Analyzes survival and reliability data using Cox’s proportional hazards model.
Synopsis
#include <imsls.h>
float *imsls_f_prop_hazards_gen_lin (int n_observations, int n_columns, float x[], int nef, int n_var_effects[], int indices_effects[], int max_class, int *ncoef, ..., 0)
The type double function is imsls_d_prop_hazards_gen_lin.
Required Arguments
int n_observations
(Input)
Number of observations.
int n_columns
(Input)
Number of columns in x.
float x[] (Input)
Array of length
n_observations *
n_columns
containing the data. When optional argument itie = 1, the
observations in x must be grouped by
stratum and sorted from largest to smallest failure time within each stratum,
with the strata separated.
int nef
(Input)
Number of effects in the model. In addition to effects
involving classification variables, simple covariates and the product of simple
covariates are also considered effects.
int n_var_effects[]
(Input)
Array of length nef containing the
number of variables associated with each effect in the model.
int indices_effects[]
(Input)
Index array of length n_var_effects[0] +
… + n_var_effects[nef-1]
containing the column indices of x associated with
each effect. The first n_var_effects[0]
elements of indices_effects
contain the column indices of x for the variables in
the first effect. The next n_var_effects[1]
elements in indices_effects
contain the column indices for the second effect, etc.
int max_class
(Input)
An upper bound on the total number of
different values found among the classification variables in x. For example, if the model consisted of two class
variables, one with the values {1, 2, 3, 4} and a second with the values {0, 1},
then then the total number of different classification values is 4+2=6, and
max_class >= 6.
int *ncoef (Output)
Number of
estimated coefficients in the model.
Return Value
Pointer to an array of length ncoef*4, coef, containing the parameter estimates and associated statistics.
|
Column |
Statistic |
|
1 |
Coefficient
estimate |
|
2 |
Estimated standard deviation of the estimated coefficient. |
|
3 |
Asymptotic normal score for testing that the coefficient is zero against the two-sided alternative. |
|
4 |
p-value associated with the normal score in column 3. |
Synopsis with Optional Arguments
#include <imsls.h>
float
*imsls_f_prop_hazards_gen_lin (int
n_observations,
int
n_columns, float
x[],
int nef, int
n_var_effects[],
int
indices_effects[],
int max_class, int
*ncoef,
IMSLS_RETURN_USER,
float cov[],
IMSLS_PRINT_LEVEL,
int iprint,
IMSLS_MAX_ITERATIONS,
int
max_iterations,
IMSLS_CONVERGENCE_EPS,
float eps,
IMSLS_RATIO,
float ratio,
IMSLS_X_RESPONSE_COL,
int irt,
IMSLS_CENSOR_CODES_COL,
int icen,
IMSLS_STRATIFICATION_COL,
int istrat,
IMSLS_CONSTANT_COL, int
ifix,
IMSLS_FREQ_RESPONSE_COL, int ifrq,
IMSLS_TIES_OPTION,
int itie,
IMSLS_MAXIMUM_LIKELIHOOD,
float algl,
IMSLS_N_MISSING,
int *nrmiss,
IMSLS_STATISTICS,
float **case,
IMSLS_STATISTICS_USER,
float case[],
IMSLS_X_MEAN,
float **xmean,
IMSLS_X_MEAN_USER, float
xmean[],
IMSLS_VARIANCE_COVARIANCE_MATRIX,
float **cov,
IMSLS_VARIANCE_COVARIANCE_MATRIX_USER,
float
cov[],
IMSLS_INITIAL_EST_INPUT,
float in_coef[],
IMSLS_UPDATE, float
**gr,
IMSLS_UPDATE_USER,
float gr[],
IMSLS_DUMMY,
int n_class_var,
int index_class_var[],
IMSLS_STRATUM_NUMBER,
int **igrp,
IMSLS_STRATUM_NUMBER_USER,
int igrp[],
IMSLS_CLASS_VARIABLES, int
**n_class_values,
float **class_values,
IMSLS_CLASS_VARIABLES_USER,
int
n_class_values[],
float
class_values[],
0)
Optional Arguments
IMSLS_RETURN_USER, float
coef[] (Output)
If
specified, coef is an array of length ncoef*4 containing the parameter estimates
and associated statistics. See Return Value.
IMSLS_PRINT_LEVEL,
int iprint (Input)
Printing
option. Default: iprint = 0.
|
iprint |
Action |
|
0 |
No printing is performed. |
|
1 |
Printing is performed, but observational statistics are not printed. |
|
2 |
All output statistics are printed. |
IMSLS_MAX_ITERATIONS,
int
max_iterations
(Input)
Maximum number of iterations. max_iterations = 30
will usually be sufficient. Use max_iterations = 0 to
compute the Hessian and gradient, stored in cov and gr, at the initial
estimates. When max_iterations = 0,
IMSLS_INITIAL_EST_INPUT
must be used.
Default: max_iterations =
30.
IMSLS_CONVERGENCE_EPS, float eps
(Input)
Convergence criterion. Convergence is assumed when the relative
change in algl
from one iteration to the next is less than eps. If eps is zero,
eps = 0.0001 is
assumed.
Default: eps = 0.0001.
IMSLS_RATIO,
float ratio
(Input)
Ratio at which a stratum is split into two strata.
Default: ratio =
1000.0.
Let
be the observation proportionality constant, where zk is the design row vector for the k-th observation and wk is the optional fixed parameter specified by xk, ifix. Let rmin be the minimum value rk in a stratum, where, for failed observations, the minimum is over all times less than or equal to the time of occurrence of the k-th observation. Let rmax be the maximum value of rk for the remaining observations in the group. Then, if rmin > ratio rmax, the observations in the group are divided into two groups at k. ratio = 1000 is usually a good value. Set ratio = −1.0 if no division into strata is to be made.
IMSLS_X_RESPONSE_COL, int irt
(Input)
Column index in x containing the
response variable. For point observations, xi,
irt contains the
time of the i-th event. For right-censored observations, xi,
irt contains the
right-censoring time. Note that because imsls_f_prop_hazards_gen_lin
only uses the order of the events, negative “times” are
allowed.
Default: irt = 0.
IMSLS_CENSOR_CODES_COL,
int icen (Input)
Column index in x containing the
censoring code for each observation. Default: A censoring code of 0
is assumed for all observations.
|
xi,icen |
Censoring |
|
0 |
Exact censoring time xi, irt. |
|
1 |
Right censored. The exact censoring time is greater than xi, irt. |
IMSLS_STRATIFICATION_COL,
int istrat
(Input)
Column number in x containing the
stratification variable. Column istrat in x contains a unique
number for each stratum. The risk set for an observation is determined by its
stratum.
Default: All observations are considered to be in one stratum.
IMSLS_CONSTANT_COL,
int ifix
(Input)
Column index in x containing a
constant, wi, to be added to the linear
response. The linear response is taken to be
where wi is the
observation constant, zi is the observation design row vector,
and
is the vector of estimated
parameters. The “fixed” constant allows one to test hypotheses about parameters
via the log-likelihoods.
Default: wi is assumed to be 0 for all
observations.
IMSLS_FREQ_RESPONSE_COL,
int ifrq
(Input)
Column index in x containing the
number of responses for each observation.
Default: A response frequency of 1
for each observation is assumed.
IMSLS_TIES_OPTION,
int itie
(Input)
Method for handling ties. Default: itie = 0.
|
itie |
Method |
|
0 |
Breslow’s approximate method. |
|
1 |
Failures are assumed to occur in the same order as the observations input in x. The observations in x must be sorted from largest to smallest failure time within each stratum, and grouped by stratum. All observations are treated as if their failure/censoring times were distinct when computing the log-likelihood. |
IMSLS_MAXIMUM_LIKELIHOOD,
float *algl
(Output)
The maximized log-likelihood.
IMSLS_N_MISSING,
int *nrmiss
(Output)
Number of rows of data in X
that contain missing values in one or more columns irt, ifrq, ifix, icen, istrat, index_class_var, or
indices_effects
of x.
IMSLS_STATISTICS,
float **case
(Output)
Address of a pointer to an array of length n_observations *
5 containing the case statistics for each observation.
|
Column |
Statistic |
|
1 |
Estimated survival probability at the observation time. |
|
2 |
Estimated observation influence or leverage. |
|
3 |
A residual estimate. |
|
4 |
Estimated cumulative baseline hazard rate. |
|
5 |
Observation proportionality constant. |
IMSLS_STATISTICS_USER,
float case[]
(Output)
Storage for case is provided by
the user. See IMSLS_STATISTICS.
IMSLS_X_MEAN,
float **xmean
(Output)
Address of a pointer to an array of length ncoef containing the
means of the design variables.
IMSLS_X_MEAN_USER,
float xmean[]
(Output)
Storage for xmean is provided by
the user. See IMSLS_X_MEAN.
IMSLS_VARIANCE_COVARIANCE_MATRIX,
float **cov
(Output)
Address of a pointer to an array of length ncoef*ncoef containing
the estimated asymptotic variance-covariance matrix of the parameters. For
max_iterations =
0, the return value is the inverse of the Hessian of the negative of the
log-likelihood, computed at the estimates input in in_coef.
IMSLS_VARIANCE_COVARIANCE_MATRIX_USER, float
cov[]
(Output)
Storage for cov is provided by the
user. See IMSLS_VARIANCE_COVARIANCE_MATRIX.
IMSLS_INITIAL_EST_INPUT,
float *in_coef
(Input)
An array of length ncoef containing the
initial estimates on input to prop_hazards_gen_lin.
Default: all initial estimates are taken to be 0.
IMSLS_UPDATE,
float **gr
(Output)
Address of a pointer to an array of length ncoef containing the
last parameter updates (excluding step halvings). For max_iterations = 0,
gr contains the
inverse of the Hessian times the gradient vector computed at the estimates input
in in_coef.
IMSLS_UPDATE_USER,
float gr[]
(Output)
Storage for gr is provided by the
user. See IMSLS_UPDATE.
IMSLS_DUMP, int
n_class_var,
int index_class_var[]
(Input)
Variable n_class_var is the
number of classification variables. Dummy variables are generated for
classification variables using the dummy_method = IMSLS_LEAVE_OUT_LAST
of the IMSLS_DUMMY option of
imsls_f_regressors_for_glm function (see
Chapter 2, Regression.). Argument index_class_var is an
index array of length n_class_var containing
the column numbers of x that are the
classification variables. (if n_class_var is is
equal to zero, index_class_var is not
used).
Default: n_class_var = 0.
IMSLS_STRATUM_NUMBER,
int **igrp
(Output)
Address of a pointer to an array of length
n_observations giving
the stratum number used for each observation. If ratio is not −1.0,
additional “strata” (other than those specified by column istrat of x) may be
generated. igrp also contains a
record of the generated strata. See the “Description” section for
more detail.
IMSLS_STRATUM_NUMBER_USER,
int igrp[]
(Output)
Storage for igrp is provided by
the user. See IMSLS_STRATUM_NUMBER.
IMSLS_CLASS_VARIABLES,
int **n_class_values, float **class_values
(Output)
n_class_values is an
address of a pointer to an array of length n_class_var containing
the number of values taken by each classification variable. n_class_values[i]
is the number of distinct values for the i-th classification
variable. class_values is an
address of a pointer to an array of length n_class_values[0] +
n_class_values[1] + …
+ n_class_values[n_class_var-1]
containing the distinct values of the classification variables. The
first n_class_values[0]
elements of class_values contain
the values for the first classification variable, the next n_class_values[1]
elements contain the values for the second classification variable, etc.
IMSLS_CLASS_VARIABLES_USER,
int n_class_values[],
float class_values[]
(Output)
Storage for n_class_values and
class_values is
provided by the user. The length of class_values will not
be known in advance, use max_class as the
maximum length of class_values.
See IMSLS_CLASS_VARIABLES.
Description
Function imsls_f_prop_hazards_gen_lin computes parameter estimates and other statistics in Proportional Hazards Generalized Linear Models. These models were first proposed by Cox (1972). Two methods for handling ties are allowed in imsls_f_prop_hazards_gen_lin. Time-dependent covariates are not allowed. The user is referred to Cox and Oakes (1984), Kalbfleisch and Prentice (1980), Elandt-Johnson and Johnson (1980), Lee (1980), or Lawless (1982), among other texts, for a thorough discussion of the Cox proportional hazards model.
Let λ(t, zi) represent the hazard rate at time t for observation number i with covariables contained as elements of row vector zi. The basic assumption in the proportional hazards model (the proportionality assumption) is that the hazard rate can be written as a product of a time varying function λ0(t), which depends only on time, and a function ƒ(zi), which depends only on the covariable values. The function ƒ(zi) used in imsls_f_prop_hazards_gen_lin is given as ƒ(zi) = exp(wi + βzi) where wi is a fixed constant assigned to the observation, and β is a vector of coefficients to be estimated. With this function one obtains a hazard rate λ(t, zi) = λ0(t) exp(wi + βzi). The form of λ0(t) is not important in proportional hazards models.
The constants wi may be known
theoretically. For example, the hazard rate may be proportional to a known
length or area, and the wi can then be determined
from this known length or area. Alternatively, the wi may be used to fix a
subset of the coefficients β (say, β1) at specified values.
When wi is used in this way,
constants
wi = β1z1i are used, while the
remaining coefficients in β are free to vary in the optimization algorithm. If
user-specified constants are not desired, the user should set ifix
to 0 so that wi = 0 will be used.
With this definition of λ(t, zi), the usual partial (or marginal, see Kalbfleisch and Prentice (1980)) likelihood becomes
where R(ti) denotes the set of indices of observations that have not yet failed at time ti (the risk set), ti denotes the time of failure for the i-th observation, nd is the total number of observations that fail. Right-censored observations (i.e., observations that are known to have survived to time ti, but for which no time of failure is known) are incorporated into the likelihood through the risk set R(ti). Such observations never appear in the numerator of the likelihood. When itie = 0, all observations that are censored at time ti are not included in R(ti), while all observations that fail at time ti are included in R(ti).
If it can be assumed that the dependence of the hazard rate upon the covariate values remains the same from stratum to stratum, while the time-dependent term, λ0(t), may be different in different strata, then imsls_f_prop_hazards_gen_lin allows the incorporation of strata into the likelihood as follows. Let k index the m = istrat strata. Then, the likelihood is given by
In imsls_f_prop_hazards_gen_lin, the log of the likelihood is maximized with respect to the coefficients β. A quasi-Newton algorithm approximating the Hessian via the matrix of sums of squares and cross products of the first partial derivatives is used in the initial iterations (the “Q-N” method in the output). When the change in the log-likelihood from one iteration to the next is less than 100*eps, Newton-Raphson iteration is used (the “N-R” method). If, during any iteration, the initial step does not lead to an increase in the log-likelihood, then step halving is employed to find a step that will increase the log-likelihood.
Once the maximum likelihood estimates have been computed, imsls_f_prop_hazards_gen_lin computes estimates of a probability associated with each failure. Within stratum k, an estimate of the probability that the i-th observation fails at time ti given the risk set R(tki) is given by
A diagnostic “influence” or “leverage” statistic is computed for each noncensored observation as:
where Hs is the matrix of second partial derivatives of the log-likelihood, and
is computed as:
Influence statistics are not computed for censored observations.
A “residual” is computed for each of the input observations according to methods given in Cox and Oakes (1984, page 108). Residuals are computed as
where dkj is the number of tied failures in group k at time tkj. Assuming that the proportional hazards assumption holds, the residuals should approximate a random sample (with censoring) from the unit exponential distribution. By subtracting the expected values, centered residuals can be obtained. (The j-th expected order statistic from the unit exponential with censoring is given as
where h is the sample size, and censored observations are not included in the summation.)
An estimate of the cumulative baseline hazard within group k is given as
The observation proportionality constant is computed as
Programming Notes
1. The
covariate vectors zki are computed from each row of the input
matrix x via
function imsls_f_regressors_for_glm (see Chapter
2, Regression.). Thus, class variables are
easily incorporated into the zki. The reader is referred to the document
for imsls_f_regressors_for_glm
in the regression chapter for a more detailed discussion.
Note that imsls_f_prop_hazards_gen_lin
calls imsls_f_regressors_for_glm
with dummy_method = IMSLS_LEAVE_OUT_LAST
of the IMSLS_DUMMY
option.
2. The average of each of the explanatory variables is subtracted from the variable prior to computing the product zkiβ. Subtraction of the mean values has no effect on the computed log-likelihood or the estimates since the constant term occurs in both the numerator and denominator of the likelihood. Subtracting the mean values does help to avoid invalid exponentiation in the algorithm and may also speed convergence.
3. Function imsls_f_prop_hazards_gen_lin allows for two methods of handling ties. In the first method (itie = 1), the user is allowed to break ties in any manner desired. When this method is used, it is assumed that the user has sorted the rows in X from largest to smallest with respect to the failure/censoring times xi, irt within each stratum (and across strata), with tied observations (failures or censored) broken in the manner desired. The same effect can be obtained with itie = 0 by adding (or subtracting) a small amount from each of the tied observations failure/ censoring times ti = xi, irt so as to break the ties in the desired manner.
The second method for handling ties (itie = 0) uses an
approximation for the tied likelihood proposed by Breslow (1974). The likelihood
in Breslow’s method is as specified above, with the risk set at time
ti
including all observations that fail at time ti, while all
observations that are censored at time ti are not included.
(Tied censored
observations are assumed to be censored immediately prior to the time
ti).
4. If IMSLS_INITIAL_EST_INPUT option is used, then it is assumed that the user has provided initial estimates for the model coefficients β in in_coef. When initial estimates are provided by the user, care should be taken to ensure that the estimates correspond to the generated covariate vector zki. If IMSLS_INITIAL_EST_INPUT option is not used, then initial estimates of zero are used for all of the coefficients. This corresponds to no effect from any of the covariate values.
5. If a linear
combination of covariates is monotonically increasing or decreasing with
increasing failure times, then one or more of the estimated coefficients is
infinite and extended maximum likelihood estimates must be computed. Such
estimates may be written as 
where ρ = ∞ at the supremum of
the likelihood so that
is the finite part of the solution. In
imsls_f_prop_hazards_gen_lin,
it is assumed that extended maximum likelihood estimates must be computed if,
within any group k, for any time t,
where ρ = ratio is specified by the user. Thus, for example, if ρ = 10000, then imsls_f_prop_hazards_gen_lin does not compute extended maximum likelihood estimates until the estimated proportionality constant
is 10000 times larger for all observations prior to t than for all
observations after t. When this occurs, imsls_f_prop_hazards_gen_lin
computes estimates for by splitting the failures in stratum
k into two strata at t (see Bryson and Johnson 1981). Censored
observations in stratum k are placed into a stratum based upon the
associated value for
The results of the splitting are returned in igrp.
The estimatesbased upon the stratified likelihood
represent the finite part of the extended maximum likelihood solution. Function
imsls_f_prop_hazards_gen_lin
does not compute 
explicitly, but an estimate formay be obtained in some circumstances
by setting ratio = −1 and optimizing the log-likelihood
without forming additional strata. The solution obtained will be such thatfor some finite value of
ρ > 0. At this solution, the Newton-Raphson algorithm will not have
“converged” because the Newton-Raphson step sizes returned in gr will be large,
at least for some variables. Convergence will be declared, however, because the
relative change in the log-likelihood
during the final iterations will be small.
Example
The following data are taken from Lawless (1982, page 287) and involve the survival of lung cancer patients based upon their initial tumor types and treatment type. In the first example, the likelihood is maximized with no strata present in the data. This corresponds to Example 7.2.3 in Lawless (1982, page 367). The input data is printed in the output. The model is given as:
where αi and γj correspond to dummy variables generated from column indices 5 and 6 of x, respectively, x1 corresponds to column index 2, x2 corresponds to column index 3, and x3 corresponds to column index 4 of x.
#include <imsls.h>
#define NOBS 40
#define NCOL 7
#define NCLVAR 2
#define NEF 5
int main ()
{
int icen = 1, iprint = 2, maxcl = 6, ncoef;
int indef[NEF] = { 2, 3, 4, 5, 6 };
int nvef[NEF] = { 1, 1, 1, 1, 1 };
int indcl[NCLVAR] = { 5, 6 };
float *coef, ratio = 10000.0;
float x[NOBS * NCOL] = {
411, 0, 7, 64, 5, 1, 0,
126, 0, 6, 63, 9, 1, 0,
118, 0, 7, 65, 11, 1, 0,
92, 0, 4, 69, 10, 1, 0,
8, 0, 4, 63, 58, 1, 0,
25, 1, 7, 48, 9, 1, 0,
11, 0, 7, 48, 11, 1, 0,
54, 0, 8, 63, 4, 2, 0,
153, 0, 6, 63, 14, 2, 0,
16, 0, 3, 53, 4, 2, 0,
56, 0, 8, 43, 12, 2, 0,
21, 0, 4, 55, 2, 2, 0,
287, 0, 6, 66, 25, 2, 0,
10, 0, 4, 67, 23, 2, 0,
8, 0, 2, 61, 19, 3, 0,
12, 0, 5, 63, 4, 3, 0,
177, 0, 5, 66, 16, 4, 0,
12, 0, 4, 68, 12, 4, 0,
200, 0, 8, 41, 12, 4, 0,
250, 0, 7, 53, 8, 4, 0,
100, 0, 6, 37, 13, 4, 0,
999, 0, 9, 54, 12, 1, 1,
231, 1, 5, 52, 8, 1, 1,
991, 0, 7, 50, 7, 1, 1,
1, 0, 2, 65, 21, 1, 1,
201, 0, 8, 52, 28, 1, 1,
44, 0, 6, 70, 13, 1, 1,
15, 0, 5, 40, 13, 1, 1,
103, 1, 7, 36, 22, 2, 1,
2, 0, 4, 44, 36, 2, 1,
20, 0, 3, 54, 9, 2, 1,
51, 0, 3, 59, 87, 2, 1,
18, 0, 4, 69, 5, 3, 1,
90, 0, 6, 50, 22, 3, 1,
84, 0, 8, 62, 4, 3, 1,
164, 0, 7, 68, 15, 4, 1,
19, 0, 3, 39, 4, 4, 1,
43, 0, 6, 49, 11, 4, 1,
340, 0, 8, 64, 10, 4, 1,
231, 0, 7, 67, 18, 4, 1
};
coef = imsls_f_prop_hazards_gen_lin (NOBS, NCOL, x, NEF,
nvef, indef, maxcl, &ncoef,
IMSLS_PRINT_LEVEL, iprint,
IMSLS_CENSOR_CODES_COL, icen,
IMSLS_RATIO, ratio,
IMSLS_DUMMY, NCLVAR, &indcl[0], 0);
}
Output
Initial Estimates
1 2 3 4 5 6 7
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Method Iteration Step size Maximum scaled Log
coef. update likelihood
Q-N 0 -102.4
Q-N 1 1.0000 0.5034 -91.04
Q-N 2 1.0000 0.5782 -88.07
N-R 3 1.0000 0.1131 -87.92
N-R 4 1.0000 0.06958 -87.89
N-R 5 1.0000 0.0008145 -87.89
Log-likelihood -87.88778
Coefficient Statistics
Coefficient Standard Asymptotic Asymptotic
error z-statistic p-value
1 -0.585 0.137 -4.272 0.000
2 -0.013 0.021 -0.634 0.526
3 0.001 0.012 0.064 0.949
4 -0.367 0.485 -0.757 0.449
5 -0.008 0.507 -0.015 0.988
6 1.113 0.633 1.758 0.079
7 0.380 0.406 0.936 0.349
Asymptotic Coefficient Covariance
1 2 3 4 5
1 0.01873 0.000253 0.0003345 0.005745 0.00975
2 0.0004235 -4.12e-005 -0.001663 -0.0007954
3 0.0001397 0.0008111 -0.001831
4 0.235 0.09799
5 0.2568
6 7
1 0.004264 0.002082
2 -0.003079 -0.002898
3 0.0005995 0.001684
4 0.1184 0.03735
5 0.1253 -0.01944
6 0.4008 0.06289
7 0.1647
Case Analysis
Survival Influence Residual Cumulative Prop.
Probability hazard constant
1 0.00 0.04 2.05 6.10 0.34
2 0.30 0.11 0.74 1.21 0.61
3 0.34 0.12 0.36 1.07 0.33
4 0.43 0.16 1.53 0.84 1.83
5 0.96 0.56 0.09 0.05 2.05
6 0.74 ............ 0.13 0.31 0.42
7 0.92 0.37 0.03 0.08 0.42
8 0.59 0.26 0.14 0.53 0.27
9 0.26 0.12 1.20 1.36 0.88
10 0.85 0.15 0.97 0.17 5.76
11 0.55 0.31 0.21 0.60 0.36
12 0.74 0.21 0.96 0.31 3.12
13 0.03 0.06 3.02 3.53 0.86
14 0.94 0.09 0.17 0.06 2.71
15 0.96 0.16 1.31 0.05 28.89
16 0.89 0.23 0.59 0.12 4.82
17 0.18 0.09 2.62 1.71 1.54
18 0.89 0.19 0.33 0.12 2.68
19 0.14 0.23 0.72 1.96 0.37
20 0.05 0.09 1.66 2.95 0.56
21 0.39 0.22 1.17 0.94 1.25
22 0.00 0.00 1.73 21.11 0.08
23 0.08 ............ 2.19 2.52 0.87
24 0.00 0.00 2.46 8.89 0.28
25 0.99 0.31 0.05 0.01 4.28
26 0.11 0.17 0.34 2.23 0.15
27 0.66 0.25 0.16 0.41 0.38
28 0.87 0.22 0.15 0.14 1.02
29 0.39 ............ 0.45 0.94 0.48
30 0.98 0.25 0.06 0.02 2.53
31 0.77 0.26 1.03 0.26 3.90
32 0.63 0.35 1.80 0.46 3.88
33 0.82 0.26 1.06 0.19 5.47
34 0.47 0.26 1.65 0.75 2.21
35 0.51 0.32 0.39 0.67 0.58
36 0.22 0.18 0.49 1.53 0.32
37 0.80 0.26 1.08 0.23 4.77
38 0.70 0.16 0.26 0.36 0.73
39 0.01 0.23 0.87 4.66 0.19
40 0.08 0.20 0.81 2.52 0.32
Last Coefficient Update
1 2 3 4 5 6
-1.296e-008 2.269e-009 -5.894e-009 -4.782e-007 -1.787e-007 1.509e-007
7
4.327e-008
Covariate Means
1 2 3 4 5 6
5.65 56.58 15.65 0.35 0.28 0.13
7
0.53
Distinct Values For Each Class Variable
Variable 1: 1 2 3 4
Variable 2: 0 1
Stratum Numbers For Each Observation
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Number of Missing Values 0
