We first load the necessary packages and set some pre-defined values needed to replicate the analysis.

Process and clean the data

The data for the analysis must come in a data-frame with required columns d1, d2 and effect for doses of two compounds and observed cell counts respectively. The effect column may represent also a type of normalized data and subsequent transformation functions should be adjusted.

We will use sample data included in the package - directAntivirals.

data("directAntivirals", package = "BIGL")
head(directAntivirals)

##   experiment   cpd1   cpd2 effect  d1  d2
## 1          1 cpd1_A cpd2_A 509.64 500 500
## 2          1 cpd1_A cpd2_A 589.67 500 500
## 3          1 cpd1_A cpd2_A 524.71 500 130
## 4          1 cpd1_A cpd2_A 575.45 500 130
## 5          1 cpd1_A cpd2_A 634.53 500  31
## 6          1 cpd1_A cpd2_A 702.35 500  31

This data consists of 11 experiments that can be processed separately. For initial illustration purposes we choose just one experiment and retain only the columns of interest. We define a simple function to do just that.

subsetData <- function(data, i) {
  ## Subset data to a single experiment and, optionally, select the necessary
  ## columns only
  subset(data, experiment == i)[, c("effect", "d1", "d2")]
}

Now let us only pick Experiment 4 to illustrate the functionality of the package.

i <- 4
data <- subsetData(directAntivirals, i)

Dose-response data for Experiment 4 will be used for a large share of the analysis presented here, therefore this subset is stored in a dataframe called data. Later, we will run the analysis for some other experiments as well.

Data transformation

If raw data is measured in cell counts, data transformation might be of interest to improve accuracy and interpretation of the model. Of course, this will depend on the model specification. For example, if a generalized Loewe model is assumed on the growth rate of the cell count, the appropriate conversion should be made from the observed cell counts. The formula used would be \[y = N_0\exp\left(kt\right)\] where $k$ is a growth rate, $t$ is time (fixed) and $y$ is the observed cell count. If such a transformation is specified, it is referred to as the biological transformation.

In certain cases, variance-stabilizing transformations (Box-Cox) can also be useful. We refer to these transformations as power transformations. In many cases, a simple logarithmic transformation can be sufficient but, if desired, a helper function optim.boxcox is available to automate the selection of Box-Cox transformation parameters.

In addition to specifying biological and power transformations, users are also asked to specify their inverses. These are later used in the bootstrapping procedure and plotting methods.

As an example, we might define a transforms list that will be passed to the fitting functions. It contains both biological growth rate and power transformations along with their inverses.

## Define forward and reverse transform functions
transforms <- list(
  "BiolT" = function(y, args) with(args, N0*exp(y*time.hours)),
  "InvBiolT" = function(T, args) with(args, 1/time.hours*log(T/N0)),
  "PowerT" = function(y, args) with(args, log(y)),
  "InvPowerT" = function(T, args) with(args, exp(T)),
  "compositeArgs" = list(N0 = 1,
                         time.hours = 72)
)

compositeArgs contains the initial cell counts (N0) and incubation time (time.hours). In certain cases, the getTransformations wrapper function can be employed to automatically obtain a prepared list with biological growth rate and power transformations based on results from optim.boxcox. Its output will also contain the inverses of these transforms.

transforms_auto <- getTransformations(data)
fitMarginals(data, transforms = transforms_auto)

## In the case of 1-parameter Box-Cox transformation, it is easy
## to retrieve the power parameter by evaluating the function at 0.
## If parameter is 0, then it is a log-transformation.
with(transforms_auto, -1 / PowerT(0, compositeArgs))

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 coefficients.

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"))

## Warning: The `transformations` argument of `fitMarginals()` will be deprecated as of
## BIGL newer versions.
## ℹ Please transform response before all analysis.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

summary(marginalFit)

## Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
## Transformations: Yes
## 
##                  Drug A Drug B
## Slope             1.253  0.843
## Maximal response  0.076  0.082
## log10(EC50)       0.915 -1.041
## 
## Common baseline at: 0.115

marginalFit 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 = 0.1
constraints <- list("matrix" = rbind(c(0, 0, 0, -1, 1, 0, 0),
                                     c(0, 0, 1, 0, 0, 0, 0)),
                    "vector" = c(0, 0.1))

## 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 0.1 and maximal responses at 0.
fitMarginals(data, transforms = transforms,
             fixed = c("m1" = 0, "m2" = 0, "b" = 0.1))

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 estimates
sigma: standard deviation of residuals
df: degrees of freedom from monotherapy curve estimates
model: 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 of constructFormula function with no inputs.
shared_asymptote: whether estimation is constrained to share the asymptote. During the estimation, this is deduced from model object.
method: method used in dose-response curve estimation which will be re-used in bootstrapping
transforms: power and biological transformation functions (and their inverses) used in monotherapy curve estimation. This should be a list in a format described above. If transforms is unspecified or NULL, no transformations will be used in statistical bootstrapping unless the user asks for it explicitly via one of the arguments to fitSurface.

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.

customMarginalFit <- 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(customMarginalFit) <- append(class(customMarginalFit), "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

Five 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 and null_model = "loewe" in response calculation.
Classical Loewe model is used if constraints are such that shared_asymptote = TRUE in the marginal fitting procedure and null_model = "loewe" in response calculation.
Highest Single Agent is used if null_model = "hsa" irrespective of the value of shared_asymptote.
Bliss independence model is used when null_model = "bliss". In the situations when maximal responses are constrained to be equal, the classical Bliss independence approach is used, when they are not equal, the Bliss independence calculation is performed on responses rescaled to the maximum range (i.e. absolute difference between baseline and maximal response).
Alternative Loewe Generalization is used when null_model = "loewe2". If the asymptotes are constrained to be equal, this reduces to the classical Loewe. Note that if shared_asymptote = TRUE constraints are used, this also reduces to classical Loewe model.

Three methods are available to control for errors

Family wise error rate is used if control = "FWER"
False coverage rate is used if control = "FCR"
Directional false coverage rate is used if control = "dFCR"

(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,
                 null_model = "loewe",
                 B.CP = 50, statistic = "none", parallel = FALSE,
                 wild_bootstrap = TRUE, wild_bootType = "normal",
                 control = "dFCR")
summary(rs)

Null model: Generalized Loewe Additivity
Variance assumption used: "equal"
Mean occupancy rate: 0.5948603

Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
Transformations: Yes

                 Drug A Drug B
Slope             1.253  0.843
Maximal response  0.076  0.082
log10(EC50)       0.915 -1.041

Common baseline at: 0.115



No test statistics were computed.
CONFIDENCE INTERVALS
Overall effect
Estimated mean departure from null response surface with 95% confidence interval:
-0.2208 [-0.2594, -0.1757]
Evidence for effects in data: Syn 

Significant pointwise effects
           estimate   lower   upper call
0.12_0.004  -0.2404 -0.4094 -0.0713  Syn
0.12_1      -0.2675 -0.4724 -0.0625  Syn
0.49_0.016  -0.3341 -0.5100 -0.1581  Syn
0.49_0.063  -0.3613 -0.5420 -0.1805  Syn
0.49_0.25   -0.2129 -0.3929 -0.0328  Syn
0.49_1      -0.3840 -0.5892 -0.1788  Syn
130_0.25    -0.1904 -0.3637 -0.0171  Syn
2_0.004     -0.4308 -0.6034 -0.2581  Syn
2_0.016     -0.5848 -0.7535 -0.4162  Syn
2_0.063     -0.7147 -0.8899 -0.5394  Syn
2_0.25      -0.5367 -0.7123 -0.3611  Syn
2_1         -0.4389 -0.6452 -0.2326  Syn
31_0.016    -0.3208 -0.4970 -0.1447  Syn
31_0.063    -0.3865 -0.5621 -0.2109  Syn
31_0.25     -0.4510 -0.6350 -0.2671  Syn
31_1        -0.4359 -0.6663 -0.2054  Syn
7.8_0.004   -0.2392 -0.4212 -0.0571  Syn
7.8_0.016   -0.6429 -0.8180 -0.4677  Syn
7.8_0.063   -0.8496 -1.0195 -0.6797  Syn
7.8_0.25    -0.8124 -0.9824 -0.6424  Syn
7.8_1       -0.6211 -0.8335 -0.4087  Syn

Pointwise 95% confidence intervals summary:
 Syn Ant Total
  21   0    49

The occupancy 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.

isobologram(rs)

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).

plot(rs, legend = FALSE, main = "")

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,
                 wild_bootstrap = TRUE, wild_bootType = "normal",
                 control = "dFCR")
summary(rsh)

Null model: Highest Single Agent with differing maximal response
Variance assumption used: "equal"

Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
Transformations: Yes

                 Drug A Drug B
Slope             1.253  0.843
Maximal response  0.076  0.082
log10(EC50)       0.915 -1.041

Common baseline at: 0.115

Exact meanR test (H0 = no synergy/antagonism):
    F(49,117) = 24.8702 (p-value < 2e-16)

Evidence for effects in data: Syn
Points with significant deviations from the null: 
               d1      d2      absR p-value call
0.12_0.25    0.12 0.25000  3.984755 0.00571  Ant
0.49_0.016   0.49 0.01600  3.654198 0.01882  Syn
2_0.004      2.00 0.00400  7.030019  <2e-16  Syn
2_0.016      2.00 0.01600 12.372232  <2e-16  Syn
2_0.063      2.00 0.06300 11.845195  <2e-16  Syn
2_0.25       2.00 0.25000  6.820106  <2e-16  Syn
31_0.063    31.00 0.06300  4.048320 0.00433  Syn
31_0.25     31.00 0.25000  5.350282   1e-05  Syn
31_1        31.00 1.00000  5.224748   1e-05  Syn
7.8_0.00024  7.80 0.00024  6.175151  <2e-16  Ant
7.8_0.016    7.80 0.01600  8.091246  <2e-16  Syn
7.8_0.063    7.80 0.06300 14.996961  <2e-16  Syn
7.8_0.25     7.80 0.25000 15.296929  <2e-16  Syn
7.8_1        7.80 1.00000  7.654020  <2e-16  Syn

Overall maxR summary:
 Call Syn Ant Total
  Syn  12   2    49

CONFIDENCE INTERVALS
Overall effect
Estimated mean departure from null response surface with 95% confidence interval:
-0.2612 [-0.3024, -0.2024]
Evidence for effects in data: Syn 

Significant pointwise effects
             estimate   lower   upper call
0.12_0.004    -0.2593 -0.4216 -0.0969  Syn
0.12_1        -0.2693 -0.4610 -0.0775  Syn
0.49_0.00024  -0.1711 -0.3225 -0.0196  Syn
0.49_0.001    -0.1987 -0.3491 -0.0484  Syn
0.49_0.004    -0.2269 -0.3893 -0.0645  Syn
0.49_0.016    -0.4097 -0.5848 -0.2346  Syn
0.49_0.063    -0.4127 -0.5837 -0.2418  Syn
0.49_0.25     -0.2350 -0.4079 -0.0622  Syn
0.49_1        -0.3913 -0.5831 -0.1996  Syn
2_0.004       -0.5442 -0.7267 -0.3617  Syn
2_0.016       -0.8954 -1.0633 -0.7276  Syn
2_0.063       -0.9172 -1.0881 -0.7462  Syn
2_0.25        -0.6241 -0.7969 -0.4513  Syn
2_1           -0.4681 -0.6599 -0.2763  Syn
31_0.016      -0.3227 -0.5051 -0.1403  Syn
31_0.063      -0.3929 -0.5753 -0.2104  Syn
31_0.25       -0.4675 -0.6500 -0.2851  Syn
31_1          -0.4603 -0.6428 -0.2779  Syn
7.8_0.004     -0.2689 -0.4490 -0.0887  Syn
7.8_0.016     -0.7481 -0.9282 -0.5679  Syn
7.8_0.063     -1.1442 -1.3244 -0.9641  Syn
7.8_0.25      -1.1104 -1.2832 -0.9376  Syn
7.8_1         -0.7267 -0.9185 -0.5350  Syn

Pointwise 95% confidence intervals summary:
 Syn Ant Total
  23   0    49

Bliss Independence

Also for the Bliss independence 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.

rsb <- fitSurface(data, marginalFit, 
                  null_model = "bliss",
                  B.CP = 50, statistic = "both", parallel = FALSE,
                 wild_bootstrap = TRUE, wild_bootType = "normal",
                 control = "dFCR")
summary(rsb)

Null model: Bliss independence with differing maximal response
Variance assumption used: "equal"

Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
Transformations: Yes

                 Drug A Drug B
Slope             1.253  0.843
Maximal response  0.076  0.082
log10(EC50)       0.915 -1.041

Common baseline at: 0.115

Exact meanR test (H0 = no synergy/antagonism):
    F(49,117) = 8.9099 (p-value < 2e-16)

Evidence for effects in data: Syn
Points with significant deviations from the null: 
               d1      d2     absR p-value call
0.49_1       0.49 1.00000 3.578273 0.02423  Syn
2_0.004      2.00 0.00400 3.568116 0.02524  Syn
2_0.016      2.00 0.01600 5.417975   2e-05  Syn
2_0.063      2.00 0.06300 7.533176  <2e-16  Syn
2_0.25       2.00 0.25000 5.089923   9e-05  Syn
31_0.016    31.00 0.01600 3.799524 0.01139  Syn
7.8_0.00024  7.80 0.00024 3.610220 0.02203  Ant
7.8_0.016    7.80 0.01600 5.931782  <2e-16  Syn
7.8_0.063    7.80 0.06300 8.103392  <2e-16  Syn
7.8_0.25     7.80 0.25000 7.244128  <2e-16  Syn
7.8_1        7.80 1.00000 4.009338 0.00526  Syn

Overall maxR summary:
 Call Syn Ant Total
  Syn  10   1    49

CONFIDENCE INTERVALS
Overall effect
Estimated mean departure from null response surface with 95% confidence interval:
-0.1734 [-0.2166, -0.1107]
Evidence for effects in data: Syn 

Significant pointwise effects
             estimate   lower   upper call
0.12_0.004    -0.2460 -0.3966 -0.0954  Syn
0.12_1        -0.2656 -0.4367 -0.0944  Syn
0.49_0.00024  -0.1558 -0.3010 -0.0106  Syn
0.49_0.001    -0.1488 -0.2958 -0.0019  Syn
0.49_0.016    -0.3423 -0.5027 -0.1818  Syn
0.49_0.063    -0.3610 -0.5230 -0.1990  Syn
0.49_0.25     -0.2018 -0.3630 -0.0406  Syn
0.49_1        -0.3702 -0.5416 -0.1989  Syn
2_0.004       -0.4096 -0.5827 -0.2365  Syn
2_0.016       -0.5498 -0.7213 -0.3783  Syn
2_0.063       -0.6520 -0.8166 -0.4874  Syn
2_0.25        -0.4539 -0.6150 -0.2927  Syn
2_1           -0.3600 -0.5299 -0.1901  Syn
31_0.016      -0.2525 -0.4075 -0.0974  Syn
31_0.063      -0.2346 -0.3846 -0.0846  Syn
31_0.25       -0.2055 -0.3543 -0.0566  Syn
7.8_0.004     -0.1875 -0.3492 -0.0259  Syn
7.8_0.016     -0.5204 -0.6784 -0.3624  Syn
7.8_0.063     -0.6312 -0.7824 -0.4799  Syn
7.8_0.25      -0.5444 -0.6944 -0.3944  Syn
7.8_1         -0.3674 -0.5232 -0.2115  Syn

Pointwise 95% confidence intervals summary:
 Syn Ant Total
  21   0    49

Alternative Loewe Generalization

Also for the Alternative Loewe Generalization 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.

rsl2 <- fitSurface(data, marginalFit, 
                  null_model = "loewe2",
                  B.CP = 50, statistic = "both", parallel = FALSE,
                 wild_bootstrap = TRUE, wild_bootType = "normal",
                 control = "dFCR")
summary(rsl2)

Null model: Alternative generalization of Loewe Additivity
Variance assumption used: "equal"

Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
Transformations: Yes

                 Drug A Drug B
Slope             1.253  0.843
Maximal response  0.076  0.082
log10(EC50)       0.915 -1.041

Common baseline at: 0.115

Exact meanR test (H0 = no synergy/antagonism):
    F(49,117) = 17.0629 (p-value < 2e-16)

Evidence for effects in data: Syn
Points with significant deviations from the null: 
               d1      d2      absR p-value call
0.49_0.016   0.49 0.01600  3.729228 0.01492  Syn
2_0.004      2.00 0.00400  5.769174  <2e-16  Syn
2_0.016      2.00 0.01600  7.898791  <2e-16  Syn
2_0.063      2.00 0.06300  9.581066  <2e-16  Syn
2_0.25       2.00 0.25000  6.165876  <2e-16  Syn
2_1          2.00 1.00000  3.865374 0.00945  Syn
31_0.063    31.00 0.06300  4.399434  0.0012  Syn
31_0.25     31.00 0.25000  5.523963  <2e-16  Syn
31_1        31.00 1.00000  4.938002 0.00013  Syn
7.8_0.00024  7.80 0.00024  4.589089 0.00053  Ant
7.8_0.016    7.80 0.01600  8.052364  <2e-16  Syn
7.8_0.063    7.80 0.06300 11.876320  <2e-16  Syn
7.8_0.25     7.80 0.25000 11.257939  <2e-16  Syn
7.8_1        7.80 1.00000  7.335702  <2e-16  Syn

Overall maxR summary:
 Call Syn Ant Total
  Syn  13   1    49

CONFIDENCE INTERVALS
Overall effect
Estimated mean departure from null response surface with 95% confidence interval:
-0.224 [-0.2669, -0.1913]
Evidence for effects in data: Syn 

Significant pointwise effects
             estimate   lower   upper call
0.12_0.004    -0.2416 -0.4030 -0.0802  Syn
0.12_1        -0.2679 -0.4499 -0.0860  Syn
0.49_0.00024  -0.1581 -0.3117 -0.0045  Syn
0.49_0.016    -0.3391 -0.5056 -0.1727  Syn
0.49_0.063    -0.3662 -0.5490 -0.1834  Syn
0.49_0.25     -0.2163 -0.4066 -0.0261  Syn
0.49_1        -0.3860 -0.5680 -0.2039  Syn
130_0.25      -0.1756 -0.3471 -0.0041  Syn
2_0.004       -0.4378 -0.6046 -0.2710  Syn
2_0.016       -0.5983 -0.7587 -0.4379  Syn
2_0.063       -0.7308 -0.9054 -0.5562  Syn
2_0.25        -0.5495 -0.7303 -0.3686  Syn
2_1           -0.4464 -0.6289 -0.2639  Syn
31_0.016      -0.3227 -0.5022 -0.1432  Syn
31_0.063      -0.3929 -0.5715 -0.2142  Syn
31_0.25       -0.4675 -0.6452 -0.2899  Syn
31_1          -0.4603 -0.6451 -0.2756  Syn
7.8_0.004     -0.2434 -0.4269 -0.0600  Syn
7.8_0.016     -0.6563 -0.8308 -0.4819  Syn
7.8_0.063     -0.8775 -1.0414 -0.7137  Syn
7.8_0.25      -0.8453 -1.0064 -0.6842  Syn
7.8_1         -0.6454 -0.8323 -0.4584  Syn

Pointwise 95% confidence intervals summary:
 Syn Ant Total
  22   0    49

Plot 2D cross section of response surface

Α 2-dimensional predicted response surface plot can be generated for a group of null models. In the plot, the points refer to the observed effects whereas the colored lines are the model-predicted responses for the different null models.

The panels correspond to concentration levels of one compound and the x-axis shows the concentration levels of the second compound. The user has the option to define which compound will be shown in the panels and in the x-axis.

nullModels <- c("loewe", "loewe2", "bliss", "hsa")
rs_list <- Map(fitSurface, null_model = nullModels, MoreArgs = list(
        data = data, fitResult = marginalFit, 
        B.CP = 50, statistic = "none", parallel = FALSE,
        wild_bootstrap = TRUE, wild_bootType = "normal",
        control = "dFCR")
)

synergy_plot_bycomp(rs_list, ylab = "Response", plotBy = "Drug A", color = TRUE)

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.

meanR test 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. The meanR test is not designed to pinpoint which combinations produce these effects nor what type of deviating effect is present.
maxR test 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,
                     wild_bootstrap = TRUE, wild_bootType = "normal",
                     control = "dFCR")

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	16.12679	0
Bootstrapped errors	16.12679	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,
                     wild_bootstrap = TRUE, wild_bootType = "normal",
                     control = "dFCR")
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 4 given the above calculations.

	Call	Syn	Ant	Total
Normal errors	Syn	12	1	49
Bootstrapped errors	Syn	10	1	49

By using the outsidePoints function, we can obtain a quick summary indicating which dose combinations in Experiment 4 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))

Non-additive points for Experiment 4
	d1	d2	absR	p-value	call
2_0.004	2.0	0.00400	6.133101	0.00	Syn
2_0.016	2.0	0.01600	8.011602	0.00	Syn
2_0.063	2.0	0.06300	10.042789	0.00	Syn
2_0.25	2.0	0.25000	6.602912	0.00	Syn
31_0.063	31.0	0.06300	4.077214	0.02	Syn
31_0.25	31.0	0.25000	4.393387	0.00	Syn
7.8_0.00024	7.8	0.00024	4.436426	0.00	Ant
7.8_0.016	7.8	0.01600	8.157618	0.00	Syn
7.8_0.063	7.8	0.06300	11.914330	0.00	Syn
7.8_0.25	7.8	0.25000	10.903613	0.00	Syn
7.8_1	7.8	1.00000	6.158407	0.00	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", "white", "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.

plot(maxR_B, color = "maxR", legend = FALSE, main = "")

Effect sizes and confidence interval

The BIGL package also yields effect sizes and corresponding confidence intervals with respect to any response surface. The overall effect size and confidence interval is output in the summary of the ResponseSurface, but can also be called directly:

summary(maxR_B$confInt)

## Overall effect
## Estimated mean departure from null response surface with 95% confidence interval:
## -0.2208 [-0.2606, -0.1899]
## Evidence for effects in data: Syn 
## 
## Significant pointwise effects
##              estimate   lower   upper call
## 0.12_0.004    -0.2404 -0.4031 -0.0776  Syn
## 0.12_1        -0.2675 -0.4649 -0.0701  Syn
## 0.49_0.00024  -0.1573 -0.3115 -0.0030  Syn
## 0.49_0.016    -0.3341 -0.5035 -0.1646  Syn
## 0.49_0.063    -0.3613 -0.5353 -0.1872  Syn
## 0.49_0.25     -0.2129 -0.3862 -0.0395  Syn
## 0.49_1        -0.3840 -0.5816 -0.1864  Syn
## 130_0.25      -0.1904 -0.3573 -0.0236  Syn
## 2_0.004       -0.4308 -0.5970 -0.2645  Syn
## 2_0.016       -0.5848 -0.7472 -0.4224  Syn
## 2_0.063       -0.7147 -0.8834 -0.5459  Syn
## 2_0.25        -0.5367 -0.7057 -0.3676  Syn
## 2_1           -0.4389 -0.6376 -0.2402  Syn
## 31_0.016      -0.3208 -0.4905 -0.1512  Syn
## 31_0.063      -0.3865 -0.5556 -0.2174  Syn
## 31_0.25       -0.4510 -0.6282 -0.2739  Syn
## 31_1          -0.4359 -0.6578 -0.2139  Syn
## 7.8_0.004     -0.2392 -0.4145 -0.0638  Syn
## 7.8_0.016     -0.6429 -0.8115 -0.4742  Syn
## 7.8_0.063     -0.8496 -1.0132 -0.6860  Syn
## 7.8_0.25      -0.8124 -0.9761 -0.6487  Syn
## 7.8_1         -0.6211 -0.8256 -0.4166  Syn
## 
## Pointwise 95% confidence intervals summary:
##  Syn Ant Total
##   22   0    49

In addition, a contour plot can be made with pointwise confidence intervals. Contour plot colouring can be defined according to the effect sizes or according to maxR results.

plotConfInt(maxR_B, color = "effect-size")

You can also customize the coloring of the contour plot and 3-dimensional predicted response surface plot based on effect sizes:

contour(
    maxR_B,
    colorPalette = c("Syn" = "blue", "None" = "white", "Ant" = "red"),
    main = paste0(" Experiment ", i, " contour plot for effect size"),
    colorBy = "effect-size",
    scientific = TRUE, digits = 3, cutoff = cutoff
)

plot(maxR_B, color = "effect-size", legend = FALSE, main = "", gradient = FALSE,
     colorPalette = c("Ant" = "red", "None" = "white", "Syn" = "blue"),
     colorPaletteNA = "white")

Analysis in case of variance heterogeneity

Starting from the package version 1.2.0 the variance can be estimated separately for on-axis (monotherapy) and off-axis points using method argument to fitSurface. The possible values for method are:

"equal", equal variances assumed (as above, default),
"unequal", variance is estimated separately for on-axis and off-axis points,
"model", the variance is modelled as a function of the mean.

Please see the methodology vignette for details. Below we show an example analysis in such case. Note that transformations are not possible if variances are not assumed equal.

marginalFit <- fitMarginals(data, transforms = NULL)
summary(marginalFit)

## Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
## Transformations: No
## 
##                  Compound 1 Compound 2
## Slope                 1.223      1.008
## Maximal response    214.310    462.322
## log10(EC50)           0.457     -1.639
## 
## Common baseline at: 3979.458

resU <- fitSurface(data, marginalFit, method = "unequal", 
    statistic = "both", B.CP = 20, B.B = 20, parallel = FALSE,
                      wild_bootstrap = TRUE, wild_bootType = "normal",
                      control = "dFCR")
summary(resU)

## Null model: Generalized Loewe Additivity
## Variance assumption used: "unequal"
## Mean occupancy rate: 0.7222181
## 
## Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
## Transformations: No
## 
##                  Compound 1 Compound 2
## Slope                 1.223      1.008
## Maximal response    214.310    462.322
## log10(EC50)           0.457     -1.639
## 
## Common baseline at: 3979.458
## 
## Bootstrapped meanR test (H0 = no synergy/antagonism):
##  F(49,117) = 7.948 (p-value < 2e-16)
## 
## Evidence for effects in data: Syn
## Points with significant deviations from the null: 
##                d1      d2     absR p-value call
## 0.12_0.004   0.12 0.00400 7.666810  <2e-16  Syn
## 0.49_0.00024 0.49 0.00024 4.896459  <2e-16  Syn
## 0.49_0.001   0.49 0.00100 4.378196  <2e-16  Syn
## 0.49_0.016   0.49 0.01600 5.902539  <2e-16  Syn
## 2_0.004      2.00 0.00400 7.032048  <2e-16  Syn
## 2_0.016      2.00 0.01600 7.318842  <2e-16  Syn
## 2_0.063      2.00 0.06300 5.169768  <2e-16  Syn
## 7.8_0.016    7.80 0.01600 4.265172  <2e-16  Syn
## 7.8_0.063    7.80 0.06300 4.378360  <2e-16  Syn
## 
## Overall maxR summary:
##  Call Syn Ant Total
##   Syn   9   0    49
## 
## CONFIDENCE INTERVALS
## Overall effect
## Estimated mean departure from null response surface with 95% confidence interval:
## -194.23 [-293.6487, 25.3107]
## Evidence for effects in data: None 
## 
## Significant pointwise effects
##               estimate     lower     upper call
## 0.12_0.004   -744.1210 -957.2780 -530.9641  Syn
## 0.49_0.00024 -465.6427 -688.9717 -242.3138  Syn
## 0.49_0.001   -427.0063 -646.9899 -207.0227  Syn
## 0.49_0.004   -425.2081 -640.8506 -209.5656  Syn
## 0.49_0.016   -636.8597 -853.7796 -419.9397  Syn
## 0.49_0.063   -363.1038 -583.2415 -142.9662  Syn
## 2_0.004      -745.5425 -962.5064 -528.5787  Syn
## 2_0.016      -781.8747 -995.8941 -567.8553  Syn
## 2_0.063      -567.9238 -783.9720 -351.8756  Syn
## 2_0.25       -297.3783 -513.0082  -81.7483  Syn
## 7.8_0.004    -263.1519 -493.1373  -33.1664  Syn
## 7.8_0.016    -483.6000 -707.7305 -259.4696  Syn
## 7.8_0.063    -475.0748 -691.5275 -258.6220  Syn
## 7.8_0.25     -355.7178 -570.5703 -140.8652  Syn
## 7.8_1        -263.4871 -482.1530  -44.8211  Syn
## 
## Pointwise 95% confidence intervals summary:
##  Syn Ant Total
##   15   0    49

For the variance model, an exploratory plotting function is available to explore the relationship between the mean and the variance.

plotMeanVarFit(data)

plotMeanVarFit(data, log = "xy") #Clearer on the log-scale

plotMeanVarFit(data, trans = "log") #Thresholded at maximum observed variance

The linear fit seems fine in this case.

resM <- fitSurface(data, marginalFit, method = "model", 
    statistic = "both", B.CP = 20, B.B = 20, parallel = FALSE,
                      wild_bootstrap = TRUE, wild_bootType = "normal",
                      control = "dFCR")

If the log transformation yielded a better fit, then this could be achieved by using the following option.

resL <- fitSurface(data, marginalFit, method = "model", trans = "log", 
    statistic = "both", B.CP = 20, B.B = 20, parallel = FALSE,
                      wild_bootstrap = TRUE, wild_bootType = "normal",
                      control = "dFCR")

Negative variances were modelled, but variance model has the smallest observed variances as minimum so we can proceed.`

summary(resM)

## Null model: Generalized Loewe Additivity
## Variance assumption used: "model"
## Mean occupancy rate: 0.7222181
## 
## Formula: effect ~ fn(h1, h2, b, m1, m2, e1, e2, d1, d2)
## Transformations: No
## 
##                  Compound 1 Compound 2
## Slope                 1.223      1.008
## Maximal response    214.310    462.322
## log10(EC50)           0.457     -1.639
## 
## Common baseline at: 3979.458
## 
## Bootstrapped meanR test (H0 = no synergy/antagonism):
##  F(49,117) = 63.5616 (p-value < 2e-16)
## 
## Evidence for effects in data: Syn
## Points with significant deviations from the null: 
##                 d1      d2      absR p-value call
## 0.49_0.016    0.49 0.01600  4.002029    0.05  Syn
## 130_0.00024 130.00 0.00024 20.154443  <2e-16  Ant
## 130_0.001   130.00 0.00100 12.860045  <2e-16  Ant
## 130_0.016   130.00 0.01600  5.446635  <2e-16  Syn
## 130_0.063   130.00 0.06300  4.270034    0.05  Syn
## 130_0.25    130.00 0.25000  4.873464  <2e-16  Syn
## 130_1       130.00 1.00000 13.713871  <2e-16  Ant
## 2_0.004       2.00 0.00400  5.370226  <2e-16  Syn
## 2_0.016       2.00 0.01600  6.836464  <2e-16  Syn
## 2_0.063       2.00 0.06300  6.749047  <2e-16  Syn
## 2_1           2.00 1.00000  8.380021  <2e-16  Ant
## 31_0.004     31.00 0.00400 17.764933  <2e-16  Ant
## 31_0.016     31.00 0.01600 11.383025  <2e-16  Syn
## 31_0.063     31.00 0.06300 10.803941  <2e-16  Syn
## 31_1         31.00 1.00000  4.043937    0.05  Syn
## 500_0.00024 500.00 0.00024  3.973185    0.05  Ant
## 500_0.001   500.00 0.00100  4.881484  <2e-16  Syn
## 500_0.063   500.00 0.06300  6.560268  <2e-16  Syn
## 500_0.25    500.00 0.25000 10.550823  <2e-16  Syn
## 500_1       500.00 1.00000  8.463668  <2e-16  Ant
## 7.8_0.016     7.80 0.01600  6.851392  <2e-16  Syn
## 7.8_0.063     7.80 0.06300 27.710960  <2e-16  Syn
## 7.8_0.25      7.80 0.25000 15.181416  <2e-16  Syn
## 7.8_1         7.80 1.00000 10.426276  <2e-16  Syn
## 
## Overall maxR summary:
##  Call Syn Ant Total
##   Syn  17   7    49
## 
## CONFIDENCE INTERVALS
## Overall effect
## Estimated mean departure from null response surface with 95% confidence interval:
## -194.23 [-296.6101, -115.3013]
## Evidence for effects in data: Syn 
## 
## Significant pointwise effects
##            estimate      lower     upper call
## 2_0.004   -745.5425 -1408.0272  -83.0579  Syn
## 2_0.016   -781.8747 -1296.3699 -267.3795  Syn
## 2_0.063   -567.9238  -916.9363 -218.9113  Syn
## 2_0.25    -297.3783  -543.5988  -51.1578  Syn
## 31_0.016  -139.2683  -253.9192  -24.6174  Syn
## 31_0.063  -160.0900  -282.7304  -37.4496  Syn
## 31_0.25   -183.3832  -356.3636  -10.4029  Syn
## 7.8_0.016 -483.6000  -755.1243 -212.0758  Syn
## 7.8_0.063 -475.0748  -579.7293 -370.4203  Syn
## 7.8_0.25  -355.7178  -521.7351 -189.7004  Syn
## 
## Pointwise 95% confidence intervals summary:
##  Syn Ant Total
##   10   0    49

Synergy analysis

2024-08-01