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## Revista de Saúde Pública

*Print version* ISSN 0034-8910

### Rev. Saúde Pública vol.45 n.6 São Paulo Dec. 2011

#### http://dx.doi.org/10.1590/S0034-89102011000600011

**ORIGINAL
ARTICLES**

**Estimation
of live birth underreporting with a capture-recapture method, Sergipe, northeastern
Brazil**

**Estimación
de subregistro de nacidos vivos por el método de captura-recaptura, Sergipe,
Noreste de Brasil, 2006**

**Bianca Schmid ^{I};
Nilza Nunes da Silva^{II}**

^{I}Instituto
Brasileiro de Geografia e Estatística. Unidade Estadual São Paulo.
São Paulo, SP, Brasil

^{II}Faculdade de Saúde Pública. Universidade de São
Paulo. São Paulo, SP, Brasil

**ABSTRACT**

**OBJECTIVE:**
Estimate the number of live births and, therefore, underreporting of live births.

**METHODS:** The databases of the Live Birth Information System and the Civil
Registry of the Brazilian Institute of Geography and Statistics, from the second
and third trimesters of 2006 in Sergipe state (Northeastern Brazil) were paired
by deterministic linkage based on the number of the Live Birth Declaration.
The geographic disaggregation utilized was mother's microregion of residence.
Huggins closed population models were used to estimate the capture probabilities
for each database and the total live births during the period, within each geographic
subdivision. MARK® software was used for the estimates.

**RESULTS:** Underregistration during the period studied was 19.3%. Application
of the capture-recapture method to estimate underregistration of live births
is possible, including for geographic disaggregations smaller than a state.
The deterministic linkage was impaired in four microregions, due to non-inclusion
of the Live Birth Declaration number in the database of the Brazilian Institute
of Geography and Statistics. Maternal age, a heterogeneity characteristic in
the population of live births, affected the probability of capture by the civil
registry.

**CONCLUSIONS:** Capture-recapture was a viable method to estimate the underregistration
of live births.

**Descriptors:
** Live Birth. Birth Certificates. Underregistration. Registries. Records
as Topic. Vital Statistics.

**RESUMEN**

**OBJETIVO:**
Estimar el número de ocurrencias de nacidos vivos, y en consecuencia,
el subregistro civil de nacidos vivos.

**MÉTODOS:** Las bases de datos del Sistema Nacional de Información
sobre Nacidos Vivos y del Registro Civil del Instituto Brasileño de Geografía
y Estadística, en los segundo y tercero trimestres de 2006 del estado
de Sergipe (Noreste de Brasil), fueron pareadas por relación determinística
a partir del número de la Declaración de Nacido Vivo. La desagregación
geográfica adoptada fue la de microregión de residencia de la
madre. Los modelos de Huggins para poblaciones cerradas fueron aplicados para
estimar las probabilidades de captura en cada base y el total de nacidos vivos
ocurrido en el período, dentro de cada desagregación geográfica.
El aplicativo utilizado para las estimaciones fue el Software MARK®.

**RESULTADOS:** El subregistro civil en el período analizado fue de
19,3%. La aplicación del método de captura-recaptura para estimar
subregistro de nacidos vivos es factible, inclusive para desagregaciones geográficas
menores que la unidad de federación. La relación deterministica
fue perjudicada en cuatro microregiones, debido a la falta de llenado del número
de la Declaración de Nacido Vivo en la base del Instituto Brasileño
de Geografía y Estadística. Se identificó que la edad de
la madre afecta la probabilidad de captura por el Registro Civil, característica
de heterogeneidad en la población de nacidos vivos.

**CONCLUSIONES:** El método de captura-recaptura se mostró
viable para la estimación de subregistro de nacidos vivos.

**Descriptores:
** Nacimiento Vivo. Certificado de Nacimiento. Omisiones de Registro. Sistema
de Registros. Registros como Asunto. Estadísticas Vitales.

**INTRODUCTION**

Underregistration
of vital events is still a reality in Brazil.^{18,}ª^{,}^{b}
According to Simões,^{c}
the lack of coverage by vital statistics is a barrier to the direct calculation
of fertility and mortality rates in Brazil.

Calculation of
the fertility and child mortality rates with direct methods, without correction
for the underreporting of births and deaths, can hide the demographic reality
of a population.ª To calculate these indicators, indirect
techniques are employed for estimates, with information sources including demographic
census and representative studies. Often, violation of the assumptions implicit
when implementing such techniques causes distortion of estimates. When estimates
are made for smaller geographic disaggregation of federal units, the problem
becomes more complex due to the small population size of many Brazilian municipalities.^{c}

Various indicators
may be calculated using statistics from the Civil Registry, such as fertility
rates, mortality coefficients and life expectancy at birth. Efforts, to understand
the negligence of civil records, attempt to remedy this situation and adhere
to the principals of efficient professional practices contained in the United
Nation's Fundamental Principles of Official Statistics.^{d}

Estimation through
the capture-recapture method seeks to use the overlap between incomplete registries
to formally measure the underestimation of these sources. This allows for the
correction of statistics and the production of indicators that better approximate
reality. These available sources (lists) may include mandatory reportable diseases,
statistics from hospitals and other health services and death records, in addition
to other sources.^{2,11,12}

In summary, the
capture-recapture method was utilized to estimate the population of France in
1793. Since the 19th century, the technique was widely used to estimate the
population of wild animals,^{3,8} and in various other application in
medicine, demography and epidemiology.

In 1984, Petersen
developed the most simply model for estimation with the capture-recapture method,
using two samples.^{8} In the 1940s, Sekar & Deming^{13}
estimated the under registration of live births and deaths in India. Using census
data, Shapiro^{14} applied the technique to calculate the under registration
of live births in the USA. In 1968, Wittes & Sidel^{19} introduced
a generalized capture-recapture method for epidemiology applications, through
use of two or more lists. Interest continued to increase in this method, and
since the 1990s there was a considerable increase in its use in epidemiologic
research.^{8}

The objective of this study was to apply the capture-recapture method to estimate live births.

**METHODS**

In ecology the
most straightforward method involves sampling the population, marking the individuals,
allowing them to mix with the remaining population, and then taking a new sample.
The marked and recaptured individuals are counted, and the total population
size is estimated based on the number of individuals exclusively contained in
the first sample, (n_{A}), exclusively in the second sample (n_{B})
and in both samples (n_{A}_{∩B}).
To use this technique, the following assumptions are necessary:^{4,8}

1. The population is closed, or in other words there are no births, deaths, or migration in the period between samples;

2. Marking is unique, meaning that each individual is identified by the mark and there is no possibility of losing it;

3. In each sample, every individual has the same probability of being sampled (equiprobability);

4. The two samples are independent, i.e. the event of one individual captured in a sample is independent from the event of one individual captured by another sample; and

5. In each sample, any individual is captured (re-captured) independently from others.

The idea is that if the population in a given area is small, a large number of individuals captured by the second sample will have been marked in the first sample. On the other hand, if the population is large, the second sample will have a small number of individuals marked by the first sample.

In epidemiology,
each available list is considered a sample of the population and "being registered
on the list" is equivalent to "being captured" in the sample. For more details
about the development of this method in epidemiology, refer to Coeli et al,^{2}
Hook & Hegal,^{4,5} *International Working Group for Disease Monitoring
and Forecasting* (IWGDMF),^{8,9} Wittes et al^{18} and Wittes
& Sidel.^{19}

Huggins^{6,7}
introduced a procedure to estimate the size of a closed population when the
capture probabilities are heterogenous, by modeling based on observed variables
such as sex, age and capture history The modeling is performed by calculating
the conditional likelihood of captured individuals in order to estimate the
parameters.

If p_{ij}
is the probability of individual *i* being captured in sample *j*,
where *i* = 1, 2, 3, ..., *N* are the individuals in the population
(N is the population size) and *j* =1, 2, ..., *t* are the individuals
sampled. The conditional likelihood of captured individuals can be expressed
in terms of:

Where equals when
z_{ij} = 0, where z_{ij} indicates a prior capture of individual
*i.* Alternatively:

Therefore, Γ_{ij}
is the probability of individual *i* being captured in sample *j*
given its capture history and given it was captured at least once during the
study.

If x_{ij}
= 1 when individual *i* is captured in sample *j* and x_{ij}
= 0 if the individual is not captured, individuals are renamed as 1, 2, 3, ...,
*n* and the non-captured individuals renamed *n*+1, *n*+2, *n*+3,
..., *N*, then the conditional likelihood is proportional to:

This only depends
on the individuals sampled. The formula for the linear adjustment according
to individual and/or environmental characteristics is a logistic function {ln[p_{ij}/(1
- p_{ij})]}.^{7} According to the author, the variables
are normally distributed and their variances can be estimated with a secondary
derivatives matrix. Various models can be adjusted based on the observed variables
and the capture history.

To estimate the
population size, the probability of individual i being captured at least once
during the study is:^{7}

Where β is the vector of the parameters associated with the adjusted model. An estimate that does not depend on the population size is:

And the variance is:

The standard error of is the square root of its variance. The 95% confidence interval is:

Data were obtained
from the Ministry of Health (preliminary data for 2006 were from the Live Birth
Information System, *Sistema de Informações sobre Nascidos Vivos*,
- SINASC) and the Brazilian Institute of Geography and Statistics (*Instituto
Brasileiro de Geografia e Estatística*, IBGE - Civil Registry of Live
Births for 2006). In 2006, the collection form used by IBGE included the number
from the Live Birth Certificate, which was used to link the two databases and
identify unreported live births. Information on data organization, standardization
and linkage between the two databases was previously described.^{e}

The total number of live births was estimated with Huggins models, adjusted for the second and third trimesters of 2006, in Sergipe state (Northeastern Brazil). Under-reporting was calculated using the estimates for total live births. Each data source was considered as a sample, or occurrence. SINASC was considered the first occurrence (first capture) and the Civil Registry was considered the second occurrence (re-capture). The geographic analysis considered the microregion of mother's residence. Each microregion was considered as a group of individuals.

Various factors
can influence under-reporting of live births, and some characteristics can be
incorporated in models, including mother's education, race, number of previous
children and the existence of piped water and sewage connection in the residence.
Nonetheless, to use this technique the individual variables included in the
linear model must be available at all captures,^{7} in this case, the
two data sources. Only the sex of the child and maternal age were available
in the two databases and were considered in the estimation models. The lack
of an official document center (*cartório*) for the registration
of people in the municipality is an institutional factor that can hinder civil
registration of live births; in Sergipe, only two municipalities did not have
this type of official document center, in 2006.^{f}
Therefore, this factor was not considered, since microregions were used in the
geographic disaggregation and official document centers were located in all
sub-divisions.

Between 4/1/2006 and 9/30/2006, SINASC captured 19,502 live births and the Civil Registry captured 17,254. The creation of pairs from the databases through use of the birth certificate number generated 15,532 pairs. Based on this pairing, the two databases included 21,224 registrations of live births from mothers residing in Sergipe. During the study period, the Civil Registry had 808 registrations with a missing birth certificate number, approximately 4.7% of the database. When the registrations with a birth certificate number are compared to the ones without a number, there is no statistical difference in the average age of the mother (p = 0.992) and in the proportion of the sex of the child (p = 0.510).

The distribution of registrations with a missing birth certificate number in the Civil Registry revealed that some microregions have more than 5% of registrations with a missing birth certificate number: Agreste de Lagarto (31.9%), Tobias Barreto (9.2%), Boquim (8.2%) and Japaratuba (7.9%). In the other microregions, the percentage missing varied from 0.4% in Nossa Senhora das Dores to 4.0% in Carira.

For the sex of child, the null hypothesis was that the proportion of girls in each health microregion was the same as the proportion for the state as a whole. For maternal age, the null hypothesis was that the mean age in the health microregion was the same as in Sergipe. The proportion of female children was not statistically different from the mean of state. Mother's age was statistically different. Therefore, only maternal age was considered for inclusion in the estimation models.

Considering *i*
= 1, 2, 3, ..., *N* as live births and *b* = the databases (SINASC,
Civil Registry), the full linear model for the capture probabilities of SINASC
and the Civil Registry were calculated as:

Where,

p* _{ib}*
is the probability of individual

*i*being in database

*b*(SINASC or Civil Registry);

β_{0}
is the intercept;

β_{k}
is the parameter estimate for group k (*k* is the microregion);

g_{k} are
the individuals that belong to group *k*;

β_{13}
is the parameter estimate for maternal age;

idmae_{i}
is the age of the mother of individual *i*, in years, and

β_{k+13}
is the parameter estimate for the interaction between group *k* and maternal
age.

In this model the probability of capture varies according to individual characteristics. The notation adopted for the full model was [p(g+idmae+g* idmae) c(g+ idmae +g* idmae)].

Each sub-model generated specific parameter estimates in accordance with the terms specified. For example, in one model the probability of capture by SINASC depends on the group (geographic disaggregation) and maternal age and the probability of capture by the Civil Registry depends only on maternal age, [p(g + idmae) c(idmae)], have different parameter estimates of another model where the probabilities of capture by SINASC and the Civil Registry do not vary across microregions and are independent of maternal age, [p(.) c(.)]. Models with more parameters respond better to the data but the precision of parameter estimates decrease.

One method to evaluate responsiveness and precision is to evaluate the models according to information criteria. One of these methods is the Akaike Information Criteria (AIC), which relates the conditional likelihood of the model to the number of parameters estimated:

AIC = -2ln(L) + 2k

Where L is the
conditional likelihood of the model and k is the number of parameters. Models
with greater responsiveness have higher conditional likelihoods, decreasing
the value [- 2ln(L)]. The additive term [+ 2k] penalizes the
AIC value. Additional parameters decrease the AIC, since the conditional likelihood
increases. However, the sum of the term [2k] balances the AIC value.
Therefore, the model with a smaller AIC is the most parsimonious in relation
to its likelihood and number of parameters.^{g}

Although it is
easy to interpret and to select the model the best fits the data, sometimes
there can be models with very similar AIC values, which makes selection difficult.
Then the models can be calibrated in a way to provide a relative plausibility
index, using the normalized Akaike weights. The weights, w_{i}, are
calculated for each model of the group of *I* candidate models, according
to the formula:

Where ∆AIC_{i}
is the difference between the AIC value of model *i* and the model with
the smallest AIC. The weight w_{i} is considered as evidence that model
*i* is the best model of all the candidate models. Greater model weight
can be interpreted to better support the data.^{f}

The models were
adjusted with MARK,^{®} which estimates the capture-recapture probabilities
in accordance with the linear model one desires to adjust for each probability.
This way various models were adjusted where the probability of capture by SINASC
was adjusted starting with a constant through the full model described in the
equation (1). The same procedure was adopted for the capture probability of
the Civil Registry, totaling 30 sub-models for the microregions. In addition,
four models were adjusted for the entire state of Sergipe, in which no group
was considered and to investigate the influence of maternal age on capture by
the databases.

The MARK^{®}
program calculated the number of parameters for each model, as well as the conditional
likelihood, the AIC value, ∆AIC,
w_{i}, the probabilities for capture (SINASC) and recapture (Civil Registry),
and also the derived estimate for total live births ().

After obtaining the estimates for total live births, the civil underreporting was calculated as a percentage l:

Where,

= The percentage of underreporting in subdivision s (microregion).

_{}=
Estimate of total live births for subdivision s.

n_{RCs}
= Number of live births captured by the Civil Registry, in subdivision s.

The cartogram was
created with Tabwin^{®} application.

**RESULTS**

Among the four adjusted models for Sergipe state as a whole, the model with greatest weight {p(idmae) c(.)} was the model where maternal age interferes in the capture of live births by the Civil Registry, with approximately 66% of the total weight among the models.

This model estimated 21,391 (95%CI 21,363;21,423) live births in the second and third trimesters of 2006 in Sergipe, with a probability of capture by SINASC (^p) estimated at 0,912 and for the Civil Registry, 0.804. By deriving the estimates from the number of live births, civil underreporting was calculated at 19.3%. By using the estimate given in (2), with a 95%CI for total live births, the variation in underreporting was estimated between 19.2% and 19.5%.

When including the microregions of maternal place of residence as groups of live births, only 5 of the 30 models fitted to the data demonstrated any relative weight when evaluating the AIC and conditional likelihood criteria. The model with the greatest weight {p(g) c(g + idmae + g*idmae)}, 67%, was selected (Table 1).

The probability of capture by SINASC was high in all microregions, varying from 0.69 in Agreste de Lagarto to 0.95 in Estância and Nossa Senhora das Dores. In Agreste de Lagarto there are a large number of records in the Civil Registry with a missing birth certificate number (more than 31%), which limited the pairing of the databases. Also, in Tobias Barreto, Japaratuba and Boquim, the percentage of records with a missing birth certificate number was greater than 5%. Besides these microregions, where matching was affected by the lack of a birth certificate number, only the Propriá microregion had a SINASC capture probability less than 0.90 (Table 2).

The capture probabilities for the Civil Registry were noticeably smaller than for SINASC. In the Civil Registry, the greatest capture probability estimated was in the Aracaju microregion (0.85) and the smallest was in Sergipana do Sertão do São Francisco (0.71), excluding the mircoregions with problematic matching (Table 2).

The total estimated
live births () was very close to the total measured in all the microregions,
due to the high overlap of the lists. Again in Agreste de Lagarto, the high
percentage of Civil Registry records with a missing birth certificate number
created low overlap (relatively low n_{S}_{∩RC})
and therefore inflated the total estimate of live births. The absolute difference
between the estimated live births and captured live births was less than 20
in almost all the microregions where the percentage of missing birth certificate
numbers in the Civil Registry was less than 5%, and reached only 2 live births
in Carira and Nossa Senhora das Dores (Table
3).

Underreporting across microregions varied from slightly more than 12% in Baixo Contiguiba to almost 27% in Sergipana do Sertão do São Francisco. Estimated underreporting in Agreste de Lagarto exceeded 40%, although this finding should be interpreted with caution. Civil underreporting was less in mircoregions located in the central part of the state, Aracaju, Baixo Contiguiba and Agreste de Itabaiana (< 15% of live births). As the microregions of maternal residence increase in distance from the central area, civil underreporting of live births increases (Figure 1).

When considering maternal age from the Civil Registry, there was a subtle decreasing trend in underreporting as maternal age increased (Figure 2). The Agreste de Lagarto microregion was not included in the analysis due to the large number of missing birth certificate numbers in the Civil Registry.

**DISCUSSION**

The findings suggest that the probability of civil underreporting increases as maternal age decreases (Figure 2); while underreporting is also influenced by proximity to the central area of the state. The microregion of Baixo Contiguiba, did not show high underreporting, although it had the lowest mean maternal age; the opposite occurred in Tobias Barreto, the microregion farthest from Aracaju, which has the highest average maternal age and also high underreporting. Nonetheless, matching in Tobias Barreto was harmed since almost 10% of records in the IBGE database were missing the birth certificate number.

It is important to discuss the assumptions in estimation by capture-recapture. It assumes a closed population where there is no migration nor births or deaths during the study period. In this study, the population to estimate was the total number of live births. The event of a birth happens once, and the number of live births is constant in the period and the geographic area used. Clearly, neonatal and/or child deaths can occur during the period analyzed and the families may have moved to another municipality of federative unit after the birth. Nonetheless, these factors do not alter the size of the "population of live births of mothers residing in Sergipe", since a death and change of address do not change the fact that the baby was born alive and the mother resided at the given location.

In regards to the unique marking, the failings in filling out the birth certificate number harmed the deterministic linkage utilized. Of the 17,254 live births present in the Civil Registry, 808 (4.7%) had a missing birth certificate number, which generates questions concerning the extent that these 808 records are able to be matched with the SINASC records.

When using the results of the model for the entire state of Sergipe, the estimated capture probability for SINASC was 0.912 and for the Civil Registry 0.804. The probability for one live birth to be included in the two databases would therefore be, 0.912 * 0.804 = 0.733. Of the 808 records with a missing birth certificate number in the Civil Registry, 0.733 * 808 = 592 would also have been captured by SINASC. Using these numbers, one can assume that the correct number of pairs formed by matching between the two databases would be 15,532 + 592 = 16,124 live births. Although it is possible to estimate the probable number of pairs, performing this relation after estimation would be imprecise, since the available variables have little discriminatory power, which implies finding more than one record with the same characteristics. The year 2006 was the first time that the birth certificate number was collected on the IBGE questionnaires, and in subsequent years data quality may improve, with less discrepancy between SINASC and the Civil Registry.

There are two issues
related to the assumption of equal probability, where each individual has the
same probability of capture in a given sample. The first issue concerns the
influence of maternal age on capture by the Civil Registry. This source of heterogeneity
can be included in the models and was found significant. The second issue concerns
the time interval used for matching the databases. The Civil Registry database
at the IBGE is organized by the year of data collection. The records and their
respective dates of birth are not lost, although they are organized according
to the year collected. Therefore, the total number of live births in 2006 will
vary as the year of data collection progresses. Evidently the variation will
not be large, since the late records will be residual with the passage of time.
As addressed by Oliveira & Simões,^{h}
the coverage by the Civil Registry increases when analyzing records one year
after the live birth. Among those born in the second trimester of 2006, close
to 94% were registered in the third trimester. Since access to data was restricted
to collection year 2006, births in the third trimester of 2006 and registered
beginning in 2007 were not detected in this study. Therefore, there was not
equal probability of capture in the Civil Registry among individuals born in
the third trimester of 2006 in relation to those born in the second trimester.
Since access to the IBGE database was restricted to collection in 2006, the
matching between SINASC and Civil Registry for live births in the third trimester
has additional limitations, in addition to missing birth certificate numbers.
It can therefore be deduced that the number of pairs presented in the 2 x 2
table for capture by n_{(A}_{∩B)},
would be larger than measured, demonstrating the large overlap between the two
databases.

In relation to
SINASC, the data are consolidated by sending information from state health secretaries
to the Ministry of Health and are eventually updated.^{i}
The number of live births available on the internet site differs from the preliminary
data available for this study, in 2007. The dynamism of the two databases should
be considered, characterized by continual changes in the total number of records.

The assumption that the capture-recapture probability of one individual does not affect the probabilities for others was not violated because the birth of a baby does not interfere with the identification of another baby by SINASC, as well as the Civil Registry. In regards to multiple births - the fact that the live births are or are not registered together - does not meet this assumption. Nonetheless, the occurrence of multiple gestations is very rare, and the number of twins born alive does not harm this assumption.

Regarding the independence
of the samples, here databases, the large overlap between them (large n_{A}_{∩B})
suggests a positive dependence. This indicates that the number of estimated
live births would not be much larger than the distinct individuals identified
in the two databases. In order to quantify the dependence between two epidemiologic
sources (lists), Brenner^{1} included a correction factor, for the probability
of an individual to be captured in the two lists. The author simulates situations
where both the capture probabilities for each source (n_{A} and n_{B})
and the correction factors that modify the probability for inclusion in the
two sources (n_{A}_{∩B})
vary, creating positive and negative dependence in order to observe the behavior
of the under- and over-estimation factor. In the case of negative dependence,
the investigator concluded that overestimation of the total population size
would be more serious when the lists have low coverage of individuals and a
small probability for including the individuals. In cases of positive dependence,
the author affirms that lists with a high inclusion probability have smaller
underestimation factors. When considering this type of dependence, estimates
of population size will still be closer to reality than simple aggregation of
the sources.^{1}

In regards to the
model selected, Tilling & Sterne^{16} and Tilling et al^{19}
applied the Huggins model for estimating epidemiologic data, demonstrating the
viability of the model for these types of data. Also, use of the conditional
likelihood for the observed individuals allows flexibility to include covariates
to model the capture probabilities with adjusted linear models. We believe the
Huggins model can continue to be applied to estimate total live births through
linkage of SINASC and the Civil Registry, as long as future studies resolve
the problem encountered with the IBGE database regarding the equal probability
of capture during the study period used to identify records.

In conclusion,
the results of the present study suggest minimal values for civil underreporting
and SINASC coverage and that it is possible to apply the capture-recapture methodology
to estimate underreporting of live births. In the case of large overlap between
two databases, the International Working Group for Disease Monitoring and Forecasting^{9}
recommends aggregation of the sources and turning them into one source. An alternative
may be the deterministic linkage of SINASC and the Civil Registry and probabilistic
association between this new database and other sources that can be used when
applying capture-recapture, such as enrollment in the Family Health Programs
and the Hospital Information System, for example.

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**Correspondence:
** Bianca Schmid

R. Frei Caneca, 443 - Apto. 102

Consolação

01307-001 São Paulo, SP, Brasil

E-mail: bika.schmid@gmail.com

Received: 12/13/2010

Approved: 7/27/2011

Article based on
the doctoral thesis by Schmid B, presented to the Faculdade de Saúde
Pública at Universidade de São Paulo in 2010.

The authors declare no conflicts of interests.

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