Sampling Weights for Analyses of Couple Data: Example of the Demographic and Health Surveys

Abstract

In some surveys, women and men are interviewed separately in selected households, allowing matching of partner information and analyses of couples. Although individual sampling weights exist for men and women, sampling weights specific for couples are rarely derived. We present a method of estimating appropriate weights for couples that extends methods currently used in the Demographic and Health Surveys (DHS) for individual weights. To see how results vary, we analyze 1912 estimates (means; proportions; linear regression; and simple and multinomial logistic regression coefficients, and their standard errors) with couple data in each of 11 DHS surveys in which the couple weight could be derived. We used two measures of bias: absolute percentage difference from the value estimated with the couple weight and ratio of the absolute difference to the standard error using the couple weight. The latter shows greater bias for means and proportions, whereas the former and a combination of both measures show greater bias for regression coefficients. Comparing results using couple weights with published results using women’s weights for a logistic regression of couple contraceptive use in Turkey, we found that 6 of 27 coefficients had a bias above 5 %. On the other hand, a simulation of varying response rates (27 simulations) showed that median percentage bias in a logistic regression was less than 3 % for 17 of 18 coefficients. Two proxy couple weights that can be calculated in all DHS surveys perform considerably better than either male or female weights. We recommend that a couple weight be calculated and made available with couple data from such surveys.

Appendices

Appendix: Derivation of Couple Weight and Relationship to Household Weight Available in DHS

Here we derive a couple weight for the general sampling design of DHS of two-stage sampling within each strata—that is, sampling of clusters within strata and then households within a selected cluster. Algebra to derive the couple weight from the normalized household weight provided with the DHS surveys is also given. Women’s, men’s, and couple weights differ from the household weight and each other only because of different response rates of these groups.

Notation

i = strata identifier

j = cluster identifier

h = household identifier

l = identifier of person or couple in the household

n^h = total number of completed households in the sample

n^w = total number of completed women in the sample

n^m = total number of completed men in the sample

n^c = total number of completed couples in the sample

I = number of strata

J_i = number of clusters in strata i that are selected

H_ij = number of households selected in cluster j in strata i

z = identifier for woman (z = w), man (z = m), or couple (z = c)

L_ijh = Number of eligible z in household ijh

We assume clusters and households are selected from all strata:

$$ {p}_{ij}^1=\Pr \left\{j\mathrm{th}\ \mathrm{cluster}\ \mathrm{in}\ \mathrm{stratum}\ i\ \mathrm{is}\ \mathrm{selected}\right\} $$

${p}_{ijh}^2=\Pr \left\{\mathrm{household}\ h\ \mathrm{in}\ \mathrm{cluster}\ j\ \mathrm{in}\ \mathrm{stratum}\ i\ \mathrm{is}\ \mathrm{selected}\ \right|\mathrm{cluster}\ j\ \mathrm{is}\ \mathrm{selected}$}

$$ {p}_{ijh}^3=\Pr \left(\begin{array}{c}\mathrm{household}\ h\ \mathrm{in}\ \mathrm{cluster}\ j\ \mathrm{in}\ \mathrm{strata}\ i\ \mathrm{is}\ \mathrm{selected}\ \mathrm{for}\ \mathrm{male}\ \mathrm{interview}\mid \mathrm{household}\ \\ {}h\ \mathrm{in}\ \mathrm{cluster}\ j\ \mathrm{in}\ \mathrm{stratum}\ i\ \mathrm{is}\ \mathrm{selected}\end{array}\right). $$

However, by design in DHS, ${p}_{ijh}^3$ = p³, a fixed constant, given that the sampling design calls for a fixed proportion of households to be sampled for the male interview in addition to the household and female interviews.

${r}_i^h=\Pr \Big\{\mathrm{household}\ \mathrm{is}\ \mathrm{completed}\ \mathrm{in}\ \mathrm{stratum}\ i$ | household is selected in strata i}

$$ {r}_i^z=\Pr \left\{\mathrm{interview}\ \mathrm{with}\ z\ \mathrm{is}\ \mathrm{completed}\ \mathrm{in}\ \mathrm{stratum}\ i\ \right|\ \mathrm{household}\ \mathrm{is}\ \mathrm{completed}\ \mathrm{in}\ \mathrm{stratum}\ i\Big\}. $$

Whereas ${p}_{ij}^1$, ${p}_{ijh}^2$, and p³ are probabilities from the sampling design, ${r}_i^h$ and ${r}_i^z$are proportions estimated ex post facto. These latter proportions are typically computed at the strata level in DHS, but clearly the equations could be modified for other designs.

The household weight is derived as follows:

$$ {x}_{ij h}^h=\frac{1}{p_{ij}^1\times {p}_{ij h}^2\ {r}_i^h\ }. $$

However, for many applications, it is desirable to have a normalized household weight (that sums to the sample size of households). So let the normalized weight be ${w}_{ijh}^h$. Let S denote the set of all households with completed questionnaires. Then define the indicator function:

1[h ∊ S] = 1 if selected household h (in stratum i and cluster j) had a completed household interview, and 0 otherwise.

Then also define

$$ {T}^h={\sum}_{i=1}^I{\sum}_{j=1}^{J_i}{\sum}_{h=1}^{H_{ij}}1\left[h\in S\right]\times {x}_{ij h}^h. $$

Thus, for completed households, ${w}_{ijh}^h=\frac{n^h\times {x}_{ijh}^h}{T^h}$.

Similarly, to derive women’s weights, men’s weights or couple weights,

$$ {x}_{ij h}^z=\frac{1}{p_{ij}^1\times {p}_{ij h}^2\times {p}^3\times {r}_i^h\times {r}_i^z}. $$

Note that for women’s weights we take p³ = 1.0.

Define B to be the set of all completed interviews of individuals/couples z, and let 1[l ∊ B] = 1 if selected household h in stratum i and cluster j had a completed interview(s) of the lth individual/couple, and 0 otherwise; this is also 0 if there is no eligible woman/male/couple in the household.

Then let

$$ {T}^z={\sum}_{i=1}^I{\sum}_{j=1}^{J_i}{\sum}_{h=1}^{H_{ij}}{\sum}_{l=1}^{Lijh}1\left[l\in B\right]\times {x}_{ij h}^z, $$

and

$$ {w}_{ijh}^z=\frac{x_{ijh}^z\ {n}^z}{T^z}. $$

Obtaining the Couple Weight From ${w}_{ijh}^h$ and ${r}_i^c$

With the DHS data files that are publicly available, the original sampling probabilities are not given. Thus, in deriving the couple weight for these data, the normalized household weight must be used as the starting point. Using this weight then,

$$ {w}_{ij h}^h=\frac{n^h\times {x}_{ij h}^h}{T^h}={n}^h\times \frac{1}{p_{ij}^1\times {p}_{ij h}^2\times {r}_i^h\ }/{T}^h. $$

As described in the text, we invert this and multiply by ${r}_i^c$ and a constant k = (p³ × n^h / T^h); that is,

$k\times {r}_i^c\times \left(1/{w}_{ij h}^h\right)={p}_{ij}^1\times {p}_{ij k}^2\times {p}^3\times {r}_i^h\times {r}_i^c=1/{x}_{ij h}^c.$So now for couples, the normalized weight will be

$$ {w}_{ijh}^c={x}_{ijh}^c\ {n}^c/{T}^c, $$

which by definition will sum across the entire completed sample of couples, to n^c.

In our analyses, we utilize women’s, men’s, and the couple weights as well as two proxy couple weights. The two proxies estimate the proportion of couples completed from information available on individual in-union persons. Specifically, following the Venn diagram of Fig. 1, information on couples in which both partners were nonrespondents is not available in the DHS survey data (other than in these 11 surveys). However, in virtually all DHS surveys, it is possible to enumerate wives whose husbands are nonrespondents and husbands whose wives are nonrespondents because spouse’s line number is given in the completed individual questionnaires. Thus, one proxy (ALT) adds these two numbers to the completed couples to form the denominator of the couple response rate—that is, the estimate of ${r}_i^c$. For the other proxy (EST), the Chandra-Sekar–Deming method is used to estimate the number of couples in which both partners were nonrespondents, assuming independence of the probabilities of nonresponse.