Org/countr_org/var-covariance/var_covar.md
In GeoBosh/Countr: Flexible Univariate Count Models Based on Renewal Processes
Computing standard errors and confidence intervals for estimated parameters is a
common task in regression analysis. These quantities allow the analyst to
quantify the certainty (*confidence*) associated with the obtained estimates. In
`Countr` two different approaches have been implemented to do this job. First, using
asymptotic MLE (Maximum Likelihood Estimator) theory, numeric computation of the
inverse Hessian matrix can be used as a consistent estimator of the
variance-covariance matrix, which in turn can be used to derive standard errors
and confidence intervals. The second option available in `Countr` is to use
bootsrap \citep{efron1979bootstrap}. In this document, we give the user an
overview of how to do to the computation in `Countr`.
Before starting our analysis, we need to load the useful packages. On top of
Countr, the dplyr package \citep{dplyr2016} will be used:
library(Countr)
library(dplyr)
Maximum Likelihood estimator (MLE)
Theory
Let (f(y, \mathbf{x}, \bm{\theta})) be the probability density function of the
renewal-count model, where (y) is the count variable, (\mathbf{x}) the vector of
covariate values and (\bm{\theta}) the vector of associated coefficients to be
estimated ((q \times 1) vector). Define the log-likelihood (\mathcal{L} =
\sum_{i=1}^{n} ln f(y_i|\mathbf{x}_i, \bm{\theta}_i)). Under regularity
conditions (see for example \citet[Section 2.3]{cameron2013regression}), the MLE
(\bm{\hat{\theta}}) is the solution of the first-order conditions,
\begin{equation}
\frac{\partial \mathcal{L}}{\partial \bm{\theta}} = \sum_{i=1}^{n} \frac{\partial lm f_i}{\partial \bm{\theta}} = 0
,
\end{equation}
where (f_i = f(y_i|\mathbf{x}_i, \bm{\theta}_i)) and (\frac{\partial
\mathcal{L}}{\partial \bm{\theta}}) is a (q \times 1) vector.
Let (\bm{\theta}_0) be the true value of (\bm{\theta}). Using MLE theory and
assuming regularity conditions, we obtain (\bm{\hat{\theta}} \xrightarrow{p}
\bm{\theta}_0) and
\begin{equation}
\sqrt{n}(\bm{\hat{\theta}}_{ML} - \bm{\theta}_0) \xrightarrow{d} \mathcal{N}[\mathbf{0}, \mathbf{V}^{-1}]
,
\end{equation}
where the (q \times q) matrix (\mathbf{V}) matrix is defined as
\begin{equation}
\mathbf{V} = - \lim_{n \rightarrow \infty} \frac{1}{n} \mathop{\mathbb{E}}
\bigg[ \sum_{i=1}^n \frac{\partial^2 \ln f_i }{\partial \bm{\theta} \partial \bm{\theta}'}|_{\bm{\theta}_0}
\bigg]
.
\end{equation}
To use this result, we need a consistent estimator of the variance matrix
(\mathbf{V}). Many options are available: the one implemented in Countr is
known as the Hessian estimator and simply evaluates Equation 3 at
(\bm{\hat{\theta}}) without taking expectation.
Implementation in Countr
The easiest way to compute the variance-covariance matrix when fitting a
renewal-count model with Countr is to set the argument computeHessian to
TRUE when calling the fitting routine renewalCount(). Note that this is the
default behaviour and it will save the matrix in the returned object (slot
vcov). Following standard practice in R, the matrix can be extracted or
recomputed using the vcov() method. We show here an example with the
fertility data.
data(fertility)
form <- children ~ german + years_school + voc_train + university + Religion +
year_birth + rural + age_marriage
gam <- renewalCount(formula = form, data = fertility, dist = "gamma",
computeHessian = TRUE,
control = renewal.control(trace = 0, method = "nlminb")
)
v1 <- gam$vcov
v2 <- vcov(gam)
all(v1 == v2)
[1] TRUE
The above vcov() method simply extracts the variance-covariance matrix if it
has been computed at fitted. Otherwise, the function will re-compute it. The
user can choose the computation method by specifying the method argument:
asymptotic for numerical hessian computation or boot for the bootstrap
method. In the latter case, user can customise the bootstrap computation as will
be discussed in Section 2 by using the ... argument.
Parameters' standard errors and confidence intervals can be computed by calls to
the generic functions se.coef() and confint(). The hessian method can be
specified by setting the argument type = "asymptotic".
se <- se.coef(gam, type = "asymptotic")
se
rate_ rate_germanyes rate_years_school
0.2523375 0.0590399 0.0265014
rate_voc_trainyes rate_universityyes rate_ReligionCatholic
0.0358390 0.1296149 0.0578032
rate_ReligionMuslim rate_ReligionProtestant rate_year_birth
0.0698429 0.0622954 0.0019471
rate_ruralyes rate_age_marriage shape_
0.0311876 0.0053274 0.0710611
ci <- confint(gam, type = "asymptotic")
ci
2.5 % 97.5 %
rate_ 1.0621317 2.0512766
rate_germanyes -0.3054815 -0.0740495
rate_years_school -0.0202566 0.0836270
rate_voc_trainyes -0.2141792 -0.0736929
rate_universityyes -0.4000943 0.1079867
rate_ReligionCatholic 0.0924771 0.3190616
rate_ReligionMuslim 0.3857453 0.6595243
rate_ReligionProtestant -0.0149604 0.2292330
rate_year_birth -0.0014721 0.0061603
rate_ruralyes -0.0056353 0.1166177
rate_age_marriage -0.0392385 -0.0183553
shape_ 1.3000440 1.5785983
One can validate the result obtained here by comparing them to what is reported
in \citet[Table 1]{winkelmann1995duration}.
Bootstrap
Theory
The type of bootstrap used in Countr is known as nonparametric or bootstrap
pairs. It is valid under the assumption that ((y_i, \mathbf{x}_i)) is iid. The
algorithm works as follows:
- Generate a pseudo-sample of size (n), ((y^_l, \mathbf{x}^_l), \ l=1, \dots,
n), by sampling with replacement from the original pairs ((y_i,
\mathbf{x}_i), \ i =1, \dots, n).
- Compute the estimator (\hat{\bm{\theta}}_1) from the pseudo-sample generated
in 1.
- Repeat steps 1 and 2 (B) times giving (B) estimates (\hat{\bm{\theta}}_1,
\dots, \hat{\bm{\theta}}_B).
- The bootstrap estimate of the variance-covariance matrix is given by
(\hat{\mathbf{V}}{Boot}[\bm{\hat{\theta}}] = \frac{1}{B -1}
\sum{b=1}^B(\hat{\bm{\theta}}b - \bar{\bm{\theta}})(\hat{\bm{\theta}}_b -
\bar{\bm{\theta}})') where (\bar{\bm{\theta}} = [\bar{\theta}_1, \dots,
\bar{\theta}_q]) and (\bar{\theta}_j) is the sample mean (\bar{\theta}_j = (1/B)
\sum{b=1}^B \hat{\bm{\theta}}_{j,b}).
Asymptotically ((B \rightarrow \infty)), the bootstrap variance-covariance
matrix and standard errors are equivalent to their robust counterpart obtained
by sandwich estimators. In practice, using (B=400) is usually recommended
\citep[Section 2.6.4]{cameron2013regression}}
Implementation in Countr
The bootstrap computations in Countr are based on the boot() function
from the package with the same name \citep{boot2017}.
The variance-covariance matrix is again computed with the renewal method for
vcov() by specifying the argument method = "boot". The computation can be
further customised by passing other options accepted by boot() other than
data and statistic which are provided by the Countr code. Note that the
matrix is only computed if it is not found in the passed renewal object. The
bootstrap sample is actually computed by a separate function
addBootSampleObject(), which computes the bootstrap sample and adds it as
component boot to the renewal object. Functions like vcov() and confint()
check if a bootstrap sample is already available and use it is. It is a good
idea to call addBootSampleObject() before attempting computations based on
bootstrapping. We show below how to update the previously fitted gamma model
with 400 bootstrap iterations using the parallel option and 14 CPUs. if (B) is
large and depending on how fast the model can be fitted, this computation may be
time consuming.
gam <- addBootSampleObject(gam, R = 400, parallel = "multicore", ncpus = 14)
Once the object is updated, the variance-covariance matrix is computed by vcov
in a straightforward way:
gam$vcov <- matrix()
varCovar <- vcov(gam, method = "boot")
varCovar
rate_ rate_germanyes rate_years_school
rate_ 0.07318764 5.0060e-03 -5.6704e-03
rate_germanyes 0.00500601 3.7300e-03 -6.9972e-04
rate_years_school -0.00567041 -6.9972e-04 7.1876e-04
rate_voc_trainyes 0.00087763 -4.3825e-04 -1.6477e-04
rate_universityyes 0.01675906 1.8472e-03 -2.2823e-03
rate_ReligionCatholic -0.00072461 -9.5183e-04 -5.0782e-05
rate_ReligionMuslim -0.00350408 1.1549e-03 1.3725e-04
rate_ReligionProtestant -0.00062891 -1.3197e-03 -4.8670e-05
rate_year_birth -0.00018173 -3.8643e-05 5.6028e-06
rate_ruralyes -0.00197542 -2.2663e-04 1.1529e-04
rate_age_marriage -0.00036158 6.7750e-05 -1.8638e-05
shape_ 0.00860625 1.1773e-05 6.3172e-05
rate_voc_trainyes rate_universityyes
rate_ 8.7763e-04 1.6759e-02
rate_germanyes -4.3825e-04 1.8472e-03
rate_years_school -1.6477e-04 -2.2823e-03
rate_voc_trainyes 1.4116e-03 1.6609e-03
rate_universityyes 1.6609e-03 1.5801e-02
rate_ReligionCatholic 5.5154e-05 5.2001e-04
rate_ReligionMuslim 2.0904e-04 4.9984e-04
rate_ReligionProtestant 9.5869e-06 5.3275e-04
rate_year_birth 9.8568e-06 3.1314e-06
rate_ruralyes 3.5200e-04 3.9912e-04
rate_age_marriage -3.0112e-06 3.0646e-05
shape_ 4.7749e-04 4.1102e-04
rate_ReligionCatholic rate_ReligionMuslim
rate_ -7.2461e-04 -3.5041e-03
rate_germanyes -9.5183e-04 1.1549e-03
rate_years_school -5.0782e-05 1.3725e-04
rate_voc_trainyes 5.5154e-05 2.0904e-04
rate_universityyes 5.2001e-04 4.9984e-04
rate_ReligionCatholic 3.1847e-03 1.6360e-03
rate_ReligionMuslim 1.6360e-03 4.6887e-03
rate_ReligionProtestant 2.7871e-03 1.4184e-03
rate_year_birth -2.4024e-06 -1.7800e-05
rate_ruralyes 2.1452e-04 3.7533e-04
rate_age_marriage -3.0908e-05 1.7879e-05
shape_ -6.7137e-05 8.1247e-05
rate_ReligionProtestant rate_year_birth rate_ruralyes
rate_ -6.2891e-04 -1.8173e-04 -1.9754e-03
rate_germanyes -1.3197e-03 -3.8643e-05 -2.2663e-04
rate_years_school -4.8670e-05 5.6028e-06 1.1529e-04
rate_voc_trainyes 9.5869e-06 9.8568e-06 3.5200e-04
rate_universityyes 5.3275e-04 3.1314e-06 3.9912e-04
rate_ReligionCatholic 2.7871e-03 -2.4024e-06 2.1452e-04
rate_ReligionMuslim 1.4184e-03 -1.7800e-05 3.7533e-04
rate_ReligionProtestant 3.7010e-03 1.6907e-06 1.8705e-04
rate_year_birth 1.6907e-06 3.7486e-06 -2.0782e-07
rate_ruralyes 1.8705e-04 -2.0782e-07 1.0425e-03
rate_age_marriage -4.3948e-05 -2.2939e-06 1.2680e-05
shape_ -2.3479e-04 -2.4092e-05 1.3058e-04
rate_age_marriage shape_
rate_ -3.6158e-04 8.6063e-03
rate_germanyes 6.7750e-05 1.1773e-05
rate_years_school -1.8638e-05 6.3172e-05
rate_voc_trainyes -3.0112e-06 4.7749e-04
rate_universityyes 3.0646e-05 4.1102e-04
rate_ReligionCatholic -3.0908e-05 -6.7137e-05
rate_ReligionMuslim 1.7879e-05 8.1247e-05
rate_ReligionProtestant -4.3948e-05 -2.3479e-04
rate_year_birth -2.2939e-06 -2.4092e-05
rate_ruralyes 1.2680e-05 1.3058e-04
rate_age_marriage 3.0042e-05 8.9316e-05
shape_ 8.9316e-05 1.2840e-02
Bootstrap standard errors are also very easy to compute by calling se.coef()
with argument type"boot"=. As discussed before, if the boot object is not
found in the renewal object, users can customise the boot() call by
passing the appropriate arguments in ....
se_boot <- se.coef(gam, type = "boot")
se_boot
rate_ rate_germanyes rate_years_school
0.2705321 0.0610739 0.0268098
rate_voc_trainyes rate_universityyes rate_ReligionCatholic
0.0375714 0.1257020 0.0564330
rate_ReligionMuslim rate_ReligionProtestant rate_year_birth
0.0684739 0.0608356 0.0019361
rate_ruralyes rate_age_marriage shape_
0.0322879 0.0054811 0.1133149
Finally bootstrap confidence intervals can also be computed by confint()
using the same logic described for se.coef(). Different types of
confidence intervals are available (default is normal) and can be selected by
choosing the appropriate type in bootType. We refer the user to the boot
package \citep{boot2017} for more information.
ci_boot <- confint(gam, level = 0.95, type = "boot", bootType = "norm")
ci_boot
Bootstrap quantiles, type = normal
2.5 % 97.5 %
rate_ 1.0105534 2.0710199
rate_germanyes -0.3116866 -0.0722812
rate_years_school -0.0176548 0.0874376
rate_voc_trainyes -0.2211511 -0.0738740
rate_universityyes -0.4072628 0.0854802
rate_ReligionCatholic 0.0985243 0.3197377
rate_ReligionMuslim 0.3871631 0.6555759
rate_ReligionProtestant -0.0133962 0.2250750
rate_year_birth -0.0016933 0.0058961
rate_ruralyes -0.0098069 0.1167594
rate_age_marriage -0.0401915 -0.0187062
shape_ 1.1860640 1.6302501
Save Image :ignoreheading:
We conclude this analysis by saving the work space to avoid re-running the
computation in future exportation of the document:
save.image()
Bibliography :ignoreheading:
\bibliographystyle{apalike}
\bibliography{REFERENCES}
GeoBosh/Countr documentation built on Jan. 26, 2024, 12:16 p.m.
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Computing standard errors and confidence intervals for estimated parameters is a
common task in regression analysis. These quantities allow the analyst to
quantify the certainty (*confidence*) associated with the obtained estimates. In
`Countr` two different approaches have been implemented to do this job. First, using
asymptotic MLE (Maximum Likelihood Estimator) theory, numeric computation of the
inverse Hessian matrix can be used as a consistent estimator of the
variance-covariance matrix, which in turn can be used to derive standard errors
and confidence intervals. The second option available in `Countr` is to use
bootsrap \citep{efron1979bootstrap}. In this document, we give the user an
overview of how to do to the computation in `Countr`.
Before starting our analysis, we need to load the useful packages. On top of
Countr, the dplyr package \citep{dplyr2016} will be used:
library(Countr)
library(dplyr)
Maximum Likelihood estimator (MLE)
Theory
Let (f(y, \mathbf{x}, \bm{\theta})) be the probability density function of the renewal-count model, where (y) is the count variable, (\mathbf{x}) the vector of covariate values and (\bm{\theta}) the vector of associated coefficients to be estimated ((q \times 1) vector). Define the log-likelihood (\mathcal{L} = \sum_{i=1}^{n} ln f(y_i|\mathbf{x}_i, \bm{\theta}_i)). Under regularity conditions (see for example \citet[Section 2.3]{cameron2013regression}), the MLE (\bm{\hat{\theta}}) is the solution of the first-order conditions,
\begin{equation} \frac{\partial \mathcal{L}}{\partial \bm{\theta}} = \sum_{i=1}^{n} \frac{\partial lm f_i}{\partial \bm{\theta}} = 0 , \end{equation}
where (f_i = f(y_i|\mathbf{x}_i, \bm{\theta}_i)) and (\frac{\partial \mathcal{L}}{\partial \bm{\theta}}) is a (q \times 1) vector.
Let (\bm{\theta}_0) be the true value of (\bm{\theta}). Using MLE theory and assuming regularity conditions, we obtain (\bm{\hat{\theta}} \xrightarrow{p} \bm{\theta}_0) and
\begin{equation} \sqrt{n}(\bm{\hat{\theta}}_{ML} - \bm{\theta}_0) \xrightarrow{d} \mathcal{N}[\mathbf{0}, \mathbf{V}^{-1}] , \end{equation}
where the (q \times q) matrix (\mathbf{V}) matrix is defined as
\begin{equation} \mathbf{V} = - \lim_{n \rightarrow \infty} \frac{1}{n} \mathop{\mathbb{E}} \bigg[ \sum_{i=1}^n \frac{\partial^2 \ln f_i }{\partial \bm{\theta} \partial \bm{\theta}'}|_{\bm{\theta}_0} \bigg] . \end{equation}
To use this result, we need a consistent estimator of the variance matrix
(\mathbf{V}). Many options are available: the one implemented in Countr is
known as the Hessian estimator and simply evaluates Equation 3 at
(\bm{\hat{\theta}}) without taking expectation.
Implementation in Countr
The easiest way to compute the variance-covariance matrix when fitting a
renewal-count model with Countr is to set the argument computeHessian to
TRUE when calling the fitting routine renewalCount(). Note that this is the
default behaviour and it will save the matrix in the returned object (slot
vcov). Following standard practice in R, the matrix can be extracted or
recomputed using the vcov() method. We show here an example with the
fertility data.
data(fertility)
form <- children ~ german + years_school + voc_train + university + Religion +
year_birth + rural + age_marriage
gam <- renewalCount(formula = form, data = fertility, dist = "gamma",
computeHessian = TRUE,
control = renewal.control(trace = 0, method = "nlminb")
)
v1 <- gam$vcov
v2 <- vcov(gam)
all(v1 == v2)
[1] TRUE
The above vcov() method simply extracts the variance-covariance matrix if it
has been computed at fitted. Otherwise, the function will re-compute it. The
user can choose the computation method by specifying the method argument:
asymptotic for numerical hessian computation or boot for the bootstrap
method. In the latter case, user can customise the bootstrap computation as will
be discussed in Section 2 by using the ... argument.
Parameters' standard errors and confidence intervals can be computed by calls to
the generic functions se.coef() and confint(). The hessian method can be
specified by setting the argument type = "asymptotic".
se <- se.coef(gam, type = "asymptotic")
se
rate_ rate_germanyes rate_years_school
0.2523375 0.0590399 0.0265014
rate_voc_trainyes rate_universityyes rate_ReligionCatholic
0.0358390 0.1296149 0.0578032
rate_ReligionMuslim rate_ReligionProtestant rate_year_birth
0.0698429 0.0622954 0.0019471
rate_ruralyes rate_age_marriage shape_
0.0311876 0.0053274 0.0710611
ci <- confint(gam, type = "asymptotic")
ci
2.5 % 97.5 %
rate_ 1.0621317 2.0512766
rate_germanyes -0.3054815 -0.0740495
rate_years_school -0.0202566 0.0836270
rate_voc_trainyes -0.2141792 -0.0736929
rate_universityyes -0.4000943 0.1079867
rate_ReligionCatholic 0.0924771 0.3190616
rate_ReligionMuslim 0.3857453 0.6595243
rate_ReligionProtestant -0.0149604 0.2292330
rate_year_birth -0.0014721 0.0061603
rate_ruralyes -0.0056353 0.1166177
rate_age_marriage -0.0392385 -0.0183553
shape_ 1.3000440 1.5785983
One can validate the result obtained here by comparing them to what is reported in \citet[Table 1]{winkelmann1995duration}.
Bootstrap
Theory
The type of bootstrap used in Countr is known as nonparametric or bootstrap
pairs. It is valid under the assumption that ((y_i, \mathbf{x}_i)) is iid. The
algorithm works as follows:
- Generate a pseudo-sample of size (n), ((y^_l, \mathbf{x}^_l), \ l=1, \dots, n), by sampling with replacement from the original pairs ((y_i, \mathbf{x}_i), \ i =1, \dots, n).
- Compute the estimator (\hat{\bm{\theta}}_1) from the pseudo-sample generated in 1.
- Repeat steps 1 and 2 (B) times giving (B) estimates (\hat{\bm{\theta}}_1, \dots, \hat{\bm{\theta}}_B).
- The bootstrap estimate of the variance-covariance matrix is given by
(\hat{\mathbf{V}}{Boot}[\bm{\hat{\theta}}] = \frac{1}{B -1} \sum{b=1}^B(\hat{\bm{\theta}}b - \bar{\bm{\theta}})(\hat{\bm{\theta}}_b - \bar{\bm{\theta}})') where (\bar{\bm{\theta}} = [\bar{\theta}_1, \dots, \bar{\theta}_q]) and (\bar{\theta}_j) is the sample mean (\bar{\theta}_j = (1/B) \sum{b=1}^B \hat{\bm{\theta}}_{j,b}).
Asymptotically ((B \rightarrow \infty)), the bootstrap variance-covariance matrix and standard errors are equivalent to their robust counterpart obtained by sandwich estimators. In practice, using (B=400) is usually recommended \citep[Section 2.6.4]{cameron2013regression}}
Implementation in Countr
The bootstrap computations in Countr are based on the boot() function
from the package with the same name \citep{boot2017}.
The variance-covariance matrix is again computed with the renewal method for
vcov() by specifying the argument method = "boot". The computation can be
further customised by passing other options accepted by boot() other than
data and statistic which are provided by the Countr code. Note that the
matrix is only computed if it is not found in the passed renewal object. The
bootstrap sample is actually computed by a separate function
addBootSampleObject(), which computes the bootstrap sample and adds it as
component boot to the renewal object. Functions like vcov() and confint()
check if a bootstrap sample is already available and use it is. It is a good
idea to call addBootSampleObject() before attempting computations based on
bootstrapping. We show below how to update the previously fitted gamma model
with 400 bootstrap iterations using the parallel option and 14 CPUs. if (B) is
large and depending on how fast the model can be fitted, this computation may be
time consuming.
gam <- addBootSampleObject(gam, R = 400, parallel = "multicore", ncpus = 14)
Once the object is updated, the variance-covariance matrix is computed by vcov
in a straightforward way:
gam$vcov <- matrix()
varCovar <- vcov(gam, method = "boot")
varCovar
rate_ rate_germanyes rate_years_school
rate_ 0.07318764 5.0060e-03 -5.6704e-03
rate_germanyes 0.00500601 3.7300e-03 -6.9972e-04
rate_years_school -0.00567041 -6.9972e-04 7.1876e-04
rate_voc_trainyes 0.00087763 -4.3825e-04 -1.6477e-04
rate_universityyes 0.01675906 1.8472e-03 -2.2823e-03
rate_ReligionCatholic -0.00072461 -9.5183e-04 -5.0782e-05
rate_ReligionMuslim -0.00350408 1.1549e-03 1.3725e-04
rate_ReligionProtestant -0.00062891 -1.3197e-03 -4.8670e-05
rate_year_birth -0.00018173 -3.8643e-05 5.6028e-06
rate_ruralyes -0.00197542 -2.2663e-04 1.1529e-04
rate_age_marriage -0.00036158 6.7750e-05 -1.8638e-05
shape_ 0.00860625 1.1773e-05 6.3172e-05
rate_voc_trainyes rate_universityyes
rate_ 8.7763e-04 1.6759e-02
rate_germanyes -4.3825e-04 1.8472e-03
rate_years_school -1.6477e-04 -2.2823e-03
rate_voc_trainyes 1.4116e-03 1.6609e-03
rate_universityyes 1.6609e-03 1.5801e-02
rate_ReligionCatholic 5.5154e-05 5.2001e-04
rate_ReligionMuslim 2.0904e-04 4.9984e-04
rate_ReligionProtestant 9.5869e-06 5.3275e-04
rate_year_birth 9.8568e-06 3.1314e-06
rate_ruralyes 3.5200e-04 3.9912e-04
rate_age_marriage -3.0112e-06 3.0646e-05
shape_ 4.7749e-04 4.1102e-04
rate_ReligionCatholic rate_ReligionMuslim
rate_ -7.2461e-04 -3.5041e-03
rate_germanyes -9.5183e-04 1.1549e-03
rate_years_school -5.0782e-05 1.3725e-04
rate_voc_trainyes 5.5154e-05 2.0904e-04
rate_universityyes 5.2001e-04 4.9984e-04
rate_ReligionCatholic 3.1847e-03 1.6360e-03
rate_ReligionMuslim 1.6360e-03 4.6887e-03
rate_ReligionProtestant 2.7871e-03 1.4184e-03
rate_year_birth -2.4024e-06 -1.7800e-05
rate_ruralyes 2.1452e-04 3.7533e-04
rate_age_marriage -3.0908e-05 1.7879e-05
shape_ -6.7137e-05 8.1247e-05
rate_ReligionProtestant rate_year_birth rate_ruralyes
rate_ -6.2891e-04 -1.8173e-04 -1.9754e-03
rate_germanyes -1.3197e-03 -3.8643e-05 -2.2663e-04
rate_years_school -4.8670e-05 5.6028e-06 1.1529e-04
rate_voc_trainyes 9.5869e-06 9.8568e-06 3.5200e-04
rate_universityyes 5.3275e-04 3.1314e-06 3.9912e-04
rate_ReligionCatholic 2.7871e-03 -2.4024e-06 2.1452e-04
rate_ReligionMuslim 1.4184e-03 -1.7800e-05 3.7533e-04
rate_ReligionProtestant 3.7010e-03 1.6907e-06 1.8705e-04
rate_year_birth 1.6907e-06 3.7486e-06 -2.0782e-07
rate_ruralyes 1.8705e-04 -2.0782e-07 1.0425e-03
rate_age_marriage -4.3948e-05 -2.2939e-06 1.2680e-05
shape_ -2.3479e-04 -2.4092e-05 1.3058e-04
rate_age_marriage shape_
rate_ -3.6158e-04 8.6063e-03
rate_germanyes 6.7750e-05 1.1773e-05
rate_years_school -1.8638e-05 6.3172e-05
rate_voc_trainyes -3.0112e-06 4.7749e-04
rate_universityyes 3.0646e-05 4.1102e-04
rate_ReligionCatholic -3.0908e-05 -6.7137e-05
rate_ReligionMuslim 1.7879e-05 8.1247e-05
rate_ReligionProtestant -4.3948e-05 -2.3479e-04
rate_year_birth -2.2939e-06 -2.4092e-05
rate_ruralyes 1.2680e-05 1.3058e-04
rate_age_marriage 3.0042e-05 8.9316e-05
shape_ 8.9316e-05 1.2840e-02
Bootstrap standard errors are also very easy to compute by calling se.coef()
with argument type"boot"=. As discussed before, if the boot object is not
found in the renewal object, users can customise the boot() call by
passing the appropriate arguments in ....
se_boot <- se.coef(gam, type = "boot")
se_boot
rate_ rate_germanyes rate_years_school
0.2705321 0.0610739 0.0268098
rate_voc_trainyes rate_universityyes rate_ReligionCatholic
0.0375714 0.1257020 0.0564330
rate_ReligionMuslim rate_ReligionProtestant rate_year_birth
0.0684739 0.0608356 0.0019361
rate_ruralyes rate_age_marriage shape_
0.0322879 0.0054811 0.1133149
Finally bootstrap confidence intervals can also be computed by confint()
using the same logic described for se.coef(). Different types of
confidence intervals are available (default is normal) and can be selected by
choosing the appropriate type in bootType. We refer the user to the boot
package \citep{boot2017} for more information.
ci_boot <- confint(gam, level = 0.95, type = "boot", bootType = "norm")
ci_boot
Bootstrap quantiles, type = normal
2.5 % 97.5 %
rate_ 1.0105534 2.0710199
rate_germanyes -0.3116866 -0.0722812
rate_years_school -0.0176548 0.0874376
rate_voc_trainyes -0.2211511 -0.0738740
rate_universityyes -0.4072628 0.0854802
rate_ReligionCatholic 0.0985243 0.3197377
rate_ReligionMuslim 0.3871631 0.6555759
rate_ReligionProtestant -0.0133962 0.2250750
rate_year_birth -0.0016933 0.0058961
rate_ruralyes -0.0098069 0.1167594
rate_age_marriage -0.0401915 -0.0187062
shape_ 1.1860640 1.6302501
Save Image :ignoreheading:
We conclude this analysis by saving the work space to avoid re-running the computation in future exportation of the document:
save.image()
Bibliography :ignoreheading:
\bibliographystyle{apalike} \bibliography{REFERENCES}
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