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A new version of the CALMAR calibration adjustment program

CALMAR2 is an updated version of the CALMAR calibration program, offering new features like simultaneous calibration with multiple levels, handling total non-response, and interactive parameter input. This program allows for calibrating data collected at different observation levels efficiently. The method involves gathering information at various levels, such as households and individuals, and performing independent or simultaneous calibration. An example scenario illustrates how weights are computed based on known auxiliary variables. CALMAR2 enables users to input sample data and calibration variables to generate calibrated weights effectively. The program streamlines the process by conducting multiple calibrations to ensure statistical consistency across different data files.

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A new version of the CALMAR calibration adjustment program

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  1. A new version of the CALMAR calibration adjustment program

  2. The CALMAR2 macros

  3. I.1. Background CALMAR = CALibration on MARgins CALMAR 1 = SAS macro program, written in 1992-1993 at France’s INSEE by Sautory Scope : implementing calibration methods developped by Deville & Särndal (JASA, 1992) CALMAR 2 = SAS macro, written in 2000 at France’s INSEE Scope : implementing generalized calibration method for handling total non-response (Deville, 1998)

  4. I.2. What’s new in CALMAR 2 • Simultaneous calibration with 2 or 3 levels • Total non- response adjustment using generalized calibration • Handling collinearities between auxiliary variables • A 5th distance function : generalized hyperbolic sine • Interactive screens to enter parameters, thanks to • CALMAR2_GUIDE

  5. Simultaneous calibration

  6. II.1. The method • Informations are collected at several levels of observation :  • households + every household’s member • or : firms + every establishment of the firms • i.e. cluster sampling survey, including questions about the clusters • households + some of their members (Kish individuals)  • i.e. two-stages sampling, including questions about the primary units (P.U.) • households + every household’s member + Kish units • + auxiliary information available at every level

  7. How performing calibration ? • Independent calibration at every level of observation   • Simultaneous calibration (or "integrated") :   - same weights for all members of a household   - consistency between statistics obtained from varied data files Simultaneous calibration method A single calibration is performed at the P.U. level, after having computed the calibration variables totals defined at the secondary levels for each P.U. (Sautory, 1996)

  8. II.2. An example • households (sM sample) • all the members of the selected households (sI sample) • one member (Kish individual) in each selected • household m, chosen by simple random sampling among • the eligible members of the household (sK sample)  Weight of the household m : Weight of the member i of the household m : Weight of the Kish-individual of the household m :

  9. Auxiliary information   = auxiliary variables vector for each household m in = vector of the known auxiliary variables totals in the households population = auxiliary variables vector for each individual (m,i) in sI = vector of the known totals in the individuals population = auxiliary variables vector for the Kish- individual in   = vector of the known totals in the Kish-units population

  10. For each household m we compute : • the totals of the individual variables : • the estimatedtotals of the Kish- individual variables : • Vector of the calibration variables for the household m : • Vector of the totals : (X , Z, V)  • Calibration equations :

  11.  weights = weight of the household m in = weight of the individual (m,i) of the household m in = weight of the Kish-individual of the household m in The 3 samples are correctly calibrated on totals X, Z et V :

  12. Calmar 2 performs such simultaneous calibrations. The user must provide the entry tables for the various levels (sample data files and calibration variables totals files) : the program performs all the required operations necessary to reduce the process to a single calibration, and creates the varied calibrated weights files.

  13. An example of simultaneous calibration

  14. The survey • Sampling design : two stages sampling • primary units = households, selected by stratified sampling with S.R.S. in the stratum • secondary units (Kish-units) = one member per selected household, withdrawn by S.R.S. among more than 14 years old members • Questionary • variables of interest are measured on Kish-units • questions about the habitation and the whole family • questions about each member of the household (age, sex, profession) • Calibration variables (xk) • Households : household size + head of household professional group + strata (~ agglomeration size) • All individuals : sex + age group • Kish individuals : sex + age group • Population totals (X) come from the sampling frame

  15. The program

  16. %CALMAR2(datamen=base.echant_menages, • marmen=base.marge_men, • poids=poids1, • ident=ident, • dataind=base.echant_indiv, • marind=base.marge_ind, • ident2=id, • datakish=base.echant_kish, • markish=base.marge_kish, • poidkish=nbelig, • m=1, • datapoi=poidsmen, • datapoi2=poidsind, • datapoi3=poidskish, • poidsfin=w3, • labelpoi=calage 3 niveaux, • poidskishfin=w3k, • labelpoikish=poids kish total, • edition=3)

  17. The output

  18. ********************************** *** PARAMÈTRES DE LA MACRO *** ********************************** TABLE(S) EN ENTRÉE : TABLE DE DONNÉES DE NIVEAU 1 DATAMEN = BASE.ECHANT_MENAGES IDENTIFIANT DU NIVEAU 1 IDENT = IDENT TABLE DE DONNÉES DE NIVEAU 2 DATAIND = BASE.ECHANT_INDIV IDENTIFIANT DU NIVEAU 2 IDENT2 = ID TABLE DES INDIVIDUS KISH DATAKISH = BASE.ECHANT_KISH PONDÉRATION INITIALE POIDS = POIDS1 FACTEUR D'ÉCHELLE ECHELLE = 1 PONDÉRATION QK PONDQK = __UN PONDÉRATION KISH POIDKISH = NBELIG TABLE(S) DES MARGES : DE NIVEAU 1 MARMEN = BASE.MARGE_MEN DE NIVEAU 2 MARIND = BASE.MARGE_IND DE NIVEAU KISH MARKISH = BASE.MARGE_KISH MARGES EN POURCENTAGES PCT = NON EFFECTIF DANS LA POPULATION : DES ÉLÉMENTS DE NIVEAU 1 POPMEN = DES ÉLÉMENTS DE NIVEAU 2 POPIND = DES ÉLÉMENTS KISH POPKISH =

  19. MÉTHODE UTILISÉE M = 1 BORNE INFÉRIEURE LO = BORNE SUPÉRIEURE UP = COEFFICIENT DU SINUS HYPERBOLIQUE ALPHA = 1 SEUIL D'ARRÊT SEUIL = 0.0001 NOMBRE MAXIMUM D'ITÉRATIONS MAXITER = 15 TRAITEMENT DES COLINÉARITÉS COLIN = NON TABLE(S) CONTENANT LA POND. FINALE DE NIVEAU 1 DATAPOI = POIDSMEN DE NIVEAU 2 DATAPOI2 = POIDSIND DE NIVEAU KISH DATAPOI3 = POIDSKISH MISE À JOUR DE(S) TABLE(S) DATAPOI(2)(3) MISAJOUR = OUI PONDÉRATION FINALE POIDSFIN = W3 LABEL DE LA PONDÉRATION FINALE LABELPOI = CALAGE 3 NIVEAUX PONDÉRATION FINALE DES UNITES KISH POIDSKISHFIN = W3K LABEL DE LA PONDÉRATION KISH LABELPOIKISH = POIDS KISH TOTAL CONTENU DE(S) TABLE(S) DATAPOI(2)(3) CONTPOI = OUI ÉDITION DES RÉSULTATS EDITION = 3 ÉDITION DES POIDS EDITPOI = NON STATISTIQUES SUR LES POIDS STAT = OUI CONTRÔLES CONT = OUI TABLE CONTENANT LES OBS. ÉLIMINÉES OBSELI = NON NOTES SAS NOTES = NON

  20. COMPARAISON ENTRE LES MARGES TIRÉES DE L'ÉCHANTILLON (PONDÉRATION INITIALE) ET LES MARGES DANS LA POPULATION (MARGES DU CALAGE) MARGE MARGE POURCENTAGE POURCENTAGE VARIABLE MODALITÉ ÉCHANTILLON POPULATION ÉCHANTILLON POPULATION NBIND 01 1525.60 1539 26.30 26.53 02 1914.37 1860 33.00 32.06 03 797.71 1000 13.75 17.24 04 930.78 885 16.05 15.26 05 365.18 361 6.30 6.22 06 267.36 156 4.61 2.69 PCSPR 1 80.70 124 1.39 2.14 2 191.78 290 3.31 5.00 3 822.81 624 14.18 10.76 4 832.34 870 14.35 15.00 5 569.41 682 9.82 11.76 6 1279.53 1237 22.06 21.32 7 1839.32 1831 31.71 31.56 8 185.11 143 3.19 2.47 STRATE 0 1453.00 1453 25.05 25.05 1 966.00 966 16.65 16.65 2 805.00 805 13.88 13.88 3 1689.00 1689 29.12 29.12 4 888.00 888 15.31 15.31

  21. MARGE MARGE POURCENTAGE POURCENTAGE VARIABLE MODALITÉ ÉCHANTILLON POPULATION ÉCHANTILLON POPULATION AGE 00-14 ans 3245.32 2857 21.46 19.52 15-24 ans 2217.86 2044 14.67 13.96 25-59 ans 6699.70 6800 44.31 46.45 60- ? ans 2957.50 2939 19.56 20.08 SEXE 1 7546.69 7108 49.91 48.55 2 7573.69 7532 50.09 51.45 AGEK A15 2155.94 2044 18.28 17.35 A25 6752.61 6800 57.25 57.71 A60 2885.84 2939 24.47 24.94 SEXEK 1 5596.30 5673 47.45 48.15 2 6198.09 6110 52.55 51.85

  22. MÉTHODE : LINÉAIREPREMIER TABLEAU RÉCAPITULATIF DE L'ALGORITHME LA VALEUR DU CRITÈRE D'ARRÊT ET LE NOMBRE DE POIDS NÉGATIFS APRÈS CHAQUE ITÉRATION CRITÈRE POIDS ITÉRATION D'ARRÊT NÉGATIFS 1 1.31960 1 2 0.00000 1

  23. MÉTHODE : LINÉAIREDEUXIÈME TABLEAU RÉCAPITULATIF DE L'ALGORITHME LES COEFFICIENTS DU VECTEUR LAMBDA DE MULTIPLICATEURS DE LAGRANGE APRÈS CHAQUE ITÉRATION VARIABLE MODALITÉ LAMBDA1 LAMBDA2 NBIND 01 -0.15325 -0.15325 NBIND 02 -0.24295 -0.24295 NBIND 03 0.00562 0.00562 NBIND 04 -0.17355 -0.17355 NBIND 05 -0.00502 -0.00502 NBIND 06 -0.44773 -0.44773 PCSPR 1 0.92036 0.92036 PCSPR 2 0.50376 0.50376 PCSPR 3 -0.18514 -0.18514 PCSPR 4 0.15354 0.15354 PCSPR 5 0.36019 0.36019 PCSPR 6 0.08424 0.08424 PCSPR 7 0.16042 0.16042 PCSPR 8 . .

  24. VARIABLE MODALITÉ LAMBDA1 LAMBDA2 STRATE 0 -0.14172 -0.14172 STRATE 1 -0.07338 -0.07338 STRATE 2 -0.12634 -0.12634 STRATE 3 -0.03106 -0.03106 STRATE 4 . . AGE 00-14 ans -0.03549 -0.03549 AGE 15-24 ans -0.65576 -0.65576 AGE 25-59 ans -0.52872 -0.52872 AGE 60- ? ans -0.64430 -0.64430 SEXE 1 -0.08395 -0.08395 SEXE 2 . . AGEK A15 0.67198 0.67198 AGEK A25 0.68366 0.68366 AGEK A60 0.74262 0.74262 SEXEK 1 0.01727 0.01727 • SEXEK 2 . .

  25. COMPARAISON ENTRE LES MARGES FINALES DANS L'ÉCHANTILLON (AVEC LA PONDÉRATION FINALE)ET LES MARGES DANS LA POPULATION (MARGES DU CALAGE) MARGE MARGE POURCENTAGE POURCENTAGE VARIABLE MODALITÉ ÉCHANTILLON POPULATION ÉCHANTILLON POPULATION NBIND 01 1539 1539 26.53 26.53 02 1860 1860 32.06 32.06 03 1000 1000 17.24 17.24 04 885 885 15.26 15.26 05 361 361 6.22 6.22 06 156 156 2.69 2.69 PCSPR 1 124 124 2.14 2.14 2 290 290 5.00 5.00 3 624 624 10.76 10.76 4 870 870 15.00 15.00 5 682 682 11.76 11.76 6 1237 1237 21.32 21.32 7 1831 1831 31.56 31.56 8 143 143 2.47 2.47 STRATE 0 1453 1453 25.05 25.05 1 966 966 16.65 16.65 2 805 805 13.88 13.88 3 1689 1689 29.12 29.12 4 888 888 15.31 15.31

  26. MARGE MARGE POURCENTAGE POURCENTAGE VARIABLE MODALITÉ ÉCHANTILLON POPULATION ÉCHANTILLON POPULATION AGE 00-14 ans 2857 2857 19.52 19.52 15-24 ans 2044 2044 13.96 13.96 25-59 ans 6800 6800 46.45 46.45 60- ? ans 2939 2939 20.08 20.08 SEXE 1 7108 7108 48.55 48.55 2 7532 7532 51.45 51.45 AGEK A15 2044 2044 17.35 17.35 A25 6800 6800 57.71 57.71 A60 2939 2939 24.94 24.94 SEXEK 1 5673 5673 48.15 48.15 2 6110 6110 51.85 51.85

  27. STATISTIQUES SUR LES RAPPORTS DE POIDS (= PONDÉRATIONS FINALES / PONDÉRATIONS INITIALES) ET SUR LES PONDÉRATIONS FINALES The UNIVARIATE Procedure Variable: _F_ (RAPPORT DE POIDS) Basic Statistical Measures Quantiles (Definition 5) Location Variability Quantile Estimate Mean 1.000000 Std Deviation 0.24564 100% Max 2.009262 Median 0.996533 Variance 0.06034 99% 1.745002 Mode 0.991339 Range 2.32886 95% 1.377982 Interquartile Range 0.21258 90% 1.278637 75% Q3 1.105492 50% Median 0.996533 25% Q1 0.892917 10% 0.749877 5% 0.613091 1% 0.251528 0% Min -0.319601 Extreme Observations -------------Lowest------------- ------------Highest----------- Value IDENT Obs Value IDENT Obs -0.3196012 1163032100 27 1.76397 5363019600 293 0.0374385 7363016270 365 1.79618 7463000450 381 0.1498661 1169040310 73 1.85813 2369004180 129 0.1872096 7269001420 348 1.97094 5463007950 326 0.2314417 7363017990 366 2.00926 5263016110 268

  28. STATISTIQUES SUR LES RAPPORTS DE POIDS (= PONDÉRATIONS FINALES / PONDÉRATIONS INITIALES) ET SUR LES PONDÉRATIONS FINALES The UNIVARIATE Procedure Variable: _F_ (RAPPORT DE POIDS) Histogram # Boxplot 2.05+* 1 * .* 1 * .* 1 * .* 3 0 .* 3 0 .** 5 0 .*** 7 0 .********* 26 | .********* 27 | .******************** 59 +-----+ .************************************* 110 | + | .******************************************* 128 *-----* 0.85+******************* 57 +-----+ .*********** 33 | .****** 17 | .*** 8 0 .** 5 0 .* 3 0 .* 2 0 .* 2 * .* 1 * . . . -0.35+* 1 * ----+----+----+----+----+----+----+----+--- * may represent up to 3 counts

  29. STATISTIQUES SUR LES RAPPORTS DE POIDS (= PONDÉRATIONS FINALES / PONDÉRATIONS INITIALES) ET SUR LES PONDÉRATIONS FINALES The UNIVARIATE Procedure Variable: __WFIN (PONDÉRATION FINALE) Basic Statistical Measures Quantiles (Definition 5) Location Variability Quantile Estimate Mean 11.60200 Std Deviation 4.62597 100% Max 29.19457 Median 10.11949 Variance 21.39957 99% 25.69548 Mode 9.57633 Range 32.03263 95% 20.11085 Interquartile Range 5.70090 90% 18.04434 75% Q3 13.98763 50% Median 10.11949 25% Q1 8.28672 10% 7.15056 5% 6.41373 1% 2.50660 0% Min -2.83806 Extreme Observations -------------Lowest------------ ------------Highest----------- Value IDENT Obs Value IDENT Obs -2.838058 1163032100 27 25.7604 5369016540 317 0.543982 7363016270 365 26.0985 7463000450 381 1.330811 1169040310 73 28.6378 5463007950 326 1.808444 7269001420 348 28.6643 8269018030 421 2.235727 7363017990 366 29.1946 5263016110 268

  30. STATISTIQUES SUR LES RAPPORTS DE POIDS (= PONDÉRATIONS FINALES / PONDÉRATIONS INITIALES) ET SUR LES PONDÉRATIONS FINALES The UNIVARIATE Procedure Variable: __WFIN (PONDÉRATION FINALE) Histogram # Boxplot 29+* 3 0 .* 1 0 .** 4 0 .*** 8 0 .**** 11 | .********* 25 | .************** 41 | .*********** 32 | 13+*********************** 67 +-----+ .********************** 64 *--+--* .********************************************* 134 +-----+ .****************************** 88 | .***** 14 | .** 4 | .* 3 | . -3+* 1 0 ----+----+----+----+----+----+----+----+----+ * may represent up to 3 counts

  31. MÉTHODE : LINÉAIRE RAPPORTS DE POIDS MOYENS (PONDÉRATIONS FINALES / PONDÉRATIONS INITIALES) POUR CHAQUE VALEUR DES VARIABLES NOMBRE D'OBSERVATIONS RAPPORT VARIABLE MODALITE DE NIVEAU 1 DE POIDS NBIND 01 133 1.00152 NBIND 02 167 0.97304 NBIND 03 69 1.24647 NBIND 04 79 0.95151 NBIND 05 31 0.99271 NBIND 06 21 0.58818 PCSPR 1 6 1.55064 PCSPR 2 15 1.52001 PCSPR 3 73 0.76645 PCSPR 4 73 1.04281 PCSPR 5 51 1.20566 PCSPR 6 111 0.96902 PCSPR 7 157 0.99429 PCSPR 8 14 0.76191 STRATE 0 100 1.00000 STRATE 1 100 1.00000 STRATE 2 100 1.00000 STRATE 3 100 1.00000 STRATE 4 100 1.00000 ENSEMBLE 500 1.00000

  32. MÉTHODE : LINÉAIRE RAPPORTS DE POIDS MOYENS (PONDÉRATIONS FINALES / PONDÉRATIONS INITIALES) POUR CHAQUE VALEUR DES VARIABLES NOMBRE D'OBSERVATIONS RAPPORT VARIABLE MODALITE DE NIVEAU 2 DE POIDS AGE 00-14 an 274 0.88664 AGE 15-24 an 184 0.93210 AGE 25-59 an 581 1.01758 AGE 60- ? an 249 0.99088 SEXE 1 640 0.94443 SEXE 2 648 0.99993 ENSEMBLE 1288 0.97235 NOMBRE D'INDIVIDUS RAPPORT VARIABLE MODALITE KISH DE POIDS AGEK A15 66 0.95043 AGEK A25 283 1.01108 AGEK A60 151 1.00090 SEXEK 1 232 0.98540 SEXEK 2 268 1.01264 ENSEMBLE 500 1.00000

  33. MÉTHODE : LINÉAIRE CONTENU DE LA TABLE poidsmen CONTENANT LA NOUVELLE PONDÉRATION w3 The CONTENTS Procedure # Variable Type Len Pos Label 1 IDENT Char 10 8 2 w3 Num 8 0 calage 3 niveaux CONTENU DE LA TABLE poidsind CONTENANT LA NOUVELLE PONDÉRATION w3 # Variable Type Len Pos Label 2 IDENT Char 10 20 1 id Char 12 8 3 w3 Num 8 0 calage 3 niveaux CONTENU DE LA TABLE poidskish CONTENANT LA NOUVELLE PONDÉRATION w3 # Variable Type Len Pos Label 2 ID Char 12 26 1 IDENT Char 10 16 3 w3 Num 8 0 calage 3 niveaux 4 w3k Num 8 8 poids kish total

  34. ********************* *** BILAN *** ********************* * * DATE : 24 AOUT 2005 HEURE : 11:12 * * ************************************* * TABLE EN ENTRÉE : BASE.ECHANT_MENAGES * ************************************* * * NOMBRE D'OBSERVATIONS DANS LA TABLE EN ENTRÉE : 500 * NOMBRE D'OBSERVATIONS ÉLIMINÉES : 0 * NOMBRE D'OBSERVATIONS CONSERVÉES : 500 * * VARIABLE DE PONDÉRATION : POIDS1 * * NOMBRE DE VARIABLES CATÉGORIELLES : 3 * LISTE DES VARIABLES CATÉGORIELLES ET DE LEURS NOMBRES DE MODALITÉS : nbind (6) pcspr (8) strate (5) * * SOMME DES POIDS INITIAUX : 5801 * TAILLE DE LA POPULATION : 5801 * * * *********************************** * TABLE EN ENTRÉE : BASE.ECHANT_INDIV * *********************************** * * NOMBRE D'OBSERVATIONS DANS LA TABLE EN ENTRÉE : 1288 * NOMBRE D'OBSERVATIONS ÉLIMINÉES : 0 * NOMBRE D'OBSERVATIONS CONSERVÉES : 1288 * * NOMBRE DE VARIABLES CATÉGORIELLES : 2 * LISTE DES VARIABLES CATÉGORIELLES ET DE LEURS NOMBRES DE MODALITÉS : * age (4) sexe (2) * SOMME DES POIDS INITIAUX : 15120 * TAILLE DE LA POPULATION : 14640 *

  35. * *********************************** * TABLE EN ENTRÉE : BASE.ECHANT_KISH * *********************************** * * NOMBRE D'OBSERVATIONS DANS LA TABLE EN ENTRÉE : 500 * NOMBRE D'OBSERVATIONS ÉLIMINÉES : 0 * NOMBRE D'OBSERVATIONS CONSERVÉES : 500 * * VARIABLE DE PONDÉRATION CONDITIONNELLE : NBELIG * NOMBRE MAXIMUM D'UNITES SECONDAIRES PAR UP : 1 * * NOMBRE DE VARIABLES CATÉGORIELLES : 2 * LISTE DES VARIABLES CATÉGORIELLES ET DE LEURS NOMBRES DE MODALITÉS : agek (3) sexek (2) * * SOMME DES POIDS INITIAUX : 11794 * TAILLE DE LA POPULATION : 11783 * * * MÉTHODE UTILISÉE : LINÉAIRE * LE CALAGE A ÉTÉ RÉALISÉ EN 2 ITÉRATIONS * IL Y A 1 POIDS NÉGATIFS * LES POIDS ONT ÉTÉ STOCKÉS DANS LA VARIABLE W3 DE LA TABLE POIDSMEN * ET DE LA TABLE POIDSIND * ET DE LA TABLE POIDSKISH * LES POIDS DES UNITES KISH ONT ÉTÉ STOCKÉS DANS LA VARIABLE W3K * DE LA TABLE POIDSKISH

  36. Handling total non-response with generalized calibration

  37. III.1. Generalized calibration Calibration functions : where  : vector of p adjustment parameters Calibration equations : Solving for  

  38. Basic result = parameter estimates of the instrumental regression of on with as instrumentalvariables, weighted by

  39. Precision = residual of the regression of Y on X in U with the instrumental variables Z Note : the instruments are equal to

  40. III.2. Calibration in case of total non-response Calibration after adjustment for non-response 1.a. Adjustment for non-response  Response probabilities (conditionnally to s) : is estimated referring to a response model and an estimation method Expansion estimator :

  41. Examples • Uniform response model : • Homogeneous response groups : • Generalized linear model : • vector of explanatory non-response variables • Note : for estimating , must be known both for respondents AND NON-RESPONDENTS

  42. 1.b. Calibration We start from corrected weights Conventional calibration :

  43. Direct conventional calibration  is equivalent to  with a uniform non-response model. Comparison between  and (Dupont, 1993) Let’s suppose : - N.R. is corrected by a GLM, in which H is one of the usual calibration functions F : - non-response variables are included into calibration set of variables . Then :  and are " similar " 

  44.  and  are identical when : (b) .N.R. is corrected by HRG model based on a categorical variable X .The sample is calibrated on the number of units in U for each X level  =  = formal post-stratification on U

  45. Directgeneralized calibration (E) Interpretation Response model : (E) can be written :

  46. So, if the were known : (E) = generalized calibration equation, with : F is defined as and such as

  47. Precision • uses the residuals in the population • uses the residuals of the instrumental regression • in r, weighted by the : • estimator for if response probabilities • were known

  48. Response probabilities are unknown  "estimate" and the residuals : i.e. instrumental regression weighted by final weights Note : looks like = estimated variance 1st phase (sample s selection) = estimated variance 2nd phase (respondents r "selection")

  49. Properties of the method • allows non–response correction even when explanatory variables are only known for respondents  • Handles the particular situation in which non-response explanatory variables are variables of interest (non ignorable response mechanism )  • reduces the bias produced by non–response thanks to variables , and reduces the variance thanks to variables  This method is performed in Calmar 2.

  50. An example of generalized calibration

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