-
knitr::opts_chunk$set(fig.path='Figs/')
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 hyx520101@gmail.com for more information.
To install the latest version from github:
install.packages("devtools")
library(devtools)
install_github("lhe17/ACEt")
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$dz)
#> T_d
#> [1,] 0.1402850 -0.3430456 1
#> [2,] 0.8588600 -0.7381698 1
#> [3,] 0.4025476 0.2794685 1
#> [4,] -0.4564107 0.2008932 1
#> [5,] -0.2458682 -3.0600677 1
#> [6,] 0.1459282 0.4588418 1
The age is distributed uniformly from 1 to 50 in both twin datasets and
the phenotypes are normally distributed with a mean equal to zero. As
discussed in He et al. (2017), before used as an input for this package,
the phenotype should be centered, for example, by using residuals from a
linear regression model lm() in which covariates for the mean function
can be included. Fitting an ACE(t) model can be done by calling the
AtCtEt function, in which users can specify a function (null, constant
or splines) for each of the A, C, and E components independently through
the mod argument.
# fitting the ACE(t) model
re <- AtCtEt(data_ace$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- NULL
In 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 an expected and an approximate observed Fisher information
matrices (shown below), which are close to each other in general and can
be used to compute pointwise CIs. Note that the expected information
matrix is always positive (semi)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 expected information 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 observed information matrix approximated by the L-BFGS algorithm
re$hessian_ap[1:8,1:8]
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 3.3066880 8.583804 0.9771341 0.00000000 0.000000 0.000000
#> [2,] 8.5838039 64.611733 43.1402778 2.22883073 0.000000 0.000000
#> [3,] 0.9771341 43.140278 144.8342062 67.91347006 2.836111 0.000000
#> [4,] 0.0000000 2.228831 67.9134701 181.14525247 67.405340 2.312405
#> [5,] 0.0000000 0.000000 2.8361108 67.40534018 155.154944 55.376537
#> [6,] 0.0000000 0.000000 0.0000000 2.31240520 55.376537 114.366468
#> [7,] 0.0000000 0.000000 0.0000000 0.00000000 1.856622 23.555961
#> [8,] 5.0462737 13.315584 2.7042919 0.02561431 0.000000 0.000000
#> [,7] [,8]
#> [1,] 0.000000 5.04627369
#> [2,] 0.000000 13.31558433
#> [3,] 0.000000 2.70429192
#> [4,] 0.000000 0.02561431
#> [5,] 1.856622 0.00000000
#> [6,] 23.555961 0.00000000
#> [7,] 11.486909 0.00000000
#> [8,] 0.000000 11.08078500
The AtCtEt function returns the minus log-likelihood evaluated at the
estimates that is needed to make inference based on LRT. For example,
the following program tests whether the A or C component has a constant
variance with respect to age, we fit the null models and calculate the
p-values based on χ2 distributions. It can be seen that the
LRT has no sufficient statistical power to reject the constancy of the C
component with this sample size (p1>0.05). In addition, we test
whether the C component can be ignored by comparing re_cc and re_cn
and compute the p-value (p3) based on a mixture of χ2
distributions.
re_cc <- AtCtEt(data_ace$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$lik), 1, lower.tail=FALSE)
p3
#> [1] 2.155026e-08
After fitting the ACE(t) model, we can plot the estimated variance
curves by calling the plot_acet function.
plot_acet(re, ylab='Var', xlab='Age (1-50)')
By default, the 95% pointwise CIs are estimated using the delta method.
Alternatively, we can choose the bootstrap method by setting boot=TRUE
and giving the number of bootstrap resampling, the default value of
which is 100.
## fitting an ACE(t) model with the CIs esitmated by the bootstrap method
re_b <- AtCtEt(data_ace$mz, data_ace$dz, mod = c('d','d','c'), knot_a = 6, knot_c = 4, boot = TRUE,
num_b = 60)
plot_acet(re_b, boot = TRUE)
Next, we plot the age-specific heritability by setting the argument
heri=TRUE in the plot_acet function. And similarly we can choose
either the delta method or the bootstrap method to generate the CIs.
## plot dynamic heritability with the CIs using the delta method
plot_acet(re_b, heri=TRUE, boot = FALSE)
## plot dynamic heritability with the CIs using the bootstrap method
plot_acet(re_b, heri=TRUE, boot = TRUE)
An ADE(t) model can be fitted and plotted similarly using the AtDtEt
function as shown below.
## fitting an ADE(t) model with the CIs esitmated by the bootstrap method
re_b <- AtDtEt(data_ace$mz, data_ace$dz, mod = c('d','d','c'), boot = TRUE, num_b = 60)
plot_acet(re_b, boot = TRUE)
An ACE(t)-p model is a more stable model, which reduces the sensitivity
to the number of knots by using P-splines. The ACE(t)-p model is
implemented in the AtCtEtp function, in which users can choose
exponential of penalized splines, a linear function or a constant to
model a certain component by setting the mod argument. Compared to the
ACE(t) model, it is not an essential problem to provide an excessive
number of knots (the default value of interior knots is 8) when using
the ACE(t)-p model as it is more important to ensure adequate knots for
curves with more fluctuation than to avoid overfitting. Below, we fit
the example dataset using the AtCtEtp function in which the A and C
components are modelled by B-splines of 8 interior knots and the E
component by a log-linear function. Similar to the AtCtEt function, we
can use the robust argument to specify the number of randomly
generated initial values, which can reduce the program’s possibility of
being stuck on a local maximum in the EM algorithm.
## fitting an ACE(t)-p model
re <- AtCtEtp(data_ace$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- numeric
The AtCtEtp function finds MLE of the variance
σβA, C, E2 using the
integrated likelihood and also provides estimates of the spline
coefficients, i.e. βA, C, E, which are based on
maximum a posteriori (MAP) estimation. For a variance component of
log-linearity (the E component in this example), β is a vector of two
elements that exp(β) are the variances of this component at the
minimum and maximum age in the dataset. To obtain the empirical Bayes
estimates of βA, 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- numeric
Given 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 (Left: the ACE(t) model. Right: the ACE(t)-p model.) 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.
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 χ2 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,] 0.2651204
The 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$p
The result (p>0.05) shows that the null hypothesis of the log-linearity is not rejected.
He, Liang, Janne Pitkäniemi, Karri Silventoinen, and Mikko J. Sillanpää. 2017. “ACEt : An r Package for Estimating 1 Dynamic Heritability and Comparing Twin Models.” Behavior Genetics 47 (6): 620–41.
He, Liang, Mikko J. Sillanpää, Karri Silventoinen, Jaakko Kaprio, and Janne Pitkäniemi. 2016. “Estimating Modifying Effect of Age on Genetic and Environmental Variance Components in Twin Models.” Genetics 202 (4): 1313–28.