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Symptom Surveys

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:

  1. Influenza-like and COVID-like illness indicators, based on reported symptoms
  2. Behavior indicators, including mask-wearing, traveling, and activities outside the home
  3. Testing indicators based on respondent reporting of their COVID test results
  4. Vaccination indicators, based on respondent reporting of COVID vaccinations and whether they would accept a vaccine
  5. Mental health indicators, based on self-reports of anxiety, depression, isolation, and worry about COVID

Table of Contents

  1. Overview
  2. Survey Text and Questions
  3. ILI and CLI Indicators
    1. Defining Household ILI and CLI
    2. Estimating Percent ILI and CLI
    3. Estimating “Community CLI”
    4. Smoothing
  4. Behavior Indicators
  5. Testing Indicators
  6. Vaccination Indicators
  7. Mental Health Indicators
  8. Survey Weighting
    1. Adjusting Household ILI and CLI
    2. Adjusting Other Percentage Estimators
  9. Appendix

Survey Text and Questions

The survey starts with the following 5 questions:

  1. 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
  2. How many people in your household (including yourself) are sick (fever, along with at least one other symptom from the above list)?
  3. 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.]
  4. What is your current ZIP code?
  5. 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 de-identified) individual survey responses for research purposes.

As of mid-August 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 COVID-like illness (fever, along with cough, or shortness of breath, or difficulty breathing) or influenza-like illness (fever, along with cough or sore throat). Using this survey data, we estimate the percentage of people who have a COVID-like illness, or influenza-like illness, in a given location, on a given day.

Signals Description
raw_cli and smoothed_cli Estimated percentage of people with COVID-like illness based on the criteria below, with no survey weighting
First Available: 2020-04-06
raw_ili and smoothed_ili Estimated percentage of people with influenza-like illness based on the criteria below, with no survey weighting
First Available: 2020-04-06
raw_wcli and smoothed_wcli Estimated percentage of people with COVID-like illness; adjusted using survey weights as described below
First Available: 2020-04-06
raw_wili and smoothed_wili Estimated percentage of people with influenza-like illness; adjusted using survey weights as described below
First Available: 2020-04-06
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: 2020-04-15
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: 2020-04-15

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 COVID-like and influenza-like illness.

Influenza-like 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).

COVID-like 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 day-to-day 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:

  1. daily, at the county level;
  2. daily, at the MSA (metropolitan statistical area) level;
  3. daily, at the HRR (hospital referral region) level;
  4. daily, at the state level.

These are possible because we have the ZIP code of the household from Q4 of the survey. Our current rule-of-thumb 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, daily-county), 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 pseudo-observation at 1/2 (treating \(m\) as fixed). The use of the pseudo-observation prevents standard error estimates of zero, and in simulations improves the quality of the standard error estimates.

The pseudo-observation 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, daily-county), 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:

\[\begin{aligned} \widehat{\mathrm{se}}(\hat{a}) &= 100 \cdot \sqrt{\frac{\frac{\hat{a}}{100}(1-\frac{\hat{a}}{100})}{m}} \\ \widehat{\mathrm{se}}(\hat{b}) &= 100 \cdot \sqrt{\frac{\frac{\hat{b}}{100}(1-\frac{\hat{b}}{100})}{m}}, \end{aligned}\]

which are the plug-in 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 fb-survey 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: 2020-09-08
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: 2020-11-24
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: 2020-04-06
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: 2020-09-08
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: 2020-09-08
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: 2020-09-08
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: 2020-09-08
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: 2020-09-08
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: 2020-09-08
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 COVID-19 in the past 14 days, regardless of their test result
First Available: 2020-09-08
B8, B10
smoothed_tested_positive_14d Estimated test positivity rate (percent) among people tested for COVID-19 in the past 14 days
First Available: 2020-09-08
B10a
smoothed_wanted_test_14d Estimated percentage of people who wanted to be tested for COVID-19 in the past 14 days, out of people who were not tested in that time
First Available: 2020-09-08
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 COVID-19 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: 2021-01-01
V3
smoothed_covid_vaccinated Estimated percentage of respondents who have already received a vaccine for COVID-19. 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: 2021-06-01
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: 2020-09-08
C8
smoothed_depressed_5d Estimated percentage of respondents who reported feeling depressed for most or all of the past 5 days
First Available: 2020-09-08
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: 2020-09-08
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 COVID-19”
First Available: 2020-09-08
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: 2020-09-08
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 state-by-age-gender 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 state-by-age-gender 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, daily-county), 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 geographical-division 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

\[w_i = a\cdot\frac1m + (1-a)\cdot w^{\text{init}}_i \leq 0.01\]

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 self-normalized importance sampling estimators above, after combining with a pseudo-observation 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 pseudo-observations. 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, daily-county), 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 self-normalized 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 self-normalized 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 previously-defined 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 plug-in 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.