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Eur J Cardiothorac Surg 2005;27:1129-1132
© 2005 Elsevier Science NL

Letter to the Editor

Reply to Kuss and Börgermann

Confidence intervals for the prediction of mortality in the logistic EuroSCORE

^a EA 3672, IFR 99, Université Victor Segalen, Bordeaux 2, Bordeaux, France
^b CCECQA, Bordeaux, France
^c CHU Fort de France, Martinique, France
^d Papworth Hospital, Cambridge, UK

Received 21 February 2005; accepted 22 February 2005.

^* Corresponding author. Tel.: +44 1480 364299; fax: +44 1480 364744. (E-mail: sam.nashef{at}euroscore.org).

Key Words: EuroSCORE • Risk modelling • Statistics

Drs Kuss and Börgermann advocate the calculation of some measure of uncertainty in the estimation of the predicted mortality in the logistic EuroSCORE. They also request examples of such calculations. This is a most interesting suggestion and we are happy to respond as follows.

Let us start, for educational purposes at the beginning. The predicted mortality is calculated according to a risk stratification model. This logistic model is made of 17 risk factors x_i [1]. The weight of each factor in the calculation of the predicted mortality was estimated using a logistic regression model, as 17 coefficients ß_i (see values of the coefficients on the EuroSCORE website (http://www.EuroSCORE.org/calc.html). According to the logit distribution, the estimated predicted mortality is calculated as:

where ß₀ is the constant of the logistic regression (equal to –4.789594 in our model).

The estimation of the 17 ß coefficients was made on the EuroSCORE sample. These estimates are subject to some uncertainty and would have been slightly different on another sample, difference negligible because of the size of our sample (13,302 patients from eight European countries). Therefore, it is conceptually intuitive to calculate confidence interval of the predicted mortality. As the authors point out, its calculation cannot be easily achieved because the response variable (mortality) has binomially distributed errors. Confidence intervals can be calculated using the variance–covariance matrix of the parameter estimators, in the following way:

with and where and is the variance–covariance matrix of the parameter estimators.

Here is an example in which calculations (at least the first three steps) can easily be done by the reader. This is a 55-year-old woman (ß coefficient of sex is 0.3304052) with four clinical risk factors: serum creatinine 250µmol/l (ß coefficient is 0.6521653), extracardiac arteriopathy (ß coefficient is 0.6558917), chronic pulmonary disease (ß coefficient is 0.4931341) and previous cardiac surgery (ß coefficient is 1.002625):

1. is equal to 0.3304052x1+0.6521653x1+0.6558917x1+0.4931341x1+1.002625x1=3.1342213

2.

3.

4. the 95% confidence interval (95% CI) of this risk is [0.0909;0.2675]

We computerized the variance–covariance matrix of the EuroSCORE parameter estimators using SAS 8.1 (proc ‘logistic’, option ‘covb’) (Table 1). However, the SAS package does not provide confidence intervals. Other commonly used packages do not even provide the matrix (e.g. SPSS) and alternative methods of calculating these intervals have been proposed [2].

View larger version (19K): [in this window] [in a new window]   Fig. 1. 95% CI of the individual predicted risk-adjusted mortality in the EuroSCORE sample.

Predicting risk can be very useful for making treatment decisions, preparing for post-intervention care, and for helping the patient and family to make informed choices about treatment options. Surgeons and other clinicians now widely use risk stratification models. However they, along with patients, always must be cautious in the interpretation of the predicted values. These estimates are indeed subject to biases (systematic errors) and random errors. As an example, risk factor measurements in a cohort tend to underestimate the risk related to factors with a high intra-individual variability. This bias is sometimes called the regression dilution bias [4]. Random errors, that are by definition neither preventable nor corrigible, arise for example when the risk estimation is calculated on a cohort that is over ten years old, when the mean patient characteristics and the practices have changed. The 95% CI around the individual prediction only makes it possible to take into account the statistical uncertainty, related for example to the sample size; it does not correct for the former errors which may be much larger than the statistical uncertainty. The 95% CI may, therefore, provide a sense of false security in decision making.

In our opinion, individual prediction is useful in evaluating the approximate level of risk of a particular patient. For a patients who is classified as low or high risk, as in our example, we do not need much precision in the size of the uncertainty for decision making: to what extent the 95% CI (between 0.10 and 0.26) will modify the decision concerning this patient with 0.16 mortality probability? We believe that the 95% CI may in fact be useful for the intermediate category, as surgeons or patients may decide that, for example, operative decision may vary according to a predefined risk threshold, say 0.3. If ‘0.3’ is within the 95% CI interval, the true value of the predicted mortality may be 0.3 and the operative decision has to be carefully considered.

In conclusion, we thank Drs Kuss and Börgermann for their interest in our work and we are happy to provide them with the information they request. We believe that the calculation of the confidence interval for individual risk prediction may be of limited value in most instances, primarily because it does not cover the total uncertainty of the measurement. Moreover, in some cases, it even may represent a small (and unknown) part of the uncertainty.

Footnotes

¹ The Editor-in-Chief has exceptionally allowed more authors, a Table and a Figure because of the specific information which was requested in the corresponding Letter to the Editor.

References

1. Roques F, Michel P, Goldstone AR, Nashef SA. The logistic EuroSCORE. Eur Heart J 2003;24(9):881-882.[Free Full Text]
2. Sofroniou N, Hutcheson GD. Confidence intervals for the predictions of logistic regression in the presence and absence of a variance–covariance matrix. Understanding Stat 2002;1(1):3-18.[CrossRef]
3. Hosmer DW, Lemeshow S. Confidence interval estimates of an index of quality performance based on logistic regression models. Stat Med 1995;14:2161-2172.[Medline]
4. Hense HW. Observations, predictions and decisions- assessing cardiovascular risk assessment. Int J Epidemiol 2004;33:235-239.[Free Full Text]

This Article Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to Personal Folders Download to citation manager Author home page(s): François Roques Samer A.M. Nashef Permission Requests Citing Articles Citing Articles via Google Scholar Google Scholar Articles by Michel, P. Articles by Nashef, S. A.M. Search for Related Content PubMed Articles by Michel, P. Articles by Nashef, S. A.M. Related Collections Education Cardiac - other