Investigating population stratification and admixture using eigenanalysis of dense genotypes

Principal components analysis of genetic data is used to avoid inflation in type I error rates in association testing due to population stratification by covariate adjustment using the top eigenvectors and to estimate cluster or group membership independent of self-reported or ethnic identities. Eigendecomposition transforms correlated variables into an equal number of uncorrelated variables. Numerous stopping rules have been developed to identify which principal components should be retained. Recent developments in random matrix theory have led to a formal hypothesis test of the top eigenvalue, providing another way to achieve dimension reduction. In this study, I compare Velicer's minimum average partial test to a test on the basis of Tracy-Widom distribution as implemented in EIGENSOFT, the most widely used implementation of principal components analysis in genome-wide association analysis. By computer simulation of vicariance on the basis of coalescent theory, EIGENSOFT systematically overestimates the number of significant principal components. Furthermore, this overestimation is larger for samples of admixed individuals than for samples of unadmixed individuals. Overestimating the number of significant principal components can potentially lead to a loss of power in association testing by adjusting for unnecessary covariates and may lead to incorrect inferences about group differentiation. Velicer's minimum average partial test is shown to have both smaller bias and smaller variance, often with a mean squared error of 0, in estimating the number of principal components to retain. Velicer's minimum average partial test is implemented in R code and is suitable for genome-wide genotype data with or without population labels.     

Design and analysis of admixture mapping studies

Admixture between populations originating on different continents can be exploited to detect disease susceptibility loci at which risk alleles are distributed differentially between these populations. We first examine the statistical power and mapping resolution of this approach in the limiting situation in which gamete admixture and locus ancestry are measured without uncertainty. We show that, for a rare disease, the most efficient design is to study affected individuals only. In a typical African American population (two-way admixture proportions 0.8/0.2, ancestry crossover rate 2 per 100 cM), a study of 800 affected individuals has 90% power to detect at P values <10(-5) a locus that generates a risk ratio of 2 between populations, with an expected mapping resolution (size of 95% confidence region for the position of the locus) of 4 cM. In practice, to infer locus ancestry from marker data requires Bayesian computationally intensive methods, as implemented in the program ADMIXMAP. Affected-only study designs require strong prior information on the frequencies of each allele given locus ancestry. We show how data from unadmixed and admixed populations can be combined to estimate these ancestry-specific allele frequencies within the admixed population under study, allowing for variation between allele frequencies in unadmixed and admixed populations. Using simulated data based on the genetic structure of the African American population, we show that 60% of information can be extracted in a test for linkage using markers with an ancestry information content of 36% at 3-cM spacing. As in classic linkage studies, the most efficient strategy is to use markers at a moderate density for an initial genome search and then to saturate regions of putative linkage with additional markers, to extract nearly all information about locus ancestry.     

TYPE I ERROR RATES FOR TESTING GENETIC DRIFT WITH PHENOTYPIC COVARIANCE MATRICES: A SIMULATION STUDY

Studies of evolutionary divergence using quantitative genetic methods are centered on the additive genetic variance–covariance matrix ( G ) of correlated traits. However, estimating G properly requires large samples and complicated experimental designs. Multivariate tests for neutral evolution commonly replace average G by the pooled phenotypic within‐group variance–covariance matrix ( W ) for evolutionary inferences, but this approach has been criticized due to the lack of exact proportionality between genetic and phenotypic matrices. In this study, we examined the consequence, in terms of type I error rates, of replacing average G by W in a test of neutral evolution that measures the regression slope between among‐population variances and within‐population eigenvalues (the Ackermann and Cheverud [AC] test) using a simulation approach to generate random observations under genetic drift. Our results indicate that the type I error rates for the genetic drift test are acceptable when using W instead of average G when the matrix correlation between the ancestral G and P is higher than 0.6, the average character heritability is above 0.7, and the matrices share principal components. For less‐similar G and P matrices, the type I error rates would still be acceptable if the ratio between the number of generations since divergence and the effective population size (t/N_e) is smaller than 0.01 (large populations that diverged recently). When G is not known in real data, a simulation approach to estimate expected slopes for the AC test under genetic drift is discussed.     

Empirical Bayes inference of pairwise F(ST) and its distribution in the genome

Populations often have very complex hierarchical structure. Therefore, it is crucial in genetic monitoring and conservation biology to have a reliable estimate of the pattern of population subdivision. F(ST)'s for pairs of sampled localities or subpopulations are crucial statistics for the exploratory analysis of population structures, such as cluster analysis and multidimensional scaling. However, the estimation of F(ST) is not precise enough to reliably estimate the population structure and the extent of heterogeneity. This article proposes an empirical Bayes procedure to estimate locus-specific pairwise F(ST)'s. The posterior mean of the pairwise F(ST) can be interpreted as a shrinkage estimator, which reduces the variance of conventional estimators largely at the expense of a small bias. The global F(ST) of a population generally varies among loci in the genome. Our maximum-likelihood estimates of global F(ST)'s can be used as sufficient statistics to estimate the distribution of F(ST) in the genome. We demonstrate the efficacy and robustness of our model by simulation and by an analysis of the microsatellite allele frequencies of the Pacific herring. The heterogeneity of the global F(ST) in the genome is discussed on the basis of the estimated distribution of the global F(ST) for the herring and examples of human single nucleotide polymorphisms (SNPs).     

Patterns of DNA sequence diversity and genetic structure after a range expansion: lessons from the infinite-island model

It has been long recognized that population demographic expansions lead to distinctive features in the molecular diversity of populations. However, recent simulation results have suggested that a distinction could be made between a pure demographic expansion in an unsubdivided population, and a range expansion in a subdivided population, both leading to a large increase in the total number of the individuals. In order to better characterize the effect of a range expansion, I introduce a simple model of instantaneous expansion under an infinite-island model, under which I derive the distribution of the number of mutation differences between pairs of genes (the mismatch distribution), the heterozygosity, the average number of pairwise difference, and the fixation index F(ST). These derivations are checked against simulations, and are shown to lead to results qualitatively similar to those one would obtain after a range expansion in a 2-dimensional stepping-stone model. I then apply these results to estimate immigration rates in hunter-gather and post-Neolithic human populations from patterns of mitochondrial (mtDNA) diversity. Potential problems with this estimation procedure are also discussed.     

Genetic Structure Within and Among Populations of Saruma henryi, an Endangered Plant Endemic to China

The endangered perennial plant Saruma henryi Oliv. is endemic to China and has phylogenetic, ecological, and medicinal value. We used 10 microsatellite (SSR) loci to investigate genetic diversity and differentiation in 16 natural populations. Genetic diversity was high at the species level (H(E) = 0.9427, h = 0.9410, I = 3.0213) and low at the population level (H(E) = 0.4441, h = 0.4307, I = 0.6822). Pronounced genetic differentiation was detected among populations (G(ST) = 0.5428, F(ST) = 0.5524), in line with the limited among-population gene flow (Nm = 0.21). The significant isolation-by-distance pattern revealed by a Mantel test (r = 0.311, P = 0.001) suggested a clear geographic tendency in the distribution of genetic variability. Bayesian assignment and principal coordinates analyses supported the clustering of 16 populations into three groups. The present SSR results were also compared with previously published ISSR results. These results have significant implications for conservation of the species.     

Development of a panel of genome-wide ancestry informative markers to study admixture throughout the Americas

Most individuals throughout the Americas are admixed descendants of Native American, European, and African ancestors. Complex historical factors have resulted in varying proportions of ancestral contributions between individuals within and among ethnic groups. We developed a panel of 446 ancestry informative markers (AIMs) optimized to estimate ancestral proportions in individuals and populations throughout Latin America. We used genome-wide data from 953 individuals from diverse African, European, and Native American populations to select AIMs optimized for each of the three main continental populations that form the basis of modern Latin American populations. We selected markers on the basis of locus-specific branch length to be informative, well distributed throughout the genome, capable of being genotyped on widely available commercial platforms, and applicable throughout the Americas by minimizing within-continent heterogeneity. We then validated the panel in samples from four admixed populations by comparing ancestry estimates based on the AIMs panel to estimates based on genome-wide association study (GWAS) data. The panel provided balanced discriminatory power among the three ancestral populations and accurate estimates of individual ancestry proportions (R2 > 0.9 for ancestral components with significant between-subject variance). Finally, we genotyped samples from 18 populations from Latin America using the AIMs panel and estimated variability in ancestry within and between these populations. This panel and its reference genotype information will be useful resources to explore population history of admixture in Latin America and to correct for the potential effects of population stratification in admixed samples in the region.     

A Comparison of Isozyme and Quantitative Genetic Variation in Pinus Contorta Ssp. Latifolia by F(st)

We employed F-statistics to analyze quantitative and isozyme variation among five populations of Pinus contorta ssp. latifolia, a wind-pollinated outcrossing conifer with wide and continuous distribution in west North America. Estimates of population differentiation (F(ST)) for six quantitative traits were compared with the overall estimate of the differentiation (F*(ST)) from 19 isozymes that tested neutral to examine whether similar evolutionary processes were involved in morphological and isozyme differentiation. While the F(ST) estimates for specific gravity, stem diameter, stem height and branch length were significantly greater than the F*(ST) estimate, as judged from the 95% confidence intervals by bootstrapping, the F(ST) estimates for branch angle and branch diameter were indistinguishable from the F*(ST) estimate. Differentiation in stem height and stem diameter might reflect the inherent adaptation of the populations for rapid growth to escape suppression by neighboring plants during establishment and to regional differences in photoperiod, precipitation and temperature. In contrast, divergences in wood specific gravity and branch length might be correlated responses to population differentiation in stem growth. Possible bias in the estimation of F(ST) due to Hardy-Weinberg disequilibrium (F(IS) & 0), linkage disequilibrium, maternal effects and nonadditive genetic effects was discussed with special reference to P. contorta ssp. latifolia.     

F Statistics in Drosophila Buzzatii: Selection, Population Size and Inbreeding

Drosophila buzzatii is confined to reproducing in a well defined patchy environment consisting of rotting cactus cladodes which are ephemeral, permitting at most three generations. Flies emerging from such rots were used to estimate the additive genetic variance within rots and the genetic variance between rots for body size and also were electrophoresed to determine their genotypes at six polymorphic loci. F statistics were estimated from body size and allozyme data. The F(ST) derived from body size was significantly larger than the allozyme F(ST). It is proposed this is due to selective differentiation of body size. The allozyme F(ST) is used to estimate effective population size: 10 < N < 50. It is suggested that the regularly observed positive F(IS)'s could be due to partial sib mating, S. If so, the estimated lower bound is S = 0.258. Experiments are identified which could support or contradict these interpretations.     

A worldwide analysis of AG molecular diversity inferred from serology.

Ten population samples from different geographic origins were tested serologically for the AG polymorphism of human beta-lipoproteins. Their haplotype frequencies were used with previously published data to perform a wide analysis of AG genetic differentiations throughout the world. Coancestry coefficients were computed from weighted F(ST)s among populations by using a matrix of molecular distances among AG haplotypes, which is here determined on the basis of DNA studies. Coancestry coefficients derived from unweighted F(ST)s and more classical Prevosti distances were computed on the same data and used for a comparison. In all cases a highly significant correlation was found between genetics and geography on a worldwide scale, while the significance of the correlation with linguistics differed. A test of significance of the pairwise F(ST)s among populations also gave different results depending on whether the molecular distance matrix among AG haplotypes was included. Globally, this study shows that in spite of being highly significantly correlated to each other, different genetic distance measures can lead to different interpretations of the same data set. Moreover, the elucidation of the molecular models related to the presently known serological polymorphisms may represent an additional tool for analyzing such polymorphisms in human population genetics studies.     

Mississippians in motion? A population genetic analysis of interregional gene flow in West-Central Illinois

Population genetic and biological distance studies of Late Woodland and Mississippian populations from west-central Illinois have provided insight into a number of prehistoric demographic processes at the regional level. However, a formal analysis of diachronic interregional gene flow has not been attempted within a population genetics framework. In this study, cranial measurements of 489 individuals from 13 skeletal samples across the central and lower Illinois valleys are analyzed to address two central issues. First, the potential impact of Cahokia's decline and associated demographic events on the population structure of west-central Illinois Mississippians is examined. Second, the Mississippian and Late Woodland interregional migration patterns are compared to determine if geographic and/or cultural boundaries affected local population structure. Following Relethford and Blangero ([1990] Hum Biol 62:5-25), R matrix methods are utilized to calculate observed and expected phenotypic variances, minimum genetic distances, and F(ST) values in order to detect patterns of differential external gene flow over time. The results indicate that Late Woodland peoples had a larger sphere of biological interaction than Mississippians. In the Mississippian period, culturally imposed barriers paralleled geographic boundaries between regions such that the geographic distribution of biological variation closely adheres to a classic isolation-by-distance model. Further, intraregional population movement was a more significant contributor to Mississippian population structure than interregional gene flow, even during periods of sociopolitical strife. Small-scale intraregional shuffling is consistent with other recent studies of prehistoric Mississippian biocultural and geographic landscapes in the southeast United States.     

Genetic Differentiation and Estimation of Gene Flow from F-Statistics under Isolation by Distance

I reexamine the use of isolation by distance models as a basis for the estimation of demographic parameters from measures of population subdivision. To that aim, I first provide results for values of F-statistics in one-dimensional models and coalescence times in two-dimensional models, and make more precise earlier results for F-statistics in two-dimensional models and coalescence times in one-dimensional models. Based on these results, I propose a method of data analysis involving the regression of F(ST)/(1 - F(ST)) estimates for pairs of subpopulations on geographic distance for populations along linear habitats or logarithm of distance for populations in two-dimensional habitats. This regression provides in principle an estimate of the product of population density and second moment of parental axial distance. In two cases where comparison to direct estimates is possible, the method proposed here is more satisfactory than previous indirect methods.     

Anthropometric variation among Bering Sea natives

Recent research indicates that anthropometrics can be used to study microevolutionary forces acting on humans. We examine the use of morphological traits in reconstructing the population history of Aleuts and Eskimos of the Bering Sea. From 1979 to 1981, W. S. Laughlin measured a sample of St. Lawrence Island Eskimos and Pribilof Island Aleuts. These samples included adult participants from St. George and St. Paul in the Pribilof Islands and from Gambell and Savoonga on St. Lawrence Island. The Relethford-Blangero method was used to examine the phylogenetic relationship between Aleuts and Eskimos. Anthropometric measurements for Native North Americans (measured by Boas and a team of trained anthropometrists in 1890-1904) and Native Mesoamericans (compiled from the literature for 1898-1952) were used for comparison. A principal components analysis of means for measurements and a neighbor-joining tree were constructed using Euclidean distances. All these tests revealed the same strong relationship among the focus populations. The R matrix from the Relethford-Blangero method clusters Aleuts and Eskimos separately and accounts for 97.3% of the variation in the data. Phenotypic variation within the population is minimal and therefore minimum F(ST) values are low. Genetic distances were compared to a Euclidean distance matrix of anthropometric measurements using a Mantel test and gave a high but not significant correlation. Our results provide evidence of a close phylogenetic relationship between Aleut and Eskimo populations in the Bering Sea. However, it is apparent that history has affected the relationship among the populations. Despite previous findings of higher European admixture in Gambell (based on blood group markers) than in Savoonga, Savoonga has greater within-group variation in anthropometric measurements. Anthropometrics reveal a close relationship between Gambell and St. Paul as a result of European admixture. The St. George population was the most divergent of the populations, indicating that it diverged from the Eskimos and St. Paul because of the compounding effects of genetic drift and limited European gene flow. These findings are in agreement with previous anthropometric and genetic studies of the Aleut and Eskimo populations and support the utility of anthropometrics in inferring population history and structure.     

Genetic structure among continental and island populations of gyrfalcons

Little is known about the possible influence that past glacial events have had on the phylogeography and population structure of avian predators in the Arctic and sub-Arctic. In this study, we use microsatellite and mitochondrial control region DNA variation to investigate the population genetic structure of gyrfalcons (Falco rusticolus) throughout a large portion of their circumpolar distribution. In most locations sampled, the mtDNA data revealed little geographic structure; however, five out of eight mtDNA haplotypes were unique to a particular geographic area (Greenland, Iceland, or Alaska) and the Iceland population differed from others based on haplotype frequency differences (F(ST)). With the microsatellite results, significant population structure (F(ST), principal components analysis, and cluster analysis) was observed identifying Greenland and Iceland as separate populations, while Norway, Alaska and Canada were identified as a single population consistent with contemporary gene flow across Russia. Within Greenland, differing levels of gene flow between western and eastern sampling locations was indicated with apparent asymmetric dispersal in western Greenland from north to south. This dispersal bias is in agreement with the distribution of plumage colour variants with white gyrfalcons in much higher proportion in northern Greenland. Lastly, because the mtDNA control region sequence differed by only one to four nucleotides from a common haplotype among all gyrfalcons, we infer that the observed microsatellite population genetic structure has developed since the last glacial maximum. This conclusion is further supported by our finding that a closely related species, the saker falcon (Falco cherrug), has greater genetic heterogeneity, including mtDNA haplotypes differing by 1-16 nucleotide substitutions from a common gyrfalcon haplotype. This is consistent with gyrfalcons having expanded rapidly from a single glacial-age refugium to their current circumpolar distribution. Additional sampling of gyrfalcons from Fennoscandia and Russia throughout Siberia is necessary to test putative gene flow between Norway and Alaska and Canada as suggested by this study.     

Effect of anthropogenic landscape features on population genetic differentiation of Przewalski's gazelle: main role of human settlement

Anthropogenic landscapes influence evolutionary processes such as population genetic differentiation, however, not every type of landscape features exert the same effect on a species, hence it is necessary to estimate their relative effect for species management and conservation. Przewalski's gazelle (Procapra przewalskii), which inhabits a human-altered area on Qinghai-Tibet Plateau, is one of the most endangered antelope species in the world. Here, we report a landscape genetic study on Przewalski's gazelle. We used skin and fecal samples of 169 wild gazelles collected from nine populations and thirteen microsatellite markers to assess the genetic effect of anthropogenic landscape features on this species. For comparison, the genetic effect of geographical distance and topography were also evaluated. We found significant genetic differentiation, six genetic groups and restricted dispersal pattern in Przewalski's gazelle. Topography, human settlement and road appear to be responsible for observed genetic differentiation as they were significantly correlated with both genetic distance measures [F(ST)/(1-F(ST)) and F'(ST)/(1-F'(ST))] in Mantel tests. IBD (isolation by distance) was also inferred as a significant factor in Mantel tests when genetic distance was measured as F(ST)/(1-F(ST)). However, using partial Mantel tests, AIC(c) calculations, causal modeling and AMOVA analysis, we found that human settlement was the main factor shaping current genetic differentiation among those tested. Altogether, our results reveal the relative influence of geographical distance, topography and three anthropogenic landscape-type on population genetic differentiation of Przewalski's gazelle and provide useful information for conservation measures on this endangered species.     

Maximum-likelihood estimation of admixture proportions from genetic data

For an admixed population, an important question is how much genetic contribution comes from each parental population. Several methods have been developed to estimate such admixture proportions, using data on genetic markers sampled from parental and admixed populations. In this study, I propose a likelihood method to estimate jointly the admixture proportions, the genetic drift that occurred to the admixed population and each parental population during the period between the hybridization and sampling events, and the genetic drift in each ancestral population within the interval between their split and hybridization. The results from extensive simulations using various combinations of relevant parameter values show that in general much more accurate and precise estimates of admixture proportions are obtained from the likelihood method than from previous methods. The likelihood method also yields reasonable estimates of genetic drift that occurred to each population, which translate into relative effective sizes (N(e)) or absolute average N(e)'s if the times when the relevant events (such as population split, admixture, and sampling) occurred are known. The proposed likelihood method also has features such as relatively low computational requirement compared with previous ones, flexibility for admixture models, and marker types. In particular, it allows for missing data from a contributing parental population. The method is applied to a human data set and a wolflike canids data set, and the results obtained are discussed in comparison with those from other estimators and from previous studies.     

If FST does not measure neutral genetic differentiation,then comparing it with QST is misleading. Or is it?

The comparison between neutral genetic differentiation (F(ST) ) and quantitative genetic differentiation (Q(ST) ) is commonly used to test for signatures of selection in population divergence. However, there is an ongoing discussion about what F(ST) actually measures, even resulting in some alternative metrics to express neutral genetic differentiation. If there is a problem with F(ST) , this could have repercussions for its comparison with Q(ST) as well. We show that as the mutation rate of the neutral marker increases, F(ST) decreases: a higher within-population heterozygosity (He) yields a lower F(ST) value. However, the same is true for Q(ST) : a higher mutation rate for the underlying QTL also results in a lower Q(ST) estimate. The effect of mutation rate is equivalent in Q(ST) and F(ST) . Hence, the comparison between Q(ST) and F(ST) remains valid, if one uses neutral markers whose mutation rates are not too high compared to those of quantitative traits. Usage of highly variable neutral markers such as hypervariable microsatellites can lead to serious biases and the incorrect inference that divergent selection has acted on populations. Much of the discussion on F(ST) seems to stem from the misunderstanding that it measures the differentiation of populations, whereas it actually measures the fixation of alleles. In their capacity as measures of population differentiation, Hedrick's G'(ST) and Jost's D reach their maximum value of 1 when populations do not share alleles even when there remains variation within populations, which invalidates them for comparisons with Q(ST) .     

Principal components analysis of population admixture

With the availability of high-density genotype information, principal components analysis (PCA) is now routinely used to detect and quantify the genetic structure of populations in both population genetics and genetic epidemiology. An important issue is how to make appropriate and correct inferences about population relationships from the results of PCA, especially when admixed individuals are included in the analysis. We extend our recently developed theoretical formulation of PCA to allow for admixed populations. Because the sampled individuals are treated as features, our generalized formulation of PCA directly relates the pattern of the scatter plot of the top eigenvectors to the admixture proportions and parameters reflecting the population relationships, and thus can provide valuable guidance on how to properly interpret the results of PCA in practice. Using our formulation, we theoretically justify the diagnostic of two-way admixture. More importantly, our theoretical investigations based on the proposed formulation yield a diagnostic of multi-way admixture. For instance, we found that admixed individuals with three parental populations are distributed inside the triangle formed by their parental populations and divide the triangle into three smaller triangles whose areas have the same proportions in the big triangle as the corresponding admixture proportions. We tested and illustrated these findings using simulated data and data from HapMap III and the Human Genome Diversity Project.     

Correcting for measurement error in individual ancestry estimates in structured association tests

We present theoretical explanations and show through simulation that the individual admixture proportion estimates obtained by using ancestry informative markers should be seen as an error-contaminated measurement of the underlying individual ancestry proportion. These estimates can be used in structured association tests as a control variable to limit type I error inflation or reduce loss of power due to population stratification observed in studies of admixed populations. However, the inclusion of such error-containing variables as covariates in regression models can bias parameter estimates and reduce ability to control for the confounding effect of admixture in genetic association tests. Measurement error correction methods offer a way to overcome this problem but require an a priori estimate of the measurement error variance. We show how an upper bound of this variance can be obtained, present four measurement error correction methods that are applicable to this problem, and conduct a simulation study to compare their utility in the case where the admixed population results from the intermating between two ancestral populations. Our results show that the quadratic measurement error correction (QMEC) method performs better than the other methods and maintains the type I error to its nominal level.     

Phylogeography, intraspecific structure and sex-biased dispersal of Dall's porpoise, Phocoenoides dalli, revealed by mitochondrial and microsatellite DNA analyses

Mitochondrial DNA (mtDNA) control-region sequences and microsatellite loci length polymorphisms were used to estimate phylogeographical patterns (historical patterns underlying contemporary distribution), intraspecific population structure and gender-biased dispersal of Phocoenoides dalli dalli across its entire range. One-hundred and thirteen animals from several geographical strata were sequenced over 379 bp of mtDNA, resulting in 58 mtDNA haplotypes. Analysis using F(ST) values (based on haplotype frequencies) and phi(ST) values (based on frequencies and genetic distances between haplotypes) yielded statistically significant separation (bootstrap values P < 0.05) among most of the stocks currently used for management purposes. A minimum spanning network of haplotypes showed two very distinctive clusters, differentially occupied by western and eastern populations, with some common widespread haplotypes. This suggests some degree of phyletic radiation from west to east, superimposed on gene flow. Highly male-biased migration was detected for several population comparisons. Nuclear microsatellite DNA markers (119 individuals and six loci) provided additional support for population subdivision and gender-biased dispersal detected in the mtDNA sequences. Analysis using F(ST) values (based on allelic frequencies) yielded statistically significant separation between some, but not all, populations distinguished by mtDNA analysis. R(ST) values (based on frequencies of and genetic distance between alleles) showed no statistically significant subdivision. Again, highly male-biased dispersal was detected for all population comparisons, suggesting, together with morphological and reproductive data, the existence of sexual selection. Our molecular results argue for nine distinct dalli-type populations that should be treated as separate units for management purposes.