bayessurvreg2           package:bayesSurv           R Documentation

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_D_e_s_c_r_i_p_t_i_o_n:

     A function to estimate a regression model with possibly clustered
     (possibly right, left, interval or doubly-interval censored) data.
     In the case of doubly-interval censoring, different regression
     models can be specified for the onset and event times.

     (Multivariate) random effects, normally distributed and  acting as
     in the linear mixed model, normally distributed, can be included
     to adjust for clusters.

     The error density of the regression model is specified as a
     mixture of Bayesian G-splines (normal densities with equidistant
     means and constant variances). This function performs an MCMC
     sampling from the posterior distribution of unknown quantities.

     For details, see Kom&#225rek (2006), and Kom&#225rek, Lesaffre and
     Legrand (2007).

     We explain first in more detail a model without doubly censoring.
     Let T[i,l], i=1,..., N, l=1,..., n[i] be event times for ith
     cluster and the units within that cluster The following regression
     model is assumed:

 log(T[i,l]) = beta'x[i,l] + b[i]'z[i,l] + epsilon[i,l], i=1,..., N, l=1,..., n[i]

     where beta is unknown regression parameter vector, x[i,l] is a
     vector of covariates. b[i] is a (multivariate) cluster-specific
     random effect vector and z[i,l] is a vector of covariates for
     random effects.

     The random effect vectors b[i], i=1,..., N are assumed to be
     i.i.d. with a (multivariate) normal distribution with the mean
     beta[b] and a~covariance matrix D. Hierarchical centring (see
     Gelfand, Sahu, Carlin, 1995) is used. I.e. beta[b] expresses the
     average effect of the covariates included in z[i,l]. Note that
     covariates included in z[i,l] may not be included in the covariate
     vector x[i,l]. The covariance matrix D is assigned an inverse
     Wishart prior distribution in the next level of hierarchy.

     The error terms epsilon[i,l], i=1,..., N, l=1,..., n[i] are
     assumed to be i.i.d. with a~univariate density g[epsilon](e). This
     density is expressed as a~mixture of Bayesian G-splines (normal
     densities with equidistant means and constant variances). We
     distinguish two, theoretically equivalent, specifications.

     _S_p_e_c_i_f_i_c_a_t_i_o_n _1 .DS B epsilon is distributed as sum[j=-K][K] w[j]
          N(mu[j], sigma^2) .DE

          where sigma^2 is the *unknown* basis variance and
          mu[j],;j=-K,..., K is an~equidistant grid of knots symmetric
          around the *unknown* point gamma  and related to the unknown
          basis variance through the relationship

              mu[j] = gamma + j*delta*sigma, j=K,..., K

          where delta is fixed constants, e.g. delta=2/3 (which has
          a~justification of being close to cubic B-splines).

          .DS B

          .DE


     _S_p_e_c_i_f_i_c_a_t_i_o_n _2 .DS B epsilon[1] is distributed as alpha + tau * V
          .DE

          where alpha is an *unknown* intercept term and tau is an
          *unknown* scale parameter. V is then standardized error term
          which is distributed according to the univariate normal
          mixture, i.e.

       V is distributed as sum[j=-K][K] w[j] N(mu[j], sigma^2)

          where mu[j], j=-K,..., K is an~equidistant grid of *fixed*
          knots (means), usually symmetric about the *fixed* point
          gamma = 0 and sigma^2 is *fixed* basis variance. Reasonable
          values for the numbers of grid points K is K=15 with the
          distance between the two knots equal to delta=0.3 and for the
          basis variance sigma^2=0.2^2. 

     Personally, I found Specification 2 performing better. In the
     paper Komarek, Lesaffre and Legrand (2007) only Specification 2 is
     described.

     The mixture weights w[j], j=-K,..., K are not estimated directly.
     To avoid the constraints 0 < w[j] < 1 and sum[j=-K][K] w[j] = 1
     transformed weights a[j], j=-K,..., K related to the original
     weights by the logistic transformation:

                  a[j] = exp(w[j])/sum[m] exp(w[m])

     are estimated instead.

     A~Bayesian model is set up for all unknown parameters. For more
     details I refer to Komarek (2006) and to Komarek, Lesafre, and
     Legrand (2007).

     If there are doubly-censored data the model of the same type as
     above can be specified for both the onset time and the
     time-to-event.

_U_s_a_g_e:

     bayessurvreg2(formula, random, formula2, random2,
        data = parent.frame(),
        na.action = na.fail, onlyX = FALSE,
        nsimul = list(niter = 10, nthin = 1, nburn = 0, nwrite = 10),   
        prior, prior.beta, prior.b, init = list(iter = 0),
        mcmc.par = list(type.update.a = "slice", k.overrelax.a = 1,
                        k.overrelax.sigma = 1, k.overrelax.scale = 1),
        prior2, prior.beta2, prior.b2, init2,
        mcmc.par2 = list(type.update.a = "slice", k.overrelax.a = 1,
                         k.overrelax.sigma = 1, k.overrelax.scale = 1),                           
        store = list(a = FALSE, a2 = FALSE, y = FALSE, y2 = FALSE,
                     r = FALSE, r2 = FALSE, b = FALSE, b2 = FALSE), 
        dir = getwd())

_A_r_g_u_m_e_n_t_s:

 formula: model formula for the regression. In the case of
          doubly-censored data, this is the model formula for the onset
          time.

          The left-hand side of the 'formula' must be an~object created
          using 'Surv'.

          In the formula all covariates appearing both in the vector
          x[i,l] and z[i,l] must be mentioned. Intercept is implicitely
          included in the model by the estimation of the error
          distribution. As a~consequence '-1' in the model formula does
          not have any effect on the model specification.

          If 'random' is used then the formula must contain an
          identification of clusters in the form 'cluster(id)', where
          'id' is a name of the variable that determines clusters, e.g.

            'Surv(time, event)~gender + cluster(id)'.

  random: formula for the `random' part of the model, i.e. the part
          that specifies the covariates z[i,l]. In the case of
          doubly-censored data, this is the 'random' formula for the
          onset time.

          If omitted, no random part is included in the model. E.g. to
          specify the model with a random intercept, say 'random=~1'.
          All effects mentioned in 'random' should also be mentioned on
          the right-hand side of 'formula'.

          When some random effects are included the random intercept is
          added by default. It can be removed using e.g. 'random=~-1 +
          gender'. 

formula2: model formula for the regression of the time-to-event in the
          case of doubly-censored data. Ignored otherwise. The same
          structure as for 'formula' applies here. 

 random2: specification of the `random' part of the model for
          time-to-event in the case of doubly-censored data. Ignored
          otherwise. The same structure as for 'random' applies here. 

    data: optional data frame in which to interpret the variables
          occuring in the 'formula', 'formula2', 'random', 'random2'
          statements. 

na.action: the user is discouraged from changing the default value
          'na.fail'. 

   onlyX: if 'TRUE' no MCMC sampling is performed and only the design
          matrix (matrices) are returned. This can be useful to set up
          correctly priors for regression parameters in the presence of
          'factor' covariates. 

  nsimul: a list giving the number of iterations of the MCMC and other
          parameters of the simulation.

          _n_i_t_e_r total number of sampled values after discarding thinned
               ones, burn-up included;

          _n_t_h_i_n thinning interval;

          _n_b_u_r_n number of sampled values in a burn-up period after
               discarding thinned values. This value should be smaller
               than 'niter'. If not, 'nburn' is set to 'niter - 1'. It
               can be set to zero;

          _n_w_r_i_t_e an interval at which information about the number of
               performed iterations is print on the screen and during
               the burn-up period an interval with which the sampled
               values are writen to files;

   prior: a~list specifying the prior distribution of the G-spline
          defining the distribution of the error term in the regression
          model given by 'formula' and 'random'. See 'prior' argument
          of 'bayesHistogram' function for more detail. In this list
          also 'Specification' as described above is specified.

          The item 'prior$neighbor.system' can only be equal to
          'uniCAR' here. 

 prior.b: a list defining the way in which the random effects involved
          in 'formula' and 'random' are to be updated and the
          specification of priors for parameters related to these
          random effects. The list is assumed to have the following
          components.

          _p_r_i_o_r._D a string defining the prior distribution for the
               covariance matrix of random effects D. It can be either
               ``inv.wishart'' or ``sduniform''.

               _i_n_v._w_i_s_h_a_r_t in that case is assumed that the prior
                    distribution of the matrix D is Inverse-Wishart
                    with degrees of freedom equal to tau and a scale
                    matrix equal to S. When D is a matrix q x q a prior
                    expectation of D is equal to (1/(tau - q - 1))S if
                    tau > q + 1. For q - 1 < tau <= q + 1 a prior
                    expectation is not finite.       Degrees of freedom
                    parameter tau does not have to be an integer. It
                    has to only satisfy a condition tau > q - 1.
                    'prior.b$df.D' gives a prior degrees of freedom
                    parameter tau and 'prior.b$scale.D' determines the
                    scale matrix D. Inverse-Wishart is also the default
                    choice.

               _s_d_u_n_i_f_o_r_m this can be used only when the random effect
                    is univariate (e.g. only random intercept in the
                    model). Then the matrix D is just a scalar and the
                    prior of sqrt(D) (standard deviation of the
                    univariate random effect) is assumed to be uniform
                    on interval (0, S). The upper limit S is given by
                    'prior.b$scale.D'. 


          _d_f._D degrees of freedom parameter tau in the case that the
               prior of the matrix D is inverse-Wishart.

          _s_c_a_l_e._D a lower triangle of the scale matrix S in the case
               that the prior of the matrix D is inverse-Wishart or the
               upper limit S of the uniform distribution in the case
               that sqrt(D) ~ Unif(0, S). 

prior.beta: prior specification for the regression parameters, in the
          case of doubly-censored data for the regression parameters of
          the onset time, i.e. it is related to 'formula' and 'random'.
          Note that the 'beta' vector contains both the fixed effects
          beta and the means of the random effects (except the random
          intercept) beta[b].

          This should be a~list with the following components:

          _m_e_a_n._p_r_i_o_r a~vector specifying a~prior mean for each 'beta'
               parameter in the model.

          _v_a_r._p_r_i_o_r a~vector specifying a~prior variance for each
               'beta' parameter.

          It is recommended to run the function bayessurvreg2 first
          with its argument 'onlyX' set to 'TRUE' to find out how the
          betas are sorted. They must correspond to a design matrix X
          taken from 'formula'. 

    init: an~optional list with initial values for the MCMC related to
          the model given by 'formula' and 'random'. The list can have
          the following components:

          _i_t_e_r the number of the iteration to which the initial values
               correspond, usually zero.

          _b_e_t_a a~vector of initial values for the regression parameters
               (both the fixed effects and means of the random
               effects). It must be sorted in the same way as are the
               columns in the design matrix. Use 'onlyX=TRUE' if you do
               not know how the columns in the design matrix are
               created.

          _a a~vector of length 2*K+1 with the initial values of
               transformed mixture weights.

          _l_a_m_b_d_a initial values for the Markov random fields precision
               parameter. 

          _g_a_m_m_a an~initial values for the middle knot gamma.

               If 'Specification' is 2, this value will not be changed
               by the MCMC and it is recommended (for easier
               interpretation of the results) to set 'init$gamma' to
               zero (default behavior).

               If 'Specification' is 1 'init$gamma' should be
               approximately equal to the mean value of the residuals.

          _s_i_g_m_a an~initial values of the basis standard deviation
               sigma.

               If 'Specification' is 2, this value will not be changed
               by the MCMC and it is recommended to set it
               approximately equal to the range of standardized data
               (let say 4 + 4) divided by the number of knots and
               multiplied by something like 2/3.

               If 'Specification' is 1 this should be approximately
               equal to the range of the residuals divided by the
               number of knots (2*K+1) and multiplied again by
               something like 2/3. 

          _i_n_t_e_r_c_e_p_t an~initial values of the intercept term alpha.

               If 'Specification' is 1 this value is not changed by the
               MCMC and the initial value is always changed to zero.

          _s_c_a_l_e an~initial value of the scale parameter tau.

               If 'Specification' is 1 this value is not changed by the
               MCMC and the initial value is always changed to one.

          _D initial value for the covariance matrix of random effects
               D. Only its lower triangle must be given in a~vector,
               e.g. 'c(d[1,1], d[2,1], d[3,1], d[2,2], d[3,2], d[3,3])'
               for a matrix 3 x 3.

          _b a~vector or matrix of the initial values of random effects
               b[i],;i=1,..., N for each cluster. The matrix should be
               of size q x N, where q is the number of random effects.
               I.e. each column of the matrix contains the initial
               values for one cluster.

          _y a~vector of length sum[i=1][N] n[i] with initial values of
               log-event-times.

          _r a~vector of length sum[i=1][N] n[i] with initial component
               labels for each residual. All values must be between -K
               and K. See argument 'init' of the function
               'bayesHistogram' for more details.

mcmc.par: a~list specifying how some of the G-spline parameters related
          to the distribution of the error term from 'formula' are to
          be updated. See 'bayesBisurvreg' for more details.

          In contrast to 'bayesBisurvreg' function argument
          'mcmc.par$type.update.a' can also be equal to '"block"' in
          which case all a coefficients are updated in 1 block using
          the Metropolis-Hastings algorithm.  

  prior2: a~list specifying the prior distribution of the G-spline
          defining the distribution of the error term in the regression
          model given by 'formula2' and 'random2'. See 'prior' argument
          of 'bayesHistogram' function for more detail. 

prior.b2: prior specification for the parameters related to the random
          effects from 'formula2' and 'random2'. This should be a~list
          with the same structure as 'prior.b'. 

prior.beta2: prior specification for the regression parameters of
          time-to-event in the case of doubly censored data (related to
          'formula2' and 'random2'). This should be a~list with the
          same structure as 'prior.beta'. 

   init2: an~optional list with initial values for the MCMC related to
          the model given by 'formula2' and 'random2'. The list has the
          same structure as 'init'. 

mcmc.par2: a~list specifying how some of the G-spline parameters
          related to 'formula2' are to be updated. The list has the
          same structure as 'mcmc.par'. 

   store: a~list of logical values specifying which chains that are not
          stored by default are to be stored. The list can have the
          following components.

          _a if 'TRUE' then all the transformed mixture weights a[k],
               k=-K,..., K, related to the G-spline (error
               distribution) of 'formula' are stored.

          _a_2 if 'TRUE' and there are doubly-censored data then all the
               transformed mixture weights a[k], k=-K,..., K, related
               to the G-spline (error distribution) of 'formula2' are
               stored.

          _y if 'TRUE' then augmented log-event times for all
               observations related to the 'formula' are stored.

          _y_2 if 'TRUE' then augmented log-event times for all
               observations related to 'formula2' are stored.

          _r if 'TRUE' then labels of mixture components for residuals
               related to 'formula' are stored.

          _r_2 if 'TRUE' then labels of mixture components for residuals
               related to 'formula2' are stored.

          _b if 'TRUE' then the sampled values of the random effects
               related to 'formula' and 'random' are stored.

          _b_2 if 'TRUE' then the sampled values of the random effects
               related to 'formula2' and 'random2' are stored.

     dir: a string that specifies a directory where all sampled values
          are to be stored. 

_V_a_l_u_e:

     A list of class 'bayessurvreg2' containing an information
     concerning the initial values and prior choices.

_F_i_l_e_s _c_r_e_a_t_e_d:

     Additionally, the following files with sampled values are stored
     in a directory specified by 'dir' argument of this function (some
     of them are created only on request, see 'store' parameter of this
     function).

     Headers are written to all files created by default and to files
     asked by the user via the argument 'store'. During the burn-in,
     only every 'nsimul$nwrite' value is written. After the burn-in,
     all sampled values are written in files created by default and to
     files asked by the user via the argument 'store'. In the files for
     which the corresponding 'store' component is 'FALSE', every
     'nsimul$nwrite' value is written during the whole MCMC (this might
     be useful to restart the MCMC from some specific point).

     The following files are created:

     _i_t_e_r_a_t_i_o_n._s_i_m one column labeled 'iteration' with indeces of MCMC
          iterations to which the stored sampled values correspond.

     _m_i_x_m_o_m_e_n_t._s_i_m columns labeled 'k', 'Mean.1',  'D.1.1', where

          *k* = number of mixture components that had probability
          numerically higher than zero;

          *Mean.1* = E(epsilon[i,l]);

          *D.1.1* = var(epsilon[i,l]);

          all related to the distribution of the error term from the
          model given by 'formula'.

     _m_i_x_m_o_m_e_n_t_{}_2._s_i_m in the case of doubly-censored data, the same
          structure as 'mixmoment.sim', however related to the model
          given by 'formula2'.      

     _m_w_e_i_g_h_t._s_i_m sampled mixture weights w[k] of mixture components
          that had probabilities numerically higher than zero. Related
          to the model given by 'formula'.

     _m_w_e_i_g_h_t_{}_2._s_i_m in the case of doubly-censored data, the same
          structure as 'mweight.sim', however related to the model
          given by 'formula2'. 

     _m_m_e_a_n._s_i_m indeces k, k in {-K, ..., K} of mixture components that
          had probabilities numerically higher than zero. It
          corresponds to the weights in 'mweight.sim'. Related to the
          model given by 'formula'.

     _m_m_e_a_n_{}_2._s_i_m in the case of doubly-censored data, the same
          structure as 'mmean.sim', however related to the model given
          by 'formula2'. 

     _g_s_p_l_i_n_e._s_i_m characteristics of the sampled G-spline (distribution
          of epsilon[i,l]) related to the model given by 'formula'.
          This file together with 'mixmoment.sim', 'mweight.sim' and
          'mmean.sim' can be used to reconstruct the G-spline in each
          MCMC iteration.

          The file has columns labeled 'gamma1', 'sigma1', 'delta1',
          'intercept1',  'scale1', The meaning of the values in these
          columns is the following:

          *gamma1* = the middle knot gamma  If 'Specification' is 2,
          this column usually contains zeros;

          *sigma1* = basis standard deviation sigma of the G-spline.
          This column contains a~fixed value if 'Specification' is 2;

          *delta1* = distance delta between the two knots of the
          G-spline. This column contains a~fixed value if
          'Specification' is 2;

          *intercept1* = the intercept term alpha of the G-spline. If
          'Specification' is 1, this column usually contains zeros;

          *scale1* = the scale parameter tau of the G-spline. If
          'Specification' is 1, this column usually contains ones;

     _g_s_p_l_i_n_e_{}_2._s_i_m in the case of doubly-censored data, the same
          structure as 'gspline.sim', however related to the model
          given by 'formula2'. 

     _m_l_o_g_w_e_i_g_h_t._s_i_m fully created only if 'store$a = TRUE'. The file
          contains the transformed weights a[k], k=-K,..., K of all
          mixture components, i.e. also of components that had
          numerically zero probabilities. This file is related to the
          error distribution of the model given by 'formula'.

     _m_l_o_g_w_e_i_g_h_t_{}_2._s_i_m fully created only if 'store$a2 = TRUE' and in
          the case of doubly-censored data, the same structure as
          'mlogweight.sim', however related to the error distribution
          of the model given by 'formula2'. 

     _r._s_i_m fully created only if 'store$r = TRUE'. The file contains
          the labels of the mixture components into which the residuals
          are intrinsically assigned. Instead of indeces on the scale
          {-K,..., K} values from 1 to (2*K+1) are stored here.
          Function 'vecr2matr' can be used to transform it back to
          indices from -K to K.

     _r_{}_2._s_i_m fully created only if 'store$r2 = TRUE' and in the case
          of doubly-censored data, the same structure as 'r.sim',
          however related to the model given by 'formula2'. 

     _l_a_m_b_d_a._s_i_m one column labeled 'lambda'. These are the values of
          the smoothing parameterlambda (hyperparameters of the prior
          distribution of the transformed mixture weights a[k]). This
          file is related to the model given by 'formula'.

     _l_a_m_b_d_a_{}_2._s_i_m in the case of doubly-censored data, the same
          structure as 'lambda.sim', however related to the model given
          by 'formula2'. 

     _b_e_t_a._s_i_m sampled values of the regression parameters, both the
          fixed effects beta and means of the random effects beta_b
          (except the random intercept which has always the mean equal
          to zero). This file is related to the model given by
          'formula'. The columns are labeled according to the
          'colnames' of the design matrix.

     _b_e_t_a_{}_2._s_i_m in the case of doubly-censored data, the same
          structure as 'beta.sim', however related to the model given
          by 'formula2'. 

     _D._s_i_m sampled values of the covariance matrix D of the random
          effects. The file has 1 + 0.5*q*(q+1) columns (q is the
          dimension of the random effect vector b_i). The first column
          labeled 'det' contains the determinant of the sampled matrix,
          additional columns labeled 'D.1.1', 'D.2.1', ..., 'D.q.1',
          ... 'D.q.q' contain the lower triangle of the sampled matrix.
          This file is related to the model specified by 'formula' and
          'random'.

     _D_{}_2._s_i_m in the case of doubly-censored data, the same structure
          as 'D.sim', however related to the model given by 'formula2'
          and 'random2'.

     _b._s_i_m fully created only if 'store$b = TRUE'. It contains sampled
          values of random effects for all clusters in the data set.
          The file has q*N columns sorted as b[1,1],..., b[1,q],...,
          b[N,1],..., b[N,q]. This file is related to the model given
          by 'formula' and 'random'.

     _b_{}_2._s_i_m fully created only if 'store$b2 = TRUE' and in the case
          of doubly-censored data, the same structure as 'b.sim',
          however related to the model given by 'formula2' and
          'random2'.  

     _Y._s_i_m fully created only if 'store$y = TRUE'. It contains sampled
          (augmented) log-event times for all observations in the data
          set.

     _Y_{}_2._s_i_m fully created only if 'store$y2 = TRUE' and in the case
          of doubly-censored data, the same structure as 'Y.sim',
          however related to the model given by 'formula2'. 

     _l_o_g_p_o_s_t_e_r._s_i_m columns labeled 'loglik', 'penalty', and 'logprw'.
          This file is related to the model given by 'formula'. The
          columns have the following meaning.

          *loglik* = -(sum[i=1][N] n[i]) * (log(sqrt(2*pi)) +
          log(sigma)) -0.5*sum[i=1][N] sum[l=1][n[i]](
          (sigma^2*tau^2)^(-1) * (y[i,l] - x[i,l]'beta - z[i,l]'b[i] -
          alpha - tau*mu[r[i,l]])^2)

          where y[i,l] denotes (augmented) _(i,l)_th true log-event
          time.

          In other words, 'loglik' is equal to the conditional
          log-density

 sum[i=1][N] sum[l=1][n[i]] log(p(y[i,l] | r[i,l], beta, b[i], G-spline));

          *penalty:* the penalty term

                      -0.5*sum[k] (Delta a[k])^2

          (not multiplied by lambda);

          *logprw* = -2*(sum[i] n[i])*log(sum[k] exp(a[k])) + sum[k[1]]
          N[k]*a[k], where N[k] is the number of residuals assigned
          intrinsincally to the kth mixture component.

          In other words, 'logprw' is equal to the conditional
          log-density sum[i=1][N] sum[l=1][n[i]] log(p(r[i,l] |
          G-spline weights)).

     _l_o_g_p_o_s_t_e_r_{}_2._s_i_m in the case of doubly-censored data, the same
          structure as 'logposter.sim', however related to the model
          given by 'formula2'. 

_A_u_t_h_o_r(_s):

     Arno&#353t Kom&#225rek komarek@karlin.mff.cuni.cz

_R_e_f_e_r_e_n_c_e_s:

     Gelfand, A. E., Sahu, S. K., and Carlin, B. P. (1995). Efficient
     parametrisations for normal linear mixed models. _Biometrika,_
     *82,* 479-488.

     Kom&#225rek, A. (2006). _Accelerated Failure Time Models for
     Multivariate Interval-Censored Data with Flexible Distributional
     Assumptions_. PhD. Thesis, Katholieke Universiteit Leuven,
     Faculteit Wetenschappen.

     Kom&#225rek, A., Lesaffre, E., and Legrand, C. (2007). Baseline
     and treatment effect heterogeneity for survival times between
     centers using a random effects accelerated failure time model with
     flexible error distribution. To appear in _Statistics in
     Medicine._

_E_x_a_m_p_l_e_s:

     ## See the description of R commands for
     ## the model with EORTC data,
     ## analysis described in Komarek, Lesaffre and Legrand (2007).
     ##
     ## R commands available in the documentation
     ## directory of this package
     ## as eortc.pdf.
     ##

