A statistical approach to validation

Interesting results in high-dimensional studies are the assays that are statistically significant at a specified false discovery rate (FDR). The FDR can be thought of as the acceptable level of false positives among a set of significant results [3]. If 100 variables are significant at an FDR ≤ 5%, then we expect no more than 0.05 × 100 = 5 false discoveries. Applying an independent technology to confirm all 100 variables and finding 5 or fewer false positives would verify this claim. However, applying independent technologies or functional assays to individual discoveries is often costly and time consuming.

Here we propose to experimentally test a random sample of significant results with an independent technology and confirm the false discovery rate with a statistical procedure. The approach consists of manually confirming a sample of n hits with the independent technology, determining the number of false positives, n _FP , and calculating the probability the true proportion of false positives, Π₀, is less than the claimed FDR of $\hat{\alpha }$. When a subset of features is claimed to be significant at a specified FDR, α, then the expected proportion of false positives among the significant results is α. The expected proportion of false positives in the validation sample Π₀, should then be approximately equivalent to the original FDR. We can then use the expected proportion of false positives in the validation sample to confirm the original FDR estimate.

If the probability $\mathrm{Pr}({\Pi }_0\le \widehat{\alpha }|n_{\mathrm{FP}},n)$ is larger than 0.5, then the validation sub-study supports the original FDR estimate, although larger values are required to strongly support validation. Using the posterior distribution, it is also possible to calculate a posterior credible interval for the false discovery rate, Π₀, as a measure of variability. This probability represents a direct measurement of concordance, unlike statistics like the correlation between the original and validation statistics which measure only agreement and depend on the scale of the measurements being taken [14].

Calculating validation probabilities

So the number of false positives, n _FP has a binomial distribution with parameter Π₀. We assume a Beta(a b) conjugate prior distribution for Π₀[15], so the posterior distribution is a Beta(a + n _FP b + n−n _FP ) [16]. This result relies on the independence of the validation experiments, but does not rely on the rate of true or false positives. The potential reasons for dependence between validation tests include batch and other technical artifacts [9]. However, methods have been developed to address these potential sources of dependence [8, 17], which lead to nearly independent hypotheses [18].

Using this posterior distribution it is straightforward to calculate the probability that the FDR (Π₀) is less than the claimed level $\left(\hat{\alpha }\right)$: $\mathrm{Pr}({\Pi }_0<\hat{\alpha }|n_{\mathrm{FP}},n)$. We can also calculate the posterior expected value of the FDR using the mean of the posterior, $\frac{a+n_{\mathrm{FP}}}{a+b+n}$, which can give us an idea of the actual FDR of our original results. For our analysis, we made the assumption that the prior distribution for Π₀ was U(0,1), by setting the parameters to the Beta function to a = b = 1. This prior is somewhat conservative, since it is likely that many of the results in the validation experiment will be true positives.

In some cases it may be useful to encode the belief that most of the results will be true positives in the prior by choosing values of a and b that put greater prior weight on higher validation probabilities. Specifically, a much less conservative prior choice would set the mean of the prior distribution to be the observed FDR for the validation targets in the original study $\hat{\alpha }=\frac{a}{a+b}$. Since this choice permits a range of solutions for a and b, one could select the choice that maximizes the prior variance $\frac{\mathrm{ab}}{{(a+b)}^2(a+b+1)}$. This could be accomplished by choosing a to be a small value like 0.01 and setting $b=\frac{1-\hat{\alpha }}{\hat{\alpha }}\times a$.

For small values of $\hat{\alpha }$ this prior may influence the prior probabilities and make it difficult to compare validation at different FDR levels. This is a particular concern given the known variability in FDR estimates, particularly for small sample sizes or low FDR levels [19]. In general, conservative and non-adaptive prior distributions will lead to less potential for bias and greater comparability across FDR levels. Our R functions for calculating the validation probabilities allow for different choices of a and b. However, the validation probability is somewhat robust to prior choice (Results).

Bootstrap confidence intervals for the validation probabilities

The bootstrap is not justified for small sample sizes, and when the validation sample size is small, these bootstrap confidence intervals may not have the appropriate coverage.

Choosing the FDR level and sample size

In other words, what is the minimum validation sample size needed to get at least the target validation probability, assuming that q × n false positives will be observed?

Here, as in any sample size calculation, we must estimate the effect size - in this case the expected number of false positives in the validation set. In our examples, we estimate the effect size as the observed FDR for the validation targets. However, our R functions allow for alternative choices of the expected FDR for each true FDR level. If a user chooses higher FDR levels than the observed values, the minimum sample size will be smaller to confirm that higher FDR threshold.

This optimization problem can be solved for any specific study based only on the set of p-values for the original analysis performed. For a fixed target probability and a fixed false discovery rate threshold, the minimum sample size will be fixed as long as the number of significant features n _sig (q) is sufficiently large. The reason is that the optimization is over only the single variable n, when q and the Target Probability are fixed.

As an example of this procedure, we use the data from the first simulated study (of 100) in the errorless validation simulation as described in the Results. Based on the p-values from that study, we calculated the minimum validation sample size needed for each FDR threshold to achieve a target validation probability of Pr(Π₀ < q|n × q,n) ≥ 0.5 (Figure 2). For an FDR of 5%, it is not possible to achieve the desired validation probability. For increasing FDR cutoffs, the required validation sample size decreases. This is not surprising, since we have shown in the previous section that validation is more likely at higher FDR thresholds. The lower the FDR threshold used for validation, the more convincing the validation may be, so the investigator can calculate this curve for any given study and use the results to decide how many results to validate based on available resources.

If the authors have designed their study using our estimates of the minimum validation sample size, and the validation probability is low, then it is likely that Π₀ is greater than the claimed FDR level. If however, they choose to validate many fewer targets than suggested by the minimum validation sample size, it is ambiguous whether the sample size was too small or the FDR did not validate.

Calculating qPCR validation costs

Manually confirming genomic results with independent technologies or functional assays can be costly and time consuming, since most validation technologies must be performed one gene, transcript, or protein at a time. There are a large number of validation technologies, but one of the most commonly used is quantitative PCR (qPCR). To compare costs associated with different strategies we use qPCR validation for gene expression studies as an example. The results presented here are representative of the results for any costly independent confirmation experiment.

We estimated the costs associated with two different qPCR technologies: SYBrGreen and TaqMan. For TaqMan we assumed that three genes and a reference gene were multiplexed in each reaction, which is theoretically possible but optimistic in practice. SYBrGreen reactions also included a reference gene, but were not assumed to be multiplexed. We calculated costs as follows: $250 for each TaqMan probe, $150 for Mastermix for each plate for both SYBrGreen and TaqMan, and $4 for each 96-well plate. We assumed each reaction was replicated three times to ensure accurate measurements - a typical approach taken in validation experiments. We also made the assumption that one research assistant, working full time and paid $40,000 per year, could run and analyze four 96-well plates per day. For the purposes of our analysis we assumed 22 working days per month.

In these equations the terms inside the floor operators ⌊·⌋ represent the number of plates needed to run the reactions, which must be multiplied by the fixed costs for those plates. From these equations, it can be seen that manual confirmation of gene expression results using either Taqman or SYBrGreen is costly and time consuming. Taqman is slightly more expensive, but slightly less time consuming.

Non-random validation can not be used to confirm a complete list of significant results

Manually confirming only the most significant results (Figure 1) is probably the most common validation strategy in genomic studies. To highlight the need for a statistical approach to validation, we consider two representative studies that use this strategy. In both cases, the implicit assumption was made that validating the most significant results was sufficient to support the accuracy of a statistical method or an entire list of significant results [21, 22]. This assumption is often made by investigators who perform global gene-set enrichment or functional analysis of significant features, after manually confirming only the most extreme results. This has led some to suggest that individual validation experiments do not reflect the quality of genomic results [4].

In the first study 388 cassette exons were identified with a Bayesian network as Nova alternative splicing targets with FDR < 1%. A convenience sample of 31 exons was validated, yielding 28 true positives [22]. We calculate the posterior distribution for the estimated FDR among the validated results using the Beta distribution derived from equation 1 (see Methods). Based on this posterior distribution we obtained a 95% posterior credible interval for the FDR of (0.04, 0.25). Our results suggest that based on the validation sample the true FDR for the original analysis is likely between 4% and 25%, substantially higher than the reported FDR. This is not surprising, since the expected number of false discoveries at an FDR of 1% is 0.01×31=0.31. Since the claimed FDR does not match the validation results, the probability of validation (see Methods) is low, $\mathrm{Pr}({\Pi }_0\le 0.01|n_{\mathrm{FP}}=3,n=31)=2.87\times 10^{-4}$.

In the second study 53 master regulator transcription factors were identified for a “mesenchymal” gene expression signature at an FDR of 5% [21]. The two most significant master regulators were confirmed functionally. The investigators did not report the FDR for the hits that were functionally confirmed, so it is not possible to directly calculate the validation probability. However, we make the conservative assumption that these hits were also significant at an FDR of 5%, although the FDR is likely much lower for the top two hits. Using the less stringent threshold of 5% the 95% credible interval for the true FDR is wide (0.01, 0.70). The reason is that although 100% of tested results were confirmed using the functional test, the sample size is too small to confirm the original false discovery rate claims. Thus, the corresponding probability of validation Pr(Π₀ ≤ 0.05|n _FP = 0,n = 2) = 0.14 is low.

In these studies, only a small fraction of the top hits were confirmed with independent technology. However, the entire list of significant results was used in each case to form a biological picture and interpretation of the results. The corresponding validation probabilities suggest that this confidence in the entire set of significant results is not sufficiently justified. These examples are illustrative of typical validation strategies and suggest the need for a new approach for supporting lists of significant results with independent measurements. In the next section, we show that manually confirming a random sample of results is more effective than manually confirming only the most significant results when the goal is to provide statistical support for validation.

Statistical validation can be used to confirm lists of significant results

To compare: (i) manually confirming only the most significant results and (ii) statistical validation, an example is needed where every single genomic feature is assayed on both the original technology and an independent validation technology. Such data sets are rare, since most measurements from high-throughput -omics studies are not confirmed with independent technology because of time or financial constraints.

To make this comparison, we obtained expression data for 805 genes from a study of brain and reference tissue measured by both RNA sequencing and an independent technology, quantitative PCR (qPCR) [23]. These data can be thought of as an experiment where every gene has been subjected to independent confirmation. Previous studies have shown that the qPCR and RNA-sequencing approaches produce comparable results in this experiment, suggesting that the validation probability should be high [23]. We use RNA-sequencing and quantitative PCR (qPCR) as a representative example of the type of independent manual confirmation that is performed in genomic experiments. However, our results easily generalize to any costly/time-consuming independent validation, whether it is with an independent technology or using a functional assay.

We compared the two strategies: (i) statistical validation by manual confirmation of a random sample of 20 genes significant at an FDR of 10% and (ii) manual confirmation of only the top 20 genes. We performed a likelihood ratio test to identify differentially expressed genes based on RNA-sequencing [23, 24]. Genes with a qPCR fold-change above 0.25 were considered true positives. The estimated FDR for the top 20 genes was 1.09×10⁻⁷. All 20 were true positives according to qPCR. The FDR threshold is extremely low so a huge number of results would need to be confirmed to convincingly support the FDR claim. It is not surprising then, that the 95% posterior credible interval (0.001,0.161) does not cover the original FDR estimate and the validation probability for the 20 best hits is low $\mathrm{Pr}({\Pi }_0\le 1.09\times 10^{-7}|n_{\mathrm{FP}}=0,n=20)=2.28\times 10^{-6}$.

For a random sample of size 20 from among the 591 genes significant at an FDR of 10%, on average 1.65 were false positives. Since this is a random sample of results with FDR < 10%, the validation probability is calculated at the 10% threshold (Methods), yielding a 95% posterior credible interval (0.02, 0.28) that covers 10% and a substantially higher value of Pr(Π₀ ≤ 0.10|n _FP = 1.65,n = 20) = 0.44.

These results suggest that choosing a random sample of results based on a higher FDR threshold made it easier to statistically support significance claims in genomic studies. In the next section, we show that when attempting to statistically validate a set of significant results, it is generally better to choose a higher FDR threshold.

Statistical validation is more likely at higher FDR thresholds

As an example, suppose that in a given study the observed number of false positives is fixed at 0.7 × FDR level × the validation sample size, so the validation probability is expected to be high. In this scenario, the validation probability increases with increasing validation sample size (Figure 3). But for a fixed validation sample size higher FDR thresholds on the original data set lead to higher validation probabilities.

The reason is if only the most significant results are chosen for manual confirmation, then the claimed FDR for these results will be very small. Even if all of the results are confirmed as true positives, it is difficult to prove the FDR claims. For example, suppose the top 10 results correspond to an FDR of 1×10⁻⁵. This suggests that for every 100,000 results there should only be one false positive. Strong evidence for this claim would require confirming a huge number, hundreds of thousands, of results. Alternatively, if 10 results are validated at a higher FDR threshold like 0.50, then we expect about 5 false positives. If only 3 or 4 are observed, this would lend reasonable support to the FDR claims.

Simulation study to evaluate the validation probability

The validation probability is a measure of how well the conclusions of the original study are supported by the validation sample. We have demonstrated the potential utility of this approach using real data sets.

We also consider three simulated scenarios to evaluate the properties of the validation probability parameter estimates. For each scenario, we calculate the validation probability using a conservative prior (a = b = 1) and the adaptive prior described in Methods (a = 0.01 and $b=\frac{1-\hat{\alpha }}{\hat{\alpha }}\times a$). In each case, we simulated 100 gene expression experiments with 1,000 genes, 300 of which were differentially expressed. In all cases, we assume that all genes which were significant at the given FDR cutoff were validated (exhaustive validation).

When the original experiment is supported by the validation data and the validation test is perfect (Table 1 Scenario: Errorless validation) the validation probabilities are high. This is true even in the more realistic setting where the independent technology is subject to random error (Table 1, Scenario: Validation subject to error). The validation probability is also larger for larger values of the FDR, consistent with observations on the real examples described above. The posterior expected value of the FDR based on the validation data is at or below the FDR from the original data set, which is consistent with expectations. In the case where the validation experiment does not support the original experiment, the validation probabilities are much lower (Table 1, Scenario: Results should not validate). Here, we can see the posterior expected value of the FDR exceeds the FDR from the original data set, again as expected.

Interestingly, the coverage probabilities for the 95% posterior credible intervals are greater for lower values of the FDR. The reason is that the estimate of the FDR is conservatively biased and this bias is stronger for higher FDR cutoffs. The bias means that the posterior credible intervals cover the true FDR, but frequently do not contain the original FDR estimate because of the conservative bias.

The adaptive prior led to slightly higher validation probabilities but lower coverage of the 95% credible intervals - suggesting that the adaptive prior may be slightly anti-conservatively biased. However, the estimates were not wildly different suggesting relative robustness of the validation probabilities to the choice of prior.

Statistical validation is cheaper and less time-consuming than manually confirming all significant results

When performing statistical validation, only a subsample of results must be confirmed, so the costs are substantially lower. Although the costs are still high in this case, they are substantially less prohibitive than manually confirming an entire list of results. The larger the list of significant hits, the more pronounced the savings from statistical validation. From the table, it is clear that validation at the FDR 5% level is both costly and time consuming, even using statistical validation. As we have shown, higher FDR thresholds lead to smaller minimum validation sample sizes (Figure 3). The results in Table 2 suggest that choosing a higher FDR threshold for statistical validation may be more economical.