ACEt: An R package for estimating dynamic heritability and twin model comparison
Liang He
2022-09-26
ACEt v1.8.2
Installation
The R package can be installed from CRAN
install.packages("ACEt")The installation requires Rcpp-0.11.1 and has been tested on R-4.1.3. The installation of the ACEt package also requires installing the BH and RcppArmadillo packages.
Please contact lianghe@health.sdu.dk for more information.
Most recent version
To install the latest version from github:
install.packages("devtools")
library(devtools)
install_github("lhe17/ACEt")Application to an example dataset
We illustrate how to utilize the ACEt R package with an example dataset that can be loaded with the following codes. More detail about the method is given in He et al. (2016) and He et al. (2017).
library(ACEt)
data(data_ace)The example dataset contains two matrices mz and
dz for MZ and DZ twins, respectively. Each matrix includes
2500 twin pairs, of which the first two columns are the quantitative
phenotype of the twin pair and the third column (T_m or
T_d) is age.
attributes(data_ace)
#> $names
#> [1] "mz" "dz"
head(data_ace$mz)
#> T_m
#> [1,] 2.5638027 4.7355457 1
#> [2,] -3.1959902 -3.0133873 1
#> [3,] 1.3924694 2.8656795 1
#> [4,] 2.4519483 1.9145994 1
#> [5,] -0.4186678 1.3470608 1
#> [6,] -1.2805044 -0.5234272 1
head(data_ace$dz)
#> T_d
#> [1,] 0.1402850 -0.3430456 1
#> [2,] 0.8588600 -0.7381698 1
#> [3,] 0.4025476 0.2794685 1
#> [4,] -0.4564107 0.2008932 1
#> [5,] -0.2458682 -3.0600677 1
#> [6,] 0.1459282 0.4588418 1The age is distributed uniformly from 1 to 50 in both twin datasets
and the phenotypes are normally distributed with a mean equal to zero.
As discussed in He et al. (2017), before
used as an input for this package, the phenotype should be centered, for
example, by using residuals from a linear regression model
lm() in which covariates for the mean function can be
included. Fitting an ACE(t) model can be done by calling the
AtCtEt function, in which users can specify a function
(null, constant or splines) for each component independently through the
mod argument.
# fitting the ACE(t) model
re <- AtCtEt(data_ace$mz, data_ace$dz, mod = c('d','d','c'), knot_a = 6, knot_c = 4)
summary(re)
#> Length Class Mode
#> n_beta_a 1 -none- numeric
#> n_beta_c 1 -none- numeric
#> n_beta_e 1 -none- numeric
#> beta_a 7 -none- numeric
#> beta_c 5 -none- numeric
#> beta_e 1 -none- numeric
#> hessian_ap 169 -none- numeric
#> hessian 169 -none- numeric
#> con 1 -none- numeric
#> lik 1 -none- numeric
#> knots_a 10 -none- numeric
#> knots_c 8 -none- numeric
#> knots_e 2 -none- numeric
#> min_t 1 -none- numeric
#> max_t 1 -none- numeric
#> boot 0 -none- NULLIn the above script, an ACE(t) model is fitted for the example
dataset. The first two arguments specify the matrices of the phenotypes
for MZ and DZ twins, respectively. The argument
mod = c('d','d','c') specifies that we allow the variances
of the A and C components to change dynamically and assume the variance
of the E component to be a constant over age. The mod
argument is a vector of three elements corresponding to the A, C and E
components that can be 'd', 'c' or 'n', in which
'n' represents the exclusion of a component. For example,
mod = c('d','n','c') indicates that we fit an AE model with
a dynamic A component and a constant E component. It should be noted
that the E component cannot be eliminated. We can also give the number
of knots for each component, which is ignored if we choose
'c' or 'n' for that component. The number of
randomly generated initial values for the estimation algorithm can be
specified using the robust argument. Multiple initial
values can be attempted to minimize the risk of missing the global
maximum. The AtCtEt function returns both analytical and
approximated Hessian matrices (shown below), which are close to each
other in general and can be used to compute pointwise CIs. Note that the
analytical Hessian is always positive definite, but the approximated one
is not necessarily positive definite. The returned value
lik is the negative log-likelihood that can be used for LRT
for the comparison of twin models.
# part of the analytical Hessian matrix
re$hessian[1:8,1:8]
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 3.3122436 8.572181 0.9774334 0.00000000 0.000000 0.000000
#> [2,] 8.5721808 64.560338 43.2299744 2.22206265 0.000000 0.000000
#> [3,] 0.9774334 43.229974 145.2021892 67.94632724 2.836022 0.000000
#> [4,] 0.0000000 2.222063 67.9463272 182.31265682 68.569776 2.334633
#> [5,] 0.0000000 0.000000 2.8360218 68.56977584 155.634291 54.989838
#> [6,] 0.0000000 0.000000 0.0000000 2.33463321 54.989838 113.226897
#> [7,] 0.0000000 0.000000 0.0000000 0.00000000 1.840752 22.878770
#> [8,] 4.9109578 13.518421 2.7632296 0.02669836 0.000000 0.000000
#> [,7] [,8]
#> [1,] 0.000000 4.91095781
#> [2,] 0.000000 13.51842124
#> [3,] 0.000000 2.76322956
#> [4,] 0.000000 0.02669836
#> [5,] 1.840752 0.00000000
#> [6,] 22.878770 0.00000000
#> [7,] 11.077766 0.00000000
#> [8,] 0.000000 11.02095804
# part the Hessian matrix approximated by the L-BFGS algorithm
re$hessian_ap[1:8,1:8]
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 3.3066880 8.583804 0.9771341 0.00000000 0.000000 0.000000
#> [2,] 8.5838039 64.611733 43.1402778 2.22883073 0.000000 0.000000
#> [3,] 0.9771341 43.140278 144.8342062 67.91347006 2.836111 0.000000
#> [4,] 0.0000000 2.228831 67.9134701 181.14525247 67.405340 2.312405
#> [5,] 0.0000000 0.000000 2.8361108 67.40534018 155.154944 55.376537
#> [6,] 0.0000000 0.000000 0.0000000 2.31240520 55.376537 114.366468
#> [7,] 0.0000000 0.000000 0.0000000 0.00000000 1.856622 23.555961
#> [8,] 5.0462737 13.315584 2.7042919 0.02561431 0.000000 0.000000
#> [,7] [,8]
#> [1,] 0.000000 5.04627369
#> [2,] 0.000000 13.31558433
#> [3,] 0.000000 2.70429192
#> [4,] 0.000000 0.02561431
#> [5,] 1.856622 0.00000000
#> [6,] 23.555961 0.00000000
#> [7,] 11.486909 0.00000000
#> [8,] 0.000000 11.08078500The AtCtEt function returns the minus log-likelihood
evaluated at the estimates that is needed to make inference based on
LRT. For example, the following program tests whether the A or C
component has a constant variance with respect to age, we fit the null
models and calculate the p-values based on \(\chi^2\) distributions. It can be seen that
the LRT has no sufficient statistical power to reject the constancy of
the C component with this sample size (p1>0.05). In
addition, we test whether the C component can be ignored by comparing
re_cc and re_cn and compute the p-value
(p3) based on a mixture of \(\chi^2\) distributions.
re_cc <- AtCtEt(data_ace$mz, data_ace$dz, mod = c('d','c','c'), knot_a = 6, knot_c = 4)
p1 <- pchisq(2*(re_cc$lik-re$lik), 4, lower.tail=FALSE)
p1
#> [1] 0.2079343
re_ac <- AtCtEt(data_ace$mz, data_ace$dz, mod = c('c','d','c'), knot_a = 6, knot_c = 4)
p2 <- pchisq(2*(re_ac$lik-re$lik), 6, lower.tail=FALSE)
p2
#> [1] 8.937498e-12
re_cn <- AtCtEt(data_ace$mz, data_ace$dz, mod = c('d','n','c'), knot_a = 6, knot_c = 4)
p3 <- 0.5*pchisq(2*(re_cn$lik-re_cc$lik), 1, lower.tail=FALSE)
p3
#> [1] 2.155026e-08After fitting the ACE(t) model, we can plot the estimated variance
curves by calling the plot_acet function.
plot_acet(re, ylab='Var', xlab='Age (1-50)')
By default, the 95% pointwise CIs are estimated using the delta
method. Alternatively, we can choose the bootstrap method by setting
boot=TRUE and giving the number of bootstrap resampling,
the default value of which is 100.
## fitting an ACE(t) model with the CIs esitmated by the bootstrap method
re_b <- AtCtEt(data_ace$mz, data_ace$dz, mod = c('d','d','c'), knot_a = 6, knot_c = 4, boot = TRUE,
num_b = 60)
plot_acet(re_b, boot = TRUE)
Next, we plot the age-specific heritability by setting the argument
heri=TRUE in the plot_acet function. And
similarly we can choose either the delta method or the bootstrap method
to generate the CIs.
## plot dynamic heritability with the CIs using the delta method
plot_acet(re_b, heri=TRUE, boot = FALSE)
## plot dynamic heritability with the CIs using the bootstrap method
plot_acet(re_b, heri=TRUE, boot = TRUE)
The ACE(t)-p model is implemented in the AtCtEtp
function, in which users can choose exponential of penalized splines, a
linear function or a constant to model a certain component by setting
the mod argument. Compared to the ACE(t) model, it is not
an essential problem to provide an excessive number of knots (the
default value of interior knots is 8) when using the ACE(t)-p model as
it is more important to ensure adequate knots for curves with more
fluctuation than to avoid overfitting. Below, we fit the example dataset
using the AtCtEtp function in which the A and C components
are modelled by B-splines of 8 interior knots and the E component by a
log-linear function. Similar to the AtCtEt function, we can
use the robust argument to specify the number of randomly
generated initial values, which can reduce the program’s possibility of
being stuck on a local maximum in the EM algorithm.
## fitting an ACE(t)-p model
re <- AtCtEtp(data_ace$mz, data_ace$dz, knot_a = 8, knot_c = 8, mod=c('d','d','l'))
summary(re)
#> Length Class Mode
#> D_a 81 -none- numeric
#> D_c 81 -none- numeric
#> D_e 4 -none- numeric
#> pheno_m 5000 -none- numeric
#> pheno_d 5000 -none- numeric
#> T_m 5000 -none- numeric
#> T_d 5000 -none- numeric
#> knot_a 12 -none- numeric
#> knot_c 12 -none- numeric
#> knot_e 2 -none- numeric
#> beta_a 9 -none- numeric
#> beta_c 9 -none- numeric
#> beta_e 2 -none- numeric
#> con 1 -none- numeric
#> lik 1 -none- numeric
#> iter 5 -none- numeric
#> var_b_a 1 -none- numeric
#> var_b_c 1 -none- numeric
#> var_b_e 1 -none- numeric
#> mod 3 -none- character
#> hessian 9 -none- numericThe AtCtEtp function finds MLE of the variance \(\sigma^{2}_{\beta^{A,C,E}}\) using the
integrated likelihood and also provides estimates of the spline
coefficients, i.e. \(\beta^{A,C,E}\),
which are based on maximum a posteriori (MAP) estimation. For a variance
component of log-linearity (the E component in this example), \(\beta\) is a vector of two elements that
\(exp(\beta)\) are the variances of
this component at the minimum and maximum age in the dataset. To obtain
the empirical Bayes estimates of \(\beta^{A,C,E}\) and the covariance matrix
using the MCMC method, we then call the acetp_mcmc function
by plugging the result from the AtCtEtp function. We can
also specify the numbers of the MCMC iterations and burn-in.
re_mcmc <- acetp_mcmc(re, iter_num = 5000, burnin = 500)
summary(re_mcmc)
#> Length Class Mode
#> beta_a_mc 9 -none- numeric
#> beta_c_mc 9 -none- numeric
#> beta_e_mc 2 -none- numeric
#> cov_mc 400 -none- numeric
#> knots_a 12 -none- numeric
#> knots_c 12 -none- numeric
#> knots_e 2 -none- numeric
#> min_t 1 -none- numeric
#> max_t 1 -none- numericGiven the esimates together with their covariance matrix, we can plot
the variance curves or dynamic heritability by calling the
plot_acet function. The boot option is ignored
for the ACE(t)-p model.
plot_acet(re_mcmc)
plot_acet(re_mcmc, heri=TRUE)
Assigning too many knots in the ACE(t)-p model is much less harmful than that in the ACE(t) model. Comparing the following two plots from the application of the two models with 10 knots for each component to the example data set, it suggests that the ACE(t) model has an overfitting problem but the ACE(t)-p model works properly.
Plots of variance curves of the example data set fitted by the ACE(t) and ACE(t)-p model with 10 interior knots for each component. Left: the ACE(t) model. Right: the ACE(t)-p model.
Finally, we give an example to test a linear or constant variance
curve. The test_acetp function is dedicated to the model
comparison for the ACE(t)-p model and returns a p-value from LRT using a
resampling method for testing log-linearity or from a \(\chi^2\) distribution for testing
constancy. First, the following code tests whether the E component is
invariant with age. Before testing, we need to fit the data using the
AtCtEtp function and obtain an AtCtEtp_model
object re. Note that when testing a constant component, the
component must be specified as log-linear when fitting the model (as
shown above).
test <- test_acetp(re, comp = 'e')
#> Model comparison:
#> [1] "Constancy (null) vs. Log-linear"
test$p
#> [,1]
#> [1,] 0.2746529The result suggests that the E component is time-invariant as the p-value is larger than 0.05. Next, we test whether a log-linear model would be fitted better for the C component.
test <- test_acetp(re, comp = 'c', sim = 100, robust = 0)
test$pThe result (p>0.05) shows that the null hypothesis of the log-linearity is not rejected.