Power calculator for detecting allelic imbalance using hierarchical Bayesian model

Abstract

Objective: Allelic imbalance (AI) is the differential expression of the two alleles in a diploid. AI can vary between tissues, treatments, and environments. Methods for testing AI exist, but methods are needed to estimate type I error and power for detecting AI and difference of AI between conditions. As the costs of the technology plummet, what is more important: reads or replicates?

Results: We find that a minimum of 2400, 480, and 240 allele specific reads divided equally among 12, 5, and 3 replicates is needed to detect a 10, 20, and 30%, respectively, deviation from allelic balance in a condition with power > 80%. A minimum of 960 and 240 allele specific reads divided equally among 8 replicates is needed to detect a 20 or 30% difference in AI between conditions with comparable power. Higher numbers of replicates increase power more than adding coverage without affecting type I error. We provide a Python package that enables simulation of AI scenarios and enables individuals to estimate type I error and power in detecting AI and differences in AI between conditions.

Keywords: Allele specific reads; Allelic imbalance; Biological replicates; Power; Simulation; Type I error.

Fig. 1

Read counts are simulated for different scenarios in two conditions. A scenario is defined as a specific number of simulations, number of allele specific reads, number of biological replicates (bioreps), level of allelic imbalance (AI) θ, and the probability of mapping an allele g1 (g2) specific read. Without loss of generality, let allele g1 be allele A and g2 be allele C (blue boxes). The number of allele specific reads (yellow reads) is the sum of unambiguously mapped reads in the experiment. Grey reads are reads that map equally well, i.e. ambiguously, to both alleles. Biological replicates in an experiment are samples from the same genotype and condition. In this example, there are k biological replicates, $12\times k$ allele specific reads, and the probability of an allele specific read is $r_{i,g1}=r_{i,g2}=0.8$. The null H1 and H2 hypotheses are allelic balance θ₁ = 0.5 in condition 1 (ex. liver) and θ₂ = 0.5 in condition 2 (ex. kidney), respectively. These cases are used to estimate the Type I error in rejecting allelic balance in conditions 1 (H1) and 2 (H2). In this example, θ₁ = 0.55 under the alternative (alt) H1 hypothesis and θ₂ = 0.55 under the alternative (alt) H2 hypothesis. These cases are used to estimate the power in rejecting allelic balance in conditions 1 (H1) and 2 (H2). θ₁ = 0.5 and θ₂ = 0.55 under the alternative (alt) H3 hypothesis, which allows estimation of the power rejecting equal levels of AI between the two conditions (H3). The null H3 hypothesis is simulated in both the complete null case: θ₁ = θ₂ = 0.5 and in the scenario where there is allelic imbalance in both conditions θ₁ = θ₂ = 0.55. These cases can be used to estimate the Type I error in rejecting equal levels of AI between the two conditions (H3)

Fig. 2

Variations in type I error (y-axis) are shown as a function of the number of biological replicates, or bioreps (x-axis) assuming different numbers of allele specific reads. H1 and H3 refer to the null hypothesis of allelic balance within a condition (H1) and the null hypothesis of equal levels of AI between the two conditions (H3). The Type I Error (y-axis) is computed as the proportion of simulations for which the Bayesian evidence against allelic balance within a condition or equal AI between conditions is $<$ 0.05. Plots a and b show eight simulated values of the number (#) of allele specific reads, which is the sum of the reads that map unambiguously to an allele in the experiment. Plots c and d show four simulated values of the number (#) of allele specific reads per bioreps, which is the number of allele specific reads divided by the number of bioreps. $\mathrm{\Delta }AI$ is deviation from the null, i.e. deviation from allelic balance in condition (${\mathrm{\Delta }AI}_1$) or the relative difference in the levels of allelic imbalance between the two conditions (${\mathrm{\Delta }AI}_3$)The probability of an allele specific read is r_i,g1 = r_i,g2 = 0.8 and there are 1000 simulations

Fig. 3

H1 refers to simulations under the alternative hypothesis of allelic imbalance within a condition and H3 refers to unequal levels of AI between the two conditions. For H1, the x-axis is the effect size, which is the relative deviation from allelic balance ${\mathrm{\Delta }AI}_1=$ $\frac{\left|\theta ,-,{\theta }_0\right|}{{\theta }_0}$, where ${\theta }_0=0.5$. For H3, the x-axis is the relative difference in levels of AI between the two conditions ${\mathrm{\Delta }AI}_3=$ $\frac{\left|{\theta }_2,-,{\theta }_1\right|}{{\theta }_1}$ where the first condition is simulated under the null hypothesis and the second under the alternative hypothesis $\theta \ne 0.5$. The power (y-axis) is computed as the proportion of simulations for which the Bayesian evidence against allelic balance within a condition or against equal levels of AI between conditions is $<$ 0.05. There are 1000 simulations and the probability of an allele specific read is r_i,g1 = r_i,g2 = 0.8. Simulations for 3, 4, 5, 6, 8, and 12 biological replicates (bioreps, x-axis) for varying numbers (#) of allele specific reads are reported