.. _bda-using-regression-predictions-survey-incentives: ======================================================================== Using regression predictions: survey incentives ======================================================================== **Part 2 · Stage 7 · 🗳️ Data Collection & Decisions** · Lesson 058 of 144 · *intermediate* :doc:`◀ Previous · Bayesian decision theory in different contexts <057-bayesian-decision-theory-in-different-contexts>` · :doc:`Next · Multistage decision making: medical screening ▶ <059-multistage-decision-making-medical-screening>` · :doc:`↑ Section ` .. important:: **✨ AI-generated content.** This page was written with the assistance of an AI language model and is provided as a learning aid. Despite careful review, it may still contain mistakes, omissions, or out-of-date information. Whether you are new to the topic, a team lead, or a senior practitioner, treat it as a starting point rather than an authoritative reference: read it critically and independently verify anything you act on (code, commands, figures, and factual claims) against official documentation and primary sources before relying on it. Should you pay respondents? ----------------------------- A survey organisation must choose whether to offer respondents an incentive, of what size, in what form, and when. The stakes are concrete: incentives cost money but raise **response rates**, and a higher response rate reduces both the number of calls needed and the nonresponse bias. This is a cost–benefit problem, and Gelman, Stevens and Chan turned it into a worked Bayesian decision analysis. Meta-analysis feeding a decision ---------------------------------- There is no single experiment that answers the question, so the analysis proceeds in two stages. First, a **hierarchical meta-analysis** of many surveys' incentive experiments estimates the effect of incentive value, timing and mode on response rate. The design variables matter: * **prepaid** incentives (sent with the request) versus **postpaid** (paid on completion); * the **value** in dollars, whose effect need not be linear; * the survey's **burden** and mode. The regression's output is a posterior for the **expected increase in response rate** as a function of the incentive — with uncertainty, and with partial pooling across the studies, since the surveys are exchangeable but not identical. From response rate to utility ------------------------------- The second stage converts that posterior into money. The utility of an incentive scheme is the **net cost per respondent**, combining the incentive paid to everyone contacted, the interviewer time saved by fewer callbacks, and the value placed on a marginal completed interview: .. code-block:: python import numpy as np # posterior draws of the incentive's effect on response rate, from the meta-analysis d_rate = idata.posterior["beta_incentive"].values.ravel() def net_cost_per_respondent(incentive, d_rate, prepaid, base_rate=0.30, call_cost=1.25, calls_per_contact=8): rate = base_rate + d_rate * incentive # prepaid: paid to everyone contacted; postpaid: paid only to respondents paid = incentive if prepaid else incentive / rate interviewing = call_cost * calls_per_contact / rate return paid + interviewing for inc in [0, 5, 10, 20]: c = net_cost_per_respondent(inc, d_rate, prepaid=True) print(inc, c.mean(), np.percentile(c, [2.5, 97.5])) # posterior cost, with uncertainty The lessons ------------- Three, and they generalise. **The decision needs the whole posterior**, because cost is a nonlinear function of the response rate and averaging the rate first would give the wrong answer. **Prepaid and postpaid differ structurally**, not just in magnitude: a prepaid incentive is paid to *everyone contacted*, a postpaid one only to *respondents*, so their cost curves diverge as the response rate falls. And the analysis is honest about what it optimises — small incentives typically repay themselves, but the recommendation depends on the dollar value assigned to a completed interview, which is a **judgement**, stated openly rather than buried. .. hint:: **Related lessons:** :doc:`Bayesian decision theory in different contexts <057-bayesian-decision-theory-in-different-contexts>` · :doc:`Hierarchical modeling applied to a meta-analysis <038-hierarchical-modeling-applied-to-a-meta-analysis>` · :doc:`Sample surveys <052-sample-surveys>` · :doc:`Conditional modeling <091-conditional-modeling>` .. seealso:: **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/11/11/using-regression-predictions-survey-incentives/ `__ (insightful-data-lab.com). .. tags:: purpose: reference, topic: data analysis, domain: bayesian, level: intermediate