propHazardsGenLin¶

Analyzes survival and reliability data using Cox’s proportional hazards model.

Synopsis¶

propHazardsGenLin (x, nVarEffects, indicesEffects, maxClass, ncoef)

Required Arguments¶

float x[[]] (Input): Array of length nObservations×nColumns 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 nVarEffects[] (Input): Array of length nef containing the number of variables associated with each effect in the model.
int indicesEffects[] (Input): Index array of length nVarEffects[0] + … + nVarEffects[nef-1] containing the column indices of x associated with each effect. The first nVarEffects[0] elements of indicesEffects contain the column indices of x for the variables in the first effect. The next nVarEffects[1] elements in indicesEffects contain the column indices for the second effect, etc.
int maxClass (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 the total number of different classification values is \(4+2=6\), and maxClass >= 6.
int ncoef (Output): Number of estimated coefficients in the model.

Return Value¶

An array of length ncoef×4, coef, containing the parameter estimates and associated statistics.

`Column`	Statistic
`1`	Coefficient estimate \(\hat{\beta}\)
`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.

Optional Arguments¶

printLevel, int (Input)

Printing option.

`printLevel`	Action
0	No printing is performed.
1	Printing is performed, but observational statistics are not printed.
2	All output statistics are printed.

Default: printLevel = 0.

maxIterations, int (Input)

Maximum number of iterations. maxIterations = 30 will usually be sufficient. Use maxIterations = 0 to compute the Hessian and gradient, stored in cov and gr, at the initial estimates. When maxIterations = 0, initialEstInput must be used.

Default: maxIterations = 30.

convergenceEps, float (Input)

Convergence criterion. Convergence is assumed when the relative change in algl from one iteration to the next is less than convergenceEps. If convergenceEps is zero, convergenceEps = 0.0001 is assumed.

Default: convergenceEps = 0.0001.

ratio, float (Input)

Ratio at which a stratum is split into two strata. Let

\[r_k = \exp\left(z_k \hat{\beta} + w_k\right)\]

be the observation proportionality constant, where \(z_k\) is the design row vector for the k-th observation and \(w_k\) is the optional fixed parameter specified by \(\text{x}_{k,xColFixedParameter}\). Let \(r_{min}\) be the minimum value \(r_k\) 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 \(r_{max}\) be the maximum value of \(r_k\) for the remaining observations in the group. Then, if \(r_{min}\) > ratio \(r_{max}\), 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.

Default: ratio = 1000.0.

xResponseCol, int (Input)

Column index in x containing the response variable. For point observations, \(\text{x}_{i,xResponseCol}\) contains the time of the i-th event. For right-censored observations, \(\text{x}_{i,xResponseCol}\) contains the right-censoring time. Note that because propHazardsGenLin only uses the order of the events, negative “times” are allowed.

Default: xResponseCol = 0.

censorCodesCol, int (Input)

Column index in x containing the censoring code for each observation. Default: A censoring code of 0 is assumed for all observations.

\(\text{x}_{i,censorCodesCol}\)
0	Exact censoring time \(\text{x}_{i,xResponseCol}\).
1	Right censored. The exact censoring time is greater than \(\text{x}_{i,xResponseCol}\).

stratificationCol, int (Input)

Column number in x containing the stratification variable. Column stratificationCol 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.

constantCol, int (Input)

Column index in x containing a constant, \(w_i\), to be added to the linear response. The linear response is taken to be \(w_i+z_i \hat{\beta}\) where \(w_i\) is the observation constant, \(z_i\) is the observation design row vector, and \(\hat{\beta}\) is the vector of estimated parameters. The “fixed” constant allows one to test hypotheses about parameters via the log-likelihoods.

Default: \(w_i\) is assumed to be 0 for all observations.

freqResponseCol, int (Input)

Column index in x containing the number of responses for each observation.

Default: A response frequency of 1 for each observation is assumed.

tiesOption, int (Input)

Method for handling ties.

`tiesOption`
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.

Default: tiesOption = 0.

maximumLikelihood (Output): The maximized log-likelihood.
nMissing (Output): Number of rows of data in X that contain missing values in one or more columns xResponseCol, freqResponseCol, ifix, censorCodesCol, stratificationCol, indexClassVar, or indicesEffects of x.
statistics (Output): An array of length nObservations×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.

xMean (Output)

An array of length ncoef containing the means of the design variables.

varianceCovarianceMatrix, float cov (Output)

An array of length ncoef*ncoef containing the estimated asymptotic variance-covariance matrix of the parameters. For maxIterations = 0, the return value is the inverse of the Hessian of the negative of the log-likelihood, computed at the estimates input in initialEstInput.

initialEstInput, float (Input)

An array of length ncoef containing the initial estimates on input to propHazardsGenLin.

Default: all initial estimates are taken to be 0.

update (Output)

An array of length ncoef containing the last parameter updates (excluding step halvings). For maxIterations = 0, update contains the inverse of the Hessian times the gradient vector computed at the estimates input in initialEstInput.

dummy (Input)

Argument dummy is an index array of length nClassVar containing the column numbers of x that are the classification variables. (If nClassVar is equal to zero, indexClassVar is not used). nClassVar is the number of classification variables. Dummy variables are generated for classification variables using the dummy = leaveOutLast of the dummy option of regressorsForGlm function (see Regression).

Default: nClassVar = 0.

stratumNumber (Output)

An array of length nObservations giving the stratum number used for each observation. If ratio is not -1.0, additional “strata” (other than those specified by column stratificationCol of x) may be generated. stratumNumber also contains a record of the generated strata. See the Description section for more detail.

classVariables, nClassValues, classValues (Output)

nClassValues is an array of length nClassVar containing the number of values taken by each classification variable. nClassValues[i] is the number of distinct values for the i-th classification variable. classValues is an array of length nClassValues[0] + nClassValues[1] + … + nClassValues[nClassVar‑1] containing the distinct values of the classification variables. The first nClassValues[0] elements of classValues contain the values for the first classification variable, the next nClassValues[1] elements contain the values for the second classification variable, etc.

Description¶

Function propHazardsGenLin 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 propHazardsGenLin. 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 \(\lambda(t,z_i)\) represent the hazard rate at time t for observation number i with covariables contained as elements of row vector \(z_i\). 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 \(\lambda_0(t)\), which depends only on time, and a function \(f(z_i)\), which depends only on the covariable values. The function \(f(z_i)\) used in propHazardsGenLin is given as \(f(z_i)= \exp(w_i+\beta z_i)\) where \(w_i\) 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 \(\lambda(t,z_i)=\lambda_0(t) \exp(w_i+\beta z_i)\). The form of \(\lambda_0(t)\) is not important in proportional hazards models.

The constants \(w_i\) may be known theoretically. For example, the hazard rate may be proportional to a known length or area, and the \(w_i\) can then be determined from this known length or area. Alternatively, the \(w_i\) may be used to fix a subset of the coefficients β (say, \(\beta_1\)) at specified values. When \(w_i\) is used in this way, constants \(w_i=\beta_1 z_{1i}\) 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 \(w_i=0\) will be used.

With this definition of \(\lambda(t,z_i)\), the usual partial (or marginal, see Kalbfleisch and Prentice (1980)) likelihood becomes

\[L = \prod_{i=1}^{n_d} \frac{\exp \left(w_i + \beta z_i\right)} {\sum\limits_{j \in R\left(t_i\right)} \exp \left(w_j + \beta z_j\right)}\]

where \(R(t_i)\) denotes the set of indices of observations that have not yet failed at time \(t_i\) (the risk set), \(t_i\) denotes the time of failure for the i-th observation, \(n_d\) is the total number of observations that fail. Right-censored observations (i.e., observations that are known to have survived to time \(t_i\), but for which no time of failure is known) are incorporated into the likelihood through the risk set \(R(t_i)\). Such observations never appear in the numerator of the likelihood. When tiesOption = 0, all observations that are censored at time \(t_i\) are not included in \(R(t_i)\), while all observations that fail at time \(t_i\) are included in \(R(t_i)\).

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, \(\lambda_0(t)\), may be different in different strata, then propHazardsGenLin allows the incorporation of strata into the likelihood as follows. Let k index the m = stratificationCol strata. Then, the likelihood is given by

\[L_s = \prod_{k=1}^{m} \left[ \prod_{i=1}^{n_k} \frac{\exp \left(w_{ki} + \beta z_{ki}\right)} {\sum_{j \in R\left(t_{ki}\right)} \exp \left(w_{kj} + \beta z_{kj}\right)} \right]\]

In propHazardsGenLin, 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*convergenceEps, 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, propHazardsGenLin 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 \(t_i\) given the risk set \(R(t_{ki})\) is given by

\[p_{ki} = \frac{\exp \left(w_{ki} + z_{ki} \beta\right)} {\sum_{j \in R\left(t_{ki}\right)} \exp \left(w_{kj} + z_{kj} \beta\right)}\]

A diagnostic “influence” or “leverage” statistic is computed for each noncensored observation as:

\[l_{ki} = -g'_{ki} H_s^{-1} g'_{ki}\]

where \(H_s\) is the matrix of second partial derivatives of the log-likelihood, and

\[g'_{ki}\]

is computed as:

\[g'_{ki} = z_{ki} - \frac{z_{ki} \exp \left(w_{ki} + z_{ki} \beta\right)} {\sum_{j \in R\left(t_{ki}\right)} \exp \left(w_{kj} + z_{kj} \beta\right)}\]

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

\[r_{ki} = \exp \left(w_{ki} + z_{ki} \hat{\beta}\right) \sum_{j \in R\left(t_{ki}\right)} \frac{d_{kj}}{\sum_{\ell \in R\left(t_{kj}\right)} \exp \left(w_{k\ell} + z_{k\ell} \hat{\beta}\right)}\]

where \(d_{kj}\) is the number of tied failures in group k at time \(t_{kj}\). 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

\[e_j = \sum_{l \leq j} \frac{1}{h-l+1}\]

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

\[\hat{H}_{k0}\left(t_{ik}\right) = \sum_{t_{kj} \leq t_{ki}} \frac{d_{kj}}{\sum_{l \in R\left(t_{kj}\right)} \exp\left(w_{kl} + z_{kl} \hat{\beta}\right)}\]

The observation proportionality constant is computed as

\[\exp \left(w_{ki} + z_{ki} \hat{\beta}\right)\]

Programming Notes¶

The covariate vectors \(z_{ki}\) are computed from each row of the input matrix x via function regressorsForGlm (see Chapter 2, Regression). Thus, class variables are easily incorporated into the \(z_{ki}\). The reader is referred to the document for regressorsForGlm in the regression chapter for a more detailed discussion.

Note that propHazardsGenLin calls regressorsForGlm with dummy = leaveOutLast of the dummy option.
The average of each of the explanatory variables is subtracted from the variable prior to computing the product \(z_{ki} \beta\). 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.
Function propHazardsGenLin allows for two methods of handling ties. In the first method (tiesOption = 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 \(\text{x}_{i,xResponseCol}\) within each stratum (and across strata), with tied observations (failures or censored) broken in the manner desired. The same effect can be obtained with tiesOption = 0 by adding (or subtracting) a small amount from each of the tied observations failure/ censoring times \(t_i= \text{x}_{i,xResponseCol}\) so as to break the ties in the desired manner.

The second method for handling ties (tiesOption = 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 \(t_i\) including all observations that fail at time \(t_i\), while all observations that are censored at time \(t_i\) are not included.

(Tied censored observations are assumed to be censored immediately prior to the time \(t_i\)).
If initialEstInput option is used, then it is assumed that the user has provided initial estimates for the model coefficients β in initialEstInput. When initial estimates are provided by the user, care should be taken to ensure that the estimates correspond to the generated covariate vector \(z_{ki}\). If initialEstInput 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.
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 \(\hat{\beta}=\hat{\beta}_f+\rho\hat{\gamma}\) where \(\rho= \infty\) at the supremum of the likelihood so that \(\hat{\beta}_f\) is the finite part of the solution. In propHazardsGenLin, it is assumed that extended maximum likelihood estimates must be computed if, within any group k, for any time t,

\[\min_{t_{ki} < t} \exp \left(w_{ki} + z_{ki} \hat{\beta}\right) > \rho \max_{t_{ki} < t} \exp \left(w_{ki} + z_{ki} \hat{\beta}\right)\]

where ρ = ratio is specified by the user. Thus, for example, if \(\rho=10000\), then propHazardsGenLin does not compute extended maximum likelihood estimates until the estimated proportionality constant

\[\exp \left(w_{ki} + z_{ki} \hat{\beta}\right)\]

is 10000 times larger for all observations prior to t than for all observations after t. When this occurs, propHazardsGenLin computes estimates for \(\hat{\beta}_f\) 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

\[\exp \left(w_{ki} + z_{ki} \hat{\beta}\right)\]

The results of the splitting are returned in stratumNumberCol.

The estimates \(\hat{\beta}_f\) based upon the stratified likelihood represent the finite part of the extended maximum likelihood solution. Function propHazardsGenLin does not compute \(\hat{y}\) explicitly, but an estimate for \(\hat{y}\) may be obtained in some circumstances by setting ratio = -1 and optimizing the log‑likelihood without forming additional strata. The solution \(\hat{\beta}\) obtained will be such that \(\hat{\beta}=\hat{\beta}_f+\rho\hat{\gamma}\) for some finite value of \(\rho>0\). 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:

\[\ln (\lambda) = \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \alpha_i + \gamma_j\]

where \(\alpha_i\) and \(\gamma_j\) correspond to dummy variables generated from column indices 5 and 6 of x, respectively, \(x_1\) corresponds to column index 2, \(x_2\) corresponds to column index 3, and \(x_3\) corresponds to column index 4 of x.

from numpy import *
from pyimsl.stat.propHazardsGenLin import propHazardsGenLin

icen = 1
iprint = 2
maxcl = 6
indef = [2, 3, 4, 5, 6]
nvef = [1, 1, 1, 1, 1]
indcl = [5, 6]
ratio = 10000.0

x = array([
    [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]])

ncoef = []
coef = propHazardsGenLin(x, nvef, indef, maxcl, ncoef,
                         printLevel=2,
                         censorCodesCol=icen,
                         ratio=ratio, dummy=indcl)

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        0.0008149          -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-05     -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         21.10          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
 -5.835e-08    1.401e-09   -8.597e-09   -2.822e-07   -4.566e-08    1.256e-07
 
          7
  1.058e-08
 
                               Covariate Means
          1            2            3            4            5            6
       5.65        56.58        15.65         0.35         0.28         0.12
 
          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