The Estimation Methodology of the 2011 National Household Survey

Transcription

1 The Estiation Methodoogy of the 2011 Nationa Househod Survey François Verret Statistics Canada, 100 Tunney s Pasture Driveway, Ottawa, Ontario, Canada, K1A 0T6 Abstract Prior to 2011, the Canadian census of popuation was conducted with a andatory ongfor sent to 20% of the househods and a andatory short-for sent to the rest of the househods. The 2011 Census was conducted with a andatory short-for sent to the entire popuation. A vountary survey caed the Nationa Househod Survey (NHS) was created to coect the inforation that used to be coected with the ong-for. To iniize the ipact of non-response, a sape of 30% was seected for the NHS and after severa wees of coection, the foow-up efforts were concentrated on a rando subsape of the reaining non-respondents. The design-weighted response rate was 77%, whie for the ast census ong-for the response rate was 94%. This paper describes the estiation ethodoogy used in the 2011 NHS. Design weights were first cacuated. A non-response weight adjustent was done using census and ined adinistrative data and by converting a nearest-neighbour and whoe-househod iputation approach to a reweighting approach. Weight caibration to any census totas was perfored. Variance was estiated with a uti-phase variance forua and Tayor inearization. Key Words: Large-scae surveys, subsaping of non-respondents, non-response adjustent, caibration 1. Introduction The Canadian census of popuation has been conducted every five years since Before 2011, one in five househods received a andatory ong-for whie the reaining househods received a andatory short-for. In 2011, a andatory short-for was sent to every househod; however, the census ong-for was repaced by a vountary survey caed the Nationa Househod Survey (NHS). The NHS is the argest vountary househod survey ever conducted by Statistics Canada. Athough they were two distinct surveys, the 2011 Census and NHS were coected at the sae tie and using the sae infrastructure. They aso share the sae reference date: May 10 th Their target popuation differ nonetheess: the NHS covers househods of a private dweings, whie the Census covers househods of a private and coective dweings. The 2011 Census short-for consisted of ten questions. The topics covered were Popuation and dweing counts Age and sex Faiies, househods and arita status Structura type of dweing and coective dweings Language. 1876

2 The NHS for consisted of the 10 census questions, 44 additiona questions about the individuas and 10 questions about the dweing. The topics covered by the 2011 NHS were The topics covered by the census (except for coective dweings) Activity iitations Ethnic origin, visibe inorities and iigration Aborigina peopes Mobiity and igration Education Labour Pace of wor and couting to wor Incoe and earnings Housing and sheter costs. The riss associated with non-response are ore iportant with a vountary survey such as the NHS than they are with andatory surveys such as the ong-for censuses. Consequenty, the saping, coection and estiation ethodoogies that were used with the andatory ong-for were changed to iniize the effect of non-response in the NHS. This paper describes the estiation ethodoogies adopted for the 2011 NHS. In section 2, the NHS saping design and corresponding design weights are presented. In section 3, adjustents due to the vountary nature of the NHS are described. In section 4, the NHS weight caibration process is outined. In section 5, the variance estiation ethodoogy is given. The paper ends with a concusion in section NHS Saping Design and Design Weights 2.1 First Phase of Saping At the first phase of saping of the NHS, the dweings were stratified by Coection Units (CU). These sa geographic entities were used to anage the coection process and contain on average 300 dweings. To each CU corresponds a questionnaire deivery ethod. These entities respect provincia and territoria boundaries as we as Census Division (CD) boundaries (a sub-provincia and sub-territoria geography). In each of these strata, a systeatic rando sape of dweings was seected. Saping fraction varied by province and territory and by questionnaire deivery ethod. The overa saping fraction was 30% and the sape size was approxiatey 4.5 iion dweings. Note that at the tie of saping it was not nown if a given dweing on the saping frae housed a househod that was in-scope for the NHS or not. The first step of the estiation process was assigning a weight to each dweing seected in the first phase NHS sape. The first phase weight assigned to a seected dweing was equa to the inverse of the probabiity of seection of the dweing and was the inverse of the saping fraction in the stratu of the dweing. 2.2 Second Phase of Saping After severa wees of coection, a rando subsape of the reaining non-responding dweings was seected and the foow-up efforts were restricted to this group of nonrespondents. This subsaping approach was deveoped by Hansen and Hurwitz (1946) and can be pictured as in Figure 1. In this figure, the saped popuation is represented by U, the first phase sape by s a, the dweings identified as respondents and non- 1877

3 respondents at the tie of subsaping by s and s a2, respectivey, and the subsape of non-respondents by s 2. U s a s s a2 s 2 Figure 1: The NHS sape and foow-up subsape In the Hansen and Hurwitz design, a the units of the subsape are assued to respond. When this happens the foowing estiator of the tota of the characteristic y can be used and is free of non-response bias: ˆ y y ty,hh, s a s2 a b sa 2 where a is the probabiity of seection of unit in s a and b s a 2 is the probabiity of seection of unit in the subsape s 2 given that s a2 is the group of non-respondents at the tie of subsaping. Fu response in the subsape was not an assuption that coud be ade in the NHS. This eans that it coud not be expected that the non-response bias woud be eiinated as in the traditiona Hansen and Hurwitz setting and that the objective of subsaping in the NHS coud not be to copetey eiinate non-response bias. The objectives were rather to iit non-response bias, to contro and concentrate foow-up costs and efforts and to target specific popuation groups. At the tie of subsaping, 37.5% of the in-scope househods of the first-phase sape were eigibe for subsaping (i.e., part of s a2 ). The dweings eigibe to be subsaped were stratified geographicay. CUs were used as the buiding bocs of the subsaping strata. The CUs were divided in two groups: those that had a hoogeneous popuation and those that had a heterogeneous popuation. When the popuation of a stratu is heterogeneous the ris of non-response bias is ore iportant than when it is hoogeneous because there is a higher ris that respondents differ fro non-respondents. To iniize this ris the subsaping fraction was chosen to be higher in heterogeneous strata. The CU was used as the subsaping stratu uness its size was too sa. If that was the case, the CU was cobined with a few others that were geographicay cose and siiar in ters of heterogeneity. In each stratu, a systeatic rando subsape of dweings was seected. The subsaping fraction was around 1/

4 In ters of estiation, the first phase weights were adjusted to tae subsaping into account as in the Hansen and Hurwitz estiator. Thus the weights of the dweings of s were unchanged, the weights of the dweings of s a2 not seected in the subsape s2 were set to zero, and the weights of the dweings that were part of the subsape were divided by the probabiity of seection in the subsape. The resuting weight wi be caed the doube-expansion weight. 2.3 Adjustent for In-scope Househods Many dweings that received a non-zero saping weight do not house a househod that is in-scope for the NHS. After the in-scope status of each dweing is deterined, the weight of out-of-scope dweings is set to 0 so that they do not contribute to the NHS estiates. In practice, the dweings with a weight of 0 after the adjustent for in-scope househods do not contribute to the estiates because they are reoved fro the disseination fies. This is equivaent to using a doain indicator for in-scope status for every study variabe of the survey. The rest of this paper is written under the assuption that such a doain indicator is part of every possibe variabe of study. 3. Further Adjustents Due to the Vountary Nature of the NHS As was expected, not every in-scope househod in the NHS subsape responded to the survey during the foow-up. Aong the in-scope househods seected in the subsape, 43% responded. This departure fro the Hansen and Hurwitz design can be easured using a weighted response rate to the survey. The weight that shoud be used is the doube-expansion weight. If a the subsape had responded to the survey as assued in Hansen and Hurwitz s theory, the weighted response rate woud be 100% and there woud be no non-response bias. The weighted response rate of the NHS is 77.2%. In coparison, the response rate of the 2006 Census ong-for was 94%. In fact, the weight adjustent corresponding to subsaping and described in section 2.2 is a for of non-response adjustent. Moreover, unie the ajority of non-response adjustents, it is free of any odeing assuptions and, when fu response is obtained in the subsape, it is the ony non-response adjustent needed to eiinate non-response bias. The weighted response rate of 77.2% of the NHS eans that further non-response adjustents are necessary to copensate for the 22.8% weighted non-response. This section describes the estiation processes beyond the saping weight adjustent of section 2.2 that were put in pace to dea with the vountary nature of the NHS. 3.1 Adjustent for Tota Non-response in the Subsape An adjustent had to be ade to copensate for the househod non-response observed in the subsape. The objectives of this adjustent were to iniize the non-response bias and to eet the overa goa of the NHS, which is to produce reiabe estiates at various geographica eves (e.g., nationa, provincia, territoria and unicipa) Auxiiary inforation In order to produce an adjustent that iniizes the non-response bias, it is crucia to have good auxiiary inforation nown for both respondents and non-respondents. In the NHS, very detaied geographica inforation was avaiabe fro the saping frae. Furtherore, the census and the NHS were ined by design and the ten questions shared by the two surveys provided reevant auxiiary inforation for the adjustent. Before 1879

5 these data were used for the non-response adjustent, severa steps were undertaen to obtain the ost copete inforation possibe: 1. Tota non-response to the census was iniized by copying the responses of househods that responded to the NHS but that did not respond to the census. Note that the non-response rate to the census was ess than 3%. 2. The rest of the househod non-response to the census was deat with by whoe househod iputation (see Doson, 2012). In this process, iputation for nonrespondents was done using the data of the nearest geographic neighbour sharing the sae househod size which coud be nown or assigned prior to iputation. 3. For a given dweing, when a response was obtained for both surveys and when it was verified that the sae househod provided a the answers, the two sets of response to the ten questions were copared and ony the ost copete set of inforation was used for both surveys. 4. Househods were defined as in-scope for the NHS if they responded to the NHS or if, according to the census, they were in-scope. When an in-scope househod did not respond to the NHS, the census inforation was copied to the NHS and used for the non-response adjustent. 5. In both surveys, ite non-response to the ten questions was deat with either deterinisticay or using the nearest-neighbour iputation ethodoogy of the CANadian Census Edit and Iputation Syste (CANCEIS) (Liu et a., 2011). The responses to the ten census questions and the geographic inforation are very reevant because they describe any diensions of the popuation that coud expain the non-response echanis, and because they are used to define a wide range of doains of interest in the NHS. However, the goa of the NHS is to easure characteristics of the Canadian popuation that were not easured by the census and there coud sti be an iportant residua non-response bias for those characteristics even if a the bias reated to the census variabes is reoved. As a consequence, to enrich the inforation used for the non-response adjustent, severa probabiistic inages of pertinent adinistrative fies were done to both the census and the NHS. Linages were done for the 2010 incoe tax data, the 2011 Indian register and the iigration database for the years 1980 to Furtherore, soe data nown at an aggregate eve was used for geographic areas corresponding to Indian reserves. For each househod of such areas, it was identified which Indian band and tribe was predoinant in its reserve Estiation options to dea with tota non-response in the subsape The first option that was considered to adjust for househod non-response in the NHS was to treat this non-response as ite non-response. It consists in copying the census data for NHS non-respondents and in iputing the issing inforation of the subsape nonrespondents using the avaiabe auxiiary inforation. Donor iputation with CANCEIS was the ony iputation ethod considered because it has been the traditiona, and successfu, iputation approach used for the Canadian census. Iputation was aso appeaing because it has been used to dea with both ite and unit non-response in the previous censuses. Therefore, using CANCEIS gave an advantage in ters of both the systes and the expertise avaiabe. Moreover, as was aready entioned, the NHS ais at producing reiabe estiates at various geographica eves and this option was attractive because it insured that there were a iniu nuber of responses for sa geographies. The Hansen and Hurwitz estiator, however, had to be adapted to tae iputation into account. The foowing estiator aes use of the doube-expansion weight: 1880

6 * ˆ y y y ty,ip, s a s2 r a b s s a2 2nr a b sa2 where s 2r and s 2nr are the sets of respondents and non-respondents of the subsape, * respectivey, and y are the iputed vaues. Another variant of this option was aso studied where the issing data of every NHS non-respondent was iputed regardess of the being seected in the subsape or not. This variant was caed ass iputation, whie the forer variant was caed partia iputation. The estiator under ass iputation is * ˆ y y y ty, ass ip. s a s2 r a sa 2 s2r a The second option that was considered to adjust for househod non-response to the NHS was to treat this non-response as unit non-response. With this option, the objective is to expain the echanis eading to househod non-response within the subsape and to increase the weight of the respondents of the subsape accordingy so that they aso represent the non-respondents of the subsape. This is appeaing because the auxiiary inforation is abundant and reevant to expain the non-response echanis. It is aso the typica strategy used to dea with unit non-response in vountary surveys. This option was naed pure weighting because it reies on weighting and not on iputation. In ters of estiation, athough no iputation is necessary, the doube-expansion weight needs to be further adjusted. When the probabiity of response of each responding househod of the subsape is estiated, the foowing estiator can be used: ˆ y y ty,wgt, (1) pˆ where subsape. s a s2r a b sa 2 s2 p ˆ s2 is the estiated probabiity of response of unit given that it is part of the Those options were evauated and copared using 2006 ong-for data of the Toronto Census Metropoitan Area (CMA) by siuating the non-response and subsaping echaniss (Verret et a., 2011). Absoute errors were cacuated as the absoute difference between the point estiator of the tota of the option considered inus the Hansen and Hurwitz point estiator of the tota. The ass iputation variant of the first option gave the argest absoute errors because nowedge of the subsaping echanis was ignored and repaced by an iputation ode, contrary to the partia iputation variant of the first option and to the pure weighting option. The partia iputation approach gave saer errors than the pure weighting approach. The differences were very sa at the CMA eve, but at the sub-cma eve they were greater (the CMA was divided into cose to a thousand sub-cma geographica units for the anaysis). Partia iputation was the first ethod ipeented on a fu scae in the NHS because of those siuation resuts, because it was the ethod with the argest effective sape size that too both phases of the design into account and because the systes to run partia iputation were ready whie those for pure weighting were not. However, pure weighting was aso ipeented on a fu scae in the NHS and was the estiation ethod used to dea with househod non-response for reasons that wi be given in the next section. 1881

7 3.1.3 Ipeenting partia iputation Partia iputation was ipeented, using CANCEIS, in the foowing fashion. For each non-responding househod of the subsape, a distance was easured between the nonrespondent and each respondent of the subsape with the sae househod size. Additionay, househods fro Indian reserves were processed independenty of househods not iving on Indian reserves. Each auxiiary variabe contributed to the distance. For a given variabe, if the vaue of the non-respondent and the vaue of the respondent were equa, then the contribution of the variabe to the distance was zero; otherwise it was a positive nuber in the interva [0, 1], coser to 1 if the two vaues were very dissiiar. If the vaues of a variabes were identica then the overa distance was zero. After each distance was cacuated a ist of the five nearest neighbours was created and a neighbour (i.e., a donor) was seected at rando fro the ist to provide a the issing vaues of the non-responding househod. The reative weight given to each auxiiary variabe in the distance easure has an ipact on which donors are seected for iputation. Indeed, athough the nuber of potentia donors in the NHS is uch arger than in ost househod surveys, it was not arge enough to guarantee that every non-respondent woud find a perfect donor (one that atches on every auxiiary variabe). Moreover, it is extreey difficut to find the ost appropriate weights using odeing because of the utivariate nature of the auxiiary data and of the data to be iputed. The ost appropriate reative weights ight vary fro subject atter to subject atter. As a coproise, reative weights that gave siiar unatched rates for every auxiiary variabe were sought. To achieve this, partia iputation was perfored severa ties adjusting the weights each tie. It was found that the non-zero unatched rates caused soe iprobabe cobinations of copied census data and iputed data at aggregate eves significanty infating the proportions of iportant outier groups. For exape, at the Canadian eve, a ot ore teenagers had PhD degrees than expected. This happened because too any teenage nonrespondents had their data iputed using those of oder respondents who had a PhD. Thus, the arge nuber of responses in the NHS had a dua effect on partia iputation: it gave a better poo of potentia donors than if the nuber of responses had been saer, but it aso ade the infation of outier groups very evident at aggregate eves. The atter was because even a very sa proportion of isatched non-respondent/donor pairs for a ey atching variabe created a uch greater nuber of outiers associated with the atching variabe than were nown to exist in the popuation. Consequenty, a second version of partia iputation was done to get rid of the outier probes. The paraeters were changed to aow the unatched rates to be ower for auxiiary variabes that were ore critica according to subject atter anaysts. Partia iputation was aso done in two passes: soe variabes were iputed with the data of the nearest responding househod, whie the rest were iputed with the data of the nearest responding person. This gave better resuts, but outier infation was sti found. For exape, outiers were created between sex and occupation because the reative weight of sex in the distance easure gave a non-respondent/donor atching rate on this variabe that was very high (we over 90%) but sti not high enough. Partia iputation created, for exape, far too any feae pubers and echanics, and any ae anicurists. Partia iputation was abandoned at this point because it seeed ipossibe to find vaues of the iputation paraeters that woud produce no critica outier infation probes within the tight production schedue. 1882

8 3.1.4 Ipeenting pure weighting The ony option eft to dea with househod non-response in the NHS at that point was pure weighting and this was the fina ethod that was ipeented. Under this estiation approach, the probe of iprobabe cobinations of census and non-census data found in partia iputation is absent because the ony data that are used for estiation are those coing fro questionnaires of the responding househods. A standard pure weighting approach is to use the scores ethod in the subsape (Etinge & Yansaneh, 1997). Under this approach, the probabiity of response of each respondent and non-respondent is first estiated with ogistic regressions. Then, based on those probabiities and using custering agoriths, response hoogeneity groups are fored. Fina probabiities of response are estiated in each group by the observed response rate in the group. Because of tie issues, the scores approach coud not be epoyed at the tie pure weighting was adopted. If there had been enough tie, any ogistic regression odes woud have been constructed at geographica eves sufficienty sa to get estiated probabiities of response that were we adapted to oca response tendencies and sufficienty arge to get sape sizes that woud have provided good power and precision in the odes. Athough the scores ethod coud not be ipeented on a fu scae in the NHS, the pure weighting approach used attepted to repicate it. As a first step, a few ogistic regressions were done at the nationa eve to judge the reative iportance of the auxiiary variabes in predicting the probabiity of response. As a second step, to get an adjustent that is good at the oca eve, nearestneighbour searches, using CANCEIS, were done based on the reative iportance of auxiiary variabes found in the first step and the weight was transferred fro each nonresponding househod of the subsape to its nearest responding neighbours in the subsape. For the first step, one ogistic regression was done for each househod size. Househods of size nine or ore were grouped with househods of size eight because there were too few of the to perfor a ogistic regression for each of their househod size. An independent set of ogistic regressions was done for househods iving on Indian reserves. A forward approach was used to incude the variabes one by one in the ogistic regression odes and the reative iportance of each variabe was easured by the increase in the R-square of the ode. Every auxiiary variabe that did not enter the ode received a reative weight of haf the weight of the ast variabe entering the ode so that each variabe considered woud contribute to the non-response adjustent. For the second step, nearest-neighbour searches were perfored again and the resuts fro the ogistic regressions were used to assign the reative weights of each auxiiary variabe in the distance function. Additionay, the nuber of donors in the best donors ist was increased fro five to twenty to iic what is usuay done in the custering agorith of the scores ethod. In this agorith, the response hoogeneity groups are fored so that they coprise a iniu nuber of responding units (e.g., twenty) to avoid extree estiated probabiities of response and extree non-response weight adjustents. The doube-expansion weight of each non-responding househod was transferred to each of its best twenty responding neighbours. To each of the twenty nearest neighbours in the ist corresponds a distance to the non-responding househod. Because the neighbours with the shortest distance are ore siiar to the non-respondent, the weight was transferred proportionay to the inverse of the distance. Moreover, each responding househod ight appear in the nearest-neighbour ist of any nonrespondents. This eans a given respondent ight receive a arge aount of weight fro 1883

9 the various transfers. The tota weight transferred to each responding househod was capped to the observed 99 th percentie to avoid extree weight adjustents. The estiator of a tota corresponding to this weighting ethodoogy, oitting the capping step, can be expressed in the foowing way. Let d represent the doubeexpansion weight: 1 a, if sa 1 d 1 a, if s b sa 2 2 0, if sa2 s2. Let D, be the distance between househods and, and denote the set of 20 nearest responding neighbours of non-respondent. Then the weight transferred fro non-respondent to respondent is given by 1 D, d, if 1 D i, i 0, otherwise. Furtherore, et s s be the set of non-respondents of the subsape 2 nr, 2 nr : who have respondent in their 20 nearest-neighbours ist. Then the tota weight transferred to respondent is given by 1 D, d. 1 D i, s2 nr, s2 nr, i The corresponding point estiator of a tota is given by. tˆ d y d y y,wgt s s2r Lessons earned In the search for the ost appropriate estiation ethod to copensate for househod non-response in the NHS and the ipeentation of those ethods, soe iportant essons were earned. With regards to ass iputation and partia iputation, it was found in the siuations that the second phase of saping shoud not be ignored. Partia iputation taes this phase into account and ass iputation does not. Knowedge of the second phase design shoud not be repaced by an iputation ode. With regards to using a whoe-househod and nearest-neighbour iputation approach to dea with househod non-response (i.e., ipute a the issing vaues of a given nonrespondent by those of one responding househod), the probabiity of getting high rates of outiers is higher when a ot of inforation is nown about the non-respondent and when a ot of inforation is iputed. To iniize this ris, one shoud have one or the other but not both. This is the case of the whoe househod iputation approach used to dea with the househod non-response in the census, where ony the geography and househod size are nown. Note that the iportance of this probe aso increases with the extent of the observed tota non-response. This esson hods even for very arge scae surveys ie the NHS for which a arge poo of respondents is avaiabe. This coud be seen as a curse 1884

10 of diensionaity. The poo ight be arge enough to find cose neighbours when a few auxiiary variabes are used in the distance function (e.g., as is seen in the reguar census and NHS ite non-response iputation processes), but as the nuber of auxiiary variabes increases, the poo size to get cose neighbours needs to get arger and arger and possiby at an exponentia rate. 3.2 Adjustent for Tota Non-response aong the Househods not Eigibe for the Subsaping In the Hansen and Hurwitz setting, every unit of s is assued to have responded when the subsape is seected. Unfortunatey in the NHS this was not the case. Aong the inscope househods seected at the first phase, 61.5% were part of s and 1.4% were nonrespondents in s. Two ain reasons expain this. Firsty, at the tie subsaping was done, fu anaysis of the questionnaires received was not copeted. The response status was based on fied inforation. A given househod coud have been identified in the fied as having responded, but the anaysis of the questionnaire coud revea the opposite. For exape, the househod coud have returned an epty questionnaire. Secondy, soe addresses were added to the saping frae just before coection started and because of operationa issues it was not possibe to provide NHS questionnaires to those addresses. A nuber of those dweings housed a househod that was in-scope for the NHS. Those househods thus had no opportunity to respond to the survey. The adjustent for non-response in s is siiar to the adjustent used to dea with nonresponse in the subsape. The pure weighting ethodoogy described previousy was appied with the difference that the nearest neighbours coud be found in the subsape or in s. The estiator of a tota can now be expressed in the foowing way (sti oitting the capping of the weight transfer step): t ˆ d y d y, y,wgt 2 sr s2r where s ar 1 and s nr are the sets of respondents and of non-respondents of s a 1 respectivey,, s s. s nr, 2 and s2 nr, nr, nr : 3.3 Adjustent for Surprise Respondents One ast adjustent deas with the vountary nature of the NHS. This tie it was needed because of extra responses to the survey. Soe househods had not responded to the survey at the tie of subsaping, were eigibe for subsaping, were not seected in the subsape but chose to respond nonetheess. Those househods were caed the surprise respondents. They represent 1.8% of the in-scope househods of the first phase sape and the Hansen and Hurwitz design do not cover the. The easiest way to dea with surprise respondents estiation-wise is to ignore their response and to not consider the in the estiates (i.e., eep their weight at zero). This is the approach that shoud iniize the non-response bias because surprise nonrespondents are units that are ore incined to respond. However, it is not acceptabe to effectivey eiinate vaid responses to the survey. Various estiation strategies coud be epoyed to assign weights greater than zero to surprise respondents. One natura 1885

11 approach woud be (ignoring the non-respondents of s for this particuar exape) to repace pˆ by pˆ b sa 2 s 1 qˆ 2 b sa 2 s2 b sa 2 sa 2 s2 in (1), where qˆ is the estiated sa 2 s2 probabiity of response of househod given it is eigibe for subsaping and it is not seected in the subsape. That is, repacing the estiated probabiity of being in the subsape and of responding by the overa estiated probabiity of responding. Under this approach, every surprise respondent receives a weight greater or equa to the first phase weight. This strategy invoves odeing in soe fashion the probabiity of response of househods of the subsape and the probabiity of response of househods eigibe for subsaping that were not subsaped. Instead of using this approach, the weight of each surprise respondent was increased to 1 and the weight of respondents in the subsape was reduced accordingy. The reason for choosing this vaue is that it is the weight cosest to 0 that guarantees each surprise respondent is at east sef-representative. This was done using again a nearest-neighbour approach. The 20 nearest responding neighbours in the subsape of each surprise respondent were found and a weight of 0.05 was transferred fro each to the surprise respondent. Let s surprise and s surprise, denote the set of surprise respondents and the set of surprise respondents who have respondent as one of their 20 nearest neighbours respectivey. The estiator of a tota, oitting the cap on the weight transfer, is then given by tˆ y,wgt d y d y y. sr s2 r ssurprise, ssurprise 4. Weight Caibration The ast step of the point estiation process is weight caibration to census totas. The goas of caibration are to obtain coherence between NHS estiated totas and corresponding census totas, as we as a reduction of the variance. To perfor caibration, the country is first spit geographicay into Weight Caibration Areas (WCAs). Caibration constraints seection and caibration on the chosen constraints is then done WCA by WCA. WCAs are fored by cobining adjacent disseination areas (DA). Whie the CUs are used to anage coection activities, DAs are used for disseination purposes. A DA contains between 400 and 700 persons. A WCA represents on average 2300 dweings and 5700 persons. There were 5884 WCAs created for the 2011 NHS. The country was spit into WCAs to enabe the iteration of constraint seection and caibration, and to insure the cacuations, such as the atrix inversions required for caibration and variance estiation, were doabe systes-wise. In the construction of WCAs, CD boundaries are respected and boundaries of ower eves of geography (census sub-divisions and the census tracts) are respected as uch as possibe. The WCAs do not vary uch in size so that a coon ist of potentia caibration constraints can be used for a WCAs. There are any caibration constraints that can be used because of the ten questions in coon between the census and the NHS. The constraints that were considered for seection within each WCA were: Nuber of househods and nuber of persons in the WCA and in each of the DAs of the WCA 1886

12 Sex and sex crossed with indicator of age ess than 15 years od 5-year age groups up to 75 years od Marita status Nuber of census faiies with or without chidren Census and econoic faiy indicators Nuber of househods iving in sei-detached houses and nuber of househods iving in apartent buidings of ess than 5 stories Househod size Coon cobinations of other tongue, officia anguages and hoe anguage. In each WCA, the caibration constraint seection process was perfored in the foowing way. The constraints were divided into three groups. First, soe constraints were identified as andatory in every WCA because it was required to have the NHS estiated tota equa to the census tota at the nationa eve. The andatory constraints were nuber of househods and nuber of persons in the WCA. Secondy, in a given WCA, there ay have been constraints that shoud have been avoided. This was the case with every constraint that had ess than 30 responses because caibrating on those constraints ight have hurt the precision of the estiates instead of iproving it. These wi be caed the iprecise constraints. The rest of the constraints ight have been seected for caibration. These wi be caed the avaiabe constraints. The constraints were chosen in a forward fashion inspired by Verret and Kevins (2010). The two andatory constraints were incuded at the first step. In the foowing step, each of the non-seected avaiabe constraint was evauated and coud have been discarded for the rest of the seection process. The best non-discarded constraint was incuded (if this set was not epty). This was repeated unti a avaiabe constraints were either incuded or discarded. An avaiabe constraint was discarded if its incusion woud have been equivaent to caibrating on an iprecise constraint or on a cobination of incuded constraints. For exape, it ight have been desirabe to avoid caibrating on nuber of aes because their responses were too few, whie tota nuber of persons was andatory and nuber of feaes was avaiabe. In this case, excusion of nuber of aes woud not be sufficient to avoid caibrating on this constraint because incusion of the other two constraints woud have been equivaent to caibrating on nuber of aes by construction. An exape of the second probe is to have tota nuber of persons andatory and nuber of aes and nuber of feaes avaiabe. In this case, ony nuber of aes or nuber of feaes woud have to be chosen but not both, to avoid redundancy. To perfor this evauation of the avaiabe constraints, a non-redundant set of the iprecise constraints was first created. Let the caibration icro-data inforation of the 0 NHS respondents be partitioned into Z, the andatory constraints, Z, the iprecise constraints and Z, the avaiabe constraints. In these atrices rows correspond to respondents and couns to constraints. Let X represent the atrix of non-redundant 0 iprecise constraints that is sought. This atrix was first set equa to Z. Then couns of Z were either added one by one to X in a forward fashion or discarded fro further seection. At each step, a regression was perfored for each coun of Z that was neither discarded nor added to X at a previous step. Each coun was regressed on X. If the R-square was too high then the constraint was discarded because it is redundant. The reaining coun (if any) with the iniu R-square was added to X. 1887

13 This was repeated unti a couns of Z were either discarded or added. After this was 0 done, the couns of Z were reoved fro X. The avaiabe constraints were then evauated. Let X denote the atrix of chosen 0 constraints. This atrix was set equa to Z at the first step of seection. At each foowing step, a regression was done for each coun of Z that was neither discarded nor seected. Each coun was regressed on X augented by X. If the R-square was high then the constraint was discarded. The reaining constraint (if any) with the iniu R- square was seected and its coun was added to X. The process ended when every coun of Z was either discarded or seected. Statistics Canada s StatMx software was used to perfor caibration on the constraints that were seected WCA by WCA. It finds the set of caibrated weights w such that c 2 w a a is iniized, subject to the caibration constraints w x x sr and the constraint w u for sr, where s r is the set of respondents, c are chosen constants, a is the uncaibrated weight and and u are ower and upper bounds on the caibrated weights. In the NHS, sr sa 1r s2r ssurprise, c 1, a is the weight after the adjustents of section 3, 1 and u Variance Estiation For variance estiation, it was desirabe to use as uch as possibe a design-based approach for sipicity. To do this, the househod non-response in s and in the subsape were treated as a phase of saping. Furtherore, surprise respondents were ignored in the variance estiation process, aso for sipicity. Since saping fractions were arge, both phases of saping and the non-response phase had to be taen into account in variance estiation. The stratified systeatic designs of the saping phases were approxiated by stratified sipe rando saping without repaceent designs. This approxiation is reasonabe because the design strata are sa and coud thus be supposed to be hoogeneous. The approxiation was aso needed to get expicit variance estiators. Moreover, the househod non-response that was observed was supposed to have originated fro a Poisson phase happening after the second phase of saping. The probabiity of response of respondents was estiated by the inverse of the non-response weight adjustent and was assued nown for variance estiation purposes. It was equa to d d for respondents of s and to d d 2 for respondents of the subsape, where a 1 and 2 are the capped version of a 1 and 2 respectivey. For exape, a respondent that had its weight doubed by the weight transfer had an estiated probabiity of response of 50%. Variance estiators were deveoped by doing cassica deveopents, expanding the Hansen & Hurwitz variance estiator to the case where Poisson saping is done. The Hansen & Hurwitz estiator can be seen as a two-phase variance estiator when s is seen as a tae-a second phase stratu. The resuting estiator thus has the for of a three-phase variance estiator. Tayor inearization was used to tae caibration into account. sr U 1888

14 StatMx was used as we to estiate the variance. However it had to be sighty adapted because StatMx estiates the variance for one or two phase designs. The odification is based on an identity siiar to one derived in Lundströ & Särnda (1999). Under a three-phase design, et s a, s b and s denote the first, second and third phase sapes; a, b and denote the corresponding first-order incusion probabiities of unit ; a, b and denote the corresponding second-order incusion probabiities of units and ; and a, b and denote the corresponding covariances between the incusion indicators of units and. In the absence of caibration, the variance estiator of the estiator of a tota of y is given by vtˆ y a y y b y y y y. s s a b a a s s b a b a b s s a b a b When the third phase is Poisson then, if, otherwise and 1, if 0, otherwise. Let z y, if s 0, if sb s. (2) The estiator of the variance can then be rewritten in ters of z as z z z z z vt sb s b a b a a sb s b b a b a b s b a b ˆ a b y a b 1. (3) The first two ters are those of the two-phase variance estiator of t ˆz. To estiate the variance of the estiator of the tota of y, the second phase sape s s2 can be subitted to StatMx with vaues of z instead of those of y and the third ter of the equation can be prograed within a specia edition of StatMx and added to the tota estiated variance. In the presence of caibration, estiator (3) shoud be used with y repaced by gse s in the definition of z given by (2), where gs is the caibration weight adjustent or the g- T T weight, e ˆ s y xb and ˆ xx x y B. It is possibe to use s a b c s a b c StatMx to cacuate this variance, provided the third ter of (3) is added to the tota variance in the progra. This tie the second phase sape can be subitted to StatMx

15 with the g-weights obtained fro the point estiation run of StatMx, c 1 and with y and x repaced by y, if s x, if s y and x 0, if sb s, 0, if sb s. 6. Concusion The NHS is the argest vountary househod survey conducted by Statistics Canada. Its vountary nature required any saping, coection and estiation innovations. This paper focused on the innovations in the estiation ethodoogies. Two-phase design weights were cacuated to atch the saping design. Before and during production, a ot of research on adjustents for househod non-response was achieved and iportant essons were earned. Pure weighting was preferred to whoe househod iputation in the end. However, the resuts of the nearest-househod-neighbour search obtained using CANCEIS drove the non-response weight adjustent. Weight caibration to nown census totas was done after seection of constraints. The seection invoved identifying iprecise constraints and avoiding caibration on the as we as on redundant constraints. Finay, three-phase variance estiators were deveoped and used. The Tayor inearization technique was used to dea with caibration in the variance estiators. References Doson, D. (2012), Errors of Non-observation: Dweing Non-response and Coverage Error in Traditiona Census, in JSM Proceedings, Survey Research Methods Section. San Diego, CA: Aerican Statistica Association Etinge, J. L., and Yansaneh, I. S. (1997), Diagnostics for Foration of Nonresponse Adjustent Ces, with an Appication to Incoe Nonresponse in the U.S. Consuer Expenditure Survey, Survey Methodoogy, 23, 1, Hansen, M. H., and Hurwitz, W. N. (1946), The Probe of Non-Response in Sape Surveys, Journa of the Aerican Statistica Association, 41, Liu, W., Crowe, S., and Aavi, A. (2011), How CANCEIS Does Edit & Iputation, and Can it be Standardized for ore Genera Use?, in 2011 Internationa Methodoogy Syposiu. Ottawa, Canada: Statistics Canada. Lundströ, S., and Särnda, C.-E. (1999), Caibration as a Standard Method for Treatent of Nonresponse, Journa of Officia Statistics, 15, Verret, F., and Kevins, C. (2010), Caage aux Marges des Poids d Enquêtes à Pan Copexe pour e Refus à Apparieent, in SSC Annua Meeting Proceedings, Survey Methods Section. Quebec, Canada: Statistica Society of Canada. Verret, F., Banier, M., Benjain, W., and Hayden, L. (2011), The Estiation Strategy of the Nationa Househod Survey (NHS), in Internationa Tota Survey Error Worshop 2011 Presentations. Quebec, Canada: Nationa Institute of Statistica Sciences. 1890