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## Revista de Saúde Pública

##
*On-line version* ISSN 1518-8787

### Rev. Saúde Pública vol.42 n.6 São Paulo Dec. 2008

**ORIGINAL
ARTICLES**

**Methods
for estimating prevalence ratios in cross-sectional studies**

**Métodos
para estimar razón de prevalencia en estudios de cohorte transversal**

**Leticia M S
Coutinho ^{I, II}; Marcia Scazufca^{II, III}; Paulo R Menezes^{I,
II}**

^{I}Departamento
de Medicina Preventiva. Faculdade de Medicina Universidade de São Paulo.
São Paulo, SP, Brasil

^{II}Núcleo de Epidemiologia. Hospital Universitário.
Universidade de São Paulo. São Paulo, SP, Brasil

^{III}Departamento de Psiquiatria. Faculdade de Medicina Universidade
de São Paulo. São Paulo, SP, Brasil

**ABSTRACT**

**OBJECTIVE:**
To empirically compare the Cox, log-binomial, Poisson and logistic regressions
to obtain estimates of prevalence ratios (PR) in cross-sectional studies.

**METHODS: ** Data from a population-based cross-sectional epidemiological
study (n = 2072) on elderly people in Sao Paulo (Southeastern Brazil), conducted
between May 2003 and April 2005, were used. Diagnoses of dementia, possible
cases of common mental disorders and self-rated poor health were chosen as outcomes
with low, intermediate and high prevalence, respectively. Confounding variables
with two or more categories or continuous values were used. Reference values
for point and interval estimates of prevalence ratio (PR) were obtained by means
of the Mantel-Haenszel stratification method. Adjusted PR estimates were calculated
using Cox and Poisson regressions with robust variance, and using log-binomial
regression. Crude and adjusted odds ratios (ORs) were obtained using logistic
regression.

**RESULTS: ** The point and interval estimates obtained using Cox and Poisson
regressions were very similar to those obtained using Mantel-Haenszel stratification,
independent of the outcome prevalence and the covariates in the model. The log-binomial
model presented convergence difficulties when the outcome had high prevalence
and there was a continuous covariate in the model. Logistic regression produced
point and interval estimates that were higher than those obtained using the
other methods, particularly when for outcomes with high initial prevalence.
If interpreted as PR estimates, the ORs would overestimate the associations
for outcomes with low, intermediate and high prevalence by 13%, almost by 100%
and fourfold, respectively.

**CONCLUSIONS:** In analyses of data from cross-sectional studies, the Cox
and Poisson models with robust variance are better alternatives than logistic
regression is. The log-binomial regression model produces unbiased PR estimates,
but may present convergence difficulties when the outcome is very prevalent
and the confounding variable is continuous.

**Descriptors:
** Cross-Sectional Studies. Estimation Techniques. Prevalence Ratio. Logistic
Models. Comparative Study.

**RESUMEN**

**OBJETIVO:**
Comparar empíricamente las regresiones de Cox, log-binomial, Poisson
y logística para estimar razones de prevalencia en estudios de cohorte
transversal.

**MÉTODOS:** Fueron utilizados datos de un estudio epidemiológico
transversal (n=2.072), de base poblacional, realizado con ancianos en la ciudad
de Sao Paulo (Sureste de Brasil), entre mayo de 2003 y abril de 2005. Diagnósticos
de demencia, posibles casos de trastorno mental común y autopercepción
de salud pésima fueron escogidos como resultados con prevalencia baja,
intermedia y alta, respectivamente. Fueron utilizadas variables de confusión
con dos o más categorías o valores continuos. Valores de referencia
para estimaciones por punto y por intervalo para las razones de prevalencia
(RP) fueron obtenidos por el método de estratificación de Mantel-Haenszel.
Estimaciones ajustadas fueron calculadas utilizando regresiones de Cox y Poisson
con varianza robusta, y regresión log-binomial. *Odds ratios* (OR)
brutos y ajustados fueron obtenidos por la regresión logística.

**RESULTADOS:** Las estimaciones por punto y por intervalo obtenidas por
las regresiones de Cox y Poisson fueron semejantes a la obtenida por la estratificación
de Mantel-Haenszel, independientemente de la prevalencia del resultado y de
las covariables del modelo. El modelo log-binomial presentó dificultad
de convergencia cuando el resultado tenía prevalencia alta y había
convariable continua en el modelo. La regresión logística produjo
estimaciones por punto y por intervalo mayores de las obtenidas por los otros
métodos, principalmente para los resultados con mayores prevalencia iniciales.
Si se interpretaban como estimaciones de RP, los OR superestimarían las
asociaciones para los resultados con prevalencia inicial baja, intermedia y
alta en 13%, casi 100% y cuatro veces mas, respectivamente.

**CONCLUSIONES:** En análisis de datos de estudios de cohorte transversal,
los modelos Cox y Poisson con varianza robusta son mejores alternativas que
la regresión logística. El modelo de regresión log-binomial
produjo estimaciones sin sesgo de la RP, pero puede presentar dificultad de
convergencia cuando el resultado es muy frecuente y la variable de confusión
es continua.

**Descriptores:**
Estudios Transversales. Técnicas de Estimación. Razón de
Prevalencias. Modelos Logísticos. Estudio Comparativo.

**INTRODUCTION**

In cross-sectional
studies with binary outcomes, the association between exposure and outcome is
estimated by means of prevalence ratios (PRs). When adjustments for potential
confounders are needed, logistic regression models are commonly used. This type
of model yields estimates of odds ratios (ORs), and frequently ORs are reported
in the same way as PR estimates are. However, ORs do not approximate well to
PRs when the initial risk is high, and in these situations, interpreting ORs
as if they were PRs may be inadequate.^{1,2,9,12}

Some alternative
statistical models that may directly estimate PRs and their confidence intervals
have been discussed in the literature.^{1,4,6,10,12,14} Cox, log-binomial
and Poisson regression models have been suggested as good alternatives for obtaining
PR estimates adjusted for confounding variables. Using data adapted from a cross-sectional
study, Barros & Hirakata^{1} (2003) showed that these models yield
adjusted PR estimates that are very similar to those obtained by means of the
Mantel-Haenszel (MH) method.

The aim of the present study was to empirically compare the Cox, log-binomial, Poisson and logistic regression models with regard to estimating adjusted PRs, comparing their results with those obtained using the MH method.

**METHODS**

The data used came
from a population-based cross-sectional study that had the aim of estimating
the prevalence of dementia and other mental health problems among elderly people
(aged 65 years or older) who were living in an economically deprived area of
the district of Butantã, in the city of Sao Paulo (SP), between May 2003
and April 2005.^{8} Standardized procedures were used to assess cognitive
functioning and psychiatric symptoms. Information on sociodemographic and socioeconomic
characteristics was obtained. A total of 2,072 participants were included in
the study.

Three outcomes
were chosen: diagnoses of dementia, possible cases of common mental disorders
(CMD) and self-rated poor health. Diagnoses of dementia were obtained by means
of a procedure developed by the 10/66 Dementia Research Group, for use in population-based
studies in developing countries, with a detailed assessment of the onset and
course of dementia.^{7} Individuals were classified as possible cases
of CMD by means of the Self-Report Questionnaire (SRQ-20), a questionnaire developed
by the World Health Organization for studies in developing countries.^{10}
The cutoff point used was 4/5, in accordance with the validation of the Brazilian
version of the SRQ-20.^{9} Self-rated health was assessed using a single
question ("On the whole, how would you classify your health over the last 30
days?"), with the following answer options: "very good", "good", "regular",
"poor" and "very poor". These were then pooled, in order to classify participants
as having self-rated good health ("very good" and "good") or self-rated poor
health ("regular", "poor" and "very poor"). The three outcomes were chosen based
on their prevalence (low for dementia, intermediate for CMD and high for self-rated
poor health). Each outcome was associated with one main exposure and two potential
confounding factors. For the outcomes of dementia and CMD, the main exposure
was educational level and the confounding variables were age and gender. For
self-rated poor health, the main exposure was the presence of depressive episodes,
diagnosed in accordance with the ICD-10 criteria for depression, and the confounding
variables were income and gender.

In relation to previous studies, we extended the application of these methods to situations with two confounding variables (some with more than two levels of exposure or measured as continuous values) in order to verify the point and interval PR estimates generated by each multivariate model. Outcomes of different frequencies were analyzed, in order to examine how, as the prevalence of the outcome increases, the Cox, log-binomial, Poisson and logistic models behave in relation to estimating PRs.

Reference values for the adjusted PR estimates and respective 95% confidence intervals (95% CI), for the associations between each outcome and the respective main exposure, were obtained by means of the Mantel-Haenszel stratification, while controlling for the effects of the potential confounders. PR estimates with the respective 95% CI were then calculated using the Cox, log-binomial and Poisson regression models, and crude and adjusted ORs (with 95% CI) were also calculated using logistic regression. Next, for each outcome of interest, one confounding variable was tested as a continuous measurement. The Cox and Poisson regressions were performed by setting the follow-up time as one for all participants and using robust variance estimators. The statistical software used for this study was Stata version 9.0.

The Poisson regression
model is generally used in epidemiology to analyze longitudinal studies in which
the response is the number of episodes of an event occurring over a given time.
For cohort studies in which all individuals have equal follow-up time, the Poisson
regression can be used with a time-at-risk value of one for each individual.
If the model adequately fits the data, this approximation provides a correct
estimate of the adjusted relative risk.^{4} In cross-sectional studies,
a value of one can be attributed to each participant's follow-up time, as a
strategy to obtain PR point estimates, since there is no real follow-up for
the participants in this type of epidemiological studies. However, when the
Poisson regression is applied to binomial data, the error for the estimated
relative risk is overestimated, because the variance of the Poisson distribution
increases progressively, while the variance of the binomial distribution has
a maximum value when the prevalence is 0.5. This problem can be corrected by
using a robust variance procedure, as proposed by Lin & Wei (1989).^{3}
The Poisson regression with robust variance does not have any convergence difficulty,
and it produces results that are very similar to those obtained using the MH
procedure, when the covariate of interest is categorical.^{6,14}

The Cox regression
model is usually used to analyze time-to-event data. In cross-sectional studies,
no time-periods are observed, but if a constant risk period is assigned to all
the individuals in the study, the hazard ratio estimated using Cox regression
equals the PR, in the same way as with the Poisson regression. However, the
use of Cox regression without any adjustment for analyzing cross-sectional studies
can also lead to errors in estimating confidence intervals, which may then be
wider than they should be. The robust variance method may also be used in such
situations.^{3}

The log-binomial
regression model is a generalized linear model in which the link function is
the logarithm of the proportion under study and the distribution of the error
is binomial. It directly models the prevalence ratio for dichotomous variables.
However, there may be a lack of convergence when trying to provide parameter
estimates. Normally, this problem is due to Newton's method, which is used to
find a minimum or maximum for this function. This method may be unable to find
a maximum likelihood estimate when the solution is at the boundary of the restricted
interval for the parameter. Peterson & Deddens^{6} (2003) suggested
the COPY method (a macro for the SAS software), which may provide an approximate
estimates and standard errors when the *Proc Genmod* command (generally
used in SAS for binomial distribution with a logarithmic link function) fails
to converge.

Logistic regression
has been widely used in epidemiological studies with binary outcomes to obtain
unbiased OR estimates adjusted for one or more confounding variables. It is
possible to calculate the PR from the OR estimate, with 95% CI, but the calculations
are complex and require computing software to calculate variance estimates using
matrix modules.^{5}

**RESULTS**

The outcome of dementia (low prevalence: 5.1%) showed statistically significant associations with educational level and age group, but not with gender (Table 1). The risk factor of educational level also showed statistically significant associations with age group (p < 0.01) and gender (p < 0.01). There was some confounding relative to age group in estimating the association between educational level and prevalence of dementia, as shown by the MH stratification (Table 2). Comparing the results from the different models consisting of the main exposure and one confounding variable with four exposure levels, the point estimates and respective 95% CI obtained using the Poisson, Cox and log-binomial models were very close to what was obtained using MH stratification (Table 2), with differences of one or two hundredths (second decimal place). The results observed when an extra potential confounding variable (gender) was added to the Cox, Poisson and log-binomial models produced point estimates of 2 or 3 hundredths (second decimal place) lower than seen in the MH point estimate, and the 95% CI was narrower than the MH confidence interval. Putting age as a continuous variable produced further adjustment for confounding, and the estimate for the association between educational level and dementia was no longer statistically significant. Logistic regression produced a point estimate approximately 13% higher, with a wider 95% CI than what was obtained with the other regression models, in all situations.

The outcome of CMD (intermediate prevalence: 37.8%) showed statistically significant associations with educational level, gender and age group (Table 1). There was some confounding due to gender and age group in estimating the association between educational level and risk of CMD, as shown by the MH stratification (Table 3). Comparing the results from the different models, both in the situation consisting of the main exposure and one confounding variable (gender) with two exposure levels, and when adding an extra potential confounding variable (age group) with four exposure levels, the point estimates and respective 95% CI obtained using the Poisson, Cox and log-binomial models were identical to those obtained using MH stratification (Table 3). When age was taken as a continuous variable, the Cox, Poisson and log-binomial models produced almost identical point estimates and respective 95% CI. Logistic regression produced point estimates almost 100% higher than those obtained using the other regression models, with wider 95% CI.

The outcome of self-rated poor health (high prevalence: 53.8%) showed statistically significant associations with depressive episodes, gender and income (Table 1). The main exposure variable ("depressive episode") was also associated with income (p = 0.04). There was almost no confounding due to income or gender in estimating the association between depressive episodes and self-rated poor health, as shown by the MH stratification (Table 4). When the results from the different models in the situation consisting of the main exposure and one confounding variable (income) with four exposure levels were compared, or when an extra potential confounding variable (gender) was added to each model, the point estimates obtained using the Poisson and Cox models and respective 95% CI were identical to those obtained using MH stratification. The point estimates obtained using the log-binomial model were closer to one than were those yielded by the other two models. When income was taken as a continuous variable, the results from the Cox and Poisson models were similar. However, it was difficult to reach convergence using log-binomial regression. Logistic regression produced point estimates that, if interpreted as PR estimates, would be more than four times greater than those obtained using the other regression models, and the 95% CI was wider.

**DISCUSSION**

A previous study^{1}
had showed that in cross-sectional studies, the Cox and Poisson regression models
with robust variance and the log-binomial regression model generate adequate
estimates for prevalence ratios and their confidence intervals, regardless of
the base prevalence. In a recent study on this question, Peterson & Deddens^{6}
(2008) advocated, based on real and simulated data, that the Poisson regression
gave better PR estimates for very frequent outcomes, in relation to the log-binomial
regression model. However, these authors suggested that log-binomial regression
would be the best method for intermediate prevalence.

We explored the performance of these methods in relation to different prevalences of outcomes of interest, more than one confounding variable and continuous covariates. We showed that the three methods generated correct point and interval estimates in all situations, although the log-binomial models presented convergence difficulty in situations of very prevalent outcomes and continuous covariates. For the three outcomes investigated, the Cox and Poisson regression models presented identical PR estimates and 95% CI estimates, and they were very similar to those obtained using our reference (MH stratification). The use of robust methods for variance estimation in the Cox and Poisson models corrected for the overestimation of the variance and produced adequate confidence intervals. The Cox and Poisson models also behaved well in relation to continuous covariates.

The log-binomial
regression models also behaved well in most of the situations tested, yielding
point and interval estimates that were close to those obtained using MH stratification.
However, when the prevalence of the outcome was high, the log-binomial model
produced estimates closer to one than were those obtained using MH stratification
or using Cox and Poisson regression. Moreover, when one of the covariates was
continuous, the log-binomial model presented convergence difficulties, as previously
described.^{1,6}

The OR estimates obtained using the logistic regression models were close to the PR estimates when the outcome prevalence was low (dementia), although even then there was a tendency for the OR to be higher than the PR. In the situation of intermediate prevalence (CMD), the OR was almost twice the PR. In other words, if the OR were interpreted as a PR, it would seem that the relative increase in the risk of CMD for individuals with lower educational level was 23% higher than the risk for those with better educational level, instead of 12% higher, as shown by the PR. The ORs obtained when the prevalence was high (self-rated poor health) were four times higher than the PR estimates obtained using MH stratification or using the Cox, Poisson and log-binomial models. This shows the inappropriateness of interpreting OR estimates as if they were PR estimates in these situations.

The present study has some limitations. The associations in the three situations examined were not strong, thus making the estimates obtained using the various methods tested close to each other. Furthermore, the confounding effects were not pronounced, apart from the effect of age on the risk of dementia. Nevertheless, the present study provides further support for the use of the modeling techniques tested, as alternatives to logistic regression. These techniques are available in most statistical packages that are used to analyze epidemiological studies.

**REFERENCES**

1. Barros AJ, Hirakata VN. Alternatives for logistic regression in cross-sectional studies: an empirical comparison of models that directly estimate the prevalence ratio. *BMC Med Res Methodol.* 2003;3:21. DOI: 10.1186/1471-2288-3-21 [ Links ]

2. Davies HT, Crombie IK, Tavakoli M. When can odds ratios mislead? *BMJ.* 1998;316(7136):989-991. [ Links ]

3. Lin DY, Wei LJ. The robust Inference for the Cox Proportional Hazards Model. *J Am Stat Assoc.* 1989;84(408):1074-8. DOI: 10.2307/2290085 [ Links ]

4. McNutt LA, Wu C, Xue X, Hafner JP. Estimating the relative risk in cohort studies and clinical trials of common outcomes. *Am J Epidemiol.* 2003;157(10):940-3. DOI: 10.1093/aje/kwg074 [ Links ]

5. Oliveira NF, Santana VS, Lopes AA. Razões de proporções e uso do método delta para intervalos de confiança em regressão logística. *Rev Saude Publica.* 1997;31(1):90-9. [ Links ]

6. Petersen MR, Deddens JA. A comparison of two methods for estimating prevalence ratios. *BMC Med Res Methodol.* 2008;8:9. DOI: 10.1186/1471-2288-8-9 [ Links ]

7. Prince M, Acosta D, Chiu H, Scazufca M, Varghese M, Dementia diagnosis in developing countries: a cross-cultural validation study. *Lancet.* 2003;361(9361):909-17. DOI: 10.1016/S0140-6736(03)12772-9 [ Links ]

8. Scazufca M, Menezes PR, Vallada HP, Crepaldi AL, Pastor-Valero M, Coutinho LMS, et al. High prevalence of dementia among older adults from poor socioeconomic backgrounds in São Paulo, Brazil. *Int Psychogeriatrics.* 2008;20(2):394-405. DOI: 10.1017/S1041610207005625 [ Links ]

9. Scazufca M, Menezes PR, Vallada H, Araya R. Validity of the self reporting questionnaire-20 in epidemiological studies with older adults. *Soc Psychiatry Psychiatr Epidemiol*. 2008; Sep 8. [Epub ahead of print] [ Links ].

10. Thompson ML, Myers JE, Kriebel D. Prevalence odds ratio or prevalence ratio in the analysis of cross sectional data: what is to be done? *Occup Environ Med.* 1998;55(4):272-7. [ Links ]

11. World Health Organization. A user's guide to the Self-Reporting Questionnaire (SRQ). Geneva; 1994. [ Links ]

12. Zhang J, Yu KF. What's the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. *JAM*A. 1998;280(19):1690-1. DOI: 10.1001/jama.280.19.1690 [ Links ]

13. Zocchetti C, Consonni D, Bertazzi PA. Relationship between prevalence rate ratios and odds ratios in cross-sectional studies. *Int J Epidemiol.* 1997;26(1):220-3. DOI: 10.1093/ije/26.1.220 [ Links ]

14. Zou G. A modified poisson regression approach to prospective studies with binary data. *Am J Epidemiol.* 2004;159(7):702-6. DOI: 10.1093/aje/kwh090 [ Links ]

** Correspondence:
** Paulo Rossi Menezes

Departamento de Medicina Preventiva

Faculdade de Medicina da Universidade de São Paulo

Av. Dr. Arnaldo 455

01246-903 São Paulo, SP, Brasil

E-mail: pmenezes@usp.br

Received: 11/27/2007

Revised: 5/13/2008

Approved: 6/4/2008

Study funded by Wellcome Trust, UK.

Menezes PR and
Scazufca M were partially funded by Conselho Nacional de Desenvolvimento Científico
e Tecnológico (CNPq; research productivity bursaries).

Article based on the master's dissertation of LMS Coutinho, which was presented
within the Postgraduate Sciences Program of the School of Medicine of the Universidade
de São Paulo, in 2007.