Title:	Statistical Framework for in Vivo Drug Combination Studies
Version:	1.0.1
Description:	A framework for evaluating drug combination effects in preclinical in vivo studies. 'SynergyLMM' provides functions to analyze longitudinal tumor growth experiments using linear mixed-effects models, perform time-dependent analyses of synergy and antagonism, evaluate model diagnostics and performance, and assess both post-hoc and a priori statistical power. The calculation of drug combination synergy follows the statistical framework provided by Demidenko and Miller (2019, <doi:10.1371/journal.pone.0224137>). The implementation and analysis of linear mixed-effect models is based on the methods described by Pinheiro and Bates (2000, <doi:10.1007/b98882>), and Gałecki and Burzykowski (2013, <doi:10.1007/978-1-4614-3900-4>).
License:	GPL (≥ 3)
Encoding:	UTF-8
RoxygenNote:	7.3.2
Imports:	magrittr (≥ 2.0.3), rlang, dplyr (≥ 1.1.4), ggplot2 (≥ 3.5.1), cowplot (≥ 1.1.3), nlme (≥ 3.1-165), nlmeU, fBasics, car, MASS, performance, lattice, marginaleffects, clubSandwich
Suggests:	lme4, knitr, rmarkdown, testthat (≥ 3.0.0)
Config/testthat/edition:	3
Depends:	R (≥ 4.0)
LazyData:	true
VignetteBuilder:	knitr
URL:	https://github.com/RafRomB/SynergyLMM
BugReports:	https://github.com/RafRomB/SynergyLMM/issues
NeedsCompilation:	no
Packaged:	2025-02-07 15:39:59 UTC; rafrombec
Author:	Rafael Romero-Becerra [aut, cre], Zhi Zhao [ctb], Tero Aittokallio [ctb]
Maintainer:	Rafael Romero-Becerra <r.r.becerra@medisin.uio.no>
Repository:	CRAN
Date/Publication:	2025-02-07 18:10:02 UTC

Pipe operator

Description

See magrittr::%>% for details.

Usage

lhs %>% rhs

Arguments

Value

The result of calling rhs(lhs).

A Priori Synergy Power Analysis Based on Variability and Drug Effect

Description

A priori power calculation for a hypothetical two-drugs combination study of synergy using linear-mixed models with varying drug combination effect and/or experimental variability.

Usage

APrioriPwr(
  npg = 5,
  time = c(0, 3, 5, 10),
  grwrControl = 0.08,
  grwrA = 0.07,
  grwrB = 0.06,
  grwrComb = 0.03,
  sd_ranef = 0.01,
  sgma = 0.1,
  sd_eval = NULL,
  sgma_eval = NULL,
  grwrComb_eval = NULL,
  method = "Bliss",
  ...
)

Arguments

Details

APrioriPwr allows for total customization of an hypothetical drug combination study and allows the user to define several experimental parameters, such as the sample size, time of measurements, or drug effect, for the power evaluation of synergy for Bliss and HSA reference models. The power calculation is based on F-tests of the fixed effects of the model as previously described (Helms, R. W. (1992), Verbeke and Molenberghs (2009), Gałecki and Burzykowski (2013)).

The focus the power analysis with APrioriPwr is on the drug combination effect and the variability in the experiment. For other a priori power analysis see also PwrSampleSize() and PwrTime().

• npg, time, grwrControl, grwrA, grwrB, grwrComb, sd_ranef and sgma are parameters referring to the initial exemplary data set and corresponding fitted model. These values can be obtained from a fitted model, using lmmModel_estimates(), or be defined by the user.

• sd_eval, sgma_eval, and grwrComb_eval are the different values that will be modified in the initial exemplary data set to fit the corresponding model and calculate the power.

Value

The functions returns several plots:

• A plot representing the hypothetical data, with the regression lines for each treatment group according to grwrControl, grwrA, grwrB and grwrComb values. The values assigned to sd_ranef and sgma are also shown.

• A plot showing the values of the power calculation depending on the values assigned to sd_eval and sgma_eval. The power result corresponding to the values assigned to sd_ranef and sgma is shown with a red dot.

• A plot showing the values of the power calculation depending on the values assigned to grwrComb_eval. The vertical dashed line indicates the value of grwrComb. The horizontal line indicates the power of 0.80.

The statistical power for the fitted model for the initial data set according to the values given by npg, time, grwrControl, grwrA, grwrB, grwrComb, sd_ranef and sgma is also shown in the console.

References

• Helms, R. W. (1992). Intentionally incomplete longitudinal designs: I. Methodology and comparison of some full span designs. Statistics in Medicine, 11(14–15), 1889–1913. https://doi.org/10.1002/sim.4780111411

• Verbeke, G. & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer New York. https://doi.org/10.1007/978-1-4419-0300-6

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

See Also

PostHocPwr,PwrSampleSize(), PwrTime().

Examples

APrioriPwr(
sd_eval = seq(0.01, 0.2, 0.01),
sgma_eval = seq(0.01, 0.2, 0.01),
grwrComb_eval = seq(0.01, 0.1, 0.005)
)

Cook's distance for the coefficient estimates

Description

CookDistance allows the user to identify those subjects with a greater influence in the estimation of the \beta (tumor growth rate) for the treatment group, based in the calculation of Cook's distances.

Usage

CookDistance(model, cook_thr = NA, label_angle = 0, verbose = TRUE)

Arguments

Details

The identification of the subjects with a greater influence in each estimated \beta representing the tumor growth is based on the calculation of Cook's distances, as described in Gałecki and Burzykowsk (2013). To compute the Cook's distance for the \beta estimates (i.e., the contribution to each subject to the coefficient of its treatment group), first a matrix containing the leave-one-subject-out estimates or \beta is calculated. Then, the Cook's distances are calculated according to:

D_i \equiv \frac{(\hat{\beta} - \hat{\beta}_{(-i)})[\widehat{Var(\hat{\beta})}]^{-1}(\hat{\beta} - \hat{\beta}_{(-i)})}{rank(X)}

where \hat{\beta}_{(-i)} is the estimate of the parameter vector \beta obtained by fitting the model to the data with the i-th subject excluded. The denominator of the expression is equal to the number of the fixed-effects coefficients, which, under the assumption that the design matrix is of full rank, is equivalent to the rank of the design matrix.

Value

A plot of the Cook's distance value for each subject, indicating those subjects whose Cook's distance is greater than cook_thr.

If saved to a variable, the function returns a vector with the Cook's distances for each subject.

References

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

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"
  ) 
# Calulate Cook's distances for each subject
CookDistance(model = lmm)
# Change the Cook's distance threshold
CookDistance(model = lmm, cook_thr = 0.15)

Observed vs predicted values and performance of the model

Description

ObsvsPred allows the user to have a straight forward idea about how the model is fitting the data, providing plots of the predicted regression lines versus the actual data points.

Usage

ObsvsPred(model, nrow = 4, ncol = 5, ...)

Arguments

Details

The function provides visual and quantitative information about the performance of the model:

• A layout of the observed and predicted values of log(relative tumor volume) vs Time for each SampleID (i.e. subject), with the actual measurements, the regression line for each SampleID, and the marginal, treatment-specific, regression line for each treatment group.

• Performance metrics of the model obtain calculated using performance::model_performance(). The maximum likelihood-based Akaike's Information Criterion (AIC), small sample AIC (AICc), and Bayesian Information Criterion, and the Nakagawa's r-squared root mean squared error (RMSE) of the residuals, and the standard deviation of the residuals (sigma) are provided.

Value

Performance metrics of the model obtain calculated using performance::model_performance() and a layout of plots of the observed vs predicted values for each SampleID.

References

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

• Lüdecke et al., (2021). performance: An R Package for Assessment, Comparison and Testing of Statistical Models. Journal of Open Source Software, 6(60), 3139. https://doi.org/10.21105/joss.03139

• Sakamoto, Y., M. Ishiguro, and G. Kitagawa. 1984. Akaike Information Criterion Statistics. Mathematics and Its Applications. Reidel.

• Nakagawa, Shinichi, and Holger Schielzeth. 2013. A General and Simple Method for Obtaining r2 from Generalized Linear Mixed-effects Models. Methods in Ecology and Evolution 4 (February): 133–42. https://doi.org/10.1111/j.2041-210x.2012.00261.x.

• Johnson, Paul C. D. 2014. Extension of Nakagawa & Schielzeth’s r 2 GLMM to Random Slopes Models. Methods in Ecology and Evolution 5 (September): 944–46. https://doi.org/10.1111/2041-210X.12225.

• Nakagawa, Shinichi, Paul C. D. Johnson, and Holger Schielzeth. 2017. The Coefficient of Determination r2 and Intra-Class Correlation Coefficient from Generalized Linear Mixed-Effects Models Revisited and Expanded. Journal of The Royal Society Interface 14 (September): 20170213. https://doi.org/10.1098/rsif.2017.0213.

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"
  )
# Obtain Observed vs Predicted plots, and model performance metrics
ObsvsPred(model = lmm, nrow = 4, ncol = 8)

Post hoc power calculation based on simulations of the synergy evaluation using LMM.

Description

PostHocPwr allows for the post hoc power analysis of the synergy hypothesis testing for Bliss and HSA refence models for a given tumor growth data fitted model.

Usage

PostHocPwr(model, nsim = 1000, method = "Bliss", pvalue = 0.05, time = NA, ...)

Arguments

Details

The post hoc power calculation relies on simulation of the dependent variable, using nlmeU::simulateY.

1. For a given fitted model of the tumor growth data, nsim simulations of the dependent variable (\log (RTV)) are done, based on the marginal distribution implied by the fitted model.

2. The model is then fitted to the new values of the dependant variable.

3. For each simulation, the new estimates from each model are then used for the synergy hypothesis testing as explained in lmmSynergy, and the p-values stored.

4. The power is returned as the proportion of simulations resulting in a significant synergy hypothesis testing (p-value < pvalue).

When time is specified, PostHocPwr refits the model using the data from the time_start time point defined in lmmModel() until time, and report the statistical power for that model. If time is not specified, the model fitted using all data points is used for statistical power calculation.

Value

Returns a numeric value of the power for the synergy calculation for the model using the method specified in method. The power is expressed as the proportion of simulations that provides a p-value below the threshold specified in pvalue.

References

Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

See Also

APrioriPwr(), PwrSampleSize(), PwrTime().

Examples

#' 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"
  )
 PostHocPwr(lmm, nsim = 50) # 50 simulations for shorter computing time
 # Using a seed to obtain reproducible results
 PostHocPwr(lmm, seed = 123, nsim = 50)
 # Calculating the power for an specific day
 PostHocPwr(lmm, nsim = 50, time = 6)

Calculates power based on a model fit

Description

This function is generic; method functions can be written to handle specific classes of objects.

Usage

Pwr(object, ...)

Arguments

Value

numeric scalar value.

Author(s)

Andrzej Galecki and Tomasz Burzykowski

See Also

Pwr.lme

Examples

data("grwth_data")
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")
Pwr(lmm,
    L = c("Time:TreatmentControl" = 1,
    "Time:TreatmentDrugA" = -1,
    "Time:TreatmentDrugB" = -1,
    "Time:TreatmentCombination" = 1))
 

Performs power calculations

Description

This is method for Pwr() generic function. It is a modified version from the one described by Galecki and Burzykowski implemented in nlmeU package (nlmeU::Pwr.lme).

Usage

## S3 method for class 'lme'
Pwr(
  object,
  ...,
  type = c("sequential", "marginal"),
  Terms,
  L,
  verbose = FALSE,
  sigma,
  ddf = numeric(0),
  alpha = 0.05,
  altB = NULL,
  tol = 1e-10
)

Arguments

Value

a data frame inheriting from class Pwr.lme

References

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

See Also

nlme::anova.lme, nlmeU::Pwr.lme

A Priori Synergy Power Analysis Based on Sample Size

Description

A priori power calculation for a hypothetical two-drugs combination study of synergy evaluation using linear-mixed models depending on the sample size per group.

Usage

PwrSampleSize(
  npg = c(5, 8, 10),
  time = c(0, 3, 5, 10),
  grwrControl = 0.08,
  grwrA = 0.07,
  grwrB = 0.06,
  grwrComb = 0.03,
  sd_ranef = 0.01,
  sgma = 0.1,
  method = "Bliss",
  ...
)

Arguments

Details

PwrSampleSize allows the user to define an hypothetical drug combination study, customizing several experimental parameters, such as the sample size, time of measurements, or drug effect, for the power evaluation of synergy for Bliss and HSA reference models. The power calculation is based on F-tests of the fixed effects of the model as previously described (Helms, R. W. (1992), Verbeke and Molenberghs (2009), Gałecki and Burzykowski (2013)).

The focus the power analysis with PwrSampleSize is on the sample size per group. The function allows for the evaluation of how the statistical power changes when the sample size per group varies while the other parameters are kept constant. For other a priori power analysis see also APrioriPwr() and PwrTime().

• time, grwrControl, grwrA, grwrB, grwrComb, sd_ranef and sgma are parameters referring to the initial exemplary data set and corresponding fitted model. These values can be obtained from a fitted model, using lmmModel_estimates(), or be defined by the user.

• npg is a vector indicating the different sample sizes for which the statistical power is going to be evaluated, keeping the rest of parameters fixed.

Value

The functions returns two plots:

• A plot representing the hypothetical data, with the regression lines for each treatment group according to grwrControl, grwrA, grwrB and grwrComb values. The values assigned to sd_ranef and sgma are also shown.

• A plot showing the values of the power calculation depending on the values assigned to npg.

The function also returns the data frame with the power for the analysis for each sample size specified in npg.

References

• Helms, R. W. (1992). Intentionally incomplete longitudinal designs: I. Methodology and comparison of some full span designs. Statistics in Medicine, 11(14–15), 1889–1913. https://doi.org/10.1002/sim.4780111411

• Verbeke, G. & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer New York. https://doi.org/10.1007/978-1-4419-0300-6

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

See Also

PostHocPwr, APrioriPwr(), PwrTime().

Examples

PwrSampleSize(npg = 1:20)

A Priori Synergy Power Analysis Based on Time

Description

A priori power calculation for a hypothetical two-drugs combination study of synergy depending on the time of follow-up or the frequency of measurements.

Usage

PwrTime(
  npg = 5,
  time = list(seq(0, 9, 3), seq(0, 21, 3), seq(0, 30, 3)),
  type = "max",
  grwrControl = 0.08,
  grwrA = 0.07,
  grwrB = 0.06,
  grwrComb = 0.03,
  sd_ranef = 0.01,
  sgma = 0.1,
  method = "Bliss",
  ...
)

Arguments

Details

PwrTime allows the user to define an hypothetical drug combination study, customizing several experimental parameters, such as the sample size, time of measurements, or drug effect, for the power evaluation of synergy for Bliss and HSA reference models. The power calculation is based on F-tests of the fixed effects of the model as previously described (Helms, R. W. (1992), Verbeke and Molenberghs (2009), Gałecki and Burzykowski (2013)).

The focus the power analysis with PwrTime is on the time at which the measurements are done. The function allows for the evaluation of how the statistical power changes when the time of follow-up varies while the frequency of measurements is keep constant. It also allows to how the statistical power changes when the time of follow-up is kept constant, but the frequency of measurements varies.

For other a priori power analysis see also APrioriPwr() and PwrSampleSize().

• npg, grwrControl, grwrA, grwrB, grwrComb, sd_ranef and sgma are parameters referring to the initial exemplary data set and corresponding fitted model. These values can be obtained from a fitted model, using lmmModel_estimates(), or be defined by the user.

• time is a list in which each element is a vector with the times at which the tumor volume measurements have been performed, and for which the statistical power is going to be evaluated, keeping the rest of parameters fixed.

Value

The functions returns two plots:

• A plot representing the hypothetical data, with the regression lines for each treatment group according to grwrControl, grwrA, grwrB and grwrComb values. The values assigned to sd_ranef and sgma are also shown.

• A plot showing the values of the power calculation depending on the values assigned to Time. If type is set to "max", the plot shows how the power varies depending on the maximum time of follow-up. If type is set to "freq", the plot shows how the power varies depending on how frequently the measurements have been performed.

The function also returns the data frame with the power for the analysis for each value specified in Time.

References

• Helms, R. W. (1992). Intentionally incomplete longitudinal designs: I. Methodology and comparison of some full span designs. Statistics in Medicine, 11(14–15), 1889–1913. https://doi.org/10.1002/sim.4780111411

• Verbeke, G. & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer New York. https://doi.org/10.1007/978-1-4419-0300-6

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

See Also

PostHocPwr, APrioriPwr(), PwrSampleSize().

Examples

# Power analysis maintaining the frequency of measurements 
# and varying the time of follow-up ('type = "max"')
PwrTime(time = list(seq(0, 9, 3), 
                    seq(0, 12, 3), 
                    seq(0, 15, 3), 
                    seq(0, 21, 3), 
                    seq(0, 30, 3)), 
                    type = "max")

# Power analysis maintaining the time of follow-up 
# and varying the frequency of measurements ('type = "freq"')
PwrTime(time = list(seq(0, 10, 1), 
                    seq(0, 10, 2), 
                    seq(0, 10, 5), 
                    seq(0, 10, 10)), 
                    type = "freq")

Helper function to calculate the relative tumor volume from an imput data frame of tumor growth

Description

getRTV is a helper function used inside lmmModel() to obtain a dataframe with a column RTV corresponding to the relative tumor volume to time time_start, and a column logRTV with the logarithm of RTV.

Usage

getRTV(data, time_start)

Arguments

Value

The function returns the original data frame of tumor growth data, with 3 additional columns, corresponding to:

• RTV: Relative tumor volume to the tumor volume at start_time.

• logRTV: Logarithm of RTV column.

• TV0: Tumor volume at start_time.

Examples

# Load example dataset
data("grwth_data")
# Change column names
colnames(grwth_data) <- c("SampleID", "Time", "Treatment", "TV")
# Calculate relative tumor volume
getRTV(data = grwth_data, time_start = 0)

Example Tumor Growth Data

Description

A long format data frame with example tumor growth data, generated with simulateTumorGrowth(), representing a simulated in vivo two-drugs combination experiment, involving 8 subjects per group, with 30 days of follow-up, taking measurement every 3 days.

Usage

data(grwth_data)

Format

A data frame with 352 rows and 4 columns:

subject

Subject IDs.

Time

Time for each measurement in an arbitrary unit, for example, days.

Treatment

Treatment group for each subject (Control, DrugA, DrugB, and Combination).

TumorVolume

Measurement representing the tumor volume in an arbitrary unit, for example, mm^3.

Linear Mixed Effect Model for Tumor Growth

Description

lmmModel() fits a linear mixed effect model from a tumor growth dataset. The input data frame must be in long format and include at least the following columns: column with the sample ids, column with the time at which each measurement has been done, a column indicating the treatment group, and a column with the tumor measurement (e.g. tumor volume).

Usage

lmmModel(
  data,
  sample_id = "SampleID",
  time = "Time",
  treatment = "Treatment",
  tumor_vol = "TV",
  trt_control = "Control",
  drug_a = "Drug_A",
  drug_b = "Drug_B",
  drug_c = NA,
  combination = "Combination",
  time_start = NULL,
  time_end = NULL,
  min_observations = 1,
  show_plot = TRUE,
  tum_vol_0 = "ignore",
  ...
)

Arguments

Details

lmmModel() relies in the assumption that tumor growth follows an exponential kinetics. Any departure from this assumption can be tested using the diagnostics functions ranefDiagnostics(), residDiagnostics(), and ObsvsPred().

The model formula for the linear mixed-effect fitted model is:

\log RTV_{i}(t) = \beta_{T_i} \cdot t + b_i \cdot t + \varepsilon_{i} (t).

where \log RTV_{i}(t) denotes the value of the logarithm of the relative tumor volume measured for subject i at time t. The coefficient \beta_{T_i} represents the fixed effectsat time t for each treatment T_i, where T_i \in \{Control, DrugA, DrugB, Combination\} in the case of two-drugs combination experiments, or T_i \in \{Control, DrugA, DrugB, DrugC, Combination\} in the case of three-drugs combination experiments, and indicates the tumor-specific growth rate for each treatment group.

Term b_i \cdot t corresponds to the subject-specific random slope that takes into account the longitudinal nature of the data, where b_i is the random effect for subject i. \varepsilon_{i}(t) is the residual random term for subject i at time t.

The implementation of the linear mixed model in lmmModel() is done using nlme::lme, which also allows for the specification of within-group correlations structures and/or unequal variances. These, and additional parameters, can be passed to the nlme::lme function through the ... argument for fitting the model (see examples below).

Value

An object of class "lme" (see nlme::lme for details) representing the linear mixed-effects model fit. If show_plot = TRUE, the plot of the tumor growth data obtained with plot_lmmModel() is also shown.

References

• Pinheiro JC, Bates DM (2000). Mixed-Effects Models in S and S-PLUS. Springer, New York. doi:10.1007/b98882 doi:10.1007/b98882.

• Pinheiro J, Bates D, R Core Team (2024). nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-166, https://CRAN.R-project.org/package=nlme.

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

Examples

data("grwth_data")
# Most simple model
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"
 )
# Changing the last time point of follow-up
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",
 time_end = 21
 )
# Adding additional parameters for model fitting
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",
 # Adding variance function to represent a different variance per subject
 weights = nlme::varIdent(form = ~1|SampleID),
 # Specifiying control values for lme Fit (useful when convergence problems appear)
 control = nlme::lmeControl(maxIter = 1000, msMaxIter = 1000, niterEM = 100, msMaxEval = 1000)
 )

# Fit a model specifying a different variance per Time
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",
 # Adding variance function to represent a different variance per Time
 weights = nlme::varIdent(form = ~1|Time)
 )

Get estimates from a linear mixed model of tumor growth data

Description

lmmModel_estimates allows the user to easily extract some of the interesting model estimates for further use in other functions, such as for power calculation.

Usage

lmmModel_estimates(model)

Arguments

Details

The model estimates provided by lmmModel_estimates include:

• Fixed effect coefficients: \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. These are shown in columns control, drug_a, drug_b, and combination, respectively.

• Standard deviation of the random effects (between-subject variance). Column sd_ranef.

• Standard deviation of the residuals (within-subject variance). Column sd_resid.

Value

A data frame with the estimated values for the coefficients of the tumor growth for each treatment, the standard deviation of the random effects, and the standard deviation of the residuals of the model. These values can be useful for the power analysis of the model using APrioriPwr().

Examples

data("grwth_data")
# Fit example 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"
  ) 
# Get the estimates
lmmModel_estimates(lmm)

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

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")

Likelihood displacements for the model

Description

logLikSubjectDisplacements allows the user to evaluate the log-likelihood displacement for each subject, indicating the influence of every subject to the model.

Usage

logLikSubjectDisplacements(
  model,
  disp_thrh = NA,
  label_angle = 0,
  var_name = NULL,
  verbose = TRUE,
  ...
)

Arguments

Details

The evaluation of the log-likelihood displacement is based in the analysis proposed in Verbeke and Molenberghs (2009) and Gałecki and Burzykowski (2013). First, a list of models fitted to leave-one-subject-out datasets are obtained. Then, for each model, the maximum likelihood estimate obtained by fitting the model to all data and the maximum likelihood estimate obtained by fitting the model to the data with the i-th subject removed are obtained and used for the log-likelihood displacement calculation. The likelihood displacement, LDi , is defined as twice the difference between the log-likelihood computed at a maximum and displaced values of estimated parameters (Verbeke and Molenberghs (2009), Gałecki and Burzykowsk (2013)):

LD_i \equiv 2 \times \Bigr[\ell_\textrm{Full}(\widehat{\Theta};\textrm{y})-\ell_\textrm{Full}(\widehat{\Theta}_{(-i)};\textrm{y})\Bigr]

where \widehat{\Theta} is the maximum-likelihood estimate of \Theta obtained by fitting the model to all data, while \widehat{\Theta}_{-i} is the maximum-likelihood estimate obtained by fitting the model to the data with the i-subject excluded.

Value

Returns a plot of the log-likelihood displacement values for each subject, indicating those subjects whose contribution is greater than disp_thrh.

References

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

• Verbeke, G. & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer New York. https://doi.org/10.1007/978-1-4419-0300-6

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"
  ) 
# Obtain log-likelihood displacement for each subject
logLikSubjectDisplacements(model = lmm)
# Modifying the threshold for log-likelihood displacement
logLikSubjectDisplacements(model = lmm, disp_thrh = 1)

# Calculating the log-likelihood contribution in a model with a variance structure specified
lmm_var <- 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",
  weights = nlme::varIdent(form = ~ 1|SampleID)
  ) 
# Calculate the log-likelihood contribution
logLikSubjectDisplacements(model = lmm, var_name = "SampleID")

Plots of Observed vs Predicted Values

Description

Visualization of observed vs predicted values by a fitted linear mixed model of tumor growth data.

Usage

plot_ObsvsPred(model, nrow = 4, ncol = 5)

Arguments

Value

A layout (arranged in nrow rows and ncol columns) of the observed and predicted values of log(relative tumor volume) vs Time for each SampleID (i.e. subject), with the actual measurements, the regression line for each SampleID, and the marginal, treatment-specific, regression line for each treatment group.

Examples

#' 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",
  show_plot = FALSE
  )
# Obtain the plots
plot_ObsvsPred(lmm, nrow = 4, ncol = 8)    

Plotting SynergyLMM results

Description

Generic function to generate different types of plots based on SynergyLMM outputs.

Usage

plot_SynergyLMM(object, plot_type = "lmmModel", ...)

Arguments

Value

Different output plots are produced depending on the plot_type selected.

See Also

plot_lmmModel(), plot_lmmSynergy(), plot_ObsvsPred(), plot_ranefDiagnostics(), plot_residDiagnostics().

Examples

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",
  show_plot = FALSE
  )
  
# Plot lmmModel
plot_SynergyLMM(lmm, plot_type = "lmmModel",
trt_control = "Control",
drug_a = "DrugA",
drug_b = "DrugB",
combination = "Combination"
)

# Plot ObsvsPred
plot_SynergyLMM(lmm, plot_type = "ObsvsPred")

# Plot ranefDiagnostics
plot_SynergyLMM(lmm, plot_type = "ranefDiagnostics")

# Plot residDiagnostics
plot_SynergyLMM(lmm, plot_type = "residDiagnostics")

# Plot lmmSynergy
lmmSyn <- lmmSynergy(lmm)
plot_SynergyLMM(lmmSyn, plot_type = "lmmSynergy")

Plotting of tumor growth data from a fitted model

Description

Vizualization of tumor growth data and linear mixed model fitted regression line for the fixed effects. This functions returns a ggplot2 plot, allowing for further personalization.

Usage

plot_lmmModel(
  model,
  trt_control = "Control",
  drug_a = "Drug_A",
  drug_b = "Drug_B",
  drug_c = NA,
  combination = "Combination"
)

Arguments

Value

A ggplot2 plot (see ggplot2::ggplot() for more details) showing the tumor growth data represented as log(relative tumor volume) versus time since treatment initiation. The regression lines corresponding to the fixed effects for each treatment group are also plotted.

Examples

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",
  show_plot = FALSE
  )
# Default plot
plot_lmmModel(lmm,
trt_control = "Control",
drug_a = "DrugA",
drug_b = "DrugB",
combination = "Combination"
)
# Adding ggplot2 elements
plot_lmmModel(lmm,
trt_control = "Control",
drug_a = "DrugA",
drug_b = "DrugB",
combination = "Combination"
) + ggplot2::labs(title = "Example Plot") + ggplot2::theme(legend.position = "top")

Plotting synergy results

Description

Visualization of synergy results obtained by lmmSynergy(). This functions returns a ggplot2 plot, allowing for further personalization.

Usage

plot_lmmSynergy(syn_data)

Arguments

Details

plot_lmmSynergy produces a ggplot2 plot with the results of the synergy calculation. Each dot represents the estimated combination index or synergy score, and the gray lines represent the 95% confidence intervals, for each day. Each dot is colored based on the - \log_{10} (p-value), with purple colors indicating a -\log_{10} (p-value) < 1.3; (p-value > 0.05), and green colors indicating a -\log_{10} (p-value) > 1.3; (p-value < 0.05).

Value

A list with ggplot2 plots (see ggplot2::ggplot() for more details) with the combination index (CI) and synergy score (SS) estimates, confidence intervals and p-values for the synergy calculation using linear mixed models.

Examples

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"
  )
# Obtain synergy results
lmmSyn <- lmmSynergy(lmm)
# Plot synergy results
plot_lmmSynergy(lmmSyn)
# Accessing to the combination index plot
plot_lmmSynergy(lmmSyn)$CI
# Accessing to only synergy score plot
plot_lmmSynergy(lmmSyn)$SS
# Accessing to the grid of both plots side by side
plot_lmmSynergy(lmmSyn)$CI_SS
# Adding ggplot2 elements
plot_lmmSynergy(lmmSyn)$CI + 
ggplot2::labs(title = "Synergy Calculation for Bliss") + 
ggplot2::theme(legend.position = "top")  

Plots for random effects diagnostics

Description

Visualization of random effects diagnostics for a fitted linear mixed model of tumor growth data.

Usage

plot_ranefDiagnostics(model)

Arguments

Value

A list with different plots for evaluating the normality and homoscedasticity of the random effects, including:

• A normal Q-Q plot of the random effects of the model.

• A normal Q-Q plot of the residuals by sample.

• Boxplots of the "raw" residuals (observed - fitted) by sample.

• Scatter plots of the normalized residuals (standardized residuals pre-multiplied by the inverse square-root factor of the estimated error correlation matrix, see nlme::residuals.lme) vs fitted values by sample. Observations with absolute standardized (normalized) residuals greater than the 1-0.05/2 quantile of the standard normal distribution are identified in the plots labelled with the time point corresponding to the observation.

Examples

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",
  show_plot = FALSE
  )
# Generate plots 
plot_ranefDiagnostics(lmm)
# Access to specific plots
plot_ranefDiagnostics(lmm)[[1]]
plot_ranefDiagnostics(lmm)[[2]]

Plots for residuals diagnostics

Description

Visualization of residuals diagnostics for a fitted linear mixed model of tumor growth data.

Usage

plot_residDiagnostics(model)

Arguments

Value

A list with different plots for evaluating the normality and homoscedasticity of the normalized residuals (standardized residuals pre-multiplied by the inverse square-root factor of the estimated error correlation matrix, see nlme::residuals.lme), including:

• A normal Q-Q plot of the normalized residuals of the model.

• A normal Q-Q plot of the normalized residuals of the model by Time.

• A normal Q-Q plot of the normalized residuals of the model by Treatment.

• A dotplot of normalized residuals vs fitted values.

• A dotplot of the normalized residuals by Time and Treatment.

Examples

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",
  show_plot = FALSE
  )
# Generate plots 
plot_residDiagnostics(lmm)
# Access to specific plots
plot_residDiagnostics(lmm)[[1]]
plot_residDiagnostics(lmm)[[2]]

Diagnostics of random effects of the linear mixed model

Description

ranefDiagnostics provides several plots as well as statistical test for the examination of the normality of the random effects of the input model.

Usage

ranefDiagnostics(model, verbose = TRUE)

Arguments

Details

One of the assumptions of the model obtained with lmmModel() (as in any other linear mixed model) is that the random effects are normally distributed:

b_i = N(0,\psi)

For the evaluation of this assumption, ranefDiagnostics provides Q-Q plots of random effects, together with statistical assessment of their normality using Shapiro-Wilk, D'Agostini and Anderson-Darling normality tests. Additionally, Q-Q plots of the normalized residuals (standardized residuals pre-multiplied by the inverse square-root factor of the estimated error correlation matrix, see nlme::residuals.lme) by sample are provided to allow for the identification of subjects which could be notably different from the others and be affecting the adequacy of the model. Additionally, boxplots of the "raw" residuals (observed - fitted) by sample and scatter plots of the normalized residuals versus fitted values by sample are provided to give information about variability of the residuals by subject and possible outlier observations. Observations with absolute standardized (normalized) residuals greater than the 1-0.05/2 quantile of the standard normal distribution are identified in the scatter plots labelled with the time point corresponding to the observation.

Value

A list with different elements for the diagnostics of the random effects are produced:

• plots: Different plots for evaluating the normality and homoscedasticity of the random effects are produced.

• Normality: List with the results from 3 different test of the normality of the random effects: Shapiro - Wilk normality test, D'Agostino normality test and Anderson - Darling normality test.

• Levene.test: results from Levene homoscedasticity test (car::leveneTest()) of the normalized residuals by SampleID (i.e., by subject).

• Fligner.test: results from Fligner homoscedasticity test (stats::fligner.test()) of the normalized residuals by SampleID (i.e., by subject).

References

• Pinheiro JC, Bates DM (2000). Mixed-Effects Models in S and S-PLUS. Springer, New York. doi:10.1007/b98882.

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

See Also

plot_ranefDiagnostics()

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"
  )
# Run random effects diagnostics
ranef_diag <- ranefDiagnostics(lmm)

#Access to individual plots

ranef_diag$Plots[1]
ranef_diag$Plots[2]

# Access to normality tests

ranef_diag$Normality

ranef_diag$Normality$Shapiro.test

# Access to homoscedasticity tests of residuals by subject

ranef_diag$Levene.test

ranef_diag$Fligner.test

Diagnostics of residuals of the linear mixed model

Description

residDiagnostics provides several plots as well as statistical test for the examination of the normality and homoscedasticity of the residuals of the input model.

Usage

residDiagnostics(model, pvalue = 0.05, verbose = TRUE)

Arguments

Details

One of the assumption of the model fit by lmmModel() is that the residuals are normally distributed. For the evaluation of this assumption, residDiagnostics provides Q-Q plots of the normalized residuals (standardized residuals pre-multiplied by the inverse square-root factor of the estimated error correlation matrix, see nlme::residuals.lme), together with statistical assessment of their normality using Shapiro-Wilk, D'Agostini and Anderson-Darling normality tests. Additionally, Q-Q plots of the normalized residuals by time point and treatment group are provided to be able to detect time points or treatment groups which could be notably different from the others and be affecting the adequacy of the model.

Scatter plots of the normalized residuals versus fitted values and normalized residuals per time and per treatment are also provided to give information about variability of the residuals and possible outlier observations. These plots are accompanied by Levene and Fligner-Killend homogeneity of variance test results.

Observations with absolute standardized (normalized) residuals greater than the 1-0.05/2 quantile of the standard normal distribution are identified and reported as potential outlier observations.

Value

A list with different elements for the diagnostics of the residuals are produced:

• plots: Different plots for evaluating the normality and homocedasticity of the residuals.

• outliers: Data frame with the identified outliers based on the Pearson residuals and the value of pval. The column resid.p contains the value of the Pearson residuals for each observation.

• Normality: List with the results from 3 different test of the normality of the normalized residuals of the model: Shapiro - Wilk normality test, D'Agostino normality test and Anderson - Darling normality test.

• Levene.test: List with the Levene homoscedasticity test results of the normalized residuals by Time and Treatment.

• Fligner.test: List with the Fligner-Killeen homoscedasticity test results of the normalized residuals by Time and Treatment.

References

• Pinheiro JC, Bates DM (2000). Mixed-Effects Models in S and S-PLUS. Springer, New York. doi:10.1007/b98882.

• Andrzej Galecki & Tomasz Burzykowski (2013) Linear Mixed-Effects Models Using R: A Step-by-Step Approach First Edition. Springer, New York. ISBN 978-1-4614-3899-1

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"
  )
  
# Residuals diagnostics
resid_diag <- residDiagnostics(model = lmm, pvalue = 0.05)

# Access outliers data frame
resid_diag$Outliers

# Access individual plots
resid_diag$Plots[1]
resid_diag$Plots[2]

# Access results of normality tests
resid_diag$Normality
resid_diag$Normality$Shapiro.test

# Access to homoscedasticity test results

resid_diag$Levene.test

resid_diag$Fligner.test

Helper function to simulate tumor growth data for a two-drug combination experiment.

Description

Helper function to simulate tumor growth data for a two-drug combination experiment.

Usage

simulateTumorGrowth(
  npg = 5,
  timepoints = c(0, 3, 5, 10),
  initial_volume = 100,
  grwrControl = 0.08,
  grwrA = 0.07,
  grwrB = 0.06,
  grwrComb = 0.04,
  sd = 0.1
)

Arguments

Details

The function simulates the tumor growth following exponential kinetics, given by

TV(t) = TV_0 \cdot e^{\beta_i \cdot t}

where TV_0 is given by initial_volume, t is given by timepoints and \beta_i are the coefficients given by grwrControl, grwrA, grwrB, and grwrComb.

The variability is simulated using the sd argument to add random noise from a normal distribution N(1, SD), with SD = sd.

Value

A data frame with tumor growth data in long format.

Examples

# This code generates the 'grwth_data' example dataset:
set.seed(123)
grwth_data <- simulateTumorGrowth(npg = 8, timepoints = seq(0,30,3), 
                                   initial_volume = 200, grwrControl = 0.08,
                                   grwrA = 0.07, grwrB = 0.065, 
                                   grwrComb = 0.04, sd = 0.2)