REFUERZO DE TALUDES

I {exp (p Z k (t))}

k e R

(8)

Estimates of the baseline hazard function, Xo^(t) can be made if desired'^^’'*^®. However, as these estimates are usually based only on the baseline values of the covariates the estimate of this hazard is of little relevance when considering time-updated covariates.

1.4 Confidence intervals and testing fo r significance

In large samples the distribution of the parameters p can be approximated by a normal distribution with mean, variances and covariances which can be estimated from the second derivative of the log of L(P). Hence confidence intervals and hypothesis tests for P can be performed in the usual way. Given a parameter estimate pj and standard error for this estimate, a 95% confidence interval can be calculated as

P j± 1 .9 6 xs.e .(p j) (9)

and the ratio of pj to its standard error can be compared to a Normal distribution in order to test whether it is significantly different from zero.

1.5 P ro p o rtio n a lity assum ption

This model relies on the fact that the ratio of hazards in equation (5) is independent of time. This assumption can be tested, both for fixed and time-updated covariates, by incorporating the interaction between the covariate of interest and the log of time in the model"^^^. If the parameter estimate for this interaction term is significantly different from zero then there is evidence that the relative hazards are not proportional over time. Alternatively, the total follow-up time can be split into intervals and the relative hazard estimated separately in each interval. These estimates can be visually inspected to see if the hazard appears to change over the different time intervals^^^.

2 MODELS FOR REPEATED MEASUREMENTS

2.1 in tro d u c tio n

Frequently, multiple measurements of the same marker are obtained from individuals at different time points. It may not be possible to control the circumstances under which measurements are taken and there may be considerable variation among individuals in

both the number and timing of observations. The resulting unbalanced data sets could be modelled using a general multivariate model. However, unless some restrictions are placed on the structure of the covariances between the repeated measurements on the same individual, a large number of parameters will be required and many of these will be poorly estimated.

Models which study this type of data by imposing some restrictions on the covariance structure have been proposed since the late Such models make no requirement of balance in the data, i.e. the number and timing of measurements can vary between individuals. Further, they usually allow for the explicit modelling and analysis of the covariance structure. Very often the covariance structure may be of interest in its own right. However, even if not, the modelling of a parsimonious error structure leads to more efficient parameter estimates.

Given an individual i, 1=1,.... ,n, with tj repeated measurements of each variable j, j=1,....,p, the general formulation for these models can be given by :

y; = X|p + ej (10)

where yi is the vector of responses for the individual i, X; is a (tj x p) flexible design matrix, p is a (p x 1 ) vector of parameters and ei is independently and identically

distributed N(0, Z j). Z i describes the covariance structure for the individual I, i=1,... ,n, and can be written as a function of the parameters 0, i.e. Z i = Z i ( 0 ) for i=1 ,...,n. The general formulation comprises of two parts, therefore, the regression part of the model in which p is estimated, and the model for the covariance structure, Zi(0), in which 0 is estimated. The specific form of the covariance structure chosen, and the interpretation of the resulting coefficients differs according to the type of model fitted.

This general structure includes marginal models, transition (or Markov) models and random or mixed effects models as special cases. In the case of marginal models the two stages of the model are modelled separately. The coefficients are interpreted as

population averages’, as if they had been estimated from a cross-sectional study. Transition models attempt to address both the regression part of the model and the covariance structure simultaneously in a common equation. Finally, in random-effects models the probability distribution for the multiple measurements has the same form for each individual, but the parameters of that distribution, p, vary over individuals. It is this class of models which are of most relevance for the data contained in this analysis, as repeated measurements of many laboratory markers are available for each patient in the

cohort and it is important to consider the within-person changes of these markers over time. Both growth curve anaiyses"^^^ and repeated measures analyses could be modelled as special forms of a general random effects model.

The first random effects models were proposed by Laird and Ware in 1982"^^^. In this paper, the general model was reformulated as :

Vi = Xia + ZiPi+eij (11)

where a is a (p x 1 ) vector of unknown population ‘fixed’ parameters and X; is the (tj x p) design matrix which links a to y;. p,- is a (k x 1) vector of unknown individual ‘random’ effects and Z; is the (tj x k) design matrix which links p; to y|. The pj are distributed as N(0, D) independently of each other, where D is a (k x k) covariance matrix, ey is distributed as N(0,Rj), where R; is a (tj x tj) covariance matrix with parameters which do not vary between individuals and is independent of the individual covariance matrix, D. Very often ejj is given by the simple form of a^I, where I is the identity matrix. In other words, after taking account of the parameters a and Pj, measurements on the same individuals at different time points are independent. In this case the model is called a

conditional-independence’ model.

The estimation of the parameters from these models has to be performed iteratively, as both the covariance structure and the population parameters need to be estimated. Either maximum likelihood or restricted maximum likelihood methods were suggested, with the EM algorithm being proposed as a means of estimating these parameters'^^ However, at the time there was no simple computer package which enabled parameters to be estimated easily. As a consequence, the use of random effects models was limited. Goldstein'^^^ reformulated this random effects model as a two-level model in which level one was taken to be the within-individual level and level 2 was taken to be the between-individual level. He showed that the within-individual covariance, eij, could be allowed to be a function of individual patient characteristics. More recently, the availability of special software for the estimation of the parameters of the model has meant that the use of multi-level modelling is becoming more widespread. The software has recently been further extended to allow for more than two levels of data, allowing for a far more complex hierarchy of data to be modelled.

APPENDIX 111 : RECENT PUBLISHED PAPERS ARISING FROM RESEARCH ON ROYAL FREE HOSPITAL HAEMOPHILIA COHORT

Papers included

Full reference of paper Page

1993

1. Sabin CA, Phillips AN, Elford J, Griffiths P, Janossy G, Lee CA.

The progression of HIV disease in a haemophilic cohort followed for 12 years.

Brit J Haematol 1993; 83: 330-333.

2. Sabin CA, Phillips AN, Elford J, Janossy G, Bofill M, Lee CA.

The incidence of HIV-related disease in a cohort of haemophilic men: natural history and changes since the introduction of pre-AIDS treatment.

Clin Lab Haematol 1993: 15: 241-251.

3. Phillips AN, Sabin CA, Elford J, Bofill M, Lee CA, Janossy G.

CDS lymphocyte counts and serum immunoglobulin A levels early in HIV infection as predictors of CD4 lymphocyte depletion during 8 years of follow-up.

A/DS 1993; 7:975-980.

4. Phillips AN, Sabin CA, Elford J, Bofill M, Janossy G, Lee CA.

Acquired immunodeficiency syndrome (AIDS) risk in recent and long­ standing human immunodeficiency virus type 1 (HIV-1 )-infected patients with similar CD4 lymphocyte counts.

Am J Epidemiol 1993; 138: 870-878. 1994

5. Sabin CA, Lee CA, Phillips AN.

The use of backcalculation to estimate the prevalence of severe immunodeficiency induced by the human immunodeficiency virus in England and Wales.

J Roy Stat SocA 1994; 157: 41-56.

177

181

192

198

228

232 6. Sabin CA, Phillips AN, Lee CA, Elford J, Timms A, Bofill M, Janossy G

Beta-2 microglobulin as a predictor of prognosis in HIV-infected men with haemophila: a proposed strategy for use in clinical care.

Brit J Haematol 1994; 86; 366-371.

7. Sabin CA, Pasi J, Phillips A, Elford J, Janossy G, Lee CA.

CD4+ counts before and after switching to monoclonal high-purity factor VIII concentrate in HIV-infected haemophilic patients.

Thromb Haem 1994; 72: 214-217.

8. Sabin CA, Phillips AN, Lee CA, Janossy G, Emery V, Griffiths PD. The effect of CMV infection on progression of human immunodeficiency virus disease in a cohort of haemophilic men followed for up to 13 years from seroconversion.

Epidemiollnfec 1994; 114: 361-372.

9. Phillips AN, Sabin CA, Elford J, Bofill M, Janossy G, Lee CA.

Use of CD4 lymphocyte count to predict long term survival free of AIDS after HIV infection.

e/WJ 1994; 309: 309-313. 1995

24Q

10. Sabin CA, Elford J, Phillips AN, Janossy G, Lee CA.

Prophylaxis for Pneumocystis carinii pneumonia: its impact on the natural history of HIV infection in men with haemophilia.

Haemophilia 1995; 1: 37-44.

B ritish lounuü of Haematologii. 1 9 9 3 . 8 3 . 3 3 0 - 3 3 3

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