vagam: Fitting generalized additive models (GAMs) using variational...
In vagam: Variational Approximations for Generalized Additive Models
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
Follows the variational approximation approach of Hui et al. (2018) for fitted generalized additive models. In this package, the term GAM is taken to be generalized linear mixed model, specifically, the nonparametric component is modeled using a P-splines i.e., cubic B-splines with a first order difference penalty. Because the penalty can be written as a quadratic form in terms of the smoothing coefficients, then it is treated a (degenerate) multivariate normal random effects distribution and a marginal log-likleihood for the resulting mixed model can be constructed.
The VA framework is then utilized to provide a fully or at least closed to fully tractable lower bound approximation to the marginal likelihood of a GAM. In doing so, the VA framework aims offers both the stability and natural inference tools available in the mixed model approach to GAMs, while achieving computation times comparable to that of using the penalized likelihood approach to GAMs.
Usage
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vagam(y, smooth.X, para.X = NULL, lambda = NULL, int.knots, family = gaussian(),
A.struct = c("unstructured", "block"), offset = NULL, save.data = FALSE,
para.se = FALSE, doIC = FALSE,
control = list(eps = 0.001, maxit = 1000, trace = TRUE, seed.number = 123,
mc.samps = 4000, pois.step.size = 0.01))
Arguments
y
A response vector.
smooth.X
A matrix of covariates, each of which are to be entered as additive smooth terms in the GAM.
para.X
An optional matrix of covariates, each of which are to be entered as parametric terms in the GAM. Please note that NO intercept term needs to be included as it is included by default.
lambda
An optional vector of length ncol(smooth.X), where each element corresponds to the smoothing parameter to be applied to the respective covariate in smooth.X. If supplied, then the GAM is fitted with the smoothing parameters held fixed at this values. If lambda=NULL, then smoothing parameters for all covariates to be smoothed are updated automatically as part of the VA algorithm.
int.knots
Either a single number of a vector of length ncol(smooth.X), corresponding to the number of interior knots to be use for the respective covariate iin smooth.X. This argument is passed to the function smooth.construct from the mgcv package (Wood, 2017) in order to construct the P-splines bases. Equally spaced knots based on quantiles are used.
family
Currently only the gaussian(link = "identity"), poisson(link = "log"), and binomial(link = "logit") corresponding to Bernoulli distributions are available.
A.struct
The assumed structure of the covariance matrix in the variational distribution of the smoothing coefficients. Currently, the two options are A.struc = "unstructured" corresponding to assuming an fully unstructured covariance matrix, and A.struc = "block" which assumes a block diagonal structure where the all covariances between different smoothing covariates are assumed to be zero (but the covariance submatrix remains unstructured within the spline basis functions for a selected smoothing covariate). The latter is sub-optimal in the sense that the most appropriate variational distribution should use a completely unstructured covariance matrix , but MAY (but is not guaranteed) save computation time especially when the number of smoothing covariates and/or the number of interior knots is very large.
offset
This can be used to specify an a-priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to length(y).
save.data
If save.date=TRUE, then the returned vagam object will also include y, smooth.X, para.X, and the full matrix of P-spline basis functions.
para.se
If para.se=TRUE, the standard errors based on the VA approach are returned for any covariates in para.X that are included as parametric terms. Note that if para.se=FALSE then a standard error for the intercept term will not be returned even though an intercept term is included by default.
doIC
If doIC=TRUE, then the AIC and BIC are returned, where the AIC is calculated as AIC = -2\times variation log-likelihood + 2\times trace(H) with trace(H) bring a measure of the degrees of freedom of the model as based on the hat-matrix arising from iterative reweighted least squares, and the BIC replaces the 2 with log(length(y)) for the model complexity penalty; please see Wood (2017) for more details. Note however that this out is largely mute as the VA approach provides an automatic method of selecting the smoothing parameters, meaning an external approach such as information criteria is not required.
control
A list controlling the finer details of the VA approach for fitting GAMs. These include:
mc.samps:This controls Monte Carlo samples for calculating variational observed information matrix using Louis' method
seed:This controls seed for starting values of the fitting algorithm in general
pois.step.size:This controls step size for penalized iterative reweighted least squares (P-IRLS) portion of the VA approach when family=poisson(). This may be tweaked to use smaller step sizes as the approach here can be a tad unstable especially if there is possible overdispersion.
Details
Please note that the package is still in its early days, and only a very basic form of GAMs with purely additive
terms and P-splines is fitted. The function borrows heavily from the excellent software available in the mgcv
package (Wood, 2017), in the sense that it uses the smooth.construct function with bs = "ps" to
set up the matrix of P-splines bases (so cubic B-splines with a first order difference penalty matrix) along with
imposing the necessary centering constraints. With these ingredients, it then maximizes the variational
log-likelihood by iteratively updating the model and variational parameters. The variational log-likelihood
is obtained by proposing a variational distribution for the smoothing coefficients (in this case, a multivariate
normal distribution between unknown mean vector and covariance matrix), and then minimizing the Kullback-Leibler
distance between this variational distribution and the true posterior distribution of the smoothing coefficients.
In turn, this is designed to be (closed to) fully tractable lower bound approximation to the true marginal
log-likelihood for the GAM, which for non-normal responses does not possess a tractable form. Note that in
contrast to the marginal log-likelihood or many approximations such the Laplace approximation and adaptive
quadrature, the variational approximation typically presents a tractable form that is relatively straightforward
to maximize. At the same time, because it takes views the GAM as a mixed model, then it also possesses nice
inference tools such as an approximate posterior distribution of the smoothing coefficients available immediately
from maximizing the VA log-likelihood, and automatic choice of the smoothing parameters. We refer to readers to
Wood (2017) and Ruppert et al. (2003) for detailed introductions to GAMs and how many of them can be set up as
mixed models; Eilers and Marx (1996) for the seminal text on P-splines, and Hui et al. (2018) for the text on
which this package is based.
Value
An object of vagam class containing one or more of the following elements:
call:The matched call.
kappa:The estimated regression coefficients corresponding to the covariates in para.X. This includes the intercept term.
a:The estimated smoothing coefficients corresponding to the (P-spline bases set up for) covariates in smooth.X. This corresponds to the mean vector of the variational distribution.
A:The estimated posterior covariance of the smoothing coefficients corresponding to the (P-spline bases set up for) covariates in smooth.X. This corresponds to the covariance matrix of the variational distribution.
lambda:The estimated smoothing parameters, or the fixed smoothing parameters if lambda was supplied.
IC:A vector containing the calculated values of and AIC and BIC if doIC=TRUE. Note this is largely a mute output.
phi:The estimated residual variance when family=gaussian().
linear.predictors:The estimated linear predictor i.e., the parametric plus nonparametric component.
logL:The maximized value of the variational log-likelihood.
no.knots:The number of interior knots used, as per int.knots.
index.cov:A vector indexing which covariate each column in the final full matrix P-spline bases belongs to.
basis.info:A list with length equal to ncol(smooth.X), with each element being the output from an application of smooth.construct to construct the P-spline for a selected covariate in smooth.X.
y, para.X, smooth.X, Z:Returned in save.data=TRUE. Note critically that Z final full matrix P-spline basis functions.
smooth.stat:A small table of summary statistics for the nonparametric component of the GAM, including an approximate Wald-type hypothesis test for the significance of each nonparametric covariate.
para.stat:If para.se=TRUE, then a small table containing summary statistics for the estimated parametric component of the GAM, including an approximate Wald-type hypothesis test for the significance of each parameteric covariate.
obs.info:If para.se=TRUE, then the estimated variational observed information matrix for the parameteric component of the GAM; please see Hui et al. (2018) for more information.
Author(s)
Han Lin Shang [aut, cre, cph] (<https://orcid.org/0000-0003-1769-6430>),
Francis K.C. Hui [aut] (<https://orcid.org/0000-0003-0765-3533>)
References
Eilers, P. H. C., and Marx, B. D. (1996) Flexible Smoothing with B-splines and Penalties. Statistical Science, 11: 89-121
Hui, F. K. C., You, C., Shang, H. L., and Mueller, S. (2018). Semiparametric regression using variational approximations, Journal of the American Statistical Association, forthcoming.
Ruppert, D., Wand M. P., and Carroll, R. J. (2003) Semiparametric Regression. Cambridge University Press.
Wood, S. N. (2017) Generalized Additive Models: An Introduction with R (2nd edition). Chapman and Hall/CRC.
See Also
summary.vagam for a basic summary of the fitted model; plot.vagam for basic plotting the component smooths; predict.vagam for basic prediction
Examples
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## Example 1: Application to wage data
data(wage_data)
south_code <- gender_code <- race_code <- union_code <- vector("numeric", nrow(wage_data))
union_code[wage_data$union == "member"] <- 1
south_code[wage_data$south == "yes"] <- 1
gender_code[wage_data$gender == "female"] <- 1
race_code[wage_data$race == "White"] <- 1
para.X <- data.frame(south = south_code, gender = gender_code, race = race_code)
fit_va <- vagam(y = union_code, smooth.X = wage_data[,c("education", "wage", "age")],
para.X = para.X,
int.knots = 8, save.data = TRUE,
family = binomial(),
para.se = TRUE)
summary(fit_va)
a <- 1
par(mfrow = c(1, 3), las = 1, cex = a, cex.lab = a+0.2, cex.main = a+0.5, mar = c(5,5,3,2))
plot(fit_va, ylim = c(-2.7, 2.7), select = 1,
xlab = "Education", ylab = "Smooth of Education", lwd = 3)
plot(fit_va, ylim = c(-2.7, 2.7), select = 2,
xlab = "Wage", ylab = "Smooth of Wage", main = "Plots from VA-GAM", lwd = 3)
plot(fit_va, ylim = c(-2.7, 2.7), select = 3,
xlab = "Age", ylab = "Smooth of Age", lwd = 3)
## Not run:
## Example 2: Simulated data with size = 50 and compare how GAMs can be fitted
## in VA and mgcv (which uses penalized quasi-likelihood)
choose_k <- 5 * ceiling(50^0.2)
true_beta <- c(-1, 0.5)
poisson_dat <- gamsim(n = 50, dist = "poisson", extra.X = data.frame(intercept = rep(1,50),
treatment = rep(c(0,1), each = 50/2)), beta = true_beta)
## GAM using VA
fit_va <- vagam(y = poisson_dat$y, smooth.X = poisson_dat[,2:5],
para.X = data.frame(treatment = poisson_dat$treatment),
int.knots = choose_k, save.data = TRUE, family = poisson(),
para.se = TRUE)
summary(fit_va)
## GAM using mgcv with default options
fit_mgcv1 <- gam(y ~ treatment + s(x0) + s(x1) + s(x2) + s(x3),
data = poisson_dat, family = poisson())
## GAM using mgcv with P-splines and preset knots;
## this is equivalent to VA in terms of the splines bases functions
fit_mgcv2 <- gam(y ~ treatment + s(x0, bs = "ps", k = round(choose_k/2) + 2, m = c(2,1)) +
s(x1, bs = "ps", k = round(choose_k/2) + 2, m = c(2,1)) +
s(x2, bs = "ps", k = round(choose_k/2) + 2, m = c(2,1)) +
s(x3, bs = "ps", k = round(choose_k/2) + 2, m = c(2,1)),
data = poisson_dat, family = poisson())
## End(Not run)
Example output
Loading required package: mgcv
Loading required package: nlme
This is mgcv 1.8-33. For overview type 'help("mgcv-package")'.
Loading required package: gamm4
Loading required package: Matrix
Loading required package: lme4
Attaching package: ‘lme4’
The following object is masked from ‘package:nlme’:
lmList
This is gamm4 0.2-6
Loading required package: mvtnorm
Loading required package: truncnorm
Lambda updated as part of VA estimation. Yeah baby!
Iteration: 1 Current VA logL: -Inf | New VA logL: -251.9948 | Difference: Inf
Iteration: 2 Current VA logL: -251.9948 | New VA logL: -249.6027 | Difference: 2.392096
Iteration: 3 Current VA logL: -249.6027 | New VA logL: -248.813 | Difference: 0.789726
Iteration: 4 Current VA logL: -248.813 | New VA logL: -248.2817 | Difference: 0.5312606
Iteration: 5 Current VA logL: -248.2817 | New VA logL: -247.9144 | Difference: 0.3673779
Iteration: 6 Current VA logL: -247.9144 | New VA logL: -247.6557 | Difference: 0.2587032
Iteration: 7 Current VA logL: -247.6557 | New VA logL: -247.4697 | Difference: 0.1859226
Iteration: 8 Current VA logL: -247.4697 | New VA logL: -247.3333 | Difference: 0.1364035
Iteration: 9 Current VA logL: -247.3333 | New VA logL: -247.2313 | Difference: 0.1020059
Iteration: 10 Current VA logL: -247.2313 | New VA logL: -247.1537 | Difference: 0.07758634
Iteration: 11 Current VA logL: -247.1537 | New VA logL: -247.0939 | Difference: 0.05988457
Iteration: 12 Current VA logL: -247.0939 | New VA logL: -247.047 | Difference: 0.04680679
Iteration: 13 Current VA logL: -247.047 | New VA logL: -247.0101 | Difference: 0.03698225
Iteration: 14 Current VA logL: -247.0101 | New VA logL: -246.9806 | Difference: 0.02949385
Iteration: 15 Current VA logL: -246.9806 | New VA logL: -246.9569 | Difference: 0.02371432
Iteration: 16 Current VA logL: -246.9569 | New VA logL: -246.9377 | Difference: 0.01920535
Iteration: 17 Current VA logL: -246.9377 | New VA logL: -246.922 | Difference: 0.01565459
Iteration: 18 Current VA logL: -246.922 | New VA logL: -246.9092 | Difference: 0.01283546
Iteration: 19 Current VA logL: -246.9092 | New VA logL: -246.8986 | Difference: 0.01058095
Iteration: 20 Current VA logL: -246.8986 | New VA logL: -246.8898 | Difference: 0.008766283
Iteration: 21 Current VA logL: -246.8898 | New VA logL: -246.8825 | Difference: 0.007297091
Iteration: 22 Current VA logL: -246.8825 | New VA logL: -246.8764 | Difference: 0.006101228
Iteration: 23 Current VA logL: -246.8764 | New VA logL: -246.8713 | Difference: 0.005123037
Iteration: 24 Current VA logL: -246.8713 | New VA logL: -246.867 | Difference: 0.00431922
Iteration: 25 Current VA logL: -246.867 | New VA logL: -246.8633 | Difference: 0.003655846
Iteration: 26 Current VA logL: -246.8633 | New VA logL: -246.8602 | Difference: 0.003106149
Iteration: 27 Current VA logL: -246.8602 | New VA logL: -246.8576 | Difference: 0.002648886
Iteration: 28 Current VA logL: -246.8576 | New VA logL: -246.8553 | Difference: 0.002267108
Iteration: 29 Current VA logL: -246.8553 | New VA logL: -246.8533 | Difference: 0.001947219
Iteration: 30 Current VA logL: -246.8533 | New VA logL: -246.8517 | Difference: 0.001678265
Iteration: 31 Current VA logL: -246.8517 | New VA logL: -246.8502 | Difference: 0.00145138
Iteration: 32 Current VA logL: -246.8502 | New VA logL: -246.849 | Difference: 0.001259362
Iteration: 33 Current VA logL: -246.849 | New VA logL: -246.8479 | Difference: 0.001096333
Iteration: 34 Current VA logL: -246.8479 | New VA logL: -246.8469 | Difference: 0.0009574839
Calculating information matrix for model parameters...
Variational approximation for GAMs
Call: vagam(y = union_code, smooth.X = wage_data[, c("education", "wage", "age")], para.X = para.X, int.knots = 8, family = binomial(), save.data = TRUE, para.se = TRUE)
Estimated regression coefficients for parametric component: -0.715371 -0.4979345 -0.7094556 -0.7259837
Estimated smoothing coefficients for nonparametric component: 0.2024977 0.2371355 0.3083756 0.3620955 0.3070319 -0.06640864 -0.271355 -0.3181048 -0.292883 -0.274131 -1.269723 1.122673 2.295864 0.7431544 0.3187937 0.2337385 -0.08007407 -0.3819107 -0.6290628 -0.6814481 -0.1070558 -0.05720291 -0.01358479 0.03131925 0.08092134 0.142305 0.1878005 0.2265736 0.2248735 0.2117996
Estimated smoothing parameters (or fixed if lambda was supplied): 10.0231 0.5683342 46.01943
Number of interior knots used: 8 8 8
Maximized value of the variational log-likelihood: -246.8469
Summary statistics for nonparametric component:
education wage age
Wald Statistic 5.24257 46.14354 1.9520
p-value 0.87440 0.00000 0.9967
Summary statistics for parametric component (if para.se = TRUE):
Intercept south gender race
Estimate -0.71537 -0.49793 -0.70946 -0.72598
Std. Error 0.29415 0.29397 0.26386 0.29681
Wald Statistic -2.43202 -1.69383 -2.68874 -2.44594
p-value 0.01501 0.09030 0.00717 0.01445
$call
vagam(y = union_code, smooth.X = wage_data[, c("education", "wage",
"age")], para.X = para.X, int.knots = 8, family = binomial(),
save.data = TRUE, para.se = TRUE)
$para.coeff
Intercept south gender race
-0.7153710 -0.4979345 -0.7094556 -0.7259837
$smooth.coeff
[1] 0.20249774 0.23713555 0.30837560 0.36209547 0.30703185 -0.06640864
[7] -0.27135505 -0.31810480 -0.29288301 -0.27413098 -1.26972325 1.12267312
[13] 2.29586353 0.74315437 0.31879374 0.23373847 -0.08007407 -0.38191068
[19] -0.62906283 -0.68144811 -0.10705576 -0.05720291 -0.01358479 0.03131925
[25] 0.08092134 0.14230505 0.18780048 0.22657360 0.22487350 0.21179957
$smooth.param
[1] 10.0231020 0.5683342 46.0194276
$phi
[1] 1
$logLik
[1] -246.8469
$family
Family: binomial
Link function: logit
$smooth.stat
education wage age
Wald Statistic 5.24257 46.14354 1.9520
p-value 0.87440 0.00000 0.9967
$para.stat
Intercept south gender race
Estimate -0.71537 -0.49793 -0.70946 -0.72598
Std. Error 0.29415 0.29397 0.26386 0.29681
Wald Statistic -2.43202 -1.69383 -2.68874 -2.44594
p-value 0.01501 0.09030 0.00717 0.01445
attr(,"class")
[1] "summary.vagam"
Lambda updated as part of VA estimation. Yeah baby!
Iteration: 1 Current VA logL: -Inf | New VA logL: -427.9932 | Difference: Inf
Iteration: 2 Current VA logL: -427.9932 | New VA logL: -330.7948 | Difference: 97.19841
Iteration: 3 Current VA logL: -330.7948 | New VA logL: -283.5286 | Difference: 47.26625
Iteration: 4 Current VA logL: -283.5286 | New VA logL: -253.0125 | Difference: 30.51608
Iteration: 5 Current VA logL: -253.0125 | New VA logL: -231.6056 | Difference: 21.40688
Iteration: 6 Current VA logL: -231.6056 | New VA logL: -215.7583 | Difference: 15.84734
Iteration: 7 Current VA logL: -215.7583 | New VA logL: -203.6006 | Difference: 12.15766
Iteration: 8 Current VA logL: -203.6006 | New VA logL: -194.0998 | Difference: 9.50088
Iteration: 9 Current VA logL: -194.0998 | New VA logL: -186.4754 | Difference: 7.62439
Iteration: 10 Current VA logL: -186.4754 | New VA logL: -180.2011 | Difference: 6.274265
Iteration: 11 Current VA logL: -180.2011 | New VA logL: -175.0458 | Difference: 5.155307
Iteration: 12 Current VA logL: -175.0458 | New VA logL: -170.6973 | Difference: 4.348472
Iteration: 13 Current VA logL: -170.6973 | New VA logL: -167.0002 | Difference: 3.697071
Iteration: 14 Current VA logL: -167.0002 | New VA logL: -163.8246 | Difference: 3.175687
Iteration: 15 Current VA logL: -163.8246 | New VA logL: -161.0074 | Difference: 2.817127
Iteration: 16 Current VA logL: -161.0074 | New VA logL: -158.5621 | Difference: 2.445371
Iteration: 17 Current VA logL: -158.5621 | New VA logL: -156.3654 | Difference: 2.196669
Iteration: 18 Current VA logL: -156.3654 | New VA logL: -154.4471 | Difference: 1.918283
Iteration: 19 Current VA logL: -154.4471 | New VA logL: -152.6919 | Difference: 1.755182
Iteration: 20 Current VA logL: -152.6919 | New VA logL: -151.0954 | Difference: 1.596569
Iteration: 21 Current VA logL: -151.0954 | New VA logL: -149.6949 | Difference: 1.400426
Iteration: 22 Current VA logL: -149.6949 | New VA logL: -148.3965 | Difference: 1.298447
Iteration: 23 Current VA logL: -148.3965 | New VA logL: -147.1911 | Difference: 1.205428
Iteration: 24 Current VA logL: -147.1911 | New VA logL: -146.0708 | Difference: 1.120304
Iteration: 25 Current VA logL: -146.0708 | New VA logL: -145.0389 | Difference: 1.031826
Iteration: 26 Current VA logL: -145.0389 | New VA logL: -144.1319 | Difference: 0.9070442
Iteration: 27 Current VA logL: -144.1319 | New VA logL: -143.2792 | Difference: 0.8526548
Iteration: 28 Current VA logL: -143.2792 | New VA logL: -142.4771 | Difference: 0.802131
Iteration: 29 Current VA logL: -142.4771 | New VA logL: -141.7221 | Difference: 0.7550214
Iteration: 30 Current VA logL: -141.7221 | New VA logL: -141.011 | Difference: 0.7110918
Iteration: 31 Current VA logL: -141.011 | New VA logL: -140.3409 | Difference: 0.6701192
Iteration: 32 Current VA logL: -140.3409 | New VA logL: -139.709 | Difference: 0.6318943
Iteration: 33 Current VA logL: -139.709 | New VA logL: -139.1127 | Difference: 0.5962212
Iteration: 34 Current VA logL: -139.1127 | New VA logL: -138.5563 | Difference: 0.5564972
Iteration: 35 Current VA logL: -138.5563 | New VA logL: -138.0674 | Difference: 0.4888532
Iteration: 36 Current VA logL: -138.0674 | New VA logL: -137.6014 | Difference: 0.4659985
Iteration: 37 Current VA logL: -137.6014 | New VA logL: -137.157 | Difference: 0.4444216
Iteration: 38 Current VA logL: -137.157 | New VA logL: -136.733 | Difference: 0.4239319
Iteration: 39 Current VA logL: -136.733 | New VA logL: -136.3286 | Difference: 0.4044801
Iteration: 40 Current VA logL: -136.3286 | New VA logL: -135.9426 | Difference: 0.3860163
Iteration: 41 Current VA logL: -135.9426 | New VA logL: -135.5741 | Difference: 0.3684918
Iteration: 42 Current VA logL: -135.5741 | New VA logL: -135.2222 | Difference: 0.3518596
Iteration: 43 Current VA logL: -135.2222 | New VA logL: -134.8861 | Difference: 0.3360743
Iteration: 44 Current VA logL: -134.8861 | New VA logL: -134.565 | Difference: 0.3210922
Iteration: 45 Current VA logL: -134.565 | New VA logL: -134.2582 | Difference: 0.3068714
Iteration: 46 Current VA logL: -134.2582 | New VA logL: -133.9648 | Difference: 0.293372
Iteration: 47 Current VA logL: -133.9648 | New VA logL: -133.6842 | Difference: 0.2805557
Iteration: 48 Current VA logL: -133.6842 | New VA logL: -133.4158 | Difference: 0.2683862
Iteration: 49 Current VA logL: -133.4158 | New VA logL: -133.159 | Difference: 0.256829
Iteration: 50 Current VA logL: -133.159 | New VA logL: -132.9132 | Difference: 0.2458511
Iteration: 51 Current VA logL: -132.9132 | New VA logL: -132.6777 | Difference: 0.2354214
Iteration: 52 Current VA logL: -132.6777 | New VA logL: -132.4543 | Difference: 0.2234448
Iteration: 53 Current VA logL: -132.4543 | New VA logL: -132.2578 | Difference: 0.196546
Iteration: 54 Current VA logL: -132.2578 | New VA logL: -132.0678 | Difference: 0.1899236
Iteration: 55 Current VA logL: -132.0678 | New VA logL: -131.8842 | Difference: 0.1835867
Iteration: 56 Current VA logL: -131.8842 | New VA logL: -131.7068 | Difference: 0.1774629
Iteration: 57 Current VA logL: -131.7068 | New VA logL: -131.5352 | Difference: 0.1715473
Iteration: 58 Current VA logL: -131.5352 | New VA logL: -131.3694 | Difference: 0.1658343
Iteration: 59 Current VA logL: -131.3694 | New VA logL: -131.2091 | Difference: 0.1603182
Iteration: 60 Current VA logL: -131.2091 | New VA logL: -131.0541 | Difference: 0.1549932
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Iteration: 283 Current VA logL: -125.5989 | New VA logL: -125.5972 | Difference: 0.001774674
Iteration: 284 Current VA logL: -125.5972 | New VA logL: -125.5954 | Difference: 0.001750674
Iteration: 285 Current VA logL: -125.5954 | New VA logL: -125.5937 | Difference: 0.001727042
Iteration: 286 Current VA logL: -125.5937 | New VA logL: -125.592 | Difference: 0.001703771
Iteration: 287 Current VA logL: -125.592 | New VA logL: -125.5903 | Difference: 0.001680855
Iteration: 288 Current VA logL: -125.5903 | New VA logL: -125.5886 | Difference: 0.001658288
Iteration: 289 Current VA logL: -125.5886 | New VA logL: -125.587 | Difference: 0.001636064
Iteration: 290 Current VA logL: -125.587 | New VA logL: -125.5854 | Difference: 0.001614176
Iteration: 291 Current VA logL: -125.5854 | New VA logL: -125.5838 | Difference: 0.001592619
Iteration: 292 Current VA logL: -125.5838 | New VA logL: -125.5822 | Difference: 0.001571386
Iteration: 293 Current VA logL: -125.5822 | New VA logL: -125.5807 | Difference: 0.001550472
Iteration: 294 Current VA logL: -125.5807 | New VA logL: -125.5791 | Difference: 0.001529872
Iteration: 295 Current VA logL: -125.5791 | New VA logL: -125.5776 | Difference: 0.00150958
Iteration: 296 Current VA logL: -125.5776 | New VA logL: -125.5761 | Difference: 0.00148959
Iteration: 297 Current VA logL: -125.5761 | New VA logL: -125.5747 | Difference: 0.001469899
Iteration: 298 Current VA logL: -125.5747 | New VA logL: -125.5732 | Difference: 0.001450499
Iteration: 299 Current VA logL: -125.5732 | New VA logL: -125.5718 | Difference: 0.001431387
Iteration: 300 Current VA logL: -125.5718 | New VA logL: -125.5704 | Difference: 0.001412557
Iteration: 301 Current VA logL: -125.5704 | New VA logL: -125.569 | Difference: 0.001394005
Iteration: 302 Current VA logL: -125.569 | New VA logL: -125.5676 | Difference: 0.001375726
Iteration: 303 Current VA logL: -125.5676 | New VA logL: -125.5662 | Difference: 0.001357715
Iteration: 304 Current VA logL: -125.5662 | New VA logL: -125.5649 | Difference: 0.001339968
Iteration: 305 Current VA logL: -125.5649 | New VA logL: -125.5636 | Difference: 0.001322481
Iteration: 306 Current VA logL: -125.5636 | New VA logL: -125.5623 | Difference: 0.001305248
Iteration: 307 Current VA logL: -125.5623 | New VA logL: -125.561 | Difference: 0.001288267
Iteration: 308 Current VA logL: -125.561 | New VA logL: -125.5597 | Difference: 0.001271532
Iteration: 309 Current VA logL: -125.5597 | New VA logL: -125.5585 | Difference: 0.001255039
Iteration: 310 Current VA logL: -125.5585 | New VA logL: -125.5572 | Difference: 0.001238785
Iteration: 311 Current VA logL: -125.5572 | New VA logL: -125.556 | Difference: 0.001222765
Iteration: 312 Current VA logL: -125.556 | New VA logL: -125.5548 | Difference: 0.001206976
Iteration: 313 Current VA logL: -125.5548 | New VA logL: -125.5536 | Difference: 0.001191414
Iteration: 314 Current VA logL: -125.5536 | New VA logL: -125.5524 | Difference: 0.001176076
Iteration: 315 Current VA logL: -125.5524 | New VA logL: -125.5513 | Difference: 0.001160956
Iteration: 316 Current VA logL: -125.5513 | New VA logL: -125.5501 | Difference: 0.001146053
Iteration: 317 Current VA logL: -125.5501 | New VA logL: -125.549 | Difference: 0.001131362
Iteration: 318 Current VA logL: -125.549 | New VA logL: -125.5479 | Difference: 0.001116881
Iteration: 319 Current VA logL: -125.5479 | New VA logL: -125.5468 | Difference: 0.001102605
Iteration: 320 Current VA logL: -125.5468 | New VA logL: -125.5457 | Difference: 0.001088531
Iteration: 321 Current VA logL: -125.5457 | New VA logL: -125.5446 | Difference: 0.001074656
Iteration: 322 Current VA logL: -125.5446 | New VA logL: -125.5435 | Difference: 0.001060977
Iteration: 323 Current VA logL: -125.5435 | New VA logL: -125.5425 | Difference: 0.001047491
Iteration: 324 Current VA logL: -125.5425 | New VA logL: -125.5415 | Difference: 0.001034195
Iteration: 325 Current VA logL: -125.5415 | New VA logL: -125.5404 | Difference: 0.001021085
Iteration: 326 Current VA logL: -125.5404 | New VA logL: -125.5394 | Difference: 0.001008159
Iteration: 327 Current VA logL: -125.5394 | New VA logL: -125.5384 | Difference: 0.000995414
Calculating information matrix for model parameters...
Redoing...
Warning message:
In sqrt(diag(solve(obs_info))) : NaNs produced
Variational approximation for GAMs
Call: vagam(y = poisson_dat$y, smooth.X = poisson_dat[, 2:5], para.X = data.frame(treatment = poisson_dat$treatment), int.knots = choose_k, family = poisson(), save.data = TRUE, para.se = TRUE)
Estimated regression coefficients for parametric component: -1.040643 0.6451899
Estimated smoothing coefficients for nonparametric component: -0.2733702 -0.2461039 -0.09914876 0.03284408 0.1684629 0.3366869 0.4947952 0.5851139 0.656716 0.5336587 0.3721285 0.2277044 -0.02135385 -0.2695109 -0.5710075 -0.7604779 -0.8174288 -1.712253 -1.563433 -1.369698 -0.9899007 -0.8411191 -0.4226654 -0.07139302 -0.2168335 0.0007790999 0.5339774 0.8247894 1.282654 2.006611 1.880903 1.258986 1.21398 1.148197 -0.751521 0.4847275 3.133529 6.330736 4.998549 3.434801 1.618585 -0.6540657 -1.360546 -1.018068 -0.4136508 -1.939511 -2.64722 -2.147782 -1.521794 -2.187518 -2.458522 -1.607145 -1.329687 0.005428171 1.0108 0.4098627 0.7475761 1.115838 0.7188687 0.7304209 1.060003 0.8690969 0.613725 0.4260507 0.4167565 0.5912033 0.3957223 0.02593618
Estimated smoothing parameters (or fixed if lambda was supplied): 6.680485 3.832892 0.2155243 1.027374
Number of interior knots used: 15 15 15 15
Maximized value of the variational log-likelihood: -125.5384
Summary statistics for nonparametric component:
x0 x1 x2 x3
Wald Statistic 358.331 840.2793 31902.05 79.84659
p-value 0.000 0.0000 0.00 0.00000
Summary statistics for parametric component (if para.se = TRUE):
Intercept treatment
Estimate -1.04064 0.64519
Std. Error NaN 0.16658
Wald Statistic NaN 3.87310
p-value NaN 0.00011
$call
vagam(y = poisson_dat$y, smooth.X = poisson_dat[, 2:5], para.X = data.frame(treatment = poisson_dat$treatment),
int.knots = choose_k, family = poisson(), save.data = TRUE,
para.se = TRUE)
$para.coeff
Intercept treatment
-1.0406428 0.6451899
$smooth.coeff
[1] -0.2733701882 -0.2461038544 -0.0991487572 0.0328440844 0.1684629451
[6] 0.3366869456 0.4947951958 0.5851139119 0.6567160275 0.5336586669
[11] 0.3721285236 0.2277043874 -0.0213538489 -0.2695109221 -0.5710074632
[16] -0.7604779036 -0.8174288063 -1.7122534105 -1.5634333694 -1.3696982975
[21] -0.9899006792 -0.8411191491 -0.4226653909 -0.0713930171 -0.2168334916
[26] 0.0007790999 0.5339773671 0.8247894171 1.2826538456 2.0066107547
[31] 1.8809033147 1.2589859418 1.2139796076 1.1481968223 -0.7515209731
[36] 0.4847274793 3.1335287755 6.3307363410 4.9985492274 3.4348007502
[41] 1.6185847776 -0.6540656573 -1.3605461824 -1.0180681483 -0.4136507929
[46] -1.9395112438 -2.6472198102 -2.1477815240 -1.5217936307 -2.1875175588
[51] -2.4585215323 -1.6071447536 -1.3296868389 0.0054281713 1.0108003112
[56] 0.4098626899 0.7475760842 1.1158378269 0.7188686895 0.7304209105
[61] 1.0600027739 0.8690969307 0.6137250079 0.4260507231 0.4167565447
[66] 0.5912032995 0.3957222731 0.0259361775
$smooth.param
[1] 6.6804854 3.8328919 0.2155243 1.0273736
$phi
[1] 1
$logLik
[1] -125.5384
$family
Family: poisson
Link function: log
$smooth.stat
x0 x1 x2 x3
Wald Statistic 358.331 840.2793 31902.05 79.84659
p-value 0.000 0.0000 0.00 0.00000
$para.stat
Intercept treatment
Estimate -1.04064 0.64519
Std. Error NaN 0.16658
Wald Statistic NaN 3.87310
p-value NaN 0.00011
attr(,"class")
[1] "summary.vagam"
vagam documentation built on Dec. 7, 2019, 1:06 a.m.
R Package Documentation
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
Follows the variational approximation approach of Hui et al. (2018) for fitted generalized additive models. In this package, the term GAM is taken to be generalized linear mixed model, specifically, the nonparametric component is modeled using a P-splines i.e., cubic B-splines with a first order difference penalty. Because the penalty can be written as a quadratic form in terms of the smoothing coefficients, then it is treated a (degenerate) multivariate normal random effects distribution and a marginal log-likleihood for the resulting mixed model can be constructed.
The VA framework is then utilized to provide a fully or at least closed to fully tractable lower bound approximation to the marginal likelihood of a GAM. In doing so, the VA framework aims offers both the stability and natural inference tools available in the mixed model approach to GAMs, while achieving computation times comparable to that of using the penalized likelihood approach to GAMs.
Usage
1 2 3 4 5 | vagam(y, smooth.X, para.X = NULL, lambda = NULL, int.knots, family = gaussian(),
A.struct = c("unstructured", "block"), offset = NULL, save.data = FALSE,
para.se = FALSE, doIC = FALSE,
control = list(eps = 0.001, maxit = 1000, trace = TRUE, seed.number = 123,
mc.samps = 4000, pois.step.size = 0.01))
|
Arguments
y |
A response vector. |
smooth.X |
A matrix of covariates, each of which are to be entered as additive smooth terms in the GAM. |
para.X |
An optional matrix of covariates, each of which are to be entered as parametric terms in the GAM. Please note that NO intercept term needs to be included as it is included by default. |
lambda |
An optional vector of length |
int.knots |
Either a single number of a vector of length |
family |
Currently only the |
A.struct |
The assumed structure of the covariance matrix in the variational distribution of the smoothing coefficients. Currently, the two options are |
offset |
This can be used to specify an a-priori known component to be included in the linear predictor during fitting. This should be |
save.data |
If |
para.se |
If |
doIC |
If |
control |
A list controlling the finer details of the VA approach for fitting GAMs. These include:
|
Details
Please note that the package is still in its early days, and only a very basic form of GAMs with purely additive
terms and P-splines is fitted. The function borrows heavily from the excellent software available in the mgcv
package (Wood, 2017), in the sense that it uses the smooth.construct function with bs = "ps" to
set up the matrix of P-splines bases (so cubic B-splines with a first order difference penalty matrix) along with
imposing the necessary centering constraints. With these ingredients, it then maximizes the variational
log-likelihood by iteratively updating the model and variational parameters. The variational log-likelihood
is obtained by proposing a variational distribution for the smoothing coefficients (in this case, a multivariate
normal distribution between unknown mean vector and covariance matrix), and then minimizing the Kullback-Leibler
distance between this variational distribution and the true posterior distribution of the smoothing coefficients.
In turn, this is designed to be (closed to) fully tractable lower bound approximation to the true marginal
log-likelihood for the GAM, which for non-normal responses does not possess a tractable form. Note that in
contrast to the marginal log-likelihood or many approximations such the Laplace approximation and adaptive
quadrature, the variational approximation typically presents a tractable form that is relatively straightforward
to maximize. At the same time, because it takes views the GAM as a mixed model, then it also possesses nice
inference tools such as an approximate posterior distribution of the smoothing coefficients available immediately
from maximizing the VA log-likelihood, and automatic choice of the smoothing parameters. We refer to readers to
Wood (2017) and Ruppert et al. (2003) for detailed introductions to GAMs and how many of them can be set up as
mixed models; Eilers and Marx (1996) for the seminal text on P-splines, and Hui et al. (2018) for the text on
which this package is based.
Value
An object of vagam class containing one or more of the following elements:
call:The matched call.
kappa:The estimated regression coefficients corresponding to the covariates in
para.X. This includes the intercept term.a:The estimated smoothing coefficients corresponding to the (P-spline bases set up for) covariates in
smooth.X. This corresponds to the mean vector of the variational distribution.A:The estimated posterior covariance of the smoothing coefficients corresponding to the (P-spline bases set up for) covariates in
smooth.X. This corresponds to the covariance matrix of the variational distribution.lambda:The estimated smoothing parameters, or the fixed smoothing parameters if
lambdawas supplied.IC:A vector containing the calculated values of and AIC and BIC if
doIC=TRUE. Note this is largely a mute output.phi:The estimated residual variance when
family=gaussian().linear.predictors:The estimated linear predictor i.e., the parametric plus nonparametric component.
logL:The maximized value of the variational log-likelihood.
no.knots:The number of interior knots used, as per
int.knots.index.cov:A vector indexing which covariate each column in the final full matrix P-spline bases belongs to.
basis.info:A list with length equal to
ncol(smooth.X), with each element being the output from an application ofsmooth.constructto construct the P-spline for a selected covariate insmooth.X.y, para.X, smooth.X, Z:Returned in
save.data=TRUE. Note critically thatZfinal full matrix P-spline basis functions.smooth.stat:A small table of summary statistics for the nonparametric component of the GAM, including an approximate Wald-type hypothesis test for the significance of each nonparametric covariate.
para.stat:If
para.se=TRUE, then a small table containing summary statistics for the estimated parametric component of the GAM, including an approximate Wald-type hypothesis test for the significance of each parameteric covariate.obs.info:If
para.se=TRUE, then the estimated variational observed information matrix for the parameteric component of the GAM; please see Hui et al. (2018) for more information.
Author(s)
Han Lin Shang [aut, cre, cph] (<https://orcid.org/0000-0003-1769-6430>), Francis K.C. Hui [aut] (<https://orcid.org/0000-0003-0765-3533>)
References
Eilers, P. H. C., and Marx, B. D. (1996) Flexible Smoothing with B-splines and Penalties. Statistical Science, 11: 89-121
Hui, F. K. C., You, C., Shang, H. L., and Mueller, S. (2018). Semiparametric regression using variational approximations, Journal of the American Statistical Association, forthcoming.
Ruppert, D., Wand M. P., and Carroll, R. J. (2003) Semiparametric Regression. Cambridge University Press.
Wood, S. N. (2017) Generalized Additive Models: An Introduction with R (2nd edition). Chapman and Hall/CRC.
See Also
summary.vagam for a basic summary of the fitted model; plot.vagam for basic plotting the component smooths; predict.vagam for basic prediction
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | ## Example 1: Application to wage data
data(wage_data)
south_code <- gender_code <- race_code <- union_code <- vector("numeric", nrow(wage_data))
union_code[wage_data$union == "member"] <- 1
south_code[wage_data$south == "yes"] <- 1
gender_code[wage_data$gender == "female"] <- 1
race_code[wage_data$race == "White"] <- 1
para.X <- data.frame(south = south_code, gender = gender_code, race = race_code)
fit_va <- vagam(y = union_code, smooth.X = wage_data[,c("education", "wage", "age")],
para.X = para.X,
int.knots = 8, save.data = TRUE,
family = binomial(),
para.se = TRUE)
summary(fit_va)
a <- 1
par(mfrow = c(1, 3), las = 1, cex = a, cex.lab = a+0.2, cex.main = a+0.5, mar = c(5,5,3,2))
plot(fit_va, ylim = c(-2.7, 2.7), select = 1,
xlab = "Education", ylab = "Smooth of Education", lwd = 3)
plot(fit_va, ylim = c(-2.7, 2.7), select = 2,
xlab = "Wage", ylab = "Smooth of Wage", main = "Plots from VA-GAM", lwd = 3)
plot(fit_va, ylim = c(-2.7, 2.7), select = 3,
xlab = "Age", ylab = "Smooth of Age", lwd = 3)
## Not run:
## Example 2: Simulated data with size = 50 and compare how GAMs can be fitted
## in VA and mgcv (which uses penalized quasi-likelihood)
choose_k <- 5 * ceiling(50^0.2)
true_beta <- c(-1, 0.5)
poisson_dat <- gamsim(n = 50, dist = "poisson", extra.X = data.frame(intercept = rep(1,50),
treatment = rep(c(0,1), each = 50/2)), beta = true_beta)
## GAM using VA
fit_va <- vagam(y = poisson_dat$y, smooth.X = poisson_dat[,2:5],
para.X = data.frame(treatment = poisson_dat$treatment),
int.knots = choose_k, save.data = TRUE, family = poisson(),
para.se = TRUE)
summary(fit_va)
## GAM using mgcv with default options
fit_mgcv1 <- gam(y ~ treatment + s(x0) + s(x1) + s(x2) + s(x3),
data = poisson_dat, family = poisson())
## GAM using mgcv with P-splines and preset knots;
## this is equivalent to VA in terms of the splines bases functions
fit_mgcv2 <- gam(y ~ treatment + s(x0, bs = "ps", k = round(choose_k/2) + 2, m = c(2,1)) +
s(x1, bs = "ps", k = round(choose_k/2) + 2, m = c(2,1)) +
s(x2, bs = "ps", k = round(choose_k/2) + 2, m = c(2,1)) +
s(x3, bs = "ps", k = round(choose_k/2) + 2, m = c(2,1)),
data = poisson_dat, family = poisson())
## End(Not run)
|
Example output
Loading required package: mgcv
Loading required package: nlme
This is mgcv 1.8-33. For overview type 'help("mgcv-package")'.
Loading required package: gamm4
Loading required package: Matrix
Loading required package: lme4
Attaching package: ‘lme4’
The following object is masked from ‘package:nlme’:
lmList
This is gamm4 0.2-6
Loading required package: mvtnorm
Loading required package: truncnorm
Lambda updated as part of VA estimation. Yeah baby!
Iteration: 1 Current VA logL: -Inf | New VA logL: -251.9948 | Difference: Inf
Iteration: 2 Current VA logL: -251.9948 | New VA logL: -249.6027 | Difference: 2.392096
Iteration: 3 Current VA logL: -249.6027 | New VA logL: -248.813 | Difference: 0.789726
Iteration: 4 Current VA logL: -248.813 | New VA logL: -248.2817 | Difference: 0.5312606
Iteration: 5 Current VA logL: -248.2817 | New VA logL: -247.9144 | Difference: 0.3673779
Iteration: 6 Current VA logL: -247.9144 | New VA logL: -247.6557 | Difference: 0.2587032
Iteration: 7 Current VA logL: -247.6557 | New VA logL: -247.4697 | Difference: 0.1859226
Iteration: 8 Current VA logL: -247.4697 | New VA logL: -247.3333 | Difference: 0.1364035
Iteration: 9 Current VA logL: -247.3333 | New VA logL: -247.2313 | Difference: 0.1020059
Iteration: 10 Current VA logL: -247.2313 | New VA logL: -247.1537 | Difference: 0.07758634
Iteration: 11 Current VA logL: -247.1537 | New VA logL: -247.0939 | Difference: 0.05988457
Iteration: 12 Current VA logL: -247.0939 | New VA logL: -247.047 | Difference: 0.04680679
Iteration: 13 Current VA logL: -247.047 | New VA logL: -247.0101 | Difference: 0.03698225
Iteration: 14 Current VA logL: -247.0101 | New VA logL: -246.9806 | Difference: 0.02949385
Iteration: 15 Current VA logL: -246.9806 | New VA logL: -246.9569 | Difference: 0.02371432
Iteration: 16 Current VA logL: -246.9569 | New VA logL: -246.9377 | Difference: 0.01920535
Iteration: 17 Current VA logL: -246.9377 | New VA logL: -246.922 | Difference: 0.01565459
Iteration: 18 Current VA logL: -246.922 | New VA logL: -246.9092 | Difference: 0.01283546
Iteration: 19 Current VA logL: -246.9092 | New VA logL: -246.8986 | Difference: 0.01058095
Iteration: 20 Current VA logL: -246.8986 | New VA logL: -246.8898 | Difference: 0.008766283
Iteration: 21 Current VA logL: -246.8898 | New VA logL: -246.8825 | Difference: 0.007297091
Iteration: 22 Current VA logL: -246.8825 | New VA logL: -246.8764 | Difference: 0.006101228
Iteration: 23 Current VA logL: -246.8764 | New VA logL: -246.8713 | Difference: 0.005123037
Iteration: 24 Current VA logL: -246.8713 | New VA logL: -246.867 | Difference: 0.00431922
Iteration: 25 Current VA logL: -246.867 | New VA logL: -246.8633 | Difference: 0.003655846
Iteration: 26 Current VA logL: -246.8633 | New VA logL: -246.8602 | Difference: 0.003106149
Iteration: 27 Current VA logL: -246.8602 | New VA logL: -246.8576 | Difference: 0.002648886
Iteration: 28 Current VA logL: -246.8576 | New VA logL: -246.8553 | Difference: 0.002267108
Iteration: 29 Current VA logL: -246.8553 | New VA logL: -246.8533 | Difference: 0.001947219
Iteration: 30 Current VA logL: -246.8533 | New VA logL: -246.8517 | Difference: 0.001678265
Iteration: 31 Current VA logL: -246.8517 | New VA logL: -246.8502 | Difference: 0.00145138
Iteration: 32 Current VA logL: -246.8502 | New VA logL: -246.849 | Difference: 0.001259362
Iteration: 33 Current VA logL: -246.849 | New VA logL: -246.8479 | Difference: 0.001096333
Iteration: 34 Current VA logL: -246.8479 | New VA logL: -246.8469 | Difference: 0.0009574839
Calculating information matrix for model parameters...
Variational approximation for GAMs
Call: vagam(y = union_code, smooth.X = wage_data[, c("education", "wage", "age")], para.X = para.X, int.knots = 8, family = binomial(), save.data = TRUE, para.se = TRUE)
Estimated regression coefficients for parametric component: -0.715371 -0.4979345 -0.7094556 -0.7259837
Estimated smoothing coefficients for nonparametric component: 0.2024977 0.2371355 0.3083756 0.3620955 0.3070319 -0.06640864 -0.271355 -0.3181048 -0.292883 -0.274131 -1.269723 1.122673 2.295864 0.7431544 0.3187937 0.2337385 -0.08007407 -0.3819107 -0.6290628 -0.6814481 -0.1070558 -0.05720291 -0.01358479 0.03131925 0.08092134 0.142305 0.1878005 0.2265736 0.2248735 0.2117996
Estimated smoothing parameters (or fixed if lambda was supplied): 10.0231 0.5683342 46.01943
Number of interior knots used: 8 8 8
Maximized value of the variational log-likelihood: -246.8469
Summary statistics for nonparametric component:
education wage age
Wald Statistic 5.24257 46.14354 1.9520
p-value 0.87440 0.00000 0.9967
Summary statistics for parametric component (if para.se = TRUE):
Intercept south gender race
Estimate -0.71537 -0.49793 -0.70946 -0.72598
Std. Error 0.29415 0.29397 0.26386 0.29681
Wald Statistic -2.43202 -1.69383 -2.68874 -2.44594
p-value 0.01501 0.09030 0.00717 0.01445
$call
vagam(y = union_code, smooth.X = wage_data[, c("education", "wage",
"age")], para.X = para.X, int.knots = 8, family = binomial(),
save.data = TRUE, para.se = TRUE)
$para.coeff
Intercept south gender race
-0.7153710 -0.4979345 -0.7094556 -0.7259837
$smooth.coeff
[1] 0.20249774 0.23713555 0.30837560 0.36209547 0.30703185 -0.06640864
[7] -0.27135505 -0.31810480 -0.29288301 -0.27413098 -1.26972325 1.12267312
[13] 2.29586353 0.74315437 0.31879374 0.23373847 -0.08007407 -0.38191068
[19] -0.62906283 -0.68144811 -0.10705576 -0.05720291 -0.01358479 0.03131925
[25] 0.08092134 0.14230505 0.18780048 0.22657360 0.22487350 0.21179957
$smooth.param
[1] 10.0231020 0.5683342 46.0194276
$phi
[1] 1
$logLik
[1] -246.8469
$family
Family: binomial
Link function: logit
$smooth.stat
education wage age
Wald Statistic 5.24257 46.14354 1.9520
p-value 0.87440 0.00000 0.9967
$para.stat
Intercept south gender race
Estimate -0.71537 -0.49793 -0.70946 -0.72598
Std. Error 0.29415 0.29397 0.26386 0.29681
Wald Statistic -2.43202 -1.69383 -2.68874 -2.44594
p-value 0.01501 0.09030 0.00717 0.01445
attr(,"class")
[1] "summary.vagam"
Lambda updated as part of VA estimation. Yeah baby!
Iteration: 1 Current VA logL: -Inf | New VA logL: -427.9932 | Difference: Inf
Iteration: 2 Current VA logL: -427.9932 | New VA logL: -330.7948 | Difference: 97.19841
Iteration: 3 Current VA logL: -330.7948 | New VA logL: -283.5286 | Difference: 47.26625
Iteration: 4 Current VA logL: -283.5286 | New VA logL: -253.0125 | Difference: 30.51608
Iteration: 5 Current VA logL: -253.0125 | New VA logL: -231.6056 | Difference: 21.40688
Iteration: 6 Current VA logL: -231.6056 | New VA logL: -215.7583 | Difference: 15.84734
Iteration: 7 Current VA logL: -215.7583 | New VA logL: -203.6006 | Difference: 12.15766
Iteration: 8 Current VA logL: -203.6006 | New VA logL: -194.0998 | Difference: 9.50088
Iteration: 9 Current VA logL: -194.0998 | New VA logL: -186.4754 | Difference: 7.62439
Iteration: 10 Current VA logL: -186.4754 | New VA logL: -180.2011 | Difference: 6.274265
Iteration: 11 Current VA logL: -180.2011 | New VA logL: -175.0458 | Difference: 5.155307
Iteration: 12 Current VA logL: -175.0458 | New VA logL: -170.6973 | Difference: 4.348472
Iteration: 13 Current VA logL: -170.6973 | New VA logL: -167.0002 | Difference: 3.697071
Iteration: 14 Current VA logL: -167.0002 | New VA logL: -163.8246 | Difference: 3.175687
Iteration: 15 Current VA logL: -163.8246 | New VA logL: -161.0074 | Difference: 2.817127
Iteration: 16 Current VA logL: -161.0074 | New VA logL: -158.5621 | Difference: 2.445371
Iteration: 17 Current VA logL: -158.5621 | New VA logL: -156.3654 | Difference: 2.196669
Iteration: 18 Current VA logL: -156.3654 | New VA logL: -154.4471 | Difference: 1.918283
Iteration: 19 Current VA logL: -154.4471 | New VA logL: -152.6919 | Difference: 1.755182
Iteration: 20 Current VA logL: -152.6919 | New VA logL: -151.0954 | Difference: 1.596569
Iteration: 21 Current VA logL: -151.0954 | New VA logL: -149.6949 | Difference: 1.400426
Iteration: 22 Current VA logL: -149.6949 | New VA logL: -148.3965 | Difference: 1.298447
Iteration: 23 Current VA logL: -148.3965 | New VA logL: -147.1911 | Difference: 1.205428
Iteration: 24 Current VA logL: -147.1911 | New VA logL: -146.0708 | Difference: 1.120304
Iteration: 25 Current VA logL: -146.0708 | New VA logL: -145.0389 | Difference: 1.031826
Iteration: 26 Current VA logL: -145.0389 | New VA logL: -144.1319 | Difference: 0.9070442
Iteration: 27 Current VA logL: -144.1319 | New VA logL: -143.2792 | Difference: 0.8526548
Iteration: 28 Current VA logL: -143.2792 | New VA logL: -142.4771 | Difference: 0.802131
Iteration: 29 Current VA logL: -142.4771 | New VA logL: -141.7221 | Difference: 0.7550214
Iteration: 30 Current VA logL: -141.7221 | New VA logL: -141.011 | Difference: 0.7110918
Iteration: 31 Current VA logL: -141.011 | New VA logL: -140.3409 | Difference: 0.6701192
Iteration: 32 Current VA logL: -140.3409 | New VA logL: -139.709 | Difference: 0.6318943
Iteration: 33 Current VA logL: -139.709 | New VA logL: -139.1127 | Difference: 0.5962212
Iteration: 34 Current VA logL: -139.1127 | New VA logL: -138.5563 | Difference: 0.5564972
Iteration: 35 Current VA logL: -138.5563 | New VA logL: -138.0674 | Difference: 0.4888532
Iteration: 36 Current VA logL: -138.0674 | New VA logL: -137.6014 | Difference: 0.4659985
Iteration: 37 Current VA logL: -137.6014 | New VA logL: -137.157 | Difference: 0.4444216
Iteration: 38 Current VA logL: -137.157 | New VA logL: -136.733 | Difference: 0.4239319
Iteration: 39 Current VA logL: -136.733 | New VA logL: -136.3286 | Difference: 0.4044801
Iteration: 40 Current VA logL: -136.3286 | New VA logL: -135.9426 | Difference: 0.3860163
Iteration: 41 Current VA logL: -135.9426 | New VA logL: -135.5741 | Difference: 0.3684918
Iteration: 42 Current VA logL: -135.5741 | New VA logL: -135.2222 | Difference: 0.3518596
Iteration: 43 Current VA logL: -135.2222 | New VA logL: -134.8861 | Difference: 0.3360743
Iteration: 44 Current VA logL: -134.8861 | New VA logL: -134.565 | Difference: 0.3210922
Iteration: 45 Current VA logL: -134.565 | New VA logL: -134.2582 | Difference: 0.3068714
Iteration: 46 Current VA logL: -134.2582 | New VA logL: -133.9648 | Difference: 0.293372
Iteration: 47 Current VA logL: -133.9648 | New VA logL: -133.6842 | Difference: 0.2805557
Iteration: 48 Current VA logL: -133.6842 | New VA logL: -133.4158 | Difference: 0.2683862
Iteration: 49 Current VA logL: -133.4158 | New VA logL: -133.159 | Difference: 0.256829
Iteration: 50 Current VA logL: -133.159 | New VA logL: -132.9132 | Difference: 0.2458511
Iteration: 51 Current VA logL: -132.9132 | New VA logL: -132.6777 | Difference: 0.2354214
Iteration: 52 Current VA logL: -132.6777 | New VA logL: -132.4543 | Difference: 0.2234448
Iteration: 53 Current VA logL: -132.4543 | New VA logL: -132.2578 | Difference: 0.196546
Iteration: 54 Current VA logL: -132.2578 | New VA logL: -132.0678 | Difference: 0.1899236
Iteration: 55 Current VA logL: -132.0678 | New VA logL: -131.8842 | Difference: 0.1835867
Iteration: 56 Current VA logL: -131.8842 | New VA logL: -131.7068 | Difference: 0.1774629
Iteration: 57 Current VA logL: -131.7068 | New VA logL: -131.5352 | Difference: 0.1715473
Iteration: 58 Current VA logL: -131.5352 | New VA logL: -131.3694 | Difference: 0.1658343
Iteration: 59 Current VA logL: -131.3694 | New VA logL: -131.2091 | Difference: 0.1603182
Iteration: 60 Current VA logL: -131.2091 | New VA logL: -131.0541 | Difference: 0.1549932
Iteration: 61 Current VA logL: -131.0541 | New VA logL: -130.9042 | Difference: 0.1498535
Iteration: 62 Current VA logL: -130.9042 | New VA logL: -130.7593 | Difference: 0.1448935
Iteration: 63 Current VA logL: -130.7593 | New VA logL: -130.6192 | Difference: 0.1401075
Iteration: 64 Current VA logL: -130.6192 | New VA logL: -130.4837 | Difference: 0.1354898
Iteration: 65 Current VA logL: -130.4837 | New VA logL: -130.3527 | Difference: 0.131035
Iteration: 66 Current VA logL: -130.3527 | New VA logL: -130.226 | Difference: 0.1267376
Iteration: 67 Current VA logL: -130.226 | New VA logL: -130.1034 | Difference: 0.1225923
Iteration: 68 Current VA logL: -130.1034 | New VA logL: -129.9848 | Difference: 0.1185939
Iteration: 69 Current VA logL: -129.9848 | New VA logL: -129.87 | Difference: 0.1147374
Iteration: 70 Current VA logL: -129.87 | New VA logL: -129.759 | Difference: 0.1110177
Iteration: 71 Current VA logL: -129.759 | New VA logL: -129.6516 | Difference: 0.1074301
Iteration: 72 Current VA logL: -129.6516 | New VA logL: -129.5476 | Difference: 0.1039699
Iteration: 73 Current VA logL: -129.5476 | New VA logL: -129.447 | Difference: 0.1006325
Iteration: 74 Current VA logL: -129.447 | New VA logL: -129.3496 | Difference: 0.09741343
Iteration: 75 Current VA logL: -129.3496 | New VA logL: -129.2553 | Difference: 0.0943085
Iteration: 76 Current VA logL: -129.2553 | New VA logL: -129.164 | Difference: 0.09131353
Iteration: 77 Current VA logL: -129.164 | New VA logL: -129.0755 | Difference: 0.08842448
Iteration: 78 Current VA logL: -129.0755 | New VA logL: -128.9899 | Difference: 0.08563749
Iteration: 79 Current VA logL: -128.9899 | New VA logL: -128.907 | Difference: 0.08294879
Iteration: 80 Current VA logL: -128.907 | New VA logL: -128.8266 | Difference: 0.08035476
Iteration: 81 Current VA logL: -128.8266 | New VA logL: -128.7487 | Difference: 0.0778519
Iteration: 82 Current VA logL: -128.7487 | New VA logL: -128.6733 | Difference: 0.07543681
Iteration: 83 Current VA logL: -128.6733 | New VA logL: -128.6002 | Difference: 0.07310626
Iteration: 84 Current VA logL: -128.6002 | New VA logL: -128.5293 | Difference: 0.07085708
Iteration: 85 Current VA logL: -128.5293 | New VA logL: -128.4607 | Difference: 0.06868625
Iteration: 86 Current VA logL: -128.4607 | New VA logL: -128.3941 | Difference: 0.06659085
Iteration: 87 Current VA logL: -128.3941 | New VA logL: -128.3295 | Difference: 0.06456807
Iteration: 88 Current VA logL: -128.3295 | New VA logL: -128.2669 | Difference: 0.06261519
Iteration: 89 Current VA logL: -128.2669 | New VA logL: -128.2062 | Difference: 0.06072961
Iteration: 90 Current VA logL: -128.2062 | New VA logL: -128.1472 | Difference: 0.05890881
Iteration: 91 Current VA logL: -128.1472 | New VA logL: -128.0901 | Difference: 0.05715038
Iteration: 92 Current VA logL: -128.0901 | New VA logL: -128.0346 | Difference: 0.05545923
Iteration: 93 Current VA logL: -128.0346 | New VA logL: -127.9808 | Difference: 0.05383573
Iteration: 94 Current VA logL: -127.9808 | New VA logL: -127.9285 | Difference: 0.05227656
Iteration: 95 Current VA logL: -127.9285 | New VA logL: -127.8777 | Difference: 0.05077858
Iteration: 96 Current VA logL: -127.8777 | New VA logL: -127.8284 | Difference: 0.04933883
Iteration: 97 Current VA logL: -127.8284 | New VA logL: -127.7805 | Difference: 0.04795453
Iteration: 98 Current VA logL: -127.7805 | New VA logL: -127.7338 | Difference: 0.04662304
Iteration: 99 Current VA logL: -127.7338 | New VA logL: -127.6885 | Difference: 0.04534191
Iteration: 100 Current VA logL: -127.6885 | New VA logL: -127.6444 | Difference: 0.04410878
Iteration: 101 Current VA logL: -127.6444 | New VA logL: -127.6015 | Difference: 0.04292144
Iteration: 102 Current VA logL: -127.6015 | New VA logL: -127.5597 | Difference: 0.04177781
Iteration: 103 Current VA logL: -127.5597 | New VA logL: -127.519 | Difference: 0.04067592
Iteration: 104 Current VA logL: -127.519 | New VA logL: -127.4794 | Difference: 0.03961388
Iteration: 105 Current VA logL: -127.4794 | New VA logL: -127.4408 | Difference: 0.03858994
Iteration: 106 Current VA logL: -127.4408 | New VA logL: -127.4032 | Difference: 0.03760241
Iteration: 107 Current VA logL: -127.4032 | New VA logL: -127.3665 | Difference: 0.0366497
Iteration: 108 Current VA logL: -127.3665 | New VA logL: -127.3308 | Difference: 0.0357303
Iteration: 109 Current VA logL: -127.3308 | New VA logL: -127.296 | Difference: 0.03484278
Iteration: 110 Current VA logL: -127.296 | New VA logL: -127.262 | Difference: 0.03398578
Iteration: 111 Current VA logL: -127.262 | New VA logL: -127.2288 | Difference: 0.033158
Iteration: 112 Current VA logL: -127.2288 | New VA logL: -127.1965 | Difference: 0.03235822
Iteration: 113 Current VA logL: -127.1965 | New VA logL: -127.1649 | Difference: 0.03158526
Iteration: 114 Current VA logL: -127.1649 | New VA logL: -127.134 | Difference: 0.03083801
Iteration: 115 Current VA logL: -127.134 | New VA logL: -127.1039 | Difference: 0.03011541
Iteration: 116 Current VA logL: -127.1039 | New VA logL: -127.0745 | Difference: 0.02941644
Iteration: 117 Current VA logL: -127.0745 | New VA logL: -127.0458 | Difference: 0.02874015
Iteration: 118 Current VA logL: -127.0458 | New VA logL: -127.0177 | Difference: 0.02808562
Iteration: 119 Current VA logL: -127.0177 | New VA logL: -126.9902 | Difference: 0.02745196
Iteration: 120 Current VA logL: -126.9902 | New VA logL: -126.9634 | Difference: 0.02683833
Iteration: 121 Current VA logL: -126.9634 | New VA logL: -126.9372 | Difference: 0.02624395
Iteration: 122 Current VA logL: -126.9372 | New VA logL: -126.9115 | Difference: 0.02566803
Iteration: 123 Current VA logL: -126.9115 | New VA logL: -126.8864 | Difference: 0.02510986
Iteration: 124 Current VA logL: -126.8864 | New VA logL: -126.8618 | Difference: 0.02456873
Iteration: 125 Current VA logL: -126.8618 | New VA logL: -126.8378 | Difference: 0.02404396
Iteration: 126 Current VA logL: -126.8378 | New VA logL: -126.8142 | Difference: 0.02353493
Iteration: 127 Current VA logL: -126.8142 | New VA logL: -126.7912 | Difference: 0.02304101
Iteration: 128 Current VA logL: -126.7912 | New VA logL: -126.7686 | Difference: 0.02256161
Iteration: 129 Current VA logL: -126.7686 | New VA logL: -126.7465 | Difference: 0.02209617
Iteration: 130 Current VA logL: -126.7465 | New VA logL: -126.7249 | Difference: 0.02164416
Iteration: 131 Current VA logL: -126.7249 | New VA logL: -126.7037 | Difference: 0.02120505
Iteration: 132 Current VA logL: -126.7037 | New VA logL: -126.6829 | Difference: 0.02077834
Iteration: 133 Current VA logL: -126.6829 | New VA logL: -126.6625 | Difference: 0.02036355
Iteration: 134 Current VA logL: -126.6625 | New VA logL: -126.6426 | Difference: 0.01996023
Iteration: 135 Current VA logL: -126.6426 | New VA logL: -126.623 | Difference: 0.01956794
Iteration: 136 Current VA logL: -126.623 | New VA logL: -126.6038 | Difference: 0.01918626
Iteration: 137 Current VA logL: -126.6038 | New VA logL: -126.585 | Difference: 0.01881477
Iteration: 138 Current VA logL: -126.585 | New VA logL: -126.5666 | Difference: 0.01845309
Iteration: 139 Current VA logL: -126.5666 | New VA logL: -126.5485 | Difference: 0.01810085
Iteration: 140 Current VA logL: -126.5485 | New VA logL: -126.5307 | Difference: 0.01775768
Iteration: 141 Current VA logL: -126.5307 | New VA logL: -126.5133 | Difference: 0.01742324
Iteration: 142 Current VA logL: -126.5133 | New VA logL: -126.4962 | Difference: 0.01709721
Iteration: 143 Current VA logL: -126.4962 | New VA logL: -126.4794 | Difference: 0.01677926
Iteration: 144 Current VA logL: -126.4794 | New VA logL: -126.4629 | Difference: 0.01646908
Iteration: 145 Current VA logL: -126.4629 | New VA logL: -126.4468 | Difference: 0.0161664
Iteration: 146 Current VA logL: -126.4468 | New VA logL: -126.4309 | Difference: 0.01587091
Iteration: 147 Current VA logL: -126.4309 | New VA logL: -126.4153 | Difference: 0.01558237
Iteration: 148 Current VA logL: -126.4153 | New VA logL: -126.4 | Difference: 0.01530051
Iteration: 149 Current VA logL: -126.4 | New VA logL: -126.385 | Difference: 0.01502508
Iteration: 150 Current VA logL: -126.385 | New VA logL: -126.3702 | Difference: 0.01475584
Iteration: 151 Current VA logL: -126.3702 | New VA logL: -126.3557 | Difference: 0.01449258
Iteration: 152 Current VA logL: -126.3557 | New VA logL: -126.3415 | Difference: 0.01423508
Iteration: 153 Current VA logL: -126.3415 | New VA logL: -126.3275 | Difference: 0.01398313
Iteration: 154 Current VA logL: -126.3275 | New VA logL: -126.3138 | Difference: 0.01373654
Iteration: 155 Current VA logL: -126.3138 | New VA logL: -126.3003 | Difference: 0.01349511
Iteration: 156 Current VA logL: -126.3003 | New VA logL: -126.287 | Difference: 0.01325867
Iteration: 157 Current VA logL: -126.287 | New VA logL: -126.274 | Difference: 0.01302706
Iteration: 158 Current VA logL: -126.274 | New VA logL: -126.2612 | Difference: 0.0128001
Iteration: 159 Current VA logL: -126.2612 | New VA logL: -126.2486 | Difference: 0.01257765
Iteration: 160 Current VA logL: -126.2486 | New VA logL: -126.2363 | Difference: 0.01235955
Iteration: 161 Current VA logL: -126.2363 | New VA logL: -126.2241 | Difference: 0.01214568
Iteration: 162 Current VA logL: -126.2241 | New VA logL: -126.2122 | Difference: 0.0119359
Iteration: 163 Current VA logL: -126.2122 | New VA logL: -126.2005 | Difference: 0.01173009
Iteration: 164 Current VA logL: -126.2005 | New VA logL: -126.1889 | Difference: 0.01152813
Iteration: 165 Current VA logL: -126.1889 | New VA logL: -126.1776 | Difference: 0.0113299
Iteration: 166 Current VA logL: -126.1776 | New VA logL: -126.1665 | Difference: 0.01113532
Iteration: 167 Current VA logL: -126.1665 | New VA logL: -126.1555 | Difference: 0.01094427
Iteration: 168 Current VA logL: -126.1555 | New VA logL: -126.1448 | Difference: 0.01075667
Iteration: 169 Current VA logL: -126.1448 | New VA logL: -126.1342 | Difference: 0.01057242
Iteration: 170 Current VA logL: -126.1342 | New VA logL: -126.1238 | Difference: 0.01039145
Iteration: 171 Current VA logL: -126.1238 | New VA logL: -126.1136 | Difference: 0.01021368
Iteration: 172 Current VA logL: -126.1136 | New VA logL: -126.1035 | Difference: 0.01003903
Iteration: 173 Current VA logL: -126.1035 | New VA logL: -126.0937 | Difference: 0.009867437
Iteration: 174 Current VA logL: -126.0937 | New VA logL: -126.084 | Difference: 0.009698838
Iteration: 175 Current VA logL: -126.084 | New VA logL: -126.0744 | Difference: 0.00953317
Iteration: 176 Current VA logL: -126.0744 | New VA logL: -126.0651 | Difference: 0.009370377
Iteration: 177 Current VA logL: -126.0651 | New VA logL: -126.0559 | Difference: 0.009210403
Iteration: 178 Current VA logL: -126.0559 | New VA logL: -126.0468 | Difference: 0.009053199
Iteration: 179 Current VA logL: -126.0468 | New VA logL: -126.0379 | Difference: 0.008898715
Iteration: 180 Current VA logL: -126.0379 | New VA logL: -126.0292 | Difference: 0.008746906
Iteration: 181 Current VA logL: -126.0292 | New VA logL: -126.0206 | Difference: 0.008597728
Iteration: 182 Current VA logL: -126.0206 | New VA logL: -126.0121 | Difference: 0.008451139
Iteration: 183 Current VA logL: -126.0121 | New VA logL: -126.0038 | Difference: 0.008307099
Iteration: 184 Current VA logL: -126.0038 | New VA logL: -125.9956 | Difference: 0.008165571
Iteration: 185 Current VA logL: -125.9956 | New VA logL: -125.9876 | Difference: 0.008026517
Iteration: 186 Current VA logL: -125.9876 | New VA logL: -125.9797 | Difference: 0.007889902
Iteration: 187 Current VA logL: -125.9797 | New VA logL: -125.972 | Difference: 0.007755691
Iteration: 188 Current VA logL: -125.972 | New VA logL: -125.9643 | Difference: 0.007623852
Iteration: 189 Current VA logL: -125.9643 | New VA logL: -125.9569 | Difference: 0.00749435
Iteration: 190 Current VA logL: -125.9569 | New VA logL: -125.9495 | Difference: 0.007367154
Iteration: 191 Current VA logL: -125.9495 | New VA logL: -125.9422 | Difference: 0.007242232
Iteration: 192 Current VA logL: -125.9422 | New VA logL: -125.9351 | Difference: 0.007119553
Iteration: 193 Current VA logL: -125.9351 | New VA logL: -125.9281 | Difference: 0.006999086
Iteration: 194 Current VA logL: -125.9281 | New VA logL: -125.9212 | Difference: 0.0068808
Iteration: 195 Current VA logL: -125.9212 | New VA logL: -125.9145 | Difference: 0.006764666
Iteration: 196 Current VA logL: -125.9145 | New VA logL: -125.9078 | Difference: 0.006650652
Iteration: 197 Current VA logL: -125.9078 | New VA logL: -125.9013 | Difference: 0.006538728
Iteration: 198 Current VA logL: -125.9013 | New VA logL: -125.8949 | Difference: 0.006428864
Iteration: 199 Current VA logL: -125.8949 | New VA logL: -125.8885 | Difference: 0.006321029
Iteration: 200 Current VA logL: -125.8885 | New VA logL: -125.8823 | Difference: 0.006215194
Iteration: 201 Current VA logL: -125.8823 | New VA logL: -125.8762 | Difference: 0.006111327
Iteration: 202 Current VA logL: -125.8762 | New VA logL: -125.8702 | Difference: 0.006009399
Iteration: 203 Current VA logL: -125.8702 | New VA logL: -125.8643 | Difference: 0.005909378
Iteration: 204 Current VA logL: -125.8643 | New VA logL: -125.8585 | Difference: 0.005811235
Iteration: 205 Current VA logL: -125.8585 | New VA logL: -125.8528 | Difference: 0.005714938
Iteration: 206 Current VA logL: -125.8528 | New VA logL: -125.8471 | Difference: 0.005620457
Iteration: 207 Current VA logL: -125.8471 | New VA logL: -125.8416 | Difference: 0.005527761
Iteration: 208 Current VA logL: -125.8416 | New VA logL: -125.8362 | Difference: 0.005436819
Iteration: 209 Current VA logL: -125.8362 | New VA logL: -125.8308 | Difference: 0.0053476
Iteration: 210 Current VA logL: -125.8308 | New VA logL: -125.8256 | Difference: 0.005260075
Iteration: 211 Current VA logL: -125.8256 | New VA logL: -125.8204 | Difference: 0.005174211
Iteration: 212 Current VA logL: -125.8204 | New VA logL: -125.8153 | Difference: 0.005089979
Iteration: 213 Current VA logL: -125.8153 | New VA logL: -125.8103 | Difference: 0.005007349
Iteration: 214 Current VA logL: -125.8103 | New VA logL: -125.8054 | Difference: 0.004926289
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Calculating information matrix for model parameters...
Redoing...
Warning message:
In sqrt(diag(solve(obs_info))) : NaNs produced
Variational approximation for GAMs
Call: vagam(y = poisson_dat$y, smooth.X = poisson_dat[, 2:5], para.X = data.frame(treatment = poisson_dat$treatment), int.knots = choose_k, family = poisson(), save.data = TRUE, para.se = TRUE)
Estimated regression coefficients for parametric component: -1.040643 0.6451899
Estimated smoothing coefficients for nonparametric component: -0.2733702 -0.2461039 -0.09914876 0.03284408 0.1684629 0.3366869 0.4947952 0.5851139 0.656716 0.5336587 0.3721285 0.2277044 -0.02135385 -0.2695109 -0.5710075 -0.7604779 -0.8174288 -1.712253 -1.563433 -1.369698 -0.9899007 -0.8411191 -0.4226654 -0.07139302 -0.2168335 0.0007790999 0.5339774 0.8247894 1.282654 2.006611 1.880903 1.258986 1.21398 1.148197 -0.751521 0.4847275 3.133529 6.330736 4.998549 3.434801 1.618585 -0.6540657 -1.360546 -1.018068 -0.4136508 -1.939511 -2.64722 -2.147782 -1.521794 -2.187518 -2.458522 -1.607145 -1.329687 0.005428171 1.0108 0.4098627 0.7475761 1.115838 0.7188687 0.7304209 1.060003 0.8690969 0.613725 0.4260507 0.4167565 0.5912033 0.3957223 0.02593618
Estimated smoothing parameters (or fixed if lambda was supplied): 6.680485 3.832892 0.2155243 1.027374
Number of interior knots used: 15 15 15 15
Maximized value of the variational log-likelihood: -125.5384
Summary statistics for nonparametric component:
x0 x1 x2 x3
Wald Statistic 358.331 840.2793 31902.05 79.84659
p-value 0.000 0.0000 0.00 0.00000
Summary statistics for parametric component (if para.se = TRUE):
Intercept treatment
Estimate -1.04064 0.64519
Std. Error NaN 0.16658
Wald Statistic NaN 3.87310
p-value NaN 0.00011
$call
vagam(y = poisson_dat$y, smooth.X = poisson_dat[, 2:5], para.X = data.frame(treatment = poisson_dat$treatment),
int.knots = choose_k, family = poisson(), save.data = TRUE,
para.se = TRUE)
$para.coeff
Intercept treatment
-1.0406428 0.6451899
$smooth.coeff
[1] -0.2733701882 -0.2461038544 -0.0991487572 0.0328440844 0.1684629451
[6] 0.3366869456 0.4947951958 0.5851139119 0.6567160275 0.5336586669
[11] 0.3721285236 0.2277043874 -0.0213538489 -0.2695109221 -0.5710074632
[16] -0.7604779036 -0.8174288063 -1.7122534105 -1.5634333694 -1.3696982975
[21] -0.9899006792 -0.8411191491 -0.4226653909 -0.0713930171 -0.2168334916
[26] 0.0007790999 0.5339773671 0.8247894171 1.2826538456 2.0066107547
[31] 1.8809033147 1.2589859418 1.2139796076 1.1481968223 -0.7515209731
[36] 0.4847274793 3.1335287755 6.3307363410 4.9985492274 3.4348007502
[41] 1.6185847776 -0.6540656573 -1.3605461824 -1.0180681483 -0.4136507929
[46] -1.9395112438 -2.6472198102 -2.1477815240 -1.5217936307 -2.1875175588
[51] -2.4585215323 -1.6071447536 -1.3296868389 0.0054281713 1.0108003112
[56] 0.4098626899 0.7475760842 1.1158378269 0.7188686895 0.7304209105
[61] 1.0600027739 0.8690969307 0.6137250079 0.4260507231 0.4167565447
[66] 0.5912032995 0.3957222731 0.0259361775
$smooth.param
[1] 6.6804854 3.8328919 0.2155243 1.0273736
$phi
[1] 1
$logLik
[1] -125.5384
$family
Family: poisson
Link function: log
$smooth.stat
x0 x1 x2 x3
Wald Statistic 358.331 840.2793 31902.05 79.84659
p-value 0.000 0.0000 0.00 0.00000
$para.stat
Intercept treatment
Estimate -1.04064 0.64519
Std. Error NaN 0.16658
Wald Statistic NaN 3.87310
p-value NaN 0.00011
attr(,"class")
[1] "summary.vagam"
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