Introduction
Principal component analysis (PCA) provides an effective statistical
approach for exploiting the patterns in CD4 count and viral load data
over time. The method defines a new variable related to adherence to
antiretroviral therapy (ART) that captures information from longitudinal
data using feature extraction properties of PCA which is demonstrated
using data from patients who acquired HIV-1 during follow-up in an ART
cohort and were subsequently followed prospectively from early
infection.
The PCA scores for each patient obtained by this method serve as
informative summary statistics for the CD4-count and viral-load
trajectories. Similar to baseline CD4 count or viral load, the first PCA
score can be interpreted as a single-value summary measure of an
individual’s overall treatment response to ART, but unlike most
single-value summaries of CD4-count or viral-load trajectories, the
first PCA score summarizes the dynamics of these quantities and reveals
specific features of the trajectories associated with the effectiveness
of the adherence of ART. Moreover, PCA scores are used as powerful
prognostic factor than other common summaries when used in predictive
analysis.
Doing a PCA under tidy principles requires running the function
PCA() from FactoMineR package on the matrix of
scaled numeric predictor variables, and then visualizing the result
nicely using the factoextra package.
In general, when performing PCA, three things are wanted: 1) Look at
the data in PC coordinates; 2) Look at the rotation matrix; and 3) Look
at the variance explained by each principal component (PC).
library(FactoMineR)
data(viral_new, package = "viruslearner")
res_pca <- PCA(viral_new, graph = FALSE)
print(res_pca)
#> **Results for the Principal Component Analysis (PCA)**
#> The analysis was performed on 87 individuals, described by 9 variables
#> *The results are available in the following objects:
#>
#> name description
#> 1 "$eig" "eigenvalues"
#> 2 "$var" "results for the variables"
#> 3 "$var$coord" "coord. for the variables"
#> 4 "$var$cor" "correlations variables - dimensions"
#> 5 "$var$cos2" "cos2 for the variables"
#> 6 "$var$contrib" "contributions of the variables"
#> 7 "$ind" "results for the individuals"
#> 8 "$ind$coord" "coord. for the individuals"
#> 9 "$ind$cos2" "cos2 for the individuals"
#> 10 "$ind$contrib" "contributions of the individuals"
#> 11 "$call" "summary statistics"
#> 12 "$call$centre" "mean of the variables"
#> 13 "$call$ecart.type" "standard error of the variables"
#> 14 "$call$row.w" "weights for the individuals"
#> 15 "$call$col.w" "weights for the variables"
To look at the data in PC coordinates requires combining the PC
coordinates with the original dataset. This is done via the
res_pca$ind$coord object. The columns containing the fitted
coordinates are called Dim.1, Dim.2, etc.
The transformation of variables shown in Figure @ref(fig:plotcoords),
in which PCA scores are prognostic factors, reflects the clinical belief
that adherence to ART may be affected by the change and other patterns
of CD4 counts and viral loads over time. The approach uses PCA with data
from the entire cohort to extract the primary structure in individual
trajectories and provides a concise summary of each trajectory (PCA
score) to reveal clinical features of the CD4 and viral load
trajectories that serve as an ART adherence transformed predictor
variable.
PC1 vs PC2 dispersion. Specific features of the
shape of ART trajectory as prognostic factor for overall
treatment.
Rotation matrix
The rotation matrix is stored as the res_pca$var$coord
object. The rotation matrix is essential for understanding the
relationship between original variables and principal components.
res_pca$var$coord
#> Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
#> cd_2018 0.27870555 -0.04062273 -0.26163183 0.873137932 0.21538696
#> cd_2019 0.64518533 0.03420379 -0.12376197 0.065834990 0.24795163
#> vl_2019 -0.07573479 -0.07003117 0.78505366 0.031053421 0.60176292
#> cd_2021 0.76117069 0.13314906 0.25218852 -0.036074808 -0.08770293
#> vl_2021 -0.34552296 0.74720041 -0.07602424 0.071507915 0.08454595
#> cd_2022 0.85936771 0.18392938 0.10428379 -0.005222839 -0.15593741
#> vl_2022 -0.17860498 0.82464513 0.11759695 0.051856613 -0.02878516
#> cd_2023 0.86609530 0.11313863 0.11020329 -0.100126342 -0.15036423
#> vl_2023 -0.30954010 -0.10341100 0.54940982 0.422680202 -0.58356162
A negative value indicates an inverse relationship with each PC.
Patients with lower baseline CD4 counts contribute more to the negative
direction of PC’s. A positive value suggests a positive relationship
with each PC. Patients with higher baseline viral loads contribute more
to the positive direction of PC’s.
Figure @ref(fig:plotrot) shows the rotation matrix in the context of
a plot. The correlation between a variable and a PC is used as the
coordinates of the variable on the PC. The correlation circle plot shows
the relationships between all variables, with positively correlated
variables shown grouped together, negatively correlated variables are
positioned on opposite sides of the plot origin (opposed quadrants), and
the distance between variables and the origin measures the quality of
the variables on the factor map.
Distribution of patients in the PC1 vs PC2
space. Contributions of baseline and longitudinal measures are reflected
by the directions and magnitudes of the arrows.
Patients with lower baseline values are positioned more towards the
negative direction of PC1 and patients with higher baseline values are
positioned more towards the positive direction of PC1. Patients with
slightly decreasing baseline values are positioned more towards the
negative direction of PC2, and those with slightly increasing baseline
values are positioned more towards the positive direction of PC2.
Patients with similar patterns in CD4 counts and viral loads are
likely to cluster together in the PC1 vs PC2 dispersion graph with the
direction of the contribution values indicating how each variable
influences the position of patients along PC1 and PC2 axes. Lower
baseline CD4 counts, higher baseline viral loads, decreasing CD4 counts,
and decreasing viral loads tend to be grouped together based on the
negative directions of PC1 and PC2. Conversely, slightly decreasing
baseline CD4 counts, slightly increasing baseline viral loads,
decreasing CD4 counts, and increasing viral loads are grouped together
based on the positive directions of PC1 and PC2.
Variance explained by each PC
The variance explained by each PC can be extracted via the function
get_eigenvalue.
eig_val <- get_eigenvalue(res_pca)
eig_val
#> eigenvalue variance.percent cumulative.variance.percent
#> Dim.1 2.8147923 31.275470 31.27547
#> Dim.2 1.3211254 14.679171 45.95464
#> Dim.3 1.1081563 12.312848 58.26749
#> Dim.4 0.9654834 10.727593 68.99508
#> Dim.5 0.8731286 9.701429 78.69651
#> Dim.6 0.6741441 7.490490 86.18700
#> Dim.7 0.5923136 6.581263 92.76826
#> Dim.8 0.4196698 4.662998 97.43126
#> Dim.9 0.2311864 2.568738 100.00000
And is shown in the context of a plot in Figure
@ref(fig:plotvarpc).
Variance captured by each PC. Together, PC1 and
PC2 provide structure into the factors influencing ART adherence and
treatment effectiveness.
The first component captures 45.2% of the variation in the data and
together with the second component approximately 80% of the variability
is captured. Eigenvalue greater than 1 indicate those PCs that account
for more variance than accounted by one of the original variables in
standarized data, so it becomes a commonly used cutoff point for which
PCs are retained.
PC’s and other variables in the data
The association of the outcome features (cd_2022 and vl_2022) of the
data with the first two principal components is shown in the scatter
plots of Figure @ref(fig:outvspc).
Outcome variables vs first two PC’s. A,
Association of PC1 with cd_2022. B, Association of PC1 with vl_2022. C,
Association of PC2 with cd_2022. D, Association of PC2 with
vl_2022.
Individual coordinates as scores
The extracted coordinates for the individual observations serve as
the transformed variable for scoring the adherence to ART. These values
get stored in the res_pca$ind$coord object.
res_pca$ind$coord
#> Dim.1 Dim.2 Dim.3 Dim.4 Dim.5
#> 1 -0.98870520 -0.5405155102 -0.383502511 0.11758005 0.3573502180
#> 2 -0.68151256 -0.3917371875 -0.536657820 0.63130683 0.2126490172
#> 3 0.57755011 0.0003973339 -0.100869418 -0.09672156 -0.0800287624
#> 4 2.43507136 0.4466142907 0.699071390 -0.52886808 -0.6855489745
#> 5 0.19217912 -0.1142184061 -0.028523841 0.21826272 -0.1810635238
#> 6 -1.51331100 -0.5031996885 -0.448863209 -0.04448105 0.2568046717
#> 7 1.40048059 -0.5320656166 -2.337653919 7.53067393 1.9670461536
#> 8 1.55749377 0.1253492882 0.284136941 -0.07535803 -0.2796376847
#> 9 -1.64727635 -0.5781447220 -0.482786717 -0.27219243 0.1612452440
#> 10 -1.67709265 -0.6461470595 -0.616179382 -0.36691068 0.4267772631
#> 11 -2.68138741 -0.7014226723 -0.533090186 -0.42818750 -0.0446504283
#> 12 -0.34904282 -0.3529301054 -0.553990858 -0.11787367 0.4174126575
#> 13 3.05438607 0.3235781005 -0.323566625 0.40517362 0.1227363909
#> 14 -1.31307839 0.5159140137 0.900200003 -0.16693948 1.1948880288
#> 15 0.41711257 -0.1168868326 -0.043978876 -0.17905349 -0.1864074227
#> 16 0.40319571 -0.1713163002 -0.557921669 0.61280572 0.6199189632
#> 17 -1.44667518 0.0311466462 -0.224201788 -0.37717503 0.1422834188
#> 18 1.32279186 0.0716430706 -0.023473630 0.11602289 -0.0850725955
#> 19 -1.18967950 -0.4688077427 -0.384614984 -0.31641020 0.0140775889
#> 20 2.44633402 0.2508076295 0.039074793 0.18016519 0.0383323379
#> 21 0.62139003 0.0502922717 0.322809044 -0.81226127 -0.7921883905
#> 22 0.38001562 -0.1303155323 -0.250956909 0.62257892 -0.0560720836
#> 23 0.48687394 0.0806233493 -0.027685311 -0.51955124 -0.3001967630
#> 24 -1.79190896 -0.5895877612 -0.429458041 -0.47314270 -0.0829786697
#> 25 0.12407424 -0.3117160968 -0.296564871 -0.04460880 0.2570393396
#> 26 1.55601245 -0.0069240447 -0.653396942 0.36426775 0.5093770890
#> 27 0.22951246 -0.1510644844 0.019957383 -0.28813007 -0.1457492868
#> 28 0.57390223 0.0046820600 0.046540072 -0.43762365 -0.3902294005
#> 29 0.73309854 -0.1101504847 -0.276041977 1.01379683 -0.0271487190
#> 30 -4.62647652 -1.5830336096 5.353325336 3.81577233 -5.0454807764
#> 31 0.53180432 -0.0395805398 0.115362542 -0.72131663 -0.4635237465
#> 32 -0.87666139 -0.4228828798 -0.276286449 -0.21336956 0.0129220628
#> 33 1.77438416 0.1777073962 -0.069467327 0.48157466 -0.0812141341
#> 34 0.74548656 -0.0367837441 -0.059557020 0.01273757 -0.0785535028
#> 35 -0.62274769 -0.3441353458 -0.202906963 -0.12698436 -0.0827287253
#> 36 2.36639454 0.2830740834 0.482457920 -0.16748537 -0.4061329146
#> 37 -0.25158190 -0.3276692868 -0.486333352 -0.01520842 0.7739732192
#> 38 -1.31337910 -0.4043508899 -0.129280826 -0.60200783 -0.1982142235
#> 39 -2.14767808 8.3465508538 0.930016658 0.52344357 -0.5234965121
#> 40 0.85515110 0.0266804074 0.265865181 -0.40060584 -0.4398120854
#> 41 0.55270440 -0.0792691160 0.007468191 -0.14163775 -0.2321123096
#> 42 -1.20220290 -0.4900904558 -0.372030506 -0.42609660 0.0352072741
#> 43 -0.83946495 -0.3412849549 -0.161729926 -0.42599418 -0.1711841934
#> 44 0.39343074 -0.1939855459 -0.346757257 -0.13461730 0.2219048244
#> 45 -1.45420619 -0.5016847594 -0.196978869 -0.57656854 -0.0177663974
#> 46 0.48486528 -0.0204741505 -0.069404591 -0.47503675 -0.0887807342
#> 47 -4.91847286 5.4350935095 -1.126008885 0.51424312 0.9472752765
#> 48 1.13479028 0.1370867673 0.264652052 -0.07473402 -0.8316267457
#> 49 2.27879028 0.2770138821 -0.120278510 -0.16561978 -0.4332859531
#> 50 2.71605796 0.2629746063 0.185099586 0.27460016 0.0956020524
#> 51 0.23240636 -0.0369567364 0.339201950 -0.74403402 -0.5957382520
#> 52 -0.40654486 -0.2129938055 0.007364296 -0.51889936 -0.4270906197
#> 53 1.24428239 0.1493784888 -0.031411269 -0.48587751 -0.2972151901
#> 54 -0.29539979 -0.2117351780 -0.012914211 -0.13535101 -0.3680289293
#> 55 -0.83044701 -0.3450922252 -0.229181356 -0.26270782 -0.0421493294
#> 56 -1.05362652 -0.4926583759 -0.453822550 0.07096787 0.1738621313
#> 57 0.01031647 -0.2307256977 -0.338203562 0.22398656 0.1209286166
#> 58 1.60733633 0.1558112991 0.141661709 -0.45410522 -0.2835030517
#> 59 -2.73214608 -0.8254271651 -0.614658499 -0.46792962 0.3046034945
#> 60 0.74642343 -0.0768789780 0.067742274 -0.19745048 -0.2340123804
#> 61 -1.22977262 -0.4115264528 -0.063764598 -0.62886181 -0.2280905536
#> 62 0.71621951 -0.0294710947 -0.012392441 -0.18788655 -0.2850253030
#> 63 -1.23238510 -0.4517404722 -0.547464672 0.07551622 0.1127870993
#> 64 -1.10159090 -0.5078295697 -0.719780641 -0.14225313 0.5863538021
#> 65 0.58735640 -0.0685325227 -0.044355792 -0.26500383 -0.0040050172
#> 66 1.00314451 -0.0288961090 0.071213248 -0.18368757 -0.0972814613
#> 67 0.79295082 0.0678617926 0.112018151 -0.30775994 -0.1565951400
#> 68 -2.71981554 -0.7944818559 -0.695617684 0.03478114 0.0283567255
#> 69 -0.16434052 -0.3126216383 -0.547949389 -0.13846149 0.2903203341
#> 70 3.93328382 0.6072039993 0.313854170 0.52356243 -0.3724959129
#> 71 -2.06398342 -0.6271244583 -0.266960841 -0.58047286 -0.1158176662
#> 72 0.40636505 -0.1103113427 -0.299305868 0.07230995 0.2284518215
#> 73 -0.30923708 0.2077146091 -0.252024774 -0.06269505 0.2561422396
#> 74 -1.90970358 -0.4929084137 -0.131379749 -0.52761419 -0.4296979594
#> 75 1.98265024 0.3976080356 0.850942882 -0.74286984 -1.1541846965
#> 76 -3.09730941 -0.8451052258 -0.605125093 0.06885556 0.3872850584
#> 77 -1.37492155 -0.5191609467 -0.374699021 -0.25565095 0.0003256777
#> 78 3.09281666 0.4757441197 0.492893156 -0.11716075 -0.5384380202
#> 79 2.20764458 0.3445328342 0.095728043 -0.07602299 -0.3573542786
#> 80 -0.86556815 -0.4551266749 0.853133716 -0.25119432 1.1167943694
#> 81 4.36106362 0.6709446328 0.494252043 0.33963788 -0.2263232092
#> 82 -0.72834646 -0.3606536878 -0.358613386 -0.15867011 0.1403901733
#> 83 0.85408000 1.0833476133 0.248846014 -0.28400861 -0.2581958862
#> 84 -0.80503884 -0.5735560973 -1.026307994 0.48184166 1.1058518237
#> 85 -0.67080711 -0.4005282405 -0.354056712 -0.37497694 0.1570103861
#> 86 -0.14576387 -0.5495637998 6.891136868 -0.16009620 5.6521120979
#> 87 1.14561551 0.1666053059 0.214964455 -0.40199106 -0.5442704268