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glmmSeq

glmmSeq logo

The aim of this package is to model gene expression with a general linear mixed model (GLMM) as described in the R4RA study [1]. The most widely used mainstream differential gene expression analysis tools (e.g Limma, DESeq2, edgeR) are all unable to fit mixed effects linear models. This package however fits negative binomial mixed effects models at individual gene level using the negative.binomial function from MASS and the glmer function in lme4 which enables random effect, as as well as mixed effects, to be modelled.

Installing from CRAN

install.packages("glmmSeq")

Installing from Github

devtools::install_github("myles-lewis/glmmSeq")

Installing Locally

Or you can source the functions individually:

functions = list.files("./R", full.names = TRUE)
invisible(lapply(functions, source))

But you will need to load in the additional libraries:

# Install CRAN packages
invisible(lapply(c("MASS", "car", "ggplot2", "ggpubr", "lme4","lmerTest",
                   "methods", "parallel", "plotly", "pbapply", "pbmcapply"),
                 function(p){
                   if(! p %in% rownames(installed.packages())) {
                     install.packages(p)
                   }
                   library(p, character.only=TRUE)
                 }))

# Install BioConductor packages
if (!requireNamespace("BiocManager", quietly = TRUE))
  install.packages("BiocManager")
invisible(lapply(c("qvalue"), function(p){
  if(! p %in% rownames(installed.packages())) BiocManager::install(p)
  library(p, character.only=TRUE)
}))

Overview

To get started, first we load in the package:

library(glmmSeq)
set.seed(1234)

This vignette will demonstrate the power of this package using a minimal example from the PEAC data set. Here we focus on the synovial biopsy RNA-Seq data from this cohort of patients with early rheumatoid arthritis.

data(PEAC_minimal_load)

This data contains:

  • metadata: which describes each sample. Including patient ID, sample time-point, and six-month EULAR response. Where EULAR is a rheumatoid arthritis response metric based on composite DAS28 scores.
  • tpm: the transcript per million RNA-seq count data

These are outlined in the following subsections.

Metadata

metadata$EULAR_binary  = NA
metadata$EULAR_binary[metadata$EULAR_6m %in%
                        c("Good", "Moderate" )] = "responder"
metadata$EULAR_binary[metadata$EULAR_6m %in% c("Non-response")] = "non_responder"

kable(head(metadata), row.names = F) %>% kable_styling()
PATID Timepoint EULAR_6m EULAR_binary
PAT300 6 Good responder
PAT209 6 Good responder
PAT219 6 Moderate responder
PAT211 6 Good responder
PAT216 6 Good responder
PAT212 6 Good responder

Count data

kable(head(tpm)) %>% kable_styling() %>%
  scroll_box(width = "100%")
S9001 S9002 S9003 S9004 S9006 S9007 S9008 S9009 S9010 S9011 S9012 S9013 S9014 S9016 S9017 S9018 S9019 S9020 S9021 S9023 S9025 S9029 S9034 S9035 S9036 S9038 S9039 S9040 S9042 S9043 S9044 S9045 S9047 S9048 S9049 S9052 S9053 S9054 S9056 S9059 S9060 S9063 S9065 S9066 S9067 S9068 S9069 S9070 S9072 S9073 S9074 S9075 S9076 S9077 S9078 S9079 S9080 S9081 S9083 S9084 S9085 S9086 S9087 S9088 S9089 S9090 S9091 S9092 S9093 S9094 S9095 S9096 S9097 S9098 S9099 S9100 S9101 S9102 S9103 S9104 S9105 S9106 S9107 S9108 S9109 S9110 S9111 S9112 S9113 S9114 S9115 S9116 S9117 S9119 S9120 S9121 S9122 S9123 S9124 S9125 S9127 S9128 S9129 S9130 S9131 S9132 S9133 S9134 S9135 S9136 S9137 S9138 S9139 S9140 S9141 S9142 S9143 S9144 S9145 S9146 S9147 S9148 S9149
MS4A1 1.622818 2.024451 0.6995153 15.742876 2.123731 57.280078 2.011160 0.580000 1.006783 0.391420 3.886618 2.746742 31.647637 20.16757 0.925393 3.4271907 3.130648 1.422480 63.14643 0.295537 12.7877279 4.817494 15.874491 0.569317 4.429667 1.060991 2.1850700 5.321541 3.542383 1.676829 5.471954 29.942195 4.5519540 7.252108 1.9540320 12.947362 73.430132 2.364819 5.2372870 12.281007 17.637846 72.557856 2.379427 0.444729 89.042394 2.4190970 226.612930 8.01254 5.328680 0.7414830 4.669566 5.218332 0.0676825 26.0579080 77.100135 23.2918620 50.306168 5.24953 1.342645 0.511810 0.2428008 7.756002 1.020052 0.602854 0.2444728 2.047680 2.110914 0.683498 24.664095 1.9789030 0.1702800 3.910493 6.5513870 0.8342158 31.7618280 1.136854 17.98745 3.369558 8.577294 1.453945 41.031738 42.991580 27.58865 14.876902 21.6405734 42.049748 75.6557000 2.501497 3.560322 32.1091249 7.008288 0.746264 10.0969886 92.3355460 0.5157370 0.8011538 5.075003 1.979931 0.2068106 2.717007 0.984857 0.7771422 0.000000 8.2769093 1.261793 2.361150 2.372850 0.6968002 43.9459380 5.0257346 0.281387 0.4190982 1.716238 16.25508 1.9852750 26.591585 27.994183 1.325606 1.323197 12.47092 2.3049434 1.4901770 0.229867
ADAM12 2.581805 1.110548 0.9573992 5.466742 2.645153 0.972897 0.658545 1.159264 0.462860 3.074193 10.947068 9.678026 1.205472 11.63376 1.772810 0.3506414 3.528288 17.416140 6.36844 4.064450 0.2666618 2.113812 0.473972 1.222569 30.012798 0.181003 2.4554526 4.136035 13.961315 4.195791 3.629029 8.362213 0.6999296 7.894137 0.3268513 2.847538 7.015717 0.717947 0.4396555 1.141057 1.069326 1.895644 0.486478 14.504778 1.897848 0.9943284 4.140489 41.30644 8.812227 0.3317777 8.699836 8.323936 0.0292791 0.9615935 2.512841 0.8960788 3.766372 7.13890 1.917924 8.893462 0.0966324 1.550900 4.760995 1.769894 4.6901815 1.200418 1.392691 0.439077 0.658254 0.3610814 0.2501964 0.610416 0.6842673 2.8749874 0.6717734 16.374299 1.36295 8.725470 0.258386 0.214470 3.573125 2.086027 13.44347 4.007794 0.0338254 1.595552 0.9645215 1.913151 5.780085 0.7098401 5.242844 1.115879 0.6499524 0.5397263 0.2299409 0.7178614 0.535170 0.282374 0.2003997 2.202211 0.310417 2.0991998 0.215329 0.1962535 25.010318 2.970828 1.461817 2.6506093 0.6501276 0.9633052 8.715446 32.4636380 4.281851 0.83041 0.9840191 2.363019 0.314493 1.179129 0.656344 1.57062 0.4597666 0.4707508 4.727155
IGHV7-4-1 0.992885 0.000000 2.2060400 26.381700 0.000000 2158.300000 0.644889 0.000000 0.000000 1.500990 4.945980 38.268000 55.570100 107.91600 0.000000 0.0000000 0.000000 1.080940 27.51650 0.000000 21.4136000 6.121120 15.190300 0.000000 0.597722 0.000000 0.0844527 1228.620000 51.561100 1.149330 1.434460 3728.700000 6.3708500 1.553570 0.0000000 165.928000 30.714000 0.000000 5.5235500 0.770965 1603.290000 53.605000 16.063900 6.209400 1021.480000 0.2965720 85.466200 9.18953 41.191800 5.0514200 47.140500 42.541900 0.4660800 835.2970000 1929.760000 57.2972000 39.990700 0.00000 0.000000 0.411695 0.0000000 26.176100 1.033220 2.273400 0.0000000 11.443300 0.272802 0.000000 81.135100 0.7083360 0.0000000 1.298760 0.5145140 0.1260250 13.9310000 2.345910 77.63540 2.181890 63.096300 0.000000 6.473680 20.211900 347.99200 39.273300 2.3200600 584.705000 2147.2900000 242.387000 14.263100 52.4861000 0.399860 0.000000 0.7025670 11.2577000 0.6126850 4.5281300 155.926000 320.086000 1.3791200 1.587010 0.167711 0.0000000 0.000000 5.6540200 0.000000 0.000000 0.560530 1.2747600 50.6122000 5.3320100 0.000000 0.0000000 0.757454 132.98900 3.4184700 18.130100 3.085550 1.183940 0.277095 135.99100 0.7799880 0.6956290 0.000000
IGHV3-49 0.000000 0.196831 8.7892000 345.493000 2.093480 858.980000 3.201300 0.000000 0.000000 0.000000 114.334000 17.150200 136.605000 1080.73000 0.196674 0.0000000 0.000000 0.931275 140.22000 0.000000 202.4950000 60.114600 145.421000 0.145099 32.808500 1.207290 3.5942600 467.534000 55.301200 0.869369 14.462700 928.660000 64.4575000 0.430536 0.0000000 80.024800 53.191800 0.000000 35.4998000 0.303541 1002.870000 127.201000 634.856000 22.119900 975.855000 0.0000000 623.330000 129.50800 180.819000 29.8933000 183.605000 170.369000 0.2201470 403.4460000 384.597000 24.9283000 426.619000 1.08364 0.591472 4768.210000 0.0000000 93.669200 9.291720 0.607172 0.0000000 9.736200 1.132510 0.000000 175.472000 1.0305600 0.0885169 0.972741 12.5531000 0.8804760 163.1520000 1.764370 37.22090 15.345300 401.861000 0.575801 11.926900 45.198600 317.65400 523.093000 171.4520000 174.163000 5414.9000000 5.045300 386.829000 1190.5200000 4.854510 0.169181 15.7114000 415.2760000 0.1668720 0.1950730 34.623200 26.949400 2.3880600 2.021500 0.000000 0.0000000 0.000000 1.5027300 0.000000 1.783170 49.278100 1.1861700 99.7745000 10.0077000 0.000000 0.1908660 3.835610 22.31070 2.4220200 22.133600 440.657000 0.000000 0.000000 22.10260 5.3278400 0.5799090 0.210089
IGHV3-23 0.715133 3.999940 87.6508000 2349.850000 5.962460 5180.460000 17.278900 0.000000 0.000000 1.437710 985.721000 123.982000 1374.860000 3568.60000 9.857240 0.5625450 2.900430 2.172310 5859.05000 0.000000 502.8400000 366.704000 1793.510000 2.294430 97.123900 11.607200 8.1870600 1080.330000 411.058000 9.718060 114.608000 2306.720000 317.0130000 7.954070 2.7908400 865.088000 3563.990000 0.123914 23.4808000 4.896920 1681.420000 1929.860000 2241.850000 26.191600 5228.450000 4.1899600 2807.770000 1213.59000 387.839000 203.5700000 374.506000 374.388000 0.3392100 1359.9900000 1535.600000 118.7600000 2708.140000 4.59234 0.607162 93.584600 0.3873680 1273.530000 1.666060 5.899820 1.8143100 18.921500 17.654300 0.567110 1593.270000 0.7579240 0.2794190 6.994180 216.3070000 4.8487200 1286.5300000 12.741600 167.65000 115.720000 2142.700000 3.020540 79.259200 6471.240000 2508.41000 4077.530000 802.0710000 1884.330000 5188.7400000 296.555000 1696.260000 3027.5700000 215.969000 0.249956 79.7680000 338.8720000 2.2993700 5.4541000 214.318000 152.085000 9.7046700 11.809000 0.300002 0.0000000 0.000000 15.5774000 0.217214 4.113890 229.482000 14.7361000 1145.9400000 35.9383000 0.613331 5.6297400 4.861760 188.38500 94.7553000 297.408000 3505.310000 19.762300 0.860007 208.00400 54.3620000 52.5728000 0.785236
ADAM12.1 2.581805 1.110548 0.9573992 5.466742 2.645153 0.972897 0.658545 1.159264 0.462860 3.074193 10.947068 9.678026 1.205472 11.63376 1.772810 0.3506414 3.528288 17.416140 6.36844 4.064450 0.2666618 2.113812 0.473972 1.222569 30.012798 0.181003 2.4554526 4.136035 13.961315 4.195791 3.629029 8.362213 0.6999296 7.894137 0.3268513 2.847538 7.015717 0.717947 0.4396555 1.141057 1.069326 1.895644 0.486478 14.504778 1.897848 0.9943284 4.140489 41.30644 8.812227 0.3317777 8.699836 8.323936 0.0292791 0.9615935 2.512841 0.8960788 3.766372 7.13890 1.917924 8.893462 0.0966324 1.550900 4.760995 1.769894 4.6901815 1.200418 1.392691 0.439077 0.658254 0.3610814 0.2501964 0.610416 0.6842673 2.8749874 0.6717734 16.374299 1.36295 8.725470 0.258386 0.214470 3.573125 2.086027 13.44347 4.007794 0.0338254 1.595552 0.9645215 1.913151 5.780085 0.7098401 5.242844 1.115879 0.6499524 0.5397263 0.2299409 0.7178614 0.535170 0.282374 0.2003997 2.202211 0.310417 2.0991998 0.215329 0.1962535 25.010318 2.970828 1.461817 2.6506093 0.6501276 0.9633052 8.715446 32.4636380 4.281851 0.83041 0.9840191 2.363019 0.314493 1.179129 0.656344 1.57062 0.4597666 0.4707508 4.727155

Dispersion

Using negative binomial models requires gene dispersion estimates to be made. This can be achieved in a number of ways. A common way to calculate this for gene i is to use the equation:

Dispersioni = (variancei - meani)/meani2

This can be calculated using:

disp <- apply(tpm, 1, function(x){
  (var(x, na.rm=TRUE)-mean(x, na.rm=TRUE))/(mean(x, na.rm=TRUE)**2)
  })

head(disp)
##     MS4A1    ADAM12 IGHV7-4-1  IGHV3-49  IGHV3-23  ADAM12.1 
##  3.789428  2.391912 11.686420 10.863156  3.262557  2.391912

Alternatively, we recommend using edgeR to estimate of the common, trended and tagwise dispersions across all tags:

disp  <- setNames(edgeR::estimateDisp(tpm)$tagwise.dispersion, rownames(tpm))

head(disp)

or with DESeq2 using the raw counts:

dds <- DESeqDataSetFromTximport(txi = txi, colData = metadata, design = ~ 1)
dds <- DESeq(dds)
dispersions <- setNames(dispersions(dds), rownames(txi$counts))

Size Factors

There is also an option to include size factors for each gene. Again this can be estimated using:

sizeFactors <- colSums(tpm)  
sizeFactors <- sizeFactors / mean(sizeFactors)  # normalise to mean = 1

head(sizeFactors)
##      S9001      S9002      S9003      S9004      S9006      S9007 
## 0.20325747 0.09330435 0.09737100 3.13456671 0.24515015 4.19419297

Or using edgeR these can be calculated from the raw read counts:

sizeFactors <- calcNormFactors(counts, method="TMM")

Similarly, with DESeq2:

sizeFactors <- estimateSizeFactorsForMatrix(counts)

Note the sizeFactors vector needs to be centred around 1, since it used directly as an offset of form log(sizeFactors) in the GLMM model.

Fitting Models

To fit a model for one gene over time we use a formula such as:

gene expression ~ fixed effects + random effects

In R the formula is defined by both the fixed-effects and random-effects part of the model, with the response on the left of a ~ operator and the terms, separated by + operators, on the right. Random-effects terms are distinguished by vertical bars (“|”) separating expressions for design matrices from grouping factors. For more information see the ?lme4::glmer.

In this case study we want to use time and response as fixed effects and the patients as random effects:

gene expression ~ time + response + (1 | patient)

To fit this model for all genes we can use the glmmSeq function. Note that this analysis can take some time, with 4 cores:

  • 1000 genes takes about 10 seconds
  • 20000 genes takes about 4 mins
results <- glmmSeq(~ Timepoint * EULAR_6m + (1 | PATID),
                   countdata = tpm,
                   metadata = metadata,
                   dispersion = disp,
                   progress = TRUE)
## 
## n = 123 samples, 82 individuals
## Time difference of 3.152688 secs
## Errors in 4 gene(s): IL12A, FGF12, VIL1, IL26

or alternatively using two-factor classification with EULAR_binary:

results2 <- glmmSeq(~ Timepoint * EULAR_binary + (1 | PATID),
                    countdata = tpm,
                    metadata = metadata,
                    dispersion = disp)
## 
## n = 123 samples, 82 individuals
## Time difference of 3.016724 secs
## Errors in 4 gene(s): IL12A, FGF12, VIL1, IL26

Outputs

This creates a GlmmSeq object which contains the following slots:

names(attributes(results))
##  [1] "info"      "formula"   "stats"     "predict"   "reduced"   "countdata" "metadata" 
##  [8] "modelData" "optInfo"   "errors"    "vars"      "class"

The variables used by the model are in the @modeldata:

kable(results@modelData) %>% kable_styling()
Timepoint EULAR_6m
0 Good
6 Good
0 Moderate
6 Moderate
0 Non-response
6 Non-response

The model fit statistics can be viewed in the @stats slot which is a list of items including fitted model coefficients, their standard errors and the results of statistical tests on terms within the model using Wald type 2 Chi square. To see the most significant interactions we can order pvals by Timepoint:EULAR_6m:

stats <- summary(results)

kable(stats[order(stats[, 'P_Timepoint:EULAR_6m']), ]) %>%
  kable_styling() %>%
  scroll_box(width = "100%", height = "400px")
Dispersion AIC logLik meanExp (Intercept) Timepoint EULAR_6mModerate EULAR_6mNon-response Timepoint:EULAR_6mModerate Timepoint:EULAR_6mNon-response se_(Intercept) se_Timepoint se_EULAR_6mModerate se_EULAR_6mNon-response se_Timepoint:EULAR_6mModerate se_Timepoint:EULAR_6mNon-response Chisq_Timepoint Chisq_EULAR_6m Chisq_Timepoint:EULAR_6m Df_Timepoint Df_EULAR_6m Df_Timepoint:EULAR_6m P_Timepoint P_EULAR_6m P_Timepoint:EULAR_6m
IGHV3-23 3.2625571 1587.9550 -785.9775 5.9461793 6.8466473 -0.0213958 0.1369681 -0.8391523 -0.3010405 -0.4047369 0.4783325 0.0906407 0.5134806 0.6320099 0.1277461 0.1624190 7.6175338 14.2631824 8.8418654 1 2 2 0.0057803 0.0007994 0.0120230
CXCL13 4.4416343 953.3999 -468.7000 2.8488723 3.5376946 -0.0405699 0.4921167 -1.4651369 -0.3181048 0.0958172 0.3701902 0.0971787 0.5619716 0.7302593 0.1464648 0.1781329 4.2900227 4.3648596 6.7344833 1 2 2 0.0383367 0.1127672 0.0344846
FGF14 0.2915518 375.0390 -179.5195 1.0784074 -0.0846561 0.1148791 -0.0436742 0.9233238 -0.0035230 -0.1731267 0.2014575 0.0467465 0.3090738 0.3176555 0.0711392 0.0783922 5.5445597 5.0815197 5.6637042 1 2 2 0.0185382 0.0788065 0.0589037
IL24 0.4177215 376.8418 -180.4209 1.0076111 0.3478977 -0.0321854 0.2217562 -0.7615200 -0.1166353 0.1298070 0.1978093 0.0500220 0.2721124 0.4427622 0.0793379 0.1003295 1.9439310 1.0333073 5.5708454 1 2 2 0.1632423 0.5965133 0.0617030
HHIP 1.0353883 537.1812 -260.5906 1.4454128 0.7998605 -0.0399783 -0.1533630 0.5663556 0.1881176 0.0332220 0.2089737 0.0569350 0.3219344 0.3888552 0.0826903 0.0975405 1.0348345 4.7912317 5.5680271 1 2 2 0.3090259 0.0911165 0.0617900
MS4A1 3.7894278 898.6829 -441.3414 2.5917399 2.7363666 0.0490839 0.1933006 -1.7237593 -0.2107974 0.1746154 0.3366868 0.0894115 0.5110345 0.6778010 0.1347788 0.1655778 0.0112466 5.6828754 5.4378747 1 2 2 0.9155427 0.0583417 0.0659448
ADAM12 2.3919119 635.7162 -309.8581 1.6675082 1.6569079 -0.1236820 -0.2683597 -0.5047247 -0.0347932 0.2699943 0.2756082 0.0750725 0.4213674 0.5491941 0.1146527 0.1351883 2.5418570 2.3034252 5.2104307 1 2 2 0.1108643 0.3160950 0.0738872
ADAM12.1 2.3919119 635.7162 -309.8581 1.6675082 1.6569079 -0.1236820 -0.2683597 -0.5047247 -0.0347932 0.2699943 0.2756082 0.0750725 0.4213674 0.5491941 0.1146527 0.1351883 2.5418570 2.3034252 5.2104307 1 2 2 0.1108643 0.3160950 0.0738872
IL2RG 0.8332758 1077.5836 -530.7918 4.3589131 3.4913720 -0.0544559 0.2592216 -0.6099765 -0.1005384 0.0741224 0.1768209 0.0432425 0.2498525 0.3251603 0.0650989 0.0795735 6.8219047 3.0208070 4.9773656 1 2 2 0.0090046 0.2208209 0.0830192
IL16 0.2419416 974.1912 -479.0956 4.5153510 3.2821832 -0.0163058 -0.1351425 -0.3377962 0.0421528 0.0983179 0.0936429 0.0242321 0.1389855 0.1814160 0.0364868 0.0447041 1.0986627 0.2976437 4.9709614 1 2 2 0.2945598 0.8617226 0.0832855
EMILIN3 2.8558506 522.1530 -253.0765 1.2381694 1.0154229 0.1211976 -2.0741731 -0.8821688 0.2905788 0.0936821 0.3076528 0.0803498 0.5615745 0.6368489 0.1317965 0.1513721 15.7455688 9.3862145 4.8709355 1 2 2 0.0000725 0.0091582 0.0875568
BLK 1.0360018 532.3068 -258.1534 1.4610649 0.9405211 0.0152304 0.2889931 -0.6186534 -0.1477598 0.0407172 0.2366609 0.0545010 0.3164350 0.4438540 0.0838780 0.1061424 0.6098286 2.2049350 4.1711539 1 2 2 0.4348524 0.3320507 0.1242354
IGHV1-69 6.1631087 1150.4929 -567.2465 3.6446738 4.5052838 0.0254377 1.0678755 0.3278992 -0.1788186 -0.3840284 0.4260744 0.1132977 0.6470204 0.8341665 0.1701410 0.2070231 2.1903776 4.5563275 3.6032048 1 2 2 0.1388753 0.1024722 0.1650342
PADI4 2.0736630 472.8051 -228.4025 1.0912063 0.2320863 0.1335275 0.3673813 0.1562910 -0.2084779 -0.1075547 0.2903619 0.0735494 0.4303320 0.5600252 0.1128728 0.1359462 0.6800223 0.2066687 3.4338579 1 2 2 0.4095790 0.9018254 0.1796169
LILRA5 0.8432028 787.5680 -385.7840 2.8996321 2.1304753 -0.0337206 0.3077720 -0.0538474 -0.0772630 0.0741083 0.1682070 0.0451020 0.2531455 0.3301611 0.0677629 0.0815717 2.3877870 0.6639482 3.3402241 1 2 2 0.1222866 0.7175059 0.1882260
IGHV5-10-1 5.5424948 1103.9302 -543.9651 3.3881684 4.8972304 -0.0222759 0.3631280 -0.0880610 -0.2792955 -0.0040168 0.6691549 0.1111489 0.6495654 0.8664366 0.1672584 0.2171123 2.8322683 0.3035545 3.0813079 1 2 2 0.0923878 0.8591797 0.2142410
IGHV3OR16-8 4.4604808 728.3564 -356.1782 1.8706896 2.9350507 -0.1123000 -0.2194363 -0.3427264 -0.1967506 0.1245432 0.4522157 0.0999647 0.5628228 0.7216137 0.1486832 0.1890461 5.6767433 2.1920958 3.0625257 1 2 2 0.0171912 0.3341892 0.2162624
PHOSPHO1 1.5924621 697.7595 -340.8798 2.1552363 1.4569268 0.0953195 0.0654148 0.0650941 -0.1432879 -0.0867006 0.2317083 0.0605645 0.3511832 0.4523094 0.0922446 0.1113749 0.4904949 1.0829818 2.4703240 1 2 2 0.4837066 0.5818801 0.2907876
S100B 3.9902943 1068.8305 -526.4153 3.4429454 2.6698463 -0.0027976 0.4409051 -0.0676109 -0.1150445 0.1545318 0.3455381 0.0918899 0.5240035 0.6768250 0.1380280 0.1673433 0.0514070 0.6091648 2.4356546 1 2 2 0.8206329 0.7374313 0.2958723
IGHV2-70D 6.4682516 857.1816 -420.5908 2.3395373 3.7263964 -0.1188104 0.3457869 -0.8728989 -0.2362598 -0.2549742 0.4369557 0.1163299 0.6635831 0.8571720 0.1751114 0.2152755 10.2774050 5.8449029 2.3829431 1 2 2 0.0013467 0.0538016 0.3037739
PPIL4 0.1705260 779.2111 -381.6056 3.2865169 2.2892504 -0.0074724 -0.0643473 -0.0248330 0.0481026 0.0498523 0.0961397 0.0244635 0.1450737 0.1868843 0.0362900 0.0440609 1.5034278 0.5850044 2.2272561 1 2 2 0.2201447 0.7463936 0.3283655
IGHJ3P 4.7990172 1085.2659 -534.6329 3.4861371 4.9831232 -0.0810362 -0.0437042 -1.6508369 -0.2109033 -0.0303276 0.3759413 0.0999966 0.5711055 0.7377189 0.1503431 0.1829906 5.8423701 9.6857792 2.1043333 1 2 2 0.0156447 0.0078842 0.3491804
IGHV7-4-1 11.6864201 1101.3138 -542.6569 3.0586094 5.3345103 -0.0825370 0.3334439 -2.6497953 -0.2951205 -0.3087896 0.5861759 0.1559168 0.8904806 1.1502771 0.2342605 0.2878086 5.5348827 16.9828417 2.0270663 1 2 2 0.0186410 0.0002052 0.3629344
LILRB3 0.4876740 945.3343 -464.6671 3.9474519 2.8953372 -0.0258468 0.2114848 -0.1435780 -0.0203999 0.0663273 0.1263695 0.0337764 0.1909106 0.2488511 0.0505310 0.0614439 0.8158759 1.2121573 1.9058332 1 2 2 0.3663887 0.5454857 0.3856147
IL36RN 32.6380418 500.2509 -242.1254 0.3962182 0.9057894 -0.3373192 -1.8591000 -1.0185519 0.5013854 0.0103691 0.9858035 0.2684523 1.5254092 1.9455086 0.4019578 0.5027249 0.7092169 0.6065049 1.7620973 1 2 2 0.3997039 0.7384127 0.4143482
IGHV3OR16-12 12.1178160 539.6728 -261.8364 0.8508034 1.5129747 -0.0122481 -0.6073245 -0.9450944 -0.3231954 -0.1338251 0.6024029 0.1602493 0.9189728 1.1916279 0.2498545 0.2984661 1.7312000 4.0960562 1.6732410 1 2 2 0.1882577 0.1289890 0.4331720
IGHV5-78 5.5798767 569.8859 -276.9429 1.1452612 1.0630323 0.0397579 -0.0398071 0.6148506 -0.1990246 -0.1458526 0.4174605 0.1105884 0.6345932 0.8092152 0.1691994 0.2013966 0.5850495 1.7295021 1.4883376 1 2 2 0.4443399 0.4211564 0.4751291
IGHV3-49 10.8631558 1271.1690 -627.5845 3.9774695 5.7635604 -0.1677969 -0.4705138 0.4518068 -0.2392902 -0.2599216 0.5652409 0.1503381 0.8587381 1.1067198 0.2259735 0.2743113 9.0576243 2.5620700 1.4849116 1 2 2 0.0026160 0.2777497 0.4759437
IGHV1OR15-4 4.1384009 737.1263 -360.5631 1.9435728 2.4925095 -0.0853148 0.8616507 -0.8866376 -0.1479020 0.0087225 0.3523523 0.0940646 0.5335425 0.6969554 0.1411093 0.1736246 4.7041753 6.3140910 1.3100148 1 2 2 0.0300894 0.0425513 0.5194382
IL2RA 1.1303114 545.7334 -264.8667 1.5164121 1.3234092 -0.1238205 0.0366304 -0.7791815 -0.0168110 0.1045126 0.2320175 0.0578359 0.3155440 0.4380156 0.0864420 0.1086981 8.0196412 2.5247446 1.2479947 1 2 2 0.0046273 0.2829819 0.5357984
EMILIN2 0.2149030 1022.9549 -503.4775 4.8723536 3.4282169 -0.0478693 0.1907900 0.2159081 0.0307185 0.0370946 0.0852933 0.0229787 0.1287306 0.1654155 0.0341139 0.0412203 3.7233248 9.7663063 1.1782527 1 2 2 0.0536574 0.0075731 0.5548118
CXCL11 4.3308929 558.9347 -271.4674 1.2009946 1.4333300 -0.0924072 -0.1022289 -1.4166330 -0.1214451 0.0706083 0.3665977 0.0986193 0.5577862 0.7597702 0.1509614 0.1891941 3.2760631 4.5591179 1.1209216 1 2 2 0.0702974 0.1023293 0.5709459
IGHV1OR21-1 7.6550929 357.7987 -170.8994 0.4791891 -0.7429909 0.1076602 1.1516429 -0.9438997 -0.2080242 -0.1730854 0.5357072 0.1379439 0.7790812 1.1726661 0.2039685 0.2979735 0.0006757 5.1554210 1.1160140 1 2 2 0.9792619 0.0759477 0.5723486
CXCL9 7.7483868 1111.5068 -547.7534 3.2358534 4.1831440 -0.1589647 -0.1505768 -1.9470604 -0.0796054 0.1738691 0.4778487 0.1271852 0.7259615 0.9396472 0.1911665 0.2327504 3.2587761 4.3825426 1.1034721 1 2 2 0.0710421 0.1117746 0.5759491
IL36G 55.6403102 592.4912 -288.2456 0.3500668 0.9983391 -0.3974202 -2.2326598 -1.8715276 0.5421133 0.1857025 1.2833185 0.3479235 1.9795707 2.5460467 0.5217738 0.6460743 0.5065694 0.7179775 1.0879014 1 2 2 0.4766277 0.6983822 0.5804505
SAA1 28.3095252 964.5999 -474.2999 1.7170262 3.3639676 -0.2882037 -1.3730549 -0.6174882 0.0034016 0.4195829 0.9130145 0.2432002 1.3883874 1.7883085 0.3665894 0.4431598 1.5819814 2.4808846 1.0261291 1 2 2 0.2084755 0.2892562 0.5986582
EAF2 0.5728653 724.8370 -354.4185 2.6574912 2.0987532 -0.0752533 0.2348372 -0.3563569 -0.0430362 0.0245403 0.1639929 0.0392425 0.2172068 0.2901849 0.0587958 0.0730532 10.6659261 3.5062734 0.9427034 1 2 2 0.0010913 0.1732297 0.6241580
IGHV7-27 14.5870583 478.5468 -231.2734 0.5865966 0.9619084 -0.4094644 -0.3547719 -0.9275209 0.0653310 0.3182143 0.6635349 0.1904674 1.0110864 1.3177309 0.2856316 0.3366545 6.4310429 0.0762873 0.9125184 1 2 2 0.0112143 0.9625747 0.6336496
CILP 3.4124368 1167.6923 -575.8462 3.9198405 3.5693809 0.1148879 -0.8956625 0.2112677 0.0958712 -0.0338922 0.4055484 0.0914020 0.5413721 0.6810569 0.1336378 0.1646109 5.5764552 2.8853109 0.7728507 1 2 2 0.0182035 0.2362994 0.6794815
IGHJ6 6.8375817 1218.9556 -601.4778 3.8318813 6.0208224 -0.0985782 -0.2121991 -0.0204935 -0.1566467 -0.1585515 0.9091690 0.1323640 0.8037477 1.0769099 0.2003718 0.2552463 2.4592121 0.9099440 0.7343401 1 2 2 0.1168374 0.6344657 0.6926919
FGFR2 4.6484751 484.1123 -234.0562 0.8968766 0.8562658 -0.0725830 -0.9483749 -1.1838893 0.1214805 0.1107619 0.3862808 0.1041300 0.6084195 0.8076027 0.1594249 0.1983454 0.0203936 3.1938868 0.6806884 1 2 2 0.8864434 0.2025146 0.7115254
IGLV4-3 12.6762412 578.4695 -281.2348 0.9284219 1.1862192 0.0748321 -0.4837409 -1.0268216 -0.1506973 -0.2101672 0.6179164 0.1638830 0.9425678 1.2285077 0.2483898 0.3082778 0.0235014 3.2907505 0.6188439 1 2 2 0.8781603 0.1929401 0.7338711
SAA2 59.8447169 871.9089 -427.9545 0.8633720 2.4771413 -0.3747888 -2.0207013 -1.8018215 0.1545014 0.3539452 1.3276104 0.3542967 2.0219601 2.6062629 0.5349296 0.6463979 1.1134331 1.1183830 0.3091668 1 2 2 0.2913369 0.5716711 0.8567720
FGFRL1 0.4980641 705.6527 -344.8264 2.5933500 1.8343691 0.0371586 -0.2538998 -0.0556572 0.0160323 0.0082694 0.1459866 0.0369363 0.2165394 0.2760841 0.0557414 0.0668920 3.1425845 1.7555806 0.0832211 1 2 2 0.0762729 0.4157005 0.9592433
FGF2 0.5313473 506.2867 -245.1434 1.5782689 0.6276046 0.0754127 0.1511292 0.0584818 0.0153345 0.0146872 0.1770025 0.0443837 0.2634760 0.3428858 0.0652600 0.0802985 8.2319678 0.9794959 0.0656301 1 2 2 0.0041159 0.6127808 0.9677175
IGHV7-40 15.9375305 492.4418 -238.2209 0.6102728 1.3965106 -0.2999772 -1.0354265 -1.8729132 0.0202320 0.0876576 0.6899486 0.1877131 1.0563185 1.3922542 0.2887295 0.3614414 4.5933524 2.9065704 0.0589747 1 2 2 0.0320962 0.2338009 0.9709432

Summary statistics on a single gene can be shown specifying the argument gene.

summary(results, gene = "MS4A1")
## Generalised linear mixed model
## Method: lme4
## Family: Negative Binomial
## Formula: count ~ Timepoint + EULAR_6m + (1 | PATID) + Timepoint:EULAR_6m
##  Dispersion         AIC      logLik     meanExp 
##    3.789428  898.682864 -441.341432    2.591740 
## 
## Fixed effects:
##                                Estimate Std. Error
## (Intercept)                     2.73637    0.33669
## Timepoint                       0.04908    0.08941
## EULAR_6mModerate                0.19330    0.51103
## EULAR_6mNon-response           -1.72376    0.67780
## Timepoint:EULAR_6mModerate     -0.21080    0.13478
## Timepoint:EULAR_6mNon-response  0.17462    0.16558
## 
## Analysis of Deviance Table (Type II Wald chisquare tests)
##                      Chisq Df Pr(>Chisq)
## Timepoint          0.01125  1    0.91554
## EULAR_6m           5.68288  2    0.05834
## Timepoint:EULAR_6m 5.43787  2    0.06594

Estimated means based on each gene’s fitted model to show fixed effects and their 95% confidence intervals can be seen in the @predict slot:

predict = data.frame(results@predict)
kable(predict) %>%
  kable_styling() %>%
  scroll_box(width = "100%", height = "400px")
y_0_Good y_6_Good y_0_Moderate y_6_Moderate y_0_Non.response y_6_Non.response LCI_0_Good LCI_6_Good LCI_0_Moderate LCI_6_Moderate LCI_0_Non.response LCI_6_Non.response UCI_0_Good UCI_6_Good UCI_0_Moderate UCI_6_Moderate UCI_0_Non.response UCI_6_Non.response
MS4A1 15.4308165 20.7152501 18.7213987 7.0949707 2.7527690 10.5360538 7.9761875 9.1362861 8.8123057 2.8391654 0.8690157 3.2873828 29.852621 46.968930 39.7728791 17.730073 8.719908 33.768026
ADAM12 5.2430737 2.4963164 4.0090258 1.5491357 3.1650953 7.6145225 3.0548055 1.2417443 2.1464867 0.6925006 1.2474757 2.9782353 8.998878 5.018421 7.4877183 3.465443 8.030480 19.468225
IGHV7-4-1 207.3711708 126.3793894 289.4417955 30.0243655 14.6540239 1.4003967 65.7332940 30.2740093 77.7802350 6.1747981 2.1061575 0.1747709 654.201241 527.573006 1077.0930810 145.990606 101.958387 11.221040
IGHV3-49 318.4802290 116.3704819 198.9486219 17.2968396 500.3797001 38.4383698 105.1815920 29.3401887 56.0305175 3.7578080 77.5070454 5.4682432 964.328970 461.554259 706.4106482 79.615738 3230.413998 270.197980
IGHV3-23 940.7216984 827.3870548 1078.8117590 155.8662025 406.4638638 31.5224184 368.3802249 216.1259312 479.9200884 50.9259112 116.6317075 4.5833076 2402.293212 3167.455819 2425.0595869 477.051319 1416.534801 216.800388
ADAM12.1 5.2430737 2.4963164 4.0090258 1.5491357 3.1650953 7.6145225 3.0548055 1.2417443 2.1464867 0.6925006 1.2474757 2.9782353 8.998878 5.018421 7.4877183 3.465443 8.030480 19.468225
FGFRL1 6.2611828 7.8249778 4.8572349 6.6833212 5.9222232 7.7778506 4.7031640 5.4424398 3.4727956 4.5342807 3.7175132 4.8018096 8.335327 11.250520 6.7935847 9.850908 9.434460 12.598367
IL36G 2.7137709 0.2500280 0.2910324 0.6933905 0.4176179 0.1172444 0.2193767 0.0099191 0.0151667 0.0211411 0.0056096 0.0010308 33.570354 6.302371 5.5845859 22.741957 31.090509 13.335847
BLK 2.5613159 2.8064025 3.4195679 1.5439398 1.3797023 1.9300642 1.6106950 1.6207458 2.1312914 0.8267426 0.6368195 0.8505196 4.072986 4.859426 5.4865537 2.883304 2.989196 4.379850
SAA1 28.9036424 5.1281536 7.3222136 1.3259118 15.5876583 34.2866094 4.8281301 0.5508635 0.9425296 0.1098707 0.7653814 1.4752343 173.031906 47.739519 56.8839540 16.001015 317.456229 796.871112
CILP 35.4946119 70.7185855 14.4937624 51.3298849 43.8444724 71.2805236 16.0307054 29.1753805 5.4851887 19.4941384 13.9480147 21.5622062 78.590894 171.415702 38.2975244 135.156375 137.821604 235.639757
EMILIN3 2.7605305 5.7122239 0.3468891 4.1037785 1.1425403 4.1476108 1.5104738 2.7596692 0.1381292 1.8195270 0.3830247 1.4660145 5.045125 11.823700 0.8711557 9.255701 3.408130 11.734315
EMILIN2 30.8216346 23.1270044 37.3005036 33.6529884 38.2492888 35.8547740 26.0766974 18.7030941 30.8774326 26.7788797 28.9721406 26.7968846 36.429965 28.597318 45.0596910 42.291673 50.497066 47.974413
IGHJ6 411.9172371 228.0017937 333.1601604 72.0438020 403.5615024 86.2759658 69.3281680 24.1760209 61.9255246 6.2351369 64.6936801 4.6733772 2447.429597 2150.263608 1792.4061715 832.429107 2517.431161 1592.754427
CXCL9 65.5716886 25.2633747 56.4055334 13.4791866 9.3566156 10.2318955 25.7017909 7.8711090 19.3246013 3.7026566 1.9161150 1.9545700 167.289757 81.086173 164.6390605 49.069760 45.689459 53.562517
IGHV3OR16-12 4.5402163 4.2185263 2.4735396 0.3305456 1.7645228 0.7345045 1.3941223 0.9711074 0.6347234 0.0547623 0.2352107 0.0838919 14.786051 18.325433 9.6394722 1.995175 13.237241 6.430857
IGHV1OR21-1 0.4756891 0.9075291 1.5047880 0.8240433 0.1850942 0.1250000 0.1664639 0.2635246 0.4965519 0.2081891 0.0239578 0.0120623 1.359334 3.125360 4.5602221 3.261686 1.430009 1.295361
IGHV1-69 90.4940260 105.4155442 263.2645407 104.8860184 125.6101523 14.6090165 39.2588920 37.3270484 101.3602448 33.2849004 30.8025681 3.3414820 208.593985 297.704679 683.7810874 330.512537 512.227108 63.870870
IL2RA 3.7562053 1.7869069 3.8963475 1.6757324 1.7232769 1.5347695 2.3837025 0.9628603 2.4156890 0.8829030 0.8100605 0.6772097 5.918976 3.316199 6.2845522 3.180507 3.666002 3.478269
IGHV2-70D 41.5291831 20.3592259 58.6849465 6.9710229 17.3483513 1.8418858 17.6363681 7.0058072 22.0499942 2.1252117 4.0882417 0.3851985 97.790715 59.164928 156.1870225 22.866033 73.617294 8.807260
IGHV5-10-1 133.9183643 117.1640246 192.5504939 31.5297033 122.6297203 104.7328705 36.0781271 25.3006880 51.2885084 7.4553695 17.4016401 7.9938842 497.091444 542.570569 722.8849865 133.343115 864.174193 1372.170766
S100B 14.4377496 14.1974270 22.4378966 11.0640242 13.4938676 33.5367082 7.3345173 6.1165212 10.3671028 4.3513003 4.3126849 10.2548275 28.420223 32.954506 48.5631535 28.132426 42.220674 109.676227
IL24 1.4160874 1.1674074 1.7676552 0.7237782 0.6612507 1.1878055 0.9609734 0.7059607 1.1821741 0.3847163 0.2982339 0.5877232 2.086742 1.930476 2.6431004 1.361665 1.466140 2.400589
IGHV3OR16-8 18.8224569 9.5950843 15.1138928 2.3662614 13.3607901 14.3792157 7.7578712 2.9290911 6.1522410 0.7459544 3.7615084 2.2117728 45.667797 31.431471 37.1295201 7.506080 47.457215 93.482406
FGFR2 2.3543526 1.5231307 0.9120056 1.2229624 0.7206342 0.9061504 1.1042379 0.5817560 0.3629698 0.4152740 0.1794752 0.2196150 5.019730 3.987801 2.2915246 3.601567 2.893513 3.738855
FGF14 0.9188282 1.8305534 0.8795628 1.7156739 2.3132829 1.6309814 0.6190849 1.2487274 0.5555778 1.1138378 1.4294500 0.9296847 1.363699 2.683473 1.3924796 2.642698 3.743592 2.861293
IGHJ3P 145.9294371 89.7390368 139.6890608 24.2346133 28.0022900 14.3550764 69.8449824 35.8887019 60.1444962 8.7705922 8.0703361 3.8967415 304.895210 224.390805 324.4358992 66.964291 97.161783 52.882188
FGF2 1.8731183 2.9449178 2.1787119 3.7554829 1.9859282 3.4099108 1.3240295 1.9939340 1.4861792 2.4858794 1.1168334 1.9938050 2.649920 4.349462 3.1939523 5.673506 3.531333 5.831810
IL16 26.6338572 24.1515439 23.2671102 27.1702545 18.9990127 31.0767442 22.1678520 19.2600942 18.9118883 21.1415003 13.9463343 22.4649453 31.999598 30.285266 28.6252968 34.918181 25.882248 42.989823
LILRA5 8.4188674 6.8767861 11.4529494 5.8846392 7.9775228 10.1650340 6.0544255 4.5387996 7.9046755 3.6963449 4.5712739 5.7288427 11.706698 10.419096 16.5939829 9.368438 13.921911 18.036438
CXCL11 4.1926376 2.4082176 3.7852093 1.0491573 1.0168372 0.8921720 2.0437751 0.9690135 1.6605102 0.3627796 0.2759225 0.2244879 8.600854 5.984965 8.6285584 3.034159 3.747276 3.545718
IGHV7-40 4.0410745 0.6680766 1.4348842 0.2678342 0.6210134 0.1737184 1.0452033 0.1166984 0.2991980 0.0345152 0.0580428 0.0110776 15.624026 3.824613 6.8813716 2.078363 6.644364 2.724252
PHOSPHO1 4.2927466 7.6052889 4.5829437 3.4367675 4.5814744 4.8246311 2.7258452 4.3937329 2.7321974 1.8230721 2.1396778 2.1835113 6.760352 13.164300 7.6873556 6.478828 9.809845 10.660382
EAF2 8.1559946 5.1925955 10.3149093 5.0726017 5.7110124 4.2127624 5.9140279 3.5052660 7.3785031 3.3609542 3.4581956 2.4307341 11.247875 7.692155 14.4199105 7.655947 9.431411 7.301238
IGHV5-78 2.8951365 3.6750970 2.7821533 1.0699636 5.3542086 2.8329389 1.2773781 1.3373735 1.0902916 0.3290647 1.3759852 0.6715812 6.561734 10.099152 7.0993643 3.479018 20.834200 11.950220
CXCL13 34.3875502 26.9578739 56.2502960 6.5388640 7.9451183 11.0678173 16.6451992 10.9322716 24.5594487 2.3809991 2.3135614 3.0653257 71.041722 66.475385 128.8341540 17.957479 27.284732 39.962010
IGHV7-27 2.6166853 0.2242760 1.8351688 0.2327794 1.0349855 0.5986291 0.7127532 0.0362055 0.4114229 0.0312680 0.1111337 0.0552486 9.606469 1.389284 8.1858470 1.732965 9.638792 6.486256
PPIL4 9.8675383 9.4349051 9.2525866 11.8069073 9.6255151 12.4124137 8.1728411 7.4612844 7.4435374 9.1550596 7.0032677 9.0262444 11.913643 11.930578 11.5013003 15.226887 13.229616 17.068894
IGLV4-3 3.2746770 5.1305456 2.0187496 1.2805470 1.1728042 0.5206835 0.9754104 1.1457196 0.5002862 0.2351746 0.1463548 0.0544524 10.993844 22.974642 8.1460375 6.972694 9.398189 4.978869
IL36RN 2.4738840 0.3268918 0.3854628 1.0315749 0.8933628 0.1256238 0.3582997 0.0269224 0.0393643 0.0708579 0.0333675 0.0029092 17.080959 3.969121 3.7745225 15.018036 23.918386 5.424689
IL2RG 32.8309600 23.6801432 42.5463280 16.7873895 17.8391493 20.0733867 23.2151011 15.4606283 29.0283527 10.4067217 10.2232165 10.8570804 46.429775 36.269495 62.3593782 27.080233 31.128681 37.113187
IGHV1OR15-4 12.0915817 7.2472411 28.6215571 7.0629254 4.9822015 3.1465857 6.0611466 3.0540928 13.0514886 2.7158823 1.5330688 0.9037637 24.121896 17.197416 62.7662910 18.367849 16.191271 10.955300
HHIP 2.2252305 1.7506563 1.9088433 4.6428720 3.9204878 3.7647380 1.4773826 1.0307931 1.1811760 2.7695040 2.0615802 1.9186382 3.351637 2.973242 3.0847926 7.783437 7.455555 7.387142
PADI4 1.2612285 2.8101880 1.8211490 1.1615614 1.4745861 1.7232541 0.7138925 1.4650901 0.9772170 0.5264347 0.5768278 0.6580193 2.228203 5.390219 3.3939069 2.562948 3.769590 4.512945
SAA2 11.9071767 1.2565988 1.5784447 0.4209318 1.9646612 1.7337013 0.8825190 0.0485258 0.0794404 0.0110056 0.0242233 0.0175383 160.654742 32.540205 31.3629807 16.099376 159.346366 171.380603
LILRB3 18.0895994 15.4909569 22.3499024 16.9343137 15.6701735 19.9782091 14.1208561 11.3557725 16.8836572 12.0341979 10.2941943 12.9525110 23.173779 21.131961 29.5858968 23.829671 23.853672 30.814785

Q-values

The q-values from each of the p-value columns can be calculated using glmmQvals. This will print a significance table based on the cut-off (default p=0.05) and add a matrix named qvals to the @stats slot:

results <- glmmQvals(results)
## 
## Timepoint
## ---------
## Not Significant     Significant 
##              29              17 
## 
## EULAR_6m
## --------
## Not Significant     Significant 
##              41               5 
## 
## Timepoint:EULAR_6m
## ------------------
## Not Significant 
##              46

glmmTMB models

The glmmTMB package is an alternative to lme4 for fitting negative binomial GLMM models. Note, that this package optimises the dispersion parameter for each GLMM model. This has the advantage that a predefined list of dispersions for each gene is not required. But it also means that model fitting is significantly slower than a negative binomial GLMM fit by lme4::glmer with family function MASS::negative.binomial for which the dispersion parameter must be pre-specified. Each glmmTMB model takes about 0.8 seconds, so that even with 8 cores, a typical transcriptome of 20,000 genes would take about 30 minutes, although this will vary depending on the complexity of the design formula. In comparison, the standard glmer pipeline with known dispersions supplied takes about 2 minutes on 8 cores.

Although glmmTMB is slower than lme::glmer with known dispersion, glmmTMB is still faster than the equivalent lme4 function glmer.nb when the dispersion is unknown.

Setting method = "glmmTMB" when calling glmmSeq() switches from using lme4::glmer (the default) to the glmmTMB package. The dispersion argument is then no longer required. The family argument can be used to specify different family functions and glmmTMB provides nbinom1 and nbinom2 (glmmSeq uses the latter by default). Other family functions e.g. poisson can be used.

Gaussian mixed effects models

An alternative to fitting negative binomial GLMM is to use transformed data and fit gaussian LMM. The lmmSeq function will fit many LMM across genes (or other data) where random effects/mixed effects models are useful. It is almost 3x faster than glmmSeq. For example, gaussian normalised DNA methylation data can be analysed using this method. In the example below the RNA-Seq gene expression data is log transformed so that it is approximately Gaussian. Alternatively variance stabilising transformation (VST) can be applied to count data through DESeq2 or the voom transformation in limma voom.

logtpm <- log2(tpm + 1)
lmmres <- lmmSeq(~ Timepoint * EULAR_6m + (1 | PATID),
                   maindata = logtpm,
                   metadata = metadata,
                   progress = TRUE)
## 
## n = 123 samples, 82 individuals
## Time difference of 0.7561669 secs
summary(lmmres, "MS4A1")
## Linear mixed model
## Formula: gene ~ Timepoint + EULAR_6m + (1 | PATID) + Timepoint:EULAR_6m
##         AIC      logLik     meanExp 
##  503.415984 -243.707992    2.590358 
## 
## Fixed effects:
##                                 Estimate Std. Error
## (Intercept)                     2.614919    0.30012
## Timepoint                       0.024789    0.06841
## EULAR_6mModerate                0.860136    0.45592
## EULAR_6mNon-response           -0.983157    0.58504
## Timepoint:EULAR_6mModerate     -0.256884    0.10233
## Timepoint:EULAR_6mNon-response -0.009612    0.12407
## 
## Analysis of Deviance Table (Type II Wald chisquare tests)
##                    Chisq Df Pr(>Chisq)
## Timepoint          2.321  1    0.12767
## EULAR_6m           6.098  2    0.04740
## Timepoint:EULAR_6m 7.134  2    0.02823

Hypothesis testing

By default, p-values for each model term are computed using Wald type 2 Chi-squared test as per car::Anova(). The underlying code for this has been optimised for speed.

For LMM via lmmSeq(), an alternative to the Wald type 2 Chi-squared test is an F-test with Satterthwaite denominator degrees of freedom, which has been implemented using the lmerTest package and is enabled using test.stat = "F". This is not available for GLMM.

However, if a reduced model formula is specified by setting reduced, then a likelihood ratio test is performed instead using anova. This will double computation time since two (G)LMM have to be fitted for each gene. For further information on inference testing on GLMM, we recommend Ben Bolker’s GLMM FAQ page.

glmmLRT <- glmmSeq(~ Timepoint * EULAR_6m + (1 | PATID),
                   reduced = ~ Timepoint + EULAR_6m + (1 | PATID),
                   countdata = tpm,
                   metadata = metadata,
                   dispersion = disp, verbose = FALSE)

summary(glmmLRT, "MS4A1")
## Generalised linear mixed model
## Method: lme4
## Family: Negative Binomial
## Formula: count ~ Timepoint + EULAR_6m + (1 | PATID) + Timepoint:EULAR_6m
##  Dispersion         AIC      logLik     meanExp 
##    3.789428  898.682864 -441.341432    2.591740 
## 
## Fixed effects:
##                                Estimate Std. Error
## (Intercept)                     2.73637    0.33669
## Timepoint                       0.04908    0.08941
## EULAR_6mModerate                0.19330    0.51103
## EULAR_6mNon-response           -1.72376    0.67780
## Timepoint:EULAR_6mModerate     -0.21080    0.13478
## Timepoint:EULAR_6mNon-response  0.17462    0.16558
## 
## Likelihood ratio test
## Reduced formula: count ~ Timepoint + EULAR_6m + (1 | PATID)
##   Chisq Df Pr(>Chisq)
##    4.85  2    0.08846

Extracting individual Genes

Similarly you can run the script for an individual gene (make sure you use drop = FALSE to maintain countdata as a matrix).

MS4A1glmm <- glmmSeq(~ Timepoint * EULAR_6m + (1 | PATID),
                     countdata = tpm["MS4A1", , drop = FALSE],
                     metadata = metadata,
                     dispersion = disp,
                     verbose = FALSE)

The (g)lmer or glmmTMB fitted object for a single gene can be obtained using the function glmmRefit():

fit <- glmmRefit(results, gene = "MS4A1")
## boundary (singular) fit: see help('isSingular')
fit
## Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) [
## glmerMod]
##  Family: Negative Binomial(0.2639)  ( log )
## Formula: count ~ Timepoint + EULAR_6m + (1 | PATID) + Timepoint:EULAR_6m
##    Data: data
##  Offset: offset
##       AIC       BIC    logLik  deviance  df.resid 
##  898.6829  921.1803 -441.3414  882.6829       115 
## Random effects:
##  Groups Name        Std.Dev.
##  PATID  (Intercept) 0       
## Number of obs: 123, groups:  PATID, 82
## Fixed Effects:
##                    (Intercept)                       Timepoint  
##                        2.73637                         0.04908  
##               EULAR_6mModerate            EULAR_6mNon-response  
##                        0.19330                        -1.72376  
##     Timepoint:EULAR_6mModerate  Timepoint:EULAR_6mNon-response  
##                       -0.21080                         0.17462  
## optimizer (bobyqa) convergence code: 0 (OK) ; 0 optimizer warnings; 1 lme4 warnings

These (g)lmer objects can be passed to other packages notably emmeans for visualising estimated marginal means. Note these are based on the model, not directly on the data.

library(emmeans)

emmeans(fit, ~ Timepoint | EULAR_6m)
## EULAR_6m = Good:
##  Timepoint emmean    SE  df asymp.LCL asymp.UCL
##          0   2.74 0.337 Inf      2.08      3.40
##          6   3.03 0.418 Inf      2.21      3.85
## 
## EULAR_6m = Moderate:
##  Timepoint emmean    SE  df asymp.LCL asymp.UCL
##          0   2.93 0.384 Inf      2.18      3.68
##          6   1.96 0.467 Inf      1.04      2.88
## 
## EULAR_6m = Non-response:
##  Timepoint emmean    SE  df asymp.LCL asymp.UCL
##          0   1.01 0.588 Inf     -0.14      2.17
##          6   2.35 0.594 Inf      1.19      3.52
## 
## Results are given on the log (not the response) scale. 
## Confidence level used: 0.95
emmip(fit, ~ Timepoint | EULAR_6m)

Refitting a different model

glmmRefit() allows a different model to be fitted using the original data. The example below shows how to refit the model without the interaction term and then perform a likelihood ratio test using anova. Note for glmmSeq() objects the LHS of the reduced formula must be "count", while for lmmSeq() objects the LHS must be "gene". For glmmTMB analyses, the GLM family can also be changed.

fit2 <- glmmRefit(results, gene = "MS4A1",
                  formula = count ~ Timepoint + EULAR_6m + (1 | PATID))

anova(fit, fit2)
## Data: data
## Models:
## fit2: count ~ Timepoint + EULAR_6m + (1 | PATID)
## fit: count ~ Timepoint + EULAR_6m + (1 | PATID) + Timepoint:EULAR_6m
##      npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)  
## fit2    6 899.53 916.41 -443.77   887.53                       
## fit     8 898.68 921.18 -441.34   882.68 4.8505  2    0.08846 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Model Plots

For variables such as time, which are matched according to an ID (the random effect), we can examine the fitted model using plots which show estimated means and confidence intervals based on coefficients for the fitted regression model, overlaid upon the underlying data. In this case the samples are matched longitudinally over time.

Plots can be viewed using either ggplot or base graphics. We can start looking at the gene with the most significant interaction IGHV3-23:

plotColours <- c("skyblue", "goldenrod1", "mediumseagreen")
modColours <- c("dodgerblue3", "goldenrod3", "seagreen4")
shapes <- c(17, 19, 18)

ggmodelPlot(results,
            geneName = "IGHV3-23",
            x1var = "Timepoint",
            x2var="EULAR_6m",
            xlab="Time",
            colours = plotColours,
            shapes = shapes,
            lineColours = plotColours, 
            modelColours = modColours,
            modelSize = 10)

Alternatively plots can be generated using base graphics, here with or without the model fit overlaid. By default p-value labels are taken from the column names of the pvals object in the stats slot of the S4 result object. These can be relabelled using the plab argument.

oldpar <- par(mfrow=c(1, 2))

modelPlot(results2,
          geneName = "FGF14",
          x1var = "Timepoint",
          x2var="EULAR_binary",
          fontSize=0.65,
          colours=c("coral", "mediumseagreen"),
          modelColours = c("coral", "mediumseagreen"),
          modelLineColours = "black",
          modelSize = 2)

modelPlot(results,
          geneName = "EMILIN3",
          x1var = "Timepoint",
          x2var = "EULAR_6m",
          colours = plotColours,
          plab = c("time", "response", "time:response"),
          addModel = FALSE)

par(oldpar)

To plot the model fits alone set addPoints = FALSE.

library(ggpubr)

p1 <- ggmodelPlot(results,
                  "ADAM12",
                  x1var="Timepoint",
                  x2var="EULAR_6m",
                  xlab="Time",
                  addPoints = FALSE,
                  colours = plotColours)

p2 <- ggmodelPlot(results,
                  "EMILIN3",
                  x1var="Timepoint",
                  x2var="EULAR_6m",
                  xlab="Time",
                  fontSize=8,
                  x2Offset=1,
                  addPoints = FALSE,
                  colours = plotColours)

ggarrange(p1, p2, ncol=2, common.legend = T, legend="bottom")

Fold change plots

The comparative fold change (for x1var variables) between conditions (x2var and x2Values variables) can be plotted using an fcPlot for all genes to highlight significance. This type of plot is most suited to look for interaction between time (x1var) and a two-level factor (x2var), looking at change between two timepoints. In the example below from the R4RA study [1], gene expression pre- and post-drug treatment is compared between two drugs (rituximab & tocilizumab), using the design formula gene_counts ~ time * drug. By setting graphics = "plotly" this can be viewed interactively.

r4ra_glmm <- glmmSeq(~ time * drug + (1 | Patient_ID), 
                       countdata = tpmdata, metadata,
                       dispersion = dispersions, cores = 8, removeSingles = T)
r4ra_glmm <- glmmQvals(r4ra_glmm)
labels = c(..)  # Genes to label
fcPlot(r4ra_glmm, x1var = "time", x2var = "drug", graphics = "plotly",
       pCutoff = 0.05, useAdjusted = TRUE,
       labels = labels,
       colours = c('grey', 'green3', 'gold3', 'blue'))

Log2 fold change between the two time points for individuals treated with rituximab on the x axis and individuals treated with tocilizumab on the y axis with each point representing a gene. Genes showing an interaction between time and drug are coloured blue or gold depending on whether their fold change is greater post-rituximab (blue) or post-tocilizumab (gold). Genes without interaction, but changing significantly over time are coloured green and tend to lie along the line of identity. See the Longitudinal tab in https://r4ra.hpc.qmul.ac.uk for an interactive version of the above plot.

MA plots

An MA plot is an application of a Bland–Altman plot. The plot visualizes the differences between measurements taken in two samples, by transforming the data onto M (log ratio) and A (mean average) scales, then plotting these values.

labels = c('MS4A1', 'FGF14', 'IL2RG', 'IGHV3-23', 'ADAM12', 'IL36G', 
           'BLK', 'SAA1', 'CILP', 'EMILIN3', 'EMILIN2', 'IGHJ6', 
           'CXCL9', 'CXCL13')
maPlots <- maPlot(results,
                  x1var="Timepoint",
                  x2var="EULAR_6m",
                  x2Values=c("Good", "Non-response"),
                  colours=c('grey', 'midnightblue',
                             'mediumseagreen', 'goldenrod'),
                  labels=labels,
                  graphics="ggplot")

maPlots$combined

Troubleshooting

Mixed effects models are tricky to fit and lme4::glmer sometimes returns errors. In fact, a significant amount of code in glmmSeq is devoted to catching and handling errors to allow parallelisation to continue. Errors in genes are stored in the errors slot.

results@errors[1]   # first gene error
##                                                                                                                         IL12A 
## "Error in (function (fr, X, reTrms, family, nAGQ = 1L, verbose = 0L, maxit = 100L,  : \n  PIRLS loop resulted in NaN value\n"

Sometimes glmmSeq returns errors in all genes. This usually means a problem with not enough samples in each timepoint or a mistake in the formula. Since version 0.5.5 glmmSeq now returns a vector of the error messages for all genes, which can be useful for debugging. In the example below, only timepoint 0 is specified.

results3 <- glmmSeq(~ Timepoint * EULAR_6m + (1 | PATID),
                    countdata = tpm[, metadata$Timepoint == 0],
                    metadata = metadata[metadata$Timepoint == 0, ],
                    dispersion = disp)
## 
## n = 72 samples, 72 individuals
## All genes returned an error. Check call. Check sufficient data in each group
##                                                                              MS4A1 
## "fixed-effect model matrix is rank deficient so dropping 3 columns / coefficients"

If there are errors which are not caught by the error checking core mechanism, this can lead to problems with grouping results after the core models have been fit. Setting returnList=TRUE when calling glmmSeq returns the list output direct from mclapply (or parLapply on windows). This can be helpful for debugging unforeseeen problems in the core loop.

Citing glmmSeq

glmmSeq was developed by the bioinformatics team at the Experimental Medicine & Rheumatology department and Centre for Translational Bioinformatics at Queen Mary University London.

If you use this package please cite as:

citation("glmmSeq")
## 
## To cite package 'glmmSeq' in publications use:
## 
##   Lewis M, Goldmann K, Sciacca E (????). _glmmSeq: General Linear Mixed Models
##   for Gene-Level Differential Expression_.
##   https://myles-lewis.github.io/glmmSeq/,
##   https://github.com/myles-lewis/glmmSeq.
## 
## A BibTeX entry for LaTeX users is
## 
##   @Manual{,
##     title = {glmmSeq: General Linear Mixed Models for Gene-Level Differential Expression},
##     author = {Myles Lewis and Katriona Goldmann and Elisabetta Sciacca},
##     note = {https://myles-lewis.github.io/glmmSeq/, https://github.com/myles-lewis/glmmSeq},
##   }

References

  1. Felice Rivellese, Anna Surace, Katriona Goldmann, Elisabetta Sciacca, Cankut Cubuk, Giovanni Giorli, … Michael Barnes, Myles J. Lewis, Costantino Pitzalis, R4RA collaborative group. Rituximab versus tocilizumab in rheumatoid arthritis: synovial biopsy-based biomarker analysis of the phase 4 R4RA randomized trial. Nature medicine 2022; 28(6): 1256-68. doi:10.1038/s41591-022-01789-0

Statistical software used in this package:

  1. lme4: Douglas Bates, Martin Maechler, Ben Bolker, Steve Walker (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 67(1), 1-48. doi: 10.18637/jss.v067.i01.

  2. car: John Fox and Sanford Weisberg (2019). An {R} Companion to Applied Regression, Third Edition. Thousand Oaks CA: Sage. URL: https://socialsciences.mcmaster.ca/jfox/Books/Companion/

  3. MASS: Venables, W. N. & Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0

  4. glmmTMB: Mollie Brooks, Ben Bolker, Kasper Kristensen, Martin Maechler, Arni Magnusson (2022). glmmTMB: Generalized Linear Mixed Models using Template Model Builder

  5. qvalue: John D. Storey, Andrew J. Bass, Alan Dabney and David Robinson (2020). qvalue: Q-value estimation for false discovery rate control. R package version 2.22.0. https://github.com/StoreyLab/qvalue

  6. lmerTest: Alexandra Kuznetsova, Per Brockhoff, Rune Christensen, Sofie Jensen. lmerTest: Tests in Linear Mixed Effects Models

  7. emmeans: Russell V. Lenth, Paul Buerkner, Maxime Herve, Jonathon Love, Fernando Miguez, Hannes Riebl, Henrik Singmann. emmeans: Estimated Marginal Means, aka Least-Squares Means