lnam: Fit a Linear Network Autocorrelation Model

In sna: Tools for Social Network Analysis

Fit a Linear Network Autocorrelation Model

Description

lnam is used to fit linear network autocorrelation models. These include standard OLS as a special case, although lm is to be preferred for such analyses.

Usage

lnam(y, x = NULL, W1 = NULL, W2 = NULL, theta.seed = NULL, 
    null.model = c("meanstd", "mean", "std", "none"), method = "BFGS", 
    control = list(), tol=1e-10)

Arguments

y	a vector of responses.
x	a vector or matrix of covariates; if the latter, each column should contain a single covariate.
W1	one or more (possibly valued) graphs on the elements of y.
W2	one or more (possibly valued) graphs on the elements of y.
theta.seed	an optional seed value for the parameter vector estimation process.
null.model	the null model to be fit; must be one of "meanstd", "mean", "std", or "none".
method	method to be used with optim.
control	optional control parameters for optim.
tol	convergence tolerance for the MLE (expressed as change in deviance).

Details

lnam fits the linear network autocorrelation model given by

y = W_1 y + X \beta + e, \quad e = W_2 e + \nu

where y is a vector of responses, X is a covariate matrix, \nu \sim N(0,\sigma^2),

W_1 = \sum_{i=1}^p \rho_{1i} W_{1i}, \quad W_2 = \sum_{i=1}^q \rho_{2i} W_{2i},

and W_{1i}, W_{2i} are (possibly valued) adjacency matrices.

Intuitively, \rho_1 is a vector of “AR”-like parameters (parameterizing the autoregression of each y value on its neighbors in the graphs of W_1) while \rho_2 is a vector of “MA”-like parameters (parameterizing the autocorrelation of each disturbance in y on its neighbors in the graphs of W_2). In general, the two models are distinct, and either or both effects may be selected by including the appropriate matrix arguments.

Model parameters are estimated by maximum likelihood, and asymptotic standard errors are provided as well; all of the above (and more) can be obtained by means of the appropriate print and summary methods. A plotting method is also provided, which supplies fit basic diagnostics for the estimated model. For purposes of comparison, fits may be evaluated against one of four null models:

1. meanstd: mean and standard deviation estimated (default).

2. mean: mean estimated; standard deviation assumed equal to 1.

3. std: standard deviation estimated; mean assumed equal to 0.

4. none: no parameters estimated; data assumed to be drawn from a standard normal density.

The default setting should be appropriate for the vast majority of cases, although the others may have use when fitting “pure” autoregressive models (e.g., without covariates). Although a major use of the lnam is in controlling for network autocorrelation within a regression context, the model is subtle and has a variety of uses. (See the references below for suggestions.)

Value

An object of class "lnam" containing the following elements:

y	the response vector used.
x	if supplied, the coefficient matrix.
W1	if supplied, the W1 array.
W2	if supplied, the W2 array.
model	a code indicating the model terms fit.
infomat	the estimated Fisher information matrix for the fitted model.
acvm	the estimated asymptotic covariance matrix for the model parameters.
null.model	a string indicating the null model fit.
lnlik.null	the log-likelihood of y under the null model.
df.null.resid	the residual degrees of freedom under the null model.
df.null	the model degrees of freedom under the null model.
null.param	parameter estimates for the null model.
lnlik.model	the log-likelihood of y under the fitted model.
df.model	the model degrees of freedom.
df.residual	the residual degrees of freedom.
df.total	the total degrees of freedom.
rho1	if applicable, the MLE for rho1.
rho1.se	if applicable, the asymptotic standard error for rho1.
rho2	if applicable, the MLE for rho2.
rho2.se	if applicable, the asymptotic standard error for rho2.
sigma	the MLE for sigma.
sigma.se	the standard error for sigma
beta	if applicable, the MLE for beta.
beta.se	if applicable, the asymptotic standard errors for beta.
fitted.values	the fitted mean values.
residuals	the residuals (response minus fitted); note that these correspond to \hat{e} in the model equation, not \hat{\nu}.
disturbances	the estimated disturbances, i.e., \hat{\nu}.
call	the matched call.

Note

Actual optimization is performed by calls to optim. Information on algorithms and control parameters can be found via the appropriate man pages.

Author(s)

Carter T. Butts [e-mail]

References

Leenders, T.Th.A.J. (2002) “Modeling Social Influence Through Network Autocorrelation: Constructing the Weight Matrix” Social Networks, 24(1), 21-47.

Anselin, L. (1988) Spatial Econometrics: Methods and Models. Norwell, MA: Kluwer.

See Also

lm, optim

Examples

## Not run: 
#Construct a simple, random example:
w1<-rgraph(100)               #Draw the AR matrix
w2<-rgraph(100)               #Draw the MA matrix
x<-matrix(rnorm(100*5),100,5) #Draw some covariates
r1<-0.2                       #Set the model parameters
r2<-0.1
sigma<-0.1
beta<-rnorm(5)
#Assemble y from its components:
nu<-rnorm(100,0,sigma)          #Draw the disturbances
e<-qr.solve(diag(100)-r2*w2,nu) #Draw the effective errors
y<-qr.solve(diag(100)-r1*w1,x%*%beta+e)  #Compute y

#Now, fit the autocorrelation model:
fit<-lnam(y,x,w1,w2)
summary(fit)
plot(fit)

## End(Not run)

sna documentation built on Sept. 11, 2024, 8:45 p.m.