weighting variables A variable earns a place in the weighting scheme only if all three hold: 1. **It predicts the outcome.** Weighting on something unrelated to the answer adds variance and removes no bias. 2. **The sample is imbalanced on it** relative to the population. If your panel already matches the census on a variable, weighting on it changes nothing except the weight spread. 3. **Reliable population totals exist** for it, measured the same way you measured it on respondents. Mismatched category definitions between your questionnaire and the census silently corrupt the weights. Adding every available demographic is a common and costly error: each extra dimension thins the cells and widens the weight distribution, buying tiny bias reductions with real precision. ## When post-stratification cannot save you - **Selection driven by something unmeasured.** If people join the panel because of an attitude you never observe, no demographic weighting reaches it. - **Zero-probability groups.** Weighting can up-weight a cell only if the cell contains someone. A group entirely absent from the frame — people who never use the internet, for a web panel — cannot be recovered by any weight. - **Selection on the outcome itself.** If willingness to answer depends directly on the answer, within-cell exchangeability fails by construction. ## What to say in an interview Give the formula, then immediately name the assumption it rests on and the variance price it charges. The signal an interviewer is listening for is that you treat weighting as a *stated assumption with a cost*, not as a button that makes a convenience sample representative. Reporting the achieved weighted composition, the weight distribution and the effective sample size alongside the estimate is the professional standard.