NAG Library Routine Document
G11CAF returns parameter estimates for the conditional logistic analysis of stratified data, for example, data from case-control studies and survival analyses.
|SUBROUTINE G11CAF (
||N, M, NS, Z, LDZ, ISZ, IP, IC, ISI, DEV, B, SE, SC, COV, NCA, NCT, TOL, MAXIT, IPRINT, WK, LWK, IFAIL)
||N, M, NS, LDZ, ISZ(M), IP, IC(N), ISI(N), NCA(NS), NCT(NS), MAXIT, IPRINT, LWK, IFAIL
||Z(LDZ,M), DEV, B(IP), SE(IP), SC(IP), COV(IP*(IP+1)/2), TOL, WK(LWK)
In the analysis of binary data, the logistic model is commonly used. This relates the probability of one of the outcomes, say
explanatory variates or covariates by
is a vector of unknown coefficients for the covariates
is a constant term. If the observations come from different strata or groups,
would vary from strata to strata. If the observed outcomes are independent then the
s follow a Bernoulli distribution, i.e., a binomial distribution with sample size one and the model can be fitted as a generalized linear model with binomial errors.
In some situations the number of observations for which
may not be independent. For example, in epidemiological research, case-control studies are widely used in which one or more observed cases are matched with one or more controls. The matching is based on fixed characteristics such as age and sex, and is designed to eliminate the effect of such characteristics in order to more accurately determine the effect of other variables. Each case-control group can be considered as a stratum. In this type of study the binomial model is not appropriate, except if the strata are large, and a conditional logistic model is used. This considers the probability of the cases having the observed vectors of covariates given the set of vectors of covariates in the strata. In the situation of one case per stratum, the conditional likelihood for
strata can be written as
is the set of observations in the
th stratum, with associated vectors of covariates
is the vector of covariates of the case in the
th stratum. In the general case of
cases per strata then the full conditional likelihood is
is the sum of the vectors of covariates for the cases in the
th stratum and
refer to the sum of vectors of covariates for all distinct sets of
observations drawn from the
th stratum. The conditional likelihood can be maximized by a Newton–Raphson procedure. The covariances of the parameter estimates can be estimated from the inverse of the matrix of second derivatives of the logarithm of the conditional likelihood, while the first derivatives provide the score function,
, which can be used for testing the significance of parameters.
If the strata are not small,
can be large so to improve the speed of computation, the algorithm in Howard (1972)
and described by Krailo and Pike (1984)
A second situation in which the above conditional likelihood arises is in fitting Cox's proportional hazard model (see G12BAF
) in which the strata refer to the risk sets for each failure time and where the failures are cases. When ties are present in the data G12BAF
uses an approximation. For an exact estimate, the data can be expanded using G12ZAF
to create the risk sets/strata and G11CAF used.
Cox D R (1972) Regression models in life tables (with discussion) J. Roy. Statist. Soc. Ser. B 34 187–220
Cox D R and Hinkley D V (1974) Theoretical Statistics Chapman and Hall
Howard S (1972) Remark on the paper by Cox, D R (1972): Regression methods J. R. Statist. Soc. B 34 and life tables 187–220
Krailo M D and Pike M C (1984) Algorithm AS 196. Conditional multivariate logistic analysis of stratified case-control studies Appl. Statist. 33 95–103
Smith P G, Pike M C, Hill P, Breslow N E and Day N E (1981) Algorithm AS 162. Multivariate conditional logistic analysis of stratum-matched case-control studies Appl. Statist. 30 190–197
- 1: N – INTEGERInput
On entry: , the number of observations.
- 2: M – INTEGERInput
: the number of covariates in array Z
- 3: NS – INTEGERInput
On entry: the number of strata, .
- 4: Z(LDZ,M) – REAL (KIND=nag_wp) arrayInput
On entry: the th row must contain the covariates which are associated with the th observation.
- 5: LDZ – INTEGERInput
: the first dimension of the array Z
as declared in the (sub)program from which G11CAF is called.
- 6: ISZ(M) – INTEGER arrayInput
: indicates which subset of covariates are to be included in the model.
If , the th covariate is included in the model.
If , the th covariate is excluded from the model and not referenced.
and at least one value must be nonzero.
- 7: IP – INTEGERInput
, the number of covariates included in the model as indicated by ISZ
number of nonzero values of ISZ
- 8: IC(N) – INTEGER arrayInput
: indicates whether the
th observation is a case or a control.
If , indicates that the th observation is a case.
If , indicates that the th observation is a control.
or , for .
- 9: ISI(N) – INTEGER arrayInput
: stratum indicators which also allow data points to be excluded from the analysis.
If , indicates that the th observation is from the th stratum, where .
If , indicates that the th observation is to be omitted from the analysis.
and more than IP
- 10: DEV – REAL (KIND=nag_wp)Output
On exit: the deviance, that is, , (maximized log marginal likelihood).
- 11: B(IP) – REAL (KIND=nag_wp) arrayInput/Output
: initial estimates of the covariate coefficient parameters
must contain the initial estimate of the coefficent of the covariate in Z
corresponding to the
th nonzero value of ISZ
in many cases an initial value of zero for
may be used. For another suggestion see Section 8
contains the estimate
of the coefficient of the covariate stored in the
th column of Z
th nonzero value in the array ISZ
- 12: SE(IP) – REAL (KIND=nag_wp) arrayOutput
On exit: is the asymptotic standard error of the estimate contained in and score function in , for .
- 13: SC(IP) – REAL (KIND=nag_wp) arrayOutput
On exit: is the value of the score function for the estimate contained in .
- 14: COV() – REAL (KIND=nag_wp) arrayOutput
: the variance-covariance matrix of the parameter estimates in B
stored in packed form by column, i.e., the covariance between the parameter estimates given in
, is given in
- 15: NCA(NS) – INTEGER arrayOutput
On exit: contains the number of cases in the th stratum, for .
- 16: NCT(NS) – INTEGER arrayOutput
On exit: contains the number of controls in the th stratum, for .
- 17: TOL – REAL (KIND=nag_wp)Input
On entry: indicates the accuracy required for the estimation. Convergence is assumed when the decrease in deviance is less than . This corresponds approximately to an absolute accuracy if the deviance is small and a relative accuracy if the deviance is large.
- 18: MAXIT – INTEGERInput
: the maximum number of iterations required for computing the estimates. If MAXIT
is set to
then the standard errors, the score functions and the variance-covariance matrix are computed for the input value of
is not updated.
- 19: IPRINT – INTEGERInput
: indicates if the printing of information on the iterations is required.
- No printing.
- The deviance and the current estimates are printed every IPRINT iterations. When printing occurs the output is directed to the current advisory message unit (see X04ABF).
- 20: WK(LWK) – REAL (KIND=nag_wp) arrayWorkspace
- 21: LWK – INTEGERInput
: the dimension of the array WK
as declared in the (sub)program from which G11CAF is called.
, where is the number of observations included in the model, i.e., the number of observations for which and is the maximum number of observations in any stratum.
- 22: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|On entry,||, for some ,|
|or||the value of IP is incompatible with ISZ,|
|or|| or .|
|or|| or ,|
|or||the number of values of is greater than or equal to , the number of observations excluding any with .|
The value of LWK
is too small.
Overflow has been detected. Try using different starting values.
The matrix of second partial derivatives is singular. Try different starting values or include fewer covariates.
Convergence has not been achieved in MAXIT
iterations. The progress towards convergence can be examined by using a nonzero value of IPRINT
. Any non-convergence may be due to a linear combination of covariates being monotonic with time.
Full results are returned.
The accuracy is specified by TOL
The other models described in Section 3
can be fitted using the generalized linear modelling routines G02GBF
The case with one case per stratum can be analysed by having a dummy response variable
for a case and
for a control, and fitting a Poisson generalized linear model with a log link and including a factor with a level for each strata. These models can be fitted by using G02GCF
G11CAF uses mean centering, which involves subtracting the means from the covariables prior to computation of any statistics. This helps to minimize the effect of outlying observations and accelerates convergence. In order to reduce the risk of the sums computed by Howard's algorithm becoming too large, the scaling factor described in Krailo and Pike (1984)
If the initial estimates are poor then there may be a problem with overflow in calculating or there may be non-convergence. Reasonable estimates can often be obtained by fitting an unconditional model.
The data was used for illustrative purposes by Smith et al. (1981)
and consists of two strata and two covariates. The data is input, the model is fitted and the results are printed.
9.1 Program Text
Program Text (g11cafe.f90)
9.2 Program Data
Program Data (g11cafe.d)
9.3 Program Results
Program Results (g11cafe.r)