Symptom Surveys
 Source name:
fbsurvey
 First issued: April 29, 2020
 Number of data revisions since May 19, 2020: 1
 Date of last change: June 3, 2020
 Available for: county, hrr, msa, state, nation (see geography coding docs)
 Time type: day (see date format docs)
 License: CC BY
Overview
This data source is based on symptom surveys run by the Delphi group at Carnegie Mellon. Facebook directs a random sample of its users to these surveys, which are voluntary. Individual survey responses are held by CMU and are sharable with other health researchers under a data use agreement. No individual survey responses are shared back to Facebook. See our surveys page for more detail about how the surveys work and how they are used outside the COVIDcast API.
We produce several sets of signals based on the survey data, listed and described in the sections below:
 Influenzalike and COVIDlike illness indicators, based on reported symptoms
 Behavior indicators, including maskwearing, traveling, and activities outside the home
 Testing indicators based on respondent reporting of their COVID test results
 Vaccination indicators, based on respondent reporting of COVID vaccinations and whether they would accept a vaccine
 Mental health indicators, based on selfreports of anxiety, depression, isolation, and worry about COVID
Table of Contents
 Overview
 Survey Text and Questions
 ILI and CLI Indicators
 Behavior Indicators
 Testing Indicators
 Vaccination Indicators
 Mental Health Indicators
 Survey Weighting
 Appendix
Survey Text and Questions
The survey starts with the following 5 questions:
 In the past 24 hours, have you or anyone in your household had any of the
following (yes/no for each):
 (a) Fever (100 °F or higher)
 (b) Sore throat
 (c) Cough
 (d) Shortness of breath
 (e) Difficulty breathing
 How many people in your household (including yourself) are sick (fever, along with at least one other symptom from the above list)?
 How many people are there in your household in total (including yourself)? [Beginning in wave 4, this question asks respondents to break the number down into three age categories.]
 What is your current ZIP code?
 How many additional people in your local community that you know personally are sick (fever, along with at least one other symptom from the above list)?
Beyond these 5 questions, there are also many other questions that follow in the survey, which go into more detail on symptoms, contacts, risk factors, and demographics. These are used for many of our behavior and testing indicators below. The full text of the survey (including all deployed versions) can be found on our questions and coding page. Researchers can request access to (fully deidentified) individual survey responses for research purposes.
As of midAugust 2020, the average number of Facebook survey responses we receive each day is about 74,000, and the total number of survey responses we have received is over 9 million.
ILI and CLI Indicators
Of primary interest for the API are the symptoms defining a COVIDlike illness (fever, along with cough, or shortness of breath, or difficulty breathing) or influenzalike illness (fever, along with cough or sore throat). Using this survey data, we estimate the percentage of people who have a COVIDlike illness, or influenzalike illness, in a given location, on a given day.
Signals  Description 

raw_cli and smoothed_cli 
Estimated percentage of people with COVIDlike illness based on the criteria below, with no survey weighting First Available: 20200406 
raw_ili and smoothed_ili 
Estimated percentage of people with influenzalike illness based on the criteria below, with no survey weighting First Available: 20200406 
raw_wcli and smoothed_wcli 
Estimated percentage of people with COVIDlike illness; adjusted using survey weights as described below First Available: 20200406 
raw_wili and smoothed_wili 
Estimated percentage of people with influenzalike illness; adjusted using survey weights as described below First Available: 20200406 
raw_hh_cmnty_cli and smoothed_hh_cmnty_cli 
Estimated percentage of people reporting illness in their local community, as described below, including their household, with no survey weighting First Available: 20200415 
raw_nohh_cmnty_cli and smoothed_nohh_cmnty_cli 
Estimated percentage of people reporting illness in their local community, as described below, not including their household, with no survey weighting First Available: 20200415 
Note that for raw_hh_cmnty_cli
and raw_nohh_cmnty_cli
, the illnesses
included are broader: a respondent is included if they know someone in their
household (for raw_hh_cmnty_cli
) or community with fever, along with sore
throat, cough, shortness of breath, or difficulty breathing. This does not
attempt to distinguish between COVIDlike and influenzalike illness.
Influenzalike illness or ILI is a standard indicator, and is defined by the CDC as: fever along with sore throat or cough. From the list of symptoms from Q1 on our survey, this means a and (b or c).
COVIDlike illness or CLI is not a standard indicator. Through our discussions with the CDC, we chose to define it as: fever along with cough or shortness of breath or difficulty breathing.
Symptoms alone are not sufficient to diagnose influenza or coronavirus infections, and so these ILI and CLI indicators are not expected to be unbiased estimates of the true rate of influenza or coronavirus infections. These symptoms can be caused by many other conditions, and many true infections can be asymptomatic. Instead, we expect these indicators to be useful for comparison across the United States and across time, to determine where symptoms appear to be increasing.
Smoothing. The signals beginning with smoothed_
estimate the same quantities as their
raw
partners, but are smoothed in time to reduce daytoday sampling noise;
see details below. Crucially, because the smoothed signals combine
information across multiple days, they have larger sample sizes and hence are
available for more counties and MSAs than the raw signals.
Defining Household ILI and CLI
For a single survey, we are interested in the quantities:
 \(X =\) the number of people in the household with ILI;
 \(Y =\) the number of people in the household with CLI;
 \(N =\) the number of people in the household.
Note that \(N\) comes directly from the answer to Q3, but neither \(X\) nor \(Y\) can be computed directly (because Q2 does not give an answer to the precise symptomatic profile of all individuals in the household, it only asks how many individuals have fever and at least one other symptom from the list).
We hence estimate \(X\) and \(Y\) with the following simple strategy. Consider ILI, without a loss of generality (we apply the same strategy to CLI). Let \(Z\) be the answer to Q2.
 If the answer to Q1 does not meet the ILI definition, then we report \(X=0\).
 If the answer to Q1 does meet the ILI definition, then we report \(X = Z\).
This can only “over count” (result in too large estimates of) the true \(X\) and \(Y\). For example, this happens when some members of the household experience ILI that does not also qualify as CLI, while others experience CLI that does not also qualify as ILI. In this case, for both \(X\) and \(Y\), our simple strategy would return the sum of both types of cases. However, given the extreme degree of overlap between the definitions of ILI and CLI, it is reasonable to believe that, if symptoms across all household members qualified as both ILI and CLI, each individual would have both, or neither—with neither being more common. Therefore we do not consider this “over counting” phenomenon practically problematic.
Estimating Percent ILI and CLI
Let \(x\) and \(y\) be the number of people with ILI and CLI, respectively, over a given time period, and in a given location (for example, the time period being a particular day, and a location being a particular county). Let \(n\) be the total number of people in this location. We are interested in estimating the true ILI and CLI percentages, which we denote by \(p\) and \(q\), respectively:
\[p = 100 \cdot \frac{x}{n} \quad\text{and}\quad q = 100 \cdot \frac{y}{n}.\]We estimate \(p\) and \(q\) across 4 aggregation schemes:
 daily, at the county level;
 daily, at the MSA (metropolitan statistical area) level;
 daily, at the HRR (hospital referral region) level;
 daily, at the state level.
These are possible because we have the ZIP code of the household from Q4 of the survey. Our current ruleofthumb is to discard any estimate (whether at a county, MSA, HRR, or state level) that is based on fewer than 100 survey responses. When our geographical mapping data indicates that a ZIP code is part of multiple geographical units in a single aggregation, we assign weights \(w_i^\text{geodiv}\) to each of these units (based on the ZIP code’s overlap with each geographical unit) and use these weights as part of the survey weighting, as described below.
In a given aggregation unit (for example, dailycounty), let \(X_i\) and \(Y_i\) denote number of ILI and CLI cases in the household, respectively (computed according to the simple strategy described above), and let \(N_i\) denote the total number of people in the household, in survey \(i\), out of \(m\) surveys we collected. Then our estimates of \(p\) and \(q\) (see the appendix for motivating details) are:
\[\hat{p} = 100 \cdot \frac{1}{m}\sum_{i=1}^m \frac{X_i}{N_i} \quad\text{and}\quad \hat{q} = 100 \cdot \frac{1}{m}\sum_{i=1}^m \frac{Y_i}{N_i}.\]Their estimated standard errors are:
\[\begin{aligned} \widehat{\mathrm{se}}(\hat{p}) &= 100 \cdot \frac{1}{m+1}\sqrt{ \left(\frac{1}{2}  \frac{\hat{p}}{100}\right)^2 + \sum_{i=1}^m \left(\frac{X_i}{N_i}  \frac{\hat{p}}{100}\right)^2 } \\ \widehat{\mathrm{se}}(\hat{q}) &= 100 \cdot \frac{1}{m+1}\sqrt{ \left(\frac{1}{2}  \frac{\hat{q}}{100}\right)^2 + \sum_{i=1}^m \left(\frac{Y_i}{N_i}  \frac{\hat{q}}{100}\right)^2 }, \end{aligned}\]the standard deviations of the estimators after adding a single pseudoobservation at 1/2 (treating \(m\) as fixed). The use of the pseudoobservation prevents standard error estimates of zero, and in simulations improves the quality of the standard error estimates.
The pseudoobservation is not used in \(\hat{p}\) and \(\hat{q}\) themselves, to avoid potentially large amounts of estimation bias, as \(p\) and \(q\) are expected to be small.
Estimating “Community CLI”
Over a given time period, and in a given location, let \(u\) be the number of people who know someone in their community with CLI, and let \(v\) be the number of people who know someone in their community, outside of their household, with CLI. With \(n\) denoting the number of people total in this location, we are interested in the percentages:
\[a = 100 \cdot \frac{u}{n} \quad\text{and}\quad b = 100 \cdot \frac{y}{n}.\]We will estimate \(a\) and \(b\) across the same 4 aggregation schemes as before.
For a single survey, let:
 \(U = 1\) if and only if a positive number is reported for Q2 or Q5;
 \(V = 1\) if and only if a positive number is reported for Q2.
In a given aggregation unit (for example, dailycounty), let \(U_i\) and \(V_i\) denote these quantities for survey \(i\), and \(m\) denote the number of surveys total. Then to estimate \(a\) and \(b\), we simply use:
\[\hat{a} = 100 \cdot \frac{1}{m} \sum_{i=1}^m U_i \quad\text{and}\quad \hat{b} = 100 \cdot \frac{1}{m} \sum_{i=1}^m V_i.\]Hence \(\hat{a}\) is reported in the hh_cmnty_cli
signals and \(\hat{b}\) in
the nohh_cmnty_cli
signals. Their estimated standard errors are:
which are the plugin estimates of the standard errors of the binomial proportions (treating \(m\) as fixed).
Note that \(\sum_{i=1}^m U_i\) is the number of survey respondents who know someone in their community with either ILI or CLI, and not CLI alone; and similarly for \(V\). Hence \(\hat{a}\) and \(\hat{b}\) will generally overestimate \(a\) and \(b\). However, given the extremely high overlap between the definitions of ILI and CLI, we do not consider this to be practically very problematic.
Smoothing
The smoothed versions of all fbsurvey
signals (with smoothed_
prefix) are
calculated using seven day pooling. For example, the estimate reported for June
7 in a specific geographical area (such as county or MSA) is formed by
collecting all surveys completed between June 1 and 7 (inclusive) and using that
data in the estimation procedures described above.
Behavior Indicators
Signal  Description  Survey Item  Introduced 

smoothed_wearing_mask 
Estimated percentage of people who wore a mask for most or all of the time while in public in the past 5 days; those not in public in the past 5 days are not counted. First Available: 20200908 
C14  Wave 4, Sept 8, 2020 
smoothed_others_masked 
Estimated percentage of respondents who say that most or all other people wear masks, when they are in public and social distancing is not possible First Available: 20201124 
C16  Wave 5, Nov 24, 2020 
smoothed_travel_outside_state_5d 
Estimated percentage of respondents who report traveling outside their state in the past 5 days First Available: 20200406 
C6  Wave 1 
smoothed_work_outside_home_1d 
Estimated percentage of respondents who worked or went to school outside their home in the past 24 hours First Available: 20200908 
C13  Wave 4, Sept 8, 2020 
smoothed_shop_1d 
Estimated percentage of respondents who went to a “market, grocery store, or pharmacy” in the past 24 hours First Available: 20200908 
C13  Wave 4, Sept 8, 2020 
smoothed_restaurant_1d 
Estimated percentage of respondents who went to a “bar, restaurant, or cafe” in the past 24 hours First Available: 20200908 
C13  Wave 4, Sept 8, 2020 
smoothed_spent_time_1d 
Estimated percentage of respondents who “spent time with someone who isn’t currently staying with you” in the past 24 hours First Available: 20200908 
C13  Wave 4, Sept 8, 2020 
smoothed_large_event_1d 
Estimated percentage of respondents who “attended an event with more than 10 people” in the past 24 hours First Available: 20200908 
C13  Wave 4, Sept 8, 2020 
smoothed_public_transit_1d 
Estimated percentage of respondents who “used public transit” in the past 24 hours First Available: 20200908 
C13  Wave 4, Sept 8, 2020 
Weighted versions of these signals, using the survey weighting described
below to be more representative of state demographics, are
also available. These have names beginning smoothed_w
, such as
smoothed_wwearing_mask
.
Testing Indicators
Signal  Description  Survey Item 

smoothed_tested_14d 
Estimated percentage of people who were tested for COVID19 in the past 14 days, regardless of their test result First Available: 20200908 
B8, B10 
smoothed_tested_positive_14d 
Estimated test positivity rate (percent) among people tested for COVID19 in the past 14 days First Available: 20200908 
B10a 
smoothed_wanted_test_14d 
Estimated percentage of people who wanted to be tested for COVID19 in the past 14 days, out of people who were not tested in that time First Available: 20200908 
B12 
These indicators are based on questions in Wave 4 of the survey, introduced on September 8, 2020.
Weighted versions of these signals, using the survey weighting described
below to be more representative of state demographics, are
also available. These have names beginning smoothed_w
, such as
smoothed_wtested_14d
.
Vaccination Indicators
Signal  Description  Survey Item 

smoothed_accept_covid_vaccine 
Estimated percentage of respondents who would definitely or probably choose to get vaccinated, if a COVID19 vaccine were offered to them today. Note: Until January 6, 2021, all respondents answered this question; beginning on that date, only respondents who said they have not received a COVID vaccine are asked this question. First Available: 20210101 
V3 
smoothed_covid_vaccinated 
Estimated percentage of respondents who have already received a vaccine for COVID19. Note: The Centers for Disease Control compiles data on vaccine administration across the United States. This signal may differ from CDC data because of survey biases and should not be treated as authoritative. However, the survey signal is not subject to the lags and reporting problems in official vaccination data. First Available: 20210601 
V1 
These indicators are based on questions added in Wave 6 of the survey, introduced on December 19, 2020; however, Delphi only enabled item V1 beginning January 6, 2021. Note: As of January 2021, vaccination items on the survey are being revised in preparation for Wave 7. We may replace the signals above with new signals (with different names) if the underlying survey items change significantly.
Weighted versions of these signals, using the survey weighting described
below to be more representative of state demographics, are
also available. They have names beginning with smoothed_w
, such as
smoothed_waccept_covid_vaccine
.
Mental Health Indicators
Signal  Description  Survey Item 

smoothed_anxious_5d 
Estimated percentage of respondents who reported feeling “nervous, anxious, or on edge” for most or all of the past 5 days First Available: 20200908 
C8 
smoothed_depressed_5d 
Estimated percentage of respondents who reported feeling depressed for most or all of the past 5 days First Available: 20200908 
C8 
smoothed_felt_isolated_5d 
Estimated percentage of respondents who reported feeling “isolated from others” for most or all of the past 5 days First Available: 20200908 
C8 
smoothed_worried_become_ill 
Estimated percentage of respondents who reported feeling very or somewhat worried that “you or someone in your immediate family might become seriously ill from COVID19” First Available: 20200908 
C9 
smoothed_worried_finances 
Estimated percentage of respondents who report being very or somewhat worried about their “household’s finances for the next month” First Available: 20200908 
C15 
Some of these questions were present in the earliest waves of the survey, but only in Wave 4 did respondents consent to our use of aggregate data to study other impacts of COVID, such as mental health. Hence, these aggregates only include respondents to Wave 4 and later waves, beginning September 8, 2020.
Weighted versions of these signals, using the survey weighting described
below to be more representative of state demographics, are
also available. These have names beginning smoothed_w
, such as
smoothed_wdepressed_14d
.
Survey Weighting
Notice that the estimates defined in the previous sections are calculated with respect to the population of US Facebook users. (To be precise, the ILI and CLI indicators reflect the population of US Facebook users and their household members). In reality, our estimates are even further skewed by the varying propensity of people in the population of US Facebook users to take our survey in the first place.
When Facebook sends a user to our survey, it generates a random ID number and sends this to us as well. Once the user completes the survey, we pass this ID number back to Facebook to confirm completion, and in return receive a weight—call it \(w_i\) for user \(i\). (The random ID number is completely meaningless for any other purpose than receiving this weight, and does not allow us to access any information about the user’s Facebook profile.)
We can use these weights to adjust our estimates so that they are representative of the US population—adjusting both for the differences between the US population and US Facebook users (according to a statebyagegender stratification of the US population from the 2018 Census March Supplement) and for the propensity of a Facebook user to take our survey in the first place.
In more detail, we receive a participation weight
\[w^{\text{part}}_i \propto \frac{1}{\pi_i},\]where \(\pi_i\) is an estimated probability (produced by Facebook) that an individual with the same statebyagegender profile as user \(i\) would be a Facebook user and take our CMU survey. The adjustment we make follows a standard inverse probability weighting strategy (this being a special case of importance sampling).
Detailed documentation on how Facebook calculates these weights is available on our survey weight documentation page.
Adjusting Household ILI and CLI
As before, for a given aggregation unit (for example, dailycounty), let \(X_i\) and \(Y_i\) denote the numbers of ILI and CLI cases in household \(i\), respectively (computed according to the simple strategy above), and let \(N_i\) denote the total number of people in the household. Let \(i = 1, \dots, m\) denote the surveys started during the time period of interest and reported in a ZIP code intersecting the spatial unit of interest.
Each of these surveys is assigned two weights: the participation weight \(w^{\text{part}}_i\), and a geographicaldivision weight \(w^{\text{geodiv}}_i\) describing how much a participant’s ZIP code “belongs” in the spatial unit of interest. (For example, a ZIP code may overlap with multiple counties, so the weight describes what proportion of the ZIP code’s population is in each county.)
Let \(w^{\text{init}}_i=w^{\text{part}}_i w^{\text{geodiv}}_i\) denote the initial weight assigned to this survey. First, we adjust these initial weights to reduce sensitivity to any individual survey by “mixing” them with a uniform weighting across all relevant surveys. This prevents specific survey respondents with high survey weights having disproportionate influence on the weighted estimates.
Specifically, we select the smallest value of \(a \in [0.05, 1]\) such that
\[w_i = a\cdot\frac1m + (1a)\cdot w^{\text{init}}_i \leq 0.01\]for all \(i\). If such a selection is impossible, then we have insufficient survey responses (less than 100), and do not produce an estimate for the given aggregation unit.
Next, we rescale the weights \(w_i\) over all \(i\) so that \(\sum_{i=1}^m w_i=1\). Then our adjusted estimates of \(p\) and \(q\) are:
\[\begin{aligned} \hat{p}_w &= 100 \cdot \sum_{i=1}^m w_i \frac{X_i}{N_i} \\ \hat{q}_w &= 100 \cdot \sum_{i=1}^m w_i \frac{Y_i}{N_i}, \end{aligned}\]with estimated standard errors:
\[\begin{aligned} \widehat{\mathrm{se}}(\hat{p}_w) &= 100 \cdot \sqrt{ \left(\frac{1}{1 + n_e}\right)^2 \left(\frac12  \frac{\hat{p}_w}{100}\right)^2 + n_e \hat{s}_p^2 }\\ \widehat{\mathrm{se}}(\hat{q}_w) &= 100 \cdot \sqrt{ \left(\frac{1}{1 + n_e}\right)^2 \left(\frac12  \frac{\hat{q}_w}{100}\right)^2 + n_e \hat{s}_q^2 }, \end{aligned}\]where
\[\begin{aligned} \hat{s}_p^2 &= \sum_{i=1}^m w_i^2 \left(\frac{X_i}{N_i}  \frac{\hat{p}_w}{100}\right)^2 \\ \hat{s}_q^2 &= \sum_{i=1}^m w_i^2 \left(\frac{Y_i}{N_i}  \frac{\hat{q}_w}{100}\right)^2 \\ n_e &= \frac1{\sum_{i=1}^m w_i^2}, \end{aligned}\]which are the delta method estimates of variance associated with selfnormalized importance sampling estimators above, after combining with a pseudoobservation of 1/2 with weight assigned to appear like a single effective observation according to importance sampling diagnostics.
The sample size reported is calculated by rounding down \(\sum_{i=1}^{m} w^{\text{geodiv}}_i\) before adding the pseudoobservations. When ZIP codes do not overlap multiple spatial units of interest, these weights are all one, and this expression simplifies to \(m\). When estimates are available for all spatial units of a given type over some time period, the sum of the associated sample sizes under this definition is consistent with the number of surveys used to prepare the estimate. (This notion of sample size is distinct from “effective” sample sizes based on variance of the importance sampling estimators which were used above.)
Adjusting Other Percentage Estimators
The household ILI and CLI estimates are complex to weight, as shown in the previous subsection, because they use an estimator based on the survey respondent and their household. All other estimates reported in the API are simply based on percentages of respondents, such as the percentage who report knowing someone in their community who is sick. In this subsection we will describe how survey weights are used to construct weighted estimates for these indicators, using community CLI as an example.
As before, in a given aggregation unit (for example, dailycounty), let \(U_i\) and \(V_i\) denote the indicators that the survey respondent knows someone in their community with CLI, including and not including their household, respectively, for survey \(i\), out of \(m\) surveys collected. Also let \(w_i\) be the selfnormalized weight that accompanies survey \(i\), as above. Then our adjusted estimates of \(a\) and \(b\) are:
\[\begin{aligned} \hat{a}_w &= 100 \cdot \sum_{i=1}^m w_i U_i \\ \hat{b}_w &= 100 \cdot \sum_{i=1}^m w_i V_i. \end{aligned}\]with estimated standard errors:
\[\begin{aligned} \widehat{\mathrm{se}}(\hat{a}_w) &= 100 \cdot \sqrt{\sum_{i=1}^m w_i^2 \left(U_i  \frac{\hat{a}_w}{100} \right)^2} \\ \widehat{\mathrm{se}}(\hat{b}_w) &= 100 \cdot \sqrt{\sum_{i=1}^m w_i^2 \left(V_i  \frac{\hat{b}_w}{100} \right)^2}, \end{aligned}\]the delta method estimates of variance associated with selfnormalized importance sampling estimators.
Appendix
Here are some details behind the choice of estimators for percent ILI and percent CLI.
Suppose there are \(h\) households total in the underlying population, and for household \(i\), denote \(\theta_i=N_i/n\). Then note that the quantities of interest, \(p\) and \(q\), are
\[p = \sum_{i=1}^h \frac{X_i}{N_i} \theta_i \quad\text{and}\quad q = \sum_{i=1}^h \frac{Y_i}{N_i} \theta_i.\]Let \(S \subseteq \{1,\dots,h\}\) denote sampled households, with \(m=S\), and suppose we sampled household \(i\) with probability \(\theta_i=N_i/n\) proportional to the household size. Then unbiased estimates of \(p\) and \(q\) are simply
\[\hat{p} = \frac{1}{m} \sum_{i \in S} \frac{X_i}{N_i} \quad\text{and}\quad \hat{q} = \frac{1}{m} \sum_{i \in S} \frac{Y_i}{N_i},\]which are an equivalent way of writing our previouslydefined estimates.
Note that we can again rewrite our quantities of interest as
\[p = \frac{\mu_x}{\mu_n} \quad\text{and}\quad q = \frac{\mu_y}{\mu_n},\]where \(\mu_x=x/h\), \(\mu_y=y/h\), \(\mu_n=n/h\) denote the expected number people with ILI per household, expected number of people with CLI per household, and expected number of people total per household, respectively, and \(h\) denotes the total number of households in the population.
Suppose that instead of proportional sampling, we sampled households uniformly, resulting in \(S \subseteq \{1,\dots,h\}\) denote sampled households, with \(m=S\). Then the natural estimates of \(p\) and \(q\) are instead plugin estimates of the numerators and denominators in the above,
\[\tilde{p} = \frac{\bar{X}}{\bar{N}} \quad\text{and}\quad \tilde{q} = \frac{\bar{X}}{\bar{N}}\]where \(\bar{X}=\sum_{i \in S} X_i/m\), \(\bar{Y}=\sum_{i \in S} Y_i/m\), and \(\bar{N}=\sum_{i \in S} N_i/m\) denote the sample means of \(\{X_i\}_{i \in S}\), \(\{Y_i\}_{i \in S}\), and \(\{N_i\}_{i \in S}\), respectively.
Whether we consider \(\hat{p}\) and \(\hat{q}\), or \(\tilde{p}\) and \(\tilde{q}\), to be more natural—mean of fractions, or fraction of means, respectively—depends on the sampling model: if we are sampling households proportional to household size, then it is \(\hat{p}\) and \(\hat{q}\); if we are sampling households uniformly, then it is \(\tilde{p}\) and \(\tilde{q}\). We settled on the former, based on both conceptual and empirical supporting evidence:

Conceptually, though we do not know the details, we have reason to believe that Facebook offers an essentially uniform random draw of eligible users—those 18 years or older—to take our survey. In this sense, the sampling is done proportional to the number of “Facebook adults” in a household: individuals 18 years or older, who have a Facebook account. Hence if we posit that the number of “Facebook adults” scales linearly with the household size, which seems to us like a reasonable assumption, then sampling would still be proportional to household size. (Notice that this would remain true no matter how small the linear coefficient is, that is, it would even be true if Facebook did not have good coverage over the US.)

Empirically, we have computed the distribution of household sizes (proportion of households of size 1, size 2, size 3, etc.) in the Facebook survey data thus far, and compared it to the distribution of household sizes from the Census. These align quite closely, also suggesting that sampling is likely done proportional to household size.