Glycan-analysis

About this vignette

This tutorial intends to showcase and explain the capabilities of the SplineOmics package by walking through a real and complete glycan analysis example, from start to finish. SplineOmics is explained in more detail in the get-started vignette, where a proteomics example is covered. This vignette is more focused on showing how glycan data can be used, and because of this, less details about the overall package are provided here.

Data Overview

This dataset originates from a time-series glycan experiment designed to study Chinese Hamster Ovary (CHO) cells. The data is compositional, meaning that the observed glycoform abundances represent parts of a whole and are constrained by a constant-sum constraint, making standard statistical analyses inappropriate without appropriate log-ratio transformations. The experiment involved cultivating cells in eight bioreactors, with four bioreactors subjected to a temperature shift after 146 hours (experimental condition) and the remaining four bioreactors maintained without a temperature shift (control condition).

Timepoints

Samples were collected at 7 distinct time points throughout the experiment, specifically: "120h", "144h", "168h", "192h", "216h", "288h", and "336h" after cultivation start. Each time point was sampled from all eight bioreactors, but E17_336 is missing, therefore resulting in a total of 55 samples.

Effects in the Experiment: Reactor and Batch

In this experiment, there are two effects to consider: Reactor and Batch.

Reactor:
- This refers to the different bioreactors used for cell cultivation, which can exhibit substantial variability.
- Each reactor was assigned a single condition: either constant temperature or temperature-shifted. As a result, condition and reactor are confounded.
- Reactor should not be treated as a fixed effect to simply remove its influence. Instead, it is treated as a random effect, which allows us to model its variability appropriately.
Batch:
- This refers to the two separate glycan analysis batches.
- Batch is considered a batch effect with respect to the condition (constant temperature vs. temperature-shifted).

Since Condition and Reactor are confounded, the variability due to the reactors cannot be directly separated from the condition. Instead, linear mixed models (LMMs) are used to attribute the reactor as a random effect, allowing us to account for its variability while isolating the effects of the condition. This approach ensures that the analysis appropriately handles the hierarchical structure of the data and avoids incorrect conclusions.

In this vignette, we will demonstrate how to use linear mixed models to address these challenges and properly account for both reactor and plate effects.

Further info

The data matrix comprises glycoforms as rows (such as G0/G0 and none/G0F) and samples as columns, providing glycoform measurements for all time points. Each glycoform as a row stands for a combination of sugars (glycans) that can be attached to the left and right side of the product antibody, that we produced in our CHO cell cultivation. For example G0/G0 means that the glycan G0 was attached to both sides, and none/G0F means at the left side, there was no glycan, and on the right side, there was the G0F glycan.

The goal of this experiment is to investigate the effect of a temperature shift during CHO cell cultivation on the antibody glycan dynamics over time.

Note: This is not the original dataset, as it has not yet been published at the time of this vignette’s creation. For demonstration purposes, the glycans have been randomly shuffled, and 2% has been randomly added or substracted off each value.

Analysis Goals

The main objectives of this analysis are:

Identify glycans with significant temporal changes: Among the glycoforms measured, the goal is to identify those that exhibit significant changes in abundance over time.
Cluster glycans based on temporal patterns: Glycoforms showing significant temporal changes (hits) will be grouped into clusters based on their time-dependent expression patterns.
Assess the impact of temperature shifts on temporal patterns: The analysis will determine whether the temporal patterns of glycoform abundance are affected by the temperature shift, i.e., whether glycoform abundance dynamics differ over time under temperature shift conditions compared to controls.

Note

The documentation of all the SplineOmics package functions can be viewed here

Load the packages

# library(SplineOmics)
library(devtools)
#> Loading required package: usethis
devtools::load_all()
#> ℹ Loading SplineOmics
library(here) # For managing filepaths
#> here() starts at /home/thomas/Documents/PhD/projects/DGTX/SplineOmics_hub/SplineOmics
library(dplyr) # For data manipulation
#> 
#> Attaching package: 'dplyr'
#> The following object is masked from 'package:testthat':
#> 
#>     matches
#> The following objects are masked from 'package:stats':
#> 
#>     filter, lag
#> The following objects are masked from 'package:base':
#> 
#>     intersect, setdiff, setequal, union
library(knitr) # For Showing the head of the data and the meta tables.
library(compositions)  # For clr transforming the glycan data
#> Welcome to compositions, a package for compositional data analysis.
#> Find an intro with "? compositions"
#> 
#> Attaching package: 'compositions'
#> The following objects are masked from 'package:SplineOmics':
#> 
#>     cor, dist
#> The following objects are masked from 'package:stats':
#> 
#>     anova, cor, cov, dist, var
#> The following object is masked from 'package:graphics':
#> 
#>     segments
#> The following objects are masked from 'package:base':
#> 
#>     %*%, norm, scale, scale.default

Load the files

data <- read.csv(
  system.file(
    "extdata",
    "glycan_data.csv",
    package = "SplineOmics"
  ),
  stringsAsFactors = FALSE
)

meta <- read.csv(
  system.file(
    "extdata",
    "glycan_meta.csv",
    package = "SplineOmics"
  ),
  stringsAsFactors = FALSE
)

# Set the first column as row names and remove it from the dataframe
rownames(data) <- data[[1]]  # Assign first column as row names
data <- data[, -1]  # Remove the first column


# Make data to a numeric matrix (required by SplineOmics)
data <- data.matrix(data)

Show top rows of data

knitr::kable(
  head(data),
  format = "markdown"
  )

	E13_120	E13_144	E13_168	E13_192	E13_216	E13_288	E13_336	E14_120	E14_144	E14_168	E14_192	E14_216	E14_288	E14_336	E15_120	E15_144	E15_168	E15_192	E15_216	E15_288	E15_336	E16_120	E16_144	E16_168	E16_192	E16_216	E16_288	E16_336	E17_120	E17_144	E17_168	E17_192	E17_216	E17_288	E18_120	E18_144	E18_168	E18_192	E18_216	E18_288	E18_336	E19_120	E19_144	E19_168	E19_192	E19_216	E19_288	E19_336	E20_120	E20_144	E20_168	E20_192	E20_216	E20_288	E20_336
none/G0F	1.0558960	1.6535404	0.6493199	0.4020102	0.2991287	0.2952674	0.4744168	1.0534692	1.093678	0.5601851	0.5350867	0.3788699	0.4035115	0.3757274	0.8211157	0.8911038	0.3801903	0.4426778	0.3843382	0.3318367	0.3398791	0.7996867	1.571295	1.4926070	0.4777732	0.6603077	0.3305801	0.2346606	0.7880709	1.0201295	0.7577885	0.1905728	0.3328567	0.2802764	0.7150925	0.7753927	0.9080968	0.3811781	0.2896830	0.3272081	0.2351659	1.3833084	0.9376732	1.1173998	0.6469200	0.3043375	0.2228878	0.3791463	1.0784873	0.8032912	0.3756631	0.3559814	0.2675088	0.3162241	0.2643130
none/G1F	0.5323374	0.1912681	0.3820570	0.1920724	0.3369441	0.6384174	0.8548695	0.5733259	0.567492	0.7163753	0.4688813	0.4682832	0.5383108	0.6551322	0.4758381	0.3954215	0.3811519	0.3056876	0.4411230	0.5827720	0.6590168	0.4662977	0.621623	0.4692532	0.4053633	0.4627089	0.5262178	0.4367194	0.2626777	0.4807747	0.4591137	0.0997751	0.2675994	0.7528157	0.3142561	0.2728702	0.4900672	0.3237094	0.3904656	0.5668250	0.4722005	0.5748934	0.6408565	0.5941058	0.3463255	0.2640809	0.4779125	0.8081467	0.3307047	0.3004908	0.2121393	0.3430701	0.3647285	0.5845660	0.6514764
none/G2F	1.5324508	1.8521343	0.8379884	0.7293874	0.4902723	0.5382350	0.6436627	1.5629840	1.289147	0.9681731	0.8381506	0.7008038	0.6532061	0.6719115	1.2138739	1.1464261	0.8146055	0.7811525	0.6912260	0.5626390	0.4371626	1.2651715	1.989103	1.5335954	0.7930346	0.9061705	0.4856366	0.4323194	1.4655321	1.3453025	1.1834043	0.5523239	0.5450371	0.5750546	1.0783679	1.1487218	1.1215321	0.6694769	0.5519309	0.5116042	0.4199132	1.7598936	1.1849219	1.3638929	0.7197493	0.5763007	0.2844577	0.5380468	1.5920806	1.3495185	0.8517591	0.6509213	0.6691747	0.6211062	0.6178701
G0/G0	12.4576400	10.2160346	10.2302200	6.5323299	4.5557592	4.6899314	4.7203765	12.3929924	10.510602	11.4393911	9.4008800	9.4694548	8.1328422	7.6767012	12.1137799	11.9553981	11.2695201	9.3657469	8.0426315	6.5983448	5.2429438	12.0570776	10.660712	11.3814227	10.8993364	9.5825590	5.9291533	4.4084289	10.3195660	8.3433472	9.5614397	7.4586037	6.6427808	8.4710449	11.0041856	9.8884089	10.6157702	8.8851596	7.7355069	6.4930713	6.3931685	10.3100238	8.9498072	9.7648715	7.6610699	6.2008262	3.8324832	3.4344767	10.5438400	8.6848225	9.8761378	8.7988318	8.1356305	7.7008655	8.5356119
G0/G0F	34.1841189	38.3372702	41.0179628	49.0643714	57.0485903	55.2748321	51.4400433	35.7867219	39.951440	41.3072610	45.0540699	45.5449446	46.7403271	46.3506974	37.0730784	39.6951101	40.6190535	46.7785275	48.3297400	55.0964653	55.9160088	34.3909667	35.142868	38.1036796	41.7931083	44.5759546	56.0226674	63.5428148	36.5324214	40.7206353	41.4229164	48.7461735	52.2302885	49.1391718	40.4162712	43.1801464	42.3152053	47.0698711	48.8438299	50.1411749	51.0158271	35.9886203	40.0629781	40.8438051	46.0248557	52.3444655	58.5929353	57.2609213	36.2618340	41.6920256	42.9619517	45.8755552	45.5816057	45.9113688	46.1812359
G0F/G0F	1.5218235	0.7862833	1.1317116	0.3237543	-0.1691198	0.9544022	1.2641091	1.8948004	1.654238	1.2341346	1.2849757	1.1123587	1.1001650	0.6284302	1.5736525	1.3480386	1.6721168	1.0828775	0.8432043	0.8164667	1.0990374	2.2216793	1.773861	1.5704807	1.3096889	1.2557005	0.8205026	0.5531342	1.1133060	0.8061578	1.1689967	0.3151786	0.2796022	-0.1211154	1.1292473	0.9209109	1.3967621	1.0388824	0.7251202	0.7248511	0.5504109	1.3079025	0.7601992	1.1123447	0.5752250	0.4825608	0.0937523	0.2893026	1.2427219	0.7371158	0.8688332	0.8953497	0.6032681	0.8568143	1.0963033

Show top rows of meta

knitr::kable(
  head(meta),
  format = "markdown"
  )

sample_name	Reactor	Time	Condition	Batch
E13_120	E13	120	constant	1
E13_144	E13	144	constant	1
E13_168	E13	168	constant	1
E13_192	E13	192	constant	1
E13_216	E13	216	constant	1
E13_288	E13	288	constant	1

Perform EDA (exploratory data analysis)

# Those fields are mandatory, because we believe that when such a report is
# opened after half a year, those infos can be very helpful.
report_info <- list(
  omics_data_type = "Glycan",
  data_description = "clr transformed timeseries fractional abundance glycoform data of CHO cells, batch corrected",
  data_collection_date = "September 2024",
  analyst_name = "Thomas Rauter",
  contact_info = "thomas.rauter@plus.ac.at",
  project_name = "DGTX"
)

report_dir <- here::here(
  "results",
  "explore_data"
)

Compositional data, such as glycan profiles expressed as relative abundances, are inherently constrained by a constant sum and therefore reside in a simplex rather than Euclidean space. This violates the assumptions of standard statistical methods. To address this, we applied a centered log-ratio (CLR) transformation, which appropriately maps the data into real space by accounting for the compositional structure. While some studies apply a simple log-transformation to stabilize variance, this approach ignores the relative nature of the data and may lead to biased or misleading results. CLR transformation is thus the more appropriate and statistically sound choice for downstream modeling.

# TRANSPOSE: make samples rows, features columns
data_t <- t(data)

# Handle zeros (simple pseudocount, crude but common)
data_t[data_t == 0] <- 1e-6

# Apply CLR
clr_data_t <- compositions::clr(data_t)

# Transpose back to original shape: features as rows, samples as columns
clr_data <- t(clr_data_t)

# splineomics now contains the SplineOmics object.
splineomics <- SplineOmics::create_splineomics(
  data = clr_data,
  meta = meta,
  report_info = report_info,
  condition = "Condition", # Column of meta that contains the levels.
  meta_batch_column = "Batch" # For batch effect removal in the plots
)

# Special print.SplineOmics function leads to selective printing
print(splineomics)
#> data:SplineOmics Object
#> -------------------
#> Number of features (rows): 10 
#> Number of samples (columns): 55 
#> Meta data columns: 5 
#> First few meta columns:
#>   sample_name Reactor Time Condition Batch
#> 1     E13_120     E13  120  constant     1
#> 2     E13_144     E13  144  constant     1
#> 3     E13_168     E13  168  constant     1
#> Condition: Condition 
#> No RNA-seq data provided.
#> No annotation provided.
#> No spline parameters set.
#> P-value adjustment method: BH

plots <- SplineOmics::explore_data(
  splineomics = splineomics, # SplineOmics object
  report_dir = report_dir
)

Here you can see the HTML report of the explore_data() function with the NOT batch-corrected data, and here the report for the batch-corrected data.

Run limma spline analysis

In this example, we are skipping finding the best hyperparameters with the screen_limma_hyperparams() function, because we already have a clear idea of what do use.

Lets define our parameters and put them into the SplineOmics object:

splineomics <- SplineOmics::update_splineomics(
  splineomics = splineomics,
  # Reactor as random effect.
  design = "~ 1 + Condition*Time + Batch + (1|Reactor)", 
  mode = "integrated", # means limma uses the full data for each condition.
  spline_params = list(
    spline_type = c("n"), # natural cubic splines (take these if unsure)
    dof = c(2L) # If you are unsure about which dof, start with 2 and increase
  )
)

The term ‘(1|Reactor)’ in the design formula are the random effects. Therefore, the linear mixed models that we use will model Reactor as a random effect.

Run the run_limma_splines() function with the updated SplineOmics object:

splineomics <- SplineOmics::run_limma_splines(
  splineomics = splineomics
)
#> Hint: The data contains negative values. This may occur if the data has been transformed (e.g., log-transformed or normalized) and is valid in such cases. Ensure that the data preprocessing aligns with your analysis requirements.
#> Column 'Batch' of meta will be used to remove the batch effect for the plotting
#> Make sure that the design formula contains no interaction between the condition and time for mode == isolated, and that it contains an interaction for mode == integrated. Otherwise, you will get an uncaught error of 'coefficients not estimable' or 'subscript out of bounds'.
#> 
#> Running Breusch-Pagan test to check for violation of homoscedasticity of each feature...
#> 
#> ------------------------------------------------------------
#> Fraction of features violating homoscedasticity (p < 0.050): 10.00%
#> ❗ Linear model assumption of homoscedasticity is likely violated.
#> ➡️  Using robust modeling strategy:
#> arrayWeights() and eBayes(robust = TRUE) to stabilize variance.
#> ------------------------------------------------------------
#> Info limma spline analysis completed successfully

Note that for our dataset, the Wilcoxon signed rank exact test revealed that we have a problem of heteroscedasticity for our data. This means that the variance between the samples is significantly higher in the stationary phase compared to the exponential phase. This is a violation for the assumption of homoscedasticity of linear models, which means that the derived p-values are not trustworthy. To adress this issue, limma has a function (arrayWeights) which downweights samples with higher variance, which we then use here automatically. Further, the eBayes function of limma with the robust = TRUE argument avoids using variance “outliers” for the variance shrinkage.

Build limma report

The topTables of all three limma result categories can be used to generate p-value histograms an volcano plots.

report_dir <- here::here(
  "results",
  "create_limma_reports"
)

plots <- SplineOmics::create_limma_report(
  splineomics = splineomics,
  report_dir = report_dir
)

You can view the generated analysis report of the create_limma_report function here.

Cluster the hits (significant features)

After we obtained the limma spline results, we can cluster the hits based on their temporal pattern (their spline shape). We define what a hit is by setting an adj. p-value threshold for every level. Hits are features (e.g. proteins) that have an adj. p-value below the threshold. Hierarchical clustering is used to place every hit in one of as many clusters as we have specified for that specific level.

adj_pthresholds <- c( # 0.05 for both levels
  0.05, # exponential
  0.05 # stationary
)

clusters <- c(
  2L,   # clusters for the exponential level
  2L    # clusters for the stationary level
)

report_dir <- here::here(
  "results",
  "clustering_reports"
)

# treatment_labels allows to place vertical dashed lines into the plots, that
# indicate a treatment, such as "temp shift" in this experiment. For each level,
# the treatment can be specified individually. For the "double spline plots", 
# where two levels are combined into one plot, treatment lines can also be 
# defined. The correct fieldname for those is {first_level}_{second_level}, 
# with first_level being the level of the two occuring first in the respective
# meta column, and second_level the one that occurs after. 
# Here, we don't want a treatment line for the constant level, since no 
# treatment was applied. To achieve that, we simply set it to NA or leave it
# out. 
treatment_labels = list(
  # constant = NA, 
  tshifted = "temp shift",  
  constant_tshifted = "temp shift" 
  )

# treatment_timepoints allows to specify the timepoint, at which the vertical
# dashed treatment line is placed. Again, for the constant level, we don't
# have a treatment, so we also do not specify a timepoint.
treatment_timepoints = list(
  # constant = NA,
  tshifted = 146,  
  constant_tshifted = 146
  )

plot_info <- list( # For the spline plots
  y_axis_label = "Glycan fractional abundance",
  time_unit = "hours", # our measurements were in minutes
  treatment_labels = treatment_labels,
  treatment_timepoints = treatment_timepoints
)

# Those are not genes for this glycan analysis here, but this argument expects
# the feature names (which usually are gene names, which is why it is called 
# like this.
genes <- rownames(data)

plot_options <- list(
  # When meta_replicate_column is not there, all datapoints are blue.
  meta_replicate_column = "Reactor", # Colors the data points based on Reactor
  cluster_heatmap_columns = FALSE # Per default FALSE, just for demonstration
)

clustering_results <- SplineOmics::cluster_hits(
  splineomics = splineomics,
  adj_pthresholds = adj_pthresholds,
  clusters = clusters,
  genes = genes,
  plot_info = plot_info,
  plot_options = plot_options,
  report_dir = report_dir,
  adj_pthresh_avrg_diff_conditions = 0.05,
  adj_pthresh_interaction_condition_time = 0.05
)

You can view the generated analysis report of the cluster_hits function here.

As discussed before, there are three limma result categories. The cluster_hits() report shows the results of all three, if they are present (category 2 and 3 can only be generated when the design formula contains an interaction effect).

Session Info

#> R version 4.3.3 (2024-02-29)
#> Platform: x86_64-pc-linux-gnu (64-bit)
#> Running under: Ubuntu 22.04.5 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/local/R-4.3.3/lib/R/lib/libRblas.so 
#> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
#> 
#> locale:
#>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
#>  [3] LC_TIME=de_AT.UTF-8        LC_COLLATE=en_US.UTF-8    
#>  [5] LC_MONETARY=de_AT.UTF-8    LC_MESSAGES=en_US.UTF-8   
#>  [7] LC_PAPER=de_AT.UTF-8       LC_NAME=C                 
#>  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
#> [11] LC_MEASUREMENT=de_AT.UTF-8 LC_IDENTIFICATION=C       
#> 
#> time zone: Europe/Vienna
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices datasets  utils     methods   base     
#> 
#> other attached packages:
#> [1] compositions_2.0-8 knitr_1.49         dplyr_1.1.4        here_1.0.1        
#> [5] SplineOmics_0.2.0  testthat_3.2.3     devtools_2.4.5     usethis_3.1.0     
#> 
#> loaded via a namespace (and not attached):
#>   [1] RColorBrewer_1.1-3       tensorA_0.36.2.1         rstudioapi_0.17.1       
#>   [4] jsonlite_1.8.9           shape_1.4.6.1            magrittr_2.0.3          
#>   [7] nloptr_2.1.1             farver_2.1.2             rmarkdown_2.29          
#>  [10] GlobalOptions_0.1.2      fs_1.6.5                 ragg_1.3.3              
#>  [13] vctrs_0.6.5              minqa_1.2.8              memoise_2.0.1           
#>  [16] base64enc_0.1-3          htmltools_0.5.8.1        progress_1.2.3          
#>  [19] broom_1.0.7              variancePartition_1.32.5 sass_0.4.9              
#>  [22] KernSmooth_2.23-22       bslib_0.9.0              htmlwidgets_1.6.4       
#>  [25] desc_1.4.3               plyr_1.8.9               pbkrtest_0.5.3          
#>  [28] zoo_1.8-13               cachem_1.1.0             mime_0.12               
#>  [31] lifecycle_1.0.4          iterators_1.0.14         pkgconfig_2.0.3         
#>  [34] Matrix_1.6-5             R6_2.6.1                 fastmap_1.2.0           
#>  [37] rbibutils_2.3            shiny_1.10.0             clue_0.3-66             
#>  [40] numDeriv_2016.8-1.1      digest_0.6.37            colorspace_2.1-1        
#>  [43] patchwork_1.3.0          S4Vectors_0.40.2         rprojroot_2.0.4         
#>  [46] pkgload_1.4.0            textshaping_1.0.0        compiler_4.3.3          
#>  [49] remotes_2.5.0            aod_1.3.3                withr_3.0.2             
#>  [52] doParallel_1.0.17        backports_1.5.0          BiocParallel_1.36.0     
#>  [55] viridis_0.6.5            dendextend_1.19.0        pkgbuild_1.4.6          
#>  [58] gplots_3.2.0             bayesm_3.1-6             MASS_7.3-60.0.1         
#>  [61] sessioninfo_1.2.3        rjson_0.2.23             corpcor_1.6.10          
#>  [64] gtools_3.9.5             caTools_1.18.3           tools_4.3.3             
#>  [67] lmtest_0.9-40            zip_2.3.2                httpuv_1.6.15           
#>  [70] remaCor_0.0.18           glue_1.8.0               nlme_3.1-164            
#>  [73] promises_1.3.2           grid_4.3.3               reshape2_1.4.4          
#>  [76] cluster_2.1.6            generics_0.1.3           gtable_0.3.6            
#>  [79] tidyr_1.3.1              hms_1.1.3                BiocGenerics_0.48.1     
#>  [82] stringr_1.5.1            ggrepel_0.9.6            foreach_1.5.2           
#>  [85] pillar_1.10.1            limma_3.58.1             later_1.4.1             
#>  [88] robustbase_0.99-4-1      circlize_0.4.16          splines_4.3.3           
#>  [91] lattice_0.22-5           renv_1.1.1               tidyselect_1.2.1        
#>  [94] ComplexHeatmap_2.18.0    miniUI_0.1.1.1           reformulas_0.4.0        
#>  [97] gridExtra_2.3            IRanges_2.36.0           svglite_2.1.3           
#> [100] RhpcBLASctl_0.23-42      stats4_4.3.3             xfun_0.50               
#> [103] Biobase_2.62.0           statmod_1.5.0            brio_1.1.5              
#> [106] matrixStats_1.5.0        DEoptimR_1.1-3-1         pheatmap_1.0.12         
#> [109] stringi_1.8.4            yaml_2.3.10              boot_1.3-29             
#> [112] evaluate_1.0.3           codetools_0.2-19         tibble_3.2.1            
#> [115] BiocManager_1.30.25      cli_3.6.4                xtable_1.8-4            
#> [118] systemfonts_1.2.1        Rdpack_2.6.2             munsell_0.5.1           
#> [121] jquerylib_0.1.4          Rcpp_1.0.14              EnvStats_3.0.0          
#> [124] png_0.1-8                parallel_4.3.3           ellipsis_0.3.2          
#> [127] pkgdown_2.1.1            ggplot2_3.5.1            prettyunits_1.2.0       
#> [130] profvis_0.4.0            urlchecker_1.0.1         bitops_1.0-9            
#> [133] lme4_1.1-36              mvtnorm_1.3-3            viridisLite_0.4.2       
#> [136] lmerTest_3.1-3           scales_1.3.0             openxlsx_4.2.8          
#> [139] purrr_1.0.4              crayon_1.5.3             fANCOVA_0.6-1           
#> [142] GetoptLong_1.0.5         rlang_1.1.5

Thomas Rauter

13 February, 2024