DeltaBeta: Delta Method Sampling Variance-Covariance Matrix for the...
In cTMed: Continuous Time Mediation

DeltaBeta

R Documentation

Delta Method Sampling Variance-Covariance Matrix for the Elements of the Matrix of Lagged Coefficients Over a Specific Time Interval or a Range of Time Intervals

Description

This function computes the delta method sampling variance-covariance matrix for the elements of the matrix of lagged coefficients \boldsymbol{\beta} over a specific time interval \Delta t or a range of time intervals using the first-order stochastic differential equation model's drift matrix \boldsymbol{\Phi}.

Usage

DeltaBeta(phi, vcov_phi_vec, delta_t, ncores = NULL)

Arguments

`phi`	Numeric matrix. The drift matrix (`\boldsymbol{\Phi}`). `phi` should have row and column names pertaining to the variables in the system.
`vcov_phi_vec`	Numeric matrix. The sampling variance-covariance matrix of `\mathrm{vec} \left( \boldsymbol{\Phi} \right)`.
`delta_t`	Vector of positive numbers. Time interval (`\Delta t`).
`ncores`	Positive integer. Number of cores to use. If `ncores = NULL`, use a single core. Consider using multiple cores when the length of `delta_t` is long.

Details

See Total().

Delta Method

Let \boldsymbol{\theta} be \mathrm{vec} \left( \boldsymbol{\Phi} \right), that is, the elements of the \boldsymbol{\Phi} matrix in vector form sorted column-wise. Let \hat{\boldsymbol{\theta}} be \mathrm{vec} \left( \hat{\boldsymbol{\Phi}} \right). By the multivariate central limit theory, the function \mathbf{g} using \hat{\boldsymbol{\theta}} as input can be expressed as:

\sqrt{n} \left( \mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) - \mathbf{g} \left( \boldsymbol{\theta} \right) \right) \xrightarrow[]{ \mathrm{D} } \mathcal{N} \left( 0, \mathbf{J} \boldsymbol{\Gamma} \mathbf{J}^{\prime} \right)

where \mathbf{J} is the matrix of first-order derivatives of the function \mathbf{g} with respect to the elements of \boldsymbol{\theta} and \boldsymbol{\Gamma} is the asymptotic variance-covariance matrix of \hat{\boldsymbol{\theta}}.

From the former, we can derive the distribution of \mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) as follows:

\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) \approx \mathcal{N} \left( \mathbf{g} \left( \boldsymbol{\theta} \right) , n^{-1} \mathbf{J} \boldsymbol{\Gamma} \mathbf{J}^{\prime} \right)

The uncertainty associated with the estimator \mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) is, therefore, given by n^{-1} \mathbf{J} \boldsymbol{\Gamma} \mathbf{J}^{\prime} . When \boldsymbol{\Gamma} is unknown, by substitution, we can use the estimated sampling variance-covariance matrix of \hat{\boldsymbol{\theta}}, that is, \hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right) for n^{-1} \boldsymbol{\Gamma}. Therefore, the sampling variance-covariance matrix of \mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) is given by

\mathbf{g} \left( \hat{\boldsymbol{\theta}} \right) \approx \mathcal{N} \left( \mathbf{g} \left( \boldsymbol{\theta} \right) , \mathbf{J} \hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right) \mathbf{J}^{\prime} \right) .

Value

Returns an object of class ctmeddelta which is a list with the following elements:

call: Function call.
args: Function arguments.
fun: Function used ("DeltaBeta").
output: A list the length of which is equal to the length of delta_t.

Each element in the output list has the following elements:

delta_t: Time interval.
jacobian: Jacobian matrix.
est: Estimated elements of the matrix of lagged coefficients.
vcov: Sampling variance-covariance matrix of estimated elements of the matrix of lagged coefficients.

Author(s)

Ivan Jacob Agaloos Pesigan

References

Bollen, K. A. (1987). Total, direct, and indirect effects in structural equation models. Sociological Methodology, 17, 37. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/271028")}

Deboeck, P. R., & Preacher, K. J. (2015). No need to be discrete: A method for continuous time mediation analysis. Structural Equation Modeling: A Multidisciplinary Journal, 23 (1), 61–75. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10705511.2014.973960")}

Ryan, O., & Hamaker, E. L. (2021). Time to intervene: A continuous-time approach to network analysis and centrality. Psychometrika, 87 (1), 214–252. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-021-09767-0")}

Examples

phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0, 0, -0.693
  ),
  nrow = 3
)
colnames(phi) <- rownames(phi) <- c("x", "m", "y")
vcov_phi_vec <- matrix(
  data = c(
    0.002704274, -0.001475275, 0.000949122,
    -0.001619422, 0.000885122, -0.000569404,
    0.00085493, -0.000465824, 0.000297815,
    -0.001475275, 0.004428442, -0.002642303,
    0.000980573, -0.00271817, 0.001618805,
    -0.000586921, 0.001478421, -0.000871547,
    0.000949122, -0.002642303, 0.006402668,
    -0.000697798, 0.001813471, -0.004043138,
    0.000463086, -0.001120949, 0.002271711,
    -0.001619422, 0.000980573, -0.000697798,
    0.002079286, -0.001152501, 0.000753,
    -0.001528701, 0.000820587, -0.000517524,
    0.000885122, -0.00271817, 0.001813471,
    -0.001152501, 0.00342605, -0.002075005,
    0.000899165, -0.002532849, 0.001475579,
    -0.000569404, 0.001618805, -0.004043138,
    0.000753, -0.002075005, 0.004984032,
    -0.000622255, 0.001634917, -0.003705661,
    0.00085493, -0.000586921, 0.000463086,
    -0.001528701, 0.000899165, -0.000622255,
    0.002060076, -0.001096684, 0.000686386,
    -0.000465824, 0.001478421, -0.001120949,
    0.000820587, -0.002532849, 0.001634917,
    -0.001096684, 0.003328692, -0.001926088,
    0.000297815, -0.000871547, 0.002271711,
    -0.000517524, 0.001475579, -0.003705661,
    0.000686386, -0.001926088, 0.004726235
  ),
  nrow = 9
)

# Specific time interval ----------------------------------------------------
DeltaBeta(
  phi = phi,
  vcov_phi_vec = vcov_phi_vec,
  delta_t = 1
)

# Range of time intervals ---------------------------------------------------
delta <- DeltaBeta(
  phi = phi,
  vcov_phi_vec = vcov_phi_vec,
  delta_t = 1:5
)
plot(delta)

# Methods -------------------------------------------------------------------
# DeltaBeta has a number of methods including
# print, summary, confint, and plot
print(delta)
summary(delta)
confint(delta, level = 0.95)
plot(delta)

cTMed documentation built on Oct. 21, 2024, 5:08 p.m.