data {Blend}R Documentation

simulated data for demonstrating the features of Blend

Description

Simulated gene expression data for demonstrating the features of Blend.

Format

The data object consists of 8 components: y, x, t, J, kn and degree.

Details

The data and model setting

Consider a longitudinal study on n subjects with J_i repeated measurements for each subject. Let Y_{ij} be the measurement for the i-th subject at each time point t_{ij}, (1 \leq i \leq n, 1 \leq j \leq J_i). We use an m-dimensional vector X_{ij} to denote the genetic factors, where X_{ij} = (X_{ij1},...,X_{ijm})^\top. Z_{ij} is a 2 \times 1 covariate associated with random effects and \zeta_{i} is a 2 \times 1 vector of random effects corresponding to the random intercept and slope model. We have the following semi-parametric quantile mixed-effects model:

Y_{ij} = \alpha_0(t_{ij}) + \sum_{k=1}^{m} \beta_{k}(t_{ij}) X_{ijk} + Z_{ij}^\top \zeta_{i} + \epsilon_{ij}, \zeta_{i} \sim N(0, \Lambda)

where the fixed effects include: (a) the varying intercept \alpha_0(t_{ij}), and (b) the varying coefficients \beta(t_{ij}).

The varying intercept and the varying coefficients for the genetic factors can be further expressed as \alpha_0(t_{ij}) and \beta(t_{ij}) = (\beta_{1}(t_{ij}), ..., \beta_{m}(t_{ij}))^\top.

For the random intercept and slope model, Z_{ij}^\top = (1, j) and \zeta_{i} = (\zeta_{i1}, \zeta_{i2})^\top.

Furthermore, Z_{ij}^\top \zeta_{i} can be expressed as (b_i^\top \otimes Z^\top_{ij}) J_2 \delta, where \zeta_{i} = \Delta b_i, \Lambda = \Delta \Delta^\top, and

b_i^\top \otimes Z^\top_{ij} = (b_{i1} Z_{ij1}, b_{i1} Z_{ij2}, b_{i2}Z_{ij1}, b_{i2} Z_{ij2})^\top.

In the simulated data,

Y = \alpha_{0}(t)+\beta_{1}(t)X_{1} + \beta_{2}(t)X_{2} + \beta_{3}(t)X_{3}+ \beta_{4}(t)X_{4}+0.8X_{5} -1.2 X_{6} + 0.7X_{7}-1.1 X_{8}+\epsilon

where \epsilon\sim N(0,1), \alpha_{0}(t)=2+\sin(2\pi t), \beta_{1}(t)=2.5\exp(2.5t-1) ,\beta_{2}(t)=3t^2-2t+2,\beta_{3}(t)=-4t^3+3 and \beta_{4}(t)=3-2t

See Also

Blend

Examples

data(dat)
length(y)
dim(x)
length(t)
length(J)
print(t)
print(J)
print(kn)
print(degree)

[Package Blend version 0.1.0 Index]