Description Usage Arguments Details Value Note Author(s) References See Also Examples
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.
1 2 3 
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 
W2 
one or more (possibly valued) graphs on the elements of 
theta.seed 
an optional seed value for the parameter vector estimation process. 
null.model 
the null model to be fit; must be one of 
method 
method to be used with 
control 
optional control parameters for 
tol 
convergence tolerance for the MLE (expressed as change in deviance). 
lnam
fits the linear network autocorrelation model given by
y = W1 %*% y + X %*% beta + e, e = W2 %*% e + nu
where y is a vector of responses, X is a covariate matrix, nu ~ Norm(0,sigma^2),
W1 = sum( rho1_i W1_i, i=1..p), W2 = sum( rho2_i W2_i, i=1..q),
and W1_i, W2_i are (possibly valued) adjacency matrices.
Intuitively, rho1 is a vector of “AR”like parameters (parameterizing the autoregression of each y value on its neighbors in the graphs of W1) while rho2 is a vector of “MA”like parameters (parameterizing the autocorrelation of each disturbance in y on its neighbors in the graphs of 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:
meanstd
: mean and standard deviation estimated (default).
mean
: mean estimated; standard deviation assumed equal to 1.
std
: standard deviation estimated; mean assumed equal to 0.
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.)
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 loglikelihood 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 loglikelihood 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 ehat in the model equation, not nuhat. 
disturbances 
the estimated disturbances, i.e., nuhat. 
call 
the matched call. 
Actual optimization is performed by calls to optim
. Information on algorithms and control parameters can be found via the appropriate man pages.
Carter T. Butts [email protected]
Leenders, T.Th.A.J. (2002) “Modeling Social Influence Through Network Autocorrelation: Constructing the Weight Matrix” Social Networks, 24(1), 2147.
Anselin, L. (1988) Spatial Econometrics: Methods and Models. Norwell, MA: Kluwer.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  ## 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)

Loading required package: statnet.common
Loading required package: network
network: Classes for Relational Data
Version 1.13.0 created on 20150831.
copyright (c) 2005, Carter T. Butts, University of CaliforniaIrvine
Mark S. Handcock, University of California  Los Angeles
David R. Hunter, Penn State University
Martina Morris, University of Washington
Skye BenderdeMoll, University of Washington
For citation information, type citation("network").
Type help("networkpackage") to get started.
sna: Tools for Social Network Analysis
Version 2.4 created on 20160723.
copyright (c) 2005, Carter T. Butts, University of CaliforniaIrvine
For citation information, type citation("sna").
Type help(package="sna") to get started.
Loading required namespace: numDeriv
Call:
lnam(y = y, x = x, W1 = w1, W2 = w2)
Residuals:
Min 1Q Median 3Q Max
0.74745 0.21060 0.01533 0.17953 0.70902
Coefficients:
Estimate Std. Error Z value Pr(>z)
X1 0.3404143 0.0079307 42.924 < 2e16 ***
X2 0.6081271 0.0087983 69.118 < 2e16 ***
X3 1.8080255 0.0094922 190.476 < 2e16 ***
X4 0.0552872 0.0076382 7.238 4.55e13 ***
X5 0.5242264 0.0104786 50.028 < 2e16 ***
rho1.1 0.2003312 0.0008426 237.754 < 2e16 ***
rho2.1 0.0972474 0.0194177 5.008 5.49e07 ***

Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Estimate Std. Error
Sigma 0.1024 0
GoodnessofFit:
Residual standard error: 0.2933 on 93 degrees of freedom (w/o Sigma)
Multiple RSquared: 0.9897, Adjusted RSquared: 0.989
Model log likelihood: 104.4 on 92 degrees of freedom (w/Sigma)
AIC: 192.8 BIC: 172
Null model: meanstd
Null log likelihood: 240.6 on 98 degrees of freedom
AIC: 485.2 BIC: 490.4
AIC difference (model versus null): 678
Heuristic Log Bayes Factor (model versus null): 662.4
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