library(goSorensen)

PRELIMINARIES.

Theoretical Framework of Reference.

Given two gene lists, $L_1$ and $L_2$, (the data) and a given set of $n$ Gene Ontology (GO) terms (the frame of reference for biological significance in these lists), goSorensen makes the required computations to answer the following question: Is the dissimilarity between the biological information in both lists negligible? In other words, are both lists functionally equivalent?

We employ the following metric derived from the Sorensen-Dice index to quantify this dissimilarity:

\[\begin{equation*} \hat{d_S} = 1 - \dfrac{2n_{11}}{2n_{11} + n_{10} + n_{01}} \\ \hat{d_S} = 1 - \dfrac{\frac{2n_{11}}{n}}{\frac{2n_{11}}{n} + \frac{n_{10}}{n} + \frac{n_{01}}{n}} \\ \hat{d_S} = 1 - \dfrac{2\widehat{p}_{11}}{2\widehat{p}_{11} + \widehat{p}_{10} + \widehat{p}_{01}} \end{equation*}\]

where:

$n_{11}$ corresponds to the number (i.e., the absolute frequency) of GO terms enriched in both gene lists. Similarly, $\hat{p}_{11}$ corresponds to the relative frequency of joint enrichment.
$n_{10}$ represents the number of GO terms enriched in list $L_1$ but not in list $L_2$, and $n_{01}$ the number of GO terms non enriched in list $L_1$ but enriched in $L_2$. In other words,$n_{10} + n_{01}$ is the absolute frequency of marginal enrichment, and $\hat{p}_{10} + \hat{p}_{01}$ the proportion of marginal enrichment.
$n_{00}$ corresponds to the absolute frequency (and $p_{00}$ to the relative frequency) of GO terms not enriched in either gene list.

The enrichment frequency can be represented in a $2 \times 2$ contingency table, as follows:

	enriched in $L_2$	non-enriched in $L_2$
enriched in $L_1$	$n_{11}$	$n_{10}$	$n_{1.}$
non-enriched in $L_1$	$n_{01}$	$n_{00}$	$n_{0.}$
	$n_{.1}$	$n_{.0}$	$n$

In Flores et al. (2022) it is shown that $d_S$ asymptotically follows a normal distribution. In cases of low joint enrichment, a sampling distribution derived from the bootstrap approach demonstrates a better fit and provides more suitable results.

Consider the following equivalence hypothesis test:

\[\begin{equation*} H_0:d_S \ge d_0 \\ H_1: d_S < d_0 \end{equation*}\]

where $d_S$ represents the “true” Sorensen dissimilarity. If this theoretical measure is equivalent to zero, it implies that the compared lists $L_1$ and $L_2$ share an important proportion of enriched GO terms, which can be interpreted as biological similarity. Equivalence is understood as an equality, except for negligible deviations, which is defined by the irrelevance limit $d_0$

$d_0$ is a value that should be fixed in advance, greater than 0 and less than 1. In Flores et al. (2022) it is shown that a not-so-arbitrary irrelevance limit is $d_0 = 0.4444$, or more restrictive $d_0=0.2857$

For the moment, the reference set of GO terms can be only all those GO terms in a given level of one GO ontology, either Biological Process (BP), Cellular Component (CC) or Molecular Function (MF).

For more details, see the reference paper.

goSorensen Installation.

goSorensen package must be installed with a working R version (>=4.4.0). Installation could take a few minutes on a regular desktop or laptop. Package can be installed from Bioconductor, then it needs to be loaded using library(goSorensen):

if (!requireNamespace("goSorensen", quietly = TRUE)) {
    BiocManager::install("goSorensen")
}
library(goSorensen)

Data Used in this Vignette.

The dataset used in this vignette, allOncoGeneLists, is based on the gene lists compiled at http://www.bushmanlab.org/links/genelists, a comprehensive set of gene lists related to cancer. The package goSorensen loads this dataset using data(allOncoGeneLists):

data("allOncoGeneLists")

allOncoGeneLists is an object of class list, containing seven character vectors with the ENTREZ gene identifiers of a gene list related to cancer.

sapply(allOncoGeneLists, length)

        atlas      cangenes           cis miscellaneous        sanger 
          991           189           613           187           450 
   Vogelstein       waldman 
          419           426

# First 15 gene identifiers of gene lists atlas and sanger:
allOncoGeneLists[["atlas"]][1:15]

 [1] "11186" "27086" "239"   "7764"  "5934"  "246"   "57124" "79663" "23136"
[10] "6281"  "1307"  "1978"  "3732"  "3636"  "7159"

allOncoGeneLists[["sanger"]][1:15]

 [1] "25"    "27"    "2181"  "57082" "10962" "51517" "27125" "10142" "207"  
[10] "208"   "217"   "238"   "57714" "324"   "23365"

Before Using goSorensen.

Before using goSorensen, the users must have adequate knowledge of the species they intend to focus their analysis on. The genomic annotation packages available in Bioconductor provide all the essential information about many species.

For the specific case of this vignette and the help pages of the package, given that the analysis will be done in the human species, the org.Hs.eg.db package must be previously installed and activated as follows:

if (!requireNamespace("org.Hs.eg.db", quietly = TRUE)) {
    BiocManager::install("org.Hs.eg.db")
}

library(org.Hs.eg.db)

Actually, the org.Hs.eg.db package is automatically installed as a dependency on goSorensen, making its installation unnecessary. However, for any other species, the user must install the corresponding genome annotation for the species to analyse, as indicated in the above code.

In addition, it is necessary to have a vector containing the IDs of the universe of genes associated with the species under study. The genomic annotation package provides an easy way to obtain this universe. The ENTREZ identifiers of the gene universe for humans, necessary for this vignette, are obtained as follows:

humanEntrezIDs <- keys(org.Hs.eg.db, keytype = "ENTREZID")

In this same way, the identifiers of the gene universe can be obtained for any other species.

Other species available in Bioconductor may include:

org.Hs.eg.db: Genome wide annotation for Humans.
org.At.tair.db: Genome wide annotation for Arabidopsis
org.Ag.eg.db: Genome wide annotation for Anopheles
org.Bt.eg.db: Genome wide annotation for Bovine
org.Ce.eg.db: Genome wide annotation for Worm
org.Cf.eg.db: Genome wide annotation for Canine
org.Dm.eg.db: Genome wide annotation for Fly
org.EcSakai.eg.db: Genome wide annotation for E coli strain Sakai
org.EcK12.eg.db: Genome wide annotation for E coli strain K12
org.Dr.eg.db: Genome wide annotation for Zebrafish
org.Gg.eg.db: Genome wide annotation for Chicken
org.Mm.eg.db: Genome wide annotation for Mouse
org.Mmu.eg.db: Genome wide annotation for Rhesus

Due to the extensive research conducted on the human species and the examples documented in goSorensen for this species, the installation of the goSorensen package automatically includes the annotation package org.Hs.eg.db as a dependency.

If you are working with other species, you must install the appropriate package to use the genomic annotation for those species.”

MATRIX OF GO TERMS ENRICHMENT.

For One List.

The first step¹ is to determine whether the GO terms for a specific ontology and GO level are enriched or not enriched across the different lists to be compared. The enrichedIn function assigns TRUE when a GO term is enriched in a gene list and FALSE when it is not.

For example, for the list atlas, which is part of allOncoGeneLists, the enrichment of GO terms in the BP ontology at GO level 4 may be obtained as follows:

enrichedAtlas <- enrichedIn(allOncoGeneLists[["atlas"]],
           geneUniverse = humanEntrezIDs, 
           orgPackg = "org.Hs.eg.db",
           onto = "BP", GOLevel = 4)
enrichedAtlas

GO:0001649 GO:0030278 GO:0030279 GO:0030282 GO:0045778 GO:0048755 GO:0060688 
      TRUE       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE 
GO:0061138 GO:0002263 GO:0030168 GO:0050866 GO:0050867 GO:0072537 GO:0001780 
      TRUE       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE 
GO:0002260 GO:0001818 GO:0002367 GO:0002534 GO:0010573 GO:0032602 GO:0032609 
      TRUE       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE 
GO:0032612 GO:0032613 GO:0032615 GO:0032623 GO:0032635 GO:0071604 GO:0071706 
      TRUE       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE 
GO:0002562 GO:0016445 GO:0002433 GO:0002443 GO:0002697 GO:0002698 GO:0002699 
      TRUE       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE 
GO:0043299 GO:0002218 GO:0034101 GO:0002377 GO:0002700 GO:0002701 GO:0002702 
      TRUE       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE 
GO:0002200 GO:0048534 GO:0002685 GO:1903706 GO:0002695 GO:0050777 GO:0050858 
      TRUE       TRUE       TRUE       TRUE       TRUE       TRUE       TRUE 
GO:1903707 
      TRUE 
 [ reached 'max' / getOption("max.print") -- omitted 340 entries ]
attr(,"nTerms")
[1] 3381

The result is a vector containing only the GO terms enriched (TRUE) in the atlas list in the BP ontology at GO level 4.

The attribute nTerms indicates the total number of GO terms, both enriched (TRUE) and non-enriched (FALSE), by the list atlas in the BP ontology at GO level 4. To obtain this vector, the logical argument onlyEnriched (which is TRUE by default) must be set to FALSE, as follows:

fullEnrichedAtlas <- enrichedIn(allOncoGeneLists[["atlas"]],
           geneUniverse = humanEntrezIDs, 
           orgPackg = "org.Hs.eg.db",
           onto = "BP", GOLevel = 4, 
           onlyEnriched = FALSE)
fullEnrichedAtlas

GO:0036349 GO:0036350 GO:0060256 GO:0060257 GO:1900735 GO:0010590 GO:0030994 
     FALSE      FALSE      FALSE      FALSE      FALSE      FALSE      FALSE 
GO:0030995 GO:1904652 GO:1905391 GO:2001042 GO:2001043 GO:0001649 GO:0030278 
     FALSE      FALSE      FALSE      FALSE      FALSE       TRUE       TRUE 
GO:0030279 GO:0030282 GO:0036072 GO:0036075 GO:0036076 GO:0036077 GO:0043931 
      TRUE       TRUE      FALSE      FALSE      FALSE      FALSE      FALSE 
GO:0043932 GO:0045778 GO:0010223 GO:0048755 GO:0060688 GO:0061137 GO:0061138 
     FALSE       TRUE      FALSE       TRUE       TRUE      FALSE       TRUE 
GO:0080181 GO:0002263 GO:0002266 GO:0007343 GO:0007407 GO:0014719 GO:0030168 
     FALSE       TRUE      FALSE      FALSE      FALSE      FALSE       TRUE 
GO:0032980 GO:0042118 GO:0044566 GO:0045321 GO:0050865 GO:0050866 GO:0050867 
     FALSE      FALSE      FALSE      FALSE      FALSE       TRUE       TRUE 
GO:0061900 GO:0071892 GO:0072537 GO:0001780 GO:0002260 GO:0033023 GO:0035702 
     FALSE      FALSE       TRUE       TRUE       TRUE      FALSE      FALSE 
GO:0036145 
     FALSE 
 [ reached 'max' / getOption("max.print") -- omitted 3331 entries ]
attr(,"nTerms")
[1] 3381

The full vector (fullEnrichedAtlas) is much larger than the vector containing only the enriched GO terms (enrichedAtlas), which implies a higher memory usage.

# number of GO terms in enrichedAtlas
length(enrichedAtlas)

[1] 390

# number of GO terms in fullEnrichedAtlas
length(fullEnrichedAtlas)

[1] 3381

The length of fullEnrichedAtlas corresponds to the total number of GO terms in the BP ontology at GO level 4. In contrast, the length of enrichedAtlas represents only the number of GO terms that are enriched in the list atlas.

For Two or More Lists.

For the seven lists of allOncoGeneLists, the matrix containing the GO terms enriched in at least one of the lists to be compared is calculated as follows:

enrichedInBP4 <- enrichedIn(allOncoGeneLists, 
                               geneUniverse = humanEntrezIDs, 
                               orgPackg = "org.Hs.eg.db", 
                               onto = "BP", GOLevel = 4) 
enrichedInBP4

           atlas cangenes   cis miscellaneous sanger Vogelstein waldman
GO:0001649  TRUE    FALSE  TRUE          TRUE   TRUE       TRUE    TRUE
GO:0030278  TRUE    FALSE FALSE          TRUE  FALSE       TRUE    TRUE
GO:0030279 FALSE    FALSE FALSE         FALSE  FALSE       TRUE    TRUE
GO:0030282  TRUE    FALSE FALSE         FALSE  FALSE       TRUE    TRUE
GO:0036075  TRUE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:0045778 FALSE    FALSE FALSE          TRUE  FALSE      FALSE    TRUE
GO:0048755  TRUE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:0060688  TRUE    FALSE FALSE          TRUE   TRUE       TRUE    TRUE
GO:0061138  TRUE    FALSE  TRUE          TRUE   TRUE       TRUE    TRUE
GO:0002263  TRUE    FALSE  TRUE          TRUE   TRUE       TRUE    TRUE
GO:0030168  TRUE    FALSE FALSE          TRUE   TRUE      FALSE    TRUE
GO:0042118 FALSE    FALSE FALSE         FALSE   TRUE       TRUE    TRUE
GO:0050866  TRUE    FALSE  TRUE          TRUE   TRUE       TRUE    TRUE
GO:0050867  TRUE    FALSE  TRUE          TRUE   TRUE       TRUE    TRUE
 [ reached 'max' / getOption("max.print") -- omitted 475 rows ]
attr(,"nTerms")
[1] 3907

To obtain the full matrix with all the GO terms in the BP ontology at GO level 4, we must set the argument onlyEnriched (which is TRUE by default) to FALSE, as follows:

fullEnrichedInBP4 <- enrichedIn(allOncoGeneLists, 
                               geneUniverse = humanEntrezIDs, 
                               orgPackg = "org.Hs.eg.db", 
                               onto = "BP", GOLevel = 4,
                               onlyEnriched = FALSE) 
fullEnrichedInBP4

           atlas cangenes   cis miscellaneous sanger Vogelstein waldman
GO:0036349 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:0036350 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:0060256 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:0060257 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:1900735 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:0010590 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:0030994 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:0030995 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:1904652 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:1905391 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:2001042 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:2001043 FALSE    FALSE FALSE         FALSE  FALSE      FALSE   FALSE
GO:0001649  TRUE    FALSE  TRUE          TRUE   TRUE       TRUE    TRUE
GO:0030278  TRUE    FALSE FALSE          TRUE  FALSE       TRUE    TRUE
 [ reached 'max' / getOption("max.print") -- omitted 3893 rows ]
attr(,"nTerms")
[1] 3907

The number of rows in the full matrix (fullEnrichedInBP4) is much larger than in the matrix containing only the GO terms enriched in at least one list (enrichedInBP4), which implies a more intensive memory usage..

# number of GO terms (rows) in enrichedInBP4
nrow(enrichedInBP4)

[1] 489

# number of GO terms (rows) in fullEnrichedInBP4
nrow(fullEnrichedInBP4)

[1] 3907

The number of rows in fullEnrichedInBP4 corresponds to the total number of GO terms in the BP ontology at GO level 4. In contrast, the number of rows in enrichedInBP4 represents only the GO terms enriched in at least one list from allOncoGeneLists, meaning that each row in this matrix contains at least one TRUE.

NOTE:

To provide users with a quick visualization, the goSorensen package includes the objects enrichedInBP4 and fullEnrichedInBP4, which can be accessed using data(enrichedInBP4) and data(fullEnrichedInBP4).

Note that gene lists, GO terms, and Bioconductor may change over time. So, consider these objects only as illustrative examples, valid exclusively for the allOncoGeneLists at a specific time. The current version of these results was generated with Bioconductor version 3.20. The same comment is applicable to other objects included in goSorensen for quick visualization, some of which are also described in this vignette.

The calculations illustrated in this vignette are based on the matrix containing GO terms enriched in at least one list (in our case, enrichedInBP4). In the illustrations provided for this vignette, there is no evidence to suggest that this matrix produces results different from the full matrix, which includes all GO terms for a specific ontology and level, including those that are not enriched in any of the lists being compared. This is very beneficial since the computational cost of processing is much lower than it could be.

ENRICHMENT CONTINGENCY TABLES

For a Specific Ontology and one GO Level.

Contingency Tables for Two Lists.

The enrichment contingency tables considered in goSorensen are the direct result of obtaining cross-frequency tables between pairs of columns (lists) of the enrichment matrices described in the Section 2 of this vignette. In general, these are internal details that the user of this package does not need to worry about.

Possibly, the only aspect to take into account here is that the main function for this task, buildEnrichTable, always calls internally the function enrichedIn with the argument onlyEnriched put at TRUE and, therefore, the obtained enrichment tables are always in their compact version: Only rows with at least one TRUE (in other words, GO terms enriched in at least one gene list).

For the specific case of two gene lists, the function buildEnrichTable computes the contingency table by accepting two vectors of the class character containing the IDs of the lists to be compared. For instance, for the BP ontology at GO level 4, the contingency table representing the enrichment of GO terms in the lists atlas and sanger is obtained as follows:

cont_atlas.sanger_BP4 <- buildEnrichTable(allOncoGeneLists$atlas, 
                                          allOncoGeneLists$sanger,
                                          listNames = c("atlas", "sanger"),
                                          geneUniverse = humanEntrezIDs, 
                                          orgPackg = "org.Hs.eg.db", 
                                          onto = "BP", 
                                          GOLevel = 4)
cont_atlas.sanger_BP4

                 Enriched in sanger
Enriched in atlas TRUE FALSE
            TRUE   201   212
            FALSE   29  3465

The result is an enrichment contingency table of class table. If the argument storeEnrichedIn of buildEnrichTable was set to TRUE (the default value), it has an attribute, enriched, with the logical matrix of enriched GO terms in these gene lists, i.e., the output of function enrichedIn (always de compact form of these matrices, only rows with almost one TRUE).

NOTE:

To provide users with a quick visualization, the goSorensen package includes the object cont_atlas.sanger_BP4, which can be accessed using data(cont_atlas.sanger_BP4).

Contingency Tables for Two or More Lists.

Given $s$ ($s \geq 2$) lists to compare, the $s(s-1)/2$ possible enrichment contingency tables can also be obtained using the function buildEnrichTable. Instead of providing two vectors as the main argument, we provide an object of the class list, containing at least two elements, each of which contains the identifiers of the different lists to be compared.

For example, for the BP ontology at GO level 4, the $7(6)/2=21$ contingency tables calculated from the 7 gene lists contained in allOncoGeneLists are obtained as follows:

cont_all_BP4 <- buildEnrichTable(allOncoGeneLists,
                                 geneUniverse = humanEntrezIDs, 
                                 orgPackg = "org.Hs.eg.db", 
                                 onto = "BP", 
                                 GOLevel = 4)

The result is an object of the class tableList, which is exclusive from goSorensen and contains all the possible enrichment contingency tables between the compared gene lists at GO level 4 for the ontology BP. Since the output is very large, it is not displayed it in this vignette.

If the argument storeEnrichedIn of buildEnrichTable was set to TRUE (its default value), an important attribute of this object is enriched, accessible via attr(cont_all_BP4, "enriched"), which contains the enrichment matrix obtained using the enrichedIn function. For this particular case, attr(cont_all_BP4, "enriched") contains exactly the same information as the object enrichedInBP4 from Section 2.2 of this vignette.

NOTE:

To provide users with a quick visualization, the goSorensen package includes the object cont_all_BP4, which can be accessed using data(cont_all_BP4).

For More than One Ontology and GO Level.

When you want to obtain contingency tables for two or more lists across multiple ontologies and more than one GO level, you can use the function allBuildEnrichTable.

For example to obtain the $7(6)/2=21$ contingency tables calculated from the 7 gene lists in allOncoGeneLists for the three ontologies (BP, CC, and MF) and for the GO levels from 3 to 10, you can use the function allBuildEnrichTable as follows:

allContTabs <- allBuildEnrichTable(allOncoGeneLists,
                                   geneUniverse = humanEntrezIDs, 
                                   orgPackg = "org.Hs.eg.db", 
                                   ontos = c("BP", "CC", "MF"), 
                                   GOLevels = 3:10)

The result is an object of the class allTableList, which is exclusive from goSorensen and contains all possible enrichment contingency tables between the compared gene lists for the BP, CC, and MF ontologies, and for GO levels 3 to 10. Since the output is very large, it is not displayed in this vignette.

The attribute enriched is present in each element of this output, meaning that for each ontology and GO level contained in this object, there is an enrichment matrix similar to the one obtained with the function enrichedIn. For instance, by running the code attr(allContTabs$BP$'level 4', 'enriched'), you can access the enrichment matrix enrichedInBP4 obtained in Section 2.2 of this vignette.

NOTE:

To provide users with a quick visualization, the goSorensen package includes the object allContTabs, which can be accessed using data(allContTabs).

EQUIVALENCE TESTS.

For a Specific Ontology and GO level.

Equivalence Test for Two Lists.

The function equivTestSorensen performs an equivalence test to detect equivalence between gene lists.

For the specific case of two gene lists, you need to provide the function equivTestSorensen with two character vectors containing the IDs of the lists to be compared.

For example, using an asymptotic normal distribution, an irrelevance limit $d_0=0.4444$ and a significance level $\alpha=0.05$ (the default parameters), an equivalence test to compare the lists atlas and sanger for the BP ontology at GO level 4 can be performed as follows:

eqTest_atlas.sanger_BP4 <- equivTestSorensen(allOncoGeneLists$atlas, 
                                             allOncoGeneLists$sanger,
                                             listNames = c("atlas", "sanger"),
                                             geneUniverse = humanEntrezIDs, 
                                             orgPackg = "org.Hs.eg.db", 
                                             onto = "BP", GOLevel = 4,
                                             d0 = 0.4444, 
                                             conf.level = 0.95)
eqTest_atlas.sanger_BP4


    Normal asymptotic test for 2x2 contingency tables based on the
    Sorensen-Dice dissimilarity

data:  tab
(d - d0) / se = -3.1, p-value = 9e-04
alternative hypothesis: true equivalence limit d0 is less than 0.4444
95 percent confidence interval:
 0.0000 0.4117
sample estimates:
Sorensen dissimilarity 
                0.3748 
attr(,"se")
standard error 
       0.02241

If the enrichment contingency table is available prior to performing the test, such as cont_atlas.sanger_BP4 determined in Section 3.1.1, the execution time for the calculation is much shorter. To use equivTestSorensen with the contingency table as input, proceed as follows:

equivTestSorensen(cont_atlas.sanger_BP4, 
                  d0 = 0.4444, 
                  conf.level = 0.95)


    Normal asymptotic test for 2x2 contingency tables based on the
    Sorensen-Dice dissimilarity

data:  cont_atlas.sanger_BP4
(d - d0) / se = -3.1, p-value = 9e-04
alternative hypothesis: true equivalence limit d0 is less than 0.4444
95 percent confidence interval:
 0.0000 0.4117
sample estimates:
Sorensen dissimilarity 
                0.3748 
attr(,"se")
standard error 
       0.02241

As you can see, both procedures produce the same result, but the last one (whenever possible) is much faster because no time is wasted internally generating the contingency table from the lists of genes and GO terms. Regardless of the procedure, the result is an object of class equivSDhtest (a specialization of class htest), which is exclusive from goSorensen.

If you want to change the calculation parameters of the test, such as using a bootstrap distribution instead of a normal distribution, setting an irrelevance limit of $d_0 = 0.2857$ instead of $d_0 =0.4444$ (or any other), or changing the significance level to $\alpha = 0.01$ instead of $\alpha = 0.05$ (or any other), one option would be to use the equivTestSorensen function again with the new parameters. However, this would require performing all the calculations again, leading to additional computational costs, which increase as more tests are performed. Instead, the function upgrade allows you to update the test output much more quickly by simply specifying the name of the object where the test results are stored and the new parameters you wish to apply, as shown below:

upgrade(eqTest_atlas.sanger_BP4, d0 = 0.2857, 
        conf.level = 0.99, boot = TRUE)


    Bootstrap test for 2x2 contingency tables based on the Sorensen-Dice
    dissimilarity (10000 bootstrap replicates)

data:  tab
(d - d0) / se = 4, p-value = 1
alternative hypothesis: true equivalence limit d0 is less than 0.2857
99 percent confidence interval:
 0.0000 0.4297
sample estimates:
Sorensen dissimilarity 
                0.3748 
attr(,"se")
standard error 
       0.02241

Equivalence Test for Two or More Lists.

Given $s$ ($s \geq 2$) lists to compare, the $s(s-1)/2$ possible equivalence tests can also be obtained using the function equivTestSorensen Instead of providing two vectors as the main argument, we provide an object of the class list, containing at least two elements, each of which contains the identifiers of the different lists to be compared.

For example, for the BP ontology at GO level 4, the $7(6)/2=21$ possible test calculated from the 7 gene lists contained in allOncoGeneLists are obtained as follows:

eqTest_all_BP4 <- equivTestSorensen(allOncoGeneLists,
                                    geneUniverse = humanEntrezIDs, 
                                    orgPackg = "org.Hs.eg.db", 
                                    onto = "BP", 
                                    GOLevel = 4,
                                    d0 = 0.4444, 
                                    conf.level = 0.95)

But remember, it is much simpler if we already have the contingency tables as an object of the class tableList. In our case, we have already calculated the contingency tables for all possible pairs of allOncoGeneLists for the ontology BP, GO level 4, in Section 3.1.2 and stored them in the object cont_all_BP4. Therefore, we can calculate the eqTest_all_BP4 object more efficiently in the following way:

eqTest_all_BP4 <- equivTestSorensen(cont_all_BP4,
                                    d0 = 0.4444, 
                                    conf.level = 0.95)

Remember that similarly to the comparison of two lists in Section 4.1.1, you can use the function upgrade to update the results by changing the parameters of the tests, such as the confidence level, irrelevance limit, and others. For instance, upgrade(eqTest_atlas.sanger_BP4, d0 = 0.2857 to update the equivalence test using an irrelevance limit $d_0=0.2857$.

Since the output contained in eqTest_all_BP4 is very large, it is not displayed in this vignette. However, goSorensen provides accessor functions that allow you to retrieve specific outputs of interest. For example, to obtain a summary of the Sorensen dissimilarities contained in the tests comparing all pairs of lists in the BP ontology at GO level 4, you can use the function getDissimilarity and retrieve them as follows:

getDissimilarity(eqTest_all_BP4, simplify = FALSE)

               atlas cangenes    cis miscellaneous sanger Vogelstein waldman
atlas         0.0000        1 0.6964        0.3937 0.3748     0.3474  0.2807
cangenes      1.0000        0 1.0000        1.0000 1.0000     1.0000  1.0000
cis           0.6964        1 0.0000        0.5503 0.5884     0.6216  0.6180
miscellaneous 0.3937        1 0.5503        0.0000 0.3065     0.3390  0.2281
sanger        0.3748        1 0.5884        0.3065 0.0000     0.1162  0.3270
Vogelstein    0.3474        1 0.6216        0.3390 0.1162     0.0000  0.2956
waldman       0.2807        1 0.6180        0.2281 0.3270     0.2956  0.0000

Another example of accessor function is the function getPvalue to obtain the p-values across the object eqTest_all_BP4:

getPvalue(eqTest_all_BP4, simplify = FALSE)

                  atlas cangenes    cis miscellaneous     sanger Vogelstein
atlas         0.000e+00      NaN 1.0000     1.362e-02  9.429e-04  3.028e-06
cangenes            NaN        0    NaN           NaN        NaN        NaN
cis           1.000e+00      NaN 0.0000     9.984e-01  1.000e+00  1.000e+00
miscellaneous 1.362e-02      NaN 0.9984     0.000e+00  1.643e-08  1.582e-05
sanger        9.429e-04      NaN 1.0000     1.643e-08  0.000e+00 1.724e-100
Vogelstein    3.028e-06      NaN 1.0000     1.582e-05 1.724e-100  0.000e+00
waldman       5.349e-18      NaN 1.0000     3.385e-26  3.222e-07  1.057e-11
                waldman
atlas         5.349e-18
cangenes            NaN
cis           1.000e+00
miscellaneous 3.385e-26
sanger        3.222e-07
Vogelstein    1.057e-11
waldman       0.000e+00

NaN values occur when the test statistic cannot be calculated due to an indeterminacy, for example when the standard error of the sample Sorensen-Dice dissimilarity cannot be calculated or is null. One of these scenarios occurs when there is no joint enrichment between two lists (i.e., when $n_{11}=0$).

In addition other accesor functions include: getSE for the standard error, getUpper for the upper bound of the confidence interval and getTable for the enrichment contingency tables.

NOTE:

To provide users with a quick visualization, the goSorensen package includes the object eqTest_all_BP4, which can be accessed using data(eqTest_all_BP4).

For More than One Ontology and GO level.

When you want to obtain the outputs of the equivalence tests to compare two or more lists across multiple ontologies and more than one GO level, you can use the function allEquivTestSorensen

For example to obtain the $7(6)/2=21$ equivalence tests calculated from the 7 gene lists in allOncoGeneLists for the three ontologies (BP, CC, and MF) and for the GO levels from 3 to 10, you can use the function allEquivTestSorensen as follows:

allEqTests <- allEquivTestSorensen(allOncoGeneLists,
                                   geneUniverse = humanEntrezIDs, 
                                   orgPackg = "org.Hs.eg.db", 
                                   ontos = c("BP", "CC", "MF"), 
                                   GOLevels = 3:10,
                                   d0 = 0.4444, 
                                   conf.level = 0.95)

But remember, it is much simpler if we already have the contingency tables as an object of the class allTableList In our case, we have already calculated the contingency tables for all possible pairs of allOncoGeneLists for the ontologies BP, CC, MF, and for the GO levels 3 to 10, in Section 3.2 and stored them in the object allContTabs Therefore, we can calculate the allEqTests object more efficiently in the following way:

allEqTests <- allEquivTestSorensen(allContTabs,
                                   d0 = 0.4444, 
                                   conf.level = 0.95)

The result is an object of the class AllEquivSDhtest, which is exclusive to goSorensen.

In a similar way to what is explained in Section 4.1.1 and 4.1.2, you can use the function upgrade to update the results by changing the parameters of the tests, such as the confidence level, irrelevance limit, sample distribution (normal or bootstrap) and others.

You can use also the accessor functions to obtain key test outputs, such as the Sorensen dissimilarities (getDissimilarity), p-values (getPvalue), enrichment contingency tables (getTable), and more.

NOTE:

To provide users with a quick visualization, the goSorensen package includes the object allEqTests, which can be accessed using data(allEqTests).

References

Flores, Pablo, Miquel Salicrú, Alex Sánchez-Pla, and Jordi Ocaña. 2022. “An Equivalence Test Between Features Lists, Based on the Sorensen–Dice Index and the Joint Frequencies of GO Term Enrichment.” BMC Bioinformatics 23 (1): 207.

Wellek, Stefan. 2010. Testing Statistical Hypotheses of Equivalence and Noninferiority. CRC Press-Chapman & Hall.

In fact, this is an internal step hidden within the function buildEnrichTable described in Section 3. However, providing a brief explanation of this process may help clarify certain details about how enrichment contingency tables are constructed. For users working with the package at a not-so-advanced level, this step can be skipped without affecting their understanding of how to use the core functions of goSorensen.↩︎

	enriched in \(L_2\)	non-enriched in \(L_2\)
enriched in \(L_1\)	\(n_{11}\)	\(n_{10}\)	\(n_{1.}\)
non-enriched in \(L_1\)	\(n_{01}\)	\(n_{00}\)	\(n_{0.}\)
	\(n_{.1}\)	\(n_{.0}\)	\(n\)

An Introduction to goSorensen R-Package

PRELIMINARIES.

Theoretical Framework of Reference.

goSorensen Installation.

Data Used in this Vignette.

Before Using goSorensen.

MATRIX OF GO TERMS ENRICHMENT.

For One List.

For Two or More Lists.

NOTE:

ENRICHMENT CONTINGENCY TABLES

For a Specific Ontology and one GO Level.

Contingency Tables for Two Lists.

NOTE:

Contingency Tables for Two or More Lists.

NOTE:

For More than One Ontology and GO Level.

NOTE:

EQUIVALENCE TESTS.

For a Specific Ontology and GO level.

Equivalence Test for Two Lists.

Equivalence Test for Two or More Lists.

NOTE:

For More than One Ontology and GO level.

NOTE:

References