Costanza, Michael

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Soz.- Präventivmed. 49 (2004) 224–2260303-8408/04/030224–3DOI 10.1007/s00038-004-3091-1© Birkhäuser Verlag, Basel, 2004Hints & Kinks in survey researchEstimating and approximating prevalence trendsMichael C. CostanzaClinical Epidemiology Division, Department of Community Medicine, University Hospitals of GenevaFigure 1 depicts gender-specific annual trends in overweight(body mass index (BMI) > 25 kg/m2) prevalence (%) fornine independent, annual, cross-sectional samples of adultmen and women (total n = 9716) who were randomly se-lected within age strata during 1993–2001. Five differenttrend “curves” are shown. These ways of estimating or ap-proximating prevalence trends, and of assessing trend P-val-ues, are discussed in this Hints & Kinks.1. Crude mean estimates (no trend model)The dark circles are annual crude sample means of the over-weight “indicator” variable: Y = 100 for overweight individ-uals, Y = 0 for non-overweight individuals. Thus, they areannual overweight sample prevalences, estimating Psurvey =annual overweight population prevalences. (For propor-tions, code Y = 1 for overweight.)Connecting crude means provides some idea of trend, but itis difficult to formalize this without a statistical model specif-ically designed to assess it.2. ANCOVA “Least squares means” approximationsConsider an analysis of covariance (ANCOVA) model forthe population overweight prevalences by age (years, con-tinuous) and survey (9 groups (not continuous)),Figure 1 Illustrations of the five approaches in Sections 1–5 for estimating or approximating annual trends in the prevalence of overweight (BMI > 25 kg/m2). The data are from nine independent, annual, cross-sectional samples of adult (35–74 years) men and women non-institutionalizedresidents of Geneva, Switzerland who were randomly selected within age strata from 1993 through 2001 (see Galobardes et al. 2003a; b). Eachparticipant appeared in a single survey, and all analyses were gender-specific. Annual trend P-values: mean logistic predicted values estimates: men:P = 0.000027; women: P = 0.141; linear regression approximations: men: P = 0.000030; women: P = 0.137225Hinks & Kinks in survey researchSoz.- Präventivmed. 49 (2004) 224–226© Birkhäuser Verlag, Basel, 2004Costanza MCEstimating and approximating prevalence trends Page, survey = b0 + (b1 ¥ age) + (b2 ¥ survey),                                       (1)which is linear in age and survey for some unknown para-meters b0, b1, b2. Should neither age nor survey have effectson being overweight (i.e., b1 = b2 = 0), then Page, survey = b0(constant). It is an “additive” (assumes no (age ¥ survey) in-teraction effect) model.Correspondingly, the open circles approximatePsurvey = Page, survey – (b1 ¥ age).                                                         (2)Specifically, they are so-called  “least squares means” (esti-mated “population marginal means”, Searle et al. 1980) ob-tained by analyzing the {Y, age, survey} data trio for each in-dividual using (e.g.) the “LSMEANS” option for survey(declared a “CLASS” (grouped) variable), in the SAS GLM(Generalized Linear Models) program (SAS Institute, Inc.1999).Connecting least squares means provides a further idea of(mean age-adjusted) annual trends (because age was in themodel). However, this ANCOVA model was designed to as-sess any differences between annual prevalences, not trend,per se.3. Mean linear predicted values approximationsIf survey is continuous, the above ANCOVA model be-comes a (multiple) linear regression model. The (generic) b0,b1, b2 parameters (also approximations, say b0, b1, b2) are notthe same as before (e.g., b2 is now a linear slope). Neverthe-less, one can analyze the same (Y, age, survey) data with(e.g.) another GLM run (no “CLASS” declaration) to ob-tain approximations (“linear predicted values”) of Page, sureveyin (1),L = b0 + (b1 ¥ age) + (b2 ¥ survey).Then, an approximate Psurvey in (2) is the mean L over indi-viduals in that survey.Connecting mean linear predicted values (“L” points) pro-vides a “smoother” idea of trend than connecting crude orleast squares means. (The “L” points are almost superim-posed on the “P” points, defined in Section 5.) However, al-though the linear regression model was designed to assesstrend (because age and survey are continuous), this still doesnot constitute a formal test for trend.4. Linear regression approximationOn the other hand, within the framework of the model inSection 3, the usual t- or F-test of H0 : b2 = 0 (no survey slope)does provide a formal assessment of (linear) trend. In the ex-ample, the annual overweight prevalences increased signifi-cantly in men (P = 0.00003), but not in women (P = 0.14).In fact, however, these trend P-values refer to another (sim-pler) way of approximating (mean age-adjusted) Psurvey in(2). Specifically, for each gender, the solid lines depict thesingle linear function of survey,b0 + (b1 ¥ mean age) + (b2 ¥ survey),i.e., the sample linear regression equation evaluated at the(overall) mean age.5.  Mean logistic regression predicted values estimatesAlthough 0 ≤ Page, survey ≤ 100 since it is a percentage, it is pos-sible to obtain a least squares mean, a mean linear predictedvalue, or even a point on the sample regression line outsidethat range because those approaches do not constrain the fi-nal approximated numerical values in any way. This is onereason why these three approaches were dubbed “approxi-mations” rather than estimates. This difficulty is avoided by a logistic regression model and analysis. In lieu of modeling Page, survey directly, the“logit” (or log odds) of Page, survey, which is the logarithm (log)of {Page, survey/(100 – Page, survey)}, is modeled instead. The cor-responding analogue of model (1) is:log{Page, survey/(100 – Page, survey)} = b0 + (b1 ¥ age) + (b2 ¥ year).(3)This model assumes the logit, not Page, year itself, is linear inage and survey (both continuous).Once again, the individual (Y, age, survey) data are analyzed(e.g., with the SAS LOGISTIC program). In addition, cor-responding likelihood ratio tests (LRT) for trend can be ob-tained. In the example, the logistic LRT P-values  were vir-tually the same as those reported in Section 4.One can also estimate, for each individual in an annual sur-vey, the corresponding individual analogue of Psurvey in (2)  byback-transforming the estimated logit of Page, survey, (say)LOGIT = {b0 + (b1 ¥ age) + (b2 ¥ survey)})in two steps as follows:  (i)  exponentiate to eLOGIT,  and (ii)compute eLOGIT/(1 + eLOGIT). Then, to estimate the corre-sponding (overall) analogue of Psurvey in (2), one can use themean eLOGIT/(1 + eLOGIT) over all individuals surveyed thatyear (“P” points).As mentioned above, in the figure the mean logistic pre-dicted values (“P”) were practically identical to the meanlinear predicted values ( “L”). So, connecting the “P” pointsalso provides a “smoother” idea of trends than connectingthe crude or least squares means. However, neither the con-nected “P” nor “L” points is as smooth as the single regres-sion line approximation of Section 4.Hinks & Kinks in survey researchSoz.- Präventivmed. 49 (2004) 224–226© Birkhäuser Verlag, Basel, 2004226 Costanza MCEstimating and approximating prevalence trends References Galobardes B, Costanza MC, Bernstein MS, Delhumeau CH, Morabia, A (2003).Trends in risk factors for the major “lifestyle-related” diseases in Geneva, Switzerland,1993–2000. Ann Epidemiol 13: 537–40.Galobardes B, Costanza MC, Bernstein MS, Delhumeau CH, Morabi, A (2003).Trends in risk factors for lifestyle-related diseases by socioeconomic position in Geneva,Switzerland, 1993–2000: health inequalities persist. Am J Public Health 93: 1302–9.Korn EL, Graubard BI (1999). Analysis of health surveys. New York: John Wiley & Sons.MathSoft, Inc. (1999). S-PLUS 2000 user’s guide.Seattle, WA: Data Analysis Products Division.SAS Institute, Inc. (1999). SAS OnlineDoc“,VERSION EIGHT. Cary, NC: SAS.Searle SR, Speed FM, Milliken GA (1980). Population marginal means in the linear model:an alternative to least squares means. Am Statist34: 216-21.Szklo M, Nieto FJ (2000). Epidemiology: beyondthe basics. Gaithersgurg, MD: Aspen Publishers,Inc.Address for correspondenceMichael C. Costanza, PhDClinical Epidemiology Division Department of Community MedicineUniversity Hospitals of Geneva (HUG)CH-1211 Geneva 14Tel.: +41 22 372 95 52Fax: +41 22 372 95 65e-mail:  Michael.Costanza@hcuge.chDiscussion and recommendationsRemember, prevalence estimates and trend P-values shouldbe obtained by analyzing the individual-level data, not theaggregated (e.g.) least squares means data. Crude and leastsquares means were stressed in the first two approaches, butother types of adjusted means, or medians, etc., also could beused. SAS programs were cited, but others, e.g., in S-PLUS2000 (MathSoft, Inc. 1999), used for Figure 1, could equallywell be employed for the analyses.In the moderately prevalent (25% to 65%) overweight ex-ample, there was little practical difference between the(technically more correct) mean logistic predicted values es-timates and the mean linear predicted values approxima-tions. Just as an observation, consistently similar degrees ofconcordance between these two approaches have occurredin our research on much less prevalent risk factors or out-comes (e.g., diabetes treatment prevalences from 0% to2%). The even simpler, single straight line regression ap-proximations were also reasonably close to both the latterapproaches for all prevalence magnitudes considered.Further examples of the least squares means and linear re-gression approximations approaches can be found in Galo-bardes et al. (2003a; b), where quarterly trends in a varietyof cardiovascular disease risk factors were assessed usingthese techniques.The range of models covered here was curtailed and delib-erately simplified. Readers requiring more depth or morecomplex models for dealing with other important issues suchas interaction effects, different methods of age-adjustment(e.g., direct standardization) or other covariate-adjustment,nonlinear trends, etc., are directed to more comprehensivereferences such as (e.g.) Szklo and Nieto (2000), or Kornand Graubard (1999).AcknowledgementsThe surveys were funded by the Swiss National Fund for Sci-entific Research (Grants No 32-31.326.91, 32-37986.93, 32-47219.96, 32-49847.96, 32-054097.98, 32-57104.99). The veryconstructive suggestions of two anonymous reviewers aregreatly appreciated.    

Estimating and approximating prevalence trends

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Estimating and approximating prevalence trends

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