How much is the error variance if the test reliability is 81? 2024

Xem How much is the error variance if the test reliability is 81? 2024

Where diag(V) is the diagonal matrix of the observed
variances of W, and αprop denotes the estimate of Cronbach’s when the items are assigned different weights. Note that the weighted version of Cronbach’s alpha may violate the underlying assumptions. For example, when different weights are assigned to different items, it does not make sense to assume that the variance of the ME is constant. However, Cronbach’s alpha still provides useful results, which explains its appeal and is what makes it so widely used in
practice. In fact, we are considering two possible estimates of the ME variance, because equation (6) yields two possible estimates: one using equation 5 (same weights) and another using equation 8
(different weights).

Another approach would be to consider Armor’s theta [20], which is a special case of Cronbach’s alpha where the weights πi are chosen such that the reliability estimate is maximized. This maximization is realized by selecting the eigenvector corresponding to the largest eigenvalue of the correlation matrix computed from the individual ancestry
proportion estimates. Armor showed that this estimate can be written as

where λ1 is the largest eigenvalue of the correlation matrix computed from the items, which in this case is the correlation matrix between the different sets of individual ancestry proportion estimates.

2.3 Estimating the ME variance by using the repeated measurement approach

The repeated measurement approach may be the most widely used approach
to estimate the ME variance. Assume that the ME model can again be written as in equation (1). Under the assumption that the measurement errors are independent of the true ancestry proportions and Var(Uj)=σU,j2, we have from equation (1) that

after relaxing the homoscedasticity assumption to allow the ME variance to vary with the informativeness content of the subset of markers that is used to estimate individual ancestry. There are several
ways to estimate the ME variance by using repeated data. The simplest approach consists in estimating the measurement error variance from each subset and pooling the results into a single estimate. This is in fact the approach taken by Carroll et al. [8] (page 71, equation 4.3). However, they assume that the ME variance is constant in all subsets. Other estimates can be defined by relaxing the
assumption made in equation (2) such that one now assumes that

such that Var(Uj)

Let W¯i=∑j=1pπjWij be the overall estimate of the ancestry proportion of the
ith individual obtained by combining the p estimates (one for each subset) into a single estimate of ancestry proportions. As mentioned above, W̄i is the estimate that will be used as a covariate in the association test. It is highly correlated with Wi, the estimate that would be obtained if all the AIMS were used to produce a single estimate. The value of using
W̄i instead of Wi is that unlike Wi, an estimate of the ME variance can be computed for W̄i. In fact, it can be shown that W̄i is the minimum variance unbiased estimator of Xi. An estimate of the ME
variance is given by

σ^U2=∑i=1n∑j
=1pλj(Wij−W¯i)2n(p−
1)

(12)

2.4 Reliability estimates for multivariate IAPE

Multivariate estimates of genetic background can arise with both individual ancestry proportions and principal components. If the admixed population is derived from k>2 ancestral populations, the estimate of individual ancestry proportions can be represented by a vector with
k′ = k − 1 elements because the k estimates sum to 1. Caribbean Hispanics, for example, exhibit various levels of Native American (mostly Taínos), European, and African ancestry [21]. Therefore, their ancestry proportion can be represented by a vector with 2 components. It should also be noted that a negative correlation is expected between these components because of the
linear constraint. It is also not uncommon to use more than one principal component to achieve the appropriate type I error control in a GWAS. In fact, most investigators use the default setting of EIGENSTRAT, which suggests that the first 10 principal components are included as covariates in their association tests. The reliability of the sample eigenvector as an estimate of the true eigenvector in the population can also be a useful measure. Paul
[11] provides a method to compute this reliability for the eigenvector corresponding to the largest eigenvalue. Kimmel et al. [22] show how a non-zero angle between true and estimated axes of variation can lead to type I error inflation in GWAS. Measurements errors in the estimation of the genetic
background variable can occur in both cases.

We focus on estimating the ME variance when individual ancestry proportion estimates are computed by using AIMs when k > 2. Previous papers on estimating reliability in the multivariate case have focused on relating s observed variables with t latent factors [23].

We propose the following ME
model

Wij=Xi+Uij;j=1,2,..,p

(13)

where
Wij is a (k′, 1) vector of ancestry proportion estimates, Xi is the true but unobserved (k′, 1) vector of ancestry proportions, and Uij is the (k′,1) vector of Measurement errors associated with the estimation of Wij. Note the we also assume that (1) Uij ~ MVN(0, Σj), (2) Xi and Uij are independent, (3)
Uij is independent of Uil for any two subsets l and j, (4) and that ∑j=(1πj)∑, where
πj is defined similarly as in equation (7). Similarly to the univariate case, these assumptions lead to Cov(Wij, Wil)=Cov(Xi, Xi)=ΣX except that ΣX is now a (k′, k′) positive definite matrix.

Equation (13) implies that

Cov(Wij)=∑X+∑j,j=1,2,..,p

(14)

Similarly to the univariate case, the (k′, 1) vector

will be used as a covariate in the association test instead of Wi. As mentioned above, W̄i is almost always perfectly correlated
with Wi, and contrary to Wi an estimate of the ME variance can be computed for W̄i. Combining (14) and (15), and using the fact that under the assumptions of the CTM model, it can
be shown that Cov(Wi, Wj) = ΣX, we have

Cov(W¯i)=∑X+
∑j=1pλj2∑j

(16)

2.4.1 Estimating the ME covariance by using the repeated measurement approach

There are a number of ways to estimate the ME covariance by using
repeated data. However, we opted to use an estimate similar to (15) in which the variable Wij is now a vector with k′ elements. The repeated measurement estimate of the ME covariance matrix is given by

∑^RM=∑i=1n∑j=1pπj(
Wij−W¯i)(Wij−W¯i)Tn(p−1)

(17)

2.4.2 Estimating the ME covariance by using the reliability approach

Following Tarkkonen and Vehkalahti [23], we can write the reliability matrix as

This estimate cannot be computed
based on equation (18) alone since the individual ancestry proportion are not directly observed. We know describe a procedure to estimate the reliability matrix. Consider the block matrix C = Cov(W1, W2,…., Wp)T, which is a (k′p, k′p) matrix, where the submatrix
on the ith row and jth column of Cov(W1, W2,…., Wp) corresponds to a (k′, k′)matrix that is defined as:

Clj={πj2n−1∑i=1n(Wij−W¯
i)(Wij−W¯i)T,l=jπlπ
jn−1∑i=1n(Wij−W¯i)(
Wij−W¯i)T,l≠j

where Wij and Wil are the (k′, 1)vectors of ancestry proportion estimated using markers on the
jth and lth subsets.

An estimate of the reliability matrix is given by

Ω^=pp−1((∑l
≠jpClj)(∑j=1p∑l=1pClj
)−1)

(19)

and an estimate of the ME covariance matrix is

∑^Rel=(Ip−Ω^)C^ov(W¯).

(20)

Ĉov (W̄) is the
observed (p, p) covariance matrix of the overall estimate of the ancestry proportion computed in equation 19 for the n individuals in the sample.

In the case in which the estimated ME covariance matrix is not a positive definite matrix, a simple correction consists of replacing it by another matrix whose minimum eigenvalue is equal to a very small positive
value. Let ζ be the minimum eigenvalue of Σ̂, where Σ̂ denotes the ME covariance matrix estimated by using either equation (17) or equation (20). If ζ ≤ 0, then
Σ̂ is not a positive definite matrix. However, the matrix Σ̂corrected = Σ̂ + (0.01 − ζ)Ik−1, where Ik−1 is the (k − 1) × (k − 1) identity matrix. The minimum eigenvalue of this new matrix is 0.01, which makes a
positive definite matrix.

3. Simulation study

We present 2 sets of simulation studies. First, we assumed an admixed population derived from intermating between exactly 2 ancestral populations. In this case, an individual’s ancestry proportion estimate can be represented by a scalar and the confounder is univariate. We also assumed that the total number of AIMs is divided into p subsets with mj,
j =1,2,…, p being the number of markers used to obtain the admixture estimate on the jth subset. Also, let M=∑j=1pmj is the total number of markers
simulated. For simplicity, we set p=22 to mimic the more intuitive case in which each estimate is computed at the autosome level. We considered 2 cases: (1) M=110 and (2) M=220. In each case, we compared the estimate of the ME variance obtained with Cronbach’s alpha to the estimate obtained when we used the repeated measurements approach. These results were compared by assuming (a) an equal allocation of the number of markers per subset and (b) an allocation
proportional to the chromosome length. Weights under scenario (b) were determined by using the Marshfield sex-averaged chromosome lengths [24].

Let prs be the allele frequency of the sth marker s =1,2,…, M in the rth ancestral population r =1,2. The allele frequencies were chosen such that each
marker was ancestry informative. In practice, a minimum absolute difference of 0.3 between allele frequencies in the 2 ancestral populations is required before a marker can qualify as an AIM. The true ancestry proportions, which is denoted by ai, i =1, 2,.., n were drawn from a beta distribution (ai ~ Beta(10, 40)) such that the expected value was 0.2, which is
close to estimates of the European genetic contribution to the African-American population [25]. We then computed the allele frequency at the sth marker for the ith individual as qis =aip1s + (1 −
ai)p2s, which served as the parameter for the binomial distribution from which the individual genotype at that marker is drawn. Finally, we applied a maximum likelihood approach [26] to provide both a global and p IAPEs (one of each subset). The difference between the true ancestry proportions, which were drawn
from the Beta distribution, and the global ancestry proportion estimates was computed and regarded as the ME variable. In practice, this difference is never observed because the true ancestry proportions are not known. However, computation of this difference allowed us to directly compute the ME variance and to evaluate the performance of each approach. The ME variance was estimated under 4 different scenarios. (1) The total number of markers was divided equally into 22 subsets of markers, and
we applied both methods assuming the IAPEs obtained on each subset carried the same weight. (2) The total number of markers was again divided into 22 subsets, but marker allocation was done proportional to chromosome length. (3) The total number of markers was divided equally into 4 subsets. (4) The total number of markers was divided into 4 subsets with proportional allocation where 10% of markers were allocated to the first subset, 20% to the second, 30% to the third, and 40% to the last. We
generated 10,000 data sets containing 1000 individuals each. The simulation results are summarized in Tables 1 to ​4.

Table 1

Five-point
summary and standard error of the distribution of the measurement error variance associated with ancestry proportion estimates for K=2 when 110 AIMS divided into 22 subsets are used.

Table 4

Five-point summary and standard error of the distribution of the measurement error variance associated with ancestry proportion estimates for K=2 when 220 AIMS divided into 4 subsets

The purpose of the second simulation study was to evaluate the performance of the two ME variance estimation approaches when the confounder is multivariate, that is, when the number of ancestral populations (k) is greater than 2. For simplicity, we focus on the case in which k=3. In this case, individual ancestry proportion estimates can be represented by a vector with 2 components. The simulation procedure when k>3
is similar to the one we just described above for the case in which k=2, except that the true ancestry proportions were drawn from a Dirichlet instead of a Beta distribution. Again, we assumed that the total number of AIMs (M) is divided into p subsets with mj, j = 1,2,…, p being the number of markers used to obtain the ancestry estimate on the jth subset. The performance of the two ME variance estimation
approaches was evaluated again by using the 4 scenarios described above.

Let prs be the allele frequency of the sth marker, s = 1,2,…, M in the rth ancestral population r = 1,2,3. Following Pfaff et al. [27], the allele frequency of the sth marker in the admixed population can be
written as

ps(adx)=m1p1s(1)
+m2p2s(2)+m3p3s(3)

(21)

where mr
is the genetic contribution of the rth ancestral population. Note that mr is such that ∑r=13mr=1. The
genotype at each marker was simulated by two independent draws from a Bernoulli (ps). The true ancestry proportions were drawn from a Dirichlet distribution with parameters (0.4, 0.3, 0.3), which is close to the ancestry proportions observed in Caribbean Hispanics such as Puerto Ricans [28]. We should note that since we generated markers to estimate the IAPE, the
distribution of the ME variable was not known. The ME variance was computed for each simulation as the difference between the simulated true ancestry proportions and the overall estimate computed by combining all the subsets into a single data set.

4. Results

We note that both approaches for estimating the ME variance implicitly assume that the average IAPE computed over the p subsets is the control variable that is used in the
association test to guard against spurious associations. We showed previously [9] that the correlation between the average estimates computed over the p subsets and the estimate obtained when all the markers are used to provide a single estimate was around 99% when we considered a real data set with over 6,000 individuals in which 1,312 AIMs based on the marker panel described in
[29] were typed.

The simulation results for the case in which k=2 are given in Tables 1 to
​4. These tables present the five-point summary of the estimate of the ME variance computed with each method. The distribution of these estimates was then compared with the true ME variance to evaluate their performance. These simulations showed that the estimates of the ME variance based on the internal reliability measures
(i.e., Cronbach’s alpha and Armor’s theta in the univariate case) seemed to outperform those computed by using the repeated measurement approach. Considering the estimates based on the internal reliability approach, we see that the error variance estimate obtained with Armor’s theta was always between the estimates computed by using Cronbach’s alpha under the two weighting schemes. This result should be expected because Armor’s theta corresponds to the maximum estimate of Cronbach’s, which is
obtained by considering all possible weightings of the IAPE computed for each subset. As can be seen in Tables 1 to ​4, the estimate with Armor’s theta was always closer to
the estimate provided by Cronbach’s alpha under equal allocation of markers. This result is in agreement with equation (6), because Armor’s theta yields the maximal value of Cronbach’s alpha when the weights are all equal. This result suggests that there might be an advantage to weighting the items before computing Cronbach’s alpha. The estimates based on the repeated measurement had less bias when
the markers were allocated equally.

We also note that the performance of the ME variance estimated by using Cronbach’s alpha varied with the total number of markers. That is, when we considered 110 AIMS, the allocation proportional to subset size worked better then when the markers were equally allocated to each subset. However, the reverse situation was observed when we considered 220 markers. Very few markers were assigned to the smaller subsets when only 110 markers were used.
Therefore, the IAPE calculated from these subsets can have strong bias, which leads to higher variance estimates. The repeated measurements approach, as expected, was more affected by these biases than were the internal reliability measures. When we considered 220 AIMS, there seemed to be enough information such that not much was gained by allocating markers proportional to subset size. Finally, the number of subsets considered appeared to be a significant predictor of the performance of these
methods. The repeated measurement approach with equal weights yielded acceptable results when 22 subsets were considered, but failed when only 4 subsets were used, with estimates that were more than twice larger than the true ME variance.

We observed similar results when we considered the case in which the number of ancestral populations was equal to 3. Because the IAPEs must sum to 1, the ME covariance matrix can be represented by a (2,2) matrix. Comparisons between each ME
covariance estimation approach and the true ME covariance are presented in Tables 5 and ​6. These tables show the error covariance matrix estimated after 10,000 replications.
The elements of each covariance matrix are shown in bold. We present the standard error associated with the estimation of each element in italic below this element. We focus on the case in which the IAPEs were computed with 110 AIMs in Table 5.
Table 6 is very similar to Table 5, except that it presents the results for the case in which 220 AIMs were used to estimate the IAPEs. Each table also compares the effect of equal vs.
allocation proportional to subset size as well as the effect of dividing the total number of AIMs in 22 vs. 4 subsets. Both tables confirm the observations made in the univariate case. That is, (1) the reliability approach leads to more accurate and less biased estimates of the true covariance matrix than the repeated measurement approach, and (2) the allocation of markers proportionally to the number of markers in the subset appears to be beneficial only when 110 AIMs are considered.
Table 7 presents the Frobenius norm, the L-2 norm for a matrix, between the estimated covariance matrices and the true covariance matrix is the 4 scenarios shown in Tables 5 and
​6. This table confirms that the internal reliability approach seemed to perform better in the context of estimating the ME associated with the estimation of individual ancestry proportions.

Table 5

Performance of the repeated measurement approach and multivariate reliability when k=3
and 110 AIMs are used to estimate the individual ancestry proportions

Table 6

Performance of the repeated measurement approach and multivariate reliability when k=3 and 220 AIMs are used to estimate the individual ancestry proportions

Table 7

Frobrenius norm between the estimated covariance using each approach and true error covariance

5. Discussion

Population stratification and admixture-induced linkage disequilibrium remain a concern in genetic association tests. These tests are often conducted under the assumption that the measure of genetic background that is used to control for the confounding effect caused by population stratification and admixture is obtained without errors. This assumption is not likely to hold, however.
Consequently, the type I error and the power of these studies are not likely to remain at their nominal levels. We presented here a simple procedure that can be applied in conjunction with well known ME correction methods to help to account for these Measurement errors.

We show that a generalization of Cronbach’s alpha can be used to estimate the ME covariance matrix. This estimate performed better than the one obtained by using the repeated measurement approach. We note that
repeated measurements are the most widely used approach to estimating the ME variance in ME correction efforts. The reliability approach relies on several assumptions. First, it is assumed that the estimate computed on each subset is measuring the same underlying latent variable, which in this case is the true individual ancestry proportion. This assumption is likely to hold when the subsets are created so that they each yield an estimate of the overall ancestry proportion. We note that this
assumption is likely violated, however, in efforts to estimate local ancestry because of local variations in the ancestry estimate. We did not evaluate the robustness of this approach under this type of violation and do not advocate its application in efforts to estimate the ME associated with local ancestry estimates. Second, we note that Carroll et al. (page 3, ref #8) for example, referred to the error contaminated variables W and the ME variable U as conditional
distributions, where the conditioning is done on the true unobserved variable (X). In this case, Uij|Xi and Uil|Xi will be independent assuming that the subsets of AIMS used to estimate the individual admixture proportions are disjoint and they are selected far enough from each other such that admixture induced LD is the only source of correlation. By conditioning on the true ancestry, the admixture induced LD is eliminated. We ran a simulation study
to evaluate the magnitude of this correlation. We only consider the univariate case for simplicity and apply the same simulation procedure described in the manuscript in section 3. Briefly, we draw the true individual ancestry (Xi) from a beta distribution, we simulate 2 sets (M = 5,10) of ancestry informative markers conditional on the underlying ancestry proportion and compute admixture proportion estimate for each set (Wi1 and
Wi2), we then obtained the ME variables (Ui1 and Ui2) as the difference between Wi1 and Wi2 and Xi and compute the correlation between them. We chose to presentation the correlations instead of the covariances simply for ease of interpretation. The average correlation after a 1000 iterations was 8×10−4 with a standard error of 5 ×10−3 for M=5 and 2
×10−3 with a standard error of 3 ×10−3 for M=10. The simulation showed that these correlations are very small, which implies that the assumption of independence is not too strong to invalidate the proposed methods. The correlation between Wij and Wil is also always positive, which means that the covariance between these two variables is also always positive.

Third, a closer look at
equation 12 shows that the repeated measurement estimate of the ME variance does not make use of these assumptions. Therefore, the resulting estimate should not be affected if these assumptions do not hold. However, the estimates based on the reliability approach may be biased and equations (3) and
(16) would have to be modified to account for the correlation in the Measurement errors.

We considered two possible ways of partitioning the AIMs into subsets. The first partition consisted of dividing the AIMs into 22 subsets. This partition can be seen as a natural way of dividing the data when an investigator seeks to obtain an estimate for each chromosome. In the second
partition, we divided the AIMs into 4 subsets. We chose this partition to evaluate whether the second method would perform better when fewer subsets with a larger number of AIMs were considered. We observed an advantage of considering fewer but larger subsets only in the case in which 110 markers were used.

In conclusion, our results offer information that can be used to enhance the estimation of the degree of ME or, conversely, the reliability of individual ancestry proportion
estimates. The estimate of ME variance can in turn be incorporated into other analyses, such as genetic association tests in candidate gene and genome-wide association studies.

​

Table 2

Five-point summary and standard error of the distribution of the measurement error variance associated with ancestry proportion estimates for K=2 when 110 AIMS divided into divided into 4 subsets are used

Table 3

Five-point summary and standard error of the distribution of the measurement error variance associated with ancestry proportion estimates for K=2 when 220 AIMS divided into 22 subsets are used.

Acknowledgments

This research was supported by R01GM077490 (JD, DTR, RJC and DBA), R01AR057106 (JD), R37CA057030 (RJC), 2P30AI027767 (DBA) and KUS-CI-016-04 made by King Abdullah University of Science and Technology (KAUST) to RJC.

Appendix

A.- Table of abbreviations

B.- Proof that equation (12) is unbiased

Let

Assumethat{E(Uij)=0∀jVar(Uij)=
σj2

Let πj be a set of weights such that πj ≥ 0 and ∑j=1pπj=
1 where p is the number of subsets of AIMs.

Define

Let

Therefore, W̄i is minimum variance unbiased linear estimator E(Xi). From equation (1) we have E(Wij) =
E(Xi)

Note that from (1) and (2), we also have

where U¯i=∑j=1pUij

Subtracting (3) from (1) we have

which leads to

∑j=1pπj(Wij−W¯i)2
=∑j=1pπj(Uij−U¯i)2=σ2
∑j=1p(Uij−U¯i)2σj2

(26)

E{∑j=1pπj(Wij−W¯i
)2}=σ2E{∑j=1p(Uij−U¯i
)2σj2}

(27)

E{∑j=1pπj
(Wij−W¯i)2}=σ2(p−1)

(28)

Finally,

E(σ^2)=1n(p−1)∑i=1n
E{∑j=1pπj(Wij−W¯i)2}=
σ2.

The same approach can be taken to show that equation 17 also provides an unbiased estimate of the ME variance when k>2.

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