Appendix A: Equivalence Validation - TSENAT vs SplicingFactory • TSENAT

Introduction

This appendix validates that TSENAT’s transcript diversity measures (Shannon entropy, Simpson index) are mathematically equivalent to SplicingFactory’s implementations. We benchmark both packages on TCGA BRCA RNA-seq data to demonstrate:

Shannon equivalence: TSENAT Tsallis q=1 ~ SplicingFactory Shannon
Simpson equivalence: TSENAT Tsallis q=2 ~ SplicingFactory Simpson
Identical statistical results: Same gene rankings, p-values, fold changes

Who needs this appendix? Anyone evaluating TSENAT for transcript diversity analysis or validating cross-package comparisons.

Document Structure

Background: Conceptual overview of transcript diversity and the Tsallis framework
Benchmarking (main section): Hands-on comparison using both packages
Results: Side-by-side analysis tables (Shannon and Simpson)
Analysis: Technical explanation of equivalence, normalization differences, and when to use each method
Conclusion

Background: Transcript Diversity Concepts

Transcript diversity measures how variable isoform expression is within genes. Two standard approaches:

Shannon entropy: Overall diversity (number and evenness of isoforms)
Simpson index: Dominance (whether one isoform monopolizes expression)

TSENAT’s approach: Generalized Tsallis entropy framework where q=1 gives Shannon and q=2 gives Simpson. This allows sensitivity analysis across the q continuum while maintaining equivalence with established methods.

Key question: Does TSENAT’s mathematical generalization produce identical statistical results?

Dataset and Methods

We use TCGA BRCA RNA-seq data as our validation dataset, which is included in the SplicingFactory package:

Public, reproducible resource.
Multiple transcript isoforms per gene.
Real biological signal.
Variable sequencing depth and isoform abundance.

Analysis pipeline:

Load transcript-level read counts.
Filter genes.
Compute entropy values.
Statistical testing (Wilcoxon rank-sum) for group differences.
Compare results (concordance, effect sizes, rankings).

Benchmarking

Importing example data

We begin by loading the required libraries and the TCGA BRCA dataset that will serve as our validation benchmark.

suppressPackageStartupMessages({
    library("SplicingFactory")
    library(TSENAT)
    library("SummarizedExperiment")
})
set.seed(12345)

# Load dataset
data(tcga_brca_luma_dataset)

# Extract gene names
genes <- tcga_brca_luma_dataset[, 1]

# Extract read count data without gene names
readcounts <- tcga_brca_luma_dataset[, -1]

Data filtering and preprocessing

We filter genes to retain only those with sufficient coverage across samples, ensuring robust diversity estimates.

tokeep <- rowSums(readcounts > 5) > 5
readcounts <- readcounts[tokeep, ]
genes      <- genes[tokeep]

# Create SummarizedExperiment object for cleaner data handling
# Important: Set gene names as rownames for compatibility with SplicingFactory
# TSENATAnalysis requires 'gene_id' in rowData
se_data <- SummarizedExperiment(
  assays = list(counts = as.matrix(readcounts)),
  rowData = DataFrame(gene_id = genes)
)
rownames(se_data) <- genes

Transcript diversity calculation

To calculate transcript diversity measures (Shannon entropy, Simpson index, and Tsallis entropy at q=1 and q=2), use:

# Create sample group classification based on sample names
# (Normal samples end with _N, Tumor samples don't)
sample_group <- ifelse(grepl("_N$", colnames(se_data)), "Normal", "Tumor")

# Add group metadata to se_data for all downstream analyses
colData(se_data)$group <- sample_group

# SplicingFactory: Calculate diversity for both Shannon (naive) and Simpson methods
sf_methods <- c("naive", "simpson")
splicing_results <- setNames(vector("list", length(sf_methods)), sf_methods)

for (method in sf_methods) {
  splicing_results[[method]] <- SplicingFactory::calculate_diversity(
    x = se_data,
    method = method,
    norm = TRUE,
    verbose = FALSE
  )
  # Add sample_type and group metadata to SplicingFactory objects
  # (sample_type matches SplicingFactory naming; group matches TSENAT naming for consistency)
  colData(splicing_results[[method]])$sample_type <- sample_group
  colData(splicing_results[[method]])$group <- sample_group
}

# TSENAT: Single config for q-value sweep
# (Configuration is identical for all q-values)
config <- TSENAT_config(condition_col = "group")

# TSENAT: Create analysis objects and compute diversity for q=1 and q=2
q_values <- c(1, 2)
tsenat_results <- setNames(vector("list", length(q_values)), paste0("q", q_values))

for (q in q_values) {
  # Create analysis object (following Bioconductor pattern: config first, then constructor)
  tsenat_results[[paste0("q", q)]] <- TSENATAnalysis(se_data, config = config)
  
  # Compute diversity with dynamic output filename using sprintf
  # Use explicit namespace to avoid SplicingFactory::calculate_diversity masking
  tsenat_results[[paste0("q", q)]] <- TSENAT::calculate_diversity(
    analysis = tsenat_results[[paste0("q", q)]],
    q = q,
    norm = TRUE,
    verbose = FALSE
  )
}

# Extract for downstream use (aliases maintain backward compatibility with existing code)
shannon_entropy <- splicing_results$naive
simpson_index <- splicing_results$simpson
tsenat_analysis_q1 <- tsenat_results$q1
tsenat_analysis_q2 <- tsenat_results$q2

Both packages return a SummarizedExperiment object, that you can investigate further with the assay function.

Differential analysis

We apply statistical testing to identify genes with significantly different transcript diversity between Normal and Tumor samples using both SplicingFactory and TSENAT frameworks.

# SplicingFactory: Differential analysis for Shannon entropy
entropy_significance <- SplicingFactory::calculate_difference(
  x = shannon_entropy,
  samples = "sample_type",
  control = "Normal",
  method = "mean",
  test = "wilcoxon",
  verbose = FALSE
)

# SplicingFactory: Differential analysis for Simpson index
simpson_significance <- SplicingFactory::calculate_difference(
  x = simpson_index,
  samples = "sample_type",
  control = "Normal",
  method = "mean",
  test = "wilcoxon",
  verbose = FALSE
)


# TSENAT: Differential analysis for Tsallis q=1 (Shannon equivalent)
# Extract the diversity results for q=1 using the results() accessor
# Returns SummarizedExperiment with diversity values (genes x samples)
div_result_q1 <- results(tsenat_analysis_q1, type = "diversity", q = 1.0, format = "se")

# Apply SplicingFactory's calculate_difference to TSENAT q=1 diversity
# Use "group" column already in colData (from se_data)
tsenat_shannon_diff <- SplicingFactory::calculate_difference(
  x = div_result_q1,
  samples = "group",
  control = "Normal",
  method = "mean",
  test = "wilcoxon",
  verbose = FALSE
)


# TSENAT: Differential analysis for Tsallis q=2 (Simpson equivalent)
# Extract the diversity results for q=2 using the results() accessor
# Returns SummarizedExperiment with diversity values (genes x samples)
div_result_q2 <- results(tsenat_analysis_q2, type = "diversity", q = 2.0, format = "se")

# Apply SplicingFactory's calculate_difference to TSENAT q=2 diversity
# Use "group" column already in colData (from se_data)
tsenat_simpson_diff <- SplicingFactory::calculate_difference(
  x = div_result_q2,
  samples = "group",
  control = "Normal",
  method = "mean",
  test = "wilcoxon",
  verbose = FALSE
)

For the benchmark comparison with TSENAT, we will compute Tsallis entropy at q=1 (equivalent to Shannon) and q=2 (equivalent to Simpson). This allows direct comparison with the SplicingFactory results shown above.

Benchmark Results Summary

Shannon Entropy (SplicingFactory Laplace vs TSENAT Tsallis q=1)

**Supplementary Table 5 | Shannon Entropy Analysis Summary Statistics.** Differential analysis comparing Normal (N=8) vs Tumor (N=8) samples using SplicingFactory’s `calculate_difference()` method.
Method	Genes Tested	Significant (padj<0.05)	Mean log2FC	SD log2FC
SplicingFactory (Shannon/naive)	214	26	-0.0009354	1.582967
TSENAT (Tsallis q=1)	213	26	0.0003618	1.586675

Interpretation: Both methods similarly identify significantly different transcript diversity between Normal and Tumor samples. The comparable number of significant genes (padj<0.05) and mean log2FC values demonstrate equivalent statistical power and effect size detection.

Top 10 Significant Genes - SplicingFactory (Laplace)

**Supplementary Table 6a | SplicingFactory method - Top 10 genes identified by Shannon entropy (naive mode).** Ranked by adjusted P-value.
Gene	Normal Mean	Tumor Mean	Mean Diff	log2FC	p-value	adj p-value
C1orf213	0.8122	0.1400	-0.6722	-2.5365	3.97e-07	8.50e-05
HAPLN3	0.7030	0.4299	-0.2731	-0.7096	4.17e-05	4.46e-03
COL1A2	0.0000	0.1643	0.1643	Inf	6.68e-05	4.77e-03
F10	0.1311	0.3720	0.2409	1.5050	1.37e-04	7.31e-03
DUSP14	0.2989	0.1302	-0.1688	-1.1995	1.89e-04	8.10e-03
MBD2	0.0027	0.0371	0.0344	3.7876	2.39e-04	8.54e-03
C1orf86	0.1479	0.0029	-0.1450	-5.6777	4.59e-04	9.85e-03
GFPT1	0.0231	0.0025	-0.0207	-3.2253	3.80e-04	9.85e-03
HNRNPR	0.5162	0.5907	0.0746	0.1947	4.60e-04	9.85e-03
OSR1	0.0497	0.2891	0.2394	2.5411	4.10e-04	9.85e-03

Top 10 Significant Genes - TSENAT (Tsallis q=1)

**Supplementary Table 6b | TSENAT method - Top 10 genes identified by Tsallis entropy at q=1 (Shannon equivalence).** Results obtained using SplicingFactory’s `calculate_difference()` method applied to TSENAT Tsallis q=1 diversity values. Ranked by adjusted P-value.
Gene	Normal Mean	Tumor Mean	Mean Diff	log2FC	p-value	adj p-value
C1orf213	0.8122	0.1400	-0.6722	-2.5365	3.97e-07	8.46e-05
HAPLN3	0.7030	0.4299	-0.2731	-0.7096	4.17e-05	4.44e-03
COL1A2	0.0000	0.1643	0.1643	Inf	6.68e-05	4.74e-03
F10	0.1311	0.3720	0.2409	1.5050	1.37e-04	7.27e-03
DUSP14	0.2989	0.1302	-0.1688	-1.1995	1.89e-04	8.06e-03
MBD2	0.0027	0.0371	0.0344	3.7876	2.39e-04	8.50e-03
C1orf86	0.1479	0.0029	-0.1450	-5.6777	4.59e-04	9.80e-03
HNRNPR	0.5162	0.5907	0.0746	0.1947	4.60e-04	9.80e-03
OSR1	0.0497	0.2891	0.2394	2.5411	4.10e-04	9.80e-03
GFPT1	0.0231	0.0025	-0.0207	-3.2253	3.80e-04	9.80e-03

Simpson Index (SplicingFactory vs TSENAT Tsallis q=2)

**Supplementary Table 8 | Simpson Index Analysis Summary Statistics.** Differential analysis using Simpson diversity metric (Gini-Simpson index) comparing Normal (N=8) vs Tumor (N=8) samples using SplicingFactory’s `calculate_difference()` method.
Method	Genes Tested	Significant (padj<0.05)	Mean log2FC	SD log2FC
SplicingFactory (Simpson)	214	26	-0.0012687	1.812772
TSENAT (Tsallis q=2)	213	26	-0.0000521	1.817060

Interpretation: Simpson index results parallel Shannon entropy findings, with both methods identifying similar numbers of significantly different isoform dominance patterns. The agreement across both diversity measures provides strong evidence for TSENAT’s mathematical and statistical equivalence to SplicingFactory.

Top 10 Significant Genes - SplicingFactory (Simpson)

**Supplementary Table 9a | SplicingFactory method - Top 10 genes by Simpson index (Gini-Simpson, emphasizes dominant isoforms).** Ranked by adjusted P-value.
Gene	Normal Mean	Tumor Mean	Mean Diff	log2FC	p-value	adj p-value
C1orf213	0.3852	0.0515	-0.3337	-2.9037	3.97e-07	8.50e-05
HAPLN3	0.4823	0.2558	-0.2266	-0.9152	5.17e-06	5.53e-04
COL1A2	0.0000	0.0626	0.0626	Inf	6.68e-05	4.77e-03
F10	0.0697	0.2494	0.1797	1.8400	1.10e-04	5.86e-03
DUSP14	0.1686	0.0617	-0.1069	-1.4499	2.11e-04	7.32e-03
HNRNPR	0.5376	0.6134	0.0758	0.1903	1.79e-04	7.32e-03
MBD2	0.0005	0.0107	0.0102	4.5028	2.39e-04	7.32e-03
CXorf40A	0.7455	0.7939	0.0484	0.0907	2.75e-04	7.34e-03
GFPT1	0.0051	0.0004	-0.0047	-3.5271	3.80e-04	8.78e-03
OSR1	0.0124	0.1156	0.1032	3.2168	4.10e-04	8.78e-03

Top 10 Significant Genes - TSENAT (Tsallis q=2)

**Supplementary Table 9b | TSENAT method - Top 10 genes by Tsallis entropy at q=2 (Simpson equivalence).** Results obtained using SplicingFactory’s `calculate_difference()` method applied to TSENAT Tsallis q=2 diversity values. Ranked by adjusted P-value.
Gene	Normal Mean	Tumor Mean	Mean Diff	log2FC	p-value	adj p-value
C1orf213	0.7703	0.1029	-0.6674	-2.9037	3.97e-07	8.46e-05
HAPLN3	0.7235	0.3836	-0.3398	-0.9152	5.17e-06	5.50e-04
COL1A2	0.0000	0.1251	0.1251	Inf	6.68e-05	4.74e-03
F10	0.1045	0.3741	0.2696	1.8400	1.10e-04	5.83e-03
HNRNPR	0.6144	0.7010	0.0866	0.1903	1.79e-04	7.28e-03
DUSP14	0.2529	0.0926	-0.1603	-1.4499	2.11e-04	7.28e-03
MBD2	0.0009	0.0213	0.0204	4.5028	2.39e-04	7.28e-03
CXorf40A	0.8387	0.8931	0.0544	0.0907	2.75e-04	7.31e-03
OSR1	0.0249	0.2312	0.2063	3.2168	4.10e-04	8.74e-03
GFPT1	0.0103	0.0009	-0.0094	-3.5271	3.80e-04	8.74e-03

Analysis: Simpson Index and Tsallis Entropy at q=2

Both methods share the same mathematical formula:

$Simpson = Tsallis_{q=2} = 1 - \sum_{i=1}^n p_i^2$

where $p_i = \frac{x_i}{\sum_i x_i}$ (transcript proportions).

Howerver, SplicingFactory returns raw values, while TSENAT normalizes by the theoretical maximum $(1-1/n)$ :

	SplicingFactory	TSENAT
Formula	$1 - \sum p_i^2$	$\frac{1 - \sum p_i^2}{1 - 1/n}$
Range	0 to ~1	0 to 1
Scale difference	Raw	~2x for 2-isoform genes

This explains observed differences like C1orf213 Normal: 0.3852 (SplicingFactory) vs 0.7703 (TSENAT). Since C1orf213 has 2 expressed isoforms, the normalization factor is $1 - 1/2 = 0.5$ . The calculation: $0.3852 / 0.5 = 0.7704$ confirms this scaling factor. Despite raw value differences, statistical analysis yields identical results:

log2 fold changes: Normalization factor cancels in ratios: $\log_2\left(\frac{S_{norm,t}}{S_{norm,n}}\right) = \log_2\left(\frac{S_{raw,t}/K}{S_{raw,n}/K}\right) = \log_2\left(\frac{S_{raw,t}}{S_{raw,n}}\right)$
P-values and rankings: Wilcoxon rank-sum tests depend only on gene ranks, not absolute values, so both methods identify the same significant genes
Statistical validity: Both approaches are correct-SplicingFactory favors unbounded diversity, TSENAT favors interpretable [0,1] scaling

Conclusion: Equivalence Validation Summary

This appendix has demonstrated that TSENAT’s implementation of transcript diversity measures is mathematically and statistically equivalent to SplicingFactory’s established methods for both Shannon entropy and Simpson index calculations.

Session Information

sessionInfo()
#> R version 4.5.2 (2025-10-31)
#> Platform: x86_64-conda-linux-gnu
#> Running under: Ubuntu 22.04.5 LTS
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#> time zone: Europe/Berlin
#> tzcode source: system (glibc)
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#> [1] stats4    stats     graphics  grDevices utils     datasets  methods  
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#> other attached packages:
#>  [1] knitr_1.51                  dplyr_1.2.0                
#>  [3] SummarizedExperiment_1.40.0 Biobase_2.70.0             
#>  [5] GenomicRanges_1.62.1        Seqinfo_1.0.0              
#>  [7] IRanges_2.44.0              S4Vectors_0.48.0           
#>  [9] BiocGenerics_0.56.0         generics_0.1.4             
#> [11] MatrixGenerics_1.22.0       matrixStats_1.5.0          
#> [13] TSENAT_0.99.0               SplicingFactory_1.18.0     
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#> [10] pkgconfig_2.0.3     pheatmap_1.0.13     Matrix_1.7-4       
#> [13] RColorBrewer_1.1-3  S7_0.2.1            desc_1.4.3         
#> [16] lifecycle_1.0.5     compiler_4.5.2      farver_2.1.2       
#> [19] stringr_1.6.0       textshaping_1.0.5   htmltools_0.5.9    
#> [22] sass_0.4.10         yaml_2.3.12         pkgdown_2.2.0      
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#> [49] evaluate_1.0.5      viridisLite_0.4.3   rlang_1.1.7        
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#> [55] svglite_2.2.2       rstudioapi_0.18.0   jsonlite_2.0.0     
#> [58] R6_2.6.1            systemfonts_1.3.2   fs_1.6.7

Appendix A: Equivalence Validation - TSENAT vs SplicingFactory

Cristobal Gallardo gallardoalba@pm.me

2026-04-15