INTRODUCTION
Empirical research consistently finds that population estimates from nonprobability sample sources—such as optin Web panels—are less accurate than those from traditional probabilitybased sources, despite declining response rates to the latter (for a detailed review, see Cornesse et al. 2020). Hybrid sampling—surveying parallel probabilitybased and nonprobability samples and then blending the resulting completes—has been proposed to balance the lower cost of nonprobability sampling with the statistical rigor of probability sampling (DiSogra et al. 2011; Fahimi et al. 2015; Robbins, GhoshDastidar, and Ramchand 2021; Wiśniowski et al. 2020).
Hybrid approaches leverage the parallel probability sample to allow adjustment on characteristics that are related to the nonprobability selection mechanism (and therefore must be weighted on to control selection bias) but lack external population benchmarks. Two common weightingbased approaches to doing so, which can be used in tandem, are

Propensity adjustment (Valliant, Dever, and Kreuter 2018): this entails estimating a statistical model predicting presence in the nonprobability versus the probability sample, in which the predictors are characteristics collected from both samples expected to be related to the nonprobability selection mechanism. This model is then used to assign an estimated selection probability to each nonprobability complete, which is transformed into a weight.

Calibration (raking) to internal benchmarks (DiSogra et al. 2011; Fahimi et al. 2015): calibration creates weights such that the weighted distribution of specified characteristics among respondents matches known benchmarks. In hybrids, this entails first calibrating the probability sample to external benchmarks (typically demographic) obtained from highquality external surveys such as the American Community Survey (ACS). The weighted probability sample is then used to produce “internal benchmarks” for nondemographic characteristics that lack external benchmarks. The joint sample is calibrated to these internal benchmarks plus the available external benchmarks. A widely used calibration algorithm is raking (Deming and Stephan 1940), which adjusts the weights to match the marginal distributions of multiple characteristics.
Prior research has found that the choice of adjustment variables matters more than the choice of weighting methods: in particular, for a given set of adjustment variables, the addition of a propensity adjustment does not yield greater bias reduction than raking alone (Mercer, Lau, and Kennedy 2018).
We extend this prior research to the special case of a recurring (repeated crosssectional) study in which the estimation of trend is of interest. We address the following questions:

Is there evidence that nonprobability selection mechanisms change over time in recurring studies? Changes in the selection mechanism imply that selection bias may vary across waves. In this case, a raking model that is effective at removing selection bias in one wave may later become ineffective, potentially disrupting trend estimates.

If so, can the addition of a propensity adjustment help correct for changes in the selection mechanism and thereby recover an accurate trend?
METHODS
As a case study, we use data from an annual tracking poll of media consumption among adults ages 18+ in a large U.S. metropolitan area, which was sponsored by a major media outlet for market research purposes. Key outcomes were the percentage of adults reading the Sunday and daily editions of two major U.S. newspapers; to maintain client confidentiality, these are referred to here as “Newspaper A” and “Newspaper B,” respectively.
From 2016 through 2020, SSRS administered the annual study using a repeated crosssectional hybrid design, with an annual sample size of approximately 2,500 completes. Table 1 shows the evolution of the probability and nonprobability sample sizes from 2016 through 2020. From 2016 through 2019, the probability completes were collected by phone via dualframe (landline and cell) random digit dialing (RDD). In 2020, a mixedmode approach was employed, combining dualframe RDD phone completes with Web completes from the SSRS Opinion Panel, a probability panel recruited via addressbased sampling. The nonprobability completes were obtained from several Web panel vendors.^{[1]} Nonprobability sample quotas were specified to match the population distribution of the target metropolitan area with respect to age, sex, and geographic subregion.
The original weighting approach (the “rakingonly weight”) relied solely on raking. The probability sample completes were assigned base weights reflecting their original selection probabilities. The probability sample was then raked to eight ACS demographic benchmarks: sex, age, Hispanic ethnicity, race, educational attainment, phone use, region, and household income. Next, the weighted probability sample was used to obtain internal benchmarks for two nondemographic characteristics: whether the respondent accessed the Internet in the past 30 days, and when the respondent last visited the website of Newspaper A using a laptop. Finally, the hybrid sample (blending the probability and nonprobability completes) was raked to the eight ACS demographic benchmarks plus the two internal nondemographic benchmarks.
For the 2019 and 2020 samples, we calculated an alternative weight (the “propensity + raking weight”), which incorporated a propensity adjustment prior to raking. The “nonprobability propensity” score was generated as a case’s predicted probability of having been sourced from a nonprobability vendor, using a random forest (Breiman 2001) in which the dependent variable was presence in the nonprobability sample (1 = nonprobability complete, 0 = probability complete).
To control weighting variability, we used a propensity stratification approach (Valliant, Dever, and Kreuter 2018) to transform this probability into a “pseudobase weight” for nonprobability completes. We divided the combined sample into deciles based on the propensity score and calculated the following adjustment factor for each decile d:
\[{NPA}_{d} = \ \frac{1  \ \left( \frac{N_{n,d}}{N_{n,d} + \ N_{p,d}} \right)}{\left( \frac{N_{n,d}}{N_{n,d} + \ N_{p,d}} \right)}\]
where N_{n,d} is the unweighted count of nonprobability completes in decile d, and N_{p,d} is the sum of the base weights of probability completes in decile d. NPA_{d} was assigned as the pseudobase weight to all nonprobability completes in decile d. This adjustment is designed to make the pseudobaseweighted nonprobability sample resemble the baseweighted probability sample, with respect to the random forest predictors, prior to raking. Raking then proceeded as with the original weight.
The random forest predictors included the demographics and nondemographic “background” characteristics used in raking, as well as all other variables that were asked of at least half the sample and were consistent across waves—including the key study outcomes (newspaper readership). By including outcomes in the propensity model, we can account for the possibility that, even after controlling for other characteristics, selection into the nonprobability sample may be directly related to outcomes. A propensity model that includes outcomes (in lieu of direct calibration on outcomes) provides a means of adjusting for such relationships without relying entirely on the probability sample to produce estimates of outcomes, which would render the nonprobability completes superfluous.
RESULTS
Do nonprobability selection mechanisms change over time?
Figure 1 shows the estimated trend in the four outcomes from 2016 through 2020 using the rakingonly weights. The trend using only the probability sample is compared to the trend using the hybrid sample. In 2020, the hybrid sample estimates sharp increases^{[2]} in the outcomes, while the probability sample generally estimates more modest increases. If the probability sample is assumed to be approximately unbiased, this suggests increased selection bias in the hybrid estimates. This, in turn, suggests a change in the nonprobability sampling mechanism that was not corrected by the raking alone.
To disentangle the drivers of this pattern, Table 2 shows coefficients from logistic regressions predicting presence in the nonprobability sample (relative to the probability sample) in the 2019 and 2020 studies. In 2019, controlling for demographics, one of the four outcomes was a marginally statistically significant predictor of having been sampled from a nonprobability source. In 2020, two of the four outcomes became strongly significant predictors of presence in the nonprobability sample, though the sampling specifications provided to the nonprobability vendors did not change. This provides additional evidence that the nonprobability selection mechanism changed over the life of this study—specifically, that it became more independently associated with substantive outcomes.
Does the addition of a propensity adjustment help correct the trend?
Figure 2 shows the same estimates as Figure 1; but for 2019 and 2020, adds the hybrid estimates using propensity + raking weights. The propensity adjustment brings the 2020 hybrid estimates much closer to the probabilityonly estimates. Thus, the propensity adjustment partially corrects for the change in the nonprobability selection mechanism, reducing the distortion in the estimated trends.
Is there a tradeoff to the addition of a propensity adjustment?
Table 3 reports the Kish (1965) unequal weighting effect (UWE) and effective sample size (ESS) for the hybrid sample with both sets of weights, compared to the probability sample alone, for 2019 and 2020. Particularly in 2020, the addition of the propensity adjustment sharply increases the UWE and therefore decreases the hybrid ESS.
Table 3 captures the tradeoff to the addition of a propensity adjustment. Effectively, the propensity adjustment transforms the increase in bias into a reduction in precision, via an increased UWE. The more accurate trend therefore comes at the expense of a smaller ESS. However, the hybrid ESS remains higher than the probabilityonly ESS, suggesting that the nonprobability completes continue to add value despite the additional weighting required to correct for the changing selection mechanism.
DISCUSSION
Though prior research has found that propensity adjustments (relative to raking alone) add little value in weighting nonprobability or hybrid samples, these results add an important caveat.
Specifically, in the case of recurring studies, it is important to consider that patterns of selection bias in the nonprobability sample may change over time. As demonstrated here, propensity adjustments using random forests can help correct for such changes and thus obtain more accurate trends than raking alone. Therefore, in recurring studies that use nonprobability or hybrid samples, the incorporation of a preraking propensity adjustment provides “insurance” against selectiondriven distortions of estimated trends.
In this case study, the utility of the propensity adjustment was driven by the fact that the random forest included a wider range of predictors than the raking. The increase in bias in the rakingonly estimates in 2020 (Figure 1) reflects the fact that selection became more strongly related to outcomes that were not included in the raking model (Table 2). The propensity adjustment provided an alternative vehicle by which such characteristics could be accounted for in the hybrid weighting and thus helped mitigate this increase in selection bias (at the cost of a higher UWE).
Two alternatives to a propensity adjustment would be to (1) reevaluate the raking model at each wave of the study to ensure that any newly relevant characteristics are included or (2) include all potentially relevant characteristics (including study outcomes) in the raking model from the first wave.
Relative to the first alternative, the operational advantage of a propensity adjustment is that it operates largely automatically—particularly if, as in this case study, it is operationalized using nonparametric machinelearning techniques such as random forests. The number of potential raking margins in a typical study is considerable, particularly if interactions between characteristics are allowed. Therefore, choosing a new raking model may require extensive time and effort to remodel the selection mechanism at each wave. In contrast, given a large set of potentially relevant predictors, random forests can automatically identify those characteristics (and interactions between characteristics) that are most relevant in differentiating between probability and nonprobability completes. These potentially complex relationships are then built into the resulting propensity scores. This flexibility implies that a random forest propensity adjustment, once built into the weighting workflow, can pick up changes in the nonprobability selection mechanism with little intervention by the user, providing an efficient form of insurance against such changes.
Similar considerations favor the propensity approach over the second alternative. Since changes in the selection mechanism are unpredictable, the incorporation of all potentially relevant raking margins from the start is likely to be impractical. A raking model that is extensive enough to control for all characteristics that might become relevant may lead to an excessively high UWE and/or convergence problems, particularly when many margins are correlated with each other (Brick, Montaquila, and Roth 2003). In contrast, nonparametric methods such as random forests are designed to deal with a large set of candidate predictors whose relative importance is unknown in advance. Therefore, a preraking propensity adjustment allows the weighting procedure to account for a larger number of potentially relevant characteristics, while maintaining a parsimonious raking model.
In particular, a propensity adjustment provides a means of adjusting on substantive outcomes (if necessary) without raking on them. As demonstrated in Table 2, substantive outcomes may well be relevant to the nonprobability selection mechanism even after controlling for other characteristics, in which case adjustment on outcomes is needed to avoid selection bias. In most cases, however, raking on outcomes is likely to be undesirable. Raking on an outcome implies that the estimate from the hybrid sample would be forced to match the estimate obtained from the probability completes alone. This implies that sampling variability in the estimate would be driven by the size of the probability sample, not the (larger) hybrid sample. This, in turn, means that measures of sampling variability calculated using the hybrid sample size may overstate the precision of the estimates. At the extreme, raking on all outcomes of substantive interest would render the nonprobability completes superfluous (Robbins, GhoshDastidar, and Ramchand 2021). In contrast, including outcomes in a propensity model allows the weighting to account for any independent influence that they may have on the nonprobability selection mechanism, without directly controlling them to the probabilitybased estimates and thus continuing to allow the nonprobability completes to contribute to the final estimates. The ability to control for a potential “missing not at random” selection mechanism (Rubin 1976) provides a general argument for a “doubly robust” approach combining propensity adjustment and calibration (Valliant, Dever, and Kreuter 2018), even in nonrecurring studies.
It is important to emphasize that this case study used a hybrid sample in which, at each wave, the full survey instrument was administered to sidebyside probability and nonprobability samples. This allowed the propensity model to include study outcomes as well as nondemographic covariates that were collected on the questionnaire. A similar propensity adjustment could be applied to a nonprobabilityonly study by using an external publicuse dataset (such as the ACS) as the “reference” probability sample (Elliott and Valliant 2017). In this case, however, the predictors would be limited to variables that are available in both the study and the external dataset. These may not be welltailored to the study’s outcomes and in practice are likely to be limited to demographics—which, as demonstrated in Table 2, are often not sufficient to account for nonprobability selection mechanisms. Therefore, the utility of a propensity adjustment could be more limited in a study that relied solely on a nonprobability sample.
The same three vendors were used in 2019 and 2020. The authors do not have information on the identity of the vendors used prior to 2019.
A separate analysis, not shown here, found that similar increases were observed within the nonprobability sample alone, ruling out the possibility that the change in the hybrid estimate was driven by the increase in the nonprobability sample share shown in Table 1.