lmmSynergy: Synergy calculation using linear-mixed models

In SynergyLMM: Statistical Framework for in Vivo Drug Combination Studies

Synergy calculation using linear-mixed models

Description

lmmSynergy allows for the calculation of synergy using 3 different references models: Bliss independence, highest single agent and response additivity. The calculation of synergy is based on hypothesis testing on the coefficient estimates from the model fitted by lmmModel().

Usage

lmmSynergy(
  model,
  method = "Bliss",
  min_time = 0,
  robust = FALSE,
  type = "CR2",
  ra_nsim = 1000,
  show_plot = TRUE,
  ...
)

Arguments

model	An object of class "lme" representing the linear mixed-effects model fitted by lmmModel().
method	String indicating the method for synergy calculation. Possible methods are "Bliss", "HSA" and "RA", corresponding to Bliss, highest single agent and response additivity, respectively.
min_time	Minimun time for which to start calculating synergy.
robust	If TRUE, uncertainty is estimated using sandwich-based robust estimators of the variance-covariance matrix of the regression coefficient estimates provided by clubSandwich::vcovCR.lme.
type	Character string specifying which small-sample adjustment should be used, with available options "CR0", "CR1", "CR1p", "CR1S", "CR2", or "CR3". See "Details" section of clubSandwich::vcovCR() for further information.
ra_nsim	Number of random sampling to calculate the synergy for Response Additivity model.
show_plot	Logical indicating if a plot with the results of the synergy calculation should be generated.
...	Additional arguments to be passed to marginaleffects::hypotheses().

Details

lmmSynergy uses the statistical description provided by Demidenko and Miller (2019) for the calculation of synergy. It is based on hypothesis testing on the coefficients estimates from the model fitted by lmmModel(): \hat{\beta}_C, \hat{\beta}_A, \hat{\beta}_B, \hat{\beta}_{AB}, which represent the estimated specific growth rates for the Control, Drug A, Drug B and Combination groups, respectively.

Bliss Indepence Model

For Bliss model, lmmSynergy test the following null hypothesis:

Two-drugs combination experiment:

H_0: \beta_{combination} = \beta_A + \beta_B - \beta_{control}

Three-drugs combination experiment:

H_0: \beta_{combination} = \beta_A + \beta_B + \beta_C - 2\beta_{control}

Highes Single Agent (HSA)

Two-drugs combination experiment: For the HSA model, lmmSynergy test the following null hypothesis:

H_0: \beta_{combination} = \min(\beta_A, \beta_B)

Three-drugs combination experiment: For the HSA model, lmmSynergy test the following null hypothesis:

H_0: \beta_{combination} = \min(\beta_A, \beta_B, \beta_C)

Response Additivity (RA)

For the RA model, lmmSynergy test the following null hypothesis:

Two-drugs combination experiment:

H_0: e^{\beta_{combination}t} = e^{\beta_At}+e^{\beta_Bt}-e^{\beta_{control}t}

Three-drugs combination experiment:

H_0: e^{\beta_{combination}t} = e^{\beta_At}+e^{\beta_Bt}+e^{\beta_Ct}-2e^{\beta_{control}t}

For Bliss and HSA models, lmmSynergy uses marginaleffects::hypotheses() to conduct hypothesis tests on the estimated coefficients of the model.

In the case of the RA model, the null hypothesis is tested comparing the area under the curve (i.e. cumulative effect from the beginning of a treatment to a time point of interest) obtained from each side of the equation for the null hypothesis, based on ra_sim random samplings from the distribution of the coefficients.

Combination Index and Synergy Score

The results obtained by lmmSynergy include the synergy score (SS) and combination index (CI) for the model, for each time point, together with their confidence interval, and the corresponding p-value. The values of SS and CI provided by lmmSynergy follow previous definitions of these metrics so they have the same interpretation:

• The SS has been defined as the excess response due to drug interaction compared to the reference model (Ianevski et al. (2017), Ianevski, Giri, and Aittokallio (2022), Mao and Guo (2023)). Following this definition, a SS>0, SS=0, and SS<0, represent synergistic, additive and antagonistic effects, respectively.

• According to the common definition of the CI, a CI<1, CI=1, and CI>1 represent synergistic, additive and antagonistic effects, respectively (Yadav et al. (2015), Demidenko and Miller (2019), Mao and Guo (2023)), and provides information about the observed drug combination effect versus the expected additive effect provided by the reference synergy model. A drug combination effect larger than the expected (CI > 1) would indicate synergism, a drug combination effect equal to the expected (CI = 1) would indicate additivity, and a lower drug combination effect than the expected (CI < 1) would indicate antagonism.

As mentioned above, the results include the synergy results for each day. This means that lmmSynergy refits the model using the data from time_start defined in lmmModel() until each time point, providing the synergy results for each of these models and for that specific time point.

Uncertainty estimation using robust estimators

If robust = TRUE, lmmSynergy deals with possible model misspecifications, allowing for cluster-robust variance estimation using clubSandwich::vcovCR.lme. When using robust = TRUE, setting type = "CR2" is recommended. See more details in clubSandwich::vcovCR().

Note: When a variance structure has been specified in the model it is recommended to use always robust = TRUE to get a better estimation.

Value

The function returns a list with two elements:

• Constrasts: List with the outputs of the linear test for the synergy null hypothesis obtained by marginaleffects::hypotheses() for each time. See marginaleffects::hypotheses() for more details.

• Synergy: Data frame with the synergy results, indicating the model of synergy ("Bliss", "HSA" or "RA"), the metric (combination index and synergy score), the value of the metric estimate (with upper and lower confidence interval bounds) and the p-value, for each time.

If show_plot = TRUE, a plot with the synergy results obtained with plot_lmmSynergy() is also shown.

References

• Demidenko, Eugene, and Todd W. Miller. 2019. Statistical Determination of Synergy Based on Bliss Definition of Drugs Independence. PLoS ONE 14 (November). https://doi.org/10.1371/journal.pone.0224137.

• Yadav, Bhagwan, Krister Wennerberg, Tero Aittokallio, and Jing Tang. 2015. Searching for Drug Synergy in Complex Dose–Response Landscapes Using an Interaction Potency Model. Computational and Structural Biotechnology Journal 13: 504–13. https://doi.org/10.1016/j.csbj.2015.09.001.

• Ianevski, Aleksandr, Liye He, Tero Aittokallio, and Jing Tang. 2017. SynergyFinder: A Web Application for Analyzing Drug Combination Dose–Response Matrix Data. Bioinformatics 33 (August): 2413–15. https://doi.org/10.1093/bioinformatics/btx162.

• Ianevski, Aleksandr, Anil K Giri, and Tero Aittokallio. 2022. SynergyFinder 3.0: An Interactive Analysis and Consensus Interpretation of Multi-Drug Synergies Across Multiple Samples. Nucleic Acids Research 50 (July): W739–43. https://doi.org/10.1093/nar/gkac382.

• Mao, Binchen, and Sheng Guo. 2023. Statistical Assessment of Drug Synergy from in Vivo Combination Studies Using Mouse Tumor Models. Cancer Research Communications 3 (October): 2146–57. https://doi.org/10.1158/2767-9764.CRC-23-0243.

• Vincent Arel-Bundock, Noah Greifer, and Andrew Heiss. Forthcoming. How to Interpret Statistical Models Using marginaleffects in R and Python. Journal of Statistical Software. https://marginaleffects.com

Examples

# Load the example data
data(grwth_data)
# Fit the model
lmm <- lmmModel(
  data = grwth_data,
  sample_id = "subject",
  time = "Time",
  treatment = "Treatment",
  tumor_vol = "TumorVolume",
  trt_control = "Control",
  drug_a = "DrugA",
  drug_b = "DrugB",
  combination = "Combination"
  ) 
# Most simple use with default values
syn <- lmmSynergy(lmm)
# Accessing to synergy results data frame
syn$Synergy
# Selecting different reference models:
## Bliss
lmmSynergy(lmm, method = "Bliss")
## HSA
lmmSynergy(lmm, method = "HSA")
## RA
lmmSynergy(lmm, method = "RA", ra_sim = 1000)

# Only calculate synergy from Time 12 onwards
lmmSynergy(lmm, min_time = 12)

# Using robust standard errors
lmmSynergy(lmm, method = "Bliss", robust = TRUE, type = "CR2")

SynergyLMM documentation built on April 4, 2025, 4:13 a.m.