Maximum A Posteriori estimation
MAP.estimation.RdMAP.estimation function is used (in local centers) to compute Maximum A Posterior (MAP) estimators of the parameters for Generalized Linear Models (GLM) and Survival models.
Usage
MAP.estimation(y,
X,
family = c("gaussian", "binomial", "survival"),
Lambda,
intercept = TRUE,
initial = NULL,
basehaz = c("weibul", "exp", "gomp", "poly", "pwexp", "unspecified"),
alpha = 0.1,
max_order = 2,
n_intervals = 4,
min_max_times,
center_zero_sample = FALSE,
zero_sample_cov,
refer_cat,
zero_cat,
control = list())Arguments
- y
response vector. If the “
binomial” family is used, this argument is a vector with entries 0 (failure) or 1 (success). Alternatively, for this family, the response can be a matrix where the first column is the number of “successes” and the second column is the number of “failures”. For the “survival” family, the response is a matrix where the first column is the survival time, named “time”, and the second column is the censoring indicator, named “status”, with 0 indicating censoring time and 1 indicating event time.- X
design matrix of dimension \(n \times p\), where \(p\) is the number of covariates or predictors. Note that the order of the covariates must be the same across the centers; otherwise, the output estimates of bfi() will be incorrect.
- family
a description of the error distribution. This is a character string naming a family of the model. In the current version of the package, the family of model can be
"gaussian"(withidentitylink function),"binomial"(withlogitlink function), or"survival". Can be abbreviated. By default thegaussianfamily is used. In case of a linear regression model,family = "gaussian", there is an extra model parameter for the variance of measurement error. While in the case of survival model,family = "survival", the number of the model parameters depend on the choice of baseline hazard functions, see ‘Details’ for more information.- Lambda
the inverse variance-covariance matrix of the Gaussian distribution that is used as prior distribution for the model parameters. The dimension of the matrix depends on the number of columns of
X, type of the covariates (continuous / dichotomous or categorical),family, and whether aninterceptis included (if applicable). However,Lambdacan be easily created byinv.prior.cov(). Seeinv.prior.covfor more information.- intercept
logical flag for fitting an intercept. If
intercept=TRUE(the default), the intercept is fitted, i.e., it is included in the model, and ifintercept=FALSEit is set to zero, i.e., it's not in the model. This argument is not used iffamily = "survival".- initial
a vector specifying initial values for the parameters to be optimized over. The length of
initialis equal to the number of model parameters and thus, is equal to the number of rows or columns ofLambda. Since the'L-BFGS-B'method is used in the algorithm, these values should always be finite. Default is a vector of zeros, except for thesurvivalfamily with thepolyfunction, where it is a vector with the first \(p\) elements as zeros for coefficients (\(\boldsymbol{\beta}\)) and -0.5 for the remaining parameters (\(\boldsymbol{\omega}\)). For thegaussianfamily, the last element of theinitialvector could also be considered negative, because the Gaussian prior was applied to \(log(\sigma^2)\).- basehaz
a character string representing one of the available baseline hazard functions;
exponential("exp"),Weibull("weibul", the default),Gompertz("gomp"),exponentiated polynomial("poly"),piecewise constant exponential("pwexp"), andunspecified baseline hazard("unspecified"). Can be abbreviated. It is only used whenfamily = "survival". If local sample size is large and the shape of the baseline hazard function is completely unknown, the “exponentiated polynomial” and “piecewise exponential” hazard functions would be preferred above the lower dimensional alternatives. However, if the local samples size is low, one should be careful using the “piecewise exponential” hazard function with many intervals.- alpha
a significance level used in the chi-squared distribution (with one degree of freedom and 1-\(\alpha\) representing the upper quantile) to conduct a likelihood ratio test for obtaining the order of the exponentiated polynomial baseline hazard function. It is only used when
family = "survival"andbasehaz = "poly". Default is 0.1. See ‘Details’.- max_order
an integer representing the maximum value of
q_l, which is the order/degree minus 1 of the exponentiated polynomial baseline hazard function. This argument is only used whenfamily = "survival"andbasehaz = "poly". Default is 2.- n_intervals
an integer representing the number of intervals in the piecewise exponential baseline hazard function. This argument is only used when
family = "survival"andbasehaz = "pwexp". Default is 4.- min_max_times
a scalar representing the minimum of the maximum event times observed in the centers. The value of this argument should be defined by the central server (which has access to the maximum event times of all the centers) and is only used when
family = "survival"andbasehaz = "pwexp".- center_zero_sample
logical flag indicating whether the center has a categorical covariate with no observations/individuals in one of the categories. Default is
center_zero_sample = FALSE.- zero_sample_cov
either a character string or an integer representing the categorical covariate that has no samples/observations in one of its categories. This covariate should have at least two categories, one of which is the reference. It is used when
center_zero_sample = TRUE.- refer_cat
a character string representing the reference category. The category with no observations (the argument
zero_cat) cannot be used as the reference in the argumentrefer_cat. It is used whencenter_zero_sample = TRUE.- zero_cat
a character string representing the category with no samples/observations. It is used when
center_zero_sample = TRUE.- control
a list of control parameters. See ‘Details’.
Value
MAP.estimation returns a list containing the following components:
- theta_hat
the vector corresponding to the maximum a posteriori (MAP) estimates of the parameters. For the
gaussianfamily, the last element of this vector is \(\sigma^2\);- A_hat
minus the curvature (or Hessian) matrix around the point
theta_hat. The dimension of the matrix is the same as the argumentLambda;- sd
the vector of standard deviation of the MAP estimates in
theta_hat, that issqrt(diag(solve(A_hat)));- Lambda
the inverse variance-covariance matrix of the Gaussian distribution that is used as prior distribution for the parameters. It's exactly the same as the argument
Lambda;- formula
the formula applied;
- names
the names of the model parameters;
- n
sample size;
- np
the number of coefficients;
- q_l
the order/degree minus 1 of the exponentiated polynomial baseline hazard function determined for the current center by the likelihood ratio test. This output argument,
q_l, is only shown whenfamily = "survival"andbasehaz = "poly", and will be used in the functionbfi();- theta_A_poly
an array where the first component is a matrix with columns representing the MAP estimates of the parameters for different
q_l's, i.e.,q_l,q_l+1, ...,max_order. The other components are minus the curvature matrices for differentq_l's, i.e.,q_l,q_l+1, ...,max_order. Therefore, the first non-NA curvature matrix is equal to the output argumentA_hat. This output argument,theta_A_poly, is only shown iffamily = "survival"andbasehaz = "poly", and will be used in the functionbfi();- lev_no_ref_zer
a vector containing the names of the levels of the categorical covariate that has no samples/observations in one of its categories. The name of the category with no samples and the name of the reference category are excluded from this vector. This argument is shown when
family = "survival"andbasehaz = "poly", and will be used in the functionbfi();- zero_sample_cov
the categorical covariate that has no samples/observations in one of its categories. It is shown when
center_zero_sample = TRUE, and can be used in the functionbfi();- refer_cat
the reference category. It is shown when
center_zero_sample = TRUE, and can be used in the functionbfi();- zero_cat
the category with no samples/observations. It is shown when
center_zero_sample = TRUE, and can be used in the functionbfi();- value
the value of minus the log-likelihood posterior density evaluated at
theta_hat;- family
the
familyused;- basehaz
the baseline hazard function used;
- intercept
logical flag used to fit an intercept if
TRUE, or set to zero ifFALSE;- convergence
an integer value used to encode the warnings and the errors related to the algorithm used to fit the model. The values returned are:
- 0
algorithm has converged;
- 1
maximum number of iterations ('
maxit') has been reached;- 2
Warning from the 'L-BFGS-B' method. See the message after this value;
- control
the list of control parameters used to compute the MAP estimates.
Details
MAP.estimation function finds the Maximum A Posteriori (MAP) estimates of the model parameters by maximizing the log-posterior density with respect to the parameters, i.e., the estimates equal the values for which the log-posterior density is maximal (the posterior mode).
In other words, MAP.estimation() optimizes the log-posterior density with respect to the parameter vector to obtain its MAP estimates.
In addition to the model parameters (i.e., coefficients (\(\boldsymbol{\beta}\)) and variance error (\(\sigma^2_e\)) for gaussian or the parameters of the baseline hazard (\(\boldsymbol{\omega}\)) for survival), the curvature matrix (Hessian of the log-posterior) is estimated around the mode.
The MAP.estimation function returns an object of class `bfi`. Therefore, summary() can be used for the object returned by MAP.estimation().
For the case where family = "survival" and basehaz = "poly", we assume that in all centers the \(q_\ell\)'s are equal.
However, the order of the estimated polynomials may vary across the centers so that each center can have different number of parameters, say \(q_\ell\)+1.
After obtaining the estimates within the local centers (by using MAP.estimation()) and having all estimates in the central server, we choose the order of the polynomial approximation for the combined data to be the maximum of the orders of the local polynomial functions, i.e., \(\max \{q_1, \ldots, q_L \}\), to approximate the global baseline hazard (exponentiated polynomial) function more accurately. This is because the higher-order polynomial approximation can capture more complex features and details in the combined data. Using the higher-order approximation ensures that we account for the higher-order moments and features present in the data while maintaining accuracy.
As a result, all potential cases are stored in the theta_A_poly argument to be used in bfi() by the central server.
For further information on the survival family, refer to the 'References' section.
To solve unconstrained and bound-constrained optimization problems, the MAP.estimation function utilizes an optimization algorithm called Limited-memory Broyden-Fletcher-Goldfarb-Shanno with Bound Constraints (L-BFGS-B), Byrd et. al. (1995).
The L-BFGS-B algorithm is a limited-memory “quasi-Newton” method that iteratively updates the parameter estimates by approximating the inverse Hessian matrix using gradient information from the history of previous iterations. This approach allows the algorithm to approximate the curvature of the posterior distribution and efficiently search for the optimal solution, which makes it computationally efficient for problems with a large number of variables.
By default, the algorithm uses a relative change in the objective function as the convergence criterion. When the change in the objective function between iterations falls below a certain threshold (`factr`) the algorithm is considered to have converged.
The convergence can be checked with the argument convergence in the output. See ‘Value’.
In case of convergence issue, it may be necessary to investigate and adjust optimization parameters to facilitate convergence. It can be done using the initial and control arguments. By the argument initial the initial points of the interative optimization algorithm can be changed, and the argument control is a list that can supply any of the following components:
maxit:is the maximum number of iterations. Default is 150;
factr:controls the convergence of the
'L-BFGS-B'method. Convergence occurs when the reduction in the objective is within this factor of the machine tolerance. Default forfactris 1e7, which gives a tolerance of about 1e-9. The exact tolerance can be checked by multiplying this value by.Machine$double.eps;pgtol:helps to control the convergence of the
'L-BFGS-B'method. It is a tolerance on the projected gradient in the current search direction, i.e., the iteration will stop when the maximum component of the projected gradient is less than or equal topgtol, wherepgtol\(\geq 0\). Default is zero, when the check is suppressed;trace:is a non-negative integer. If positive, tracing information on the progress of the optimization is produced. Higher values may produce more tracing information: for the method
'L-BFGS-B'there are six levels of tracing. To understand exactly what these do see the source code ofoptimfunction in the stats package;REPORT:is the frequency of reports for the
'L-BFGS-B'method if'control$trace'is positive. Default is every 10 iterations;lmm:is an integer giving the number of
BFGSupdates retained in the'L-BFGS-B'method. Default is 5.
References
Jonker M.A., Pazira H. and Coolen A.C.C. (2024). Bayesian federated inference for estimating statistical models based on non-shared multicenter data sets, Statistics in Medicine, 43(12): 2421-2438. <https://doi.org/10.1002/sim.10072>
Pazira H., Massa E., Weijers J.A.M., Coolen A.C.C. and Jonker M.A. (2024). Bayesian Federated Inference for Survival Models, arXiv. <https://arxiv.org/abs/2404.17464>
Jonker M.A., Pazira H. and Coolen A.C.C. (2024b). Bayesian Federated Inference for regression models with heterogeneous multi-center populations, arXiv. <https://arxiv.org/abs/2402.02898>
Byrd R.H., Lu P., Nocedal J. and Zhu C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing, 16, 1190-1208. <https://doi.org/10.1137/0916069>
Author
Hassan Pazira and Marianne Jonker
Maintainer: Hassan Pazira hassan.pazira@radboudumc.nl
See also
bfi, inv.prior.cov and summary.bfi
Examples
###--------------###
### y ~ Gaussian ###
###--------------###
# Setting a seed for reproducibility
set.seed(11235)
# model parameters: coefficients and sigma2 = 1.5
theta <- c(1, 2, 2, 2, 1.5)
#----------------
# Data Simulation
#----------------
n <- 30 # sample size
p <- 3 # number of coefficients without intercept
X <- data.frame(matrix(rnorm(n * p), n, p)) # continuous variables
# linear predictor:
eta <- theta[1] + theta[2] * X$X1 + theta[3] * X$X2 + theta[4] * X$X3
# inverse of the link function ( g^{-1}(\eta) = \mu ):
mu <- gaussian()$linkinv(eta)
y <- rnorm(n, mu, sd = sqrt(theta[5]))
# Load the BFI package
library(BFI)
#-----------------------------------------------
# MAP estimations for theta and curvature matrix
#-----------------------------------------------
# MAP estimates with 'intercept'
Lambda <- inv.prior.cov(X, lambda = c(0.1, 1), family = "gaussian")
(fit <- MAP.estimation(y, X, family = "gaussian", Lambda))
#> Error in MAP.estimation(y, X, family = "gaussian", Lambda): The algorithm did not converge.
class(fit)
#> Error: object 'fit' not found
summary(fit, cur_mat = TRUE)
#> Error: object 'fit' not found
# MAP estimates without 'intercept'
Lambda <- inv.prior.cov(X, lambda = c(0.1, 1), family = 'gaussian', intercept = FALSE)
(fit1 <- MAP.estimation(y, X, family = 'gaussian', Lambda, intercept = FALSE))
#> Error in MAP.estimation(y, X, family = "gaussian", Lambda, intercept = FALSE): The algorithm did not converge.
summary(fit1, cur_mat = TRUE)
#> Error: object 'fit1' not found
###-----------------###
### Survival family ###
###-----------------###
# Setting a seed for reproducibility
set.seed(112358)
#-------------------------
# Simulating Survival data
#-------------------------
n <- 100
beta <- 1:4
p <- length(beta)
X <- data.frame(matrix(rnorm(n * p), n, p)) # continuous (normal) variables
## Simulating survival data from Weibull with a predefined censoring rate of 0.3
y <- surv.simulate(Z = list(X), beta = beta, a = 5, b = exp(1.8), u1 = 0.1,
cen_rate = 0.3, gen_data_from = "weibul")$D[[1]][, 1:2]
##
## MAP estimations with "weibul" function
##
Lambda <- inv.prior.cov(X, lambda = c(0.1, 1), family = 'survival', basehaz = "weibul")
fit2 <- MAP.estimation(y, X, family = 'survival', Lambda = Lambda, basehaz = "weibul")
fit2
#> $theta_hat
#> X1 X2 X3 X4 omega_1 omega_2
#> 1.216502 2.433061 3.305429 4.895585 1.411903 1.905060
#>
#> $A_hat
#> X1 X2 X3 X4 omega_1 omega_2
#> X1 54.813643 8.232601 -3.813501 -5.643886 7.220449 -49.8473
#> X2 8.232601 85.409611 -5.039660 -16.200600 15.679748 -130.8181
#> X3 -3.813501 -5.039660 57.371520 4.560337 -2.868448 -186.1929
#> X4 -5.643886 -16.200600 4.560337 56.028421 23.120562 -262.4108
#> omega_1 7.220449 15.679748 -2.868448 23.120562 77.588076 -229.2848
#> omega_2 -49.847299 -130.818053 -186.192918 -262.410849 -229.284799 2656.6802
#>
#> $sd
#> X1 X2 X3 X4 omega_1 omega_2
#> 0.15896950 0.19980633 0.26038338 0.36632933 0.14912596 0.06994841
#>
#> $Lambda
#> X1 X2 X3 X4 omega_1 omega_2
#> X1 0.1 0.0 0.0 0.0 0 0
#> X2 0.0 0.1 0.0 0.0 0 0
#> X3 0.0 0.0 0.1 0.0 0 0
#> X4 0.0 0.0 0.0 0.1 0 0
#> omega_1 0.0 0.0 0.0 0.0 1 0
#> omega_2 0.0 0.0 0.0 0.0 0 1
#>
#> $formula
#> [1] "Survival(time, status) ~ X1 + X2 + X3 + X4"
#>
#> $names
#> [1] "X1" "X2" "X3" "X4" "omega_1" "omega_2"
#>
#> $n
#> [1] 100
#>
#> $np
#> [1] 4
#>
#> $zero_sample_cov
#> NULL
#>
#> $refer_cat
#> NULL
#>
#> $zero_cat
#> NULL
#>
#> $value
#> [1] -61.88167
#>
#> $family
#> [1] "survival"
#>
#> $basehaz
#> [1] "weibul"
#>
#> $intercept
#> [1] FALSE
#>
#> $convergence
#> [1] 0
#>
#> $control
#> $control$maxit
#> [1] 100
#>
#>
#> attr(,"class")
#> [1] "bfi"
summary(fit2, cur_mat = TRUE)
#>
#> Summary of the local model:
#>
#> Formula: Survival(time, status) ~ X1 + X2 + X3 + X4
#> Family: ‘survival’
#> Baseline: ‘weibul’
#>
#> Coefficients:
#>
#> Estimate Std.Dev CI 2.5% CI 97.5%
#> X1 1.2165 0.1590 0.9049 1.5281
#> X2 2.4331 0.1998 2.0414 2.8247
#> X3 3.3054 0.2604 2.7951 3.8158
#> X4 4.8956 0.3663 4.1776 5.6136
#> omega_1 1.4119 0.1491 1.1196 1.7042
#> omega_2 1.9051 0.0699 1.7680 2.0422
#>
#> log Lik Posterior: 61.88
#> Convergence: 0
#>
#> Minus the Curvature Matrix:
#>
#> X1 X2 X3 X4 omega_1 omega_2
#> X1 54.8136 8.2326 -3.8135 -5.6439 7.2204 -49.8473
#> X2 8.2326 85.4096 -5.0397 -16.2006 15.6797 -130.8181
#> X3 -3.8135 -5.0397 57.3715 4.5603 -2.8684 -186.1929
#> X4 -5.6439 -16.2006 4.5603 56.0284 23.1206 -262.4108
#> omega_1 7.2204 15.6797 -2.8684 23.1206 77.5881 -229.2848
#> omega_2 -49.8473 -130.8181 -186.1929 -262.4108 -229.2848 2656.6802
##
## MAP estimations with "poly" function
##
Lambda <- inv.prior.cov(X, lambda = c(0.1, 1), family = 'survival', basehaz = 'poly')
fit3 <- MAP.estimation(y, X, family = "survival", Lambda = Lambda, basehaz = "poly")
# Degree of the exponentiated polynomial baseline hazard
fit3$q_l + 1
#> [1] 3
# theta_hat for (beta_1, ..., beta_p, omega_0, ..., omega_{q_l})
fit3$theta_A_poly[,,1][,fit3$q_l+1] # equal to fit3$theta_hat
#> X1 X2 X3 X4 omega_0 omega_1 omega_2
#> 0.4587444 0.9065738 1.2387261 1.8013492 -2.0743547 4.7662933 -1.0941826
# A_hat
fit3$theta_A_poly[,,fit3$q_l+2] # equal to fit3$A_hat
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> X1 67.2578299 3.698096 -6.328249 -0.830014 7.296302 1.005843
#> X2 3.6980959 85.034720 3.130474 -6.311998 15.832762 -1.055534
#> X3 -6.3282486 3.130474 73.449859 11.351621 -2.661869 -25.299425
#> X4 -0.8300140 -6.311998 11.351621 68.043358 23.430052 -8.113705
#> omega_0 7.2963020 15.832762 -2.661869 23.430052 90.074528 43.191997
#> omega_1 1.0058432 -1.055534 -25.299425 -8.113705 43.191997 57.713003
#> omega_2 -0.7626327 -7.194974 -42.325003 -19.174732 48.713003 71.822389
#> [,7]
#> X1 -0.7626327
#> X2 -7.1949742
#> X3 -42.3250033
#> X4 -19.1747319
#> omega_0 48.7130029
#> omega_1 71.8223891
#> omega_2 122.9402820
summary(fit3, cur_mat = TRUE)
#>
#> Summary of the local model:
#>
#> Formula: Survival(time, status) ~ X1 + X2 + X3 + X4
#> Family: ‘survival’
#> Baseline: ‘poly’
#>
#> Coefficients:
#>
#> Estimate Std.Dev CI 2.5% CI 97.5%
#> X1 0.4587 0.1242 0.2153 0.7022
#> X2 0.9066 0.1159 0.6793 1.1338
#> X3 1.2387 0.1364 0.9713 1.5061
#> X4 1.8013 0.1460 1.5153 2.0874
#> omega_0 -2.0744 0.1647 -2.3971 -1.7516
#> omega_1 4.7663 0.2867 4.2044 5.3282
#> omega_2 -1.0942 0.1826 -1.4520 -0.7364
#>
#> log Lik Posterior: -1.084
#> Convergence: 0
#>
#> Minus the Curvature Matrix:
#>
#> X1 X2 X3 X4 omega_0 omega_1 omega_2
#> X1 67.2578 3.6981 -6.3282 -0.8300 7.2963 1.0058 -0.7626
#> X2 3.6981 85.0347 3.1305 -6.3120 15.8328 -1.0555 -7.1950
#> X3 -6.3282 3.1305 73.4499 11.3516 -2.6619 -25.2994 -42.3250
#> X4 -0.8300 -6.3120 11.3516 68.0434 23.4301 -8.1137 -19.1747
#> omega_0 7.2963 15.8328 -2.6619 23.4301 90.0745 43.1920 48.7130
#> omega_1 1.0058 -1.0555 -25.2994 -8.1137 43.1920 57.7130 71.8224
#> omega_2 -0.7626 -7.1950 -42.3250 -19.1747 48.7130 71.8224 122.9403
##
## MAP estimations with "pwexp" function with 3 intervals
##
# Assume we have 4 centers
Lambda <- inv.prior.cov(X, lambda = c(0.1, 1), family = 'survival',
basehaz = 'pwexp', n_intervals = 3)
# For this baseline hazard ("pwexp"), we need to know
# maximum survival times of the 3 other centers:
max_times <- c(max(rexp(30)), max(rexp(50)), max(rexp(70)))
# Minimum of the maximum values of the survival times of all 4 centers is:
min_max_times <- min(max(y$time), max_times)
fit4 <- MAP.estimation(y, X, family = "survival", Lambda = Lambda, basehaz = "pwexp",
n_intervals = 3, min_max_times=max(y$time))
#>
#> No. observations in the intervals : 73 18 9
#>
summary(fit4, cur_mat = TRUE)
#>
#> Summary of the local model:
#>
#> Formula: Survival(time, status) ~ X1 + X2 + X3 + X4
#> Family: ‘survival’
#> Baseline: ‘pwexp’
#>
#> Coefficients:
#>
#> Estimate Std.Dev CI 2.5% CI 97.5%
#> X1 0.3310 0.1353 0.0658 0.5961
#> X2 0.6246 0.1242 0.3812 0.8680
#> X3 0.7795 0.1377 0.5097 1.0493
#> X4 1.1811 0.1588 0.8699 1.4924
#> omega_1 -0.3968 0.1787 -0.7471 -0.0465
#> omega_2 1.4806 0.2910 0.9104 2.0509
#> omega_3 1.4282 0.6395 0.1747 2.6817
#>
#> log Lik Posterior: -27.06
#> Convergence: 0
#>
#> Minus the Curvature Matrix:
#>
#> X1 X2 X3 X4 omega_1 omega_2 omega_3
#> X1 58.2386 2.0588 -8.1905 -0.7741 7.5573 -0.4890 0.2409
#> X2 2.0588 74.7939 1.2829 -3.3496 16.9213 -0.5449 -0.5155
#> X3 -8.1905 1.2829 65.1478 9.3205 9.5898 -10.3783 -1.8274
#> X4 -0.7741 -3.3496 9.3205 64.3901 31.2096 -6.2708 -1.4469
#> omega_1 7.5573 16.9213 9.5898 31.2096 54.3965 0.0000 0.0000
#> omega_2 -0.4890 -0.5449 -10.3783 -6.2708 0.0000 14.5194 0.0000
#> omega_3 0.2409 -0.5155 -1.8274 -1.4469 0.0000 0.0000 2.5716
##
## Semi-parametric Cox model
##
Lambda <- inv.prior.cov(X, lambda = c(0.1), family = 'survival', basehaz = "unspecified")
fit5 <- MAP.estimation(y, X, family = 'survival', Lambda = Lambda, basehaz = "unspecified")
summary(fit5, cur_mat = TRUE)
#>
#> Summary of the local model:
#>
#> Formula: Survival(time, status) ~ X1 + X2 + X3 + X4
#> Family: ‘survival’
#> Baseline: ‘unspecified’
#>
#> Coefficients:
#>
#> Estimate Std.Dev CI 2.5% CI 97.5%
#> X1 1.3040 0.1934 0.9249 1.6831
#> X2 2.4894 0.2806 1.9394 3.0394
#> X3 3.2086 0.3418 2.5387 3.8785
#> X4 5.0264 0.5237 4.0001 6.0528
#>
#> log Lik Posterior: -143.5
#> Convergence: 0
#>
#> Minus the Curvature Matrix:
#>
#> X1 X2 X3 X4
#> X1 51.7496 3.0205 -6.6147 -10.8334
#> X2 3.0205 57.6207 -11.4154 -21.3849
#> X3 -6.6147 -11.4154 34.9165 -12.4757
#> X4 -10.8334 -21.3849 -12.4757 23.3777