In earlier work, we have developed and evaluated an alternative approach to the analysis of GWAS data, based on a statistic called the PPLD. More recently, motivated by a GWAS for genetic modifiers of the X-linked Mendelian disorder Duchenne Muscular Dystrophy (DMD), we adapted the PPLD for application to time-to-event (TE) phenotypes. Because DMD itself is relatively rare, this is a setting in which the very large sample sizes generally assembled for GWAS are simply not attainable. For this reason, statistical methods specially adapted for use in small data sets are required. Here we explore the behavior of the TE-PPLD via simulations, comparing the TE-PPLD with Cox Proportional Hazards analysis in the context of small to moderate sample sizes. Our results will help to inform our approach to the DMD study going forward, and they illustrate several respects in which the TE-PPLD, and by extension the original PPLD, offer advantages over regression-based approaches to GWAS in this context.
Citation: Vieland VJ, Seok S-C (2021) The PPLD has advantages over conventional regression methods in application to moderately sized genome-wide association studies. PLoS ONE 16(9): e0257164. https://doi.org/10.1371/journal.pone.0257164
Editor: Ayse Ulgen, Girne American University - Karmi Campus: Girne Amerikan Universitesi, CYPRUS
Received: May 4, 2021; Accepted: August 24, 2021; Published: September 22, 2021
Copyright: © 2021 Vieland, Seok. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: This manuscript presents only simulated data. All computer code used in generating and analyzing the data is available at https://sourceforge.net/projects/kelvin-linkage-disequilibrium/files/.
Funding: This work was supported by National Institutes of Health grant (NINDS) NS085238 to VJV. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
In previous work we have developed and evaluated a statistic called the Posterior Probability of Linkage Disequilibrium (PPLD) as a measure of evidence for or against trait-SNP association [1–3], and we have extended the PPLD to accommodate time-to-event (TE) data, yielding the TE-PPLD . In this paper we compare and contrast the TE-PPLD with the more familiar regression-based approach to handling time-to-event phenotypes using the Cox Proportional Hazards (CPH) model, via simulations and with a focus on small to moderate sample sizes. Using simulations, rather than real data, allows us to compare and contrast the behavior of the two statistics under conditions in which the true underlying genetic model is known, so that it is possible to definitively distinguish correct from incorrect results.
This work was motivated by a search for genes that modify the Duchenne muscular dystrophy (DMD) phenotype. DMD is an X-linked recessive disorder affecting ≈ 1 in 5,000 live male births [5, 6]. DMD involves progressive muscle tissue loss with replacement by fat and fibrotic tissue, and is currently without a cure. Patients typically become reliant on wheelchairs by early to mid-adolescence, but some maintain ambulation substantially longer, and age at loss of ambulation (LOA) is an important clinical indicator of disease progression. A great deal is known about the gene (DMD) that causes DMD, including the fact that modifier genes influence the rate of disease progression in a DMD mouse model [7, 8]; evidence for modifiers exists in humans as well [9–12]. The discovery of modifier genes in humans has implications both for therapeutics and for the design of DMD clinical trials. Thus far there have been two published GWASs for LOA, based on sample sizes of 170  and 253  individuals, respectively.
Using data from the United Dystrophinopathy Project, a multisite consortium [13–15], we are engaged in a search for modifier genes under a GWAS design. The sample currently comprises ≈ 400 DMD patients, with ongoing efforts to increase the sample size to ≈ 800 DMD patients. (These patients are conservatively selected to exclude mutations in DMD itself that are known or suspected to affect disease course.) There is one binary covariate in the model: steroid use prior to LOA, which is known to increase LOA by about 3 years on average. The immediate motivation for the current work is to inform design decisions regarding the analysis of our DMD data, while illustrating issues arising for GWAS in smaller data sets which could be of relevance to other studies as well. The paper can also serve to illustrate key features of the PPLD for GWAS investigators unfamiliar with the method.
The remainder of the paper is organized as follows. In Methods we present the simulation methods and the data analytic methods used in what follows. In Results we begin by considering the distributions of the TE-PPLD and the CPH p-value under the null hypothesis H0 of “no SNP-trait association,” under various conditions that can affect those distributions. We then consider the behavior of both statistics under a variety of models under the alternative hypothesis HA of “SNP-trait association.” Finally, we briefly contrast the PPLD’s use of Bayesian sequential updating to confirm findings with the standard requirement of independent replication, again with a particular focus on small sample sizes.
Finally, in Conclusions we perform an experiment in which we loosely mimic an entire genome scan in an initial data set, following up at any findings in a separate data set. This section illustrates the practical implications of many of the results of the preceding sections, and can be read first, or even independently, for an overview of the effects the choice between CPH and the TE-PPLD can have on GWAS results.
In this section we describe the simulation methods and models used to evaluate the behavior of the different statistical analyses, and the statistical methods used to analyze the simulated data.
In order to mimic features of our DMD data set, our base model uses a sample size of N = 400 individuals, half of whom come from each of two covariate levels (y = 1, 2); for some purposes we also consider sample sizes of N = 200 (similar to some potential “replication” data sets currently available for DMD) and N = 800 (our target sample size over the next few years), as noted in context. We simulated only unrelated individuals. Details of the simulation methods follow. All simulations were conducted in Matlab using built-in cdf and inverse cdf functions.
Individuals were randomly assigned a genotype for a 2-allele locus (with alleles 1, 2) as a function of the Minor Allele Frequency (MAF). Unless otherwise noted, we set MAF = 0.5; for some comparisons we also vary MAF as noted in context. We generated genotypes under Hardy Weinberg Equilibrium (HWE). (The impact on the PPLD of violations of HWE and the impacts of population stratification have been investigated previously  and are not further considered here.)
Under the null model H0 (no SNP-trait association; Model 1 in Table 1), event time te was simulated via a random draw from a normal distribution Ny = 1(μ, σ) for individuals with y = 1, and Ny = 2(μ+3, σ) for y = 2. Under the alternative model HA (SNP-trait association), we simulated data under 7 different baseline Models 2–8 (Table 1), in which te was randomly drawn from a mixture of normal (MoN) distributions in the form Ny = 1,k(μk, σk), for given genotype k = 11, 12, 22 and y = 1, and in the form Ny = 2,k (μk+3, σk) for y = 2 (except for Model 8). Model 2 creates a simple additive mixture model for age-at-event while maintaining comparable and s.d. at the population level. The remaining models vary effect size by increasing the genotypic variances (Model 3), introducing dominance (Models 4, 5), and by generating genotypic effects on variances as well as means (Model 6) or solely on variances (Model 7). Model 8 complicates the covariate effect. The models were chosen to illustrate a range of possible trait distributions, and are by no means intended to exhaustively cover what we might find in a real application. We simulated 1,000,000 replicates under Model 1, and 1,000 replicates per model under Models 2–8.
For each generating model, an individual was simulated based on a random draw of te(xi) from the corresponding age-at-event (AE) distribution and an independent random draw of to(xi) from an age-at-observation (AO) distribution. If te(xi) < to(xi), the individual was considered uncensored with failure time tfail(xi) = te(xi); otherwise, the individual was considered censored with censoring time tcens(xi) = to(xi). AO was simulated under a negative binomial distribution with r = 10, p = 0.4 in order to roughly mimic the censoring distribution in the real data. This yields a censoring rate ≈ 40%.
For some purposes, as noted in context below, we varied the baseline models. In considering robustness to the form of the underlying survival distribution, we also generated data (1,000,000 replicates per model under H0, and 1,000 replicates per model under HA) from Weibull (WB), Birnbaum-Saunders (BS), and Gamma (GM) distributions. This was done in each case by finding parameters of the distribution that matched the mean and standard deviation of the corresponding baseline model as shown in Table 1, and using these parameters as the generating values. We also considered covariate x genotype interactions; those models are described in context below.
Data analysis methods
In this section we give a brief overview of the TE-PPLD; for additional details see . All TE-PPLD calculations were done in the software package Kelvin . We also describe the CPH analyses used in what follows, and we summarize some key differences between the TE-PPLD and CPH, which are relevant when comparing and contrasting results between the two methods. When referring to general features of Kelvin’s association statistic, we use “PPLD;” when discussing features that are (or may be) specific to the use of the PPLD with time-to-event data, we say “TE-PPLD.”
The PPLD is based on the Bayes ratio (BR), defined as where LR is a likelihood ratio representing “trait-marker association” in the numerator and “no association” in the denominator , and the single integral stands in for multiple integration over the vector γ = μ11,μ12,μ22,σ11,σ12,σ22, the means and standard deviations of three normal distributions, one for each of the three SNP genotypes . Note that the user does not specify a mode of inheritance, since this parameterization encompasses recessive, additive and dominant models. For present purposes additional parameters of the likelihood are fixed as follows: recombination fraction θ = 0; standardized linkage disequilibrium (LD) parameter D′ = 1 (see ); admixture parameter α = 1 (see ); disease minor allele frequency (MAF) = SNP MAF. These simplifications allow us to model genotypic effects of the SNP itself (whether direct effects or indirect through LD) on either μ or σ or both. (Kelvin can also be run in “means only” mode, which fixes the genotypic variances equal to one another, in case the user prefers to detect effects on means only, or as a check to see the extent to which effects on variances are driving the signal at any given SNP.) The underlying likelihood is based on the Elston-Stewart pedigree peeling algorithm  so that it can accommodate unrelated individuals as well as mixtures of pedigree structures. (This feature is helpful in our DMD study, because the dataset includes some pedigrees; however, we do not further consider it here.) The BR is proportional to a likelihood for the marker data conditioned on the trait data, and for reasons having to do with ascertainment corrections [16, 21] it is integrated as a unit, rather than separately in the numerator and denominator like a Bayes factor , using highly accurate non-stochastic numerical methods .
Let π be the probability that a randomly selected SNP is within detectable LD distance of a trait locus. The PPLD is a simple rescaling of the BR onto the (0,..,1) interval: . Thus PPLD < π indicates (some degree of) evidence in favor of H0, while PPLD > π indicates (some degree of) evidence in favor of HA; this remains true regardless of the value chosen for π. We set π = 0.0004, based in part on empirical calculations . By design, and in stark contrast with p-values, P[(PPLD > π) | H0] → 0 as N → ∞. In accumulating evidence for or against association across data sets, Bayesian sequential updating can be used by first multiplying the BRs across data sets and then applying the PPLD transformation to the resulting updated BR.
One limitation of Kelvin is that its models do not currently include any direct mechanisms for handling covariates. Our general approach to covariates is to preprocess the phenotype by performing regression analysis to make the covariate adjustments, and then to use the regression residuals as the input phenotype for subsequent analysis. In the context of linear regression, these residuals maintain the scale of the primary phenotype, and can be interpreted as estimates of how unusual is an individual’s phenotype given the individual’s covariate status. In the context of survival analysis, however, standard forms of residual (Martingale or deviance) do not maintain scale and do not have this “ordinary” interpretation . For this reason, we developed a new Ordinary Time-to-Event (OTE) residual, so-called because it maintains scaling vis a vis the primary phenotype and the interpretation of an ordinary linear regression residual, as a measure of how unusual the individual’s phenotype is given covariates. The OTE residuals then replace the underlying primary phenotype as input to TE-PPLD analysis.
In order to estimate OTE residuals for TE-PPLD analysis in what follows we use the procedures described and evaluated in . For each simulated data set, the estimated survival curve , as a function of age t, is obtained via maximum likelihood estimation of a pair of 2-parameter Weibull distributions, one for each level of y, based on time-at-event for uncensored individuals and censoring time for censored individuals. OTE residuals for each individual are then calculated using the formula in .
PPLDs are reported to 2 decimal places for values ≥ 0.01 but to 4 decimal places for values < 0.01, in order to display whether very small values are greater than or less than the prior probability π = 0.0004, indicating evidence for or against association, respectively. In general we display results in terms of the TE-PPLD itself. However, because of the very low prior probability the PPLD scale is highly compressed at the low end. Thus for visualization purposes, particularly in considering the null distribution, we sometimes display log10BR instead.
CPH regression was performed using the built-in function of the Survival package in R . Regression analyses included the covariate y as well as genotypes as predictors; in considering models with covariate x genotype interactions we also include a y x genotype interaction term, as noted in context. Unless otherwise noted, we performed CPH analysis coding the genotypes to reflect the correct (generating) mode of inheritance (recessive, additive or dominant); under H0 we assumed an additive model. CPH results are reported as P = –log10(p-value) for the genotypic coefficient unless otherwise noted, annotated as CPH-P.
Before proceeding to compare CPH with the PPLD, it is worth noting that the two approaches are in several respects incommensurate. The CPH p-value represents the probability of the data, or data more extreme, assuming H0, under the conditions imposed by the regression model; the TE-PPLD represents the posterior probability of HA, given the actual data only, under the assumptions described above. The p-value is not a measure of evidence strength , rather, it is considered significant when it crosses some preselected threshold. In GWAS contexts this threshold is conventionally set to 5x10−8, or P ≥ –log10(5x10−8) = 7.3, in order to adjust for multiple tests on a study-wide basis; in what follows we also consider a less stringent threshold of P ≥ 5.
By contrast, the BR is designed as a LR-based evidence measure [26–29]. As a result, the PPLD provides an estimated rank-ordering of SNPs in terms of strength of evidence for or against trait-SNP association. Its calculation is not in itself a decision-making procedure, that is, there is no cutoff above which we declare significance; and, because it is not an error probability, it is not subject to multiple testing corrections. In practice, one prioritizes SNPs for follow up by sorting from largest to smallest PPLD and working down the list. This leaves open the question of how large a PPLD warrants follow up. With experience we have developed certain heuristics for prioritizing SNPs for further attention in situations similar to our DMD study, with PPLDs ≥ 10% being of interest for follow up, and PPLDs ≥ 40% being of particular interest for follow up.
In what follows, these heuristics are evaluated and found to work quite well. They are, however, context-dependent, much like the principle that it is fine to leave the house without a jacket whenever the temperature exceeds 70°F. This is a reasonable norm, but one which might be modified, if, say, one’s primary interest were in showing off a new jacket. In the statistical context, one needs to develop a sense of “how big is big enough” (similarly to “how hot is hot enough?”–a question that has no single answer outside of a specific context of application), and this will depend upon several factors, including the costs and benefits of following up on false positive findings or failing to follow up on true positive findings. Sample size also plays a role (see below), and it may be helpful to use custom-tailored simulations to develop heuristics for any particular application. Interpreting the PPLD may therefore seem to be more onerous than relying upon a fixed threshold for the empirical p-value. However, as we demonstrate below, there are very practical advantages to the shift in paradigm.
Note too that the distinctive behavior of the PPLD under H0 as a function of sample size means that, even if we were to treat the PPLD as a standard test statistic, as we do below, the significance level associated with any given PPLD threshold for declaring evidence against H0 decreases as the sample size increases. Thus, by contrast with CPH-P, there is no fixed significance level associated with any given value of the PPLD; rather, the associated error probabilities are a function of sample size and would need to be established separately for different sample sizes, via custom simulations. That said, in applications of the PPLD it is not necessary to establish the significance level associated with a particular result. Researchers have become accustomed to considering the empirical Type 1 error rate as the measure of evidence against H0. However, as is well known, this practice ignores the Type II error rate (or power) and also the prior probability of H0. Hence even a small p-value can be associated with a high posterior probability that H0 is actually true under some circumstances. Since what we really want to know is the posterior probability of H0 (and/or of HA), the PPLD, which is a direct estimate of the posterior probability of association, is therefore a more informative statistic than the p-value alone, and its interpretation need not be reduced to an associated significance level.
In what follows we evaluate the behavior of the TE-PPLD and CPH in application to GWAS analysis in the context of our intended genetic application, using simulated data. We have chosen the topics for the subsections to highlight some salient differences between the 2 methods in the GWAS setting, as well as to assist us in making practical decisions regarding how best to approach the analysis and interpret the results of our DMD study, or studies like the DMD study, with a focus on achievable sample sizes for relatively rare disorders. Except where specifically noted, we consider sample size N = 400.
Behavior of TE-PPLD and CPH regression under the null hypothesis
In this section we contrast the behavior of the TE-PPLD and CPH-P under the null hypothesis H0 of no association (Table 1, Model 1). Specifically, we consider: fundamental differences in their sampling distributions; effects of the form of the true underlying survival function S; and the effects of varying MAF.
Baseline behavior under H0: “no association”.
Fig 1 shows scatter plots of the the TE-PPLD compared with CPH-P as a function of sample size N, under H0. As can be seen, when there is no association the distribution of TE-PPLD moves leftward as the sample size increases, making large scores less and less likely. The distribution of the CPH-P is essentially constant as a function of sample size, as theoretically expected. In addition, the replicates with larger TE-PPLDs are not always the same as the replicates with larger CPH-Ps. As previously noted, the mathematical frameworks underlying calculation of CPH-P and the TE-PPLD are different, and this leads to different results not only in terms of the scales of the two statistics, but also in terms of rank-ordering.
Effects of changing the underlying time-to-event distributions under H0.
In the previous section the generating distribution for time-to-event was normally distributed, as described above. Here we use 3 additional generating models: WB, BS and GM (see Methods). Fig 2 compares the sampling distributions of each statistic across the different generating models. In this view, both log10BR (and therefore the TE-PPLD) and CPH-P appear to be relatively robust to the underlying form of S, although in both cases some pairs of generating distributions appear to differ at the upper end of the (respective) scales.
Shown here are QQ plots for all possible pairs of generating distributions for S (Mixture of Normals (MoN), Weibull (WB), Birnbaum-Saunders (BS), Gamma (GM), for N = 400, based on 1,000,000 replicates per generating condition.
Below we will show some power calculations, and this would in principle require adjusting the significance thresholds under the different generating distributions were there an effect on the upper tail of the distribution. Adopting a significance threshold of P ≥ 5 returns 32 replicates above the threshold under the original MoN generating condition. (We use the lower significance threshold of 5 here because at this sample size there are no replicates with CPH-P ≥ 7.3 under any of these generating conditions; see also below.) Table 2 shows the significance thresholds corresponding to the top 32 replicates under the other generating conditions, along with what would be corresponding cutoffs (i.e., demarcating the top 32 scores) for the TE-PPLD based on its null sampling distribution, were we to treat it as a test statistic in the conventional way. For both CPH-P and the TE-PPLD, the variation in thresholds across generating distributions is small, and with 1,000,000 replicates, very small differences cannot be estimated with high precision. Hence in what follows we utilize nominal Type 1 cutoffs without adjustment.
Effects of lower MAFs.
One issue of particular concern when using smaller sample sizes for GWAS is the effect of low MAF on the distribution of the test statistic under the null hypothesis. Regression analysis in general requires a sufficient number of individuals (say, at least 10–15) in the subsets created by division based on covariates: here that would entail requiring adequate numbers of individuals in each of the subgroups created by stratifying on genotype and the covariate y (3 x 2 = 6 subgroups). In small data sets many SNPs may fail to meet this bar; and the impact would be most pronounced under a recessive model. Here we consider the impact of lowering the MAF while assuming recessive inheritance for CPH analysis, and compare this with the corresponding impact on the TE-PPLD, for which the mode of inheritance is not specified (Fig 3). Note that with 400 individuals and a MAF = 10%, we only expect to see 4 individuals on average homozygous for the rare allele, which means only 2 on average in each covariate subgroup.
Shown here are QQ plots comparing sampling distributions (N = 400) under H0 for minor allele frequency (MAF) = 0.4, 0.3, 0.2, 0.1 compared to the corresponding behavior with MAF = 0.5, for CPH-P and log10BR.
As the MAF decreases, the distribution of CPH-P becomes increasingly inflated, showing increasing (erroneous) larger P in favor of association. By contrast, for positive values (corresponding to TE-PPLD > π), there does not appear to be any inflationary effect on log10BR, at least until the MAF is quite small, and even then our interpretation of the results would not be materially affected: for instance, on the PPLD scale, the highest TE-PPLD is 0.96 for MAF = 0.1 and 0.64 for MAF = 0.5; in both cases, these would be clear cut “false-positive” results using our usual heuristics. The more notable effect on the TE-PPLD distribution is the progressive depression of log10BR for negative values as the MAF decreases, indicating increasingly larger (correct) evidence against association.
Another way to look at this is by noting the threshold that demarcates the top 32 scores, as a function of MAF, corresponding to a CPH-P cutoff of 5 with MAF = 0.50 (Table 3). As can be seen, substantially higher thresholds are needed to control the CPH-P Type 1 error rate at even moderate MAF. By contrast, while some increase in the corresponding TE-PPLD thresholds is seen, even at MAF = 0.1 the threshold remains well below our least stringent heuristic cutoff of 0.10.
The built-in R routine for CPH returns NaN (“not a number”) under the recessive model when there are 0 individuals homozygous for the rare allele. Under the MAF = 0.1 condition, CPH returned a NaN for 18,242 SNPs. We confirmed that there is also little to no systematic bias against the null hypothesis when the TE-PPLD is applied to those replicates that were dropped by CPH (mean PPLD = 0.0003 < π (s.d. = 0.0019; max PPLD = 0.17)).
Of course, power to detect association will also be very low for SNPs with very low MAF, because there will be insufficient variability in genotypes to detect anything. It is common to drop SNPs with MAF below some threshold (say, 1–3%), with the threshold set higher for smaller data sets. Thus there are separate reasons for ignoring low MAF SNPs in the course of a genome scan. For CPH, one could additionally forego analysis under a recessive model in order to avoid this problem; even at MAF = 0.10, there were 951 SNPs with CPH-P ≥ 5 under the recessive model, but there were only 59 such SNPs under the additive model (and the additive results included the 18,242 SNPs dropped from the recessive analysis; note that this may also represent some inflation of scores under the additive model at low MAF, since we would expect to see on average 10 SNPs with CPH-P ≥ 5). But this of course risks missing a true recessive association, which has been suggested for some DMD modifier loci . The TE-PPLD, which does not incur the same upward bias under the null in small samples that plagues CPH under these conditions, does not force this choice.
Behavior of TE-PPLD and CPH regression under the alternative hypothesis
In this section we explore the behavior of TE-PPLD and CPH-P under generating models in which there is an association between SNP genotypes and the time-to-event phenotype. We first show baseline comparisons for Models 2–8 (Table 1); we then explore robustness to different forms of generating distributions; and we consider an additional set of generating models involving epistasis, as described in that section. Finally, we consider challenges to independent replication as a gold standard for GWAS when only small to moderate sample sizes are attainable.
Baseline results under the alternative hypothesis HA: “SNP-trait association”.
Table 4 shows TE-PPLD and CPH-P results for the baseline alternative models (Models 2–8 in Table 1). Expected TE-PPLDs and CPH-Ps each vary as a function of generating conditions, with increasing means as sample size increases, as one would expect. Note that Model 7 involves effects on variances only; we would not expect CPH to detect association under this model. These baseline models were originally chosen in  to vary the mode of inheritance and the expected TE-PPLD while maintaining some reasonable ability to detect association at N = 400. In Table 4 these appear to be fairly strong effect sizes, in the sense that by N = 800 both methods are on average able to clearly detect association. However, recall that here we have used a generating MAF of 0.50; with lower MAFs average scores would be lower for both methods, in most cases appreciably so (see below). Note too that CPH analysis is run here under the generating mode of inheritance, which is generally unknown in applications to real data. This overestimates the performance of CPH relative to the common practice of relying on the additive model. For example, under the additive model, the results for Models 4, 5 (N = 400) would be just 5.34 (2.07) and 5.16 (2.05), respectively.
One salient feature of Table 4 is the large standard deviations across the 1,000 replicates per generating condition. Even though the generating model in each case represents a straight-forward genetic association model, and not, for example, a complex mixture of loci with different effects, nevertheless, both the TE-PPLD and CPH-P can vary widely from replicate to replicate in samples of this size; under mixture models standard deviations would be even larger. Fig 4 illustrates the extent of variability for each statistic on its own and in comparison with one another. We return to some implications of this level of variability below.
Effects of changing the underlying time-to-event distributions under HA.
Table 5 illustrates that under the alternative hypothesis the TE-PPLD is highly robust to the form of the underlying time-to-event distribution, across the range of distributions considered here. Thus neither the PPLD’s native “mixture of normals” assumption for a quantitative trait nor the use of the WB distribution for estimation of residuals complicates the interpretation of TE-PPLDs even when the underlying distribution violates these assumptions. By contrast, for some generating models the average CPH-P can drop considerably under some generating distributions.
Another way to look at this is from the point of view of conventional power calculations (Table 6). Here we used nominal thresholds of P ≥ 5 for all CPH analysis, with corresponding thresholds for the TE-PPLD, as described above. As can be seen, the power of the TE-PPLD is affected very little by the underlying time-to-event distribution, across the range of generating distributions considered here. By contrast, CPH-P can suffer a loss of power, in many cases, a quite substantial loss, depending on the form of the underlying distribution.
Genotype x covariate interactions.
In studying the effects of modifier genes in the context of DMD, we are interested in the possibility that modifiers of the DMD phenotype might work by affecting response to treatment with steroids. The PPLD’s procedure for adjusting the residuals for covariate effects is done by “preprocessing” the data once, independently of genotype; by contrast, in a regression framework the covariate adjustment would be done separately for each SNP. In  we speculated that, as a result, the TE-PPLD might not be well powered to detect genotype x covariate interactions.
To investigate further, we simulated data under a variety of models involving covariate x genotype interactions as shown in Table 7. We note that the particular type of interaction we are considering here is a form of classical epistasis, in which the effect of genotype on time-to-event may be masked by the absence of steroid exposure. Mathematically, this is only indirectly related to interaction in the usual statistical sense [31–33]. It is, however, precisely the form of interaction of biological interest for the DMD study. In anticipation of the following section, where variable rates of covariate classification become important in the context of replication of findings, instead of simply setting Ny = 1 = Ny = 2 = 0.5, here we randomly draw the proportion α of individuals with y = 2 within each replicate from a N(0.7, 0.1) distribution.
For the TE-PPLD, data were analyzed in two ways: (i) with each replicate (N = 400) “pooled,” that is, considered as a single data set; or (ii) dividing each replicate by covariate status and analyzing each of the two resulting data sets separately. For comparative purposes, we also analyzed the data under CPH, both without and with a covariate x genotype interaction term included in the model. Table 8 summarizes results. For the TE-PPLD, it is clear that in the presence of epistasis “pooled” analysis is much less effective than separate analysis in the y = 2 group, which is to be expected since the pooled group is a mixture of individuals, only some of whom represent any (detectable) genotypic effect. Also note that the TE-PPLD does an excellent job of distinguishing evidence for association from evidence against association, as reflected in the fact that for models Epi 1–4, in which there is no genotypic effect in the y = 1 group, TE-PPLD(y = 2) > TE-PPLD(pooled). For Epi 5–6, where there is some genotypic effect in the y = 1 group, TE-PPLD(y = 2) < TE-PPLD(pooled), but still TE-PPLD(y = 2) is considerably larger than TE-PPLD(y = 1); of course differences in sub-sample sizes also complicate interpretation in this case. Thus by comparing the pooled results with the subset results we are able to infer whether or not there is evidence of interaction, at least for Epi 1–4. This works precisely because the TE-PPLD, by contrast with CPH-P, is able to indicate evidence for H0. Note, however, that here we are considering any difference in the scores as some evidence of interaction, without specifying how large a difference should be considered decisive. (See also Conclusions, below.) Interestingly, under CPH analysis it seems preferable to not include the interaction term whether there is epistasis or not. Moreover, the interaction coefficient p-value is not a reliable indicator of whether or not epistasis exists.
Thus our earlier speculation in  that the TE-PPLD might not be useful for detecting covariate x genotype interactions appears to have been misplaced. For a binary covariate such interactions can apparently be detected by dividing the data set into two groups based on the covariate, then analyzing data with all of the data “pooled” (using y-adjusted residuals) and again separately in the two subsets; for the DMD application, where the interest is in a possible effect of steroid exposure (which would correspond to y = 2), this procedure would be applied only to the y = 2 group. SNPs with large TE-PPLDs in which TE-PPLD(pooled) > TE-PPLD(y = 2) would then be indicative of association in the absence of an epistatic interaction, while SNPs with TE-PPLD(y = 2) > TE-PPLD(pooled), or even in some cases TE-PPLD(y = 2) > TE-PPLD(y = 1), would indicate an association involving interaction. In Conclusions we revisit this approach in the context of a full genome scan.
Independent replication vs. sequential updating based on small samples.
As noted above, all of the generating conditions show strikingly high levels of variability across replicates in samples of the sizes considered here. This alone would suggest that clear-cut independent replication of a true signal might be unlikely, particularly when there might be only a few independent studies to use for comparison, each of which would also have a modest number of subjects, perhaps fewer than the initial study. To illustrate some of the issues involved in trying to replicate any DMD findings, we consider a situation we are likely to face, with access only to smaller replication samples for the time being.
In considering independent replication criteria based on p-values, there are many choices one could make regarding significance thresholds for the replication data set, and no clear answer as to which choice is correct. Here we use the NHGRI-EBI GWAS (https://www.ebi.ac.uk/gwas/) replication criteria: an association finding is considered to be replicated if either (Criterion 1) both of 2 studies gives P ≥ 5, or (Criterion 2) pooling the 2 data sets (or “mega-analysis”) gives P ≥ 5.
For purposes of illustration, we consider the additive Model Epi 1 (Table 7), and an initial data set of size NInit = 400. Among the 1,000 replicates generated for Table 7 under this model, there were 190 with additive CPH-P ≥ 5. We selected 2 replicates to serve, respectively, as initial Candidate SNPs (CandSNP): one with CPH-P over the threshold of 5 (CandSNP#1CPH-P = 5.70), and another with CPH-P meeting conventional genome-wide significance (CandSNP#2CPH-P = 7.3). Note that the proportion of individuals with y = 2 was α = 65.8% in the CandSNP#1CPH data set and 80.0% for CandSNP#2CPH. The higher value of α for the CandSNP#2 data set is an artifact of selecting the SNP based on a more stringent significance criterion, under conditions of variable α. We then attempted to replicate these signals in 1,000 independent replication data sets (RepSets) of size N = 200 under HA (Model Epi 1), and in another 1,000 RepSets under H0 (Model 1, Table 1). Note that although the CandSNPs were drawn from a model involving a true association, here were are interested in our ability to replicate a finding: in this context it does not matter whether the initial finding is a true positive or a false positive; all that matters is the magnitude of the initial CPH-P, along with whether the RepSet itself comes from HA or H0.
Just 3.7% of RepSets satisfied Criterion 1 under HA (0% satisfied Criterion 1 under H0). That is, the probability of achieving statistical significance based on any given RepSet is negligible for this model at this sample size. Of course, with a larger RepSets this probability would increase; however, the large standard deviations, combined with the “winner’s curse” effect on α might still make clear-cut independent replication problematic.
Hence the only real possibility of satisfying the replication criterion under these circumstances comes from pooling the initial and replication data sets. Table 9 shows the percent of RepDS for which the pooled CPH-P exceeded replication Criterion 2. As can be seen, when the RepDS is generated under HA, our power to replicate the CandSNP is high under this Criterion for both CandSNPs #1 and #2. However, there is also a high false positive replication rate when following up with data generated under H0. Moreover, the larger the initial signal we are attempting to replicate, the more unreliable is Criterion 2, because the more the data set in which the SNP was originally detected will dominate the pooled analysis, regardless of whether the replication data set itself supports association or fails to support association. Again, results would be different if the replication data set were larger than the initial one. Our point here is not to establish general power to replicate, but rather simply to illustrate some of the challenges of relying on replication to separate true from false positive findings in this setting.
For comparison, we selected separate replicates such that CandSNP#3TE-PPLD = 0.53, which is large enough to satisfy our usual heuristics, and CandSNP#4TE-PPLD = 0.88. Table 9 compares the pooled (N = 600) TE-PPLD with each of the initial CandSNP TE-PPLDs. What we see is that most of the time when the RepDS is generated under HA, the pooled TE-PPLD is larger than the initial CandSNP TE-PPLD, and the percentage of RepDSs with this feature is not dependent on the stringency of the criterion for selection of the CandSNP in the first place. Moreover, when the RepDS comes from H0, the pooled TE-PPLD is consistently << CandSNP TE-PPLD. Thus the TE-PPLD appears, at least based on this one set of generating conditions, to be a better approach to replication than CPH-P when replication involves simple pooling of the initial and replication data sets.
However, it could be argued that the pooled result is not really what we want, because it is driven to a large extent by the initial data set, which was selected specifically for the size of its signal at the CandSNP. This would be true for either CPH-P or the TE-PPLD, despite the very different implications in terms of sampling behavior under replication. When relying on p-values, this issue is hard to circumvent, because the only alternative is to use some version of Criterion 1, and as we have seen, even requiring less than conventional genome-wide significance can be very hard to achieve in a small follow-up data set. (One might consider some form of meta-analysis, but this also fails for very much the same reasons that pooling the data fails .)
In the PPLD framework, we do have an alternative, namely, to focus instead on the accumulation of evidence strength as new data are accrued, via the mathematically rigorous technique of Bayesian sequential updating (see Methods, above). The basic idea is quite simple: Because the PPLD can detect evidence both for HA and also for H0 (which the p-value cannot), when a follow-up data set supports “association” then on average we will have BR > 1 (or equivalently, PPLD > π); whereas when the follow-up data set supports “no association” we will have BR < 1 (i.e., PPLD < π). Sequential updating ensures that when the replication data set supports HA the PPLD increases upon consideration of the new data, while when the replication data set supports H0, the PPLD decreases.
Applying sequential updating in the current experiment, we find that 34.2% of RepDSs have TE-PPLD > π under HA, while 4% have TE-PPLD> π under H0. This tells us the probability (34.2%) that the sequentially updated TE-PPLD will correctly increase, relative to the initially selected CandSNP, regardless of the size of the initial TE-PPLD. While a 34% success rate may seem quite low, it is arguably a more realistic assessment of the likelihood of agreement between two datasets at any given associated SNP under the conditions simulated here.
In this paper we have evaluated the sampling behavior of the TE-PPLD in small to moderate samples sizes, and compared this behavior with the sampling behavior of CPH p-values, using simulations. We have noted a number of contrasts between the the TE-PPLD and CPH-P. Some of what we have found will be specific to time-to-event data, but most findings will apply to any application of the PPLD in the GWAS setting with small to moderate sample sizes.
We selected the sample sizes and topics covered here based on design questions facing a GWAS-based search for genetic modifiers of DMD, in order to inform our approach to analyzing and interpreting our own DMD data; and we have selected generating models to illustrate key points. We do not claim to have been exhaustive either in covering all possible topics or in covering all possible underlying genetic models. Nevertheless, the results presented above suggest several ways in which the TE-PPLD is a better choice than CPH in our setting.
In this final section we synthesize the implications of what we have found by loosely mimicking what might happen in a real GWAS for DMD modifiers. We assume 1,000,014 independent SNPs (no SNP-SNP linkage disequilibrium), 1,000,000 of which represent unassociated SNPs, simulated under H0; and 14 of which represent associated SNPs, simulated under HA and generated 1 each from the 14 HA generating models (Table 1, Models 2–8; Table 7 models Epi 1–Epi 7). We generated a single initial data set (InitDS) of size N = 400, and we followed up on selected SNPs in a single replication data set (RepSet) of N = 200. These simulations vary in 2 regards from those in the previous sections: (i) we generated y from a N(0.7, 0.1) distribution, as described above, for all SNPs, separately in the InitDS and the RepSet; (ii) we used the MAF distribution from our actual DMD data set (Illumina Infinium Omni2.5Exome-8 v1.4, omitting SNPs with MAF < 3%; mean MAF = 0.23, s.d. 0.14). The RepSet data were generated from the same Model and using the same MAF that gave rise to each CandSNP in turn. Also here we consider a single replicate (at each sample size, N = 400, N = 200 respectively). This last experiment, therefore, is subject to “luck of the draw,” just as any single real study would be.
For CPH analysis we assumed additive inheritance for all SNPs. We also repeated the experiment maximizing over the mode of inheritance at each SNP, but this approach resulted in far lower true positive rates; see below. We applied 2 significance criteria for selecting initial CandSNPs: either CPH-P ≥ 5.0, or CPH-P ≥ 7.3. We then followed up on all CandSNPs in the RepSet, again applying the NHGRI-EBI GWAS replication criteria as described above: a CandSNP was considered to be replicated if CPH-P(RepSet) ≥ 5 or if pooling the initial and replication data sets yielded CPH-P ≥ 5.
For the TE-PPLD we considered 3 thresholds for determining CandSNPs: 0.0430, which corresponded in this data set to a CPH-P threshold of 5 under H0, and our usual heuristic thresholds of 0.10, 0.40. (Note that the threshold of 0.0430 is larger than the corresponding threshold of 0.0201 in Table 2. This is because Table 2 was generated with MAF = 0.5, while the current simulation involves variable MAFs, and it is consistent with the slight and largely inconsequential inflation of TE-PPLD scores under low MAFs as noted above.) We considered a CandSNP to be confirmed by the RepSet if the sequentially updated TE-PPLD > original (N = 400) CandSNP TE-PPLD, or in other words, if the CandSNP yielded TE-PPLD > π in the RepSet alone.
Table 10 summarizes the overall performance of the two methods. In the InitSet, using a threshold of 7.3 for CPH-P, 2 SNPs cross the threshold, one of which represents HA, and this was the CPH condition yielding the highest True Positive Rate (TPR) = 1/2 = 50%. The highest TPR for the TE-PPLD was 60%, which occurred when using the heuristic threshold of 0.4, and yielded 3 truly associated SNPs, compared to the 1 association correctly identified under CPH. It is also interesting to note that in the initial genome scan, using CPH-P criterion of 5 and the equivalent TE-PPLD, which yielded the same number of “false positive” signals by design, led to the correct identification of twice as many true positives under TE-PPLD analysis compared to CPH analysis. Note too that filtering out SNPs with MAF < 0.10 does not affect the number of true positive findings; however, it does reduce the number of false positive findings with the lower thresholds, from 29 to 23 for PPLD using 0.0430 (TRP = 14%), and from 29 to 17 for CPH-P using 5 (TPR = 11%).
CPH-P successfully replicated both CandSNPs crossing the threshold of 5 in the initial data set, including the 1 CandSNP that initially crossed the threshold of 7.3. However, replication was also seen for 4 of the 29 H0 SNPs initially crossing the threshold of 5, though not for the single H0 SNP initially crossing 7.3. Hence with the more stringent criterion for selecting CandSNPs, the True Replication Rate (TRR) was 100%; however, only 1 truly associated SNP was identified; at the lower threshold 2 truly associated SNPs were identified, but these made up just 2/6 SNPs satisfying the replication criteria.
By contrast, the TE-PPLD found confirmatory evidence for 3/4 HA CandSNPs identified at the lower threshold of 0.0430 SNPs, and at none of the H0 CandSNPs, for a True Confirmation Rate (TCR) of 100%; using the thresholds of 0.10 or 0.40 for selection of the initial CandSNPs, the TCRs were also 100% in both cases, with 3/3 HA SNPs confirmed and 0 H0 SNPs, at both thresholds. This means that one true positive SNP, identified in the initial data set, was not confirmed in the smaller, lower power follow-up data set. Hence failure to confirm, even in the context of these relatively simple generating models, does not necessarily mean that the initial finding was a false positive. But overall, the TE-PPLD identified more truly associated SNPs than did CPH-P, and with 0 false-positive confirmations using a threshold of 0.10 or higher for selection of CandSNPs in the initial data set.
Note too that when maximizing over mode of inheritance (recessive, additive, dominant), at a threshold of 5 CPH-P picked up one additional true positive finding (the recessive Model 4, which was missed by additive analysis) and 439 false positives, for TPR = 1%; when dropping SNPs with MAF < 10% the TPR improved only slightly, to 2%. At the higher threshold (7.3) the TPR was 4% and 10%, respectively, with and without the low MAF SNPs included. Thus allowing for dominance in CPH analysis by maximizing over the mode of inheritance appears to perform very poorly, in contrast to relying on a single analysis under the additive model. As noted above, however, the additive model will tend to have reduced power in the presence of dominance. The PPLD, which does not require specification of the mode of inheritance, does not face this issue.
Also informative is a comparison of rank-ordering of the TE-PPLDs or CPH-Ps (additive) (Table 11), of the 14 SNPs generated under HA among all 1,000,014 SNPs. In this particular replicate, Model 5 yielded the top score for either method. The 2nd largest TE-PPLD was obtained under Model 4 and ranked 3rd (that is, the 2nd highest TE-PPLD was from H0), while the 2nd highest CPH-P was ranked 30th (Model 2). The 3rd highest TE-PPLD was from model Epi 3 and ranked 5th; while the 3rd highest CPH-P was also from Epi 3 but ranked 62nd. Thus even under these conditions, in which it was apparently quite difficult to detect signals at most of the HA models, the “true” positives tended to cluster closer to the top of the TE-PPLD rankings than the CPH-P rankings.
As a final experiment, we attempted to use stratification on y to detect covariate x genotype epistasis using the TE-PPLD, per section 3.2.3 above, particularly hoping to detect Models Epi 1–4 (Table 7). We considered two ways of doing this: (i) Based on the initial N = 400 data set, we rescanned the genome separately in the y = 2 group, accepting as evidence of epistasis any SNP at which the y = 2 TE-PPLD crossed a threshold (0.0430, 0.10, or 0.40) and also for which TE-PPLD(y = 2) > TE-PPLD(N = 400), that is, in which the subset-specific PPLD exceeded the “pooled” (across y) PPLD; (ii) performing the separate y = 2 analysis only for those SNPs crossing the given threshold in the initial “pooled” genome scan. Table 12 shows the results. In both cases, the single true positive result occurred for Model Int 3. Rescanning the entire genome in y = 2 did not detect any additional true positive signals, compared to following up only on those SNPs already detected in the pooled analysis, which is somewhat surprising given the results in Table 8; however, it did produce additional false positive signals, no doubt due to the smaller sample size. (Recall that under H0, the PPLD is a less reliable indicator of evidence in favor of H0 the smaller the sample size.) The highest true confirmation rate (0.50) occurred when performing the separate y = 2 analysis only at SNPs crossing the 0.40 threshold in the initial pooled analysis. It is also notable that we did not erroneously infer epistasis for any of the non-epistatic HA models.
In summary. We have illustrated several advantages of the PPLD over regression analysis in the context of GWAS with small to moderate sample sizes, both for identifying candidate SNPs and for confirming them in follow-up data sets (Table 13). We considered a range of models for genotypic effects on a time-to-event phenotype, and found that these tend to have low to moderate power in the sample sizes considered here, particularly using a realistic minor allele frequency distribution as would be found on a standard SNP array. In addition, these models showed very high variability across replicates, leaving a large role for chance both in terms of which truly associated SNPs can be detected in any given study and also in terms of our ability to find the same SNPs in follow-up data sets.
While there is no way to know in advance what sorts of models underlie true modifier effects in our DMD study, still, our results suggest overall that increasing sample sizes will be important, not only for the reliable detection of modifier genes but also for identification secondary effects such as covariate x genotype interactions. At the same time, however, our results support the overall reliability of TE-PPLD findings even in a data set of just N = 400; and they strongly support our contention that it is not necessary to aim for sample sizes in the the thousands or tens of thousands in order to reliably detect genes under the GWAS design, provided one applies statistical methods that are well-adapted to inference in small data sets.
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