Chapter 3 Applications
3.1 Application I: Individual-level data
The dataset used here consists of a total of 14 metabolites (N = 14) and two groups (Liyanage et al. 2024). The 12 samples in the two groups had been extracted using the same method in a controlled experiment but run separately in two different instruments: Liquid Chromatography-Mass Spectrometry (LC/MS) and Gas Chromatography-Mass Spectrometry (GC/MS), leading to two separate datasets. Thus, in this example, each separate dataset from each instrument forms a different ‘study’ (\(K = 2\)), which can be integrated using meta-analysis methods. We will use `MetaHD’ for combining summary estimates obtained from these multiple metabolomic studies.
This GC-LC dataset is available in the `MetaHD’ package and can be loaded using the following command.
As explained in Section 3.1 `MetaHDInput’ function can then be used to calculate the observed effect sizes and their associated variances and within-study covariance matrices as shown below.
## met1 met2 met3 met4 met5 met6
## GC -0.12216161 0.2408217 0.532996014 -0.06407204 -0.46280293 0.39930861
## LC 0.08730353 0.1074026 -0.002691788 -0.06451353 -0.02644625 0.08896313
## met7 met8 met9 met10 met11 met12 met13
## GC -0.03582016 0.3558101 -0.33696133 -0.5623149 0.3591404 0.6540438 0.5257092
## LC 0.08684274 -0.1067551 -0.07022999 0.6563386 0.8378621 0.4823036 0.6567706
## met14
## GC -0.6275364
## LC 0.5955336## met1 met2 met3 met4 met5
## met1 0.66816536 0.0342995006 -0.13023460 0.042994441 0.0410169309
## met2 0.03429950 0.0219976435 0.04471369 0.007700712 -0.0003259779
## met3 -0.13023460 0.0447136868 0.99714813 0.028561626 -0.0153806902
## met4 0.04299444 0.0077007122 0.02856163 0.027139572 0.0142878276
## met5 0.04101693 -0.0003259779 -0.01538069 0.014287828 0.0456121062
## met6 0.06753777 0.0191195280 0.07398638 0.015472822 0.0042490167
## met6 met7 met8 met9 met10 met11
## met1 0.067537772 0.06644299 0.19414892 0.042673455 -0.001689788 0.047069456
## met2 0.019119528 0.01098242 0.02419960 0.001167182 -0.006991485 0.022944009
## met3 0.073986377 0.02910402 0.02191034 -0.013459772 -0.029697299 0.142481398
## met4 0.015472822 0.01730561 0.03445267 0.012254275 0.005002414 0.022998932
## met5 0.004249017 0.01644365 0.03498665 0.019199078 0.015381516 0.009998191
## met6 0.073516461 0.02205751 0.05878542 0.005020851 -0.010286200 0.044605360
## met12 met13 met14
## met1 -0.0200050944 0.005789995 0.07014588
## met2 0.0075920184 0.011643333 0.01405622
## met3 0.0777879265 0.080134515 0.08293142
## met4 -0.0003712348 0.006499256 0.03996221
## met5 -0.0099810545 -0.003704279 0.04585254
## met6 0.0151338588 0.021938095 0.01626860MetaHD model can now be fitted as follows:
# Multivariate high dimensional meta analyis.
model1 <- MetaHD(Y, Slist, method = "reml", bscov = "unstructured")
model1$estimate## [1] 0.075866139 0.102629970 -0.008317808 -0.049244565 -0.089172625
## [6] 0.088244078 0.059983684 -0.092853540 -0.070839356 0.054232031
## [11] 0.556648094 0.428719822 0.540946192 0.298754741## met1 met2 met3 met4 met5 met6
## 2.464928e-03 1.681099e-01 9.279186e-01 3.846982e-01 4.302016e-01 3.140623e-01
## met7 met8 met9 met10 met11 met12
## 4.237150e-01 5.205543e-02 2.005473e-01 9.049333e-01 4.354933e-02 2.102174e-03
## met13 met14
## 1.611432e-05 3.884817e-01When both within and between study correlations are set to zero, MetaHD reduces to the usual random effects model analysis of individual metabolites (i.e. univariate meta-analyses). Additionally, when the between-study variances are also set to zero, then MetaHD reduces to a fixed-effects model univariate meta-analysis. We can perform univariate random-effects and univariate fixed-effects meta-analysis in MetaHD as follows:
diag_Slist <- list()
K <- nrow(Y)
for (k in 1:K) {
diag_Slist[[k]] <- diag(diag(Slist[[k]]))
}
# Univariate random-effects meta-analysis
model2 <- MetaHD(Y, diag_Slist, method = "reml", bscov = "diag")
model2$estimate## [1] 0.086997460 0.154229024 0.003813175 -0.064452935 -0.198813436
## [6] 0.145163312 0.063695130 -0.104116978 -0.152139261 -0.045908911
## [11] 0.687629243 0.611359068 0.634467616 0.095250338## [1] 5.365076e-03 7.921502e-02 9.723565e-01 2.909377e-01 3.513227e-01
## [6] 2.245657e-01 4.851326e-01 4.208121e-02 2.162800e-01 9.392322e-01
## [11] 1.964117e-03 1.895315e-04 4.440892e-16 8.741418e-01# Univariate fixed-effects meta-analysis
model3 <- MetaHD(Y, diag_Slist, method = "fixed")
model3$estimate## [1] 0.086997997 0.154228997 0.003810829 -0.064452935 -0.074409448
## [6] 0.132726314 0.063695809 -0.104119067 -0.100490404 -0.431983086
## [11] 0.726651494 0.611359399 0.634468201 0.426236574## [1] 5.324401e-03 7.921483e-02 9.723685e-01 2.909362e-01 2.933119e-01
## [6] 1.923816e-01 4.851192e-01 4.199529e-02 9.265717e-02 9.209603e-03
## [11] 1.630553e-05 1.895119e-04 4.440892e-16 4.421984e-02When missing data are available, we can use MetaHD to handle missing values by setting the argument “impute.na = TRUE”. This replaces the missing effect sizes and variances by zero and a large constant value respectively, so that the missing outcome is allocated a lower weight in the meta-analysis.
3.2 Application II: Summary-level data
In this example, we use a simulated dataset as described by (Liyanage et al. 2024). Here we have prepared separate data frames for observed effect sizes and within-study covariance matrices in the ‘simdata.1’ dataset as follows..
## metabolite_1 metabolite_2 metabolite_3 metabolite_4 metabolite_5
## study_1 0.231755235 -0.9377271 -3.306843 0.28142576 -0.5547623
## study_2 0.305408170 -1.2421302 -3.547565 0.66254064 -0.3162607
## study_3 -0.006523691 -1.6762418 -3.631345 0.68023162 -0.1846545
## study_4 0.283789796 -1.2533024 -3.545591 0.08533545 -1.0310925
## study_5 -0.748073950 -2.5502749 -4.386687 -1.08561122 -1.7532039
## study_6 -0.805069877 -2.2028554 -3.684388 -0.42021699 -1.3547757
## metabolite_6 metabolite_7 metabolite_8 metabolite_9 metabolite_10
## study_1 2.822603 -1.268180 2.593808 -0.3936202 -0.6887844
## study_2 2.660019 -1.434220 2.558514 -0.2623882 -0.0245477
## study_3 2.421983 -1.085327 3.099015 -0.2417435 0.3493618
## study_4 2.245691 -1.448434 2.231622 -0.6266361 -0.4590747
## study_5 1.245282 -2.156295 1.503325 -1.4695383 -1.3845496
## study_6 2.730196 -1.792440 1.812767 -0.4075956 -0.9052131
## metabolite_11 metabolite_12 metabolite_13 metabolite_14 metabolite_15
## study_1 1.25723073 -0.1669118 1.4819259 2.067361 -0.3992064
## study_2 1.50515960 -0.1288290 1.6942841 1.696545 -0.5495592
## study_3 1.02398696 -0.4566538 1.2831249 2.635289 -0.6367507
## study_4 1.01206691 -0.4866803 1.2184663 2.340882 -0.8044879
## study_5 0.05619328 -1.0199653 0.6820576 0.597100 -1.3620158
## study_6 0.11930101 -0.9310851 1.1849203 1.426632 -1.2488842
## metabolite_16 metabolite_17 metabolite_18 metabolite_19 metabolite_20
## study_1 2.035879 1.6594976 1.4785488 2.025255 1.6325979
## study_2 1.757267 1.5286110 1.4712913 2.068347 1.7831811
## study_3 2.208511 1.2437205 1.7993070 1.877336 1.0564191
## study_4 1.971764 0.6971900 1.3849356 2.472191 1.4691468
## study_5 1.162320 0.5003652 0.3460756 1.408792 0.6553110
## study_6 1.383772 1.3982598 0.8289669 1.587887 0.7490209
## metabolite_21 metabolite_22 metabolite_23 metabolite_24 metabolite_25
## study_1 -0.31458426 0.24703204 3.177385 1.5929012 -0.005622678
## study_2 -0.59096083 -0.18343696 2.512141 1.2888813 0.139087964
## study_3 -0.24805374 -0.08827114 3.071959 0.7913664 0.724998374
## study_4 -0.08455623 -0.18155226 2.534846 1.0723134 -0.258109359
## study_5 -1.20477797 -1.12187511 1.513707 0.2390821 -1.276384331
## study_6 -0.84545412 -0.58702267 2.308492 0.3502428 -0.838241155
## metabolite_26 metabolite_27 metabolite_28 metabolite_29 metabolite_30
## study_1 -2.648980 -0.8436625 -0.904995 3.653576 5.543804
## study_2 1.797236 -0.4594717 -1.428674 3.077779 5.607509
## study_3 -3.313190 -1.0387204 -1.281120 3.045154 5.590184
## study_4 -3.528483 -0.8659648 -1.185520 4.045979 5.222089
## study_5 -4.118482 -2.0616843 -2.203919 2.531608 5.210625
## study_6 -3.853180 -0.9640716 -1.899273 3.069246 5.242326## metabolite_1 metabolite_2 metabolite_3 metabolite_4 metabolite_5
## metabolite_1 0.2051971 0.2360390 0.1451389 0.2538377 0.2159263
## metabolite_2 0.2360390 0.3914241 0.2185096 0.2999849 0.2919311
## metabolite_3 0.1451389 0.2185096 0.1429655 0.2067593 0.1722842
## metabolite_4 0.2538377 0.2999849 0.2067593 0.3705385 0.2909224
## metabolite_5 0.2159263 0.2919311 0.1722842 0.2909224 0.3120427
## metabolite_6 0.3196974 0.3981609 0.2568697 0.4433588 0.3949941
## metabolite_6 metabolite_7 metabolite_8 metabolite_9 metabolite_10
## metabolite_1 0.3196974 0.1667637 0.2076705 0.08103280 0.11133998
## metabolite_2 0.3981609 0.2405715 0.2527721 0.12311977 0.15263329
## metabolite_3 0.2568697 0.1591633 0.1504863 0.07664409 0.09779216
## metabolite_4 0.4433588 0.2554422 0.2650113 0.11012275 0.15222817
## metabolite_5 0.3949941 0.2104929 0.2520871 0.09973932 0.14457419
## metabolite_6 0.5994060 0.3069871 0.3358156 0.13793029 0.19313481
## metabolite_11 metabolite_12 metabolite_13 metabolite_14
## metabolite_1 0.1579164 0.2122468 0.1870629 0.2068235
## metabolite_2 0.2037668 0.2946624 0.2341660 0.3079808
## metabolite_3 0.1147814 0.1891951 0.1460312 0.1852916
## metabolite_4 0.1918580 0.2831199 0.2571253 0.2892108
## metabolite_5 0.1842593 0.2537483 0.2475945 0.2650591
## metabolite_6 0.2576985 0.3589582 0.3438236 0.3801845
## metabolite_15 metabolite_16 metabolite_17 metabolite_18
## metabolite_1 0.2090655 0.1991912 0.11785788 0.2532525
## metabolite_2 0.2931316 0.2709879 0.16230953 0.2834488
## metabolite_3 0.1698101 0.1665578 0.09688847 0.1828497
## metabolite_4 0.2479776 0.2599880 0.15284705 0.3259542
## metabolite_5 0.2702558 0.2520948 0.13001982 0.3052326
## metabolite_6 0.3523785 0.3548082 0.19915187 0.4357530
## metabolite_19 metabolite_20 metabolite_21 metabolite_22
## metabolite_1 0.11496091 0.1299937 0.1487927 0.2316234
## metabolite_2 0.14725054 0.1725120 0.2002373 0.3218278
## metabolite_3 0.09486947 0.1084088 0.1201728 0.1890557
## metabolite_4 0.15936720 0.1823390 0.1959123 0.2944990
## metabolite_5 0.13993093 0.1637614 0.1944129 0.2788441
## metabolite_6 0.20217217 0.2420832 0.2604766 0.4135079
## metabolite_23 metabolite_24 metabolite_25 metabolite_26
## metabolite_1 0.2108364 0.2635543 0.2442990 0.2897116
## metabolite_2 0.3002607 0.3586782 0.3305478 0.3275086
## metabolite_3 0.1783501 0.2206520 0.1977199 0.2085774
## metabolite_4 0.2774928 0.3359515 0.2996849 0.3678765
## metabolite_5 0.2557165 0.3522710 0.2888747 0.3518957
## metabolite_6 0.3512251 0.4720034 0.3845858 0.4956480
## metabolite_27 metabolite_28 metabolite_29 metabolite_30
## metabolite_1 0.1090710 0.2335329 0.1896202 0.1358149
## metabolite_2 0.1323510 0.2954209 0.2487014 0.2008984
## metabolite_3 0.0808657 0.1835814 0.1600222 0.1192158
## metabolite_4 0.1407640 0.2946390 0.2564001 0.1816158
## metabolite_5 0.1330913 0.2770655 0.2196668 0.1739497
## metabolite_6 0.1689191 0.3889943 0.3132906 0.2299339Now, we can carry out the meta-analysis using MetaHD as follows.
## Multivariate random-effects meta analyis
model_MetaHD <- MetaHD(Y=Observed_effects,
Slist=Slist,
method = "reml",
bscov = "unstructured")
print(model_MetaHD$estimate) ## [1] -0.1653862 -1.9799547 -3.7332264 -0.3679426 -1.2794924 2.2045394
## [7] -1.8353620 1.9753857 -0.7671977 -0.9399812 0.6337954 -0.8684518
## [13] 1.0743271 1.3247917 -1.2438876 1.4045603 1.2003022 0.8966010
## [19] 1.7545069 1.1164275 -0.8513849 -0.6180309 2.4147547 0.5117378
## [25] -0.8791242 -3.6862738 -1.2616082 -1.7269992 3.0219733 5.5122717# TRUE VALUES ARE:
true_values<-c(-0.1675687 , -1.9658875 ,-3.7501346, -0.3722893, -1.2669714, 2.1815949, -1.8274545, 2.0032794, -0.7985332 ,-0.9362461 , 0.6539242 ,-0.8254938, 1.1240729, 1.3267165, -1.2057946,1.3967554, 1.1916929 , 0.9041837, 1.7934879, 1.1444330, -0.8233060, -0.5886543, 2.4371479, 0.4882229, -0.8903039, -3.6956073, -1.2576506, -1.7221614, 3.0298406, 5.4704779)
print(true_values)## [1] -0.1675687 -1.9658875 -3.7501346 -0.3722893 -1.2669714 2.1815949
## [7] -1.8274545 2.0032794 -0.7985332 -0.9362461 0.6539242 -0.8254938
## [13] 1.1240729 1.3267165 -1.2057946 1.3967554 1.1916929 0.9041837
## [19] 1.7934879 1.1444330 -0.8233060 -0.5886543 2.4371479 0.4882229
## [25] -0.8903039 -3.6956073 -1.2576506 -1.7221614 3.0298406 5.4704779In cases where the within-study correlations are not available, these could be estimated from the data as follows:
# If within-study correlations are not available, use only the variances and enter these values in a K x N matrix
Smat <- matrix(0, nrow = K, ncol = N)
for (i in 1:K) {
Smat[i, ] <- diag(Slist[[i]])
}
model_MetaHD_est.wscor<- MetaHD(Observed_effects ,
Smat,
method = "reml",
est.wscor = TRUE)Univariate random-effects meta analysis can be carried out as follows:
#Create matrices containing variances
diag_Slist <- list()
for (k in 1:K) {
diag_Slist[[k]] <- diag(diag(Slist[[k]]))
}
# Univariate random-effects meta-analysis
model_random <- MetaHD(Observed_effects ,
diag_Slist,
method = "reml",
bscov = "diag")Univariate fixed-effects meta analysis can be carried out as follows:
To compare the estimated values with the true values:
# Create a data frame of the differences
df <- data.frame(
Model = factor(rep(c("MetaHD", "MetaHD_est.wscor", "Random", "Fixed"),
each = N),
levels=c("MetaHD", "MetaHD_est.wscor", "Random", "Fixed")),
Difference = c( true_values- model_MetaHD$estimate,
true_values- model_MetaHD_est.wscor$estimate,
true_values -model_random$estimate,
true_values -model_fixed$estimate))
# Create the boxplots
ggplot(df, aes(x = Model, y = Difference, fill = Model)) +
geom_boxplot() +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_fill_manual(values = c("darkgreen", "red", "blue", "purple")) +
labs(title = "Differences between true and estimated values",
x = "Model",
y = "Difference (True value - Estiamted values)")
References
Liyanage, Jayamini, Luke Prendergast, Robert Staudte, and Alysha De Livera. 2024. “MetaHD: A Multivariate Meta Analysis Model for Metabolomics Data.” Submitted for Review (2024)- Article Available on Request to Authors.