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Choosing normalisation and imputation methods

Sergio Ciordia

Source: vignettes/choosing-methods.Rmd
choosing-methods.Rmd

Introduction

Normalisation and imputation are not merely technical details. Normalisation changes the scale on which samples are compared, whereas imputation determines which values replace the measurements that were not observed. Both choices can therefore affect the proteins identified as differentially abundant.

There is no method that is best for every dataset. A useful choice should be based on the characteristics of the experiment, the missing-value pattern and diagnostics calculated from the data. This vignette follows that reasoning in four steps:

  1. describe the normalisation methods available in NADIA;
  2. compare candidate normalisation methods using quantitative metrics and diagnostic plots;
  3. describe and compare the available imputation strategies; and
  4. apply the selected combination and examine how alternative choices affect the differential abundance results.

NADIA exposes 13 normalisation labels and 20 imputation strategies. At the level of selectable methods, this gives 13 × 20 = 260 nominal normalisation–imputation combinations. Running and interpreting every combination is possible, but it is rarely an efficient first step. The metrics in this vignette provide a staged screen: first rank plausible normalisation methods using replicate agreement and condition separation, then compare imputation methods on the selected—or a small number of plausible—normalised assays. The resulting ranks reduce the search space; they do not by themselves prove that the highest-ranked combination is biologically correct.

The value 260 is a catalogue-level count rather than 260 fully interchangeable completed matrices: log2 and log2Norm are numerically redundant, and limpa and none have special downstream interpretations. The count nevertheless illustrates why a principled shortlist is preferable to choosing a pipeline by trial and error.

The examples use nadia_dia, a preprocessed DIA dataset included with NADIA. The comparison is performed without an external ground truth. If an experiment contains spike-in proteins with known abundances or expected fold changes, the best normalisation–imputation combination is easier to establish. Such an experiment supplies an external reference against which the complete pipeline can be benchmarked, rather than relying only on internal properties of the processed matrix. Proteins added at known different abundances show whether the pipeline recovers the expected changes. Proteins kept at the same abundance, by contrast, show how often the pipeline reports a difference when none is expected. In that setting, the benchmarking results should take priority over the internal ranks described here; see vignette("benchmarking").

Installation and packages

While NADIA is under development, it can be installed from GitHub:

if (!requireNamespace("remotes", quietly = TRUE))
    install.packages("remotes")
remotes::install_github("sciordia/NADIA", build_vignettes = TRUE)

Only NADIA and SummarizedExperiment need to be attached for the code in this vignette. Method-specific packages are called internally when required.

Normalisation

What normalisation does

Protein intensities may contain systematic differences caused by sample preparation, injected amount, instrument response or other non-biological sources. Normalisation aims to reduce these unwanted differences while preserving the biological variation that the experiment was designed to measure.

An effective method should therefore satisfy two goals at the same time:

  • replicate samples from the same condition should become more comparable;
  • genuine differences between conditions should remain visible.

Making every sample look identical would reduce technical dispersion, but it could also remove biological signal. For this reason, no single diagnostic is sufficient on its own.

Normalisation methods available in NADIA

NADIA provides 13 normalisation options, which are selected with the norm_method argument. Regardless of the option chosen, the resulting protein-abundance matrix is expressed on the log2 scale.

norm_method Principle
log2 Applies the log2 transformation and no additional normalisation. It is the natural baseline for comparison.
log2Norm Applies the same log2 transformation. It is retained as a named benchmark method for compatibility.
GlobalMedian Divides each sample by its total intensity and rescales the totals to their median before log2 transformation.
GlobalMean Divides each sample by its total intensity and rescales the totals to their mean before log2 transformation.
eqmedians Log2-transforms the data and shifts every sample so that all sample medians are equal. This approach is based on the equal-median normalisation used by MSstats.
medianNorm Divides raw intensities by the sample median, rescales to the mean of the sample medians and then applies log2.
meanNorm Divides raw intensities by the sample mean, rescales to the mean of the sample means and then applies log2.
vsn Uses variance-stabilising normalisation from the vsn package to reduce the dependence between variance and mean intensity.
quantile Maps the empirical distribution of every sample to a common quantile distribution using limma.
quantile.robust Uses the median, rather than the mean, across sample quantiles to construct a reference distribution that is less sensitive to extreme samples.
Rlr Fits a robust linear regression between each sample and the protein-wise median profile, then removes the fitted offset and scale. This approach is based on the global robust linear regression normalisation implemented in NormalyzerDE.
MAD Aligns both the median and the median absolute deviation of every sample to the corresponding global values.
cycloess Applies cyclic loess normalisation with limma, correcting intensity-dependent differences between samples.

log2 and log2Norm produce the same values. Consequently, nm_run_normalizations(methods = "all") retains the log2 baseline and skips the redundant log2Norm assay, leaving 12 distinct assays for comparison.

How normalisation methods are evaluated

NADIA combines quantitative measures of within-condition agreement with a measure of between-condition separation. Four of the measures follow the intragroup evaluation used by the PRONE package: PCV, PMAD, PEV and intragroup correlation.

Rank What NADIA calculates Preferred direction
Rank_PCV For each protein and condition, the coefficient of variation is calculated as 100×SD/|mean|100 \times SD / |mean|. Values are averaged across conditions and summarised by the median across proteins. Lower
Rank_PMAD The median absolute deviation is calculated within each condition for every protein, averaged across conditions and summarised by the median across proteins. Lower
Rank_PEV The within-condition variance is calculated for every protein, averaged across conditions and summarised by the median across proteins. Lower
Rank_Cor Pairwise correlations are calculated between replicate samples within each condition and summarised by their median. Higher
Rank_Sep PCA is performed on complete protein profiles. A one-way ANOVA F-ratio measures separation of the PC1 sample scores by condition relative to variation within conditions. Higher

Rank_Final is the mean of these five ranks. The lowest value is preferred. This combined criterion rewards agreement among replicates without allowing dispersion alone to decide the result. That qualification is important: the PRONE evaluation showed that reducing intragroup variation does not necessarily identify the method that performs best in downstream analyses.

The ranking should be read together with the diagnostic plots returned by normalization_metrics():

  • boxplot and density compare sample intensity distributions;
  • pcv, pmad and pev show within-condition dispersion across proteins;
  • correlation shows agreement between replicate samples;
  • pca and mds show whether replicates cluster and conditions separate;
  • scatter compares two samples directly, and qq assesses one sample against a normal distribution;
  • metrics displays additional multivariate diagnostics; and
  • final_ranking displays the combined ranking.

These plots answer different questions. A method should not be selected from a single attractive PCA or a single low dispersion value.

Comparing normalisation methods with NADIA

The first step builds a baseline SummarizedExperiment. The second applies the candidate methods, storing each result as a separate assay. Four methods are used here to keep the example concise; omitting methods runs all distinct benchmark assays.

se_norm <- nm_prepare_se(nadia_dia, verbose = FALSE)
se_norm <- nm_run_normalizations(
    se_norm,
    methods = c("log2Norm", "cycloess", "Rlr", "quantile"),
    verbose = FALSE)

assayNames(se_norm)
#> [1] "log2"     "cycloess" "Rlr"      "quantile"

normalization_metrics() calculates the diagnostics and returns both tables and plots. output_dir = NULL prevents files from being written to disk.

nm <- normalization_metrics(
    se_norm,
    output_dir = NULL,
    verbose = FALSE)

nm$final_rank
#>     Method Rank_PCV Rank_PMAD Rank_PEV Rank_Cor Rank_Sep Rank_Final
#> 1 cycloess        1         1        2        1        2        1.4
#> 2      Rlr        2         2        1        3        3        2.2
#> 3 quantile        3         3        3        4        1        2.8
#> 4     log2        4         4        4        2        4        3.6

The two similarly named elements have different roles but contain the same ranking. nm$final_rank is the numerical table: its five component columns are Rank_PCV, Rank_PMAD, Rank_PEV, Rank_Cor and Rank_Sep, and Rank_Final is their mean. nm$final_ranking is the bar-chart representation of Rank_Final from that table; it does not calculate an additional ranking. The separately returned pc1_rank and mds1_rank tables are optional diagnostics and are not included in Rank_Final.

The first row is the preferred method under the combined criterion. In this example, cycloess has the lowest Rank_Final: it ranks first for PCV, PMAD and correlation while retaining good separation between conditions. Rlr is second, and the unnormalised log2 baseline ranks last.

The final ranking plot provides the same comparison visually. Shorter bars indicate a better mean rank.

nm$final_ranking

Final ranking of candidate normalisation methods

The PCA remains useful as a complementary check because it shows the individual samples rather than reducing the comparison to one score.

nm$pca

PCA of samples after each candidate normalisation method

The selected normalisation method can be extracted directly:

norm_winner <- nm$final_rank$Method[1]
norm_winner
#> [1] "cycloess"

Imputation

Why values are missing

Imputation replaces missing entries with estimates so that analyses requiring a complete matrix can be performed. The estimate should reflect a plausible reason for the value being absent. NADIA uses the following practical distinction:

  • MAR (missing at random): the protein is present, but its intensity was not quantified successfully—for example because of poor chromatography, precursor interference or a missed match. Under this assumption, the gap is not itself evidence that the unknown intensity was low. Its value can therefore be estimated from patterns in the available measurements, such as similar protein profiles or related samples.
  • MNAR (missing not at random): the protein is absent or its abundance is below the detection limit. Missingness depends on the unobserved abundance being low and therefore carries biological or measurement information. Replacing such a gap with a typical neighbouring value may erase that evidence, so a low-value or detection-aware method is generally more plausible.
Two-panel diagram comparing MAR and MNAR missing values. MAR shows a missing replicate among similar observed intensities and recommends structure-based imputation. MNAR shows low values falling below a detection limit and recommends low-value or detection-aware imputation. A note states that an individual missing value does not reveal its cause and that both mechanisms can coexist.

Practical distinction between MAR and MNAR missing values in proteomics. MAR values are reconstructed from observed data structure, whereas MNAR values call for a low-value or detection-aware model.

Click the figure to enlarge it; press Esc or click outside the image to close it.

An individual NA does not reveal why the value is missing. MAR and MNAR are possible explanations that must be evaluated using the experimental design, the overall distribution of missing values and their relationship with observed intensity. Both mechanisms may occur in the same dataset.

How NADIA represents missing intensities

By the time the data enter the analysis, an unreported intensity is represented as NA. How that NA is created depends on the input format. In a long Spectronaut report, a protein that was not quantified in a particular run is usually absent from the report; when NADIA constructs the wide protein_quant table, the corresponding protein–run cell becomes NA. Some wide reports use zero instead to mark an unreported intensity. Before normalisation, normalize_proteomics() converts these zeros to NA, so downstream functions handle both situations consistently.

Imputation methods available in NADIA

NADIA provides 20 selectable strategies. The grouping below describes their main intent and does not imply that every missing value can be classified unambiguously.

Group imp_method Principle
Structure-based, generally used for MAR bpca Bayesian principal-component imputation using pcaMethods.
knn Estimates a value from similar proteins using k-nearest neighbours.
mice Multiple imputation by chained equations; NADIA averages the completed matrices.
missForest Non-parametric random-forest imputation using relationships among variables.
Impseq Sequential model-based imputation from rrcovNA.
Impseqrob Robust sequential imputation that reduces sensitivity to outlying profiles.
MLE Maximum-likelihood imputation under a multivariate normal model.
nbavg Replaces a gap with the mean of the nearest observed positions in the same protein profile.
Low-value, generally used for MNAR QRILC Quantile-regression imputation for left-censored data.
MinDet Uses a low, sample-specific quantile; the default is the first percentile.
MinProb Draws low values around a sample-specific low quantile.
PI Perseus-style sampling from a narrowed, down-shifted normal distribution.
min Replaces every gap with the global observed minimum.
halfmin Uses half the global minimum on the original scale, equivalent to one unit below the minimum on the log2 scale; this convention is used by DIA-NN.
zero Replaces every gap with zero on the log2 scale.
with Replaces every gap with a user-supplied constant.
Hybrid combo Classifies missing cells from condition-wise presence patterns, then applies separate MAR and MNAR methods. Defaults are Impseqrob and min.
softHybrid Blends estimates from MAR and MNAR methods continuously using protein missingness and mean intensity.
Detection model limpa Estimates a detection-probability curve and returns abundance estimates with uncertainty for downstream analysis with de_method = "limpa".
No imputation none Leaves missing values unchanged.

limpa and none require special interpretation. limpa is intended to carry its estimated uncertainty into a limpa differential abundance model, whereas none deliberately does not produce a complete matrix. They should not be treated as ordinary constant-fill methods.

How imputation methods are evaluated

When the true missing values are unknown, NADIA follows the classic computational evaluation framework used by NAguideR:

  1. retain proteins that contain no missing values;
  2. hide a known subset of their observed cells;
  3. impute the artificial gaps with every candidate method; and
  4. compare the estimates with the values that were hidden.

NADIA provides two ways to decide how many values are hidden and where the artificial gaps are placed:

  • With pattern = "random", the user sets the proportion explicitly through na_prop. For example, na_prop = 0.20 hides 20% of the values in the complete-case matrix. The selected cells do not depend on intensity and are treated as MAR for the benchmark. This option is useful for a controlled experiment or for comparing performance at several predefined missingness levels.
  • With pattern = "from_data", na_prop is ignored. NADIA estimates the proportion of proteins affected by missing values and the distribution of gaps across samples from the original assay, then reproduces those features when masking the complete-case matrix. The resulting percentage of artificial gaps is therefore approximately the percentage observed in the experiment; small differences can occur because cell counts must be rounded.

The second option follows the data-derived principle used in the NAguideR evaluation, in which missing values were generated in a complete matrix at a proportion similar to that of the original dataset. For selecting an imputation method for one particular experiment, pattern = "from_data" is generally the more representative starting point because it preserves the experiment’s overall missingness burden and its distribution across samples. In nadia_dia, for example, the original assay contains approximately 8.0% missing cells and pattern = "from_data" generates approximately 8.1% artificial gaps, whereas pattern = "random", na_prop = 0.20 generates 20%.

Both options hide values that were originally observed, so the true values are known and imputation error can be calculated. Consequently, neither option directly reproduces MNAR missingness caused by low abundance or a detection limit. pattern = "from_data" is more representative of the observed NA frequency and sample distribution, but it is not a direct benchmark of MNAR imputation.

NADIA calculates four metrics:

Metric Interpretation Preferred direction
NRMSE Normalised root mean squared error over the artificially masked cells. Lower
SOR Sum of per-protein error ranks. A method receives the worst rank for a protein if it fails to impute its artificial gaps. Lower
PSS Procrustes sum of squares comparing the PCA structures of the true and imputed matrices. Lower
ACC_OI Pearson correlation between the true and imputed values pooled over the artificially masked cells. Higher

Because the four metrics have different scales, NADIA does not average their raw values. Instead, it orders the methods from best to worst for each metric and assigns rank 1 to the best method, rank 2 to the second, and so on. Rank_Mean is the mean of these four ranks, with each metric contributing equally. The method with the lowest Rank_Mean therefore has the best overall performance for the simulated gaps.

Occasionally, a metric cannot be calculated for a particular method. In that case, NADIA calculates Rank_Mean from the available ranks without assigning the best or worst rank to the missing metric. Therefore, before selecting a method solely on the basis of Rank_Mean, check whether any of its metric values are NA. A favourable mean rank based on fewer metrics may not be directly comparable with one calculated from the complete set of metrics.

ACC_OI compares the true and imputed values only at the cells that NADIA hid for the simulation. This prevents the many unchanged cells from making the correlation appear artificially high. Pearson correlation requires variation in both sets of values. A constant method such as min assigns the same value to every hidden cell, so its imputed values have no variation and ACC_OI cannot be calculated. In this situation, ACC_OI = NA means not evaluable; it does not indicate either good or poor imputation.

Comparing imputation methods with NADIA

Imputation must be evaluated on a normalised assay. The code below uses the normalisation selected above and compares representative structure-based, low-value and hybrid strategies. It uses a controlled 20% random mask for this illustration. For an experiment-specific evaluation, use pattern = "from_data" and omit na_prop from the call.

se_imp <- im_prepare_se(
    nadia_dia,
    norm_method = norm_winner,
    verbose = FALSE)

im <- imputation_metrics(
    se_imp,
    assay_name = norm_winner,
    methods = c("Impseqrob", "knn", "MinDet", "QRILC", "min"),
    combo_methods = list(
        Impseqrob_min = list(
            mar_method = "Impseqrob",
            mnar_method = "min")),
    na_prop = 0.20,
    pattern = "random",
    output_dir = NULL,
    verbose = FALSE)

im$metrics_table[, c(
    "Method", "NRMSE", "SOR", "PSS", "ACC_OI", "Rank_Mean")]
#>          Method     NRMSE    SOR         PSS     ACC_OI Rank_Mean
#> 1     Impseqrob 0.1258250 2668.0 0.008732265 0.99206183  1.250000
#> 2           knn 0.1290966 2802.0 0.004316998 0.99165565  2.000000
#> 3 Impseqrob_min 0.3999569 2701.5 0.009098402 0.92411201  2.750000
#> 4        MinDet 2.0437765 5463.0 0.195061890 0.07321024  4.000000
#> 5         QRILC 2.5574417 6685.0 0.205056432 0.02033958  5.250000
#> 6           min 3.2130875 8009.5 0.199977570         NA  5.666667

Impseqrob has the lowest Rank_Mean in this example. It produces the smallest NRMSE, the best SOR and the highest correlation with the hidden values. knn preserves the multivariate structure particularly well, as indicated by its low PSS, but ranks second overall. The low-value MNAR methods perform poorly because the simulation hides values at random rather than preferentially hiding low intensities. That result should not be used to conclude that MNAR imputation is unnecessary in the original matrix.

The metrics plot shows the values on their original scales:

im$metrics

Imputation quality metrics for the candidate methods

The heatmap shows one imputation method per row and one evaluation metric per column. The number printed in each cell is the method’s rank for that metric: rank 1 is the best result. Lower ranks are shown in blue and higher ranks in red; a grey cell marked NA indicates that the metric could not be calculated. The final column, Rank_Mean, is the average of the available metric ranks. Methods are ordered by this value, so the method in the first row has the lowest Rank_Mean and the best overall result for the simulated gaps.

im$ranking

Heatmap of imputation method ranks

The winner can again be extracted from the first row:

imp_winner <- im$metrics_table$Method[1]
imp_winner
#> [1] "Impseqrob"

Applying the selected combination

The selected normalisation and imputation methods can now be passed to the main workflow. For this example the data-driven combination is cycloess followed by Impseqrob.

res_selected <- process_proteomics(
    nadia_dia,
    norm_method = norm_winner,
    imp_method = imp_winner,
    verbose = FALSE)

table(res_selected$DEPs_results$Change)
#> 
#>        Up      Down No Change 
#>       964       759      4268

The normalised matrix and the imputed matrix are retained as separate assays, so the processing history can be inspected without overwriting earlier stages.

assayNames(res_selected$se_proc)
#> [1] "raw"       "log2"      "cycloess"  "Impseqrob"

Effect on differential abundance results

A method-selection metric is a guide, not a guarantee that every downstream result is correct. A useful final check is to repeat the analysis with plausible alternative pipelines and compare their consequences.

variants <- list(
    "cycloess + Impseqrob" = list(
        norm_method = "cycloess", imp_method = "Impseqrob"),
    "cycloess + combo" = list(
        norm_method = "cycloess", imp_method = "combo"),
    "cycloess + knn" = list(
        norm_method = "cycloess", imp_method = "knn"),
    "quantile + combo" = list(
        norm_method = "quantile", imp_method = "combo"),
    "log2 + min" = list(
        norm_method = "log2", imp_method = "min"))

# Some imputation methods print progress on stdout; capture.output keeps it out.
invisible(capture.output(
    counts <- do.call(rbind, lapply(names(variants), function(label) {
        fit <- do.call(
            process_proteomics,
            c(list(nadia_dia, verbose = FALSE), variants[[label]]))
        tab <- table(factor(
            fit$DEPs_results$Change,
            levels = c("Up", "Down", "No Change")))
        data.frame(
            Pipeline = label,
            Up = tab[["Up"]],
            Down = tab[["Down"]],
            No_Change = tab[["No Change"]])
    }))))

counts
#>               Pipeline  Up Down No_Change
#> 1 cycloess + Impseqrob 964  759      4268
#> 2     cycloess + combo 985  937      4069
#> 3       cycloess + knn 872  179      4940
#> 4     quantile + combo 902 2825      2264
#> 5           log2 + min 761  778      4452

The input matrix, differential abundance method and decision thresholds are the same in every run, but the numbers called Up and Down change substantially. These are protein–comparison results: a protein tested in several contrasts can contribute more than one row. The comparison does not establish which pipeline is biologically correct. It shows that normalisation and imputation are consequential modelling choices and must be evaluated, reported and reproduced.

Practical decision guide

For a dataset without external ground truth, the following sequence provides a defensible starting point:

  1. inspect sample distributions and the missing-value pattern;
  2. compare several plausible normalisation methods using both the final rank and the diagnostic plots;
  3. examine whether missingness appears predominantly scattered or related to low abundance and condition-wise dropout;
  4. compare imputation methods, remembering that the artificial-masking benchmark assesses reconstruction of MAR values rather than true MNAR missingness caused by low abundance or a detection limit;
  5. apply the selected combination and repeat the differential abundance analysis with reasonable alternatives as a sensitivity analysis; and
  6. report the method names, relevant parameters, package version and session information.

When a spike-in or another known reference is available, it should take priority over these proxy criteria because it permits evaluation of the expected biological changes directly.

References and software

Session information 

sessionInfo()
#> R version 4.6.1 (2026-06-24)
#> Platform: x86_64-pc-linux-gnu
#> Running under: Ubuntu 24.04.4 LTS
#> 
#> Matrix products: default
#> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
#> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
#> 
#> locale:
#>  [1] LC_CTYPE=C.UTF-8       LC_NUMERIC=C           LC_TIME=C.UTF-8        LC_COLLATE=C.UTF-8    
#>  [5] LC_MONETARY=C.UTF-8    LC_MESSAGES=C.UTF-8    LC_PAPER=C.UTF-8       LC_NAME=C             
#>  [9] LC_ADDRESS=C           LC_TELEPHONE=C         LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C   
#> 
#> time zone: UTC
#> tzcode source: system (glibc)
#> 
#> attached base packages:
#> [1] stats4    stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#>  [1] SummarizedExperiment_1.42.0 Biobase_2.72.0              GenomicRanges_1.64.0       
#>  [4] Seqinfo_1.2.0               IRanges_2.46.0              S4Vectors_0.50.2           
#>  [7] BiocGenerics_0.58.1         generics_0.1.4              MatrixGenerics_1.24.0      
#> [10] matrixStats_1.5.0           NADIA_0.99.1                BiocStyle_2.40.0           
#> 
#> loaded via a namespace (and not attached):
#>  [1] sandwich_3.1-3      permute_0.9-10      rlang_1.3.0         magrittr_2.0.5     
#>  [5] otel_0.2.0          compiler_4.6.1      mgcv_1.9-4          systemfonts_1.3.2  
#>  [9] vctrs_0.7.3         stringr_1.6.0       pkgconfig_2.0.3     fastmap_1.2.0      
#> [13] backports_1.5.1     XVector_0.52.0      labeling_0.4.3      rmarkdown_2.32     
#> [17] ragg_1.5.2          highcharter_0.9.5   purrr_1.2.2         xfun_0.60          
#> [21] cachem_1.1.0        jsonlite_2.0.0      gmm_1.9-1           DelayedArray_0.38.2
#> [25] broom_1.0.13        parallel_4.6.1      cluster_2.1.8.2     R6_2.6.1           
#> [29] bslib_0.12.0        stringi_1.8.9       RColorBrewer_1.1-3  limma_3.68.5       
#> [33] rlist_0.4.6.2       rrcov_1.7-7         lubridate_1.9.5     jquerylib_0.1.4    
#> [37] Rcpp_1.1.2          bookdown_0.48       assertthat_0.2.1    knitr_1.51         
#> [41] zoo_1.9-0           Matrix_1.7-5        splines_4.6.1       timechange_0.4.0   
#> [45] tidyselect_1.2.1    abind_1.4-8         yaml_2.3.12         vegan_2.7-6        
#> [49] curl_8.0.0          lattice_0.22-9      tibble_3.3.1        quantmod_0.4.29    
#> [53] withr_3.0.3         S7_0.2.2            evaluate_1.0.5      tmvtnorm_1.7       
#> [57] desc_1.4.3          rrcovNA_0.5-3       xts_0.14.2          norm_1.0-11.1      
#> [61] pillar_1.11.1       BiocManager_1.30.27 pcaPP_2.0-5         TTR_0.24.4         
#> [65] ggplot2_4.0.3       scales_1.4.0        glue_1.8.1          tools_4.6.1        
#> [69] robustbase_0.99-7   data.table_1.18.6.1 reactable_0.4.5     imputeLCMD_2.1     
#> [73] fs_2.1.0            mvtnorm_1.4-2       grid_4.6.1          impute_1.86.0      
#> [77] tidyr_1.3.2         nlme_3.1-169        cli_3.6.6           textshaping_1.0.5  
#> [81] S4Arrays_1.12.0     viridisLite_0.4.3   dplyr_1.2.1         pcaMethods_2.4.0   
#> [85] gtable_0.3.6        DEoptimR_1.2-1      sass_0.4.10         digest_0.6.39      
#> [89] SparseArray_1.12.2  htmlwidgets_1.6.4   farver_2.1.2        htmltools_0.5.9    
#> [93] pkgdown_2.2.1       lifecycle_1.0.5     statmod_1.5.2       MASS_7.3-65