Statistical Bioinformatics as simple as possible

Guillermo Ayala

5/11/23

Introduction

Abstract

This vignette is concerned with the use of tami. We propose different ways to do the same things. All functionalities are used. This is our purpose.

Differential expression analysis for microarrays

Packages

The following code download and install the package.

url_tami = "http://www.uv.es/ayala/docencia/tami/tami_1.0.tar.gz"
install.packages(url_tami,repos=  NULL,type="source")

We load the package.

pacman::p_load(tami)

We will use data in the package tamidata. This package can be installed with

url_tamidata = "http://www.uv.es/ayala/docencia/tami/tamidata_0.6.tar.gz"
install.packages(url_tamidata,repos=  NULL,type="source")

We need some additional packages.

pacman::p_load("Biobase","hgu133a.db","ggplot2","ggfortify")

How to create an ExpressionData starting from scratch

An ExpressionDatais an S4 class with three slots. The first one is matrix exprm and we use the attribute rownames of the matrix (rownames(exprm)) as annotation. The second slot is the phenotypic variable, covariable or experimental factor with two levels indicating the classification of each column (sample) in one experimental group.

Let us make an ExpressionData starting from scratch. The expression matrix is just a matrix. The expression values (no real interest) will be random values generated according a standard normal distribution (null mean and standard deviation one)

exprm0 = matrix(rnorm(100),ncol=5)

Now exprm0 has no attribute rownames.

rownames(exprm0)
NULL

We are going to define as rownames

rownames(exprm0) = letters[1:20]

This is the (small) expression matrix.

exprm0
          [,1]        [,2]          [,3]        [,4]        [,5]
a  0.912429987  0.18357188 -0.5906215751 -0.66245248 -0.78564388
b -0.393507292  0.64174861  0.1742275846  2.28722170  1.29144219
c -0.769680386  0.02086337  0.0697678897 -0.05563645 -1.81856579
d -0.681712677 -1.03851086  0.0614321152  0.17682770 -1.63319765
e  1.260810294 -0.90913089 -0.1716689779  0.84897174 -0.13361822
f  0.013202248  0.03490041 -0.4233372313  0.09812043 -0.90597519
g -1.598859435  1.29914331  2.0639561795 -0.11062524 -0.85839740
h -1.168896396  0.41962553  1.1986641934  1.15267216 -1.85138511
i  0.419976917  0.10578242 -1.8354921745  0.11221312 -0.52511800
j -0.107866714 -1.32826469 -0.1397170596 -1.32538625  1.68500373
k -1.410776970 -1.22709323  0.0001126676 -0.89490852  1.07693021
l -0.002355853 -0.38864084  0.6286181407  0.93479589  1.16614375
m -0.695121614 -1.07693244 -1.0318277128 -0.24253178 -1.35602674
n  0.613943043  1.23847677  0.0760794079 -1.38758856 -2.15035804
o -1.403770805  0.67199708 -1.0384458187  0.19940505  1.83607465
p -1.583231985  0.73830642  0.1581352999  0.37206905 -0.01226084
q  2.139984088 -0.20339940 -0.8081791569  0.83064594  0.12901228
r -0.593026452 -1.40923521 -0.9778623541 -1.29962627  1.55971546
s  0.183719855  1.24887173 -0.2142631148 -0.18468580  1.73575691
t  0.651892334 -1.45577762 -1.6266126573 -0.82949227  0.17471778

The second slot is the covariable groups, the experimental factor with two levels. The first level will be labelled “A” and the second levels as “B”. The samples corresponding to group “A” occupies the columns 1, 4 and 5. The other columns (samples) corresponding to group “B”.

groups0 = factor(c(1,2,2,1,1),levels = 1:2,labels=c("A","B"))

Note that the variable or factor groups0 takes numerical values 1 and 2 but R knows that it represents categories or groups. The labels of the categories will be used in plots and summaries.

Now we can construct the ExpressionData. We have two possibilities. The simplest and preferred is (we use the constructor).

x  = new("ExpressionData",exprm = exprm0,groups=groups0)

A simple summary can be obtained with

x
An object of class ExpressionData
The primary identifiers are 
 a b c d e f 
The expression matrix is 
 0.91243 -0.3935073 0.1835719 0.6417486 -0.5906216 0.1742276 -0.6624525 2.287222 -0.7856439 1.291442 
The experimental factor is 
 1 2 2 1 1

The other equivalent procedure to make the ExpressionData is

x = new("ExpressionData",exprm = exprm0, groups = groups0)

From ExpressionSet to ExpressionData

Usually, we will start our analysis with a data set organized using a Biobase::ExpressionSet. For instance, tamidata::gse21942. It is easy to construct an ExpressionData. First we load tamidata::gse21942.

data(gse21942,package="tamidata")
x = ExpressionData(exprm = exprs(gse21942), 
                     groups = pData(gse21942)[,"FactorValue..DISEASE.STATE."],
                     type="microarray")

Using ExpressionData

We can recover the different slots using the corresponding accessors.

head(exprm(x)) ## to limit the output
          GSM545846.CEL GSM545845.CEL GSM545844.CEL GSM545843.CEL GSM545842.CEL
1007_s_at      6.701185      6.770585      6.896115      6.913170      7.055230
1053_at        6.904241      7.160595      6.986157      7.252437      6.820581
117_at         8.347887      8.309832      8.042841      7.973542      7.905076
121_at         7.527261      7.732676      7.534203      7.582493      7.624345
1255_g_at      2.741237      2.785420      2.705875      2.712574      2.874974
1294_at        8.512001      8.697264      8.471296      8.609792      8.932694
          GSM545841.CEL GSM545840.CEL GSM545839.CEL GSM545838.CEL GSM545837.CEL
1007_s_at      7.090447      7.161786      6.990521      7.143398      7.019538
1053_at        6.983571      6.801948      7.131738      6.941465      6.829286
117_at         7.656960      7.661755      8.849415      7.781422      7.431034
121_at         7.462491      7.757673      7.771086      7.728931      7.507087
1255_g_at      2.731671      3.032517      2.957291      2.908258      2.801843
1294_at        8.466908      8.613721      8.996203      8.312664      8.444103
          GSM545836.CEL GSM545835.CEL GSM545834.CEL GSM545833.CEL GSM545832.CEL
1007_s_at      7.263345      7.183820      7.068422      7.018434      7.134651
1053_at        7.227540      6.779245      7.141372      6.908670      7.484561
117_at         8.128334      7.638504      8.336013      8.466568      8.495304
121_at         7.604226      7.721856      7.660064      7.735517      7.514174
1255_g_at      2.949537      2.747506      2.634829      2.995889      2.631472
1294_at        9.093238      9.048817      9.141685      8.805013      8.557823
          GSM545831.CEL GSM545830.CEL GSM545829.CEL GSM545828.CEL GSM545827.CEL
1007_s_at      7.164015      7.079536      6.879690      6.577880      6.928483
1053_at        6.997132      6.890387      7.179972      7.303770      7.410144
117_at         8.036675      8.228937      7.237569      8.042661      8.141817
121_at         7.553159      7.623796      7.507285      7.698743      7.696693
1255_g_at      2.708215      2.840438      2.723045      2.651039      2.906055
1294_at        8.761252      8.737342      8.528587      8.587533      8.761697
          GSM545826.CEL GSM545825.CEL GSM545824.CEL GSM545823.CEL GSM545822.CEL
1007_s_at      6.653279      6.900510      6.436211      7.008971      6.834951
1053_at        7.120752      7.109658      7.085381      7.046270      7.324719
117_at         8.033017      7.955324      8.457568      7.863781      7.965000
121_at         7.522826      7.725928      7.733425      7.846924      7.534118
1255_g_at      2.803210      2.889357      2.781139      2.868890      2.632601
1294_at        8.669705      8.470591      8.596620      8.660778      8.730843
          GSM545821.CEL GSM545820.CEL GSM545819.CEL GSM545818.CEL
1007_s_at      6.747209      7.052194      6.785745      6.715101
1053_at        7.176329      7.224084      7.369654      7.277721
117_at         7.776154      7.825469      8.342164      7.998972
121_at         7.443340      7.714347      7.434026      7.699178
1255_g_at      2.947213      2.765574      2.898040      2.709309
1294_at        8.648626      8.640482      8.730575      8.606553
groups(x)
 [1] multiple sclerosis multiple sclerosis multiple sclerosis multiple sclerosis
 [5] multiple sclerosis multiple sclerosis multiple sclerosis multiple sclerosis
 [9] multiple sclerosis multiple sclerosis multiple sclerosis multiple sclerosis
[13] multiple sclerosis multiple sclerosis healthy            healthy           
[17] healthy            healthy            healthy            healthy           
[21] healthy            healthy            healthy            healthy           
[25] healthy            healthy            healthy            healthy           
[29] healthy           
Levels: healthy multiple sclerosis

Plots for ExpressionData

The first plot shows the mean versus the standard deviation of the gene expression profile.

plot(x,1)

The second plot is the median versus the interquantile range of the expression profile.

plot(x,2)

The third plot displays the two first principal components. The package ggfortify has to be loaded.

library(ggfortify)
plot(x,3)

The fourth plot displays the Tukey mean-difference plot to compare the mean expressión of the gene per condition. We consider the mean expression per condition for each gene: \((x_i,y_i)\) where \(i\) is the gene and \(x_i\) and \(y_i\) and the corresponding means per condition. The x axis is the mean \(\frac{x_i+y_i}{2}\) and the y axis is the difference \(x_i - y_i\). The left plot corresponds to the original scale and the right plot uses the natural logarithms of the original data. Perhaps, it is more usual in the omics data literature to use the logarithm with base 2 but it is more natural our choice.

plot(x,4)

Differential Expression Input

The S4 class DifferentialExpressionInput provides two additional slots to control the statistical analysis to be performed.

We are going to apply a t-test per gene with the p-values adjusted using the Benjamini-Hochberg correction. Also the q-values from the original (raw) p-values are calculated.

The first option is to apply the method rowtt to an ExpressionData.

We have to define an DifferentialExpressionInput object. There are two options.

x_dei = DifferentialExpressionInput(exprm =exprm(x),
                 groups =  groups(x),association ="pvalue",
                 correction = "BH")
x_dei = new("DifferentialExpressionInput",exprm =exprm(x),
                 groups =  groups(x),association ="pvalue",
                 correction = "BH")

Again a short review of the object is returned typing its name.

x_dei
An object of class DifferentialExpressionInput
The primary identifiers are 
 1007_s_at 1053_at 117_at 121_at 1255_g_at 1294_at 
The expression matrix is 
 6.701185 6.770585 6.896115 6.91317 7.05523 7.090447 7.161786 6.990521 7.143398 7.019538 7.263345 7.18382 7.068422 7.018434 7.134651 7.164015 7.079536 6.87969 6.57788 6.928483 6.653279 6.90051 6.436211 7.008971 6.834951 6.747209 7.052194 6.785745 6.715101 
The experimental factor is 
 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
The method to correct for multiple comparisons is 
 BH 
The test of the statistical method is 
 
x_t = rowt(x= exprm(x),y=groups(x))
x_t2 = rowtest(x= exprm(x),y=groups(x),test = rowt,correction = "BH")

The value is a data.frame.

head(x_t2)
            statistic        rawp       adjp       qval
1007_s_at -2.29869931 0.029492008 0.11049371 0.05708447
1053_at    3.44084102 0.001901206 0.01513959 0.00782158
117_at    -0.08505071 0.932848609 0.96675733 0.49945673
121_at    -0.53362792 0.597965094 0.76778314 0.39666051
1255_g_at -1.01536731 0.318943716 0.53372043 0.27573648
1294_at   -1.05030996 0.302886126 0.51746083 0.26733627

Note that the rows are identified using the original Affymetrix identifiers. They are the row names of the data frame.

head(rownames(x_t2))
[1] "1007_s_at" "1053_at"   "117_at"    "121_at"    "1255_g_at" "1294_at"

They will be used as primary gene identifiers. Instead of the t-tests where the standard error of the difference of means is estimated using the profile expression of each gene, other possible procedure is contained in the Limma R package where an hierarchical model is used. Again the same results can be obtained using two procedures implemented in tami.

x_tm = rowtmod(x= exprm(x),y=groups(x))
head(x_tm)
    statistic        rawp
1 -2.31854450 0.027119463
2  3.40762889 0.001819455
3 -0.08980406 0.929015315
4 -0.49077813 0.627007830
5 -0.93447609 0.357214419
6 -1.06854428 0.293449716

The same analysis and the adjusted p-values and q-values can be obtained using tami::rowtest.

x_tm2 = rowtest(x= exprm(x),y=groups(x),test = rowtmod,correction = "BH")

The first rows of the resulting data frame can be inspected with

head(x_tm2)
            statistic        rawp      adjp        qval
1007_s_at -2.31854450 0.027119463 0.1023791 0.053520473
1053_at    3.40762889 0.001819455 0.0138995 0.007266206
117_at    -0.08980406 0.929015315 0.9653334 0.504644933
121_at    -0.49077813 0.627007830 0.7885372 0.412221609
1255_g_at -0.93447609 0.357214419 0.5729244 0.299506288
1294_at   -1.06854428 0.293449716 0.5089248 0.266049350

Similarly we can applied the rowt option.

x_tm2 = rowtest(x= exprm(x),y=groups(x),test = rowt,correction = "BH")

The first rows of the resulting data frame can be inspected with

head(x_tm2)
            statistic        rawp       adjp       qval
1007_s_at -2.29869931 0.029492008 0.11049371 0.05708447
1053_at    3.44084102 0.001901206 0.01513959 0.00782158
117_at    -0.08505071 0.932848609 0.96675733 0.49945673
121_at    -0.53362792 0.597965094 0.76778314 0.39666051
1255_g_at -1.01536731 0.318943716 0.53372043 0.27573648
1294_at   -1.05030996 0.302886126 0.51746083 0.26733627

Differential Expression Output

We have now a data frame with the annotation data and a data frame with the statistical results of the differential expresion analysis. We create a DifferentialExpressionOutput. We can made it with two procedures.

x_t2_deo = DifferentialExpressionOutput(GeneData=fData(gse21942),
                                        GeneStat = x_t2,
                                        foutput ="gse21942_rowtt",fdr = .2)

We can access or modify the different slots using

GeneData(x_t2_deo)
GeneStat(x_t2_deo)
foutput(x_t2_deo)
fdr(x_t2_deo)
x_t2_deo = new("DifferentialExpressionOutput",
                        GeneData = fData(gse21942), GeneStat = x_t2,
                        foutput = "gse21942_rowtt",fdr = .2)

We generate the corresponding tidy report where only genes with an adjusted p-value lower than FDR are included in the report.

tami::tidy(x_t2_deo)

We can see the report at this file.

However, the report with all genes can be obtained with

tami::augment(x_t2_deo)

The corresponding report can be found here.

Using rowttmod

We can repeat the previous analysis by replacing tami::rowt with the moderated t-tests (from limma package). Note that we have to modify the statistical results contained in the data frame gse21942_rowt by the data frame gse21942_rowtmod.

x_tm2_deo = new("DifferentialExpressionOutput",
                        GeneData = fData(gse21942), GeneStat = x_tm2,
                        foutput = "gse21942_rowttmod",fdr = .2)

Generate the report.

tami::tidy(x_tm2_deo)

The results are included in this new file

dema(x=gse21942,y="FactorValue..DISEASE.STATE.",correction = "BH",test = rowt,
     foutput = "gse21942",fdr = .01)

Gene set analysis

We have to choose a gene set collection. The package EnrichmentBrowser have some useful functions to construct these gene set collections. Note that we are going to analyze human and yeast data sets.

KEGG

First, we can use the gene sets from KEGG. We download the collections and save them. The species names can be found at .

pacman::p_load(EnrichmentBrowser)
hsa_kegg = get.kegg.genesets("hsa")
sce_kegg = get.kegg.genesets("sce")

Gene Ontology

hsa_go = get.go.genesets(org="hsa", onto="BP", mode="GO.db")
sce_go = get.go.genesets(org="sce", onto="BP", mode="GO.db")

As we are going to see the genes are coded using the ENTREZID. ## Exploring a gene set collection

What kind of data we have?

class(hsa_go)
[1] "list"

We can use generic functions for list.

length(hsa_go)
[1] 22934

We can access to each element in the list using the position

hsa_go[1]
$`GO:0000002`
 [1] "5428"   "6742"   "11232"  "55186"  "56652"  "84275"  "92667"  "201973"
 [9] "1763"   "7157"   "9093"   "10891"  "80119"  "83667"  "201163" "142"   
[17] "1890"   "2021"   "3980"   "4358"   "4976"   "6240"   "7156"   "10000" 
[25] "50484"  "64863"  "219736" "4205"   "9361"   "291"

or the name

hsa_go["GO:0000002_mitochondrial_genome_maintenance"]
$<NA>
NULL

If we want the elements then

hsa_go[[1]]
 [1] "5428"   "6742"   "11232"  "55186"  "56652"  "84275"  "92667"  "201973"
 [9] "1763"   "7157"   "9093"   "10891"  "80119"  "83667"  "201163" "142"   
[17] "1890"   "2021"   "3980"   "4358"   "4976"   "6240"   "7156"   "10000" 
[25] "50484"  "64863"  "219736" "4205"   "9361"   "291"

or

hsa_go[["GO:0000002_mitochondrial_genome_maintenance"]]
NULL

The first elements of the list.

head(hsa_go)
$`GO:0000002`
 [1] "5428"   "6742"   "11232"  "55186"  "56652"  "84275"  "92667"  "201973"
 [9] "1763"   "7157"   "9093"   "10891"  "80119"  "83667"  "201163" "142"   
[17] "1890"   "2021"   "3980"   "4358"   "4976"   "6240"   "7156"   "10000" 
[25] "50484"  "64863"  "219736" "4205"   "9361"   "291"   

$`GO:0000003`
   [1] "49"        "167"       "190"       "268"       "301"       "367"      
   [7] "638"       "699"       "701"       "993"       "994"       "995"      
  [13] "1060"      "1654"      "1761"      "2072"      "2177"      "2348"     
  [19] "2350"      "2352"      "2488"      "2492"      "2515"      "2528"     
  [25] "2622"      "3010"      "3024"      "3248"      "3267"      "3364"     
  [31] "3619"      "3973"      "4342"      "4361"      "4438"      "4439"     
  [37] "5021"      "5048"      "5139"      "5617"      "5620"      "5660"     
  [43] "5819"      "5888"      "5889"      "5892"      "6117"      "6406"     
  [49] "6407"      "6847"      "6865"      "6869"      "6870"      "6954"     
  [55] "7141"      "7142"      "7153"      "7155"      "7222"      "7225"     
  [61] "7272"      "7283"      "7432"      "7783"      "7784"      "8438"     
  [67] "8468"      "8521"      "8653"      "8747"      "8748"      "8749"     
  [73] "9232"      "9319"      "9576"      "9700"      "9918"      "9985"     
  [79] "10018"     "10111"     "10370"     "10388"     "10426"     "10655"    
  [85] "10744"     "10844"     "10942"     "11022"     "11055"     "11057"    
  [91] "11085"     "11086"     "11105"     "11144"     "22862"     "22917"    
  [97] "23291"     "23310"     "23424"     "23626"     "25788"     "25858"    
 [103] "26108"     "26255"     "26271"     "26528"     "26998"     "27175"    
 [109] "27229"     "27443"     "29781"     "29893"     "43847"     "50511"    
 [115] "51298"     "51314"     "54514"     "54558"     "54586"     "54742"    
 [121] "54763"     "54970"     "55521"     "56154"     "56155"     "56158"    
 [127] "56159"     "56165"     "56907"     "56979"     "57113"     "57151"    
 [133] "57587"     "57820"     "57829"     "58524"     "58531"     "63948"    
 [139] "63950"     "63951"     "64100"     "64753"     "79084"     "79703"    
 [145] "79846"     "79925"     "80010"     "80198"     "80217"     "81626"    
 [151] "81833"     "83449"     "83540"     "83639"     "83893"     "83990"    
 [157] "84057"     "84071"     "84223"     "84225"     "84229"     "84464"    
 [163] "84501"     "84690"     "84944"     "85376"     "85378"     "85438"    
 [169] "89765"     "89869"     "90780"     "91646"     "114791"    "117144"   
 [175] "117155"    "119710"    "124626"    "124783"    "124817"    "124912"   
 [181] "126549"    "128153"    "128497"    "130951"    "131375"    "132141"   
 [187] "135458"    "135927"    "136242"    "139212"    "140894"    "145645"   
 [193] "146378"    "146849"    "146956"    "147650"    "147872"    "150221"   
 [199] "150365"    "151246"    "152015"    "154313"    "157695"    "157855"   
 [205] "158401"    "163589"    "164045"    "168391"    "170370"    "171169"   
 [211] "171482"    "171483"    "171484"    "197342"    "201254"    "203102"   
 [217] "221400"    "221711"    "246777"    "254528"    "255101"    "255220"   
 [223] "257044"    "257062"    "283129"    "283417"    "283847"    "284067"   
 [229] "284071"    "284359"    "284680"    "285498"    "285588"    "286151"   
 [235] "286207"    "286826"    "317761"    "339168"    "339345"    "339834"   
 [241] "340069"    "340719"    "342977"    "346673"    "347732"    "349152"   
 [247] "374768"    "375337"    "378708"    "378807"    "387885"    "388649"   
 [253] "389320"    "389852"    "390243"    "400629"    "440804"    "494551"   
 [259] "644186"    "728637"    "729201"    "768239"    "100125288" "100130988"
 [265] "100131137" "100507650" "100996631" "101928601" "107983988" "93426"    
 [271] "91"        "397"       "480"       "796"       "928"       "1235"     
 [277] "1392"      "1393"      "2516"      "2661"      "2693"      "3965"     
 [283] "4838"      "4880"      "5047"      "5864"      "6469"      "6662"     
 [289] "6736"      "7490"      "9126"      "10497"     "10635"     "22999"    
 [295] "51738"     "57647"     "64750"     "66037"     "84868"     "130399"   
 [301] "255061"    "284382"    "373861"    "406949"    "2"         "18"       
 [307] "51"        "90"        "92"        "100"       "113"       "117"      
 [313] "133"       "150"       "151"       "174"       "181"       "182"      
 [319] "183"       "207"       "249"       "269"       "285"       "330"      
 [325] "338"       "361"       "374"       "383"       "409"       "412"      
 [331] "472"       "493"       "538"       "546"       "551"       "552"      
 [337] "558"       "572"       "578"       "581"       "582"       "595"      
 [343] "596"       "598"       "599"       "604"       "639"       "646"      
 [349] "652"       "653"       "654"       "655"       "657"       "659"      
 [355] "666"       "668"       "675"       "682"       "694"       "718"      
 [361] "734"       "771"       "780"       "811"       "824"       "835"      
 [367] "836"       "867"       "875"       "881"       "898"       "952"      
 [373] "989"       "991"       "1027"      "1028"      "1047"      "1069"     
 [379] "1164"      "1268"      "1271"      "1364"      "1390"      "1399"     
 [385] "1459"      "1495"      "1499"      "1508"      "1539"      "1543"     
 [391] "1545"      "1588"      "1602"      "1617"      "1618"      "1620"     
 [397] "1621"      "1636"      "1642"      "1718"      "1730"      "1731"     
 [403] "1738"      "1739"      "1785"      "1788"      "1812"      "1816"     
 [409] "1822"      "1828"      "1869"      "1906"      "1909"      "1910"     
 [415] "1912"      "2026"      "2056"      "2057"      "2067"      "2099"     
 [421] "2120"      "2175"      "2176"      "2182"      "2185"      "2189"     
 [427] "2192"      "2241"      "2253"      "2254"      "2255"      "2288"     
 [433] "2296"      "2353"      "2354"      "2475"      "2560"      "2583"     
 [439] "2591"      "2593"      "2620"      "2623"      "2660"      "2662"     
 [445] "2674"      "2683"      "2695"      "2706"      "2707"      "2709"     
 [451] "2735"      "2741"      "2743"      "2796"      "2827"      "2908"     
 [457] "2956"      "2959"      "3014"      "3037"      "3066"      "3073"     
 [463] "3074"      "3148"      "3169"      "3171"      "3172"      "3205"     
 [469] "3206"      "3207"      "3209"      "3235"      "3236"      "3239"     
 [475] "3291"      "3295"      "3298"      "3301"      "3305"      "3306"     
 [481] "3309"      "3371"      "3400"      "3417"      "3480"      "3482"     
 [487] "3485"      "3488"      "3490"      "3516"      "3549"      "3552"     
 [493] "3553"      "3566"      "3623"      "3625"      "3633"      "3640"     
 [499] "3643"      "3645"      "3673"      "3675"      "3678"      "3688"     
 [505] "3690"      "3691"      "3726"      "3753"      "3791"      "3814"     
 [511] "3815"      "3856"      "3857"      "3880"      "3948"      "3952"     
 [517] "3955"      "3976"      "3985"      "4033"      "4086"      "4089"     
 [523] "4090"      "4142"      "4179"      "4188"      "4201"      "4216"     
 [529] "4221"      "4240"      "4254"      "4292"      "4313"      "4318"     
 [535] "4323"      "4327"      "4436"      "4487"      "4488"      "4521"     
 [541] "4627"      "4678"      "4683"      "4693"      "4735"      "4751"     
 [547] "4808"      "4809"      "4824"      "4846"      "4851"      "4861"     
 [553] "4882"      "4889"      "4914"      "4920"      "4926"      "4948"     
 [559] "4956"      "4957"      "4986"      "4987"      "4988"      "4991"     
 [565] "5000"      "5010"      "5016"      "5017"      "5020"      "5023"     
 [571] "5028"      "5050"      "5066"      "5079"      "5087"      "5111"     
 [577] "5154"      "5156"      "5159"      "5176"      "5224"      "5228"     
 [583] "5232"      "5238"      "5241"      "5266"      "5268"      "5270"     
 [589] "5324"      "5327"      "5347"      "5360"      "5367"      "5368"     
 [595] "5469"      "5515"      "5518"      "5535"      "5591"      "5608"     
 [601] "5618"      "5619"      "5719"      "5724"      "5727"      "5729"     
 [607] "5730"      "5743"      "5781"      "5806"      "5887"      "5901"     
 [613] "5914"      "5916"      "5925"      "5926"      "5932"      "5972"     
 [619] "5997"      "6045"      "6194"      "6196"      "6198"      "6305"     
 [625] "6382"      "6414"      "6422"      "6423"      "6461"      "6477"     
 [631] "6498"      "6513"      "6522"      "6532"      "6573"      "6595"     
 [637] "6615"      "6647"      "6654"      "6670"      "6674"      "6676"     
 [643] "6677"      "6690"      "6714"      "6715"      "6716"      "6751"     
 [649] "6752"      "6753"      "6768"      "6777"      "6781"      "6788"     
 [655] "6789"      "6790"      "6794"      "6795"      "6812"      "6833"     
 [661] "6874"      "6875"      "6895"      "6950"      "7004"      "7013"     
 [667] "7016"      "7026"      "7043"      "7046"      "7048"      "7054"     
 [673] "7056"      "7067"      "7068"      "7073"      "7079"      "7080"     
 [679] "7098"      "7103"      "7110"      "7130"      "7182"      "7203"     
 [685] "7226"      "7257"      "7258"      "7301"      "7314"      "7320"     
 [691] "7337"      "7345"      "7349"      "7351"      "7372"      "7421"     
 [697] "7425"      "7458"      "7473"      "7474"      "7476"      "7484"     
 [703] "7528"      "7536"      "7704"      "7707"      "7855"      "8061"     
 [709] "8086"      "8204"      "8322"      "8398"      "8399"      "8434"     
 [715] "8528"      "8531"      "8546"      "8549"      "8614"      "8626"     
 [721] "8633"      "8654"      "8743"      "8751"      "8820"      "8852"     
 [727] "8879"      "8894"      "8924"      "8932"      "9104"      "9133"     
 [733] "9134"      "9148"      "9156"      "9184"      "9191"      "9241"     
 [739] "9271"      "9289"      "9389"      "9403"      "9420"      "9444"     
 [745] "9468"      "9509"      "9510"      "9514"      "9519"      "9612"     
 [751] "9667"      "9724"      "9825"      "9897"      "10049"     "10051"    
 [757] "10096"     "10097"     "10116"     "10134"     "10155"     "10179"    
 [763] "10184"     "10403"     "10461"     "10468"     "10481"     "10491"    
 [769] "10524"     "10549"     "10560"     "10574"     "10575"     "10576"    
 [775] "10592"     "10653"     "10657"     "10661"     "10694"     "10735"    
 [781] "10765"     "10818"     "10881"     "10919"     "10927"     "10935"    
 [787] "10959"     "10991"     "11189"     "11218"     "11251"     "11315"    
 [793] "11331"     "22836"     "22933"     "22948"     "22994"     "23064"    
 [799] "23157"     "23205"     "23230"     "23304"     "23345"     "23353"    
 [805] "23394"     "23397"     "23399"     "23411"     "23414"     "23492"    
 [811] "23542"     "23598"     "23609"     "23633"     "23641"     "23705"    
 [817] "23762"     "24145"     "24149"     "25777"     "25809"     "25831"    
 [823] "25911"     "25932"     "25945"     "25976"     "26003"     "26064"    
 [829] "26165"     "26206"     "26471"     "27030"     "27125"     "27127"    
 [835] "27136"     "27285"     "27306"     "29844"     "29924"     "29974"    
 [841] "29988"     "49855"     "50487"     "50846"     "51087"     "51247"    
 [847] "51343"     "51361"     "51460"     "51465"     "51542"     "51665"    
 [853] "51742"     "51807"     "53340"     "54106"     "54361"     "54407"    
 [859] "54457"     "54466"     "54585"     "54851"     "54852"     "54888"    
 [865] "54993"     "55064"     "55120"     "55124"     "55231"     "55329"    
 [871] "55342"     "55366"     "55585"     "55636"     "55706"     "55723"    
 [877] "55811"     "55815"     "55818"     "55840"     "55870"     "55905"    
 [883] "56163"     "56729"     "56848"     "56956"     "57054"     "57055"    
 [889] "57095"     "57097"     "57135"     "57178"     "57599"     "57728"    
 [895] "57731"     "57828"     "59272"     "59338"     "59343"     "59350"    
 [901] "63894"     "63946"     "63978"     "64224"     "64321"     "64395"    
 [907] "64396"     "64478"     "64591"     "64645"     "64847"     "64848"    
 [913] "65010"     "65110"     "79582"     "79727"     "79747"     "79820"    
 [919] "79893"     "79969"     "79977"     "79989"     "80000"     "80025"    
 [925] "81539"     "81616"     "81623"     "81671"     "81892"     "81930"    
 [931] "83700"     "83890"     "83943"     "84056"     "84132"     "84152"    
 [937] "84159"     "84168"     "84172"     "84215"     "84221"     "84687"    
 [943] "84688"     "84694"     "84812"     "84930"     "85315"     "85413"    
 [949] "85417"     "89766"     "89887"     "91746"     "91978"     "93649"    
 [955] "113746"    "114112"    "115948"    "116369"    "117154"    "120892"   
 [961] "121355"    "122042"    "122258"    "122664"    "124404"    "125972"   
 [967] "128637"    "130497"    "132243"    "132612"    "133558"    "135138"   
 [973] "135935"    "139886"    "143471"    "143689"    "144195"    "144535"   
 [979] "146310"    "146852"    "147912"    "148229"    "148327"    "149685"   
 [985] "150159"    "151056"    "151195"    "151449"    "152006"    "152586"   
 [991] "157506"    "157777"    "158062"    "161829"    "164091"    "164395"   
 [997] "164684"    "166378"    "169981"    "170690"    "192670"    "199720"   
[1003] "200232"    "200373"    "202051"    "203523"    "204474"    "219670"   
[1009] "219793"    "219938"    "221656"    "221823"    "222698"    "225689"   
[1015] "259266"    "261734"    "283471"    "283629"    "283677"    "284338"   
[1021] "285335"    "286128"    "286234"    "317719"    "338879"    "339906"   
[1027] "340784"    "344758"    "353189"    "373863"    "387712"    "388799"   
[1033] "389730"    "389761"    "389762"    "389763"    "431707"    "440699"   
[1039] "440822"    "441452"    "449520"    "474343"    "619556"    "642623"   
[1045] "642636"    "644150"    "644890"    "645832"    "645961"    "646480"   
[1051] "647060"    "727830"    "727905"    "728132"    "728137"    "728395"   
[1057] "728403"    "729967"    "100130958" "100289087" "102724560" "1396"     
[1063] "1594"      "2626"      "2627"      "3714"      "5310"      "6092"     
[1069] "6299"      "6586"      "7022"      "7482"      "8433"      "8510"     
[1075] "8644"      "8909"      "9353"      "10046"     "10361"     "10732"    
[1081] "22803"     "22873"     "26292"     "29127"     "54456"     "56977"    
[1087] "84073"     "116832"    "132625"    "138474"    "140732"    "140801"   
[1093] "3624"      "4762"      "5810"      "6627"      "140947"    "497189"   
[1099] "226"       "708"       "1672"      "1767"      "2013"      "2295"     
[1105] "2529"      "2697"      "3479"      "6658"      "6691"      "6926"     
[1111] "7076"      "7417"      "7422"      "8701"      "8890"      "8892"     
[1117] "8893"      "8912"      "9083"      "9898"      "10265"     "10699"    
[1123] "10734"     "10916"     "23639"     "25790"     "25836"     "25981"    
[1129] "27019"     "27161"     "51673"     "51759"     "55036"     "55743"    
[1135] "55779"     "57119"     "57122"     "57697"     "60675"     "64220"    
[1141] "79816"     "80314"     "84074"     "84660"     "84733"     "85360"    
[1147] "89876"     "122402"    "131118"    "144132"    "144406"    "146845"   
[1153] "148281"    "150921"    "151648"    "154197"    "159686"    "199223"   
[1159] "219990"    "253943"    "286464"    "338323"    "339829"    "346288"   
[1165] "347688"    "377630"    "378948"    "401024"    "402573"    "406950"   
[1171] "442867"    "442868"    "100506013" "109"       "142"       "351"      
[1177] "406"       "583"       "585"       "658"       "676"       "1051"     
[1183] "1080"      "1435"      "1525"      "1811"      "1843"      "1958"     
[1189] "2054"      "2069"      "2263"      "2625"      "2649"      "2678"     
[1195] "2712"      "2801"      "2879"      "3622"      "3953"      "3975"     
[1201] "4036"      "4184"      "4192"      "4603"      "4753"      "4867"     
[1207] "4878"      "5049"      "5104"      "5125"      "5127"      "5350"     
[1213] "5414"      "5566"      "5764"      "5798"      "5872"      "5950"     
[1219] "5990"      "6098"      "6652"      "6665"      "6774"      "6901"     
[1225] "6943"      "7042"      "7584"      "7589"      "7917"      "8085"     
[1231] "8195"      "8243"      "8320"      "8372"      "8382"      "9025"     
[1237] "9421"      "9425"      "9575"      "9633"      "9665"      "9698"     
[1243] "9702"      "9940"      "10038"     "10409"     "10420"     "10519"    
[1249] "10733"     "11020"     "11063"     "11077"     "22887"     "23139"    
[1255] "23198"     "23213"     "23236"     "23315"     "23318"     "23617"    
[1261] "23627"     "26038"     "26091"     "26140"     "26330"     "27120"    
[1267] "28981"     "29118"     "29947"     "30812"     "51441"     "51547"    
[1273] "51574"     "51668"     "51804"     "54760"     "54890"     "54937"    
[1279] "55063"     "55726"     "55810"     "56262"     "56339"     "56603"    
[1285] "56776"     "57721"     "58494"     "63979"     "64147"     "64207"    
[1291] "64518"     "78995"     "79173"     "79645"     "79670"     "79733"    
[1297] "81492"     "81629"     "83447"     "83853"     "83942"     "83983"    
[1303] "84072"     "84236"     "84515"     "84519"     "84678"     "84691"    
[1309] "90410"     "90853"     "91603"     "94107"     "126206"    "127579"   
[1315] "133308"    "136332"    "136991"    "143678"    "144455"    "147700"   
[1321] "149095"    "150280"    "151254"    "153218"    "157680"    "160762"   
[1327] "161142"    "161514"    "161931"    "162540"    "162979"    "164714"   
[1333] "170506"    "200162"    "200558"    "201164"    "202500"    "203074"   
[1339] "221481"    "245711"    "254394"    "255626"    "256006"    "256710"   
[1345] "326340"    "338773"    "341277"    "341567"    "346653"    "375189"   
[1351] "375341"    "388336"    "388553"    "391714"    "399949"    "402381"   
[1357] "431705"    "441161"    "642658"    "643376"    "646799"    "730249"   
[1363] "101927581" "56"        "247"       "259"       "283"       "996"      
[1369] "1081"      "1394"      "1538"      "2302"      "2692"      "4117"     
[1375] "5069"      "5670"      "5675"      "5885"      "6013"      "7356"     
[1381] "7455"      "8605"      "8697"      "8881"      "9130"      "10393"    
[1387] "10658"     "10983"     "23742"     "23780"     "25847"     "25906"    
[1393] "26256"     "29882"     "29945"     "51433"     "51434"     "51529"    
[1399] "55339"     "56648"     "56853"     "57082"     "57446"     "64682"    
[1405] "79400"     "80237"     "84203"     "84750"     "93185"     "113451"   
[1411] "116138"    "117285"    "119504"    "219771"    "246184"    "256126"   
[1417] "344018"    "406991"    "441531"    "728695"    "100137049" "116"      
[1423] "841"       "983"       "1017"      "1082"      "1307"      "1586"     
[1429] "1833"      "2491"      "2834"      "3293"      "3593"      "3620"     
[1435] "3972"      "4012"      "4486"      "4543"      "5073"      "5467"     
[1441] "5568"      "5571"      "5669"      "5671"      "5672"      "5673"     
[1447] "5676"      "5678"      "5680"      "5737"      "5744"      "5858"     
[1453] "5890"      "5987"      "6019"      "6046"      "6159"      "6696"     
[1459] "6732"      "6863"      "6866"      "7005"      "7156"      "7216"     
[1465] "7516"      "7812"      "7932"      "7993"      "8031"      "8239"     
[1471] "8287"      "8607"      "8900"      "8999"      "9082"      "9085"     
[1477] "9125"      "9183"      "9210"      "9426"      "9463"      "10007"    
[1483] "10017"     "10149"     "10343"     "10406"     "10407"     "10522"    
[1489] "10566"     "10609"     "10766"     "10863"     "10876"     "23764"    
[1495] "26476"     "26609"     "26664"     "30014"     "51207"     "51686"    
[1501] "53405"     "80705"     "94027"     "203611"    "253175"    "728712"   

$`GO:0000009`
[1] "79087" "55650"

$`GO:0000010`
[1] "23590" "57107"

$`GO:0000012`
 [1] "54840"     "55775"     "1161"      "2074"      "3981"      "7141"     
 [7] "7515"      "100133315" "142"       "23411"     "200558"    "7014"     

$`GO:0000014`
 [1] "2021"  "2072"  "4361"  "6419"  "9941"  "2067"  "10111" "64421" "7515" 
[10] "5932"

Using lapply and sapply

If we want to know the number of genes for each gene set.

ngs = lapply(hsa_go,length)

What is ngs?

class(ngs)
[1] "list"

Probably a better choice to calculate the lengths would be

ngs = sapply(hsa_go,length)

Now ngs is

class(ngs)
[1] "integer"

A summary of the lengths is

summary(ngs)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    1.0     2.0     6.0    90.3    24.0 19518.0

Perhaps a kernel density estimator could show the gene set length distribution.

df = data.frame(ngs)
ggplot(data=df,aes(x=ngs)) + geom_density()