bayesBisurvreg           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 bivariate (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.

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

     For details, see Komarek (2006) and Komarek and Lesaffre (2006).

     We explain first in more detail a model without doubly censoring.
     Let T[i,l], i=1,..., N, l=1, 2 be event times for ith cluster and
     the first and the second unit. The following regression model is
     assumed:

     log(T[i,l]) = beta'x[i,l] + epsilon[i,l], i=1,..., N, l=1,2

     where beta is unknown regression parameter vector and x[i,l] is a
     vector of covariates. The bivariate error terms epsilon[i] =
     (epsilon[i,1], epsilon[i,2])', i=1,..., N are assumed to be i.i.d.
     with a~bivariate density g[epsilon](e[1], e[2]). This density is
     expressed as a~mixture of Bayesian G-splines (normal densities
     with equidistant means and constant variance matrices). We
     distinguish two, theoretically equivalent, specifications.

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

          where sigma[1]^2, sigma[2]^2 are *unknown* basis variances
          and mu[(j[1],j[2])] = (mu[1,j[1]], mu[2,j[2]])' is
          an~equidistant grid of knots symmetric around the *unknown*
          point (gamma[1], gamma[2])'  and related to the unknown basis
          variances through the relationship

 mu[1,j[1]] = gamma[1] + j[1]*delta[1]*sigma[1], j[1]=-K[1],..., K[1]

 mu[2,j[2]] = gamma[2] + j[2]*delta[2]*sigma[2], j[2]=-K[2],..., K[2]

          where delta[1], delta[2] are fixed constants, e.g.
          delta[1]=delta[2]=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],,epsilon[2])' is distributed as
          (alpha[1], alpha[2])' + S (V[1], V[2])' .DE

          where (alpha[1], alpha[2])' is an *unknown* intercept term
          and S is a diagonal matrix with tau[1] and tau[2] on a
          diagonal, i.e. tau[1], tau[2] are *unknown* scale parameters.
          (V[1], V[2])' is then standardized bivariate error term which
          is distributed according to the bivariate normal mixture,
          i.e.

 (V[1], V[2])' is distributed as sum[j[1]=-K[1]][K[1]] sum[j[2]=-K[2]][K[2]] w[j[1],j[2]] N(mu[(j[1],j[2])], diag(sigma[1]^2, sigma[2]^2))

          where mu[(j[1],j[2])] = (mu[1,j[1]], mu[2,j[2]])' is
          an~equidistant grid of *fixed* knots (means), usually
          symmetric about the *fixed* point (gamma[1], gamma[2])' = (0,
          0)' and sigma[1]^2, sigma[2]^2 are *fixed* basis variances.
          Reasonable values for the numbers of grid points K[1] and
          K[2] are K[1]=K[2]=15 with the distance between the two knots
          equal to delta=0.3 and for the basis variances
          sigma[1]^2=sigma[2]^2=0.2^2. 

     Personally, I found Specification 2 performing better. In the
     paper Komarek and Lesaffre (2006) only Specification 2 is
     described.

     The mixture weights w[j[1],j[2]], j[1]=-K[1],..., K[1],
     j[2]=-K[2],..., K[2] are not estimated directly. To avoid the
     constraints 0 < w[j[1],j[2]] < 1 and
     sum[j[1]=-K[1]][K[1]]sum[j[2]=-K[2]][K[2]]w[j[1],j[2]]=1
     transformed weights  a[j[1],j[2]], j[1]=-K[1],..., K[1],
     j[2]=-K[2],..., K[2] related to the original weights by the
     logistic transformation:

 a[j[1],j[2]] = exp(w[j[1],j[2]])/sum[m[1]]sum[m[2]] exp(w[m[1],m[2]])

     are estimated instead.

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

     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:

     bayesBisurvreg(formula, formula2, data = parent.frame(),
        na.action = na.fail, onlyX = FALSE,
        nsimul = list(niter = 10, nthin = 1, nburn = 0, nwrite = 10),
        prior, prior.beta, 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, 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),
        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. Data are assumed to be sorted according to subjects and
          within subjects according to the types of the events that
          determine the bivariate survival distribution, i.e. the
          response vector must be
          t[1,1],t[1,2],t[2,1],t[2,2],t[3,1],t[3,2],...,t[n,1],t[n,2].
          The rows of the design matrix with covariates must be sorted
          analogically.

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

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

    data: optional data frame in which to interpret the variables
          occuring in the formulas. 

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'. See 'prior' argument of
          'bayesHistogram' function for more detail. In this list also
          'Specification' as described above is specified. 

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'.

          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 bayesBisurvreg 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'. 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. 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~matrix of size (2*K[1]+1) x (2*K[2]+1) with the initial
               values of transformed mixture weights.

          _l_a_m_b_d_a initial values for the Markov random fields precision
               parameters. According to the chosen prior for the
               transformed mixture weights, this is either a~number or
               a~vector of length 2.

          _g_a_m_m_a a~vector of length 2 of initial values for the middle
               knots gamma[1], gamma[2] in each dimension.

               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 for all dimensions (default behavior).

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

          _s_i_g_m_a a~vector of length~2 of initial values of the basis
               standard deviations sigma[1], sigma[2].

               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 in each
               margin 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 in each margin and multiplied again by
               something like 2/3. 

          _i_n_t_e_r_c_e_p_t a~vector of length~2 of initial values of the
               intercept terms alpha[1], alpha[2].

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

          _s_c_a_l_e a~vector of length~2 of initial values of the scale
               parameters tau[1], tau[2].

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

          _y a~matrix with 2 columns and N rows with initial values of
               log-event-times for each cluster in rows.

          _r a~matrix with 2 columns and N rows with initial component
               labels for each bivariate residual in rows. All values
               in the first column must be between -K[1] and K[1] and
               all values in the second column must be between -K[2]
               and K[2]. See argument 'init' of the function
               'bayesHistogram' for more details. .in -5

mcmc.par: a~list specifying how some of the G-spline parameters related
          to 'formula' are to be updated. The list can have the
          following components (all of them have their default values):

          _t_y_p_e._u_p_d_a_t_e._a G-spline transformed weights a can be updated
               using one of the following algorithms:

               _s_l_i_c_e slice sampler of Neal (2003)

               _a_r_s._q_u_a_n_t_i_l_e adaptive rejection sampling of Gilks and
                    Wild (1992) with starting abscissae being quantiles
                    of the envelop at the previous iteration

               _a_r_s._m_o_d_e adaptive rejection sampling of Gilks and Wild
                    (1992) with starting abscissae being the mode
                    plus/minus 3 times estimated standard deviation of
                    the full conditional distribution

               Default is 'slice'.

          _k._o_v_e_r_r_e_l_a_x._a if 'type.update.a == "slice"' some updates are
               overrelaxed. Then every 'k.overrelax.a'th iteration is
               not overrelaxed. Default is 'k.overrelax.a = 1', i.e. no
               overrelaxation

          _k._o_v_e_r_r_e_l_a_x._s_i_g_m_a G-spline basis standard deviations are
               updated using the slice sampler of Neal (2003). At the
               same time, overrelaxation can be used. Then every
               k.overrelax.sigma th update is not overrelaxed. Default
               is 'k.overrelax.sigma = 1', i.e. no overrelaxation

          _k._o_v_e_r_r_e_l_a_x._s_c_a_l_e G-spline scales are updated using the slice
               sampler of Neal (2003). At the same time, overrelaxation
               can be used. Then every k.overrelax.scale th update is
               not overrelaxed. Default is 'k.overrelax.scale = 1',
               i.e. no overrelaxation

  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'. See 'prior' argument of
          'bayesHistogram' function for more detail.  

prior.beta2: prior specification for the regression parameters of
          time-to-event in the case of doubly censored data (related to
          'formula2'). 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'. 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[1],k[2]], k[1]=-K[1],..., K[1], k[2]=-K[2],...,
               K[2], related to the G-spline of 'formula' are stored.

          _a_2 if 'TRUE' and there are doubly-censored data then all the
               transformed mixture weights a[k[1],k[2]],
               k[1]=-K[1],..., K[1], k[2]=-K[2],..., K[2], related to
               the G-spline 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.

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

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

     A list of class 'bayesBisurvreg' 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', 'Mean.2', 'D.1.1',
          'D.2.1', 'D.2.2', where

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

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

          *Mean.2* = E(epsilon[i,2]);

          *D.1.1* = var(epsilon[i,1]);

          *D.2.1* = cov(epsilon[i,1], epsilon[i,2]);

          *D.2.2* = var(epsilon[i,2]);

          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[1],k[2]] 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[1], k[2], k[1] in {-K[1], ..., K[1]}, k[2] in
          {-K[2], ..., K[2]} 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,1], epsilon[i,2])') 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', 'gamma2', 'sigma1',
          'sigma2', 'delta1', 'delta2', 'intercept1', 'intercept2',
          'scale1', 'scale2'. The meaning of the values in these
          columns is the following:

          *gamma1* = the middle knot gamma[1] in the first dimension.
          If 'Specification' is 2, this column usually contains zeros;

          *gamma2* = the middle knot gamma[2] in the second dimension.
          If 'Specification' is 2, this column usually contains zeros;

          *sigma1* = basis standard deviation sigma[1] of the G-spline
          in the first dimension. This column contains a~fixed value if
          'Specification' is 2;

          *sigma2* = basis standard deviation sigma[2] of the G-spline
          in the second dimension. This column contains a~fixed value
          if 'Specification' is 2;

          *delta1* = distance delta[1] between the two knots of the
          G-spline in the first dimension. This column contains a~fixed
          value if 'Specification' is 2;

          *delta2* = distance delta[2] between the two knots of the
          G-spline in the second dimension. This column contains
          a~fixed value if 'Specification' is 2;

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

          *intercept2* = the intercept term alpha[2] of the G-spline in
          the second dimension. If 'Specification' is 1, this column
          usually contains zeros;

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

          *scale2* = the scale parameter tau[2] of the G-spline in the
          second dimension. '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[1],k[2]],
          k[1]=-K[1],..., K[1], k[2]=-K[2],..., K[2] of all mixture
          components, i.e. also of components that had numerically zero
          probabilities. This file is related to 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 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 double indeces (k[1],
          k[2]), values from 1 to (2*K[1]+1)*(2*K[2]+1) are stored
          here. Function 'vecr2matr' can be used to transform it back
          to double indeces.

     _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 either one column labeled 'lambda' or two columns
          labeled 'lambda1' and 'lambda2'. These are the values of the
          smoothing parameter(s) lambda (hyperparameters of the prior
          distribution of the transformed mixture weights
          a[k[1],k[2]]). 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 beta 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'. 

     _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' or 'penalty1'
          and 'penalty2', 'logprw'. This file is related to the model
          given by 'formula'. The columns have the following meaning.

          *loglik* = -N(log(2*pi) + log(sigma[1]) + log(sigma[2]))
          -0.5*sum[i=1][N]( (sigma[1]^2*tau[1]^2)^(-1) * (y[i,1] -
          x[i,1]'beta - alpha[1] - tau[1]*mu[1,r[i,1]])^2 +
          (sigma[2]^2*tau[2]^2)^(-1) * (y[i,2] - x[i,2]'beta - alpha[2]
          - tau[2]*mu[2,r[i,2]])^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] log(p((y[i,1], y[i,2]) | r[i], beta,
          G-spline));

          *penalty1:* If 'prior$neighbor.system' = '"uniCAR"': the
          penalty term for the first dimension not multiplied by
          'lambda1';

          *penalty2:* If 'prior$neighbor.system' = '"uniCAR"': the
          penalty term for the second dimension not multiplied by
          'lambda2';

          *penalty:* If 'prior$neighbor.system' is different from
          '"uniCAR"': the penalty term not multiplied by '\lambda';

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

          In other words, 'logprw' is equal to the conditional
          log-density sum[i=1][N] log(p(r[i] | 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 'lambda.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:

     Gilks, W. R. and Wild, P. (1992). Adaptive rejection sampling for
     Gibbs sampling. _Applied Statistics,_ *41*, 337 - 348.

     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. and Lesaffre, E. (2006). Bayesian semi-parametric
     accelerated failure time model for paired doubly interval-censored
     data. _Statistical Modelling_, *6*, 3-22.

     Neal, R. M. (2003). Slice sampling (with Discussion). _The Annals
     of Statistics,_ *31*, 705 - 767.

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

     ## See the description of R commands for
     ## the population averaged AFT model
     ## with the Signal Tandmobiel data,
     ## analysis described in Komarek and Lesaffre (2006),
     ##
     ## R commands available in the documentation
     ## directory of this package as
     ## tandmobPA.pdf, tandmobPA.R.

