Appendix A

Notation and Crosswalk

One notation for weights, and a dictionary between the survey, econometrics, and causal-inference literatures.

You will need this appendix when a symbol in a chapter is not the one your field uses, or when a referee from another field uses a word you do not: when an economist's "cluster" has to be matched to a survey statistician's "PSU," when [pweight=] and [aweight=] give the same coefficient and different standard errors and you need to know which one asserts what, or when you want the book's nine weight symbols on one page. It is a reference, not a chapter: read the table you need and go back.

One notation runs through the booklet, and it is stricter than software. Stata will accept the ACS person weight as [pweight=perwt], [aweight=perwt], or [fweight=perwt] and return the same regression coefficient three times with three different standard errors, because the three keywords assert three different things about what the column means. This book gives each of those meanings its own symbol and never reuses one for another.

The notation contract

Nine symbols cover every weight the booklet uses. The first four are the stages of a survey weight, from the design's inclusion probability to the number shipped on the file; the next three are the other things software calls a weight; the last two are weights that an analysis constructs rather than receives.

Symbol	Name	What it is	Where it does its work
$\pi_i$	inclusion probability	the probability that unit $i$ is selected under the design	Chapters 0 and 2
$d_i = 1/\pi_i$	base or design weight	undoes unequal selection probabilities	Chapter 2
$g_i$	adjustment factor	nonresponse and calibration adjustment: raking, poststratification, trimming	Chapter 2
$w_i = d_i g_i$	final survey weight	the weight shipped on the file	Chapters 2 and 3
$a_i$	precision weight	inversely proportional to a row's variance, as for rows that are group means	Chapters 2 and 4
$f_i$	frequency weight	the number of identical observations a row stands for	Chapter 2
$w_i^{(r)}$	replicate weight	the weight of unit $i$ in replicate $r = 1, \dots, R$	Lab 3
$s_a$	standardization weight	the share of age group $a$ in a fixed standard population	Chapter 3
$\omega_i$	implicit weight	the weight a regression places on unit $i$ 's conditional effect	Chapters 4 and 5

A frequency weight changes the number of observations; a precision weight changes the variance attributed to each observation; a survey weight changes the population a sum represents. The three can give the same point estimate and three different standard errors, which is why $w$ is never overloaded.

The design and the regression each need a few more letters. A stratified two-stage survey has strata indexed by $h$ and PSUs by $j$ ; a cluster-robust variance has $G$ clusters of average size $\bar m$ ; a treatment-effect regression has cells $x$ with a treated share and a contrast in each.

Symbol	Meaning
$U$ , $N$ ; $s$ , $n$	the finite population and its size; the sample and its size
$h = 1, \dots, H$	strata
$j = 1, \dots, n_h$	primary sampling units (PSUs) sampled in stratum $h$
$G$	clusters in a cluster-robust variance
$\rho$	intraclass correlation
$\bar m$	average cluster size
$\DEFF(\hat\theta)$	design effect of the estimator $\hat\theta$ : its design-based variance over its variance under simple random sampling of the same size
$\nKish$ , $\ndesign$ , $\nDDC$	the three scalar effective sample sizes of the Chapter 0 box
$D_i$ , $X_i$	a binary treatment and covariates
$p_x$ , $\tau_x$	the treated share in covariate cell $x$ ; the treated-minus-untreated contrast in that cell
$\PATE$ , $\ATT$ , $\ATU$	the population average treatment effect; the average effect among the treated; among the untreated

Estimands and populations

Three objects are kept apart throughout. The estimand is the population quantity an analysis targets, defined without reference to any data. The estimator is the rule that maps a sample to a number. The estimate is the number the rule returns on the sample in hand. A weighted and an unweighted mean of the same file are two estimators; whether they target the same estimand depends on the design.

Five populations are named because each gap between them is a place where an analysis can drift:

The target population is the one the result is meant to describe.
The frame population is the list, area sample, or telephone-number bank the survey draws from.
The sampled population is the part of the frame with a positive probability of selection.
The respondent population is those in the sampled population who would respond if selected.
The analytic population is who remains after the analyst's restrictions and missing-data rules.

The sample, the units actually selected, and the respondents, the selected units that answered, are realizations of the sampled and respondent populations, not populations of their own. Chapter 1 follows all five populations through one estimate.

Coverage honesty follows from the list: when a release omits a state, the target population is the participating jurisdictions, and the result is labeled that way (Chapter 2, Chapter 8).

What software calls a weight

IPUMS's Stata example for the CPS ASEC declares the replicate weights as [iweight=], not [pweight=], because some of them are negative and Stata refuses a negative probability weight. That is the clearest sign that a weight keyword is an assertion about the column, not a formatting choice. The table maps each keyword the booklet's three languages use to the symbol it asserts.

Software syntax	Symbol in this book	What the software assumes
Stata `[pweight=]`	$w_i$	sampling weights; implies a robust (linearized) variance
Stata `[aweight=]`	$a_i$	analytic weights inversely proportional to a row's variance; rescaled to sum to the number of rows
Stata `[fweight=]`	$f_i$	integer frequency weights; the sample size becomes their sum
Stata `[iweight=]`	none	importance weights with no fixed statistical meaning; IPUMS uses them for CPS replicate designs because some replicate weights are negative
R `survey::svydesign(weights =)`	$w_i$	sampling weights inside a declared design
R `lm(weights =)`	$a_i$	precision weights: weighted least squares with error variance proportional to one over the weight
Python `svy.Design(wgt =)`	$w_i$	sampling weights inside a declared design
Python `pyfixest.feols(weights =)`	$a_i$ by default	analytic weights (`weights_type="aweights"`), or frequency weights with `"fweights"`; paired with a robust or cluster-robust variance, the coefficients equal the survey-weighted ones
Python `statsmodels` `WLS(weights =)`	$a_i$	precision weights

The table's lesson is narrow. A weighted regression coefficient is the same number whichever of $w_i$ , $a_i$ , or $f_i$ the software thinks it was given; the standard error is not. Which variance is right depends on what the weight means and on how the data were drawn, which is the subject of Chapter 3 and Chapter 6.

The master crosswalk

A survey statistician's primary sampling unit, an econometrician's cluster, and a trialist's unit of assignment are one object described in three seminar rooms; a stratum, by contrast, has no standard econometric name at all, which is one reason the stratification gain goes missing from cluster-robust standard errors. Each row of the table names one such object in the three dialects. The last column states the condition under which the translation holds; where a condition is missing, the translation is an analogy, not an identity.

Survey statistics	Econometrics	Causal inference	Condition or caveat
finite-population quantity (total, mean, ratio, census regression coefficient)	population parameter; probability limit of an estimator	estimand (PATE, ATT, CATE)	name the population and the unit before the estimator
inclusion probability $\pi_i$	sampling rate; selection probability	propensity score $e(x)$	the causal analogy holds for inverse weighting, not for design-based variance
design weight $d_i = 1/\pi_i$	probability weight (`pweight`)	inverse-propensity weight $1/e(x)$	$\pi_i$ is known by design; $e(x)$ is estimated from a model
nonresponse and calibration adjustment $g_i$ (raking, poststratification)	reweighting to known margins; inverse probability weighting for missing data (Wooldridge 2007)	balancing weights	calibrated weights are no longer inverse probabilities; their validity rests on a response or outcome model
Horvitz–Thompson estimator of a total	inverse-probability-weighted sum	inverse-probability-weighted (Horvitz–Thompson) estimator of mean potential outcomes	design-unbiased for a finite-population total when every $\pi_i \gt 0$ and is known (Horvitz and Thompson 1952)
Hájek (ratio) estimator of a mean	weighted mean; WLS on a constant	normalized inverse-probability weighting	approximately unbiased and consistent, not exactly unbiased
stratum	none standard (sampling stratification)	block in a blocked design	strata reduce variance when they predict the outcome
primary sampling unit (PSU)	cluster	cluster of jointly assigned units	clustering is a property of the sampling or assignment design, not of correlated errors alone (Abadie et al. 2023)
intra-PSU correlation	intraclass correlation; the Moulton problem	intraclass correlation of outcomes	the Moulton problem also needs a regressor that is correlated within clusters
Taylor linearization over PSU totals within strata, with replacement	cluster-robust sandwich (CRV1)	design-based variance for cluster-randomized designs	identical for a weighted mean with one stratum; with informative strata CRV1 omits the stratification gain (Appendix B)
design effect $\DEFF(\hat\theta)$	ratio of the correct variance to an iid variance	none standard	specific to an estimator and an outcome
clustering component of a design effect for a mean, $1 + (\bar m - 1)\rho_y$	Moulton factor for a regression coefficient, $1 + (\bar m - 1)\rho_x\rho_u$ with equal cluster sizes	none standard	Moulton is a special-case regression analogue of the clustering component of a design effect, not the design effect itself (Moulton 1990)
design degrees of freedom: PSUs minus strata (Taylor); replicates minus one (ACS and CPS replicates)	$G - 1$ in few-cluster inference	number of randomization units	a common rule, not a law; NCHS notes it is not directly applicable to the NHIS design
replicate weights (BRR, Fay, jackknife, successive difference, bootstrap)	resampling variance estimators	randomization inference is a different repetition	replicate weights are pre-packaged repeated sampling; the five methods are distinct procedures, and none is "a bootstrap" in general (Fay and Train 1995)
declare the design before analyzing	cluster at the level of sampling or assignment	analyze as randomized	identify every mechanism that induces common uncertainty; the "coarser level" rule holds for nested hierarchies (Abadie et al. 2023; Cameron and Miller 2015)
domain (subpopulation) estimation on the full design	subgroup estimate with full-sample clustering; interaction with a group indicator	conditional average effect for a subgroup	deleting rows gives the same point estimate and can change the variance when strata lose PSUs
frequency weight $f_i$	`fweight`	none standard	changes the number of observations
precision weight $a_i$	`aweight`; WLS for heteroskedasticity or group means	inverse-variance weights in meta-analysis	the precision motive for weighting (Solon, Haider, and Wooldridge 2015; Dickens 1990)
implicit weights $\omega_i$	variance weighting of OLS under heterogeneity (Angrist 1998)	implied regression weights (Aronow and Samii 2016; Chattopadhyay and Zubizarreta 2023)	nonnegative with saturated controls; can be negative with linear controls
standardization weights $s_a$ (direct age standardization)	reweighting to a reference distribution	standardization to a target covariate distribution	redefines the comparison population rather than repairing representation
design-based inference: $y$ fixed, the sample random	sampling-based inference: draws from a population	design-based inference: assignment random	which repetition is intended decides which variance is right (Abadie et al. 2020)
finite population correction	sampled-cluster fraction	finite-sample (Neyman) variance	when most clusters are observed, conventional cluster-robust SEs can be conservative
Kish effective sample size $\nKish$	effective sample size of a weighted estimator	effective sample size of balancing weights	prices weight dispersion only, under stated conditions (Kish 1965)

A note on the seed crosswalk

The booklet began from an article that compared survey statistics and econometrics in a seven-row table. Its table was right about the direction of the analogy and wrong in several of its equivalences; the corrections are binding everywhere in the booklet, and a reader does not need the article to use this appendix.

The seven corrections, and one about NHANES

The seed said	The crosswalk says	Why
Weights get the point estimate right; the design gets the uncertainty right	The weights help define the estimator; the sampling design and the weight construction determine repeated-sampling variance	final weights are estimated, and replication that repeats the weighting captures that
The weighted estimator is design-unbiased, however convoluted the sampling	Correct inclusion weights give design-unbiased or consistent estimators for specified finite-population quantities	the Horvitz–Thompson total is unbiased; the Hájek mean is consistent; calibrated weights are not inverse probabilities
Design effect = Moulton factor $1 + (\bar m - 1)\rho$	Moulton is a special-case regression analogue of the clustering component of a design effect	a design effect also reflects strata and weighting, and the Moulton factor involves the regressor's clustering
Replicate weights are the survey world's pre-packaged cluster bootstrap	Replicate weights are pre-packaged repeated sampling; BRR, Fay, jackknife, SDR, and bootstrap are distinct	the ACS and CPS use successive difference replication, not a bootstrap
Taylor linearization is CRV1 with strata as an extra refinement; weights plus clustering on the PSU get you most of the way	The two coincide exactly for a weighted mean in one stratum; with informative strata CRV1 omits the stratification gain and is conservative	stated with its conditions in Appendix B
df = PSUs minus strata	A common rule for Taylor linearization; replicate designs use the producer's value	ACS replicates carry 79 degrees of freedom whatever the number of strata
"Cluster at the level of treatment variation"	Identify every sampling or assignment mechanism that induces common uncertainty; the coarsest relevant level governs within a nested hierarchy	the full statement is in Chapter 6

The seed also described NHANES as roughly thirty strata of two PSUs each, with about fifteen degrees of freedom. For the August 2021–August 2023 cycle the public file has 15 pseudo-strata and 30 pseudo-PSUs, so the design degrees of freedom are 15.