A statistical framework for assessing pharmacological responses and biomarkers using uncertainty estimates

High-throughput testing of drugs across molecular-characterised cell lines can identify candidate treatments and discover biomarkers. However, the cells’ response to a drug is typically quantified by a summary statistic from a best-fit dose-response curve, whilst neglecting the uncertainty of the curve fit and the potential variability in the raw readouts. Here, we model the experimental variance using Gaussian Processes, and subsequently, leverage uncertainty estimates to identify associated biomarkers with a new Bayesian framework. Applied to in vitro screening data on 265 compounds across 1074 cancer cell lines, our models identified 24 clinically established drug-response biomarkers, and provided evidence for six novel biomarkers by accounting for association with low uncertainty. We validated our uncertainty estimates with an additional drug screen of 26 drugs, 10 cell lines with 8 to 9 replicates. Our method is applicable to any dose-response data without replicates, and improves biomarker discovery for precision medicine.


Introduction
The failure rate for new drugs entering clinical trials is in excess of 90%, with more than a quarter of drugs failing due to lack of efficacy (Arrowsmith and Miller, 2013;Cook et al., 2014). The rapid development of technologies for deep molecular characterisation of clinical samples holds the promise to uncover molecular biomarkers that stratify patients towards more efficacious drugs, a cornerstone of precision medicine. In oncology, we can identify potential biomarkers of drug response in high-throughput screens (HTS) of patient-derived cell lines; these biomarkers need to be then validated in patients.
Assessment of cell line drug response typically involves treatment with multiple concentrations of the compound, followed by measurement of the amount of viable cells after a fixed period of time for each dose, and derivation of a dose-response curve. The drug response is commonly then summarised by measurements taken from this curve, most often the concentration required to reduce cell viability by half that is IC 50 , or the area under the curve that is AUC. Currently the two largest in vitro drug screening studies, the Genomics of Drug Sensitivity in Cancer (GDSC) (Garnett et al., 2012;Iorio et al., 2016) and the Cancer Therapeutics Response Portal (CTRP) Rees et al., 2016 have shown that some clinically-actionable biomarkers of drug response can be concordantly discovered Seashore-Ludlow et al., 2015), and that different properties and mechanisms of drug response are best captured by different metrics dependent on the dose-response curve (Fallahi-Sichani et al., 2013).
Most HTS efforts focus on increasing throughput Seashore-Ludlow et al., 2015) and thereby often neglect experimental replicates, which renders it impossible to correct for experimental noise, resulting in uncertainty for the estimated drug-response metrics (e.g. IC 50 value). Extrapolating IC 50 values beyond the tested drug concentration range is particularly challenging and often unaccounted for in quality control metrics (Haibe-Kains et al., 2013;Haverty et al., 2016). Most published studies using machine learning algorithms or mechanistic models for predicting drug response and biomarkers assume that the measured drug responses are precise (Costello et al., 2014;Keshava et al., 2019;Menden et al., 2019;Silverbush et al., 2017). If this assumption is not met and there is high uncertainty in the measured drug-response values, the utility of these methods for enhancing drug development may be severely limited (Costello et al., 2014;Menden et al., 2019;Silverbush et al., 2017). Experimental noise can be reduced by adding experimental replicates, however, this either reduces the throughput of the screen or increases the cost. Most current models for curve fitting and describing dose-response data have primarily assumed that cell viability has a sigmoidal relationship to the logarithm of the dose concentrations of the drug (Dawson et al., 2012;Wang et al., 2010). Whilst some models are more flexible by allowing many inflection points in the dose-response curve (Di Veroli et al., 2016;Vis et al., 2016), their main output is a single drug-response value that does not fully capture the uncertainty in the measurements (Fallahi-Sichani et al., 2013).
Gaussian Processes (GP) are a flexible, probabilistic modelling technique that has been successfully used to measure uncertainty in noisy gene expression datasets (Lopez-Lopera and Alvarez, 2019) and has been incorporated into machine learning prediction of cell fates (Boukouvalas et al., 2018). This technique has been shown to cope well with regression tasks on dependent data and high dimensional covariates (Rasmussen and Williams, 2005;Shi and Choi, 2011). Instead of fitting a single function to the data, GPs allow for a flexible range of beliefs about the function underlying the data (Tian et al., 2017). In the case of cell line drug responses, this can be conceptualised as fitting a range of curves that have equivalently strong fit to the data. We can sample from the inferred posterior distribution over functions, that is the variance between these curves, to generate uncertainty estimates of quantities of interest, in our case, properties of the dose-response such as IC 50 .
GPs have been recently utilised to identify and guide experimental validation of compounds, on top of being applied to protein engineering and imputing gene expression values (Hie et al., 2020). GPs have also been used in conjunction with neural networks to model dose-response curves as a function of molecular markers (Tansey et al., 2018). The main objective in this work was to predict drug response using the molecular measurements, and the non-linear nature of the prediction model makes interpretation for the purpose of biomarker detection challenging. By contrast, we aimed to develop a model that could provide interpretable summary statistics with uncertainty estimates that can be flexibly used to improve biomarker detection.
In this study, we therefore introduce a new GP regression approach for describing dose-response relationships in cancer cell lines that quantifies the uncertainty of the model fitted to measured responses for each single experiment, and we show that estimates of IC 50 values within the tested concentration range correlates with confidence intervals obtained experimentally from replicate experiments. Subsequently, we use our new dose-response model to identify genetic sensitivity and resistance biomarkers in standard statistical tests (e.g. ANOVA). We demonstrate how the flexibility of the GP dose-response modelling can be further exploited in a Bayesian framework to identify novel biomarkers. We also describe the variation in the level of drug response uncertainty across cancer types and drug classes. By accounting for the uncertainty in dose-response experiments, detection of clinically-actionable biomarkers can be enhanced.

A probabilistic framework for measuring dose-response and predicting biomarkers
We analysed in vitro screening data on 265 compounds across 1,074 cell lines . In those experiments, we quantified the amount of cytotoxicity after four days of compound treatments at each dose compared to controls ( Figure 1A). The relationship between the dose and response (decrease in cell viability) was first described using a dose-response curve derived with a sigmoidal Figure 1. Workflow for fitting of Gaussian Process models to dose-response curves and estimating their uncertainty. (A) Large-scale drug screens test cell lines with different drugs and at different doses are used to obtain dose-response data. (B) Typically, for each drug tested in a cell line, the sigmoid model is fit to the drug-response data and (C) the overall measures of response (IC 50 , AUC, etc.) are extracted. (D) For each drug tested in a cell line, we fit a GP model to the dose-response data. The GP allows us to sample from a distribution of possible dose-response curves, obtaining a measure of uncertainty. (E) From these curves, we can extract overall measures of response, such as IC 50 , and importantly, their 95% confidence intervals. (F) Mutation markers for each cell line can be determined based on presence/absence of single nucleotide polymorphisms (SNPs) in key genes. Both the drug-response estimates and the mutation markers are used to compute (G) the F-statistic for ANOVA, and (H) Bayesian test for biomarker association. The drug-response summary measure g i for cell line i is modelled via a cell line-specific mean m i and standard error s i . The mean is defined as a linear effect b of the biomarker status z i and a further effect g from any remaining covariates x i , such as tissue type. The parameter s* is the standard deviation of m i . (I) Boxplots illustrate the differences in the estimated mean IC 50 of ERBB2 amplified and non-amplified breast cancer cell lines treated with afatinib. An ANOVA test was used to test this difference in means but did not consider uncertainty in each IC 50 estimate. (J) We estimated posterior distributions of gene association using the Bayesian model, that is the effect of a genetic mutation on the IC 50 measurement of drug response. Distributions centred on zero indicate no effect whilst distributions on either side of zero indicate positive or negative effects of mutations on drug response. function ( Figure 1B and C). This assumes that the number of viable cells decreases at an exponential rate, then slows down and eventually plateaus at a lower limit. Since it was costly to test all possible doses, the sigmoid function was used to extrapolate the response at concentrations that had not been tested and to estimate overall measures of response, such as IC 50 or AUC values, for downstream analysis. However, considering that each experiment tested only between five and nine dosage concentrations per experiment in GDSC, and a maximum of 16 in CTRP, the tightness of fit of the dose-response curve to the data points and therefore the level of uncertainty about the inferred response may vary. We utilised the probabilistic nature of GP models to quantify the uncertainty in the dose-response experiments as an alternate approach ( Figure 1D). We sampled from the fitted GP and used the posterior distribution to quantify the uncertainty in curve fits for each experiment. We again generated summary statistics, IC 50 and AUC values, by taking the average of the GP samples and also quantified the level of uncertainty for these statistics ( Figure 1E). The GP model has the advantage that it models outliers at higher doses as one component of a two-component Beta mixture in the model (see Materials and methods). Such outliers are typically the result of an experimental failure, and cannot be modelled using simple Gaussian noise without over-estimating the noise parameter.
After fitting the dose-response data using the sigmoid and GP models, we tested various biomarker hypotheses by examining the association between the overall response statistics from the models with genetic variants detected in the cell lines using a frequentist and a Bayesian approach ( Figure 1F-H). For one biomarker hypothesis, as an example, we examined copy number alterations and point mutations in breast cancer cell lines in relation to the measured drug response of afatinib in those cells. The GP and sigmoid estimated IC 50 from cell lines treated with afatinib were significantly different in cases with and without ERBB2 amplification (ANOVA q-value = 4.12e-9; Figure 1I). The GP models provided an added benefit of providing uncertainty estimates that were incorporated into a Bayesian hierarchical model to further verify the association between ERBB2 amplification and afatinib sensitivity (posterior probability = 0.001; Figure 1J).

Gaussian Processes provide estimates of dose-response uncertainty for single experiments
Both GP and sigmoid curve fitting produced comparable IC 50 and AUC estimates. Precursor sigmoid curve fitting methods based on Markov Chain Monte Carlo simulations enabled error estimates in IC 50 values (Garnett et al., 2012), however, this was neglected in the state-of-the-art sigmoid curve fitting  due to missing propagation to biomarker identification. Here, we introduce the added benefit of sampling from the GP posterior, which provides the models in-build uncertainty obtained for these IC 50 estimates. This is important for high-throughput drug screening experiments where there is often a high number of drugs and samples tested but very few replicate experiments. By applying the GP model to each experiment, we estimated the standard deviation for each IC 50 or AUC value based only on data points from that single experiment. These single sample standard deviations were compared to the standard deviations measured from here provided replicate experiments, that is the same drug tested multiple times on the same cell line and at the same concentration. We applied our GP estimation method to data from replicate experiments of 26 drugs on 10 cell lines, which contained 260 test conditions and 8 to 9 replicates for each condition. We wanted to see if an estimate of the uncertainty of the summary statistic, such as the standard deviation of the IC 50 posterior samples, would be correlated with the dispersion between replicates. Here, we refer to the variability between (mean) estimates for replicates as the observation uncertainty, and the variability in the estimate for a single replicate as the estimation uncertainty.
We compared observation and estimation uncertainty across replicate experiments of all 260 conditions ( Figure 2A). When the estimation uncertainty is large, we will have less confidence in the estimated IC 50 in an experiment. Measurement errors for individual points in a dose-response curve will generally result in larger estimation uncertainty, whereas greater variation between biological replicates will result in larger observation uncertainty. We found two trends in the relationship between observation and estimation uncertainty. First, for experiments where the estimated IC 50 lies within the concentration range tested, the estimation uncertainty is positively correlated (Pearson correlation = 0.84, 95% CI [0.76, 0.89]) with the observation uncertainty. Second, for experiments where the estimated IC 50 lies beyond the maximum tested concentration, we observed a negative correlation (Pearson correlation = À0.39, 95% CI [À0.51,-0.25]). We note that the latter experiments require extrapolation to estimate the IC 50 beyond the concentration range, which increases the estimation uncertainty, but does not generally affect the observational uncertainty. However, we observed that the estimation uncertainty from our GPs for dabrafenib (BRAF inhibitor) tested in two independent studies on the same cell lines were comparable both within and beyond the concentration range ( Figure 2B).
Since the replicate experiments were conducted in batches over a period of several months, we verified that the observed trends held regardless of batches (Figure 2-figure supplement 1). Additionally, we examined the relationship between estimation uncertainty and observation uncertainty in a number of edge cases where IC 50 was estimated within and beyond the maximum concentration tested ( Figure 2C-E). In the case of olaparib tested on PC-14, the uncertainty for the IC 50 within each replicate experiment was high, and this level of uncertainty was consistent across all replicates even beyond the max concentration ( Figure 2C and F). In other replicate experiments, both estimation and observation uncertainty were low ( Figure 2D and G), or varied depending on whether the batch reported mostly IC 50 values beyond the concentration range. Talazoparib tested in colorectal cancer line HCT-15 is a case where observation uncertainty was high, even though estimation uncertainty was low, and experiments in different batches showed different estimated IC 50 s from very different dose-response curves ( Figure 2E and H).
In order to examine the diversity of uncertainty estimates across experiments further, we described the relationship between AUC value of GP fits with their corresponding estimation uncertainty ( Figure 3). We decided to use AUC here due to the greater uncertainty of estimating IC 50 s beyond the maximum dose concentration. Since AUCs were computed within the tested concentration range, the estimation uncertainty for AUC was not substantially higher for cases where IC 50 s were estimated within compared to beyond the maximum concentration ( Figure 3-figure supplement 1A). The difference between the AUC estimates from the GP compared to the published GDSC sigmoid curve fits was greatest for experiments showing a partial response (AUC between 0.4 and 0.9), whilst at the same time these experiments also had the highest estimation uncertainty ( Figure 3A). Our visual examination of the raw dose-response data from those experiments revealed evidence of poor quality readouts, for instance, where cell viability increases with increasing drug dose (Figure 3-figure supplement 1B). We were able to quantify the quality of these readouts by estimating the Spearman correlation coefficient based on the raw cell viability counts and the dose concentrations ( Figure 3B). A negative Spearman correlation indicates that cell viability decreases as dosage increases (as expected) whilst a positive Spearman correlation indicates the opposite. The experiments with high estimation uncertainty from our GPs were also the experiments with high Spearman correlation pointing to poor quality.
Next, we investigated whether there were any attributes of experiments that would correspond to high estimation uncertainty and poor quality results. Labelling of experiments based on cell culture conditions, dose and cancer type revealed no obvious associations with estimation uncertainty (  in two independent studies (GDSC and CTD2). Estimation uncertainty (error bars and grey shading) were larger beyond the max concentration in both GDSC (dashed line) and CTD2 (grey line). The point estimates of the IC 50 s from the GPs (black dots) were also comparable to the published IC 50 s (red dots). (C-E) Three sets of replicate experiments, representing different amounts of estimation and observation uncertainty. Each density represents the distribution of IC 50 values from the Gaussian process samples from each replicate experiment. The colours represent different experimental batches. Narrow distributions demonstrate low estimation uncertainty and overlapping distributions demonstrate low observation uncertainty. The thick black line represents the density obtained by pooling samples from all replicates and the dashed line shows the maximal dosage tested. GP-curve fits corresponding to the three sets of replicate experiments showing IC 50 estimates with (F) high uncertainty, (G) low uncertainty, and (H) mix of uncertainties depending on whether estimates are made within or beyond the max concentration. The blue areas represent the 95% confidence interval in the curve fits and extrapolated GP curves (light grey lines) are displayed up to five times the maximum concentration, where the uncertainty will be extremely high. The online version of this article includes the following figure supplement(s) for figure 2: experiments showing resistance and sensitivity, the average estimation uncertainties varied across target pathways. Interestingly, similar target pathways (e.g. chromatin histone methylation and chromatin histone acetylation) had very different levels of estimation uncertainty. Within each of these target pathways, we also see different distributions of estimation uncertainties ( Figure 3D). Most    target pathways have a bi-modal distribution representing compounds that have low uncertainty in the cases of clear sensitivity or resistance, and high uncertainty in the cases of partial responders ( Figure 3E). Both chromatin histone methylation drugs in particular had a much longer right tail towards higher estimation uncertainties that are associated with poor experimental readouts, or possibly off-targets.

Curve fits using Gaussian Processes can help identify clinically relevant biomarkers
The IC 50 values are highly conconcordant for sigmoid and GP-curve fittings, showing an average weighted Pearson correlation of 0.88 (95% CI [0.85; 0.91]) across individual drugs, and cancer types ( Figure 4A). Strong agreement is found when true responding cell lines were observed in the screen ( Figure 4B). For example, if >10% of cell lines responded within the concentration range, that is IC 50 <maximum tested concentration, then a weighted Pearson correlation >0.75 was consistently achieved for all drugs. We  Tamura and Fukuoka, 2005;Yang et al., 2012). ERBB2(HER2) amplification in breast cancer was also recapitulated as a biomarker of sensitivity to the dual EGFR/ERBB2 inhibitor lapatinib (Figure 4-figure supplement 3H; Konecny et al., 2006). Among the 26 clinical biomarkers, we consistently found drug resistance of TP53 mutants to MDM2 inhibition with nutlin-3a in five different cancer types ( Figure 4E, Figure 4-figure supplement 3I-L). Overall, the majority of expected clinical and preclinical biomarkers are reproduced, regardless of the drug-response curve fitting method.
We concordantly and significantly identified six novel (not yet clinically established) drug sensitivity biomarkers (0.1% FDR) regardless of the applied drug-response curve fitting method. Investigating two different curve fitting algorithms, and retrieving the same biomarkers can be considered as a test of robustness, which in our case concordantly highlighted non-gold standard associations for prioritising experimental validation. For example, daporinad (also known as FK866 and APO866) is a small-molecule inhibitor of nicotinamide phosphoribosyltransferase leading to inhibition of NAD+ biosynthesis. It has been clinically tested in melanoma (ClinicalTrials.gov Identifier: NCT00432107), Refractory B-CLL (NCT00435084) and Cutaneous T-cell Lymphoma (NCT00431912), whilst showing anti-proliferative effect in glioblastoma cell lines (Zhang et al., 2012). Therapeutic potential when combining with other drugs used to treat gliomas (Lucena-Cacace et al., 2019;Lucena-Cacace et al., 2017) has been suggested, whilst we additionally and concordantly identify EGFR amplification as a biomarker ( Figure 4F).
Another novel and concordant identified biomarker is doramapimod response (also known as BIRB-796) in ARID2 mutant melanoma cell lines ( Figure 4G). Doramapimod is a small-molecule p38    MAPK inhibitor and has been reported in different cancer types (in combination with other drugs) including cervical cancer, paracrine tumours and myeloma (Jin et al., 2016;Yasui et al., 2007). ARID2 is part of chromatin remodelling complex and is involved in DNA repair in hepatocellular carcinoma cells (Oba et al., 2017) and enriched in melanomas (Ding et al., 2014;Hodis et al., 2012).
In conclusion, different curve fitting approaches lead to concordantly and novel identified biomarkers, thereby increasing the robustness in those findings, and consequently enabling to prioritise hypotheses.

Improved biomarker detection by taking into account uncertainty in a Bayesian framework
Since both Bayesian and frequentist methods can be used to prioritise biomarkers for further testing, we compared association statistics (posterior probabilities and q-values) from both statistical methods. We observed a number of cases where the Bayesian and ANOVA tests disagree ( Figure 5A; Supplementary file 2). For instance, BRAF mutations in colorectal cancer were detected as a sensitivity biomarker for dabrafenib by the Bayesian test, but less significant by the ANOVA test. This association had been repeatedly reported in in vitro models Rees et al., 2016) and also found in melanoma cases (Chapman et al., 2011), whilst not in colorectal cancer patients due to feedback activation of ERK-signalling mediated via EGFR (Corcoran et al., 2018;Prahallad et al., 2012). We note in Figure 5B that the Bayesian test takes advantage of the additional information that sensitive mutant cell lines have low estimation uncertainty, whilst the small number of resistant mutant cell lines have high estimation uncertainty, causing them to have less influence on the biomarker detection. On the other hand, the ANOVA model detected the KRAS copy number alteration as a resistance biomarker for lenalidomide (immunomodulatory drug) partial sensitivity in skin cutaneous melanoma (SKCM), whilst not detected by our Bayesian approach. Whilst on the linear IC 50 scale there is some difference between the small number of mutant cell lines and wildtypes, the Bayesian model considered that the estimated responses of the mutant cell lines had high uncertainty ( Figure 5C). Additionally, a comparison of the uncertainty estimates for the GP and the Sigmoid curve fitting methods revealed that both display concordant results ( Figure 5B and C; Figure 5-figure supplement 1); However, the Sigmoid curve fitting method (Materials and methods; Vis et al., 2016) underestimates variance of non-responding cell lines rendering the GP approach superior. The dosages within Figure 5B and C were rescaled to prevent the need for adapting the length-scale hyperparameter to the maximum dosage. IC 50 values were back-transformed to the log10 drug dosage scale to make comparisons with ) (see Materials and methods). Whilst discrepancies between Bayesian and ANOVA tests have to be taken with caution, they may highlight novel biological insights which would be missed when applying only a single model.

Discussion
The GP approach developed in this research has several advantages compared to the traditional approach of fitting sigmoidal drug-response curves. Firstly, these flexible, non-parametric models can be used to fit a wider variety of dose-response curves than the parametric sigmoidal models, for example curves of unexpected shapes may reflect biological signals of off-target effects. Secondly, the GP models provide straightforward uncertainty quantification of any summary statistic that can be calculated on a dose-response curve, a fact that we take advantage of in developing our hierarchical Bayesian model for biomarker testing. Thirdly, the GP model can deal with outlying measurements better than a sigmoidal model, due to formulating it as a mixture model with one component representing the latent GP process of the drug response, and the second component accounting for outliers.
In contrast to other GP-based models in Tansey et al., 2018, our approach is highly interpretable, as we do not integrate the biomarkers into the model estimation in a non-linear fashion, but instead proceed in a two-step approach that first fits our Gaussian process model to the doseresponse curves, and then uses the derived summary statistics and uncertainty measures to perform biomarker detection. Thus, we can take advantage of the flexibility of the Gaussian process without the complexity of fitting a non-linear neural network to enable prediction from molecular measurements.   The increased flexibility of the GP model comes at a price. Most notably, because we do not impose a specific functional form, there are few constraints on the behaviour of the curve outside the range of observed dosages. This leads to the counter-intuitive behaviour that the posterior mean estimate of drug response can go up when extrapolating beyond the maximum dosage. Note, however, that this goes along with a commensurate increase in the posterior variance ( Figure 5D,E). In other words, the model is highlighting that extrapolation beyond the observed dosage range is highly uncertain, and the posterior mean estimate should not be relied on. It would be possible to constrain this behaviour by introducing artificial data points at a high concentration, or less crudely by imposing monotonicity constraints via virtual derivative observations (Riihimäki and Vehtari, 2010). However, these methods would limit the flexibility of our method and lead us to underestimate the uncertainty of the posterior mean. An alternative approach is to constrain the Gaussian process using generalised analytic slice sampling (Tansey et al., 2019), which integrates the constraints into the sampling process. Whilst theoretically appealing, this approach is not compatible with the variational inference method that we have chosen for our work, and would lead to an unacceptable increase in computational burden for fitting the dose-response curves.
We have systematically compared the application of GP to sigmoid models across a pan-cancer drug screen. We demonstrated that our GP estimates of the IC 50 values and their subsequently predicted biomarkers using ANOVA are reliable when compared to estimates from the sigmoid models. In addition, the GP models provide useful information about the uncertainty associated with the drug-response quantification. However, there is a crucial difference between estimation uncertainty on a single experiment and observational uncertainty across multiple replicates of the same experiment, which incorporates measurement error, technical and biological variation. We are interested in the former to assess the quality of the fit, and therefore the reliability of the estimated IC 50 . We hypothesized that estimation uncertainty characterises observational uncertainty within the dose concentration range tested. However, extrapolating beyond the concentration range would be challenging due to the uncertainty in the behaviour of the dose-response curve in unobserved concentrations. Imposing monotonicity may not be the best path in this case, but we avoid making this assumption. Instead, our method defines a very large confidence interval for drug-response statistics extrapolated beyond the maximum dose tested and we would additionally need to take the observation uncertainty between replicate experiments into account. We have verified this by applying our estimation method to a replication data set of 26 drugs tested on 10 different cell lines, with 8 to 9 replicates for each drug-cell line experiment. We conclude that whilst estimation uncertainty is a useful indicator for within-concentration IC 50 values, it cannot be used as a proxy for observation uncertainty when the IC 50 is extrapolated beyond the tested concentration range. Indeed, overall drug responses and biomarkers from independent drug screens were consistent when comparing similar dose ranges (Haverty et al., 2016). Any difference between replicate experiments may be due to batch effects or other unobserved factors that are not necessarily reflected in the estimation error. Whilst previous studies have attempted to capture uncertainty by measuring the spread of the residuals from the fitted curves, such as root mean square error, they were not able to capture these false positive biomarkers by setting strict cutoffs (Cokelaer et al., 2018).
Whilst Bayesian posterior probabilities and ANOVA q-values are different statistical quantities for measuring biomarker associations that should not be compared in absolute terms, we compared these quantities in relative terms to prioritise biomarkers of response for further testing. Our Bayesian biomarker model extends the classical ANOVA testing, since it is able to leverage the estimation uncertainty of the IC 50 values. We showed that taking estimation uncertainty into account in the Bayesian model can lead to both inclusion and exclusion of putative biomarkers. For example, the Bayesian model highlighted the association between BRAF mutation in colorectal cancer and BRAF inhibitor response. Targeting BRAF signalling has recently been confirmed as a viable option for metastatic colorectal cancer cases with BRAF mutations (Kopetz et al., 2019). In contrast, the Bayesian model excluded a suggestion from ANOVA of association between KRAS mutation with lenalidomide response in melanoma. Lenalidomide has thus far had no clinical success in KRAS mutant cases nor melanoma (Gandhi et al., 2013;Glaspy et al., 2009).
Although we systematically tested for drug-biomarker associations, we did observe common behaviour for certain cell types or classes of drugs. The high uncertainty in the response estimates of chromatin histone methylation targeting compounds for instance may be due to the large number of factors contributing to epigenetic regulation of cells (Luo, 2015). It would be straightforward to extend the GP model to allow for sharing information across drugs or cell lines of similar class, by using either shared hyperparameters or a hyperprior on the hyperparameters. We have not implemented this approach in our work here as our aim was to show the advantage of fitting individual drug-response using GPs, and extending the method to fitting multiple curves jointly would increase the memory and computational requirements significantly. It is our hope to continue expanding the suite to multiple dimensions of dose-response and biomarker prediction needed for drug combinations, which is predominantly based on synergy modelling with either Loewe Additivity or Bliss Independence (Di Veroli et al., 2016;Vlot et al., 2019). In cases where multiple statistical models converge to concordant biomarkers, this increases the reproducibility of the evidence, potential for clinical translatability and ultimately enables precision medicine.
The increasing utilisation of high-throughput drug screening for identifying effective new treatments will necessitate the use of more powerful statistical and machine learning methods . We have introduced an approach for quantifying the uncertainties of doseresponse using Gaussian Processes and further described how these uncertainties can be integrated into statistical testing of biomarkers. For cancer treatments, our approach can help estimate the uncertainty of dose-responses reported in the numerous drug screening studies by academic (Ghandi et al., 2019;Holbeck et al., 2017;Iorio et al., 2016) and pharmaceutical laboratories O'Neil et al., 2016). This can provide more robust metrics for comparing drug responses to identify the most potent ones and highlight sensitivity biomarkers that are more likely to succeed clinically because they are associated with low uncertainty. The approach is also generalisable beyond cancer to any disease and any dose-response measures. We hope that by considering response uncertainty and providing a probabilistic view of drug biomarkers, the risks associated with drug development can be better balanced and smarter decisions can be made.

Drug screening
We analysed 1074 cancer cell lines tested with 265 compounds from a high-throughput screen resulting in 225,384 experiments that were previously published . Cell line data was retrieved and is publically available via the GDSC website (Key Resources Table). All cell lines were authenticated. Details for each cell line can be found at: https://www.cancerrxgene.org/help. Compounds were tested with 5 to 9 titration points, whilst either diluted with 4-or 2-fold, respectively. Cells were seeded on day zero, left in the microtiter plate for 24 hr to retain linear growth, and consecutively treated for 3 days. After those 3 days of treatment, cellTiterGlo staining is used to quantify ATP levels within each well. In parallel, untreated cells and blank wells were also measured to estimate and normalise cell viability.
Compounds within the replicate study were screened across a seven point dose-response curve with a half-log dilution and 1000 fold range. The duration of drug treatment was 72 hr and cell viability was measured using CellTiter-Glo (Promega). Each cell line and compound pair was screened in technical triplicate, three assay plates generated simultaneously, and across three biological replicates with 46 and 44 days between the first to second and second to third replicates respectively. Cell viability measurements for these experiments can be found in Supplementary file 3.

Preprocessing
Prior to analysis, we scaled the raw observed fluorescent intensities for each drug/cell line combination using the observations from the blank and negative control wells as follows. Let R ¼ r 1 ; r 2 :::; r n f g be the observed intensities for n dosages. Let B be the mean of the intensities for the blank wells on the same plate as the experiment, and C be the mean of the intensities of the negative control wells (no drug added). Then the relative cell viability V can be calculated as: Relative cell viability values below 0 (n = 2646, Figure 5-figure supplement 2) were set to 0. For the purpose of fitting the Gaussian process models, we additionally rescale the dosages to avoid having to adapt the length-scale hyperparameter to the maximum dosage. We rescale the log 2 -transformed dosages d ¼ d 1 ; d 2 :::; d n f g as follows: Note that IC 50 values have been back-transformed to the log10 drug dosage scale for comparability with those reported in Iorio et al., 2016. Sigmoid drug-response model The GDSC estimates in Iorio et al., 2016 were obtained using a sigmoid fit to the drug-response curve, using the same pre-processing of the fluorescent intensities as described above. The particular sigmoid model used is the one described in Vis et al., 2016. In brief, if we have shape parameter s i and position parameter p ij for cell line i and drug j , then cell viability can be represented as a function of dosage d: Note that this allows for cell line/drug specific position parameters, but shape parameters that only vary by cell line and are shared across drugs. The position parameter p ij corresponds to the estimated IC 50 for cell line i and drug j. For full details, see Vis et al., 2016. To estimate the uncertainty of the Sigmoid curve fitting, a random bootstrap sampling of 80% of all treated cell lines available for each drug over 100 iterations was performed. The Sigmoid curve fitting model from GDSC (Vis et al. 2013) estimates one scale parameter per drug across all treated cell lines, thus the sampling creates variance in the response data. The standard deviation of the log (IC 50 ) estimates was computed to assess the model's variance.

Gaussian process drug-response model
For simplicity, we drop the subscripts ij and present the combination. We model the drug response y via a two-component Beta mixture such that: Pðyjf; s 1 ; 2 ; s 2 ; pÞ ¼ p Beta ðyjF À1 ðfÞ; s 1 þ ð1 þ pÞBeta ðyj 2 ; s 2 ÞÞ where Beta is the reparameterization of the Beta distribution in terms of the mean m and a scale parameter s, and F À1 is the probit function (the inverse of the standard normal cumulative distribution function). Component one represents the drug response, which is driven by a latent Gaussian process f, whilst component two represents outliers that deviate from the overall dose-response trend. We set the scale parameters s 1 ¼ 50 and s 2 ¼ 11 and specify 2 ¼ 0:9 to reflect our belief that outliers will mostly be erroneous measurements of resistance. We set p ¼ 0:999 as we believe that outliers are rare.
We place a standard Gaussian process prior on f, such that: where m is the mean drug response, and C É ðd; d 0 Þ is a covariance function with hyperparameters É; in practice we choose a combined linear-Matern3/2 as a flexible option, which avoids the excessive smoothness of restrictions of the commonly used RBF kernel. Stein (1999) argues that this is a more realistic representation for physical processes (Stein, 2012). Information sharing across drugs and cell lines can be achieved via shared hyperpriors in a hierarchical model. For the application in this paper joint inference with shared hyperpriors would be computationally difficult, and we choose to instead empirically set the variance and length-scale parameters for the Matern to 0.2 and 0.3, respectively, and the variance parameter for the linear kernel to 0.1. Inference is performed using variational learning (Hensman et al., 2013), via the GPFlow software (Matthews AG de and van der Wilk, 2017). We choose variational learning over alternatives such as Markov chain Monte Carlo due to its speed, which allows us to process large drug-response panels in a realistic time frame. Hyperparameters for the GP model were determined by manual tuning; however, for other datasets, we could also envision a Bayesian model selection procedure which places the variational inference in a variational-within-MCMC scheme where the MCMC moves update the hyperparameters. If fixed hyperparameters are desired, one could use the maximum a posteriori values. To avoid massive computational complexity, the MCMC scheme could be run on a representative subsample of cell lines.

Calculation of summary statistics
Summary statistics of drug response can be calculated straightforwardly by sampling from the posterior of the Gaussian process (Supplementary file 4). Generally, let gðd; yÞ be a function that calculates a summary statistic t from a dose-response curve with dosages d and responses y, then we can obtain a posterior estimate of the mean of the summary statistic by first sampling N doseresponse curves from the posterior of the GP model, and then calculating the average: A similar procedure can be used to calculate the posterior estimate of the standard deviation. Although we can extract other response statistics from our curve fits, the most common are the IC 50 and the area under the drug-response curve (AUC). On the log 2 dosage scale the dosages are equally spaced, and hence AUC can be straightforwardly estimated by the mean function: where m indexes over the n dosages. For the IC 50 , estimation for a single curve is complicated by the fact that the curve may not cross the 50% viability threshold within the observed dosage range (non-crossing sample). We therefore extrapolate the GP samples to 10 times the maximum (log 2 ) experimental dosage and specify g IC50 ðd; yÞ as: g IC50 ðd; yÞ ¼ d m such that y m ¼ 0:5 if 9y m 0:5 Note that this ignores samples where for all dosages, y m 0:5; one could devise a multivariate sufficient statistic that takes this information into account, but we have found that in general there is a reasonable amount of correlation between g IC50 ðd; yÞ and the number of non-crossing samples for a given cell line/drug combination.

Comparison of GP and sigmoid IC 50 values
Concordance between IC 50 values based on sigmoid and GP-curve fitting is quantified with Pearson correlation for each drug. To account for tissue specificity and the varying number of cell lines assessed per tissue type, we employed the average weighted Pearson correlation (pw) of the sigmoid-curve versus GP-curve fitted IC 50 values for the individual cancer types (i).
The weight for a given cancer type i was denoted as ffiffiffiffiffiffiffiffiffiffiffiffi n i À 1 p , where n i is the total number of cell lines treated with the drug within this tissue type. The following metric was applied, where p i is unweighted Pearson correlation within a cancer type (i) and a total number of tested cancer types is N ¼ 30. For a given drug and tissue type combination, at least 10 cell lines need to be treated (n i ! 10). Differences in IC 50 values for each drug-response value j were consistently defined as with a total number of tested cell line and drug combinations equalling to N j ¼ 171; 937.

Bayesian biomarker testing
Standard statistical approaches for testing the influence of biomarkers on drug response mostly rely on analysis of variance (ANOVA) testing. An ANOVA can be understood as a linear model of the dependent variable i (in this case, a summary measure of drug response such as IC 50 ): where x i is an indicator variable denoting the group membership of data point i. In our application, the data points are cell lines, z i indicates group membership, for example the mutation status of a given SNP, and x i indicates any other covariates that we wish to correct for, such as tissue type. The parameter a captures the global mean of the drug response, whilst b captures the effect of mutation status on the drug response, g is the effect of covariates, and i is independent Gaussian noise. This model, whilst useful, fails to account for the fact that our Gaussian process model provides estimates s i of the uncertainty (or standard error) associated with the mean IC 50 estimates g i . In order to make use of these uncertainty estimates, we take an idea from Bayesian meta-analysis, and integrate them via a hierarchical model: where i is the mean drug-response estimate for cell line i, and s Ã2 is the variance across cell lines (the variance of i in the ANOVA example). Note that this model can be reduced to: We further specify a Gaussian prior b~Nð0; 0:1Þ on the effect size parameter to discourage false positives and reflect our prior belief that most mutations are not associated with drug response. We also place an exponential prior s Ã2~E xpð10Þ to regularize the variance parameter. Finally, a~Nð0; t 2 Þ is a Gaussian prior on the global mean with standard error t~Gammað1; 1Þ. Early exploratory results showed that using the estimates of s i directly placed too much weight on experiments with very low estimation uncertainty, leading to unrealistic posterior estimates of the effect size b.
To attenuate this, we used a transformed estimate s c i , where the effect of parameter c was explored over the range [0,1], and empirically set to 0.25 for the results reported in this paper. The main tuneable hyperparameter is the scaling parameter c, as the model is robust to changes to the parameters for the sparse priors on b and s Ã2 . Setting this hyperparameter is straightforward, as we can use a simple line search to find a value that optimally trades off between disregarding the uncertainty estimates (c = 0) and placing too much weights on estimates with low uncertainty (c >= 1). One way to determine the optimal value for c is to randomly permute the biomarker labels, and reduce c until the false positive rate is below some acceptable threshold. Inference in this model is performed using Hamiltonian Monte Carlo via the Stan software package Carpenter, 2017. We report the posterior mode of b as well as the posterior probability of observing b>0 (if the posterior mode is positive) or b<0 (if the posterior mode is negative).