The power of the Transmission Disequilibrium Test in the presence of population stratification

Abstract

Introduction

Case–control analysis using singletons has been shown to be generally more powerful than any other design per genotyped individual for detecting genes associated with disease.¹ However, this design is vulnerable to excessive false-positive findings in the presence of population stratification.^{2, 3, 4, 5, 6, 7} If a study population comprises two subpopulations, and the disease prevalence is greater in one subpopulation compared to the other, then cases will typically be over-sampled from the subpopulation with higher disease prevalence. Spurious associations will be observed between the disease and any genetic marker where the allele frequencies differ in both subpopulations.

The Transmission Disequilibrium Test (TDT) proposed by Spielman et al⁸ compares the rate of transmission of each allele from a heterozygous parent to an affected offspring. The TDT maintains the desired Type I error rate in the presence of population stratification, as non-transmitted parental alleles from heterozygous parents serve, in effect, as the control population. Therefore, the power of the TDT is heavily dependent on the number of heterozygotes (informative parents), who may or may not transmit the allele of interest. It is well appreciated that population stratification results in a loss of heterozygosity compared to that expected assuming Hardy–Weinberg Equilibrium (HWE). This is known as the Wahlund Effect, which results in less informative parents for the TDT. However, no guidelines currently exist for sample size calculations for the TDT in the presence of population stratification.

In this paper, we lay out a method of estimating the two parent genotype patterns (mating types) seen in the presence of population stratification using the parental allele frequencies and Wright's coefficient of inbreeding F. The estimates of the mating types are then used for power calculations using the method provided by Knapp.⁹ We then compare the sample sizes assumed using HWE to those calculated using our new method and show how these differences arise.

Methods

The effect of population stratification on mating type frequencies

Let Δ_i be the difference between the allele A frequency in subpopulation i and the allele A frequency in the entire population, where Δ_i=p_i−$\bar{p}$, for i=1, 2,…G. Furthermore, define the central moments of the allele frequency distribution, for j=1, 2, 3, 4. Therefore,

Then,

This yields

Similarly,

This yields

All of the six mating types can now be re-written in terms of the average risk allele frequency $\bar{p}$, Wright's coefficient of inbreeding F, ϑ₃ and ϑ₄ as shown in Table 1.

Table 1

Mating type frequencies in the presence of population stratification parameterized in terms of the central moments of the allele frequency distribution

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Change in the relative proportion of informative families because of population stratification

Three of the six mating types are informative for the TDT because they contain at least one heterozygous parent – AA × AB, AB × AB and AB × BB (m₂, m₄ and m₅). However, as shown above, Wright's coefficient of inbreeding, F, is insufficient to appropriately calculate all mating type frequencies without ϑ₃ and ϑ₄. Under HWE the relative proportions of the informative mating types m₂:m₄:m₅ are clearly $\bar{p}$²:$\bar{p}$ (1−$\bar{p}$):(1−$\bar{p}$)²; however, these proportions no longer hold in the presence of population stratification. To illustrate the changes in the relative proportions of the mating types, we consider a study population comprising two equal-sized subpopulations, where F=0 and 0.10, ϑ₃=0 and ϑ₄ is allowed to vary for a multiplicative MOI where γ=2 (only $\bar{p}$, F and ϑ₃ are required to completely describe a two-subpopulation model). The changes in the relative proportions of the informative mating types AA × AB, AB × AB and AB × BB are shown in Figure 1.

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Figure 1

Relative proportion of the informative mating types for a multiplicative MOI where γ=2.

The power of the TDT and sample size calculations in the presence of population stratification

The TDT tests the null hypothesis of no association between a marker and disease in the presence of linkage. Knapp provided a method for reliably calculating the power of the TDT in a homogeneous population assuming HWE for affected child trios (ACTs).⁹ We retain his symbols for ease of comparison with our method. At the heart of Knapp's method is the characterization of family types (genotypes of both parents as well as the affected offspring) for the TDT and the calculation of each family type probability (s₁, s₂,…, s₁₀), as shown in Table 2. Seven family types are informative for the TDT (s₁, s₂,…, s₇) and Knapp showed how their multinomial frequencies could be calculated based only on ψ₁, ψ₂ and $\bar{p}$.⁹ However, if population stratification exists, the average frequency of the risk allele A in the stratified population ($\bar{p}$) is not sufficient to calculate the mating type proportions. The true family type proportions (s^*₁, s^*₂,…, s^*₁₀) can be calculated from the correct mating type proportions based on the method described earlier (using $\bar{p}$, F, ϑ₃ and ϑ₄) and are shown in Table 2. However, ϑ₃ and ϑ₄ are almost always unknown, and though there are several published papers about F values expected in practice, there are no published reports of ϑ₃ and ϑ₄ values expected in practice. Yasuda¹⁴ showed that the higher central moments (terms in ϑ₃ and ϑ₄) can be ignored if $\bar{p}$>F and 1−$\bar{p}$>F. Therefore, (s₁, s₂,…, s₇) can be estimated by (ŝ^*₁, ŝ^*₂,…, ŝ^*₇) using $\bar{p}$ and F as shown in Table 2 and these estimates can then be used in power calculations. GRRs are set assuming γ=2.0 and 1.5 for multiplicative, additive, dominant and recessive models. To evaluate our method for estimating sample sizes, we considered a stratified study population comprising two smaller subpopulations. In the first example, 70% of the trios are from subpopulation 1, and 30% of the trios are derived from subpopulation 2. In the second example, the study population comprises equal proportions of trios from both subpopulations. We then calculated the true sample size requirements to achieve 80% power, using a Type I error rate of α=1 × 10⁻⁷ at various values of the average allele frequency, $\bar{p}$ (0.1, 0.3, 0.5 and 0.8), and at various levels of population stratification measured by F (0.01, 0.05 and 0.10). The true sample size estimates were compared to our estimates using only $\bar{p}$ and F, as well as compared to the sample size calculations assuming HWE. These sample size calculations are shown in Tables 3 and ​and4.4. A Type I error rate of α=1 × 10⁻⁷ is used for comparison with Knapp,⁹ and for comparision with sample size estimates from genomewide association studies.

Table 2

Family type probabilities given affected child

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Table 3

Sample size necessary to gain 80% power in TDT with singletons (α=10⁻⁷, γ=2), comparing Knapp's first approximation assuming HWE to the sample size estimates assuming population stratification

				Multiplicative MOI			Additive MOI			Recessive MOI			Dominant MOI
$\bar{p}$	F	ϑ3	ϑ4	A	B	C	A	B	C	A	B	C	A	B	C
0.1	0.01	2.3 × 10−5	1.4 × 10−6	689	692 (100.4%)	692	689	692 (100.4%)	692	45 071	39 168 (86.9%)	39 354	949	978 (103.1%)	977
		0	8.1 × 10−7			692			692			39 168			978
0.1	0.05	2.6 × 10−4	3.5 × 10−5	689	702 (101.9%)	704	689	702 (101.9%)	702	45 071	24 102 (53.5%)	25 141	949	1110 (117.0%)	1101
		0	2.0 × 10−5			702			732			24 102			1110
0.1	0.10	7.5 × 10−4	1.4 × 10−4	689	714 (103.6%)	720	689	714 (103.6%)	720	45 071	14 797 (32.8%)	16 273	949	1324 (139.5%)	1292
		0	8.1 × 10−5			714			714			14 798			1324
0.3	0.01	8.4 × 10−5	7.8 × 10−6	349	352 (100.9%)	352	349	352 (100.9%)	352	2546	2513 (98.7%)	2519	907	927 (102.2%)	925
		0	4.4 × 10−6			352			352			2513			927
0.3	0.05	9.4 × 10−4	1.9 × 10−4	349	364 (104.3%)	365	349	364 (104.3%)	365	2546	2383 (93.6%)	2449	907	1015 (111.9%)	999
		0	1.1 × 10−4			364			364			2383			1015
0.3	0.10	2.7 × 10−3	7.8 × 10−4	349	380 (108.9%)	384	349	380 (108.9%)	384	2546	2227 (87.5%)	2406	907	1150 (126.8%)	1097
		0	4.4 × 10−4			380			380			2227			1149
0.5	0.01	1.1 × 10−4	1.1 × 10−5	338	342 (101.2)	342	338	342 (101.2)	342	957	969 (101.3%)	971	1839	1855 (100.9%)	1852
		0	6.3 × 10−6			342			342			969			1855
0.5	0.05	1.2 × 10−3	2.8 × 10−4	338	358 (105.9%)	359	338	358 (105.9%)	359	957	1017 (106.3%)	1037	1839	1923 (104.6%)	1881
		0	1.6 × 10−4			358			358			1017			1923
0.5	0.10	3.5 × 10−3	1.1 × 10−3	338	380 (112.4%)	384	338	380 (112.4%)	384	957	1084 (113.3%)	1147	1839	2017 (109.7%)	1889
		0	6.3 × 10−4			380			380			1084			2017
0.8	0.01	5.6 × 10−5	4.5 × 10−6	634	643 (101.4%)	643	634	643 101.4%)	643	851	872 (102.5%)	872	21 998	20 879 (94.9%)	20 879
		0	2.6 × 10−6			643			643			872			20 879
0.8	0.05	6.2 × 10−4	1.1 × 10−4	634	682 (107.6%)	683	634	682 (107.6%)	683	851	965 (113.4%)	975	21 998	17 047 (77.5%)	16 435
		0	6.4 × 10−5			682			682			965			17 047
0.8	0.10	1.8 × 10−3	4.5 × 10−4	634	736 (116.1%)	741	634	736 (116.1%)	741	851	1111 (130.6%)	1145	21 998	13 379 (60.8%)	12 178
		0	2.6 × 10−4			736			736			1111			13 378

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A is Knapp's first approximation assuming HWE.

B is estimate in presence of population stratification, based on $\bar{p}$ and F only.

C is true sample size requirement.

Table 4

Sample size necessary to gain 80% power in TDT with singletons (α=10⁻⁷, γ=1.5), comparing Knapp's first approximation assuming HWE to the sample size estimates assuming population stratification

				Multiplicative MOI			Additive MOI			Recessive MOI			Dominant MOI
$\bar{p}$	F	ϑ3	ϑ4	A	B	C	A	B	C	A	B	C	A	B	C
0.1	0.01	2.3 × 10−5	1.4 × 10−6	2210	2224 (100.6%)	2225	1755	1738 (99.0%)	1739	174 695	151 448 (86.7%)	152 181	2897	2983 (103.0%)	2981
		0	8.1 × 10−7			2224			1738			151 448			2983
0.1	0.05	2.6 × 10−4	3.5 × 10−5	2210	2283 (103.3%)	2287	1755	1671 (95.2%)	1683	174 695	92 276 (52.8%)	96 328	2897	3375 (116.5%)	3349
		0	2.0 × 10−5			2283			1671			92 277			3375
0.1	0.10	7.5 × 10−4	1.4 × 10−4	2210	2361 (106.8%)	2371	1755	1591 (90.7%)	1623	174 695	55 922 (32.0%)	61 637	2897	4011 (138.5%)	3916
		0	8.1 × 10−5			2361			1591			55 924			4011
0.3	0.01	8.4 × 10−5	7.8 × 10−6	1037	1046 (100.9%)	1047	608	611 (100.5%)	611	9097	8963 (98.5%)	8986	2440	2493 (102.2%)	2490
		0	4.4 × 10−6			1046			611			8963			2493
0.3	0.05	9.4 × 10−4	1.9 × 10−4	1037	1084 (104.5%)	1086	608	621 (102.1%)	626	9097	8443 (92.8%)	8686	2440	2730 (111.9%)	2690
		0	1.1 × 10−4			1084			621			8443			2730
0.3	0.10	2.7 × 10−3	7.8 × 10−4	1037	1135 (109.5%)	1142	608	633 (104.1%)	649	9097	7821 (86.0%)	8473	2440	3092 (126.7%)	2956
		0	4.4 × 10−4			1135			633			7822			3092
0.5	0.01	1.1 × 10−4	1.1 × 10−5	947	957 (101.1%)	957	464	469 (101.1%)	470	3099	3133 (101.1%)	3138	4568	4610 (100.9%)	4602
		0	6.3 × 10−6			957			469			3133			4610
0.5	0.05	1.2 × 10−3	2.8 × 10−4	947	999 (105.5%)	1001	464	492 (106.0%)	495	3099	3279 (105.8%)	3343	4568	4789 (104.8%)	4687
		0	1.6 × 10−4			999			492			3279			4788
0.5	0.10	3.5 × 10−3	1.1 × 10−3	947	1057 (111.6%)	1063	464	523 (112.7%)	535	3099	3479 (112.3%)	3690	4568	5034 (110.2%)	4724
		0	6.3 × 10−4			1057			523			3479			5033
0.8	0.01	5.6 × 10−5	4.5 × 10−6	1658	1678 (101.2%)	1679	698	710 (101.7%)	710	2356	2415 (102.5%)	2417	50 826	48 284 (95.0%)	48 116
		0	2.6 × 10−6			1678			710			2415			48 284
0.8	0.05	6.2 × 10−4	1.1 × 10−4	1658	1767 (106.6%)	1769	698	763 (109.3%)	766	2356	2680 (113.8%)	2707	50 826	39 566 (77.8%)	38 166
		0	6.4 × 10−5			1767			763			2680			39 566
0.8	0.10	1.8 × 10−3	4.5 × 10−4	1658	1891 (114.1%)	1899	698	840 (120.3%)	852	2356	3095 (131.4%)	3194	50 826	31 208 (61.4%)	28 448
		0	2.6 × 10−4			1891			840			3095			31 206

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A is Knapp's first approximation assuming HWE.

B is estimate in presence of population stratification, based on $\bar{p}$ and F only.

C is true sample size requirement.

Results

Population stratification alters the frequency of the mating types when compared to those calculated assuming HWE. The change in the distribution of the mating type frequencies directly changes the family type frequencies (mother–father–offspring genotype combinations). When HWE is assumed for TDT power calculations, the expected value and variance of the test statistic under the alternative hypothesis as well as the variance of the test statistic under the null hypothesis differ from that calculated when accounting for population stratification. The variance of the TDT in the presence of stratification under the null hypothesis could be larger or smaller than that calculated assuming HWE depending on the mating type parameters ($\bar{p}$, F, ϑ₃, ϑ₄). The difference between the variance of the TDT under the alternative hypothesis accounting for population stratification and the variance of the TDT assuming HWE depends on the model (ψ₂, ψ₁) and mating type parameters ($\bar{p}$, F, ϑ₃, ϑ₄). The current methods existing for calculating TDT sample sizes ignore the fact that parental genotypes used in the TDT statistic are no longer in HWE if there is population stratification. Furthermore, these methods ignore the fact that the parents cannot be considered independent as population stratification induces non-random mating, when considered on a whole-population basis.

Knapp showed that the power of the TDT is a function of the genetic model, genotypic risk parameters and the risk allele frequency.⁹ Our model for TDT sample size calculations in the presence of population stratification based on estimations of the mating type frequencies using the average allele frequency, $\bar{p}$ and Wright's coefficient of inbreeding F allows sampling from any number of subpopulations with any ascertainment scheme, and is generalizeable to any stratified study population. In addition, we show that the change in the power of the TDT in a stratified population is due to three main factors: (1) the loss in heterozygosity, which is 100 (1−F)% of that predicted assuming HWE; (2) the change in the distribution of mating type frequencies; and (3) the change in the relative proportion of the informative mating types to each other. These factors should not be ignored if there is the possibility of even mild population stratification.

Interestingly, and somewhat contradictory to initial expectation, population stratification does not always result in decreased power of the TDT because of the loss in heterozygosity. Tables 3 and ​and44 show that the power of the TDT can be increased in the presence of population stratification because of the change in the relative proportion of the informative mating types and the disease model. If the disease model is a multiplicative MOI, then sample sizes calculated assuming HWE are generally smaller than those actually required. For a dominant, additive or recessive model, the relative frequencies of the AA × AB, AB × BB and AB × AB mating types dictate the change in sample size requirements as the relative information content for these three mating types may not be in the ratio 1:1:2. For example, if the disease is inherited in a dominant MOI, the mating type AA × AB has no information and the mating type AB × BB has the most information.

In a stratified population, the risk allele frequency and mating type frequencies in a single subpopulation may cause subjects from that subpopulation to be more informative than subjects from other subpopulations. This results in one subpopulation that heavily influences the overall TDT sample size calculations. The subjects from other subpopulations may not be as informative and add very little to the overall TDT statistic. This suggests that it may be useful to partition a study population into its component subpopulations prior to analysis.

Larger values of Wright's F were correlated larger discrepancies in the sample size requirements compared to those calculated assuming HWE. For example, in Table 3 where F=0.01 and γ=2, estimates of the sample size required varied from 3% larger to approximately 15% smaller than the sample size calculated using Knapp's first approximation. However, when F=0.1 and γ=2, estimates of the sample size required varied from 39% larger to about 68% smaller than the sample size calculated using Knapp's first approximation.

Wright's F cannot capture all the information about population stratification pertaining to the distortion in mating type frequencies; however, reasonable estimates of the mating type frequencies can be made using F and average allele frequency $\bar{p}$ when F≤min ($\bar{p}$, 1−$\bar{p}$).

To estimate sample sizes for the TDT we ignore the third and higher order risk allele frequency moments (ϑ₃ and ϑ₄). However, small changes in ϑ₃ and ϑ₄ are important and result in sample size discrepancies when our estimates are compared to the true sample size estimates for a fixed F, as seen in Tables 3 and ​and44.

One potential limitation of our method is that it does not take into account population admixture, as we assume both parents are sampled from the same subpopulation. Population admixture occurs when the study population comprises multiple subpopulations, but there is mating within and between subpopulations. Population admixture is complex, and the rate of admixture depends on socioeconomic, racial, ethnic, linguistic, migratory factors and several other factors. Additional work is needed to assess how the power of the TDT changes in recently admixed populations.

Knapp uses the disease prevalence as a normalizing factor so that the sum of the probabilities of all family types with an affected offspring sums to 1.⁹ We note that the disease prevalence, calculated assuming HWE denoted by R, is different from the disease prevalence $\bar{R}$ calculated assuming population stratification. The disease prevalence in the stratified population, is the disease prevalence in subpopulation i. We also note that the normalization factor, $\bar{R}$ in the stratified population is greater in magnitude than the normalization factor in the population in HWE, R, if ψ₂+1>2ψ₁, however, if ψ₂+1<2ψ₁ the normalization factor in the stratified population, $\bar{R}$ is smaller in magnitude than that of the population in HWE. The difference in the normalization factors between the stratified population and the population in HWE, F$\bar{p}$$\bar{q}$ (ψ₂−2ψ₁+1), depends on the genetic model chosen and the GRRs. The difference is due to the altered proportion of risk genotypes because of population stratification.

Distortion in the mating type frequencies can be seen when there is underlying population stratification. In a randomly mating population in HWE, F=ϑ₃=ϑ₄=0. Distortion in the mating type frequencies can also be observed in the parents of an affected child. For example, the parents of a child with a rare recessive disease will most likely be both heterozygous at the disease locus (AB × AB). Similarly, for a rare dominant disorder, the parental genotypes of affected children are most likely a heterozygote and a wild type (AB × BB). This distortion in mating type frequencies becomes most apparent when the risk allele is rare (less than 1% frequency in the population) but confers a substantial risk of disease and the allele explains the majority of the variability seen in the disease and the disease can be considered a simple Mendelian disease from a genetic perspective. The method proposed accounts for the genetic MOI, and takes into account the distortions in the mating type frequencies seen because the parents are the parents of an affected child. Our method is suitable for complex diseases (where the GRR conferred by any single polymorphism is generally less than 3), and the risk allele is relatively common (ie the risk allele frequency in the population is greater than 1%).

Discussion

The TDT was proposed as the solution to the challenge of finding a disease susceptibility gene in a stratified study population. This method was proposed primarily because of its robustness to population stratification and admixture, which made it superior to the traditional case–control tests, which are susceptible to false-positive results. Typically, in the design of family-based studies, HWE is assumed when calculating adequate sample sizes required for a pre-specified power level. However, when there is population stratification, there is an increase in homozygosity beyond that expected by HWE. The homozygous parents in the study population do not contribute to the TDT statistic.¹⁵ For example, in a sample of 500 ACTs, in which the estimated disease allele frequency in the parents is 0.1, 180 heterozygous parents would be expected assuming HWE. However, if there is a large degree of population substructure where F=0.25, only 135 heterozygous parents are expected under the null hypothesis (the locus of interest is not associated with the disease of interest).

Although there are several methods for calculating the power of the TDT,^{9, 16, 17} none of these methods takes into consideration the power of the TDT in the presence of population stratification. Despite the fact that the TDT maintains the correct Type I error rate, the power of the TDT is significantly affected in the presence of population stratification. This is extremely important, because the TDT is often used when population stratification is possible. Studies with sample sizes calculated ignoring population stratification might be underpowered, and fail to detect putative disease genes. Our method is the first method that proposes incorporating Wright's coefficient of inbreeding, F as a measure of population stratification to approximate the mating type frequencies in the presence of population stratification for TDT sample size calculation.

One of our major points is that in the presence of population stratification, there is considerable variation in the power of the TDT to detect an association between a locus and a disease. The power of the TDT depends on the underlying genetic structure of the study population. We suggest that in future, researchers present estimates of the parameters required to describe the population structure ($\bar{p}$, F, ϑ₃ and ϑ₄) as well as the standard errors of these parameter estimates with all TDT findings. Studies using the TDT to replicate previously published findings may be more challenging than previously thought, as two studies with the same number of subjects could have very different powers to detect the same genetic association depending on the underlying population substructure. This finding may shed light on the cornucopia of studies that have failed to replicate previously published positive findings of association at certain loci. However, one must note that the initial finding may have been spurious.

In addition to possibly losing power because of population stratification (as a result of altered proportions and relative ratios of informative family types), the power of the TDT can be reduced if there is genetic risk heterogeneity between the subpopulations. Genetic risk heterogeneity occurs if the relative risk (RR) of disease conferred by the putative deleterious allele varies from one subpopulation to another. Quantitative risk heterogeneity occurs when the effect of the deleterious allele is not homogenous across all subpopulations, but is more profound in some subpopulations. A classical example for quantitative risk heterogeneity exists between ApoE and Alzheimer's disease, where the association exists pan-ethnically but is strongest in Caucasians and Asians, and weaker in Hispanics and African-Americans.¹⁸ The results of the TDT remain valid when there is quantitative risk heterogeneity, but the power of the TDT to detect this association with a given sample size will vary depending on the ethnic composition of the study sample.

Qualitative risk heterogeneity occurs when one allele is deleterious in one subpopulation (RR>1), but is protective in another (RR<1). An important point to be emphasized is that the power of the TDT diminishes greatly if there is qualitative risk heterogeneity. In fact, the case–control genomic control method proposed by Devlin and Roeder³ and the population stratification model (STRAT) proposed by Pritchard et al¹⁹ are both more powerful than the TDT per genotyped individual in the presence of qualitative risk heterogeneity as illustrated in Table 2 of Pritchard and Donnelly.⁶

In summary, statistical geneticists and genetic epidemiologists should carefully identify their study population and based on a conservative level of population stratification, follow the guidelines proposed when calculating sample sizes in anticipation of genetic analysis using the TDT and other family-based tests.

Acknowledgments

Notes

The authors declare no conflict of interest.

References