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IDL Analyst Reference Guide: Survival Analysis |
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The IMSL_SURVIVAL_GLM function analyzes censored survival data using a generalized linear model and estimates survival probabilities and hazard rates for the various parametric models.
| Note This routine requires an IDL Analyst license. For more information, contact your ITT Visual Information Solutions sales or technical support representative. |
Result = IMSL_SURVIVAL_GLM(n_class, n_continuous, model, x
[, /CASE_ANALYSIS=variable] [, CLASS_VALS=variable] [, COEF_STAT=variable] [, COVARIANCES=variable] [, CRITERION=variable] [, /DOUBLE] [, EPS=value] [, EST_DELTA=value] [, EST_NOBS=value] [, EST_NPT=value] [, EST_PROB=variable] [, EST_TIME=value] [, EST_XBETA=variable] [, ICEN=value] [, IFIX=value] [, IFREQ=value] [, ILT=value] [, INDICIES_EFFECTS=array] [, INIT_EST=array] [, IRT=value] [, ITERATIONS=variable] [, ITMAX=value] [, LAST_STEP=variable] [, LP_MAX=value] [, MAX_CLASS=value] [, MEANS=variable] [, N_CLASS_VALS=variable] [, NMISSING=variable]
[, /NO_INTERCEPT] [, OBS_STATUS=variable] [, VAR_EFFECTS=array])
An integer value indicating n_coefficients, where n_coefficients is the number of estimated coefficients in the model.
Specifies the model used to analyze the data.
See the Discussion section for more information about these models.
Number of classification variables.
Number of continuous variables.
Two-dimensional array of size n_observations by ((n_class + n_continuous) + m) containing data for the independent variables, dependent variable, and optional parameters where n_observations is the number of observations and the optional parameters correspond to keywords Icen, Ilt, Irt, Ifreq, and Ifix.
The columns must be ordered such that the first n_class columns contain data for the class variables, the next n_continuous columns contain data for the continuous variables, and the next column contains the response variable. The final (and optional) m - 1 columns contain optional parameters.
Named variable into which a two-dimensional array of size n_observations by 5 containing the case analysis below is stored:
Named variable into which one-dimensional array of length:
containing the distinct values of the classification variables in ascending order is stored. The first N_Class_Vals(0) elements of Class_Vals contain the values for the first classification variables, the next N_Class_Vals(1) elements contain the values for the second classification variable, etc.
Named variable into which a two-dimensional array of size n_coefficients by 4 containing the parameter estimates and associated statistics is stored:
When present in the model, the first coefficient in Coef_Stat is the estimate of the "nuisance" parameter, and the remaining coefficients are estimates of the parameters associated with the "linear" model, beginning with the intercept, if present. Nuisance parameters are as follows:
Named variable into which a two-dimensional array of size n_coefficients by n_coefficients containing the estimated asymptotic covariance matrix of the coefficients is stored. For Itmax = 0, this is the Hessian computed at the initial parameter estimates.
Named variable into which the optimized criterion is stored. The criterion to be maximized is a constant plus the log-likelihood.
If present and nonzero, double precision is used.
Convergence criterion. Convergence is assumed when maximum relative change in any coefficient estimate is less than Eps from one iteration to the next or when the relative change in the log-likelihood, criterion, from one iteration to the next is less than Eps/100.0. Default: Eps = 0.001
Increment between time points on the time grid. Keywords Est_Delta, Est_Nobs, Est_Time, Est_Npt, and Est_Prob must be used together.
Number of observations for which estimates are to be calculated. Est_Nobs must be positive. Keywords Est_Nobs, Est_Time, Est_Npt, Est_Delta, and Est_Prob must be used together.
Number of points on the time grid for which survival probabilities are desired. Est_Npt must be positive. Keywords Est_Npt, Est_Nobs, Est_Time, Est_Delta, and Est_Prob must be used together.
Named variable into which a two-dimensional array of size Est_Npt by (2*n_observations + 1) containing the estimated survival probabilities for the covariate groups specified in x is stored. Column 0 contains the survival time. Columns 1 and 2 contain the estimated survival probabilities and hazard rates, respectively, for the covariates in the first row of x. In general, the survival and hazard row i of x is contained in columns 2i – 1 and 2i, respectively, for i = 1, 2, ..., Est_Npt. Keywords Est_Prob, Est_Nobs, Est_Time, Est_Npt, and Est_Delta must be used together.
Beginning of the time grid for which estimates are desired. Survival probabilities and hazard rates are computed for each covariate vector over the grid of time points Est_Time + i*Est_Delta for i = 0, 1, ..., Est_Npt -1. Keywords Est_Time, Est_Nobs, Est_Npt, Est_Delta, and Est_Prob must be used together.
Named variable into which an one-dimensional array of length n_observations containing the estimated linear response:
for each row of x is stored. To use keyword Est_Xbeta, you must also use keywords Est_Nobs, Est_Time, Est_Npt, Est_Delta, and Est_Prob.
The column in x containing the censoring code for each observation. Possible values are shown in Table 22-1.
Column number in x containing a fixed parameter for each observation that is added to the linear response prior to computing the model parameter. The "fixed" parameter allows one to test hypothesis about the parameters via the log-likelihoods.
The column number of x containing the frequency of response for each observation.
The column number of x containing the upper endpoint of the failure interval for interval- and left-censored observations.
One-dimensional index array of length Var_Effects(0) + Var_Effects(1) + ... + Var_Effects(n_effects - 1). The first Var_Effects(0) elements give the column numbers of x for each variable in the first effect. The next Var_Effects(1) elements give the column numbers for each variable in the second effect. The last Var_Effects(n_effects - 1) elements give the column numbers for each variable in the last effect. Keywords Indicies_Effects and Var_Effects must be used together.
One-dimensional array containing the initial estimates of the parameters (which requires that you know the number of coefficients in the model prior to the use of IMSL_SURVIVAL_GLM). See output keyword Coef_Stat for a description of the "nuisance" parameter, which is the first element of array Init_Est. By default, un-weighted linear regression is used to obtain initial estimates.
The column number of x containing the lower endpoint of the failure interval for interval- and right-censored observations.
Named variable into which a two-dimensional array of size, n by 5 containing information about each iteration of the analysis is stored, where n is equal to the number of iterations. This is shown in Table 22-2.
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Column
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Statistic
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Method of iteration
Q-N Step = 0 N-R Step = 1 |
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Iteration number
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Step size
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Maximum scaled coefficient update
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Log-likelihood
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Maximum number of iterations. Use Itmax = 0 to compute the Hessian, stored in Covariances, and the Newton step, stored in Last_Step, at the initial estimates (The initial estimates must be input. Use keyword Init_Est). See Example 3. Default: Itmax = 30
Named variable into which an one-dimensional array of length n_coefficients containing the last parameter updates (excluding step halvings) is stored. Keyword Last_Step is computed as the inverse of the matrix of second partial derivatives times the vector of first partial derivatives of the log-likelihood. When Itmax = 0, the derivatives are computed at the initial estimates.
Remove a right- or left-censored observation from the log-likelihood whenever the probability of the observation exceeds 0.995. At convergence, use linear programming to check that all removed observations actually have infinite linear response:
Obs_Status(i) is set to 2 if the linear response is infinite (See keyword Obs_Status). If not all removed observations have infinite linear response, recompute the estimates based upon the observations with finite:
Keyword Lp_Max is the maximum number of observations that can be handled in the linear programming. Setting Lp_Max = n_observations is always sufficient. By default, the function iterates without checking for infinite estimates. Default: No infinity checking; Lp_Max = 0
An upper bound on the sum of the number of distinct values taken on by each classification variable. Internal workspace usage can be significantly reduced with an appropriate choice of Max_Class. Default: Max_Class = n_observations * n_class
Named variable into which an one-dimensional array containing the means of the design variables is stored. The array is of length n_coefficients – k if keyword No_Intercept is used, and of length n_coefficients – k – 1 otherwise. Here, k is equal to 0 if model = 0, and equal to 1 otherwise.
Named variable into which an one-dimensional array of length n_class containing the number of values taken by each classification variable is stored; the i-th classification variable has N_Class_Vals(i).
Named variable into which the number of rows of data that contain missing values in one or more of the following vectors or columns of x is stored: Icen, Ilt, Irt, Ifreq, Ifix, or Indicies_Effects.
If present and nonzero, there is no intercept in the model. By default, the intercept is automatically included in the model.
Named variable into which an one-dimensional array of length n_observations indicating which observations are included in the extended likelihood is stored.
One-dimensional array of length n_effects containing the number of variables associated with each effect in the model, where n_effects is the number of effects (sources of variation) in the model. Keywords Var_Effects and Indicies_Effects must be used together.
The IMSL_SURVIVAL_GLM function computes the maximum likelihood estimates of parameters and associated statistics in generalized linear models commonly found in survival (reliability) analysis. Although the terminology used will be from the survival area, the methods discussed have applications in many areas of data analysis, including reliability analysis and event history analysis. These methods can be used anywhere a random variable from one of the discussed distributions is parameterized via one of the models available in IMSL_SURVIVAL_GLM. Thus, while it is not advisable to do so, standard multiple linear regression can be performed by routine IMSL_SURVIVAL_GLM. Estimates for any of 10 standard models can be computed. Exact, left-censored, right-censored, or interval-censored observations are allowed (note that left censoring is the same as interval censoring with the left endpoint equal to the left endpoint of the support of the distribution).
Let h = xTb be the linear parameterization, where x is a design vector obtained by IMSL_SURVIVAL_GLM via IMSL_REGRESSORS from a row of x, and b is a vector of parameters associated with the linear model. Let T denote the random response variable and S(t) denote the probability that T > t. All models considered also allow a fixed parameter wi for observation i (input in column Ifix of x). Use of this parameter is discussed below. There also may be nuisance parameters q > 0, or s > 0 to be estimated (along with b) in the various models. Let F denote the cumulative normal distribution. The survival models available in IMSL_SURVIVAL_GLM are listed in Table 22-3.
Note that the log-least-extreme-value model is a re-parameterization of the Weibull model. Moreover, models 0, 1, 2, 4, 6, 8, and 10 require that T > 0, while all of the remaining models allow any value for T, –¥ < T < ¥.
Each row vector in the data matrix can represent a single observation; or, through the use of vector frequencies, each row can represent several observations. Also note that classification variables and their products are easily incorporated into the models via the usual regression-type specifications.
The constant parameter Wi is input in x and may be used for a number of purposes. For example, if the parameter in an exponential model is known to depend upon the size of the area tested, volume of a radioactive mass, or population density, and so on, then a multiplicative factor of the exponential parameter l= exp (xb) may be known beforehand. This factor can be input in Wi (Wi is the log of the factor).
An alternate use of Wi is as follows: It may be that l = exp (x1b1 + x2b2), where b2 is known. Letting Wi = x2b2, estimates for b1 can be obtained via IMSL_SURVIVAL_GLM with the known fixed values for b2. Standard methods can then be used to test hypothesis about b1 via computed log-likelihoods.
The computations proceed as follows:
Models 2-10:
at the upper limit of each failure interval are obtained. (Because upper limits are used, interval- and left-censored data are assumed to be exact failures at the upper endpoint of the failure interval.) The Kaplan-Meier estimate is computed under the assumption that all failure distributions are identical (i.e., all b's but the intercept, if present, are assumed to be zero).
where t' is computed at the upper endpoint of each failure interval, t' = t in models 3, 5, 7, and 9, and t' = ln (t) in models 2, 4, 6, 8, and 10, and wi is the fixed constant, if present.
If there is no intercept in the model, then a is fixed at zero, and the model:
is fit instead. In this model, the coefficients b are used in place of the location estimate a above. Here:
is estimated from the simple linear regression with a = 0.
and the initial estimate of the intercept is assumed to be:
In the Weibull model:
and the intercept is assumed to be:
Initial estimates of all parameters b, other than the intercept, are assumed to be zero.
If there is no intercept in the model, the scale parameter is estimated as above, and the estimates:
from Step 2 are used as initial estimates for the b's.
Models 0 and 1:
For the exponential models (model = 0 or 1), the "average total time on" test statistic is used to obtain an estimate for the intercept. Specifically, let Tt denote the total number of failures divided by the total time on test. The initial estimates for the intercept is then ln(Tt). Initial estimates for the remaining parameters b are assumed to be zero, and if model = 1, the initial estimate for the linear hazard parameter q is assumed to be a small positive number. When the intercept is not in the model, the initial estimate for the parameter q is assumed to be a small positive number, and initial estimates of the parameters b are computed via multiple linear regression as in Part A.
where l'ia j is the partial derivative of the i-th term in the log-likelihood with respect to the parameter aj, and aj denotes one of the parameters to be estimated.
When the relative change in the log-likelihood from one iteration to the next is 0.1 or less, exact second partial derivatives are used for the Hessian so the Newton-Rapheson iteration is used.
If the initial step size results in an increase in the log-likelihood, the full step is used. If the log-likelihood decreases for the initial step size, the step size is halved, and a check for an increase in the log-likelihood performed. Step-halving is performed (as a simple line search) until an increase in the log-likelihood is detected, or until the step size becomes very small (the initial step size is 1.0).
As an example of a situation in which infinite estimates can occur, suppose that observation j is right-censored with tj > 15 in a normal distribution model in which the mean is:
where xj is the observation design vector. If the design vector xj for parameter bm is such that xjm = 1 and xim = 0 for all i ¹ j, then the optimal estimate of bm occurs at:
leading to an infinite estimate of both bm and hj. In IMSL_SURVIVAL_GLM, such estimates can be "computed."
In all models fit by IMSL_SURVIVAL_GLM, infinite estimates can only occur when the optimal estimated probability associated with the left- or right-censored observation is 1. If infinity checking is on, left- or right-censored observations that have estimated probability greater than 0.995 at some point during the iterations are excluded from the log-likelihood, and the iterations proceed with a log-likelihood based on the remaining observations. This allows convergence of the algorithm when the maximum relative change in the estimated coefficients is small and also allows for a more precise determination of observations with infinite:
At convergence, linear programming is used to ensure that the eliminated observations have infinite hi. If some (or all) of the removed observations should not have been removed (because their estimated hi's must be finite), then the iterations are restarted with a log-likelihood based upon the finite hi observations. See Clarkson and Jennrich (1988) for more details.
By default, or when not using keyword Lp_Max (see keyword Lp_Max), no observations are eliminated during the iterations. In this case, the infinite estimates occur, some (or all) of the coefficient estimates:
will become large, and it is likely that the Hessian will become (numerically) singular prior to convergence.
denote the derivative of Ii with respect to h = xTb, H denote the Hessian, and E denote expectation. Then the columns of Case_Analysis are:
This quantity is a one-step approximation of the change in the estimates when the i-th observation is deleted (ignoring the nuisance parameters).
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The IMSL_SURVIVAL_GLM function computes estimates of survival probabilities and hazard rates for the parametric survival/reliability models when using the Est_* keywords.
Let h = xTb be the linear parameterization, where x is the design vector corresponding to a row of x (IMSL_SURVIVAL_GLM generates the design vector using IMSL_REGRESSORS), and b is a vector of parameters associated with the linear model. Let T denote the random response variable and S(t) denote the probability that T > t. All models considered also allow a fixed parameter w (input in column Ifix of x). Use of the keyword is discussed in above. There also may be nuisance parameters q > 0 or s > 0. Let l(t) denote the hazard rate at time t. Then l(t) and S(t) are related at:
Models 0, 1, 2, 4, 6, 8, and 10 require that T > 0 (in which case assume l(s) = 0 for s < 0), while the remaining models allow arbitrary values for T, -¥ < T < ¥. The computations proceed in IMSL_SURVIVAL_GLM when using the keywords Est_* as follows:
Indicator (dummy) variables are created for the classification variables using IMSL_REGRESSORS (Chapter 3, Regression) using keyword Dummy_Method.
The following four examples all use the array x, defined as follows:
x1 = [[1.0, 0.0, 7.0, 64.0, 5.0, 411.0, 0.0] , $ [1.0, 0.0, 6.0, 63.0, 9.0, 126.0, 0.0] , $ [1.0, 0.0, 7.0, 65.0, 11.0, 118.0, 0.0] , $ [1.0, 0.0, 4.0, 69.0, 10.0, 92.0, 0.0] , $ [1.0, 0.0, 4.0, 63.0, 58.0, 8.0, 0.0] , $ [1.0, 0.0, 7.0, 48.0, 9.0, 25.0, 1.0] , $ [1.0, 0.0, 7.0, 48.0, 11.0, 11.0, 0.0] , $ [2.0, 0.0, 8.0, 63.0, 4.0, 54.0, 0.0] , $ [2.0, 0.0, 6.0, 63.0, 14.0, 153.0, 0.0] , $ [2.0, 0.0, 3.0, 53.0, 4.0, 16.0, 0.0]] ; , $ x2 = [[2.0, 0.0, 8.0, 43.0, 12.0, 56.0, 0.0] , $ [2.0, 0.0, 4.0, 55.0, 2.0, 21.0, 0.0] , $ [2.0, 0.0, 6.0, 66.0, 25.0, 287.0, 0.0] , $ [2.0, 0.0, 4.0, 67.0, 23.0, 10.0, 0.0] , $ [3.0, 0.0, 2.0, 61.0, 19.0, 8.0, 0.0] , $ [3.0, 0.0, 5.0, 63.0, 4.0, 12.0, 0.0] , $ [4.0, 0.0, 5.0, 66.0, 16.0, 177.0, 0.0] , $ [4.0, 0.0, 4.0, 68.0, 12.0, 12.0, 0.0] , $ [4.0, 0.0, 8.0, 41.0, 12.0, 200.0, 0.0] , $ [4.0, 0.0, 7.0, 53.0, 8.0, 250.0, 0.0]] ;, $ x3 = [[4.0, 0.0, 6.0, 37.0, 13.0, 100.0, 0.0] , $ [1.0, 1.0, 9.0, 54.0, 12.0, 999.0, 0.0] , $ [1.0, 1.0, 5.0, 52.0, 8.0, 231.0, 1.0] , $ [1.0, 1.0, 7.0, 50.0, 7.0, 991.0, 0.0] , $ [1.0, 1.0, 2.0, 65.0, 21.0, 1.0, 0.0] , $ [1.0, 1.0, 8.0, 52.0, 28.0, 201.0, 0.0] , $ [1.0, 1.0, 6.0, 70.0, 13.0, 44.0, 0.0] , $ [1.0, 1.0, 5.0, 40.0, 13.0, 15.0, 0.0] , $ [2.0, 1.0, 7.0, 36.0, 22.0, 103.0, 1.0] , $ [2.0, 1.0, 4.0, 44.0, 36.0, 2.0, 0.0]]; , $ x4 = [[2.0, 1.0, 3.0, 54.0, 9.0, 20.0, 0.0] , $ [2.0, 1.0, 3.0, 59.0, 87.0, 51.0, 0.0] , $ [3.0, 1.0, 4.0, 69.0, 5.0, 18.0, 0.0] , $ [3.0, 1.0, 6.0, 50.0, 22.0, 90.0, 0.0] , $ [3.0, 1.0, 8.0, 62.0, 4.0, 84.0, 0.0] , $ [4.0, 1.0, 7.0, 68.0, 15.0, 164.0, 0.0] , $ [4.0, 1.0, 3.0, 39.0, 4.0, 19.0, 0.0] , $ [4.0, 1.0, 6.0, 49.0, 11.0, 43.0, 0.0] , $ [4.0, 1.0, 8.0, 64.0, 10.0, 340.0, 0.0] , $ [4.0, 1.0, 7.0, 67.0, 18.0, 231.0, 0.0]] x = [[x1], [x2], [x3], [x4]] x = TRANSPOSE(x)
This example is taken from Lawless (1982, p. 287) and involves the mortality of patients suffering from lung cancer. An exponential distribution is fit for the model:
h = m + a i + g k + b 6 x3 + b 7x4 + b 8x5
where ai is associated with a classification variable with four levels, and gk is associated with a classification variable with two levels. Note that because the computations are performed in single precision, there will be some small variation in the estimated coefficients across different machine environments.
.RUN PRO print_results, cs PRINT, ' Coefficient Satistics' PRINT, ' Coefficient s.e z p' PM, cs, FORMAT = '(4F14.4)' END n_class = 2 n_continuous = 3 model = 0 icen = 6 irt = 5 lp_max = 40 n_coef = IMSL_SURVIVAL_GLM(n_class, n_continuous, model, x, $ ICEN = icen, IRT = irt, LP_MAX = lp_max, COEF_STAT = cs) print_results, cs Coefficient Satistics Coefficient s.e z p -1.1027 1.3091 -0.8423 0.3998 -0.3626 0.4446 -0.8156 0.4149 0.1271 0.4863 0.2613 0.7939 0.8690 0.5861 1.4825 0.1385 0.2697 0.3882 0.6948 0.4873 -0.5400 0.1081 -4.9946 0.0000 -0.0090 0.0197 -0.4594 0.6460 -0.0034 0.0117 -0.2912 0.7710
This example uses the same array x defined in Example 1, but more optional arguments are demonstrated.
.RUN PRO print_results, cs, iter, crit, nmiss PRINT, ' Coefficient Satistics' PRINT, ' Coefficient s.e z p' PM, cs, FORMAT = '(4F14.4)' PRINT PRINT, ' Iteration Information' PRINT, 'Method Iteration Step Size Coef Update ', $ 'Log-Likelihood' PM, iter, FORMAT = '(I3, I10, 2F14.4, F14.1)' PRINT PRINT, 'Log-Likelihood =', crit PRINT PRINT, 'Number of Missing Value = ', nmiss, $ FORMAT = '(A26, I3)' END n_class = 2 n_continuous = 3 model = 0 icen = 6 irt = 5 lp_max = 40 n_coef = IMSL_SURVIVAL_GLM(n_class, n_continuous, model, x, $ ICEN = icen, IRT = irt, LP_MAX = lp_max, $ N_CLASS_VALS = ncv, CLASS_VALS = cv, $ COEF_STAT = cs, CRITERION = crit, $ MEANS = means, CASE_ANALYSIS = ca, $ ITERATIONS = iter, OBS_STATUS = os, NMISSING = nmiss) print_results, cs, iter, crit, nmiss Coefficient Satistics Coefficient s.e z p -1.1027 1.3091 -0.8423 0.3998 -0.3626 0.4446 -0.8156 0.4149 0.1271 0.4863 0.2613 0.7939 0.8690 0.5861 1.4825 0.1385 0.2697 0.3882 0.6948 0.4873 -0.5400 0.1081 -4.9946 0.0000 -0.0090 0.0197 -0.4594 0.6460 -0.0034 0.0117 -0.2912 0.7710 Iteration Information Method Iteration Step Size Coef Update Log-Likelihood 0 0 NaN NaN -224.0 0 1 1.0000 0.9839 -213.4 1 2 1.0000 3.6034 -207.3 1 3 1.0000 10.1238 -204.3 1 4 1.0000 0.1430 -204.1 1 5 1.0000 0.0117 -204.1 Log-Likelihood = -204.139 Number of Missing Value = 0
In this example, the same data and model as example 1 are used, but Itmax is set to zero iterations with model coefficients restricted such that m = -1.25, b6 = -0.6, and the remaining six coefficients are equal to zero. A chi-squared statistic, with 8 degrees of freedom for testing the coefficients is specified as above (versus the alternative that it is not as specified), can be computed, based on the output, as:
where:
is output in Covariances. The resulting test statistic, c2 = 6.107, based upon no iterations is comparable to likelihood ratio test that can be computed from the log-likelihood output in this example (-206.683) and the log-likelihood output in Example 2 (-204.139).
.RUN
PRO print_results, cs, means, cov, crit, ls
PRINT, ' Coefficient Satistics'
PRINT, ' Coefficient s.e z p'
PM, cs, FORMAT = '(4F14.4)'
PRINT
PRINT, ' Covariate Means'
PRINT, means, FORMAT = '(7F8.2)'
PRINT
PRINT, ' Hessian'
PM, cov, FORMAT = '(8F8.4)'
PRINT
PRINT, 'Log-Likelihood =', crit
PRINT
PRINT, ' Newton_Raphson Step'
PRINT, ls, FORMAT = '(8F8.4)'
END
n_class = 2
n_continuous = 3
model = 0
icen = 6
irt = 5
lp_max = 40
itmax = 0
init_est = [-1.25, 0.0, 0.0, 0.0, 0.0, -0.6, 0.0, 0.0]
n_coef = IMSL_SURVIVAL_GLM(n_class, n_continuous, model, x, $
ICEN = icen, IRT = irt, ITMAX = itmax, $
LP_MAX = lp_max, INIT_EST = init_est, $
COEF_STAT = cs, CRITERION = crit, $
COVARIANCES = cov, MEANS = means, LAST_STEP = ls)
print_results, cs, means, cov, crit, ls
Coefficient Satistics
Coefficient s.e z p
-1.2500 1.3773 -0.9076 0.3643
0.0000 0.4288 0.0000 1.0000
0.0000 0.5299 0.0000 1.0000
0.0000 0.7748 0.0000 1.0000
0.0000 0.4051 0.0000 1.0000
-0.6000 0.1118 -5.3652 0.0000
0.0000 0.0215 0.0000 1.0000
0.0000 0.0109 0.0000 1.0000
Covariate Means
0.35 0.28 0.12 0.53 5.65 56.58 15.65
Hessian
1.8969 -0.0906 -0.1641 -0.1681 0.0778 -0.0818 -0.0235 -0.0012
-0.0906 0.1839 0.0996 0.1191 0.0358 -0.0005 -0.0008 0.0006
-0.1641 0.0996 0.2808 0.1264 -0.0226 0.0104 0.0005 -0.0021
-0.1681 0.1191 0.1264 0.6003 0.0460 0.0193 -0.0016 0.0007
0.0778 0.0358 -0.0226 0.0460 0.1641 0.0060 -0.0040 0.0017
-0.0818 -0.0005 0.0104 0.0193 0.0060 0.0125 0.0000 0.0003
-0.0235 -0.0008 0.0005 -0.0016 -0.0040 0.0000 0.0005 -0.0001
-0.0012 0.0006 -0.0021 0.0007 0.0017 0.0003 -0.0001 0.0001
Log-Likelihood = -206.683
Newton_Raphson Step
0.1706 -0.3365 0.1333 1.2967 0.2985 0.0625 -0.0112 -0.0026
This example is a continuation of the first example above. Keywords Est_* are used in IMSL_SURVIVAL_GLM to compute the parameter estimates. The example is taken from Lawless (1982, p. 287) and involves the mortality of patients suffering from lung cancer.
.RUN PRO print_results, ep PRINT, ' Survival and Hazard Estimates' PRINT, ' Time S1 H1 S2 H2' PM, ep, FORMAT = '(F7.2, F10.4, F13.6, F10.4, F13.6)' END n_class = 2 n_continuous = 3 model = 0 icen = 6 irt = 5 lp_max = 40 time = 10.0 npt = 10 delta = 20.0 n_coef = IMSL_SURVIVAL_GLM(n_class, n_continuous, model, x, $ ICEN=icen, IRT=irt, LP_MAX=lp_max, N_CLASS_VALS=nvc, $ CLASS_VALS=cv, COEF_STAT=cs, CRITERION=crit, MEANS=means, $ CASE_ANALYSIS=ca, OBS_STATUS=os, ITERATIONS=iter, $ EST_NOBS=2, EST_TIME=time, EST_NPT=npt, $ EST_DELTA=delta, EST_PROB=ep, EST_XBETA=xb) print_results, ep Survival and Hazard Estimates Time S1 H1 S2 H2 10.00 0.9626 0.003807 0.9370 0.006503 30.00 0.8921 0.003807 0.8228 0.006503 50.00 0.8267 0.003807 0.7224 0.006503 70.00 0.7661 0.003807 0.6343 0.006503 90.00 0.7099 0.003807 0.5570 0.006503 110.00 0.6579 0.003807 0.4890 0.006503 130.00 0.6096 0.003807 0.4294 0.006503 150.00 0.5649 0.003807 0.3770 0.006503 170.00 0.5235 0.003807 0.3310 0.006503 190.00 0.4852 0.003807 0.2907 0.006503
STAT_CONVERGENCE_ASSUMED_1—Too many step halvings. Convergence is assumed.
STAT_CONVERGENCE_ASSUMED_2—Too many step iterations. Convergence is assumed.
STAT_NO_PREDICTED_1—"estimates(0)" > 1.0. The expected value for the log logistic distribution ("model" = 4) does not exist. Predicted values will not be calculated.
STAT_NO_PREDICTED_2—"estimates(0)" > 1.0. The expected value for the log extreme value distribution("model" = 8) does not exist. Predicted values will not be calculated.
STAT_NEG_EIGENVALUE—The Hessian has at least one negative eigenvalue. An upper bound on the absolute value of the minimum eigenvalue is # corresponding to variable index #.
STAT_INVALID_FAILURE_TIME_4—"x(#)("Ilt"= #)" = # and "x(#)
("Irt"= #)" = #. The censoring interval has length 0.0. The censoring code for this observation is being set to 0.0.
STAT_MAX_CLASS_TOO_SMALL—The number of distinct values of the classification variables exceeds "Max_Class" = #.
STAT_TOO_FEW_COEF—Init_Est is specified, and "Init_Est" = #. The model specified requires # coefficients.
STAT_TOO_FEW_VALID_OBS—"n_observations" = # and "Nmissing" = #. "n_observations"("Nmissing" must be greater than or equal to 2 in order to estimate the coefficients.
STAT_SVGLM_1—For the exponential model ("model" = 0) with "n_effects" = # and no intercept, "n_coef" has been determined to equal 0. With no coefficients in the model, processing cannot continue.
STAT_INCREASE_LP_MAX—Too many observations are to be deleted from the model. Either use a different model or increase the workspace.
STAT_INVALID_DATA_8—"Class_Vals(#)" = #. The number of distinct values for each classification variable must be greater than one.