Analysis
Once dose-response dataframe is correctly set up, we may proceed onto synergy analysis. We will use transforms as defined above with a logarithmic transformation. If not desired, transforms can be set to NULL and would be ignored.
Synergy analysis is quite modular and is divided into 3 parts:
- Determine marginal curves for each of the compounds. These curves are computed based on monotherapy data, i.e. those observations where one of the compounds is dosed at 0.
- Compute expected effects for a chosen null model given the previously determined marginal curves at various dose combinations.
- Compare the expected response with the observed effect using statistical testing procedures.
Fitting marginal (on-axis) data
The first step of the fitting procedure will consist in treating marginal data only, i.e. those observations within the experiment where one of the compounds is dosed at zero. For each compound the corresponding marginal doses are modelled using a 4-parameter logistic model.
The marginal models will be estimated together using non-linear least squares estimation procedure. Estimation of both marginal models needs to be simultaneous since it is assumed they share a common baseline that also needs to be estimated. The fitMarginals function and other marginal estimation routines will automatically extract marginal data from the dose-response data frame.
Before proceeding onto the estimation, we get a rough guess of the parameters to use as starting values in optimization and then we fit the model. marginalFit, returned by the fitMarginals routine, is an object of class MarginalFit which is essentially a list containing the main information about the marginal models, in particular the estimated coeffcients.
The optional names argument allows to specify the names of the compounds to be shown on the plots and in the summary. If not defined, the defaults (“Compound 1” and “Compound 2”) are used.
## Fitting marginal models
marginalFit <- fitMarginals(data, transforms = transforms, method = "nls",
names = c("Drug A", "Drug B"))
summary(marginalFit)## Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
## Transformations: Yes
##
## Drug A Drug B
## Slope 2.132 1.201
## Maximal response 0.088 0.091
## log10(EC50) 1.008 0.977
##
## Common baseline at: 0.132marginalFit object retains the data that was supplied and the transformation functions used in the fitting procedure. It also has a plot method which allows for a quick visualization of the fitting results.
## Plotting marginal models
plot(marginalFit) + ggtitle(paste("Direct-acting antivirals - Experiment" , i))
Note as well that the fitMarginals function allows specifying linear constraints on parameters. This provides an easy way for the user to impose asymptote equality, specific baseline value and other linear constraints that might be useful. See help(constructFormula) for more details.
## Parameter ordering: h1, h2, b, m1, m2, e1, e2
## Constraint 1: m1 = m2. Constraint 2: b = 10.
constraints <- list("matrix" = rbind(c(0, 0, 0, -1, 1, 0, 0),
c(0, 0, 1, 0, 0, 0, 0)),
"vector" = c(0, 10))
## Parameter estimates will now satisfy equality:
## constraints$matrix %*% pars == constraints$vector
fitMarginals(data, transforms = transforms,
constraints = constraints)The fitMarginals function allows an alternative user-friendly way to specify one or more fixed-value constraints using a named vector passed to the function via fixed argument.
## Set baseline at 1 and maximal responses at 0.
fitMarginals(data, transforms = transforms,
fixed = c("m1" = 0, "m2" = 0, "b" = 1))By default, no constraints are set, thus asymptotes are not shared and so a generalized Loewe model will be estimated.
Optimization algorithms
We advise the user to employ the method = "nlslm" argument which is set as the default in monotherapy curve estimation. It is based on minpack.lm::nlsLM function with an underlying Levenberg-Marquardt algorithm for non-linear least squares estimation. This algorithm is known to be more robust than method = "nls" and its Gauss-Newton algorithm. In cases with nice sigmoid-shaped data, both methods should however lead to similar results.
method = "optim" is a simple sum-of-squared-residuals minimization driven by a default Nelder-Mead algorithm from optim minimizer. It is typically slower than non-linear least squares based estimation and can lead to a significant increase in computational time for larger datasets and bootstrapped statistics. In nice cases, Nelder-Mead algorithm and non-linear least squares can lead to rather similar estimates but this is not always the case as these algorithms are based on different techniques.
In general, we advise that in automated batch processing whenever method = "nlslm" does not converge fast enough and/or emits a warning, user should implement a fallback to method = "optim" and re-do the estimation. If none of these suggestions work, it might be useful to fiddle around and slightly perturb starting values for the algorithms as well. By default, these are obtained from the initialMarginal function.
nlslmFit <- tryCatch({
fitMarginals(data, transforms = transforms,
method = "nlslm")
}, warning = function(w) w, error = function(e) e)
if (inherits(nlslmFit, c("warning", "error")))
optimFit <- tryCatch({
fitMarginals(data, transforms = transforms,
method = "optim")
})Note as well that additional arguments to fitMarginals passed via ... ellipsis argument will be passed on to the respective solver function, i.e. minpack.lm::nlsLM, nls or optim.
Custom marginal fit
While BIGL package provides several routines to fit 4-parameter log-logistic dose-response models, some users may prefer to use their own optimizers to estimate the relevant parameters. It is rather easy to integrate this into the workflow by constructing a custom MarginalFit object. It is in practice a simple list with
coef: named vector with coefficient estimatessigma: standard deviation of residualsdf: degrees of freedom from monotherapy curve estimatesmodel: model of the marginal estimation which allows imposing linear constraints on parameters. If no constraints are necessary, it can be left out or assigned the output ofconstructFormulafunction with no inputs.shared_asymptote: whether estimation is constrained to share the asymptote. During the estimation, this is deduced frommodelobject.method: method used in dose-response curve estimation which will be re-used in bootstrappingtransforms: power and biological transformation functions (and their inverses) used in monotherapy curve estimation. This should be a list in a format described above. Iftransformsis unspecified orNULL, no transformations will be used in statistical bootstrapping unless the user asks for it explicitly via one of the arguments tofitSurface.
Other elements in the MarginalFit are currently unused for evaluating synergy and can be disregarded. These elements, however, might be necessary to ensure proper working of available methods for the MarginalFit object.
As an example, the following code generates a custom MarginalFit object that can be passed further to estimate a response surface under the null hypothesis.
marginalFit <- list("coef" = c("h1" = 1, "h2" = 2, "b" = 0,
"m1" = 1.2, "m2" = 1, "e1" = 0.5, "e2" = 0.5),
"sigma" = 0.1,
"df" = 123,
"model" = constructFormula(),
"shared_asymptote" = FALSE,
"method" = "nlslm",
"transforms" = transforms)
class(marginalFit) <- append(class(marginalFit), "MarginalFit")Note that during bootstrapping this would use minpack.lm::nlsLM function to re-estimate parameters from data following the null. A custom optimizer for bootstrapping is currently not implemented.
Compute expected response for off-axis data
Three types of null models are available for calculating expected response surfaces.
- Generalized Loewe model is used if maximal responses are not constrained to be equal, i.e.
shared_asymptote = FALSE, in the marginal fitting procedure andnull_model = "loewe"in response calculation - Classical Loewe model is used if constraints are such that
shared_asymptote = TRUEin the marginal fitting procedure andnull_model = "loewe"in response calculation - Highest Single Agent is used if
null_model = "hsa"irrespectively of the value ofshared_asymptote.
(Generalized) Loewe model
If transformation functions were estimated using fitMarginals, these will be automatically recycled from the marginalFit object when doing calculations for the response surface fit. Alternatively, transformation functions can be passed by a separate argument. Since the marginalFit object was estimated without the shared asymptote constraint, the following will compute the response surface based on the generalized Loewe model.
rs <- fitSurface(data, marginalFit,
## transforms = transforms,
null_model = "loewe",
B.CP = 50, statistic = "none", parallel = FALSE)
summary(rs)Null model: Generalized Loewe Additivity
Variance assumption used: "equal"
Mean occupancy rate: 0.6732745
Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
Transformations: Yes
Drug A Drug B
Slope 2.132 1.201
Maximal response 0.088 0.091
log10(EC50) 1.008 0.977
Common baseline at: 0.132
No test statistics were computed.The occcupancy matrix used in the expected response calculation for the Loewe models can be accessed with rs$occupancy.
For off-axis data and a fixed dose combination, the Z-score for that dose combination is defined to be the standardized difference between the observed effect and the effect predicted by a generalized Loewe model. If the observed effect differs significantly from the prediction, it might be due to the presence of synergy or antagonism. If multiple observations refer to the same combination of doses, then a mean is taken over these multiple standardized differences.
The following plot illustrates the isobologram of the chosen null model. Coloring and contour lines within the plot should help the user distinguish areas and dose combinations that generate similar response according to the null model. Note that the isobologram is plotted by default on a logarithmically scaled grid of doses.
The plot below illustrates the above considerations in a 3-dimensional setting. In this plot, points refer to the observed effects whereas the surface is the model-predicted response. The surface is colored according to the median Z-scores where blue coloring indicates possible synergistic effects (red coloring would indicate possible antagonism).
Highest Single Agent
For the Highest Single Agent null model to work properly, it is expected that both marginal curves are either decreasing or increasing. Equivalent summary and plot methods are also available for this type of null model.
rsh <- fitSurface(data, marginalFit,
null_model = "hsa",
B.CP = 50, statistic = "both", parallel = FALSE)
summary(rsh)Null model: Highest Single Agent with differing maximal response
Variance assumption used: "equal"
Mean occupancy rate: 0.6732745
Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
Transformations: Yes
Drug A Drug B
Slope 2.132 1.201
Maximal response 0.088 0.091
log10(EC50) 1.008 0.977
Common baseline at: 0.132
Exact meanR test (H0 = no synergy/antagonism):
F(49,117) = 18.178 (p-value < 2e-16)
Evidence for effects in data: Syn
Points with significant deviations from the null:
d1 d2 R p-value call
3 2.0 0.12 -4.274679 0.00173 Syn
10 2.0 0.49 -4.422648 0.00092 Syn
17 2.0 2.00 -4.664814 0.00025 Syn
18 7.8 2.00 -9.654355 <2e-16 Syn
24 2.0 7.80 -4.767067 0.00015 Syn
25 7.8 7.80 -22.667069 <2e-16 Syn
32 7.8 31.00 -8.024914 <2e-16 Syn
40 31.0 130.00 -3.498738 0.03177 Syn
47 31.0 500.00 -3.725266 0.01442 Syn
Overall maxR summary:
Call Syn Ant Total
Syn 9 0 49Occupancy estimates provided with HSA response surface still rely on the (generalized) Loewe model.
Statistical testing
Presence of synergistic or antagonistic effects can be formalized by means of statistical tests. Two types of tests are considered here and are discussed in more details in the methodology vignette as well as the accompanying paper.
meanRtest evaluates how the predicted response surface based on a specified null model differs from the observed one. If the null hypothesis is rejected, this test suggests that at least some dose combinations may exhibit synergistic or antagonistic behaviour. ThemeanRtest is not designed to pinpoint which combinations produce these effects nor what type of deviating effect is present.maxRtest allows to evaluate presence of synergistic/antagonistic effects for each dose combination and as such provides a point-by-point classification.
Both of the above test statistics have a well specified null distribution under a set of assumptions, namely normality of Z-scores. If this assumption is not satisfied, distribution of these statistics can be estimated using bootstrap. Normal approximation is significantly faster whereas bootstrapped distribution of critical values is likely to be more accurate in many practical cases.
meanR
Here we will use the previously computed CP covariance matrix to speed up the process.
- normal errors
meanR_N <- fitSurface(data, marginalFit,
statistic = "meanR", CP = rs$CP, B.B = NULL,
parallel = FALSE)- non-normal errors
The previous piece of code assumes normal errors. If we drop this assumption, we can use bootstrap methods to resample from the observed errors. Other parameters for bootstrapping, such as additional distribution for errors, wild bootstrapping to account for heteroskedasticity, are also available. See help(fitSurface).
meanR_B <- fitSurface(data, marginalFit,
statistic = "meanR", CP = rs$CP, B.B = 20,
parallel = FALSE)Both tests use the same calculated F-statistic but compare it to different null distributions. In this particular case, both tests lead to identical results.
| F-statistic | p-value | |
|---|---|---|
| Normal errors | 4.855372 | 0 |
| Bootstrapped errors | 4.855372 | 0 |
maxR
The meanR statistic can be complemented by the maxR statistic for each of available dose combinations. We will do this once again by assuming both normal and non-normal errors similar to the computation of the meanR statistic.
maxR_N <- fitSurface(data, marginalFit,
statistic = "maxR", CP = rs$CP, B.B = NULL,
parallel = FALSE)
maxR_B <- fitSurface(data, marginalFit,
statistic = "maxR", CP = rs$CP, B.B = 20,
parallel = FALSE)
maxR_both <- rbind(summary(maxR_N$maxR)$totals,
summary(maxR_B$maxR)$totals)Here is the summary of maxR statistics. It lists the total number of dose combinations listed as synergistic or antagonistic for Experiment 1 given the above calculations.
| Call | Syn | Ant | Total | |
|---|---|---|---|---|
| Normal errors | Syn | 3 | 0 | 49 |
| Bootstrapped errors | Syn | 2 | 0 | 49 |
By using the outsidePoints function, we can obtain a quick summary indicating which dose combinations in Experiment 1 appear to deviate significantly from the null model according to the maxR statistic.
outPts <- outsidePoints(maxR_B$maxR$Ymean)
kable(outPts, caption = paste0("Non-additive points for Experiment ", i))| d1 | d2 | R | p-value | call | |
|---|---|---|---|---|---|
| 25 | 7.8 | 7.8 | -9.353602 | 0 | Syn |
| 32 | 7.8 | 31.0 | -4.558810 | 0 | Syn |
Synergistic effects of drug combinations can be depicted in a bi-dimensional contour plot where the x-axis and y-axis represent doses of Compound 1 and Compound 2 respectively and each point is colored based on the p-value and sign of the respective maxR statistic.
contour(maxR_B,
## colorPalette = c("blue", "black", "black", "red"),
main = paste0(" Experiment ", i, " contour plot for maxR"),
scientific = TRUE, digits = 3, cutoff = cutoff)
Previously, we had colored the 3-dimensional predicted response surface plot based on its Z-score, i.e. deviation of the predicted versus the observed effect. We can also easily color it based on the computed maxR statistic to account for additional statistical variation.
