## Brief Review Of Approximate Optimal Design Theory

An optimal design is one which minimizes (or maximizes) some function of the covariances of the parameter estimates. There are many optimal design criteria. Often they lead to complicated (or intractable) expressions to optimize. As a way of dealing with the problem of complicated algebraic expressions (like the one above), Kiefer5 proposed the concept of approximate designs. Although there was controversy at the time, it is now an accepted way of solving a design problem. Generally, a design...

## Acknowledgement

Kate Bull is supported by the British Heart Foundation. 1. Kirlin, J. W., Blackstone, E. H., Tchervenkov, C. I., Casteneda, A. R. and the Congenital Heart Surgeons Society. 'Clinical outcomes after the arterial switch operation for transposition patient, support, procedural and institutional risk factors', Circulation, 86, 1501-1515 (1992). 2. Hanley, F. L., Sade, R. M., Blackstone, E. H., Kirlin, J. W., Freedom, R. M. and Nanda, N. C., 'Outcomes in neonatal pulmonary atresia and intact...

## Acknowledgements

This work was supported by The World Health Organization ARI Programme, and for F. Harrell, Research Grants HS-06830 and HS-07137 from the Agency for Health Care Policy and Research, Rockville, Maryland, U.S.A., and grants from the Robert Wood Johnson Foundation, Princeton, NJ, U.S.A. F. Harrell wishes to dedicate his work on this project to the memory of his dear colleague L. Richard Smith whose critical reading of this paper resulted in significant 1. Walker, S. H. and Duncan, D. B....

## Alternatives To The Summary Measures Approach

Finney has provided a useful review of approaches to repeated measures 5 , These include the following 1. Multivariate analysis. Here the repeated measures over time are analysed using multivariate analysis of variance (MANOVA) or some equivalent technique. For example, for the case with two treatment groups, the repeated measures could be compared using Hotelling's T2 statistic. This approach makes no use of the sequential ordering of the measurements through time, since permuting the time...

## Analysing Missing Data In Longitudinal Clinical Trials

Missing data is a major problem in longitudinal clinical trials. In the drug addiction trial more than half the subjects have missing observations by the fourth week of follow-up. The missing data in this study is due to both drop-out as well as intermittently missing data. In addition, the proportion of missed observations was different by treatment arm patients randomized to the Methadone arm had higher proportions of missed observations than those randomized to Buprenorphine. This clearly...

## Appendix

Below are examples of computer code used to generate two of the figures presented in this manuscript. The first is an S-plus program that plots observed Kaplan-Meier curves overlaid on the average estimated patient-specific survival function. The second is an SAS program for calculating confidence intervals for odds ratios and hazard ratios. Example 1 note comments and documentation in S-plus begin with a Plot observed and predicted survival curve km.object - A Kaplan-Meier object from a model...

## C

(With covariates) individual capture history (Without covariates) both types of data Log-linear models Sample coverage approach Ecological multiplicative and logistic models i 5 is suggested for heterogeneous All estimators are independent of the ordering of the lists Some estimators depend on the ordering of the lists Note that when t 3, (16) reduces to (12) because tD - Z+n + Z1+1 + Z+l . where is similarly defined as those in (13a). Standard error estimate and confidence interval can be...

## Case Study

Consider the 506-patient prostate cancer dataset from Byar and Green67 which has also been analysed in references 68 and 69. The data are listed in reference 70, Table 46, and are available by Internet at utstat.toronto.edu in the directory pub data-collect. These data were from a randomized trial comparing four treatments for stage 3 and 4 prostate cancer, with almost equal numbers of patients on placebo and each of three doses of oestrogen. Four patients had missing values on all of the...

## Conclusion

Bayesian monitoring, as illustrated here, is very simple to implement. It helps to put into perspective one major inherent problem in the early termination of trials, namely the risk that the results will be regarded by sceptical clinicians as inconclusive. Thus it should help to ensure that trials only stop early when the results to date are sufficiently conclusive. Furthermore, our experience has been that in this context clinicians find Bayesian concepts intuitively appealing the idea of...

## Conclusions

Interval-censored data often occur in medical applications. Although only two of the major statistical packages (SAS1 and S-plus2) have procedures for analysing these data using parametric models, some non-parametric methods are easily programmed and their use should be considered. In particular, Turnbull's14 method for non-parametric estimation of the survival distribution, Kooperburg and Stone's20 logspline estimates of the survival function and Finkelstein's16 test for covariates are...

## Covrml XXr1 xjy fti y X XTX

U( J) DTF1(y- (y)) 0 where the N x K matrix D (N being the overall length of the data vector y and K the length of the parameter vector P) contains the derivatives dE(yij) d ik, and V cov(y) is the covariance matrix of the yi s, which contains known functions of E(yi ) and possibly other unknown parameters. Note that, for our logistic regression model, E(yij) ni (1 + e XJ )1, where Xy denotes the column vector of covariate values xijk. These estimating equations are the weighted least squares...

## Data Reduction

Multivariable statistical models when developed carefully are excellent tools for making prognos tic predictions. However, when the assumptions of a model are grossly violated or when a model is used unwisely for a given patient sample, the performance of the model may be poor. For example, when the analyst has fitted not only real trends that further data would support, but in addition has fitted idiosyncrasies in the particular dataset by analysing too many variables, the model may predict...

## Description Of Methods

The most straightforward approach, which can be implemented directly in SAS1 (and the survreg routine in S-plus,2 although we will not go into details for this here), is to assume a parametric model for the failure times. The SAS1 procedure LIFEREG provides a way of fitting accelerated failure time (AFT) models for a variety of distributions to interval censored data. The AFT model is defined by the transformation where Tz is the failure time random variable for an individual with covariate z...

## Discussion

We have used an example from coronary artery disease to demonstrate issues in the use of observational data, for both a treatment comparison analysis and for supplying information to the medical decision making process. We used a 'treatment strategy' approach to develop a statistical model that assesses long-term survival for patients with coronary artery disease following an initial treatment decision among MED, PTCA and CABG. Using a data framework that creates a treatment assignment and...

## Dsh i

They also showed that pms and pmodhon apply when the outcome variable is censored. They recommended using the minimum of these two adjusted p-values, since the formulae tend to give conservative corrections.5 3. CASE STUDY 1 TREATMENT FOR UNRESPONSIVE LYMPHOMA 3.1. Treatment regimen and rationale for categorization A standard approach to the treatment of patients with lymphoma that has not responded to conventional-dose chemotherapy is to administer it in high doses. Since high-dose (HD)...

## Effects Of Severe Sparseness

This section summarizes some special considerations and results when the data are severely sparse, such as effects of centres containing certain patterns of empty cells and effects of modifying the data such as by adding constants to empty cells or combining centres. Table V is an example of such data 60 . This table was shown to the first author a few years back by an attendee of a short course on categorical data analysis. It shows results for five centres of a clinical trial designed to...

## Errors

The principle of the information-sandwich method is the same in each of its specific applications and we describe it in the context of the generic set of estimating equations given by the quasi-likelihood specification (7) where n E(y). As discussed, these reduce in special cases to the likelihood score equations (3) as well as the other specifications considered in Section 3. The solution (i of these equations provides unbiased estimates of the regression parameters, . To derive an expression...

## F 0 E

Where k is the largest j such that tj < t, and the survival function is S(t) 1 F(t). This recursive method for calculating the survival function and the cumulative incidence function begins at time ii and we define S(t0) S 0) 1. The traditional Kaplan-Meier method assumes that the time scale for failure times is time on study or some other function of calendar time. It can, however, be modified for use with a survival age time scale where S is the probability of surviving beyond age A. Notice...

## Generalizations For Stratified Data

Typically, one studies dose-response relations while controlling for factors that could influence the relationship. For instance, one might display the relationship separately for men and for women, for different age groups, for different centres from which the data are obtained, or for different stages or levels of severity of the medical condition being treated. To illustrate, Table V is a stratified version of Table I that classifies subjects according to the trauma severity at the time of...

## 3c L

For relatively low sample coverage data, we feel the data do not contain sufficient information to accurately estimate the population size. In this case, the following 'one-step' estimator A'i is suggested (the estimator is called 'one-step' because it is obtained by one iterative step from the adjustment formula (10) with ytf s being replaced by (11)) M 5 + (Zi+o + Z+io)712 + (zio+ + Z+oi)y13 + (Z0i+ + Z0+i)y23

## Inference About Effects

For the fixed and random effects logit models, standard methods yield inferences about the treatment effect. For instance, the likelihood-ratio test statistic is minus twice the difference in maximized log-likelihoods between model (1) or (4) with P 0 and the model with unrestricted p. It has a null chi-squared distribution with d.f. 1, as does the Wald statistic, which is the squared ratio of the estimate to its standard error. The standard error is obtained from the inverse information...

## Info

Chi-squared statistics (or, taking square roots, z statistics) testing whether j8D 0 using the likelihood-ratio, Wald, or score approaches, in the same way as just discussed for two-way tables. To illustrate, applying the simpler model with a linear dose effect and dose scores (1, 2, 3, 4) to Table V, we get 0 0 205 (ASE 0 058). The Wald chi-squared statistic equals 12-5, and the likelihood-ratio statistic comparing this model to the simpler one without the dose effect equals the difference in...

## Interim Analyses

When a trial is being monitored there will be interim, and less precise, estimates of survx and surv2 which are based upon the survival experience of patients currently recruited to the trial and followed up until the time of the analysis. These estimates allow calculation of the data-based log hazard ratio log(hd), as will be shown later. The role of the interim analysis, and indeed the application of methods described in this paper, is to assess the currently available information from the...

## Introduction

The purpose of many epidemiological surveillance studies is to estimate the size of a population by merging several incomplete lists of names in the target population. Some examples are as follows 1. An outbreak of the hepatitis A virus (HAV) occurred in and around a college in northern Taiwan from April to July 1995 1 , Cases of students in that college were * Correspondence to Anne Chao, Institute of Statistics, National Tsing Hua University, Hsin-Chu 30043, Taiwan t E-mail chao...

## Justification For Our Choice Of Prior Distributions

We have described three prior distributions, although there are many potential distributions that could be considered Parmar et al.2 and Spiegelhalter et al.x discuss a few other possibilities. The difficulty of making an objective and non-controversial selection is, of course, one of the reasons why frequentist statisticians often object to Bayesian techniques. Thus, for example, if the outcome of a clinical trial is being reported using a Bayesian approach, there is always the anxiety that...

## K ak 82 2k ak

Logit( 1k ) ak + ft 2, logit( 2k ) ak ft 2 For binary data, random effects models are most commonly used with logit or probit link functions. A structural defect exists with the log and identity links in treating ak as normally distributed for any parameter values with a> 0, with positive probability a particular realization of the random effect corresponds to nik outside 0,1 . 2.3. Treatment-by-centre interaction Even if a model that is additive in centre and treatment effects fits sample...

## Km

Agelast STATUS dead(l) PRINT TABLE PLOT SURVIVAL. Section 5.2 and Table V outcome after 1 year. One factor at a time. COMPUTE filter. (adfol 1). FILTER BY filter_ . CROSSTABS TABLES paanat agepresx BY dedlyrpp FORMAT AVALUE NOINDEX BOX LABELS TABLES CELLS COUNT ROW. Section 7.1 and Table V. Outcome after 1 year. More than 1 explanatory variable. METHOD ENTER paanat agepresx FILTER OFF. EXECUTE. Section 6.1 and Figure 6. Survival with one fixed explanatory variable KM followup STRATA paanat...

## Longitudinal Methods For Gaussian Data

Longitudinal Gaussian outcomes are common in clinical trials. The primary interest in the IPPB trial, for example, is the change in lung volume function (measured by FEVj) over time. The study was designed to examine the average change in slopes across the treatment arms. Various approaches have been proposed for modelling longitudinal Gaussian data. Approaches can be divided into four major categories simple univariate methods multivariate methods including traditional growth curve modelling...

## Longitudinal Methods For Recurrent Events

Many clinical trials compare recurrent events across treatment arms. For example, clinical exacerbations are compared in the MS clinical trial and seizure occurrence is compared in the epilepsy clinical trial. There are two basic modelling approaches. One approach is to model the repeated times between recurrent events. The second approach is to model the number of events over a fixed follow-up interval. We discuss these two approaches in turn. Various authors have proposed Poisson regression...

## Macro Description

In this section we describe our SAS macro practical incidence estimators (PIE) which provides estimates of age-specific incidence rates, crude and age-adjusted incidence rates, estimates of the unadjusted cumulative incidence and cumulative incidence rates adjusted for competing risk of death. The macro also provides the remaining lifetime risk of developing the event of interest conditional on survival to selected ages. We describe the macro parameters and the modules (sub-macros) it calls....

## Methods

In this section we establish notation and provide the formulae that are operationalized in the macro. The description of the SAS macro follows in Section 5. Alzheimer's disease develops insidiously over time as compared to other events, such as stroke, that occur suddenly. For this reason, its development is reported using a year (rather than a specific dale) of diagnosis. To be consistent, we use years as the measure of lime for all analyses. Consider the following three subjects

## Model Fitting And Estimating Effects

Most major software packages are not equipped to fit generalized linear models with random effects. Version 7 of SAS includes PROC NLMIXED, which can provide a good approximation to ML using adaptive Gauss-Hermite quadrature. The linearization approximations 26,27 are available in earlier versions with a SAS macro, called GLIMMIX, that uses iterative calling of PROC MIXED. Most other specialized programs for hierarchical models with random effects likewise use various normal approximations to...

## Model Fitting For Table I

We now apply these methods to Table I. For these data the sample success rates vary markedly among centres both for the control and drug treatments, but in all except the last centre that rate is higher for drug. Normally in using models with random centre and possibly random treatment effects, one would prefer to have more than K 8 centres keeping in mind the difficulty particularly of getting good variance component estimates with such a small value of K, we use these data to illustrate the...

## Models And Estimators

This approach specifies various forms of capture probabilities based on empirical investigations of animal ecology. Although most authors in this field did not aim to model dependence between samples, dependence is induced when some special types of capture probabilities are formulated. Two types of probabilities have been proposed multiplicative and logistic. The multiplicative class of models was first proposed by Pollock 33 and was fully discussed in the two monographs by Otis et al. 34 and...

## Models And Summaries Of Effects

LogitOu) k + P 2, logit( 2k) ak - P 2 , k 1, , K log(rcik ) ak + 2 , log( 2k) ak - 2 With this model, exp( ) n1k n2k is a ratio of success rates, analogous to a relative risk in each centre. (Here, we use notation rather than ft to reflect the effect having a different meaning than in model (1) likewise, the intercept also refers to a different scale, but we use common ak notation for simplicity since this parameter is not the main focus of interest.) Model (2) has the structural disadvantage...

## Motivation

The estimation of yearly incidence is relatively straightforward however, in prospective studies such as the Framingham Study, there are several issues that make the estimation of cumulative incidence difficult. In order to generate a valid estimate of future risk, including the lifetime risk of developing AD, we must address the following. First, we must consider that individuals are followed for different periods of time. Second, the time origin must be defined such that individuals are...

## N

Pr (Zi zu ,Z zN X1 xu ,XN xN) n e(x l - efo) 1-2'. The propensity score is the 'coarsest function' of the covariates that is a balancing score, where a balancing score, b(X), is defined as 'a function of the observed covariates X such that the conditional distribution of X given b(X) is the same for treated (Z 1) and control (Z 0) units'.22 For a specific value of the propensity score, the difference between the treatment and control means for all units with that value of the propensity score...

## O

Medicine Only 30-day Strategy 45-day Strategy Medicine Only 30-day Strategy 45-day Strategy Note 30-day, 45-day, and 60-day Strategies are nearly indistinguishable Figure 4. Kaplan-Meier survival comparison among methods for assigning patients to MED. 30 days of catheterization. An additional 3651 patients underwent CABG as a first procedure, 75 per cent within 14 days and 81 per cent within 30 days of catheterization. The overall 75th percentile for time to procedure for CABG and PTCA patients...

## P 1 Pi

A smoothed plot (for example, using the moving linear regression algorithm in lowess49) of Xim versus v m provides a non-parametric estimate of how Xm relates to the log relative odds that Y l Xm. For ordinal Y, we just need to compute binary model partial residuals for all cut-offs j then to make a plot for each m showing smoothed partial residual curves for all j, looking for similar shapes and slopes for a given predictor for all j. Each curve provides an estimate of how Xm relates to the...

## Prognostic Clinical Prediction Models

Development of Health Risk Appraisal Functions in the Presence of Multiple Indicators The Framingham Study Nursing Home Institutionalization Model. R B. D'Agostino, Albert J. Belanger, Elizabeth W. Markson, Maggie Kelly-Hayes and Philip A. Wolf 209 Multivariable Prognostic Models Issues in Developing Models, Evaluating Assumptions and Adequacy, and Measuring and Reducing Errors. Frank E. Harrell Jr., Kerry L. Lee and Daniel B. Mark 223 Development of a Clinical Prediction Model for an Ordinal...

## References

Large sample variance of kappa in the case of different sets of raters. Psychological Bulletin 1979 86 974-977. 2. Scott WA. Reliability of content analysis the case of nominal scale coding. Public Opinion Quarterly 1955 321-325. 3. Cohen J. A coefficient of agreement for nominal scales. Educational and Psychological Measurement 1960 20 37-46. 4. Feinstein AR. A bibliography of publications on observer variability. Journal of Chronic Diseases 1985 38 619-632. 5....

## Relations

The tests in Section 2 are fine for detecting evidence against the null hypothesis in the direction of a positive trend. However, they do not lend much insight about the form of the relationship. A model-based perspective is superior for this purpose. A good-fitting model describes the nature of the association, provides parameters for describing the strength of the relationship, provides predicted probabilities for the response categories at any dose, and helps us to determine the optimal...

## Remarks And Discussion

Three classes of capture-recapture models have been reviewed in this tutorial ecological models log-linear models, and the sample coverage approach. Most ecological models allowing for heterogeneous capture probabilities are recommended only when there are at least five trapping samples, whereas the other two approaches are mainly useful for two to five samples. We have focused on the latter two models for epidemiological applications and demonstrated the use of the program CARE developed by...

## Sample Size And Power

Whitehead49 discussed sample size formulae for an ordered categorical response with the proportional odds model, though only for the case of two groups (for example, two doses). Suppose we want power 1 p in an a-level test for detecting an effect of size 0 in that model. The sample is to be allocated to the two groups in the ratio A to 1, and pj denotes the anticipated marginal proportion in response category '. Whitehead49 stated that the required sample size for a two-sided test is then...

## Significance Tests For A Monotone Dose Response Relationship

This section reviews significance tests for detecting monotone dose-response relationships. Section 3 discusses related tests for models for the relationship. Non-model-based inference, though less informative, is often considered simpler from a regulatory perspective because the tests do not need to validate any modelling assumptions 14 however, we shall note in Section 3 that some tests from this section are equivalent to tests for certain models. This section presents four approaches (i)...

## Smallsample And Sparsedata Inference

The test statistics presented in this article are large-sample statistics. For chi-squared statistics, the convergence to chi-squared distributions tends to be faster for statistics having smaller values of d.f., such as the single-degree-of-freedom statistics. For any particular statistic referring to the two-way contingency table of dose by ordinal response, one can construct a small-sample 'exact' test using the generalized hypergeometric distribution that results from conditioning on the...

## Soy qTxpix2

These predicted survival times are then plotted against the observed covariate values to look for monotonicity, a cutpoint, or a region in which a cutpoint search should be performed. Code for finding these predicted survival times is not straightforward, and is available from the author of reference 9. Whether or not graphical examinations of the data suggest an obvious cutpoint, a systematic search further aids in the cutpoint selection process. In this approach, all observed values of the...

## Stopping Criteria

The table showing the probabilities associated with various target improvements can be reviewed by the Data Monitoring Committee. However, although the Bayesian framework is useful for interpreting the current trial's results in an informal manner, it is still useful to have guidelines for when seriously to consider stopping. If the trial is between two treatment arms, then a reasonable criterion is to demand that the posterior probability of one treatment being better, in the light of a...

## Summary

Capture-recapture methodology, originally developed for estimating demographic parameters of animal populations, has been applied to human populations. This tutorial reviews various closed capture-recapture models which are applicable to ascertainment data for estimating the size of a target population based on several incomplete lists of individuals. Most epidemiological approaches merging different lists and eliminating duplicate cases are likely to be biased downwards. That is, the final...

## Summary Comments And Recommendations

Similarities and differences in substantive results For the examples in this paper, we reached similar conclusions about the treatment effect whether we used fixed effects or random effects models. Our experience with a variety of examples indicates that the fixed effects model and the random effects model assuming no interaction tend to provide similar results about the common treatment effect. Those results are also similar to the ones for the mean of the treatment effects for the random...

## Summary Of Modelling Strategy

Assemble accurate, pertinent data and as large a sample as possible. For survival time data, follow-up must be sufficient to capture enough events as well as the clinically meaningful phases if dealing with a chronic disease. 2. Formulate focused clinical hypotheses that lead to specification of relevant candidate predictors, the form of expected relationships, and possible interactions. 3. Discard observations having missing Y after characterizing whether they are missing at random. See...

## Survival Analysis In Observational Studies

1 Cardiothoracic Unit, Hospital for Sick Children, Great Ormond Street, London WC1N 3JH, U.K. 2 MRC Biostatistics Unit, Institute of Public Health, Forvie Site, Robinson Way, Cambridge CB2 2SR, U.K. Multi-centre databases are making an increasing contribution to medical understanding. While the statistical handling of randomized experimental studies is well documented in the medical literature, the analysis of observational studies requires the addressing of additional important issues relating...

## The Data And Research Questions

The Adolescent Health Cohort Study was a longitudinal study of teenagers in the state of Victoria carried out between August 1992 and July 1995. The basic design was to measure a cohort of approximately 2000 adolescents on six occasions or waves of data collection, at six-monthly intervals. Such studies are sometimes called 'multiwave' or 'panel' designs because the same 'panel' of participants is assessed at each wave of the study another term commonly used is 'repeated measures' cohort study....

## Using Approximations To Simplify The Model

It is tempting to use P-values and stepwise methods to develop a parsimonious prediction model. Besides invalidating confidence limits and causing measures of predictive accuracy such as adjusted R2 to be optimistic, there are many other reasons not to rely on stepwise techniques (see Harrell et al.20 for citations). We follow Spiegelhalter's advice to use full model fits in conjunction with shrinkage.56 Parsimonious models can be developed, however, by approximating predictions from the model...

## Validating The Model

Most analysts validate a fitted model using held-back data, but this method has severe drawbacks.20 The bootstrap technique59 allows the analyst to derive bias (overfitting) - corrected estimates of predictive accuracy without holding back valuable data during the model development phase. The steps required for using the bootstrap to bias-correct indexes such as Dxy and calibration error was summarized in Harrell et al.20 For the full CR model which was fitted using PMLE, we used 150 bootstrap...

## Variable Clustering

Expert clinical judgement was used to enumerate a list of clinical variables to collect, including 47 clinical signs. The list reflects the content of an expert paediatric examination. As a first step in * Proportion of deaths for BC+ , BC - , CSF+ , CSF- , Sa02 < 90 per cent, Sa02 > 90 per cent were, respectively, 0 30, 0-08, 0-29, 0-05, 0-25, 0-04 coding the predictor variables, all questionnaire items that were connected (for example, using skip rules such as 'if condition was present,...

## Volume 1 Statistical Methods in Clinical Studies

Copyright 2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Email (for orders and customer service enquiries) cs-books wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright,...

## What Should One Do When More Than One Baseline Measurement

In many clinical trials more than one baseline measurement will be available. For example, it is not uncommon for clinical trials to have a run-in period prior to randomization. It is then usual to have at least two baseline measurements available for patients one when first selected for entry into the trial and another just prior to randomization. There may also have been intervening measurements. Given that it has been agreed that these baseline measurements should be exploited in an analysis...

## X

Three dose response curves linear, quadratic and threshold The steps in conducting a dose-response study (in idealized form) consist of the following (a) Assume a form or model for the curve (for example, linear, quadratic or threshold). (b) Select the dose metameter (dose or log (dose)). (c) Design the study so 'good' information is obtained this includes obtaining estimates of the model coefficients with small standard errors and having the ability to test for model failures (such...

## Data Preparation And Data Reduction

The data analysis for the nursing home institutionalization function consisted of two phases. First, there was the data preparation and reduction phase and second there was the development of the risk appraisal functions separately for each sex. The objective of the first phase was to insure the availability of stable, reliable variables for inclusion in the model. We now discuss the As noted already, there were a large number of variables or indicators, many measuring the same underlying...

## Software For Analysing Longitudinal Data

Unfortunately, most published methodology for analysing longitudinal data has not been incorporated into commercial software. For most of the methods discussed in this paper one needs to either develop new software or write to the authors for their research programs. Fortunately, there are some notable exceptions. Programs for fitting models to Gaussian data (that is, linear and non-linear mixed models and models with time-series error structure) and GEE for continuous and discrete data are...

## Examples

Tions about when the events took place, for example at the beginning, midpoint or end of each The breast cancer data set Table I , described more fully in Finkelstein and Wolfe,8 consists of 94 observations from a retrospective study looking at the time to cosmetic deterioration. Information is available on one covariate, the type of therapy - either radiation alone coded 0 , or in combination with chemotherapy coded 1 . Of the 94 observations, 56 are interval-censored The AIDS data9 are...

## Model Validation Methods

As mentioned before, examination of the apparent accuracy of a multivariable model using the training dataset is not very useful. The most stringent test of a model and of the entire data collection system is an external validation - the application of the 'frozen' model to a new population. It is often the case that the failure of a model to validate externally could have been predicted from an honest unbiased 'internal' validation. In other words, it is likely that many clinical models which...