/
theo.R
378 lines (340 loc) · 14.6 KB
/
theo.R
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
########################
## Theoretical genotype frequencies from Huang et al (2019)
########################
################
## The following are functions from Supplementary Material S3 of Huang et al (2019)
## https://doi.org/10.1534/g3.119.400132
## That I have converted from C code to R code
################
GFG4_ii <- function(a1, pi) {
(pi*(a1*(3.0-2.0*pi)+2.0*pi))/(2.0+a1)
}
GFG4_ij <- function(a1, pi, pj) {
-((4.0*(-1.0+a1)*pi*pj)/(2.0+a1))
}
GFG6_iii <- function(a1, pi) {
pi2 <- pi*pi
a12 <- a1*a1
(pi*(81.0*pi2-27.0*a1*pi*(-5.0+4.0*pi)+a12*(20.0-45.0*pi+27.0*pi2)))/(81.0+27.0*a1+2.0*a12)
}
GFG6_iij <- function(a1, pi, pj) {
a12 <- a1*a1
(9.0*(-3.0+a1)*pi*(-9.0*pi+a1*(-5.0+9.0*pi))*pj)/(81.0+27.0*a1+2.0*a12)
}
GFG8_iiii <- function(a1, a2, pi) {
pi2 <- pi*pi
pi3 <- pi2*pi
a12 <- a1*a1
a13 <- a12*a1
a14 <- a13*a1
a22 <- a2*a2
a23 <- a22*a2
a24 <- a23*a2
C1 <- (-35.0+210.0*pi-336.0*pi2+160.0*pi3)
-(1.0/((34.0+a1+a2)*(8.0+a1+2.0*a2)^2*(a1+2.0*(6.0+a2))))*pi*(a14*C1+a13*(-2.0*(140.0-301.0*pi+112.0*pi2+80.0*pi3)+7.0*a2*C1)+2.0*a12*(-4.0*pi*(1519.0-2632.0*pi+1264.0*pi2)+9.0*a22*C1-2.0*a2*(455.0-945.0*pi+336.0*pi2+240.0*pi3))+8.0*(-3264.0*pi3+32.0*a2*pi*(-105.0-147.0*pi+181.0*pi2)-2.0*a23*(175.0-343.0*pi+112.0*pi2+80.0*pi3)+a24*C1-4.0*a22*(210.0+609.0*pi-1316.0*pi2+632.0*pi3))+4.0*a1*(32.0*pi2*(-357.0+283.0*pi)+5.0*a23*C1-2.0*a22*(490.0-987.0*pi+336.0*pi2+240.0*pi3)-8.0*a2*(105.0+1064.0*pi-1974.0*pi2+948.0*pi3)))
}
GFG8_iiij <- function(a1, a2, pi, pj)
{
pi2 <- pi*pi
a12 <- a1*a1
a13 <- a12*a1
a14 <- a13*a1
a22 <- a2*a2
a23 <- a22*a2
C1 <- (7.0-42.0*pi+40.0*pi2)
-(1.0/((34.0+a1+a2)*(8.0+a1+2.0*a2)^2*(a1+2.0*(6.0+a2))))*8.0*pi*(2.0*a14*C1+a13*(-7.0-56.0*pi-80.0*pi2+14.0*a2*C1)+16.0*(-1.0+a2)*(a22*(28.0-70.0*pi)+4.0*a2*(147.0-158.0*pi)*pi+816.0*pi2+a23*C1)+4.0*a12*(9.0*a22*C1-2.0*a2*(-7.0+42.0*pi+60.0*pi2)-2.0*(119.0-658.0*pi+632.0*pi2))+4.0*a1*(8.0*pi*(-357.0+566.0*pi)+a22*(77.0-168.0*pi-240.0*pi2)+10.0*a23*C1-4.0*a2*(133.0-987.0*pi+948.0*pi2)))*pj
}
GFG8_iijj <- function(a1, a2, pi, pj) {
a12 <- a1*a1
a13 <- a12*a1
a14 <- a13*a1
a22 <- a2*a2
a23 <- a22*a2
a24 <- a23*a2
C1 <- (49.0-84.0*pj+12.0*pi*(-7.0+20.0*pj))
C2 <- (7.0+30.0*pj)
C3 <- (-329.0+948.0*pj)
-(1.0/((34.0+a1+a2)*(8.0+a1+2.0*a2)^2*(a1+2.0*(6.0+a2))))*4.0*pi*pj*(a14*C1+a13*(329.0-56.0*pj-8.0*pi*C2+7.0*a2*C1)+2.0*a12*(-1134.0+2632.0*pi+2632.0*pj-7584.0*pi*pj+9.0*a22*C1-a2*(-833.0+168.0*pj+24.0*pi*C2))+8.0*(-4896.0*pi*pj+24.0*a2*(-70.0-49.0*pi-49.0*pj+362.0*pi*pj)+a24*C1-a23*(7.0*(-25.0+8.0*pj)+8.0*pi*C2)-2.0*a22*(497.0-658.0*pj+2.0*pi*C3))+4.0*a1*(5.0*a23*C1-a22*(7.0*(-97.0+24.0*pj)+24.0*pi*C2)+24.0*(-119.0*pj+pi*(-119.0+566.0*pj))-4.0*a2*(532.0-987.0*pj+3.0*pi*C3)))
}
GFG10_iiiii <- function(a1, a2, pi) {
pi2 <- pi*pi
pi3 <- pi2*pi
pi4 <- pi3*pi
a12 <- a1*a1
a13 <- a12*a1
a14 <- a13*a1
a15 <- a14*a1
a22 <- a2*a2
a23 <- a22*a2
a24 <- a23*a2
a25 <- a24*a2
C1 <- (a1+2.0*(-5.0+a2))
C2 <- (20.0+a2)
(pi*(3125.0*pi4*C1*(-1.0+a1+a2)*(25000.0+a1*(-11050.0+9.0*a1*(75.0+7.0*a1))-21300.0*a2+a1*(2650.0+357.0*a1)*a2+8.0*(325.0+84.0*a1)*a22+420.0*a23)-750.0*pi*(45.0*a13*(-992.0+a1*(43.0+4.0*a1))+2.0*a1*(-133000.0+a1*(-104590.0+a1*(8039.0+780.0*a1)))*a2+2.0*(-255500.0+a1*(-154580.0+a1*(24727.0+2670.0*a1)))*a22+40.0*(-3468.0+5.0*a1*(334.0+45.0*a1))*a23+8.0*(4181.0+930.0*a1)*a24+2400.0*a25)+504.0*(a1+2.0*a2)*(25.0+2.0*a1+4.0*a2)*(9.0*a13+42.0*a12*a2+30.0*a22*C2+7.0*a1*a2*(50.0+9.0*a2))-18750.0*pi3*C1*(27.0*a14+6.0*a13*(59.0+30.0*a2)+10.0*(-1.0+a2)*a2*(-625.0+a2*(173.0+18.0*a2))+a12*(-4290.0+7.0*a2*(256.0+63.0*a2))+a1*(7500.0+a2*(-12395.0+9.0*a2*(327.0+52.0*a2))))+375.0*pi2*(1161.0*a15+234.0*a14*(34.0+43.0*a2)+9.0*a13*(-36655.0+a2*(7172.0+3827.0*a2))+2.0*a1*a2*(1510375.0+3.0*a2*(-430805.0+42236.0*a2+7998.0*a22))+10.0*a12*(129375.0+a2*(-162814.0+15.0*a2*(1289.0+387.0*a2)))+20.0*a2*(109375.0+a2*(50850.0+a2*(-64813.0+6126.0*a2+774.0*a22))))))/(2.0*(125.0+a1+a2)*(25.0+2.0*a1+4.0*a2)^2*(50.0+3.0*a1+6.0*a2)*(3.0*a1+5.0*C2))
}
GFG10_iiiij <- function(a1, a2, pi, pj) {
pi2 <- pi*pi
pi3 <- pi2*pi
a12 <- a1*a1
a13 <- a12*a1
a14 <- a13*a1
a15 <- a14*a1
a22 <- a2*a2
a23 <- a22*a2
C1 <- (a1+2.0*(-5.0+a2))
-(25.0*pi*(-625.0*pi3*C1*(-1.0+a1+a2)*(25000.0+a1*(-11050.0+9.0*a1*(75.0+7.0*a1))-21300.0*a2+a1*(2650.0+357.0*a1)*a2+8.0*(325.0+84.0*a1)*a22+420.0*a23)+84.0*(a1+2.0*a2)*(25.0+2.0*a1+4.0*a2)*(9.0*a13+3.0*a12*(-75.0+14.0*a2)+a1*a2*(-415.0+63.0*a2)+10.0*a2*(-125.0+a2*(-4.0+3.0*a2)))+2250.0*pi2*C1*(27.0*a14+6.0*a13*(59.0+30.0*a2)+10.0*(-1.0+a2)*a2*(-625.0+a2*(173.0+18.0*a2))+a12*(-4290.0+7.0*a2*(256.0+63.0*a2))+a1*(7500.0+a2*(-12395.0+9.0*a2*(327.0+52.0*a2))))-15.0*pi*(1593.0*a15+6.0*a14*(1165.0+2301.0*a2)+a13*(-442875.0+a2*(62840.0+47259.0*a2))+10.0*a12*(189375.0+a2*(-215540.0+a2*(20443.0+7965.0*a2)))+2.0*a1*a2*(2181875.0+a2*(-1683475.0+2.0*a2*(71830.0+16461.0*a2)))+20.0*a2*(109375.0+a2*(67250.0+a2*(-82765.0+2.0*a2*(3695.0+531.0*a2))))))*pj)/(2.0*(125.0+a1+a2)*(25.0+2.0*a1+4.0*a2)^2*(50.0+3.0*a1+6.0*a2)*(3.0*a1+5.0*(20.0+a2)))
}
GFG10_iiijj <- function(a1, a2, pi, pj) {
pi2 <- pi*pi
a12 <- a1*a1
a13 <- a12*a1
a14 <- a13*a1
a15 <- a14*a1
a22 <- a2*a2
a23 <- a22*a2
C1 <- (a1+2.0*(-5.0+a2))
C2 <- (27.0*a14+6.0*a13*(59.0+30.0*a2)+10.0*(-1.0+a2)*a2*(-625.0+a2*(173.0+18.0*a2))+a12*(-4290.0+7.0*a2*(256.0+63.0*a2))+a1*(7500.0+a2*(-12395.0+9.0*a2*(327.0+52.0*a2))))
C3 <- (20.0+a2)
(25.0*pj*(3.0*pi*(-125.0*pi2*C1*C2+3.0*(-216.0*a15-9.0*a14*(475.0+208.0*a2)-160.0*a22*C3*(-175.0+2.0*a2*(-23.0+9.0*a2))-10.0*a12*a2*(-23605.0+2.0*a2*(4477.0+540.0*a2))-6.0*a13*(-8025.0+a2*(5345.0+1068.0*a2))+4.0*a1*a2*(74375.0+a2*(91775.0-2.0*a2*(13765.0+1116.0*a2))))+5.0*pi*(729.0*a15+a14*(8922.0+6318.0*a2)+a13*(-216915.0+a2*(66256.0+21627.0*a2))+10.0*a12*(69375.0+a2*(-110088.0+a2*(18227.0+3645.0*a2)))+2.0*a1*a2*(838875.0+a2*(-901355.0+2.0*a2*(54878.0+7533.0*a2)))+20.0*a2*(109375.0+a2*(34450.0+a2*(-46861.0+4862.0*a2+486.0*a22)))))+5.0*pi*(648.0*a15+9.0*a14*(-161.0+624.0*a2)+125.0*pi2*C1*(-1.0+a1+a2)*(25000.0+a1*(-11050.0+9.0*a1*(75.0+7.0*a1))-21300.0*a2+a1*(2650.0+357.0*a1)*a2+8.0*(325.0+84.0*a1)*a22+420.0*a23)+480.0*(-1.0+a2)*a22*(-1025.0+a2*(97.0+18.0*a2))+6.0*a13*(-28245.0+a2*(-427.0+3204.0*a2))+12.0*a1*a2*(167875.0+a2*(-97765.0+4238.0*a2+2232.0*a22))+30.0*a12*(30000.0+a2*(-26363.0+2.0*a2*(277.0+540.0*a2)))-225.0*pi*C1*C2)*pj))/((125.0+a1+a2)*(25.0+2.0*a1+4.0*a2)^2*(50.0+3.0*a1+6.0*a2)*(3.0*a1+5.0*C3))
}
#' Theoretical frequencies at equilibrium.
#'
#' These return gamete and genotype frequencies as calculated in
#' Huang et al (2019). Only supported for ploidies less than or equal to 10.
#'
#' @param alpha A numeric vector containing the double reduction parameter(s).
#' This should be a
#' vector of length \code{floor(ploidy/4)} where \code{alpha[i]}
#' is the probability of exactly \code{i} pairs of IBDR alleles
#' being in the gamete. Note that \code{sum(alpha)} should be less than
#' 1, as \code{1 - sum(alpha)} is the probability of no double reduction.
#' @param ploidy The ploidy of the species. This should be an even positive
#' integer.
#' @param r The allele frequency of the reference allele.
#'
#' @references
#' \itemize{
#' \item{Huang, K., Wang, T., Dunn, D. W., Zhang, P., Cao, X., Liu, R., & Li, B. (2019). Genotypic frequencies at equilibrium for polysomic inheritance under double-reduction. G3: Genes, Genomes, Genetics, 9(5), 1693-1706. \doi{10.1534/g3.119.400132}}
#' }
#'
#' @return A list with the following elements
#' \describe{
#' \item{\code{p}}{The gamete frequencies at equilibrium.}
#' \item{\code{q}}{The genotype frequencies at equilibrium.}
#' }
#'
#' @author David Gerard
#'
#' @noRd
theofreq <- function(alpha, r, ploidy) {
stopifnot(length(r) == 1L, length(ploidy) == 1L)
stopifnot(ploidy %% 2 == 0)
stopifnot(ploidy > 1, ploidy < 11)
stopifnot(length(alpha) == floor(ploidy / 4))
stopifnot(sum(alpha) <= 1)
stopifnot(alpha > -sqrt(.Machine$double.eps))
alpha[alpha < 0] <- 0
if (ploidy == 2) {
hout <- c(r, 1-r)
} else if (ploidy == 4) {
hout <- c(
GFG4_ii(a1 = alpha, pi = 1 - r),
GFG4_ij(a1 = alpha, pi = 1 - r, pj = r),
GFG4_ii(a1 = alpha, pi = r)
)
} else if (ploidy == 6) {
hout <- c(
GFG6_iii(a1 = alpha, pi = 1 - r),
GFG6_iij(a1 = alpha, pi = 1 - r, pj = r),
GFG6_iij(a1 = alpha, pi = r, pj = 1 - r),
GFG6_iii(a1 = alpha, pi = r)
)
} else if (ploidy == 8) {
hout <- c(
GFG8_iiii(a1 = alpha[[1]], a2 = alpha[[2]], pi = 1 - r),
GFG8_iiij(a1 = alpha[[1]], a2 = alpha[[2]], pi = 1 - r, pj = r),
GFG8_iijj(a1 = alpha[[1]], a2 = alpha[[2]], pi = 1 - r, pj = r),
GFG8_iiij(a1 = alpha[[1]], a2 = alpha[[2]], pi = r, pj = 1 - r),
GFG8_iiii(a1 = alpha[[1]], a2 = alpha[[2]], pi = r)
)
} else if (ploidy == 10) {
hout <- c(
GFG10_iiiii(a1 = alpha[[1]], a2 = alpha[[2]], pi = 1 - r),
GFG10_iiiij(a1 = alpha[[1]], a2 = alpha[[2]], pi = 1 - r, pj = r),
GFG10_iiijj(a1 = alpha[[1]], a2 = alpha[[2]], pi = 1 - r, pj = r),
GFG10_iiijj(a1 = alpha[[1]], a2 = alpha[[2]], pi = r, pj = 1 - r),
GFG10_iiiij(a1 = alpha[[1]], a2 = alpha[[2]], pi = r, pj = 1 - r),
GFG10_iiiii(a1 = alpha[[1]], a2 = alpha[[2]], pi = r)
)
}
## Fix numerical issues
stopifnot(hout > -sqrt(.Machine$double.eps))
hout[hout < 0] <- 0
hout <- hout / sum(hout)
q <- stats::convolve(x = hout, y = rev(hout), type = "open")
q[q < 0] <- 0
q <- q / sum(q)
retlist <- list(q = q,
p = hout)
return(retlist)
}
#' log-likelihood used in hwelike
#'
#' @param par first element is r, rest are alpha
#' @param nvec the counts
#' @param which_keep A logical vector the same length as nvec, indicating
#' which genotypes to aggregate.
#'
#' @author David Gerard
#'
#' @noRd
like_obj <- function(par, nvec, which_keep = NULL) {
if (is.null(which_keep)) {
which_keep <- rep(TRUE, length(nvec))
}
ploidy <- length(nvec) - 1
r <- par[[1]]
alpha <- par[-1]
fq <- theofreq(alpha = alpha, r = r, ploidy = ploidy)
q <- fq$q
if (!all(which_keep)) {
nvec <- c(nvec[which_keep], sum(nvec[!which_keep]))
q <- c(q[which_keep], sum(q[!which_keep]))
}
return(stats::dmultinom(x = nvec, prob = q, log = TRUE))
}
#' log-likelihood used in hwelike
#'
#' @param alpha double reduction parameter
#' @param r allele frequency
#' @param nvec the counts
#' @param which_keep A logical vector the same length as nvec, indicating
#' which genotypes to aggregate.
#'
#' @author David Gerard
#'
#' @noRd
like_obj2 <- function(alpha, r, nvec, which_keep = NULL) {
like_obj(par = c(r, alpha), nvec = nvec, which_keep = which_keep)
}
#' Maximum likelihood approach for equilibrium testing and double reduction
#' estimation.
#'
#' Genotype frequencies from Huang et al (2019) are used to implement a
#' likelihood procedure to estimate double reduction rates and to test
#' for equilibrium while accounting for double reduction. This approach
#' is only implemented for ploidies 4, 6, 8, and 10.
#'
#' @inheritParams hweustat
#' @param thresh The threshold for ignoring the genotype. We keep
#' genotypes such that \code{nvec >= thresh}.
#' Setting this to \code{0} uses all genotypes.
#'
#' @author David Gerard
#'
#' @references
#' \itemize{
#' \item{Huang, K., Wang, T., Dunn, D. W., Zhang, P., Cao, X., Liu, R., & Li, B. (2019). Genotypic frequencies at equilibrium for polysomic inheritance under double-reduction. G3: Genes, Genomes, Genetics, 9(5), 1693-1706. \doi{10.1534/g3.119.400132}}
#' }
#'
#' @return A list with some or all of the following elements:
#' \describe{
#' \item{\code{alpha}}{The estimated double reduction parameter(s).
#' In diploids, this value is \code{NULL}.}
#' \item{\code{r}}{The estimated allele frequency.}
#' \item{\code{chisq_hwe}}{The chi-square test statistic for testing
#' against the null of equilibrium.}
#' \item{\code{df_hwe}}{The degrees of freedom associated with
#' \code{chisq_hwe}.}
#' \item{\code{p_hwe}}{The p-value against the null of equilibrium.}
#' }
#'
#' @export
#'
#' @examples
#' thout <- hwefreq(alpha = 0.1, r = 0.3, ploidy = 6)
#' nvec <- c(stats::rmultinom(n = 1, size = 100, prob = thout))
#' hwelike(nvec = nvec)
#'
hwelike <- function(nvec,
thresh = 5,
effdf = FALSE) {
ploidy <- length(nvec) - 1
stopifnot(ploidy %% 2 == 0, ploidy >= 4, ploidy <= 10)
stopifnot(nvec >= 0)
stopifnot(length(thresh) == 1, thresh >= 0)
stopifnot(is.logical(effdf), length(effdf) == 1)
ibdr <- floor(ploidy / 4)
omethod <- ifelse(ibdr == 1, "Brent", "L-BFGS-B")
## Choose which groups to aggregate ----
which_keep <- choose_agg(x = nvec, thresh = thresh, like = TRUE)
numkeep <- sum(which_keep)
if (numkeep >= ploidy) {
## aggregating one group = no aggregation at all.
which_keep <- rep(TRUE, ploidy + 1)
numkeep <- ploidy + 1
}
## Return early if too few groups ----
if (sum(which_keep) - ibdr <= 0) {
return(
list(
alpha = rep(NA_real_, ibdr),
r = NA_real_,
chisq_hwe = NA_real_,
df_hwe = sum(which_keep) - ibdr - 1,
p_hwe = NA_real_
)
)
}
## Find MLE under null ----
rhat <- sum(0:ploidy * nvec) / (ploidy * sum(nvec))
minval <- 0.0001
upper_alpha <- drbounds(ploidy = ploidy)
oout <- stats::optim(par = rep(minval, ibdr),
fn = like_obj2,
method = omethod,
lower = rep(minval, ibdr),
upper = upper_alpha,
control = list(fnscale = -1),
nvec = nvec,
r = rhat,
which_keep = which_keep)
alphahat <- oout$par
ll_e <- oout$value
## MLE under alternative ----
if (all(which_keep)) {
q_u <- nvec / sum(nvec)
ll_u <- stats::dmultinom(x = nvec, prob = q_u, log = TRUE)
} else {
ntemp <- c(nvec[which_keep], sum(nvec[!which_keep]))
q_temp <- ntemp / sum(ntemp)
ll_u <- stats::dmultinom(x = ntemp, prob = q_temp, log = TRUE)
}
## Test statistic ----
chisq_hwe <- -2 * (ll_e - ll_u)
## Find degrees of freedom ----
if (all(which_keep)) {
df_hwe <- ploidy - ibdr - 1
} else {
df_hwe <- sum(which_keep) - ibdr - 1 ## unconstrained as sum(which_keep), alpha is ibdr, r is 1.
}
TOL <- sqrt(.Machine$double.eps)
if (effdf) {
dfadd <- sum((abs(alphahat - minval) < TOL) | (abs(alphahat - upper_alpha) < TOL))
} else {
dfadd <- 0
}
df_hwe <- df_hwe + dfadd
## Run test ----
if (df_hwe > 0) {
p_hwe <- stats::pchisq(q = chisq_hwe, df = df_hwe, lower.tail = FALSE)
} else {
p_hwe <- NA_real_
}
retlist <- list(alpha = alphahat,
r = rhat,
chisq_hwe = chisq_hwe,
df_hwe = df_hwe,
p_hwe = p_hwe)
return(retlist)
}