In population genetics, a population is usually related to random mating. Random mating leads to homogeneity. Homogeneity makes it possible to define genotype and allele frequencies.

In Sewall Wright's four-volume book, Evolution and the Genetics of Population, Vol 2 is called "The Theory of Gene Frequencies" (1969).

Inhomogeneity (heterogeneity) is one the two major difficutlies in association analysis.

Various types of heterogeneity in case group:

1. Locus heterogeneity: genes at more than one chromosomal locations are related to the disease
   This is a problem for both linkage analysis and association analysis
2. Allelic heterogeneity: even if only one gene is related to the disease, different forms of mutation at different within-the-gene locations damage the gene.
   This is not a problem for linkage analysis, but a problem for association analysis

Various types of heterogeneity in control group:

• Different ethnic groups, sub-populations, migrations
• Causes of heterogeneity: genetic drift (allele frequency changes by chance in an isolated population), mutation (historical event being frozen in the individual), selection (certain allele takes a larger proportion of the population for some reason)

See: http://en.wikipedia.org/wiki/Simpson's_paradox

Considering the following two populations:

In general, if severe heterogeneity is not removed, an association analysis result cannot be trusted.

The heterogeneity in this example is very easy to detect: the control group allele frequency p_a is changed from 90% in population-1 to 30% in population-2. Similarly, the case group p_a is changed from 60% in population-1 to 10% in population-2.

• Do not combine samples from different ethnic groups (In particularly, Africans tend to have different allele frequencies as Caucasians)
• If samples of different ethnic backgrounds are included, there are program to identify them as being different from the rest of the samples: e.g. CHECKHET program: http://www.mds.qmw.ac.uk/statgen/dcurtis/software.html
• While individual person from different ethnic group can be separated ("stratified samples"), ancestral components from different ethnic group in one person cannot be separated ("admixtured samples").
• Keep the heterogeneity as it is, but "correct" it. e.g. using unlinked markers to confirm the different proportion of the "neutral" marker allele frequencies in case and control group.

  JK Pritchard, NA Rosenberg (1999), "Use of unlinked genetic markers to detect population stratification in association studies", American Journal of Human Genetics, 65:220-228. PDF
  B Devlin, K Roeder (1999), "Genomic control for association studies", Biometrics, 55:997-1004.
  JK Pritchard, M Stephens, NA Rosenberg, P Donnelly (2000), "Association mapping in structured populations", American Journal of Human Genetics, 67:170-181. PDF

• Use the untransmitted allele as a "pseudo control" allele, whereas the allele actually transmitted to an affected offspring as the "case allele". This is the so-called "family-based association"
• Pick samples from an isolated population so that both the locus heterogeneity and the allelic heterogeneity in the samples is reduced.

The most famous examples are the psychiatric disorders. There is no other solution but to restrict/purify/stratify the phenotype definition.

• Must have an affected offspring and his/her parents
• If we know one of the parents is not the source of the mutation, one parent is enough.
• Since usually we do not know which parent contributes to the disease susceptibility mutation, both parents are usually used: a collection of trios
• The advantage of family-based association: both the transmitted "case allele" and the untransmitted "pseudo control allele" are from the same parent, so same source. There is no population stratification. [COMMENT: what if the parent him/herself is admixtured?]
• The disadvantage includes: hard to get parents' DNA; three persons need genotyping as compared to two persons in a case-control analysis (more costly).
• New mathematics: since the transmitted and untransmitted alleles always come in a pair, it is the matched case-control analysis.
• For matched case-control analysis, if the case allele is the same as the control allele, this pair does not contribute. As a result, only when the parent of the affected offspring is heterozygous, can this pair be included in the sample.

Measuring the level/amount of subpopulations/stratifications:
Wright's F-statistics or Wright's F_st value.

Subpopulation structure increases the homozygosity frequency, but decrease the heterozygosity frequency. (Inbreeding has the similar effect.)
F_st is defined as 1- 0.25/0.255 = 0.01960784...

Another example: p₁=0.1, p₂=0.4, p = (0.1+0.4)/2= 0.25. The heterozygosity considering the subpopulation structure is (0.18+ 0.48)/2=0.33, whereas the heterozygosity without considering the subpopulation is 2*0.25*0.75=0.375.
F_st = 1- 0.33/0.375= 0.12.

If allele "a" remains as the minor allele, the maximum F_st is 0.33333...: p₁=0, p₂=0.5, p = (0.1+0.4)/2= 0.25. The heterozygosity considering the subpopulation structure is (0+ 0.5)/2=0.25, whereas the heterozygosity without considering the subpopulation is 2*0.25*0.75=0.375. And F_st = 1- 0.25/0.375= 1-2/3=1/3.

If the allele "a" is allowed to switch from a minor allele in one subpopulation to a major allele in another subpopulation, maximum F_st can be 1.

Examples: three subpopulations: European-Americans, African-Americans, Asians of Japanese/Chinese ancestry: F_st =0.145.
two subpopulations: Japanese and Chinese F_st =0.013.
33 Irish counties : F_st =0.0132.
27 Finnish districts : F_st =0.005.
11 Icelandic regions : F_st =0.00338, 0.00048, 0.00017, 0.00137.

[J Marchini, LR Cardon, MS Phillips, P Donnelly (2004), "The effects of human population structure on large genetic association studies", Nature Genetics, 36:512-517. PDF
A Helgason, B Yngvadóttir, B Hrafnkelsson, J Gulcher, KStefánsson (2005), "An Icelandic example of the impact of population structure on association studies", Nature Genetics, 37:90-95. PDF ]

Different ways to describe an isolated population:

• Not many migrations that may introduce a difference allele frequency
• Fewer founders
• The fact that there are fewer founders can be due to the fact that this population is formed by an expansion followed by a "bottleneck". Bottleneck is to reduce the number of founders. Expansion is an indication that all samples collected can be traced to the few founders (so expansion by natural growth, not expansion by immigration).

Why? Suppose a disease susceptibility gene has an effect only if an environmental factor is present (e.g. smoking):

The simplest situation ("hydrogene atom") for gene-environment interaction is the 2-by-2-by-3 table:

See: LD Botto, MJ Khoury (2001), "Facing the challenge of gene-environment interaction: the two-by-four table and beyond", American Journal of Epidemiology, 153:1016-1020. PDF