Secondary association in populations due to heterogeneity

Bull. Org. mond. Sante 1972, 47, 83-86 Bull. Wld Hith Org. Secondary association in populations due to heterogeneity R. MACLENNAN' & S. P. H. MANDEL2 This paper discusses the phenomenon of spurious or secondary association between factors in heterogeneous study populations. The mathematical basis of this phenomenon is set out and some numerical examples are given. The relevance of the findings to the interpretation of epidemiological studies is examined. In epidemiological studies it is often assumed that the examination of a total population, or at least a random sample, will result in statistically respectable data from which it will be possible to draw valid conclusions regarding any associations that may be found. In a recent study (Rossi- Espagnet et al., 1968) of blood pressure and the sickle cell trait in Negroes, the possibility of spurious associations being found as a result of stratification of genetically determined characteristics within a population due to interracial outbreeding led us to consider again the interpretation of statistical associations, particularly in studies of heterogeneous populations. The difficulties of interpreting associations ob- served between two factors are well known to epi- demiologists; MacMahon & Pugh (1970), for example, discuss the problems of distinguishing be- tween causal and secondary associations. However, although long noted by statisticians (e.g., Yule & Kendall, 1958), the fact that heterogeneity in a population can lead to spurious evidence of asso- ciation between factors in that population appears not to have been widely recognized. Discussions of the limitations in interpretation of associations found in hospital studies (Berkson, 1946) have led to the assumption that data derived from the general population are free from such problems. This assumption may not be valid, and it is the purpose of this paper to discuss the phenomenon of asso- 1 Medical Officer, Epidemiology, International Agency for Research on Cancer, Lyon, France. ' Chief, Mathematics-Statistics, Division of Research in Epidemiology and Communications Science, World Health Organization, Geneva, Switzerland. ciation and to give some examples of noncausal or secondary associations that can occur in total populations. Let us consider first the following simple but illustrative example. In Alabama in 1960 there were 80 846 live births and 2 616 infant deaths (under 1 year of age), while in Kentucky in the same year there were 72 208 live births and 2 015 infant deaths (US Department of Health, Education, and Wel- fare, Public Health Service, 1963). The crude in- fant mortality rates can be calculated from these data as 32.4 infant deaths per 1 000 live births in Alabama and 27.9 infant deaths per 1 000 live births in Kentucky. One might conclude from these figures that there is an association between place of birth and failure to survive the first year of life and that the risk of death in the first year is higher in Alabama than in Kentucky. However, if one examines these data in subpopulations the situation changes. Thus Table 1 shows the distri- bution by race of live births and infant deaths in the two states in 1960. Stratification of each population by race actually reverses the sense of the association apparent in the unstratified population; a white infant born in Alabama is actually at lower risk of death in the first year of life than a white infant born in Kentucky, and the same is true of nonwhite infants. The sense of the association observed in the unstratified popu- laiton is clearly a reflection of the higher proportion of high risk nonwhite live births in Alabama (37.1 %) than in Kentucky (8.7 %). Further stratification of the population according to additional characteris- tics (e.g., socioeconomic level) might well again alter the sense of the association between place of birth and failure to survive the first year of life. 2876 -83- R. MACLENNAN & S. P. H. MANDEL Table 1. Infant mortality by race in Alabama and Kentucky, 1960 * White Nonwhite All races Alabama Kentucky Alabama Kentucky Alabama Kentucky Live births 50 828 65 960 30 018 6 248 80 846 72 208 Infants deaths (<1 yearof age) 1 266 1 713 1 350 302 2616 2015 Infants deaths per 1 000 live births 24.9 26.0 45.0 48.3 32.4 27.9 * Data from US Department of Health, Education, and Welfare, Public Health Service (1963). The following is a simple mathematical formulation of the proposition that two factors may appear to be associated with each other when studied in an unstratified population while at the same time they are distributed independently of each other in sub- populations of that population. Suppose we consider a population consisting ofN individuals stratified into k subpopulations in such a way that there are ni individuals in the ith sub- population (i = 1, 2, . . . k), so that 2nj = N. Sup- pose we consider further two factors X and Y that are distributed in such a way that the prevalence ofX in the ith sub-population is ri and the prevalence ofY in the ith sub-population is si, and that within each subpopulation these two factors are distributed inde- pendently of each other in the statistical sense. In the unstratified population, the total number of individuals who are positive and who are negative for factor X will be £nirj and 2ni(I - ri), respec- tively; similarly the total number of individuals who are positive and who are negative for factor Y will be .nsiS and Zni(1 -si), respectively. Moreover, since we have assumed independence of the factors X andY in each subpopulation, it follows that the total number of individuals who are positive for both factors X and Y will be )nirisj, while the total num- ber of individuals who are positive for X but nega- tive for Y will be 2niri(I -si). All the summations here run over i from 1 up to k. Thus the overall prevalence in the unstratified population of factor X among those individuals who are positive for factor Y is Z(njrtsi)/2(njsj), while the overall prevalence in the unstratified pop- ulation of factor X among those individuals who are negative for Y is L[nirt(1- sj)]/7nj(1 -si). It is easy to see that these two quantities are in general unequal, since in general N2njrjsj * (2nirO)(?nisd). However, in any given subpopulation (say the ith), by assumption, the prevalence of X among Y posi- tive individuals and the prevalence of X among Y negative individuals are equal, being ri in each case. The phenomenon exemplified in Table 1 may be looked at from a slightly different point of view using a hypothetical numerical example. Suppose we stratify the total population into only two sub- populations, one of which consists of those indivi- duals who possess some characteristic C, while the other consists of those individuals who lack the characteristic C. Suppose further that the presence of C causes a high prevalence of both factors X and Y, while the absence of C gives rise to only low prevalences ofX and Y, and that X and Y are distri- buted independently of each other in those sub- populations in which C is present (subpopulation 1) and where C is absent (subpopulation 2). In such a situation we might have n, = n2 = 100 and ri= 0.8, r2 = 0.1 sL = 0.7, s2 = 0.2 (C present) (C absent) Then it can be verified that, in the total unstrati- fied population, the prevalence of factor X among those individuals who are positive for Y is 64.4%, while the prevalence of factor X among those indi- viduals who are negative for Y is 29.1 %. Thus X and Y appear to be associated with each other in the total unstratified population. In this case, this is clearly a result of the fact that each factor is controlled to a large extent by the presence or absence of the characteristic C. But if the charac- teristic C is fixed, i.e., if only those individuals who have the characteristic C, or only those who lack it, are considered, then by hypothesis X and Y are not associated. Thus, there is no direct relation- ship between X and Y, but there is an indirect 84 SECONDARY ASSOCIATION IN POPULATIONS DUE TO HETEROGENEITY relationship induced by their mutual dependence on C. This was the argument invoked by Fisher (1958a, 1958b) when he challenged the suggestion that there is a direct cause and effect relationship between smoking and lung cancer; in this situation X and Y would be the disposition to smoke and lung cancer respectively, while the characteristic C, Fisher supposed, could be something as intrinsic and immutable as a single gene, or indeed the whole genotype of an individual. An alternative way of looking at the above situation, which may also be illuminating, is the following. If we examine an individual who is Y positive the chances are that he has factor C and therefore he also has a high probability of being positive for factor X as well. If, on the other hand, he is negative for factor Y, the chances are that he lacks C, and therefore he has a much lower probability of being positive for factor X. DISCUSSION Mantel & Haenzel (1959) have emphasized the importance of obtaining data that are representative of the universe defined for investigation in retro- spective studies. Berkson (1946, 1955) has considered the special case of associations in hospital popula- tions in which association between two diseases may be determined both by the type of control group chosen and by the probabilities of patients with each disease reaching hospital. In a population given as an example, no association was present between two factors (diseases) but an association could be detected in the subgroups in hospital. In the foregoing we have shown that the reverse situa- tion can also occur-two factors may be associated in a total population but not in its subgroups. Such subgroups may or may not be identifiable. The possibility of their presence may limit the extent to which studies of association in natural populations can be interpreted. These considerations may well apply to blood pressure and the sickle cell trait in Negroes in the USA where considerable outbreeding with other races has occurred at some time in the past. It is likely that within the group defined in the USA as Negro, there are substrata with varying degrees of admixture with other races. In these subgroups genetically determined factors such as X and Y could occur with varying prevalences. Total popu- lation data based on the characteristics of indivi- duals could suggest associations that may not neces- sarily be present in the substrata. This lack of asso- ciation would not be detected if it were impossible or not feasible to identify the substrata, and this is pro- bably the situation in regard to the US Negro. Such possible spurious associations due to strati- fication as a result of suspected genetic characteris- tics could theoretically be avoided by a study of sibships in which sibs with a factor were compared with a group of control sibs who lack that factor. For example, a sickle cell positive sib might have a sickle cell negative sib as control, the level of blood pressure being compared in the two. This technique was in fact used by Fisher (1958a, 1958b) who went one step further and studied identical twin pairs. Motulsky (1968) suggests that the implied asso- ciation between G6PD deficiency and regional enteritis reported by Sheehan et al. (1967) could well be due to genetic factors that may be concen- trated among Lithuanian Jews. Motulsky stratifies the cases and controls in the studied population according to their geographic origin and finds that the proportion of Jews of north-eastern European extraction is much higher among the cases than among the controls. The apparent association be- tween the two factors G6PD deficiency and regional enteritis that is found in the unstratified population may not be present in such a cogent form in sub- populations stratified by geographical origins. There may well be other examples where an association that is apparent in an unstratified population no longer exists in subpopulations stratified out accord- ing to suitably chosen characteristics. oUMt ASSOCIATION SECONDAIRE DUE A L'HETEROGtNEITE AU SEIN DE POPULATIONS Les auteurs exposent comment l'etude de populations heterogenes-en raison de facteurs genetiques ou autres- peut conduire A d6couvrir des associations fausses ou secondaires entre diverses variables. Le ph6nomene fait l'objet d'une analyse mathematique succinte, illustr6e par un exemple portant sur des donn6es num6riques reelles. Son importance en ce qui conceme l'interpr6- tation des etudes epidemiologiques est examin&e. 85 86 R. MACLENNAN & S. H. P. MANDEL REFERENCES Berkson, J. (1946) Biomet. Bull., 2, 47-53 Berkson, J. (1955) Proc. Mayo Clin., 30, 319-348 Fisher, R. A. (1958a) Nature (Lond.), 182, 108 Fisher, R. A. (1958b) Nature (Lond.), 182, 596 MacMahon, B. & Pugh, T. F. (1970) Epidemiology: Principles and Methods, Boston, Mass., Little, Brown Mantel, N. & Haenzel, W. (1959) J. nat. Cancer Inst., 22, 719-748 Motulsky, A. G. (1968) New Engl. J. Med., 278, 281-282 Rossi-Espagnet, A. et al. (1968) Amer. J. Epidem., 88, 33-44 Sheehan, R. G. et al. (1967) New Engl. J. Med., 277, 1124-1126 US Department of Health, Education, and Welfare, Public Health Service (1963) Vital statistics of the United States, 1960, Washington, D.C. Yule, G. V. & Kendall, M. G. (1958) An introduction to the theory of statistics, London, Charles Griffin