| Title: | Regression of Network Responses |
| Version: | 1.0.1 |
| Description: | Regress network responses (both directed and undirected) onto covariates of interest that may be actor-, relation-, or network-valued. In addition, compute principled variance estimates of the coefficients assuming that the errors are jointly exchangeable. Missing data is accommodated. Additionally implements building and inversion of covariance matrices under joint exchangeability, and generates random covariance matrices from this class. For more detail on methods, see Marrs, Fosdick, and McCormick (2017) <doi:10.48550/arXiv.1701.05530>. |
| Depends: | R (≥ 3.3.0) |
| Imports: | Matrix, stats, graphics, utils |
| VignetteBuilder: | knitr |
| Suggests: | knitr, knitcitations |
| License: | MIT + file LICENSE |
| Encoding: | UTF-8 |
| LazyData: | true |
| RoxygenNote: | 6.0.1 |
| NeedsCompilation: | no |
| Packaged: | 2018-08-01 21:32:03 UTC; fmarrs3 |
| Author: | Frank W. Marrs [aut, cre], Bailey K. Fosdick [aut], Tyler H. McCormick [aut] |
| Maintainer: | Frank W. Marrs <frank.marrs@colostate.edu> |
| Repository: | CRAN |
| Date/Publication: | 2018-08-01 23:10:02 UTC |
Perform GEE estimate / IRWLS of coefficients
Description
Perform GEE estimate / IRWLS of coefficients
Usage
GEE.est(row.list, Y, X, n, directed = TRUE, beta_start = NULL,
missing = FALSE, dyads = NULL, tol.in = 1e-06, maxit = 10000,
ndstop = TRUE, verbose = FALSE)
Perform GEE estimate / IRWLS of coefficients for temporal data
Description
TO DO: eigenvalues / ndstop fix for missingness
Usage
GEE_est_time(Y, X, n, tmax, directed = TRUE, type = "EE",
write_dir = NULL, missing = FALSE, tol.in = 1e-06, maxit = 10000,
ndstop = TRUE, verbose = FALSE)
Generate list indicator matrix of overlapping dyads
Description
Generate list indicator matrix of overlapping dyads
Usage
Sigma.ind(n.tot, directed, sta_flag = F)
build list of time blocks that are correlated based on the maximum time and type of temporal model #'
Description
build list of time blocks that are correlated based on the maximum time and type of temporal model #'
Usage
build_blocklist(tmax, type = "EI")
Build an exchangeable matrix of sparseMatrix class
Description
Build an exchangeable matrix of sparseMatrix class
Usage
build_exchangeable_matrix(n, phi, directed = TRUE, dyads = NULL)
Arguments
n |
Number of actors in the network, scalar numeric. |
phi |
Appropriate-length vector of parameters, must be length 5 or 6 for directed= |
directed |
Optional logical indicator of whether input data is for a directed network, default is |
dyads |
Optional numeric vector of dyads to subset the matrix to. |
Details
This function builds a covariance matrix in the exchangeable class from the vector of parameters input. See Marrs et.al. (2017).
Value
out |
Exchangeable matrix. |
References
Marrs, F. W., Fosdick, B. K., & McCormick, T. H., (2017). Standard errors for regression on relational data with exchangeable errors. arXiv preprint arXiv:1701.05530.
See Also
rphi, invert_exchangeable_matrix
Examples
n <- 5
build_exchangeable_matrix(n, rphi(n, seed=1))
Build intermediate C(phi,n) matrix in inversion of Exchangeable variance matrix
Description
Build intermediate C(phi,n) matrix in inversion of Exchangeable variance matrix
Usage
build_phi_matrix(n, phi, directed = T, sta_flag = F)
calculate parameter estimates for different types of matrices, i.e. 6a, 6b, etc.
Description
calculate parameter estimates for different types of matrices, i.e. 6a, 6b, etc.
Usage
calculate_matrix_params(ilist, e, tmax, type = "EI")
Invert matrix parameters based on inputs.
Description
Invert matrix parameters based on inputs.
Usage
calculate_parameter_inverse(n, tmax, params, type = "EI")
Coef S3 generic for class lmnet
Description
Coef S3 generic for class lmnet
Usage
## S3 method for class 'lmnet'
coef(object, ...)
Arguments
object |
lmnet object |
... |
ignored |
Find all possible combinations of elements in two vectors, or all combinations of all elements in one without repeats
Description
Find all possible combinations of elements in two vectors, or all combinations of all elements in one without repeats
Usage
combine(v1, v2 = NA)
Dyad map from nodes i,j –> dyad d
Description
Dyad map from nodes i,j –> dyad d
Usage
dyad(i.in, j.in, n.tot, directed = T)
Eigenvalues of exchangeable matrices if calcall == TRUE, then output eigenvalues with multiplicities Outputs eigenvectors when directed==FALSE
Description
Eigenvalues of exchangeable matrices if calcall == TRUE, then output eigenvalues with multiplicities Outputs eigenvectors when directed==FALSE
Usage
eigen_exch(n, phi, directed = FALSE, calcall = TRUE)
Compute eigenvalues of covariance matrices of jointly exchangeable errors with repeated observations
Description
Compute eigenvalues of covariance matrices of jointly exchangeable errors with repeated observations
Usage
eigen_exch_time(n, tmax, params, directed, type)
Input preprocessing
Description
Prepare covariates and optional response in adjacency matrix form. If undirected, the values are drawn from the lower triangle of the adjacency matrices.
Usage
inputs_lmnet(Xlist, Y = NULL, directed = TRUE, add_intercept = TRUE,
time_intercept = FALSE)
Arguments
Xlist |
List of |
Y |
Optional |
directed |
Optional logical indicator of whether input data is for a directed network, default is |
add_intercept |
Optional logical indicator of whether intercept should be added to X, default is |
time_intercept |
Optional logical indicator of whether separate intercept should be added to X for each observation of the relational matrix, default is |
Details
This function takes a list of network covariates (in adjacency matrix form) and prepares them for the regression code lmnet. Accomodates 3-dimensional relational arrays with tmax repeated observations of the network (over time or context). Typical network data with a single observation may be input as matrices, i.e. tmax = 1.
Value
A list of:
Y |
Vector of responses (column-wise vectorization order) of appropriate length. |
X |
Matrix of covariates (column-wise vectorization order) of appropriate size. |
nodes |
2-column matrix (or 3-column for repeated observations) indicating directed relation pairs to which each entry in |
See Also
Examples
# tmax = 1
set.seed(1)
n <- 10
Xlist <- list(matrix(rnorm(n^2),n,n), matrix(sample(c(0,1), n^2, replace=TRUE),n,n))
Xlist$Y <- matrix(rnorm(n^2), n, n)
Xlist$Y[1:5] <- NA
r <- inputs_lmnet(Xlist)
r
lmnet(r$Y,r$X,nodes=r$nodes)
# tmax = 4
set.seed(1)
n <- 10
tmax <- 4
X1 <- array(rnorm(n^2*tmax),c(n,n,tmax))
X2 <- array(sample(c(0,1), n^2*tmax, replace=TRUE), c(n,n,tmax))
Xlist <- list(X1, X2)
Xlist$Y <- array(rnorm(n^2)*tmax, c(n, n, tmax))
Xlist$Y[1:5] <- NA
r <- inputs_lmnet(Xlist)
head(r$nodes)
Social interaction data set
Description
A synthetic data set of standardized, directed interactions between 25 students in a seventh grade class.
Format
A data set with four variables. Includes the true parameters.
- interactions
25 x 25 x 5 numeric array of directed relations
- xbinary
25 x 25 x 5 numeric array of binary indictors
- xabs
25 x 25 x 5 numeric array of standardized absolute difference in indicated interest in each subject area
- betatrue
Numeric vector of length 7 that contains true coefficients. The first two (value 1) pertain to
shared_projectandgrade_difference_abs. The last three are separate intercepts for each observation of the network.- Omegatrue
3000 x 3000 numeric matrix: the true covariance matrix of the errors.
- phitrue
2x6 numeric matrix: true parameters of covariance matrix.
Details
We generated a symthetic data set form a true linear model with jointly exchangeable errors. The interactions (the outcomes) between 25 students represent normalized, directed relations between them in 5 different contexts (texts). The observation could be, for instance, the standardized number of characters texted from one student to another over a month pertaining to five subjects: school, friends, family, significant others, and popular culture. The first covariate, xbinary, indicates whether both students indicated in a survey that they were interested in each topic. The second covariate, xabs, measures the absolute, standardized difference in number of characters in total texts of each student of each subject area.
Examples
data("interactions")
Invert an exchangeable matrix
Description
Invert an exchangeable matrix
Usage
invert_exchangeable_matrix(n, phi, directed = TRUE)
Arguments
n |
Number of actors in the network, scalar numeric. |
phi |
Appropriate-length vector of parameters, must be length 5 or 6 for directed= |
directed |
Optional logical indicator of whether input data is for a directed network, default is |
Details
This function inverts a covariance matrix of the exchangeable class in a manner much faster than the direct inverse, and the computational cost does not scale with n. See Marrs et. al. (2017). This approach will only work for complete networks.
Value
out |
Parameters of inverted matrix of exchangeable class. |
References
Marrs, F. W., Fosdick, B. K., & McCormick, T. H., (2017). Standard errors for regression on relational data with exchangeable errors. arXiv preprint arXiv:1701.05530.
See Also
rphi, build_exchangeable_matrix
Examples
n <- 10
phi <- rphi(n, seed=1)
p <- invert_exchangeable_matrix(n, phi)
I1 <- build_exchangeable_matrix(n, phi) %*% build_exchangeable_matrix(n, p)
range(I1 - diag(n*(n-1))) # it works
Linear regression for network response
Description
This function takes X and Y values and fits the multiple linear regression Y = X \beta + \epsilon and returns standard errors.
Usage
lmnet(Y, X, directed = TRUE, tmax = 1, nodes = NULL, reweight = FALSE,
type = "exchangeable", tol = 1e-06, maxit = 10000, ndstop = TRUE,
verbose = FALSE)
Arguments
Y |
Vector of relations to be regress, of length |
X |
Matrix of covariates to be regressed upon, including intercept if intercept is desired, must have |
directed |
Optional logical indicator of whether input data is for a directed network, default is |
tmax |
Optional numeric of third dimension of relational data array, default is |
nodes |
Optional |
reweight |
Optional logical indicator of whether iteratively reweighted least squares should be used to compute estimate of |
type |
Optional character specifying degree of exchangeability of third dimension of array (when present, i.e. in temporal relational arrays). Default is |
tol |
Optional numeric, tolerance of stopping criteria of iteratively reweighted least squares estimate of |
maxit |
Optional numeric, maximum number of iterations for iteratively reweighted least squares estimate of |
ndstop |
Optional logical indicator of whether negative definite weighting matrix in iteratively reweighted least squares should stop the descent. Default is |
verbose |
Optional logical indicator of whether information from iteratively reweighted least squares estimate of |
Details
This function takes X and Y values and fits the multiple linear regression Y = X \beta + \epsilon by ordinary least squares or iteratively reweighted least squares as indicated by the input. The covariance structure is exchangeable from that of Marrs et. al. (2017). The standard errors and test statistics are based on the same paper.
The three dimensional relational array case, i.e. temporal relational data, requires a specification of the type of exchangeability in this third dimension. We may assume that different time periods are independent. On the other hand, we might assume each repeated observation is exchangeable (for example decomposing trade networks into sectors of trade: goods vs. services). See Figure 6a of Marrs et. al. (2017) for the exchangeable case and the surrounding discussion for the independent case.
Value
fit |
An |
References
Marrs, F. W., Fosdick, B. K., & McCormick, T. H., (2017). Standard errors for regression on relational data with exchangeable errors. arXiv preprint arXiv:1701.05530.
See Also
Examples
set.seed(1)
n <- 10
d <- n*(n-1)
X <- cbind(1, rnorm(d), sample(c(0,1), d, replace=TRUE))
betatrue <- rep(1,3)
Y <- X %*% betatrue + rnorm(d)
fit <- lmnet(Y,X)
fit
fit2 <- lmnet(Y,X,reweight=TRUE)
fit2
Replace negative eigenvalues with zeros in variance matrix
Description
Replace negative eigenvalues with zeros in variance matrix
Usage
make.positive.var(V.test)
Matricize a network vector (without diagonal)
Description
Matricize a network vector (without diagonal)
Usage
mat.net(V, directed = T)
Calculate DC meat using rows of X, e
Description
Calculate DC meat using rows of X, e
Usage
meat.DC.row(row.list, X, e)
Calculate E meat using rows of X, e
Description
Calculate E meat using rows of X, e
Usage
meat.E.row(row.list, X, e)
Matrix product of A^TBC where B is a short list of parameters A and C must be matrices B is parameterized by phi, row.list, and assumed symmetric without repeats, with phi[1] along diagonal
Description
Matrix product of A^TBC where B is a short list of parameters A and C must be matrices B is parameterized by phi, row.list, and assumed symmetric without repeats, with phi[1] along diagonal
Usage
meatABC(row.list, phi, A, C, directed)
model.matrix S3 generic for class lmnet
Description
model.matrix S3 generic for class lmnet
Usage
## S3 method for class 'lmnet'
model.matrix(object, ...)
Arguments
object |
lmnet object |
... |
ignored |
Generate node pairs for complete network
Description
Generate node pairs for complete network
Usage
node.gen(n, directed = T)
Generate node sets of various overlapping dyad pairs
Description
Generate node sets of various overlapping dyad pairs
Usage
node.set(n.tot, directed = T)
Make complete node indices for temporal relational data
Description
Make complete node indices for temporal relational data
Usage
node_gen_time(n, directed, tmax)
Pre-processes data for ordering etc.
Description
Pre-processes data for ordering etc.
Usage
node_preprocess(Y, X, directed, nodes, subtract = NULL)
Pre-processes data for ordering, FOR TEMPORAL DATA, etc.
Description
TO-DO: missing data
Usage
node_preprocess_time(Y, X, directed, nodes, tmax, type, subtract = NULL)
Calculate parameter estimates using rows of e
Description
Calculate parameter estimates using rows of e
Usage
param_est(row.list, e)
Given matrix of time blocks and a particular exchangeable parameter set (within each block), calculate a single parameter/phi. ASSUMES NO MISSING DATA
Description
Given matrix of time blocks and a particular exchangeable parameter set (within each block), calculate a single parameter/phi. ASSUMES NO MISSING DATA
Usage
param_est_single_ilist(ilist, blockmat, e, d.tot)
Plot S3 generic for class lmnet
Description
Plot S3 generic for class lmnet
Usage
## S3 method for class 'lmnet'
plot(x, ...)
Arguments
x |
lmnet object |
... |
ignored |
Print S3 generic for class lmnet
Description
Print S3 generic for class lmnet
Usage
## S3 method for class 'lmnet'
print(x, ...)
Arguments
x |
lmnet object |
... |
ignored |
Print S3 generic for class summary.lmnet
Description
Print S3 generic for class summary.lmnet
Usage
## S3 method for class 'summary.lmnet'
print(x, ...)
Arguments
x |
summary.lmnet object |
... |
ignored |
Print S3 generic for summary.vnet object
Description
Print S3 generic for summary.vnet object
Usage
## S3 method for class 'summary.vnet'
print(x, ...)
Arguments
x |
summary.vnet object |
... |
ignored |
Print S3 generic for vnet object
Description
Print S3 generic for vnet object
Usage
## S3 method for class 'vnet'
print(x, ...)
Arguments
x |
vnet object |
... |
ignored |
Generate row list based on nodes input with missingness
Description
Generate row list based on nodes input with missingness
Usage
row_list_missing(nodes, dyads, directed, subtract)
Make row list for complete temporal relational data
Description
Make row list for complete temporal relational data
Usage
row_list_time(n, directed, tmax, type)
Generate positive definite phi set
Description
Generate positive definite phi set
Usage
rphi(n, seed = NULL, phi6 = FALSE, directed = TRUE)
Arguments
n |
Number of actors in the network, scalar numeric. |
seed |
Optional numeric seed to set, default is |
phi6 |
Optional logical indicator of whether sixth parameter |
directed |
Optional logical indicator of whether input data is for a directed network, default is |
Details
This function generates a set of 5 (or 6, as appropriate) parameters that corresponds to positive definite exchangeable covariance matrix for a network of size n. See Marrs et. al. (2017).
Value
phi |
Vector of parameters. |
References
Marrs, F. W., Fosdick, B. K., & McCormick, T. H., (2017). Standard errors for regression on relational data with exchangeable errors. arXiv preprint arXiv:1701.05530.
See Also
build_exchangeable_matrix, invert_exchangeable_matrix
Examples
rphi(10, seed=1)
Summary S3 generic for class lmnet
Description
Summary S3 generic for class lmnet
Usage
## S3 method for class 'lmnet'
summary(object, ...)
Arguments
object |
lmnet object |
... |
ignored |
Summary S3 generic for vnet object
Description
Summary S3 generic for vnet object
Usage
## S3 method for class 'vnet'
summary(object, ...)
Arguments
object |
vnet object |
... |
ignored |
Compute symmetric square root of A, assuming it is real, symmetric, positive definite
Description
Compute symmetric square root of A, assuming it is real, symmetric, positive definite
Usage
symm_square_root(A, prec = 15)
vcov S3 generic for class lmnet
Description
vcov S3 generic for class lmnet
Usage
## S3 method for class 'lmnet'
vcov(object, ...)
Arguments
object |
lmnet object |
... |
ignored |
Vectorize a network matrix (without diagonal)
Description
Vectorize a network matrix (without diagonal)
Usage
vec.net(A, directed = T)
Variance computation for linear regression of network response
Description
Stand-alone estimation of exchangeable variance matrix based on residuals and design matrix.
Usage
vnet(e = NULL, X = NULL, directed = TRUE, nodes = NULL,
type = "exchangeable", tmax = 1, fit = NULL)
Arguments
e |
Optional vector of residuals, of length |
X |
Optional matrix of covariates from regression, must have |
directed |
Optional logical indicator of whether input data is for a directed network, default is |
nodes |
Optional |
type |
Optional string indicating whether the ‘meat’ in the sandwich variance estimator is estimated using exchangeable theory (see Marrs et. al. (2017)) or using dyadic clustering (Fafchamps and Gubert (2007)). |
tmax |
Optional numeric of third dimension of relational data array, default is |
fit |
Optional fitted model object. One of either |
Details
This function takes X and e values computes the variance-covariance matrix of \hat{\beta} that resulted in the residuals e = Y - X \hat{\beta} assuming that the errors are exchangeable, as based on Marrs et. al. (2017) when type = "exchangeable". When type = "dyadic clustering", the theory from Fafchamps and Gubert (2007) is implemented.
Value
A an object of class vhat containing summary information:
vhat |
Estimated variance-covariance matrix of cofficient estimates |
phi |
Vector of variance-covariance parameter estimates. |
corrected |
Logical of whether variance-covariance matrix was corrected from negative definite to positive semi-definite. |
type |
See inputs. |
tmax |
See inputs. |
References
Marrs, F. W., Fosdick, B. K., & McCormick, T. H., (2017). Standard errors for regression on relational data with exchangeable errors. arXiv preprint arXiv:1701.05530.
Fafchamps, M., & Gubert, F. (2007). Risk sharing and network formation. American Economic Review, 97(2), 75-79.
See Also
Examples
set.seed(1)
n <- 10
d <- n*(n-1)
X <- cbind(1, rnorm(d), sample(c(0,1), d, replace=TRUE))
e <- rnorm(d)
vnet(e=e,X=X)
Wolf network data set
Description
A data set measuring dominance and its behavioral measures in a captive wolf pack.
Format
A data set with three variables
- wolf
16 x 16 numeric matrix of dominance measures
- wolf_age_diff
16 x 16 numeric matrix of difference in ages (column less row)
- wolf_same_sex
16 x 16 numeric matrix of indicators of same sex
Details
This is data on a captive family of wolves in Arnheim, Germany. The 16 wolves studied here were housed in a large wooded enclosure and observed in 1978. This matrix displays deference acts. The number in a cell represents the number of occasions on which the row wolf was seen to exhibit a "low posture" display directed toward the column wolf. The behavior could involve approach or retreat, but the fact that it was performed in "low posture" suggests that it was deferent. Data obtained March 20, 2018 from http://moreno.ss.uci.edu/data.html#wolf.
Source
http://moreno.ss.uci.edu/data.html#wolf
References
Jan A. R. A. M. van Hooff and Joep A. B. Wensing, "Dominance and its behavioral measures in a captive wolf pack," Chapter 11 in Harry Frank, ed., Man and Wolf. Dordrecht: Junk, 1987, pp. 219-252.
Examples
data("wolf")