lnam {sna}R Documentation

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())

Arguments

y a vector of responses.
x a vector or matrix of covariates; if the latter, each column should contain a single covariate.
W1 a (possibly valued) graph on the elements of y.
W2 another (possibly valued) graph on the elements of y.
theta.seed an optional seed value for optim.
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.

Details

lnam fits the linear network autocorrelation model given by

y = rho1 * W1 %*% y + X %*% beta + e, e = rho2 * W2 %*% e + nu

where y is a vector of responses, X is a covariate matrix, W1 and W2 are (possibly valued) adjacency matrices, and nu ~ Norm(0,sigma^2). Intuitively, rho1 is an ``AR''-like parameter (parameterizing the autoregression of each y value on its neighbors in W1) while rho2 is an ``MA''-like parameter (parameterizing the autocorrelation of each disturbance in y on its neighbors in W2). 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 matrix.
W2 if supplied, the W2 matrix.
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 e-hat in the model equation, not nu-hat.
disturbances the estimated disturbances, i.e., nu-hat.
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 buttsc@uci.edu

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

#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)

[Package sna version 1.2 Index]