Bayesian analysis of serial dilution assays.

TLDR: A Bayesian method for jointly estimating the calibration curve and the unknown concentrations using all the data and a method for determining the "effective weight" attached to each measurement, based on a local linearization of the estimated model.

Abstract: In a serial dilution assay, the concentration of a compound is estimated by combining measurements of several different dilutions of an unknown sample. The relation between concentration and measurement is nonlinear and heteroscedastic, and so it is not appropriate to weight these measurements equally. In the standard existing approach for analysis of these data, a large proportion of the measurements are discarded as being above or below detection limits. We present a Bayesian method for jointly estimating the calibration curve and the unknown concentrations using all the data. Compared to the existing method, our estimates have much lower standard errors and give estimates even when all the measurements are outside the "detection limits." We evaluate our method empirically using laboratory data on cockroach allergens measured in house dust samples. Our estimates are much more accurate than those obtained using the usual approach. In addition, we develop a method for determining the "effective weight" attached to each measurement, based on a local linearization of the estimated model. The effective weight can give insight into the information conveyed by each data point and suggests potential improvements in design of serial dilution experiments.

Figures

Figure 3. Comparison of classical estimates with our Bayesian procedure for the model fit to data from a single plate. The estimated concentrations are similar under the two methods, but our approach gives smaller standard errors. Our estimates also perform much better in cross-validation: The two estimates using just the first or second set of dilutions for each unknown are much closer under our method than with the classical approach.

Table 1 Typical setup of a plate with 96 wells for a serial dilution assay. The first two columns are dilutions of “standards” with known concentrations, and the other columns are 10 different “unknowns.” The goal of the assay is to estimate the concentrations of the unknowns, using the standards as calibration.

Figure 1. Data from a single plate of a serial dilution assay. The large graph shows the calibration data, and the 10 small graphs show the data for the unknown compounds. The goal of the analysis is to figure out how to scale the x-axes of the unknowns so they will line up with the curve estimated from the standards. (The curves shown on these graphs are estimated from the model as described in Section 3.2.)

Figure 4. Data and estimated curve (based on posterior medians of the parameters) from one of the two plates of the validation study. The 10 unknowns start with data that are mostly “above detection limit” and gradually decrease in concentration until all measurements are “below detection limit.”

Figure 5. Classical estimates and Bayesian posterior medians for the 10 unknown concentrations from each of the two plates of the validation study. The Bayesian estimates also show 50% error bars. The graph is on a logarithmic scale, and the dotted lines, with slope −log(4), display the pattern that the true concentrations are in the ratio 1 : 14 : 116 : · · · : 149 . The Bayesian estimates are closer to the line for a much wider range of concentrations for both plates, with the classical method failing by giving no estimate (as in the last two samples of plate 1) or a highly inaccurate estimate (as in the first and last four samples of plate 2).

Table 2 Example of some measurements y from a plate as analyzed by the standard software used for dilution assays. The standards data are used to estimate the calibration curve, which is then used to estimate the unknown concentrations. The measurements indicated by asterisks are labeled as “below detection limit.” However, information is present in these low observations, as can be seen by noting the decreasing pattern of the measurements from dilutions 1 to 1/3 to 1/9.

Frequently asked questions

Q1. What are the contributions in "Bayesian analysis of serial dilution assays" ?

The authors present a Bayesian method for jointly estimating the calibration curve and the unknown concentrations using all the data. Their estimates are much more accurate than those obtained using the usual approach. The effective weight can give insight into the information conveyed by each data point and suggests potential improvements in design of serial dilution experiments. 

Q2. What have the authors stated for future works in "Bayesian analysis of serial dilution assays" ?

6. 3 Further Work An important direction of future research is to study which aspects of the curves vary between samples and which are stable, to allow the possibility of more accurate calibration. The model can potentially be improved in various ways, most notably by generalizing the function ( 1 ) of expected measurements. This represents a problem with both classical and Bayes estimates, and the authors suspect it is the reason why estimates from dilution assays are in practice much more variable than would be suggested by even the classical estimates in Figure 3. 

Q3. What is the way to improve the accuracy of the assay?

By yielding more accurate estimates and quantifying inferential uncertainties (especially in the cases previously deemed outside detection limits), the Bayesian approach sets the stage for more systematic studies of model and design innovations, which the authors hope will lead to an even broader extension of the range of concentrations to which assays can be applied. 

Q4. What are the weights for each unknown sample?

The weights depend on the unknown parameters θ, β, α, and so when the authors are fitting the model, the authors compute the set of weights for each of the unknown samples and normalize each set to sum to 1, for each posterior simulation draw. 

Q5. What is the common design for estimating the concentrations of compounds in biological samples?

A common design for estimating the concentrations of compounds in biological samples is the serial dilution assay, in which measurements are taken at several different dilutions of a sample, giving several opportunities for an accurate measurement. 

Q6. What is the advantage of the Bayesian approach?

The second advantage of the Bayesian approach is that it can incorporate several sources of variation without requiring point estimation or linearization, either of which can cause uncertainties to be underestimated in this nonlinear errors-in-variables model (Davidian and Giltinan, 1995; Dellaportas and Stephens, 1995). 

Q7. What is the standard method used to estimate the concentrations of unknown samples?

the standards data are used to estimate the curve relating concentrations to measurements—typically assumed to be a four-parameter logistic function—using least squares. 

Q8. What is the purpose of the hierarchical distribution for unknown concentrations?

When using the model to estimate unknown concentrations θ1, . . . , θJ , the authors fit a hierarchical model of the form,log θj ∼ N ( µθ, σ 2 θ ) , for j = 1, . . . , J. 

Q9. What is the definition of a dilution assay?

Assays are performed using microtiter plates (for example, see Table 1) that contain two sorts of data: unknowns, which are the samples to be measured and their dilutions; and standards, which are dilutions of a known compound, used to calibrate the measurements. 

Q10. What is the posterior median of the calibration curve?

The posterior median estimates of the parameters of the calibration curve are β̂1 = 14.8 (with a posterior 50% interval of [14.7, 15.0]), β̂2 = 94.3[89.8, 99.0], β̂3 = 0.048[0.044, 0.052], and β̂4 = 1.41[1.37, 1.46]. 

Q11. How do the authors use the weights to understand the information in existing data?

5.4 Using the Weights to Understand the Information in Existing DataFor any given unknown sample, the authors now have a weight for each measurement, and these can be normalized to sum to 1. 

Q12. How can the authors estimate unknown concentrations using Bayesian methods?

The authors have also programmed the inference using the Metropolis algorithm (see, e.g., Gilks, Richardson, and Spiegelhalter, 1996) directly in R.Section 3.2 shows how Bayesian methods allow estimation of unknown concentrations with reasonable accuracy using much less data than needed by conventional methods, even with measurements that at first appear to be outside “detection limits.” 

Q13. How many samples are available for the estimation of unknown concentrations?

In Section 4, the authors show that it is possible to obtain accurate estimates of concentrations varying by a factor of 46 or 47 (that is, ranging from sample 1 to sample 7 or 8 in Figure 5) and reasonable estimates over the entire range of 49, or 260,000.