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Some more specifics:

Network autocorrelation can, if strong enough, cause arbitrarily large changes in both the direction of effects, and the significance levels. 

You should be able to explore this via simulation. For example:

x <- discrep_net_prop_more.W
fit <- lm(y ~ x)
lambda <- 0
y_sim <- fit$coefficients%*%cbind(1,x) + Solve(diag(1, length(x)) - lambda* w.list.trunc)) %*% rnorm(length(x), 0, summary(fit)$sigma)
summary(fit_sim <- lm(y_sim ~ x))

(w.list.trunc is an n x n matrix, right?)

For lambda = 0, there should be no significant differences between fit_sim and fit. However, as you change lambda (making it larger in magnitude, both positive and negative), you can see how the results of the linear model will change from network autocorrelation. 

What the network autocorrelation model does is fit this extra "lambda" parameter that is the strength of the network autocorrelation, and this "subtracts out" the effect of the network autocorrelation thereby "leaving behind" only the "pure" effect of the explanatory variables on the response variable (ideally). 

(Note that conceptually, for the simulation you should probably be using the fitted network autocorrelation model---as it has "subtracted out" existing network autocorrelation--rather than the fitted linear model which still has the network autocorrelation present, but just to understand what network autocorrelation can do in general, it's not really relevant what data you use since the original fitted model can act as your "truth" for referencing back to. If you're interested in how network autocorrelation may be acting in your specific data, then use the coefficients and estimated sigma from spautolm instead of lm(y~x).)

You can also see how different network autocorrelation structure would change the results by putting in different weights matrices. 

But of course, every model comes with assumptions. Network autocorrelation models assume (1) the effect of network autocorrelation is constant throughout the network, and (2) you are using the correct weights matrix to measure potential network autocorrelation (the correct choice is not necessarily obvious! See Leenders 2002, "Modeling social influence through network autocorrelation: constructing the weight matrix"). Personally, these assumptions are a bit too heavy for my taste, but additionally, network autocorrelation models can only (at best) control for all network dependencies at once; they cannot substantively describe (estimate and perform inference on) specific dependencies, which is what most network modeling is concerned with. 

If you want to continue with a network autocorrelation approach, George's suggestion of looking into spatial econometrics is an excellent one. The weights matrix for network autocorrelation attempts to capture network dependencies rather than spatial ones, but otherwise, the modeling process is very similar and there is a lot more work with spatial errors and types of spatial regression models. Luc Anselin's work, in particular, has some great exemplars of spatial regression in practice. 

On Mon, Mar 12, 2018 at 3:06 PM, George Vega Yon <[log in to unmask]> wrote:
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Dear Matthew,

Those two models are very different in both the underlying assumptions and the specifications. The first one is a SAR model which assumes that the errors are multivariate normal distributed, hence, no iid observations, and thus, the likelihood function is different from what you would observe in a plain OLS. Also, the spautolm includes the autocorrelation term when you specify listwd, so, if discrep_net_prop_more.W is your exposure terms, it means that you are including it twice but in the wrong way. Instead, you should type something like:

spautolm(formula = BLQFMHEDWM.W ~ 1, listw = w.list.trunc, 
    family = "SAR", zero.policy = TRUE)

For example. The autocorrelation term is the Lambda coefficient which is reported afterward.

The lm, on the other hand, runs an OLS model in which the errors are assumed to be iid making that type of model (including an exposure term) not valid as your estimates will be biased by construction (unless you are using a lagged exposure). A good reference on spatial econometrics is here: (LeSage, 2008)


On Tue, Feb 20, 2018 at 2:43 PM, Meisel, Matthew <[log in to unmask]> wrote:
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Dear all,

My mentor has a large sociocentric network dataset (N = 1342) and I am trying to analyze the data using network autocorrelation models. When I run a model in R using the spautolm function and then I run the same model using the lm function, I get incredibly different results (the effects are significant but in different directions), and I have no idea on why these two methods would provide such vastly different results. See the models below. FYI, the predictor is dichotomous and the outcome is continuous. If you have any thoughts, please let me know, cause I do not know which results to trust.  


Call: spautolm(formula = BLQFMHEDWM.W ~ discrep_net_prop_more.W, listw = w.list.trunc, 
    family = "SAR", zero.policy = TRUE)

     Min       1Q   Median       3Q      Max 
-5.71857 -1.19085 -0.49541  0.55372 15.64806 

                        Estimate Std. Error z value  Pr(>|z|)
(Intercept)              1.64796    0.40402  4.0789 4.524e-05
discrep_net_prop_more.W -1.30912    0.18460 -7.0916 1.325e-12

Lambda: 0.83807 LR test value: 425.13 p-value: < 2.22e-16 
Numerical Hessian standard error of lambda: 0.030919 

Log likelihood: -2680.358 
ML residual variance (sigma squared): 5.0308, (sigma: 2.2429)
Number of observations: 1178 
Number of parameters estimated: 4 
AIC: 5368.7

Call: lm(formula = BLQFMHEDWM.W ~ discrep_net_prop_more.W, data = subject)

    Min      1Q  Median      3Q     Max 
-2.2335 -1.7280 -1.7280  0.7665 18.2720 

                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)              1.72799    0.08878  19.465   <2e-16 ***
discrep_net_prop_more.W  0.50554    0.23578   2.144   0.0322 *  
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.823 on 1176 degrees of freedom
Multiple R-squared:  0.003894, Adjusted R-squared:  0.003047 
F-statistic: 4.597 on 1 and 1176 DF,  p-value: 0.03223

Matthew K. Meisel, Ph.D.
Assistant Professor
Center for Alcohol and Addiction Studies
Department of Behavioral and Social Sciences
Brown University | School of Public Health
Box G-S121-4 Providence, RI 02903
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