-
Notifications
You must be signed in to change notification settings - Fork 1
/
JAMAP_PRS.Rmd
1218 lines (1016 loc) · 49.9 KB
/
JAMAP_PRS.Rmd
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
---
title: "Rmarkdown for: *Polygenic risk scores – from basic science to clinical application: a primer* (Published in JAMA Psychiatry) by Wray, Lin, Austin, Hickie, McGrath, Murray & Visscher"
author: "by Tian Lin & Naomi Wray (naomi.wray@uq.edu.au) - `r format(Sys.time(), '%d %B %Y')`"
output:
html_document:
toc: TRUE
code_folding: "hide"
number_sections: TRUE
#output:
# pdf_document:
# toc: true
# highlight: tango
# number_sections: true
---
The purpose of this Supplementary Material is to provide R code for making polygenic risk score (PRS) theory calculations and to generate the Figures provided in the manuscript (and Figures that didnt make it into the manuscript). Hopefully, this is useful for students and for teaching material. The is the pdf output from the RMarkdown script. The Rmarkdown file can be downloaded from https://cnsgenomics.com/content/software or https://github.com/CNSGenomics
\pagebreak
```{r, message=FALSE, warning=FALSE, eval = T}
# You need these libraries to run this template:
library(rmarkdown) # install.packages("rmarkdown")
#library(epuRate) # devtools::install_github("holtzy/epuRate", force=TRUE)
library(dplyr)
library(scales)
library(tidyverse)
library(gridExtra)
library(grid)
library(viridis)
library(emojifont)
load.fontawesome()
library(ggplot2)
library(ggpubr)
library(patchwork)
```
# Baseline calculations for Figure 1; n alleles example
## Basic model
In Figure 1, we visualise polygenic disease assuming that 900 independent causal DNA variants contribute to disease risk, for a disease of lifetime risk 1% and heritability 70%. The code to generate the data for that Figure is provided in Section 3.
Here we provide some basic calculations:
We assume
$n$ = 900 : number of DNA variants (these could be single nucleoptide polymorphisms, SNPs, or other variants)
$p$ = 0.1 : frequency of each risk associated variant, risk allele frequency; hence the protective allele frequency is (1-p)=0.9. Risk and protective alleles are relative terms
$h^2$ = 0.7 : total proportion of variance in liability explained by the 900 risk alleles
$r^2$ = 0.1 : total proportion of variance in liability explained by the PRS
We assume 900 DNA variants contribute to disease, this is just a convenient number as it allows a 30x30 square.
Most polygenic diseases have 1000's of DNA variants contributing, obviously the higher the number of contributing DNA variants the lower the expected effect size per variant. We assume that the frequency of the risk allele at each DNA variant is 0.1 and that the effect of a DNA variant is the same for all DNA variants. Again, this is for convenience. If we were to make our toy example more general by allowing risk allele frequencies to differ and effect sizes to differ, then the take-home messages are the same. Moreover in genetic studies, the key parameter is the variance explained by the DNA variant, which is a function of the allele frequency ($p$) and the effect size ($\beta$) together ($2p(1-p)\beta^2$); many combination of allele frequency and efect size give the same variance explained.
At each DNA variant, people in the population can be homozygous for the protective alleles with frequency $(1-p)^2$, or 81% when p=0.1, or heterozygous with frequency $2p(1-p)$, or 18% when p=0.1, or homozygous for the risk alleles with $p^2$, or 1% when p=0.1
In Figure 1, the red dots are the DNA variants where a person is homozygous for the risk alleles.
In 900 DNA variants we expect (that is, on average in the population), each person to be homozygous for risk alleles at 9 of the 900 DNA variants.
## Mean and variance of number of risk alleles
Using binomial distribution theory, we can work out that an average person in the population carries $2np$ = 2x 900 x 0.1 = 180 risk alleles, the two comes in because we each have two chromosomes. The variance is $2np(1-p)$ = 2 x 900 x 0.1 x 0.9 = 162, and so the standard deviation of number of risk alleles in this toy example is sqrt(162) = 13, and 95% confidence interval, i.e the likely range of the count of the risk allele variants in the population is: 155 to 205 risk alleles. Of course 2.5% of the population have more risk alleles than this, and these people are at particuarly high risk of disease, for a disease with lifetime risk of 1% (hence they feature in the top row of Figure 1). This code provides this calculations (for the rest of the pdf the code has been mostly hidden).
```{r, message=FALSE, warning=FALSE, eval = T, fig.height=8}
#Parameters to define the disease
n = 900
p = 0.1
h2 = 0.7
r2= 0.1
K = 0.01
# Population values
Meana= 2*n*p
Va= 2*n * p * (1-p)
SDa= sqrt(Va)
L95 = Meana-1.96*SDa
U95 = Meana+1.96*SDa
LSD = Meana-3.5*SDa
USD = Meana+3.5*SDa
```
Mean number or risk alleles is `r Meana` when there are `r n` DNA variants each of frequency `r p`.
Variance in number of risk alleles is `r Va` and standard deviation `r round(SDa,1)`.
95% CI range `r round(L95,0)` to `r round(U95,0)`.
Maximum range +/- 3.5SD `r round(LSD,0)` to `r round(USD,0)`.
## Within family variance
In human genetics one of the most under-recognised features of polygenic traits is that the genetic variance within families is half of the genetic variance in the population. This can be shown by considering the genetic values of a child (Achild). Across children from the same pair of parents, their average genetic value is the mean of their parents, i.e. 0.5(Amum + Adad). But the genetic value of an individual child, can be described as this mean value, plus the segregation value for the particular child, which reflects the deviation from this mean (Aseg-child).
Achild = 0.5(Amum + Adad) + Aseg-child
If we then think about the genetic variance of a generation of children from different families:
V(Achild) = 0.25Var(Amum) + 0.25Var(Adad) + Var(Aseg-child)
the terms are assumed to be independent - certainly the Aseg-child deviation is, by definition, independent of the parental values. The parental genetic values are correlated in the context of assortative mating, that doesnt impact the discussion here,which is about within family variance (this is not affected by assortative mating, but is impacted by inbreeding).
Under basic assumptions, the genetic variance of the child generation should be the same at the genetic variance of the parental generations, and the genetic variation in females is the same as males, then V(Achild) = V(Amum) = V(Adad) = V(A), where V(A) is genetic variation in the population, so then
V(Aseg-child) = V(Achild) - 0.25Var(Amum) + 0.25Var(Adad) = 0.5V(A)
There is a lot of genetic variation hidden in our genomes, half of the genetic variation in population is found between the children of any pair of parents.
Validation of this theory and modelling has been demonstrated by selection experiments (that exploit the variation generated through segregation) which selected for change in mean phenotype over generations and which were designed to see if response to selection fitted expectations predicted by polygenic models.
```{r, message=FALSE, warning=FALSE, eval = T, fig.height=8}
# Within-family variance
Vaw= 0.5*Va
SDaw= sqrt(Vaw)
L95aw=Meana-1.96*SDaw
U95aw=Meana+1.96*SDaw
LSDaw = Meana-3.5*SDaw
USDaw = Meana+3.5*SDaw
UL95 = U95-L95
UL95aw = U95aw-L95aw
ULSD = USD-LSD
ULSDaw = USDaw-LSDaw
```
95%CI range in population when n.allele =`r n` and risk allele frequency is `r p`: `r round(UL95,0)`.
95%CI range *within* families when n.allele=`r n` and risk allele frequency is `r p`: `r round(UL95aw,0)`.
+/-3 SD range in population when n.allele=`r n` and risk allele frequency is `r p`: `r round(ULSD,0)`.
+/-3 SD range *within* families when n.allele=`r n` and risk allele frequency is `r p`: `r round(ULSDaw,0)`.
## Simulating disease status for an individual
This code just shows how to simulate the disease status for an individual, for a disease underpinned by $n$ DNA risk variants each of frequency $p$, for a disease of lifetime risk $K$ and heritability $h^2$.
```{r, message=FALSE, warning=FALSE, eval = T, fig.height=8, linewidth=60}
Va=2*n*p*(1-p) # genetic variance, we use VA not VG out of convention because we are only
# considering addition genetic effects
Vp=Va/h2 # since h2= VA/VP, hence VP=VA/h2; h2 was an input parameter
Ve=Vp-Va # residual variance; by convention we use VE, with E for environment, but this
# mostly non-indentifiable random effect
tr=-qnorm(K) # normal threshold for K proportion of
# population having disease
```
Simulated individual, for disease of lifetime risk `r K*100`%, and heritability `r h2*100`%.
Count of homozygous protective, heterozygous, homozygous risk:
```{r, linewidth=60}
D=0
disease_status="UNAFFECTED"
allele = rbinom(n,2,p) # simulate n DNA variants with risk alleles having frequency p,
# 2 chromosomes
tab_allele=table(allele)
tab_allele
S=tab_allele[2]+2*tab_allele[3]
E=rnorm(1,0,sqrt(Ve)) # residual value for individual
ESDU=E/sqrt(Ve) # residual value in residual SD units
ASDU=(S-2*n*p)/sqrt(Va) # genetic value in genetic SD units
APSDU=ASDU*sqrt(h2) # genetic value in phenotypic SD units
EPSDU=ESDU*sqrt(1-h2) # residual value in phenotypic SD units
PSDU=(S+E-2*n*p)/sqrt(Vp) # phenotypic value in SD units
if(PSDU>tr){D=1;disease_status="AFFECTED"}
# assign disease status to be 1 id the Phenotypic liability is greater than the threshold
```
This individual has `r S` risk alleles across `r n` DNA variants (`r tab_allele[2]` +2* `r tab_allele[3] `), where risk alleles have frequency `r p`. Scaled in terms of liability SD units, this genetic value is `r round(ASDU,2)` genetic liability SD units or `r round(APSDU,2)` phenotypic SD units.
The non-genetic/unique environment/residual value for the individual is `r round(ESDU,2)` residual SD units, or `r round(EPSDU,2)` phenotypic SD units, so the phenotypic liability is `r round(PSDU,2)`, (i.e.,`r round(APSDU,2)` + `r round(EPSDU,2)`) phenotypic SD units.
This individual has disease status `r disease_status`. When the code is re-run, these numbers will change.
# Normal distribution theory to link different PRS prediction statistics; needed for Figure 4
Using normal distribution theory, assuming a ~N(0,1) distribution of liability in the population and assuming a known lifetime risk of disease of $K$ and a known variance explained by the PRS of $r^2$, we calculate various expected PRS statistics. This code is used to generate Figure 4 below.
With real data, the relationship between the different evaluation statistics should be chekced empirically.
The code is annotated to explain how each variable is derived.
First, we assume that the phenotypic liability of disease (P) has variance 1, and the PRS has variance $r^2$. The covariance of P with PRS is $r^2$. Amongst cases (i.e. P > T, where T is the liability threshold that bisects the normal distribution for lifetime probability of disease, K), the variance in PRS is $r^2$(1-$k$ * $r^2$), where k is the variance reduction factor $k= i*(i-T)$, where i is the mean phenotypic liability of cases which is z/K, where z is the height of the normal curve at threshold T. The variance in PRS in controls is calculated similarly with variance reduction factor of $k = v*(1-v)$, where $v= -i*K/(1-K)$.
```{r, message=FALSE, warning=FALSE, eval = T, fig.height=8}
risk_stat<-function(K,r2,x){
# K = Probability of disease (lifetime risk of disease)
# r2 = variance explained by PRS (or any predictor)
# x = must be between 0 and 1; proportion of population ranked on PRS,
# e.g. x=0.10 means top decile
T0 = qnorm(1-K) #threshold for K
z = dnorm(T0) #height of normal distrubution at threshold
i = z/K # mean phenotypic liability of cases
v = -i*K/(1-K) #mean phenotypic liability of controls
vcase=r2*(1-r2*i*(i-T0)) # variance in PRS in cases
vcont=r2*(1-r2*v*(v-T0)) # variance in PRS in controls
# Consider top x*100% of the population based on PRS`ranking
tx=qnorm(1-x,0,sqrt(r2))
pcase_popx=1-pnorm(tx,i*r2,sqrt(vcase)) # prop of cases captured when taking the top x*100% of the population ranked on PRS
pcont_popx=1-pnorm(tx,v*r2,sqrt(vcont)) # prop of controls captured when taking the top x*100% of the population ranked on PRS
oddscase=(pcase_popx/(1-pcase_popx))/(pcont_popx/(1-pcont_popx)) #odds of being a case in the top x*100% of pop ranked ib PRS
oddscase2=pcase_popx/pcont_popx
prop_casex=pcase_popx*K/x # proportion of the top x*100% ranked on PRS that are cases
#odds relative to median
sL=0.45
tL=qnorm(sL,i*r2,sqrt(vcase))
ppop_caseL=pnorm(tL,0,sqrt(r2)) # prop of pop screened to capture 45% of cases
sH=0.55
tH=qnorm(sH,i*r2,sqrt(vcase))
ppop_caseH=pnorm(tH,0,sqrt(r2)) # prop of pop screened to capture 55% of cases
# calculate AUC for variance explained on liability scaled
# probability of a case being higher ranked than a control
auc=pnorm((i-v)*r2/(sqrt(vcase+vcont)))
return(list(auc=auc,
pcase_popx=pcase_popx,
oddscase=oddscase,
oddscase2=oddscase2,
pcase_popx=pcase_popx,
pcont_popx=pcont_popx,
prop_casex=prop_casex))
}
```
## Proportion of cases captured when ranking on PRS
If prevention is targetted at the top x proportion of the population (x-axis), what proportion of cases will be captured (y-axis) is this targetted proportion.
```{r, message=FALSE, warning=FALSE, eval = T, fig.width = 4, fig.height=4}
#opar<-par(mfrow=c(2,2))
K=0.01
r2=0.10
xx=c(0,1)
xlabel="Proportion of population screened, ranked on PRS"
ylabel="Proportion of the cases in the screened set"
main_label=paste("lifetime disease risk=",K)
plot(xx,xx,ty="n",xlab=xlabel,ylab=" ",main=" ")
mtext(ylabel,side=2,line=2,col=1)
mtext(main_label,side=3,line=0,col=1)
curve(risk_stat(K,0.10,x)$pcase_popx,from=0.01,to=1,col=1,lty=1,lwd=4,add=TRUE)
curve(risk_stat(K,0.20,x)$pcase_popx,from=0.01,to=1,col=2,lty=1,lwd=4,add=TRUE)
curve(risk_stat(K,0.5,x)$pcase_popx,from=0.01,to=1,col=3,lty=1,lwd=4,add=TRUE)
leg1=paste("r2=0.1;auc =",format(round(risk_stat(K,0.10,0.5)$auc,2),nsmall=2))
leg2=paste("r2=0.2;auc =",format(round(risk_stat(K,0.20,0.5)$auc,2),nsmall=2))
leg3=paste("r2=0.5;auc =",format(round(risk_stat(K,0.50,0.5)$auc,2),nsmall=2))
legend(0.2,0.35,legend=c(leg1,leg2,leg3),col=c(1,2,3),
lty=c(1,1,1),lwd=4)
```
## Relationship between increasing proportion of variance explained by PRS and odds of being a case amongst those in the top x% of subjects based on PRS ranking
Given the variance explained by PRS on the liability scale (x-axis), what is the odds of being a case (y-axis) in the top percentile groups (lines)?
```{r, message=FALSE, warning=FALSE, eval = T, fig.width = 4, fig.height=4}
xlabel="population variance explained by PRS (r2)"
ylabel="Odds of being a case"
main_label=paste("lifetime disease risk=",K)
plot(c(0,0.25),c(1,20),ty="n",xlab="",ylab=" ",main="")
mtext(ylabel,side=2,line=2,col=1)
mtext(xlabel,side=1,line=2,col=1)
mtext(main_label,side=3,line=0,col=1)
curve(risk_stat(K,x,0.01)$oddscase,from=0.01,to=0.25,col=1,lty=1,lwd=4,add=TRUE)
curve(risk_stat(K,x,0.05)$oddscase,from=0.01,to=0.25,col=2,lty=1,lwd=4,add=TRUE)
curve(risk_stat(K,x,0.10)$oddscase,from=0.01,to=0.25,col=3,lty=1,lwd=4,add=TRUE)
curve(risk_stat(K,x,0.20)$oddscase,from=0.01,to=0.25,col=4,lty=1,lwd=4,add=TRUE)
leg1=paste("top 1%")
leg2=paste("top 5%")
leg3=paste("top 10%")
leg4=paste("top 20%")
legend(0,20,legend=c(leg1,leg2,leg3,leg4),col=c(1,2,3,4),
lty=c(1,1,1,1),lwd=4,cex=0.8)
```
## Decile plot
Odds ratio of cases status in deciles compared to first decile .
```{r, message=FALSE, warning=FALSE, eval = T, fig.width = 4, fig.height=4}
nt=10
d_case=c(rep(0,nt)); d_cont=c(rep(0,nt))
dec_case=c(rep(0,nt));dec_cont=c(rep(0,nt))
K=0.01
r2=0.10
for(i in 1:nt){
p=(1-(i/nt))
if(i==10){p=0.00000000001}
Q=risk_stat(K,r2,p)
d_case[i]=Q$pcase_popx
d_cont[i]=Q$pcont_popx
}
for (i in nt:2){
dec_case[i]=d_case[i-1]-d_case[i]
dec_cont[i]=d_cont[i-1]-d_cont[i]
}
dec_case[1]=1-d_case[1]
dec_cont[1]=1-d_cont[1]
odds=dec_case/dec_cont
oddsR=odds/odds[1]
main_label=paste("lifetime disease risk=",K, "r2=",r2)
plot(c(1:nt),oddsR,col=1,pch=17,ylim=c(0,30),xlab="decile",ylab="odds",main=main_label)
axis(side=1,at=c(1:nt), labels=c(1:nt))
lines(c(0,(nt+1)),c(1,1),lwd=1,lty=2)
```
Values for the 10 deciles compared to first decile:
`r round(oddsR,2)`
## Odds table and AUC
The top set of numbers are odds of being a case in the top proportion of people based on PRS ranke (defined by the cut proportion). AUC is the area under the receiver operator characteristic curve and can be interpreted as the probability that a case ranks higher than a control.
For the code, go to the Rmd file.
```{r, message=FALSE, warning=FALSE, eval = T, fig.width = 4, fig.height=4}
Ks=c(0.01,0.15)
cuts=c(0.01,0.05,0.10,0.20,0.50)
r2s=c(0.10,0.20)
out=matrix(c(rep(0,6*4)),nrow=6,ncol=4)
for (i in 1:5){
for(k in 1:2){
for (jj in 1:2){
j=(k-1)*2+jj
out[i,j]=risk_stat(Ks[k],r2s[jj],cuts[i])$oddscase2
if(i==1){out[6,j]=risk_stat(Ks[k],r2s[jj],cuts[i])$auc;
}
}
}
}
row.names(out)=c("cut0.01", "cut0.05","cut0.10","cut0.20","cut0.50","AUC")
colnames(out)=c("K=0.01,r2=0.1", "K=0.01,r2=0.2","K=0.10,r2=0.1","K=0.10,r2=0.2")
format(round(out,2),nsmall=2)
```
# Code for Figure 1
## Generate random genotype files
This code users a random number generator so the Figures can change with different runs of the code.
Data files for cases and controls are generated. If the files are present in your folder (i.e. you have run the code before) they will be used to generate the Figures. Delete (or rename) those files to generate new versions of the Figure.
```{r,warning=FALSE}
# this is to decide how many loci you like.
# It need to be a number that can be square rooted.
n.allele = 900
p = 0.1 #allele frequency
h2 = 0.7 #heritability
K = 0.01 #lifetime risk
```
```{r, eval = T,warning=FALSE}
## this is the function to generate a panel.
visua.profile = function(status, n.allele,p,h2,K) {
#n.allele n number of loci
# --> must be a perfect square number like 4, 16, 49, 100...
#p allele freq per locus
#h2 proprotion of variance that is genetic
#K lifetime risk of disease
n=n.allele #n number of loci
m=n.allele #total number of alleles
Ncas=5 #Number of cases
caus=c(rep(0,m)) # assign causal loci
while(sum(caus)!=n){caus=rbinom(m,1,(n/m))}
d = status
# A function that return a set of alleles for all the loci.
# d is 1 for case and 0 for control
get_riskalleles=function(n,p,h2,K,d,m,caus){
#generating cases/controls
VG=2*n*p*(1-p)
VP=VG/h2
VE=VP-VG
tr=-qnorm(K)
#case
repeat{
if(d==1){
D=0
while(D==0){
allele = rbinom(m,2,p)
S=sum(allele[caus==1])
E=rnorm(1,0,sqrt(VE))
P=(S+E-2*n*p)/sqrt(VP)
if(P>tr){D=1}
}}else{
#control
D=1
while(D==1){
allele = rbinom(m,2,p)
S=sum(allele[caus==1])
E=rnorm(1,0,sqrt(VE))
P=(S+E-2*n*p)/sqrt(VP)
if(P<tr){D=0}
}}
score = sum(allele)
return(allele)
if(P > -3 & P < 3) break
}
}
repeat{
# Run the function to get alleles
allele_vector=get_riskalleles(n,p,h2,K,d,m,caus)
score = sum(allele_vector)
for (i in 2:Ncas){
new_vector = get_riskalleles(n,p,h2,K,d,m,caus)
allele_vector=c(allele_vector, new_vector)
score = c(score, sum(new_vector))
}
#Create the complete data frame
data.case = data.frame(
patient = rep( paste0("count RV = ", score), each=m ) ,
sick = c(rep("case",(Ncas*m))),
snp = rep( paste0("snp",seq(1,m)) , (Ncas)),
allele = allele_vector,
caus=caus,
PRS = rep(score, each = m)
)
data.case$patient = as.factor(data.case$patient)
#To plot each allele I need to give a X and a Y coordinate to each allele
mydim=sqrt(m)
data.case = data.case %>%
mutate( X = rep ( rep( seq(1, mydim), each=mydim), (Ncas))) %>%
mutate( Y = rep ( rep( seq(1, mydim), mydim), (Ncas)))
if ( nlevels(data.case$patient) == 5 ) break
}
return(data.case)
}
```
```{r, eval = T, warning=FALSE}
## cases profiles
status = 1
file.name.1 = paste0("random_generated_with_", n.allele,
"_alleles_p_h2_K_", p, "_", h2,"_",
K, "_in_status", status, ".txt")
if(file.exists(file.name.1) == F) {
write.table(visua.profile(status, n.allele,p,h2,K), file =file.name.1, sep ="\t", row.names = F )
}
## controls profiles
status = 0
file.name.0 = paste0("random_generated_with_", n.allele,
"_alleles_p_h2_K_", p, "_", h2, "_",
K, "_in_status", status, ".txt")
if(file.exists(file.name.0) == F) {
write.table(visua.profile(status, n.allele,p,h2,K), file = file.name.0, sep ="\t", row.names = F )
}
```
## Plot the genomic profiles
```{r, fig.width = 8, fig.height = 4.5, warning=FALSE}
visua.plot = function(status, n.allele, data.case ){
mydim=sqrt(n.allele)
data.case$allele = as.factor(data.case$allele)
data.case$patient = factor(data.case$patient, levels = unique(data.case$patient))
anno1 = data.frame(x1 = -2*mydim/30 +1,
x2 = 1 + 5*mydim/40,
x3 = mydim - 5*mydim/40,
x4 = mydim + 2*mydim/30,
y1 = mydim + 2*mydim/30,
y2 = 0,
y3 = -8*mydim/30,
patient = factor(data.case$patient,
levels = paste0("count RV = ", unique(data.case$PRS))))
ann_text = data.frame(patient =c( paste0("count RV = ", unique(data.case$PRS))) ,
Y = -3*mydim/30, X = mydim/2+0.5,
label.col = c("a", "b", "c", "d", "e", "f",
"g", "h", "i", "j")[(6-5*status):(10-5*status)]
)
figure.cases1 = ggplot(data.case,aes(x=X,y=Y,color=allele))+
geom_point(size = 0.4*(30/mydim)^2) +
facet_grid(~patient,
scales="free",
space="free",
switch="y",
labeller = label_wrap_gen()) +
scale_y_reverse() +
theme(panel.border = element_blank(),
panel.grid.major.x = element_blank(),
panel.grid.minor.x = element_blank(),
panel.grid.major.y = element_blank(),
panel.grid.minor.y = element_blank(),
legend.position = "none",
axis.title.x=element_blank(),
axis.text.x=element_blank(),
axis.ticks.x=element_blank(),
axis.title.y=element_blank(),
axis.text.y=element_blank(),
axis.ticks.y=element_blank(),
strip.text=element_blank()) +
scale_colour_manual(values=c("grey", "blue","red")) +
geom_segment(data = anno1, aes(x = x1, xend = x1,
y = y1, yend = y2),
colour = "gray38") +
geom_segment(data = anno1, aes(x = x4, xend = x4,
y = y1, yend = y2),
colour = "gray38") +
geom_segment(data = anno1, aes(x = x1, xend = x4,
y = y1, yend = y1),
colour = "gray38")+
geom_segment(data = anno1, aes(x = x2, xend = x3,
y = y3, yend = y3),
colour = "gray38") +
geom_segment(data = anno1, aes(x = x1, xend = x2,
y = y2, yend = y3),
colour = "gray38")+
geom_segment(data = anno1, aes(x = x3, xend = x4,
y = y3, yend = y2),
colour = "gray38")
figure.cases = figure.cases1 +
geom_text(data = ann_text,
label = ann_text$patient ,
y = 3.5*mydim/30,
colour = "black",
size = 4)
}
## input the data
data.in.cases = read.table(paste0("random_generated_with_", n.allele, "_alleles_p_h2_K_",
p, "_", h2, "_", K, "_in_status1.txt"),
header = T)
data.in.controls = read.table(paste0("random_generated_with_", n.allele,
"_alleles_p_h2_K_", p, "_", h2, "_", K,
"_in_status0.txt"),
header = T)
## get the labels
mydim=sqrt(n.allele)
x.of.10 = unique(rbind(data.in.cases, data.in.controls)[,c("patient", "PRS")])
x.of.10$label.col = c("a", "b", "c", "d", "e", "f", "g", "h", "i", "j")
x.of.10$standardized.prs = round((x.of.10$PRS-2*n.allele*p)/sqrt(2*n.allele*p*(1-p)) , 2)
x.of.10$allele = NA
## combine the two panels of case and control
vis.fig1 = ggarrange( visua.plot(1, n.allele, data.in.cases) +
ggtitle("Affected over lifetime"),
visua.plot(0, n.allele, data.in.controls) +
ggtitle("Not affected over lifetime") ,
ncol = 1, nrow = 2)
ggsave(vis.fig1,
file = paste0("Figure1_", n.allele, "_alleles_p_h2_K",
p,"_",h2,"_",K,".pdf"),
width = 8,
height =4)
vis.fig1
```
## Histogram with the 10 samples marked
In this version of the Figure we tried to show where the 10 individuals were placed on a genetic liability distribution, and to illustrate within family variance.
```{r, fig.width=10, fig.height = 4, eval = T, warning=FALSE}
xl=-3
xh=3
shade_curve <- function(MyDF, zstart, zend, fill = "red", alpha = .5){
geom_area(data = subset(MyDF, x >= mean.1 + zstart*sd.1
& x < mean.1 + zend*sd.1),
aes(y=y), fill = fill, color = NA, alpha = alpha)
}
mean.1 <-0
sd.1 <- 1
zstart <- -3
zend <- 3
x = seq(from = - 3, to = 3, by = .01)
norm.plot <- ggplot(data = data.frame(x = c(xl,xh)), aes(x)) +
stat_function(fun = dnorm, args = list(mean = 0, sd = 1)) +
ylab("") +
xlab("genetic liability") +
ylim(0,0.5) +
theme(axis.text.x=element_blank(),
panel.grid.major.x = element_blank(),
panel.grid.minor.x = element_blank(),
panel.grid.major.y = element_blank(),
panel.grid.minor.y = element_blank()) +
shade_curve(MyDF = data.frame(x = x,
y = dnorm(x, mean = 0, sd = 1)),
zstart = -3,
zend = 3,
fill = "white",
alpha = .7) +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[1,]$standardized.prs,
xend = x.of.10[1,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[1,]$standardized.prs, sd = 1, mean = 0)+0.05),
color = "red") +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[2,]$standardized.prs,
xend = x.of.10[2,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[2,]$standardized.prs, sd = 1, mean = 0)+0.05),
color = "red") +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[3,]$standardized.prs,
xend = x.of.10[3,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[3,]$standardized.prs, sd = 1, mean = 0)+0.05),
color = "red") +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[4,]$standardized.prs,
xend = x.of.10[4,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[4,]$standardized.prs, sd = 1, mean = 0)+0.05),
color = "red") +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[5,]$standardized.prs,
xend = x.of.10[5,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[5,]$standardized.prs, sd = 1, mean = 0)+0.05),
color = "red") +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[6,]$standardized.prs,
xend = x.of.10[6,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[6,]$standardized.prs, sd = 1, mean = 0)),
color = "blue") +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[7,]$standardized.prs,
xend = x.of.10[7,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[7,]$standardized.prs, sd = 1, mean = 0)),
color = "blue") +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[8,]$standardized.prs,
xend = x.of.10[8,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[8,]$standardized.prs, sd = 1, mean = 0)),
color = "blue") +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[9,]$standardized.prs,
xend = x.of.10[9,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[9,]$standardized.prs, sd = 1, mean = 0)),
color = "blue") +
geom_segment(data = data.frame(x = c(xl,xh)),
aes(x = x.of.10[10,]$standardized.prs,
xend = x.of.10[10,]$standardized.prs,
y = 0,
yend = dnorm(x = x.of.10[10,]$standardized.prs, sd = 1, mean = 0)),
color = "blue") +
geom_text(data = x.of.10[x.of.10$label.col %in% c("h", "i"),],
label = x.of.10[x.of.10$label.col %in% c("h", "i"),]$label.col,
x = x.of.10[x.of.10$label.col %in% c("h", "i"),]$standardized.prs,
y = c((dnorm(x = x.of.10[x.of.10$label.col =="h",]$standardized.prs,
sd = 1, mean = 0)+ 0.05),
(dnorm(x = x.of.10[x.of.10$label.col =="i",]$standardized.prs,
sd = 1, mean = 0)+ 0.05)),
size = 5) +
ggtitle("in population")
## plot of h and i
mean.h.i = mean(x.of.10[x.of.10$label.col%in%c("h", "i"),]$standardized.prs)
parent.norm = ggplot(data = data.frame(x = c(xl,xh)), aes(x)) +
stat_function(fun = dnorm,
args = list(mean = mean.h.i,
sd = sqrt(0.5))) +
ylab("") +
xlab("genetic liability") +
ylim(0,0.8) +
theme(axis.text.x=element_blank(),
panel.grid.major.x = element_blank(),
panel.grid.minor.x = element_blank(),
panel.grid.major.y = element_blank(),
panel.grid.minor.y = element_blank())+
shade_curve(MyDF = data.frame(x = x, y = dnorm(x, mean = mean.h.i, sd = sqrt(0.5))) ,
zstart = -3,
zend = 1.5,
fill = "white",
alpha = .7) +
shade_curve(MyDF = data.frame(x = x, y = dnorm(x, mean = mean.h.i, sd = sqrt(0.5))) ,
zstart = 1.5,
zend = 3,
fill = "pink",
alpha = .7) +
geom_segment(data = data.frame(x = c(xl, xh)), aes(x = x.of.10[8,]$standardized.prs,
xend = x.of.10[8,]$standardized.prs,
y = 0.3,
yend = 0.07 +
dnorm(x = x.of.10[8,]$standardized.prs,
sd = sqrt(0.5),
mean =mean.h.i )),
linetype= "dotted") +
geom_segment(data = data.frame(x = c(xl,xh)), aes(x = x.of.10[9,]$standardized.prs,
xend = x.of.10[9,]$standardized.prs,
y = 0.3,
yend = 0.07 +
dnorm(x = x.of.10[9,]$standardized.prs,
sd = sqrt(0.5),
mean = mean.h.i)),
linetype= "dotted") +
geom_text(data = x.of.10[x.of.10$label.col %in% c("h", "i"),],
label = x.of.10[x.of.10$label.col %in% c("h", "i"),]$label.col,
x = c(x.of.10[x.of.10$label.col=="h",]$standardized.prs ,
x.of.10[x.of.10$label.col=="i",]$standardized.prs ),
y = c(dnorm(x = x.of.10[x.of.10$label.col =="h",]$standardized.prs,
sd = sqrt(0.5),
mean = mean.h.i)+ 0.12,
dnorm(x = x.of.10[x.of.10$label.col =="i",]$standardized.prs,
sd = sqrt(0.5),
mean = mean.h.i)+ 0.12),
size = 5) +
ggtitle("children of h and i")
## make labeled vis.fig1
ann_text.cases = data.frame(patient =c( paste0("count RV = ", unique(data.in.cases$PRS))) ,
Y = -3*mydim/30, X = mydim/2+0.5,
label.col = c("a", "b", "c", "d", "e", "f",
"g", "h", "i", "j")[1:5]
)
ann_text.controls = data.frame(patient =c( paste0("count RV = ", unique(data.in.controls$PRS))) ,
Y = -3*mydim/30, X = mydim/2+0.5,
label.col = c("a", "b", "c", "d", "e", "f",
"g", "h", "i", "j")[6:10]
)
vis.fig1 = ggarrange( visua.plot(1, n.allele, data.in.cases) +
ggtitle("Affected over lifetime") +
geom_text(data =ann_text.cases,
label = ann_text.cases$label.col,
x = 0,
y = 6*mydim/30,
colour = c("blue","red")[2],
size = 6),
visua.plot(0, n.allele, data.in.controls) +
ggtitle("Not affected over lifetime") +
geom_text(data =ann_text.controls,
label = ann_text.controls$label.col,
x = 0,
y = 6*mydim/30,
colour = c("blue","red")[1],
size = 6),
ncol = 1, nrow = 2)
## combine
norm.both = ggarrange(norm.plot, parent.norm, ncol = 1)
final.fig1 = ggarrange(vis.fig1, norm.both, widths = c(7,2))
final.fig1
## save it.
ggsave(final.fig1,
file = paste0("Figure1_", n.allele, "_alleles_p_h2_K",
p,"_",h2,"_",K,"_with_dnorm.pdf"),
width = 10,
height =4)
```
# Code for Figure 4
## Calculate the risk fold and AUC
We consider a disease for which we know lifetime risk and variance explained by PRS currently and in the future. From this information we can calculate a number of statistics, which follow from normal distribution theory. These calculations hold for polygenic disease. The assumptions break down when there are common variants of very large effect (as these impact the uniomodality assumption), more complex modelling is needed in those cases.
```{r, message=FALSE, warning=FALSE, eval = T, fig.height=8}
## SCZ
KSCZ=0.01 #lifetime risk schziophrenia
r2SCZ=0.11 #r2 explained by PRS currently for SCZ approx
r2SCZF=0.25 #r2 explained by PRS future for SCZ approx
nameSCZ="schizophrenia"
SCZ10=risk_stat(KSCZ,r2SCZ,0.10)$prop_casex # SCZ now top 10%
SCZ1=risk_stat(KSCZ,r2SCZ,0.01)$prop_casex # SCZ now top 1%
SCZ10F=risk_stat(KSCZ,r2SCZF,0.10)$prop_casex # SCZ future top 10%
SCZ1F= risk_stat(KSCZ,r2SCZF,0.01)$prop_casex # SCZ future top 1%
SCZ.risk.table = data.frame(matrix(c(
r2SCZ,
r2SCZF,
SCZ10,
SCZ10F,
SCZ1,
SCZ1F,
risk_stat(KSCZ,r2SCZ,0.10)$auc ,
risk_stat(KSCZ,r2SCZF,0.10)$auc,
risk_stat(KSCZ,r2SCZ,0.10)$prop_casex/KSCZ , # SCZ now
risk_stat(KSCZ,r2SCZ,0.01)$prop_casex/KSCZ , # SCZ now
risk_stat(KSCZ,r2SCZF,0.10)$prop_casex/KSCZ, # SCZ future
risk_stat(KSCZ,r2SCZF,0.01)$prop_casex/KSCZ # SCZ future
),
ncol = 2, byrow = T))
row.names(SCZ.risk.table) = c("r2- var explained by PRS",
"prop of cases in top 10% PRS",
"prop of cases in top 1%" ,
"AUC",
"Risk fold in top 10% PRS",
"Risk fold in top 1% PRS")
colnames(SCZ.risk.table) = c("current", "future")
```
With lifetime risk of `r nameSCZ` of `r KSCZ*100`% and variance explained by PRS currently of `r r2SCZ*100`% and variance explained by PRS in future of `r r2SCZF*100`%
```{r}
round(SCZ.risk.table, 2)
```
```{r, message=FALSE, warning=FALSE, eval = T, fig.height=8}
## MD
nameMD="major depression"
KMD=0.15 #lifetime major depression
r2MD=0.04 #r2 explained by PRS currently for MD approx
r2MDF=0.12 #r2 explained by PRS future for MD approx
MD10=risk_stat(KMD,r2MD,0.10)$prop_casex # MD now top 10%
MD1=risk_stat(KMD,r2MD,0.01)$prop_casex # MD now top 1%
MD10F=risk_stat(KMD,r2MDF,0.10)$prop_casex # MD future top 10%
MD1F=risk_stat(KMD,r2MDF,0.01)$prop_casex # MD future top 1%
MD.risk.table = data.frame(matrix(c(
r2MD,
r2MDF,
MD10,
MD10F,
MD1,
MD1F,
risk_stat(KMD,r2MD,0.10)$auc ,
risk_stat(KMD,r2MDF,0.10)$auc,
risk_stat(KMD,r2MD,0.10)$prop_casex/KMD , # MD now
risk_stat(KMD,r2MD,0.01)$prop_casex/KMD , # MD now
risk_stat(KMD,r2MDF,0.10)$prop_casex/KMD, # MD future
risk_stat(KMD,r2MDF,0.01)$prop_casex/KMD # MD future
), ncol = 2, byrow = T))
row.names(MD.risk.table) = c("R2",
"prop of cases in top 10% PRS",
"prop of cases in top 1%" ,
"AUC",
"Risk fold in top 10% PRS",
"Risk fold in top 1% PRS")
colnames(MD.risk.table) = c("current", "future")
```
With lifetime risk of `r nameMD` of `r KMD*100`% and "variance explained by PRS currently of `r r2MD*100`% and variance explained by PRS in future of `r r2MDF*100`%
```{r, echo= FALSE}
round(MD.risk.table, 2)
```
## Plotting subroutine
```{r, message=FALSE, warning=FALSE, eval = T, fig.width = 15, fig.height = 12}
#setwd("~/Documents/Naomi/Collaborators/F-K/GrahamMurray/PRS/Figures")
## define the plot function
incidence.plot = function(percentage){
## generage a table
n.case = 100*percentage
n.control = 100 - n.case
sample = data.frame(matrix(NA, nrow = 100, ncol =3))
colnames(sample) = c("row", "column", "color")
sample$row = rep(c(1:10), 10)
sample$column = sort(rep(c(1:10), 10))
sample$color = as.factor(c(rep(1, n.case), rep(2, n.control)))
## generate the plot
sample$labs = fontawesome('fa-male')
example.risk.plot = ggplot(sample,aes(x=row,y=column,color=color))+
geom_text(aes(label=labs),
family='fontawesome-webfont',
size=8,
colors = c("red","blue"))+
scale_y_continuous(labels = comma)+
scale_y_reverse() +
theme(panel.border = element_blank(),
panel.grid.major.x = element_blank(),
panel.grid.minor.x = element_blank(),
panel.grid.major.y = element_blank(),
panel.grid.minor.y = element_blank(),
legend.position = "none",
axis.title.x=element_blank(),
axis.text.x=element_blank(),
axis.ticks.x=element_blank(),
axis.title.y=element_blank(),
axis.text.y=element_blank(),
axis.ticks.y=element_blank()) +
scale_colour_manual(values=c("blue", "red"))
#print (example.risk.plot)
}
```
## One disease only
```{r, message=FALSE, warning=FALSE, eval = T, fig.width = 15, fig.height = 12, echo= FALSE}
label.size = 7
## prepare the labels
ann_text1 = data.frame(row = 7.5,column = 8.5,
notes = "3.3 fold")
#labels
lab2=paste(round(risk_stat(KSCZ,r2SCZ,0.10)$prop_casex/KSCZ,1),"fold")
lab3=paste(round(risk_stat(KSCZ,r2SCZ,0.01)$prop_casex/KSCZ,1),"fold")
lab4=paste(round(risk_stat(KSCZ,r2SCZF,0.10)$prop_casex/KSCZ,1),"fold")
lab5=paste(round(risk_stat(KSCZ,r2SCZF,0.01)$prop_casex/KSCZ,0),"fold")
## make the 5 plots with different liability
plot1 = incidence.plot(round(KSCZ,2))
plot2 = incidence.plot(round(SCZ10,2)) +
geom_label(data = ann_text1, label = lab2, color = "black", size = label.size +1 )
plot3 = incidence.plot(round(SCZ1,2)) +
geom_label(data = ann_text1, label = lab3, color = "black", size = label.size +1 )
plot4 = incidence.plot(round(SCZ10F,2)) +
geom_label(data = ann_text1, label = lab4, color = "black", size = label.size +1 )
plot5 = incidence.plot(round(SCZ1F,2)) +
geom_label(data = ann_text1, label = lab5, color = "black", size = label.size +1 )
## 1 ####
my_text11 = "Approximately\nrepresentative of"
my_text12 = paste0(nameSCZ, '\nlife-time risk ', KSCZ*100 ,"%")
note1 = ggplot() +
lims(x = c(0,10), y = c(0,10)) +
annotate('text', x = 5, y = 5.5, label = my_text11, size = label.size)+
annotate('text', x = 5, y = 3.5, label = my_text12, size = label.size, fontface="bold")+
theme_bw() +
theme(panel.border = element_blank(),
panel.grid.major.x = element_blank(),
panel.grid.minor.x = element_blank(),
panel.grid.major.y = element_blank(),
panel.grid.minor.y = element_blank(),
legend.position = "none",
axis.title.x=element_blank(),
axis.text.x=element_blank(),
axis.ticks.x=element_blank(),
axis.title.y=element_blank(),
axis.text.y=element_blank(),
axis.ticks.y=element_blank())
## 2 ####
my_text21 = "Approximately\nrepresentative of"
my_text22 = paste0(nameMD, "\nlife-time risk ", KMD*100 ,"%")
note2 = ggplot() +
lims(x = c(0,10), y = c(0,10)) +
annotate('text', x = 5, y = 5.5, label = my_text21, size = label.size)+
annotate('text', x = 5, y = 3.5, label = my_text22, size = label.size, fontface="bold")+
theme_bw() +
theme(panel.border = element_blank(),
panel.grid.major.x = element_blank(),
panel.grid.minor.x = element_blank(),
panel.grid.major.y = element_blank(),
panel.grid.minor.y = element_blank(),
legend.position = "none",
axis.title.x=element_blank(),
axis.text.x=element_blank(),
axis.ticks.x=element_blank(),
axis.title.y=element_blank(),
axis.text.y=element_blank(),
axis.ticks.y=element_blank())
## 3-5 #############################################
text3 = data.frame(row =1,column = 1, notes = "100 people\n random from\n population" )
note3 = ggplot(data = text3, aes(x =row, y = column)) +
geom_text(aes(label=notes), size = label.size) +
theme_bw() +
theme(panel.border = element_blank(),
panel.grid.major.x = element_blank(),
panel.grid.minor.x = element_blank(),