Spatial simultaneous autoregressive model estimation by maximum likelihood
The lagsarlm function provides Maximum likelihood estimation of spatial simultaneous autoregressive lag and spatial Durbin (mixed) models of the form:
y = rho W y + X beta + e
where rho is found by optimize() first, and beta and other parameters by generalized least squares subsequently (one-dimensional search using optim performs badly on some platforms). In the spatial Durbin (mixed) model, the spatially lagged independent variables are added to X. Note that interpretation of the fitted coefficients should use impact measures, because of the feedback loops induced by the data generation process for this model. With one of the sparse matrix methods, larger numbers of observations can be handled, but the interval= argument may need be set when the weights are not row-standardised.
Maximum likelihood estimation of spatial simultaneous autoregressive error models of the form:
y = X beta + u, u = lambda W u + e
where lambda is found by optimize() first, and beta and other parameters by generalized least squares subsequently. With one of the sparse matrix methods, larger numbers of observations can be handled, but the interval= argument may need be set when the weights are not row-standardised. When etype is “emixed”, a so-called spatial Durbin error model is fitted.
Maximum likelihood estimation of spatial simultaneous autoregressive “SAC/SARAR” models of the form:
y = rho W1 y + X beta + u, u = lambda W2 u + e
where rho and lambda are found by nlminb or optim() first, and beta and other parameters by generalized least squares subsequently.
lagsarlm(formula, data = list(), listw, na.action, Durbin, type, method="eigen", quiet=NULL, zero.policy=NULL, interval=NULL, tol.solve=.Machine$double.eps, trs=NULL, control=list()) errorsarlm(formula, data=list(), listw, na.action, weights=NULL, Durbin, etype, method="eigen", quiet=NULL, zero.policy=NULL, interval = NULL, tol.solve=.Machine$double.eps, trs=NULL, control=list()) sacsarlm(formula, data = list(), listw, listw2 = NULL, na.action, Durbin, type, method="eigen", quiet=NULL, zero.policy=NULL, tol.solve=.Machine$double.eps, llprof=NULL, interval1=NULL, interval2=NULL, trs1=NULL, trs2=NULL, control = list()) ## S3 method for class 'Sarlm' summary(object, correlation = FALSE, Nagelkerke = FALSE, Hausman=FALSE, adj.se=FALSE, ...) ## S3 method for class 'Sarlm' print(x, ...) ## S3 method for class 'summary.Sarlm' print(x, digits = max(5, .Options$digits - 3), signif.stars = FALSE, ...) ## S3 method for class 'Sarlm' residuals(object, ...) ## S3 method for class 'Sarlm' deviance(object, ...) ## S3 method for class 'Sarlm' coef(object, ...) ## S3 method for class 'Sarlm' vcov(object, ...) ## S3 method for class 'Sarlm' fitted(object, ...)
formula |
a symbolic description of the model to be fit. The details
of model specification are given for |
data |
an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called. |
listw, listw2 |
a |
na.action |
a function (default |
weights |
an optional vector of weights to be used in the fitting process. Non-NULL weights can be used to indicate that different observations have different variances (with the values in weights being inversely proportional to the variances); or equivalently, when the elements of weights are positive integers w_i, that each response y_i is the mean of w_i unit-weight observations (including the case that there are w_i observations equal to y_i and the data have been summarized) - |
Durbin |
default FALSE (spatial lag model); if TRUE, full spatial Durbin model; if a formula object, the subset of explanatory variables to lag |
type |
(use the ‘Durbin=’ argument - retained for backwards compatibility only) default "lag", may be set to "mixed"; when "mixed", the lagged intercept is dropped for spatial weights style "W", that is row-standardised weights, but otherwise included; “Durbin” may be used instead of “mixed” |
etype |
(use the ‘Durbin=’ argument - retained for backwards compatibility only) default "error", may be set to "emixed" to include the spatially lagged independent variables added to X; when "emixed", the lagged intercept is dropped for spatial weights style "W", that is row-standardised weights, but otherwise included |
method |
"eigen" (default) - the Jacobian is computed as the product
of (1 - rho*eigenvalue) using |
quiet |
default NULL, use !verbose global option value; if FALSE, reports function values during optimization. |
zero.policy |
default NULL, use global option value; if TRUE assign zero to the lagged value of zones without
neighbours, if FALSE (default) assign NA - causing |
interval |
default is NULL, search interval for autoregressive parameter |
tol.solve |
the tolerance for detecting linear dependencies in the columns of matrices to be inverted - passed to |
llprof |
default NULL, can either be an integer, to divide the feasible ranges into a grid of points, or a two-column matrix of spatial coefficient values, at which to evaluate the likelihood function |
trs1, trs2 |
default NULL, if given, vectors for each weights object of powered spatial weights matrix traces output by |
interval1, interval2 |
default is NULL, search intervals for each weights object for autoregressive parameters |
trs |
default NULL, if given, a vector of powered spatial weights matrix traces output by |
control |
list of extra control arguments - see section below |
object |
|
correlation |
logical; if 'TRUE', the correlation matrix of the estimated parameters including sigma is returned and printed (default=FALSE) |
Nagelkerke |
if TRUE, the Nagelkerke pseudo R-squared is reported |
Hausman |
if TRUE, the results of the Hausman test for error models are reported |
adj.se |
if TRUE, adjust the coefficient standard errors for the number of fitted coefficients |
x |
|
digits |
the number of significant digits to use when printing |
signif.stars |
logical. If TRUE, "significance stars" are printed for each coefficient. |
... |
further arguments passed to or from other methods |
The asymptotic standard error of rho is only computed when
method=“eigen”, because the full matrix operations involved would be costly
for large n typically associated with the choice of method="spam" or "Matrix". The same applies to the coefficient covariance matrix. Taken as the
asymptotic matrix from the literature, it is typically badly scaled, and with the elements involving rho (lag model) or lambda (error model) being very small,
while other parts of the matrix can be very large (often many orders
of magnitude in difference). It often happens that the tol.solve
argument needs to be set to a smaller value than the default, or the RHS variables can be centred or reduced in range.
Versions of the package from 0.4-38 include numerical Hessian values where asymptotic standard errors are not available. This change has been introduced to permit the simulation of distributions for impact measures. The warnings made above with regard to variable scaling also apply in this case.
Note that the fitted() function for the output object assumes that the response
variable may be reconstructed as the sum of the trend, the signal, and the
noise (residuals). Since the values of the response variable are known,
their spatial lags are used to calculate signal components (Cressie 1993,
p. 564). This differs from other software, including GeoDa, which does not use
knowledge of the response variable in making predictions for the fitting data.
Refer to the help page of predict.Sarlm for discussions and
references.
Because numerical optimisation is used to find the values of lambda and rho in sacsarlm, care needs to be shown. It has been found that the surface of the 2D likelihood function often forms a “banana trench” from (low rho, high lambda) through (high rho, high lambda) to (high rho, low lambda) values. In addition, sometimes the banana has optima towards both ends, one local, the other global, and conseqently the choice of the starting point for the final optimization becomes crucial. The default approach is not to use just (0, 0) as a starting point, nor the (rho, lambda) values from gstsls, which lie in a central part of the “trench”, but either four values at (low rho, high lambda), (0, 0), (high rho, high lambda), and (high rho, low lambda), and to use the best of these start points for the final optimization. Optionally, nine points can be used spanning the whole (lower, upper) space.
the desired accuracy of the optimization - passed to optimize() (default=square root of double precision machine tolerance, a larger root may be used needed, see help(boston) for an example)
(error model) default TRUE, return the Vo matrix for a spatial Hausman test
(error model) default 250, if returnHcov=TRUE and the method is not “eigen”, pass this order to powerWeights as the power series maximum limit
default NULL, then set to (method != "eigen") internally; use fdHess to compute an approximate Hessian using finite differences when using sparse matrix methods; used to make a coefficient covariance matrix when the number of observations is large; may be turned off to save resources if need be
default FALSE, use fdHess from nlme, if TRUE, use optim to calculate Hessian at optimum
default “optimHess”, may be “nlm” or one of the optim methods
default FALSE; logical value used in the log likelihood function to choose compiled code for computing SSE
default 2; used for preparing the Cholesky decompositions for updating in the Jacobian function
if NULL (default), set to FALSE to use a simplicial decomposition for the sparse Cholesky decomposition and method “Matrix_J”, set to as.logical(NA) for method “Matrix”, if TRUE, use a supernodal decomposition
default 5; highest power of the approximating polynomial for the Chebyshev approximation
default 16; number of random variates
default 30; number of products of random variates matrix and spatial weights matrix
default “MMD”, alternative “RCM”
default 0.1, coefficient value for initial Cholesky decomposition in “spam_update”
default “MC”, used with method “moments”; alternatives “mult” and “moments”, for use if trs is missing, trW
default TRUE, used with method “moments” to compute the Smirnov/Anselin correction term
default TRUE, used with method “moments” to truncate the Smirnov/Anselin correction term
default “LU”, may be “MC”
default 200, as in SE toolbox; the size of the first stage lndet grid; it may be reduced to for example 40
default 2000, as in SE toolbox; the size of the second stage lndet grid
default TRUE; if the method is not “eigen”, use asymmetric covariances rather than numerical Hessian ones if n <= small
default 1500; threshold number of observations for asymmetric covariances when the method is not “eigen”
default NULL, may be used to pass a pre-computed SE toolbox style matrix of coefficients and their lndet values to the "SE_classic" and "SE_whichMin" methods
default FALSE; used in “LU_prepermutate”, note warnings given for lu method
default NULL; may be used to pass a pre-computed vector of eigenvalues
default 1; used to set the sign of the final component to negative if -1 (alpha times ((sigma squared) squared) in Ord (1975) equation B.1).
default “nlminb”, may be set to “L-BFGS-B” to use box-constrained optimisation in optim
default list(), a control list to pass to nlminb or optim
default NULL, for which five trial starting values spanning the lower/upper range are tried and the best selected, starting values of rho and lambda
default integer 4L, four trial points; if not default value, nine trial points
default NULL; may be used to pass pre-computed vectors of eigenvalues
Roger Bivand Roger.Bivand@nhh.no, with thanks to Andrew Bernat for contributions to the asymptotic standard error code.
Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion; Ord, J. K. 1975 Estimation methods for models of spatial interaction, Journal of the American Statistical Association, 70, 120-126; Anselin, L. 1988 Spatial econometrics: methods and models. (Dordrecht: Kluwer); Anselin, L. 1995 SpaceStat, a software program for the analysis of spatial data, version 1.80. Regional Research Institute, West Virginia University, Morgantown, WV; Anselin L, Bera AK (1998) Spatial dependence in linear regression models with an introduction to spatial econometrics. In: Ullah A, Giles DEA (eds) Handbook of applied economic statistics. Marcel Dekker, New York, pp. 237-289; Nagelkerke NJD (1991) A note on a general definition of the coefficient of determination. Biometrika 78: 691-692; Cressie, N. A. C. 1993 Statistics for spatial data, Wiley, New York; LeSage J and RK Pace (2009) Introduction to Spatial Econometrics. CRC Press, Boca Raton.
Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. https://www.jstatsoft.org/v63/i18/.
Bivand, R. S., Hauke, J., and Kossowski, T. (2013). Computing the Jacobian in Gaussian spatial autoregressive models: An illustrated comparison of available methods. Geographical Analysis, 45(2), 150-179.
data(oldcol, package="spdep") listw <- spdep::nb2listw(COL.nb, style="W") ev <- eigenw(listw) W <- as(listw, "CsparseMatrix") trMatc <- trW(W, type="mult") COL.lag.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw=listw, method="eigen", quiet=FALSE, control=list(pre_eig=ev, OrdVsign=1)) (x <- summary(COL.lag.eig, correlation=TRUE)) coef(x) ## Not run: COL.lag.eig$fdHess COL.lag.eig$resvar # using the apparent sign in Ord (1975, equation B.1) COL.lag.eigb <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw=listw, method="eigen", control=list(pre_eig=ev, OrdVsign=-1)) summary(COL.lag.eigb) COL.lag.eigb$fdHess COL.lag.eigb$resvar # force numerical Hessian COL.lag.eig1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw=listw, method="Matrix", control=list(small=25)) summary(COL.lag.eig1) COL.lag.eig1$fdHess # force LeSage & Pace (2008, p. 57) approximation COL.lag.eig1a <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw=listw, method="Matrix", control=list(small=25), trs=trMatc) summary(COL.lag.eig1a) COL.lag.eig1a$fdHess COL.lag.eig$resvar[2,2] # using the apparent sign in Ord (1975, equation B.1) COL.lag.eigb$resvar[2,2] # force numerical Hessian COL.lag.eig1$fdHess[1,1] # force LeSage & Pace (2008, p. 57) approximation COL.lag.eig1a$fdHess[2,2] ## End(Not run) system.time(COL.lag.M <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="Matrix", quiet=FALSE)) summary(COL.lag.M) impacts(COL.lag.M, listw=listw) ## Not run: system.time(COL.lag.sp <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw=listw, method="spam", quiet=FALSE)) summary(COL.lag.sp) COL.lag.B <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, spdep::nb2listw(COL.nb, style="B"), control=list(pre_eig=ev)) summary(COL.lag.B) COL.mixed.B <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, spdep::nb2listw(COL.nb, style="B"), type="mixed", tol.solve=1e-9, control=list(pre_eig=ev)) summary(COL.mixed.B) COL.mixed.W <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, type="mixed", control=list(pre_eig=ev)) summary(COL.mixed.W) COL.mixed.D00 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, Durbin=TRUE, control=list(pre_eig=ev)) summary(COL.mixed.D00) COL.mixed.D01 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, Durbin=FALSE, control=list(pre_eig=ev)) summary(COL.mixed.D01) COL.mixed.D1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, Durbin= ~ INC + HOVAL, control=list(pre_eig=ev)) summary(COL.mixed.D1) f <- CRIME ~ INC + HOVAL COL.mixed.D2 <- lagsarlm(f, data=COL.OLD, listw, Durbin=as.formula(delete.response(terms(f))), control=list(pre_eig=ev)) summary(COL.mixed.D2) COL.mixed.D1a <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, Durbin= ~ INC, control=list(pre_eig=ev)) summary(COL.mixed.D1a) try(COL.mixed.D1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, Durbin= ~ inc + HOVAL, control=list(pre_eig=ev))) try(COL.mixed.D1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, Durbin= ~ DISCBD + HOVAL, control=list(pre_eig=ev))) NA.COL.OLD <- COL.OLD NA.COL.OLD$CRIME[20:25] <- NA COL.lag.NA <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD, listw, na.action=na.exclude) COL.lag.NA$na.action COL.lag.NA resid(COL.lag.NA) COL.lag.NA1 <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD, listw, Durbin=~INC) # https://github.com/r-spatial/spatialreg/issues/10 COL.lag.NA1$na.action COL.lag.NA2 <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD, listw, Durbin=~INC, na.action=na.exclude) COL.lag.NA2$na.action # https://github.com/r-spatial/spatialreg/issues/11 COL.lag.NA3 <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD, listw, control=list(pre_eig=ev)) COL.lag.NA3$na.action ## End(Not run) ## Not run: data(boston, package="spData") gp2mM <- lagsarlm(log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + I(RM^2) + AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT), data=boston.c, spdep::nb2listw(boston.soi), type="mixed", method="Matrix") summary(gp2mM) W <- as(spdep::nb2listw(boston.soi), "CsparseMatrix") trMatb <- trW(W, type="mult") gp2mMi <- lagsarlm(log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + I(RM^2) + AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT), data=boston.c, spdep::nb2listw(boston.soi), type="mixed", method="Matrix", trs=trMatb) summary(gp2mMi) ## End(Not run) COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, quiet=FALSE, control=list(pre_eig=ev)) summary(COL.errW.eig) COL.errW.eig_ev <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, control=list(pre_eig=ev)) all.equal(coefficients(COL.errW.eig), coefficients(COL.errW.eig_ev)) COL.errB.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, spdep::nb2listw(COL.nb, style="B")) summary(COL.errB.eig) COL.errW.M <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="Matrix", quiet=FALSE, trs=trMatc) summary(COL.errW.M) COL.SDEM.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, etype="emixed", control=list(pre_eig=ev)) summary(COL.SDEM.eig) ## Not run: COL.SDEM.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, Durbin=TRUE, control=list(pre_eig=ev)) summary(COL.SDEM.eig) COL.SDEM.eig <- errorsarlm(CRIME ~ DISCBD + INC + HOVAL, data=COL.OLD, listw, Durbin=~INC, control=list(pre_eig=ev)) summary(COL.SDEM.eig) summary(impacts(COL.SDEM.eig)) NA.COL.OLD <- COL.OLD NA.COL.OLD$CRIME[20:25] <- NA COL.err.NA <- errorsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD, listw, na.action=na.exclude) COL.err.NA$na.action COL.err.NA resid(COL.err.NA) print(system.time(ev <- eigenw(similar.listw(listw)))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="eigen", control=list(pre_eig=ev)))) ocoef <- coefficients(COL.errW.eig) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="eigen", control=list(pre_eig=ev, LAPACK=FALSE)))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="eigen", control=list(pre_eig=ev, compiled_sse=TRUE)))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="Matrix_J", control=list(super=TRUE)))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="Matrix_J", control=list(super=FALSE)))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="Matrix_J", control=list(super=as.logical(NA))))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="Matrix", control=list(super=TRUE)))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="Matrix", control=list(super=FALSE)))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="Matrix", control=list(super=as.logical(NA))))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="spam", control=list(spamPivot="MMD")))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="spam", control=list(spamPivot="RCM")))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="spam_update", control=list(spamPivot="MMD")))) print(all.equal(ocoef, coefficients(COL.errW.eig))) print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, method="spam_update", control=list(spamPivot="RCM")))) print(all.equal(ocoef, coefficients(COL.errW.eig))) ## End(Not run) COL.sacW.eig <- sacsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, control=list(pre_eig1=ev, pre_eig2=ev)) summary(COL.sacW.eig) set.seed(1) summary(impacts(COL.sacW.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE) COL.msacW.eig <- sacsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, type="sacmixed", control=list(pre_eig1=ev, pre_eig2=ev)) summary(COL.msacW.eig) set.seed(1) summary(impacts(COL.msacW.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE) COL.msacW1.eig <- sacsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, Durbin=TRUE, control=list(pre_eig1=ev, pre_eig2=ev)) summary(COL.msacW1.eig) set.seed(1) summary(impacts(COL.msacW1.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE) COL.msacW2.eig <- sacsarlm(CRIME ~ DISCBD + INC + HOVAL, data=COL.OLD, listw, Durbin= ~ INC, control=list(pre_eig1=ev, pre_eig2=ev)) summary(COL.msacW2.eig) summary(impacts(COL.msacW2.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE) ## Not run: COL.mix.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, type="mixed", method="eigen") summary(COL.mix.eig, correlation=TRUE, Nagelkerke=TRUE) COL.mix.M <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw, type="mixed", method="Matrix") summary(COL.mix.M, correlation=TRUE, Nagelkerke=TRUE) COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, spdep::nb2listw(COL.nb, style="W"), method="eigen") summary(COL.errW.eig, correlation=TRUE, Nagelkerke=TRUE, Hausman=TRUE) ## End(Not run)
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