We distinguish two kinds of repeated experiment. When the same IGR is spotted onto an array in more than one location, we term these measurements 'duplicates,' and we consider them as two spatially separated parts of the same 'spot'. Though other approaches have been described 14, for simplicity we average duplicate signals before analyzing them, giving us a single value that is less susceptible to physical blemishes on the slide. When the same IGRs are spotted onto two or more distinct microarrays, we term them 'replicates.' We consider each replicate as an independent measurement of the binding affinity or 'occupancy' of the IGRs.

It is common to replicate genome-wide experiments several times, to improve confidence in the results, which may be degraded by array imperfections or by handling errors. Additional replicates can compensate for random error in individual measurements, and the typical number of replicates is likely to increase as the cost of microarrays falls 1. Sometimes the most significantly enhanced IGRs are those with low signal-to-noise ratio, yet applying log-ratios to such signals has the potential to introduce many false positives because minor variations in a small denominator value can have a large effect on a ratio. A single-array error model can account for this variation in calculating significance for each IGR. The log-ratios themselves are difficult to interpret, however, because two IGRs with the same log-ratio may differ in significance, and a greater log-ratio does not indicate increased significance. An alternative approach, the method of variance stabilization, was described by two groups 1516 and made available as part of the BioConductor project 17 in the package 'vsn' 15. It uses a regression algorithm that is robust to outliers to scale and offset each channel independently, in such a way that the variance between channels is independent of signal strength. The transformation of the signal y_i in the ith channel (i = 1 for IP, or i = 2 for control) can be expressed as:

where α_i and λ_i represent the background and noise in the ith channel, respectively. Because ln(a) - ln(b) = ln(a/b), the difference between the two transformed channels (Δh ≡ h_i - h₂) is then a generalized log-ratio that is asymptotically equivalent to the log-ratio of the original channels when both are high (y_i >> α_i), yet transforms smoothly to the difference between channels when both are low. This allows direct comparison between any two datapoints, even when they belong to opposite ends of the microarray's dynamic range. Two IGRs with the same Δh are equally significant, and greater Δh implies a more significantly bound IGR.

Binding of protein to DNA is a dynamic, stochastic process in equilibrium. While every TF is likely to be bound to every IGR at least some fraction of the time, our goal here is to perform binary classification of the IGRs. We therefore consider IGRs to fall into two categories: those that are specifically bound by the TF and those that are not. We wish to compute a p value that expresses our degree of surprise at seeing a particular Δh score for a given IGR, under the null hypothesis that the IGR is not bound. The 'vsn' package can be used to variance-stabilize each array separately, or all of them simultaneously; we used the former method. Having computed the inter-channel variance-stabilized difference (Δh) for each spot, we may plot a histogram of all scores from a chip. We expect that most regions are not bound. Therefore, the distribution of Δh scores should be largely determined by random binding and measurement errors 18. A smaller number of regions are bound, and those will tend to have positive scores, indicating higher occupancies in the IP channel than the whole-cell extract/mock control. Measurements in the negative portion of the Δh distribution should, therefore, be more completely dominated by unbound IGRs. By fitting a parametric curve to the region of the observed Δh distribution left of the mode, we obtain an estimate of the null distribution in the positive region of the Δh distribution. This is an essential feature of our method, because it allows us to estimate the distribution expected of unbound IGRs without performing an external control experiment in which an identical mixture is examined in both channels of a separate microarray. It is this null distribution that permits calculation of significance for each observed Δh value. The symmetric nature of the null distribution is an assumption of our model, and is based on our own experience and that of others 19.

Specifically, a parametric distribution is fitted by minimizing the negative log-likelihood of the data to the left of the mode (found after smoothing the data using gaussian kernel-based density estimation) 2021. Three possible distributions were initially considered (normal, Cauchy, and Gumbel), but the normal distribution consistently obtains the best log-likelihood score. Goodness-of-fit for the fitted normal distributions was verified with a χ² test, and all passed with p < 10^-20. The Δh scores from all replicates are standardized (centered to have zero mean and re-scaled to have unit variance) yielding a score z_i = (Δh_i - μ_i)/σ_i, where μ_i and σ_i represent the mean and standard deviation, respectively, of the Δh values obtained from replicate i. Figure 1a-c shows Δh distributions for three replicates 22. We expect the distribution of Δh scores to be centered about zero; as shown by the vertical dotted lines in Figure 1, this is true to a very good approximation. Variance stabilization attempts to transform the data such that measurement error is uniform for each spot on a given array, and if replicate arrays were identical, one would expect to see the same variance in each array; large discrepancies between arrays might indicate problems with the quality of some of the arrays. Standardization is necessary to account for minor (on the order of 10%) differences in variance between arrays. Standardized scores are averaged to give an overall score (), the distribution of which is shown in Figure 1d. This distribution is again smoothed with a gaussian kernel, and fitted as described above. Finally, a p value for each IGR is computed on the score, according to the null hypothesis that all IGRs are described by this fitted normal distribution, that is, they are not bound by the TF.

The distribution of computed p values is shown in Figure 2a. It clearly shows near-ideal behavior: uniform distribution across most of the interval (0,1) arising from the vast majority of unbound IGRs, and a peak close to p = 0, arising from bound IGRs. Figure 2b shows the distribution of q values. As expected, most IGRs have a high q value, consistent with the assumption that most are unbound. False discovery rates, as represented by q values 23, are particularly useful when the goal is discovery of TF-bound IGRs. For example, the q values for Sko1 (see Additional data file 1) indicate that scientists willing to accept a list of targets in which 33% are false positives should examine the top 224 entries using a more-accurate experimental method, while those only willing to tolerate a false-positive rate of 20% should restrict themselves to the top 91.

We independently validated 35 target genes spread widely across the top 350 in our list using targeted ChIP analysis. Considering only the 35 targets for which follow-up testing was performed, ranking of IGRs by the p values of Lee et al. 2 (see Additional data file 4) shows an ability similar to our method ('Chipper') at placing true positives above false positives. When considering all IGRs, however, there is little correlation between rank by our method and rank by the Lee et al. approach. In other words, top-ranking targets by one method are not top-ranking by the other. Thus, although our validation experiments are consistent with Chipper achieving the same sensitivity at a lower false-positive rate, it is also possible that the two methods are each adept at identifying different subsets of targets. The discrepancy may be due to some systematic error in determination of the parameters of the error model. As the error model parameters are not provided explicitly with their data, we could not investigate this possibility further. Inaccurate determination of error-model parameters can lead to unjustified confidence in differences based on noisy measurements. Therefore, in the task of ranking IGRs by the likelihood of being TF-bound, Chipper is on par and complementary to the Lee et al. approach and may outperform it. Furthermore, the Chipper algorithm uses an internally determined error model and thus is not subject to systematic errors that may arise via the separate control experiments required of the methods in Lee et al. 2. Below we show that Chipper allows increased sensitivity at a given significance threshold.

Chip² experiments cannot distinguish the strand on which binding occurs, only the location at which it takes place. When binding is assigned to an IGR less than 2,000 nt in size, which happens to separate two genes on opposite strands, it is not possible to determine, on the basis of Chip² alone, which one is the target of a TF. For example, as illustrated in Table 1, FAA1 and COT1 are divergently transcribed genes separated by a 1,800 nt IGR. The IGR is split into FAA1-proximal and COT1-proximal IGR segments. The primers used for targeted ChIP (about 200 nt) are smaller than the sheared fragments used in the microarray experiments (500 nt), which gives them a greater spatial resolution. As the primers are designed for a specific promoter, and amplified by polymerase chain reaction, they are strand-specific. Only FAA1 is found to bind Sko1 in a targeted ChIP experiment, yet because both IGR segments overlap Sko1-bound fragments in the Chip² experiment, a spurious positive result is generated for COT1. We score correctly identified IGRs as true positives, even when only a single gene is verified in the targeted experiment. The Sko1 data, along with further study of Sko1 targets, are published elsewhere in the context of a focused study of Sko1 22.

A common measure of significance used in hypothesis testing is the p value. In large-scale experiments like these, random chance can cause some IGRs to have p values that will be considered significant. Multiple hypothesis corrections (that is, corrections for the fact that a hypothesis is being tested multiple times, once for each IGR) are a popular approach in which the significance threshold is raised (or the p value lowered) as a function of the number of IGRs. Bonferroni-type 24 corrections are often conservative, in that many positives may be classified as non-significant ('false negatives'). This is borne out in our analysis of Sko1 Chip² data, in which, after multiple-hypothesis correction, only a small number of IGRs (<10) were significant, at an experimentwise p value = 0.05 or lower (equivalent to p = 1.06 × 10^-5 before multiple-hypothesis correction). However, the motivation of most Chip² users is not to cautiously establish a list of binding sites that are known with near-certainty. The attraction of Chip² is its high-throughput nature, which allows the experimentalist to rapidly generate a list of potential binding sites for subsequent study. A relatively recent alternative to the p value is the q value, which is a measure of false discovery rate (FDR) that has proven useful when the aim of an experiment is hypothesis generation rather than hypothesis testing 232526. Despite the fact that Chip² experiments are typically used for hypothesis generation, no previously reported analysis of Chip² experiments has employed an FDR approach. Figure 3 shows that the q values computed from our p values (broken line) agree quite well with our empirical FDR (solid line). As the first verified false positive ranks just above 100, our empirical FDR is zero to that point. Thereafter, it tracks the computed FDR quite closely until all true positives have been discovered.

We obtained the raw data used by Lee et al. 2 and compared the p values produced by our algorithm with the published p values. The 7,200 IGRs were ranked using the appropriate score for each method, and the ranked lists were evaluated for the presence of targets annotated as bound by the TF of interest in the Yeast Proteome Database (YPD) 2728. Data for two TFs (Ino4 and Sko1) are shown in Figure 4, and analysis of another six TFs is shown in Additional data file 5. In Figure 4a we show the receiver-operating characteristic (ROC) curve for Ino4, which tracks the sensitivity of an algorithm (its ability to find true positives (TPs)) as a function of its tendency to turn up false positives (FPs). An optimal algorithm would rank all TPs at the top. Its ROC curve would begin at the lower left-hand corner (FP = 0, TP = 0), move vertically to the upper left-hand corner (FP = 0, TP = 1), and then across the top of the chart to the upper right-hand corner (FP = 1, TP = 1). As this is a hypothesis-generation technique, only those targets near the top of a ranked list are likely to be of interest; we therefore show only the region from FP = 0 to FP = 0.1. The ranking performance of each algorithm is good in this case, and there appears little to choose between methods: either one can achieve a sensitivity of almost 1.0 with a false-positive rate of about 0.05.

In practice, however, it is common to consider only those IGRs passing a standard threshold of significance (p < 10^-3 in Lee et al. 2 and Harbison et al. 8). Therefore, we evaluated the same data, but rather than focusing on simple ranking ability, we examined the p value of each call (results for Ino4 shown in Figure 4b). We constructed the graph by choosing a significance threshold (α) and asking what fraction of the known true positives exceed the threshold (that is, have p values less than α). At α = 1, any algorithm will have perfect sensitivity because it calls all IGRs significant; this comes at the cost of specificity, as it is unable to distinguish between true and false positives. The p values reported by Lee et al. 2 are shown in green, those by our method are shown in black. The vertical dotted line indicates a threshold α₇ = 10^-3 at which we would expect approximately 7 out of 7,200 intergenic regions to achieve significant scores purely by chance, even if none were bound by the TF. The vertical dashed line indicates the threshold α₁ = 1.6 × 10^-4, which we expect to be exceeded by chance for only one out of 7,200 IGRs. The unshaded area to the right of α₁ indicates the region in which fewer than one IGR would be expected to exceed the threshold by chance. The higher an algorithm's sensitivity in this region (that is, the more true positives it puts here), the better. As we decrease the threshold, the sensitivity decreases slowly at first, for both methods. For the p values of Lee et al. 2, there is then a rapid reduction in sensitivity. At an α threshold such that only one false positive is expected, our method can recover more than half the known targets while Lee et al. 2 find none.

In Figure 4c, we show an ROC curve for the transcription factor Sko1, for which nine targets are annotated in the YPD. The error model of Lee et al. 2 ranks the targets slightly better than our method of average z scores. Yet, as shown in Figure 4d, for any given significance threshold, our algorithm returns more of those targets. Ino4 showed the most striking improvement in sensitivity (Figure 4b) for all TFs examined. However, for each of the eight TFs we examined (Figure 4 and Additional data file 5) our method called an equal or greater number of targets significant at the level of α₁ than did the method of Lee et al. 2. Thus, for all TFs examined, our method yields sensitivity either markedly better than or similar to that of the de facto standard method.