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Estimators of \(\tau\) can often be written as \[ \hat\tau = \sum_{i \in \mathcal{S}} w_iy_i, \] where \(w_i\) is the survey weight for the \(i\)-th element in the sample. Where does \(w_i\) come from?
The sampling design. If there is no adjustment/re-weighting (see below) then \(w_i = 1/\pi_i\) where \(\pi_i\) is the inclusion probability of the \(i\)-th element. See this lecture on inclusion probabilities.
Calibration. Certain estimators such as ratio and (generalized) regression estimators effectively adjust or re-weight the weights based on known totals for one or more auxiliary variables. This also includes post-stratification and raking. See this lecture on how weights on re-weighting and calibration.
Non-response/detectability. Further adjustment or re-weighting may be done to account for known (or estimated) probabilities of response or detection. See the lectures on detectability and non-response.
There are a couple of ways that we can estimate \(\mu\) using survey weights. One is \[ \hat\mu = \frac{\sum_{i \in \mathcal{S}} w_iy_i}{\text{number of elements in the population}}, \] if the denominator is known. If not, then we can use \[ \hat\mu = \frac{\sum_{i \in \mathcal{S}} w_iy_i}{\sum_{i \in \mathcal{S}} w_i}, \] since \(\sum_{i \in \mathcal{S}} w_i\) is an estimator of the number of elements in the population.
Once the weights are known, the user can define \(y_i\) depending on what they want to estimate.
\(y_i\) can be the value of some target variable of interest.
\(y_i\) can be defined as equal to one for all elements in the population. Then \(\hat\tau = \sum_{i \in \mathcal{S}} w_iy_i = \sum_{i \in \mathcal{S}} w_i\) is an estimator of the number of elements in the population.
\(y_i\) can be defined as \[ y_i = \begin{cases} 1, & \text{if the $i$-th element is in a given domain}, \\ 0, & \text{otherwise}, \end{cases} \] so that \(\hat\tau\) is an estimator of the number of elements in the domain and \(\hat\mu\) is an estimator of the proportion of elements in the domain.
\(y_i\) can be defined as \[ y_i = \begin{cases} y_i, & \text{if the $i$-th element is in a given domain}, \\ 0, & \text{otherwise}, \end{cases} \] so that \(\hat\tau\) is an estimator of the total for the sub-population of elements in the domain and \(\hat\mu\) is an estimator of the mean for the sub-population of elements in the domain.
The advantage of survey weights is that they make it relatively easy for non-specialists to compute estimates using only the survey weights and the (specified) target variable values. (However computing the (estimated) variance of an estimator to compute the bound on the error of estimation still requires more information and expertise.)
Example: The following table shows the value of the target variable and the corresponding survey weights for a sample of five elements obtained using some unknown probability sampling design.| Target Variable | Survey Weight |
|---|---|
| 3 | 17.2 |
| 9 | 18.8 |
| 5 | 27.8 |
| 5 | 20.4 |
| 4 | 20.6 |
How do we compute estimates of \(\tau\) and \(\mu\) using only this information?
Here I will demonstrate the use of sampling weights from the European Social Survey (ESS). Survey data from the ESS are available for download. For this example we will use data from the ninth round of the survey that was conducted from 2018 to 2020.
essdata <- read.csv("ESS9e03_2.csv")You can download the data yourself. Registration is required but is free and not restrictive.
The data require a little bit of formatting before we can use it. Here I re-code the missing responses, select the variables we want to use (just to keep the data to a manageable size), and drop elements with missing data. I am also going to create some age groups based on the age of the respondent. This can be done many different ways in R. I will use the dplyr and tidyr packages.
library(dplyr)
library(tidyr)
ukdata <- essdata %>% filter(cntry == "GB") %>%
select(psu, stratum, dweight, pspwght, agea, hmsacld) %>%
mutate(agea = ifelse(agea == 999, NA, agea)) %>%
mutate(hmsacld = ifelse(hmsacld %in% c(7,8,9), NA, hmsacld)) %>%
drop_na() %>%
mutate(agegroup = cut(agea, breaks = c(min(agea) - 1,
quantile(agea, c(0.25, 0.5, 0.75)), max(agea))))Here you can see the first 20 responses.
head(ukdata, 20) psu stratum dweight pspwght agea hmsacld agegroup
1 12304 1588 2.0375460 2.5500326 19 1 (14,37]
2 12445 1440 1.0187730 1.0932655 42 1 (37,53]
3 12337 1560 1.5281595 1.9591819 15 1 (14,37]
4 12255 1554 0.5093865 0.4731422 66 1 (53,67]
5 12401 1417 1.0187730 1.2063588 52 2 (37,53]
6 12330 1558 1.0187730 1.3198249 21 2 (14,37]
7 12147 1523 2.0375460 1.4712680 76 4 (67,90]
8 12392 1402 0.5093865 0.4618744 84 3 (67,90]
9 12161 1478 0.5093865 0.6377798 38 1 (37,53]
10 12446 1444 1.0187730 0.7758576 64 2 (53,67]
11 12242 1560 1.5281595 1.7110010 46 2 (37,53]
12 12233 1539 0.5093865 0.4552839 35 2 (14,37]
13 12309 1576 0.5093865 0.4271142 63 2 (53,67]
14 12331 1488 0.5093865 0.6530607 25 1 (14,37]
15 12376 1585 1.0187730 0.8542284 62 1 (53,67]
16 12388 1589 0.5093865 0.3951095 82 2 (67,90]
17 12133 1452 0.5093865 0.4164525 73 4 (67,90]
18 12396 1413 1.0187730 1.2063588 40 1 (37,53]
19 12207 1527 1.0187730 1.0036734 70 2 (67,90]
20 12143 1544 1.0187730 0.8849896 62 4 (53,67]The variables psu and stratum identify the
primary sampling unit and the stratum. I believe the survey uses a kind
of stratified multi-stage cluster sampling design. The variables
dweight and pspwght are the design weight and
survey weight, respectively. The latter has been adjusted through a
re-weighting method (post-stratification) using raking with age, gender,
education, and region as the auxiliary variables. The variable
agea is the age of the respondent, which I have used to
form four age groups (agegroup) based on quartiles. The
target variable for this demonstration will be hmsacld
which is a response to the statement “Gay male and lesbian couples
should have the same rights to adopt children as straight couples.” The
response was on a 5-point scale: agree strongly (1), agree (2), neither
agree nor disagree (3), disagree (4), disagree strongly (5).
To estimate the mean response for the population, we simply need to
multiply the target variable (hmsacld) by the survey weight
(pspwght), add these up across the sampled elements, and
divide by the sum of weights for the sampled elements. The formula is
\[
\hat\mu = \frac{\sum_{i \in \mathcal{S}} w_iy_i}{\sum_{i \in
\mathcal{S}}w_i}
\] where \(y_i\) is the target
variable (hmscald) and \(w_i\) is the survey weight
(pspwght). Here’s how we could do this in R.
ukdata %>% summarize(y = sum(hmsacld * pspwght) / sum(pspwght)) y
1 2.194636We can also estimate the mean response for each age group by doing this separately for each part of the sample. Technically we can do this by defining \(y_i\) as equal to zero if the elements is not in the domain of interest, but we can also do this simply by doing the calculation separately for each domain.
ukdata %>% group_by(agegroup) %>% summarize(y = sum(hmsacld * pspwght) / sum(pspwght))# A tibble: 4 × 2
agegroup y
<fct> <dbl>
1 (14,37] 1.76
2 (37,53] 2.10
3 (53,67] 2.39
4 (67,90] 2.96Note that this did not require any knowledge of the sampling design and the calibration. It also does not require sophisticated software. Everything that was done could be done with any statistical package or a spreadsheet program. But we can confirm that the estimates are consistent with what would be obtained using specialized software like the survey package for R.
library(survey)
ukdesign <- svydesign(ids = ~psu, strata = ~stratum, weights = ~pspwght, data = ukdata)
svymean(~hmsacld, design = ukdesign) mean SE
hmsacld 2.1946 0.0338svyby(~hmsacld, by = ~agegroup, design = ukdesign, FUN = svymean) agegroup hmsacld se
(14,37] (14,37] 1.760756 0.05010013
(37,53] (37,53] 2.100307 0.06365683
(53,67] (53,67] 2.390973 0.05664507
(67,90] (67,90] 2.959859 0.05751275Same estimates (aside from differences in rounding), but the survey package is also capable of estimating standard errors given correct information about the sampling design. Here that information is communicated through the identity of the primary sampling units, the identity of the strata, as well as the weights.
We can also treat hmscald as a categorical variable if
we want to estimate the proportion of people in the population that
would respond in a certain way.
ukdata <- ukdata %>%
mutate(hmsacld = factor(hmsacld, levels = 1:5,
labels = c("strongly agree", "agree", "neither",
"disagree", "disagree strongly")))
ukdesign <- svydesign(ids = ~psu, strata = ~stratum, weights = ~pspwght, data = ukdata)
svymean(~hmsacld, design = ukdesign) mean SE
hmsacldstrongly agree 0.338782 0.0135
hmsacldagree 0.342761 0.0114
hmsacldneither 0.158160 0.0101
hmsaclddisagree 0.105632 0.0082
hmsaclddisagree strongly 0.054665 0.0063confint(svymean(~hmsacld, design = ukdesign)) 2.5 % 97.5 %
hmsacldstrongly agree 0.31228061 0.36528393
hmsacldagree 0.32044338 0.36507824
hmsacldneither 0.13845528 0.17786513
hmsaclddisagree 0.08963606 0.12162831
hmsaclddisagree strongly 0.04225585 0.06707323