Regression models standard applications

In regression models for the GBSG-2 data, all analyses were adjusted for hormonal treatment.

Table 2.1 shows results of univariate Cox models as well as the results of the full model and the model selected by BE(0.05). For continuous covariates, we use both approaches, assuming either a linear effect (A) or using cutpoints to categorise the variables (B). Cutpoints were chosen independently from the data as predefined in the original analysis of the GBSG-2 study; for more details, see Schumacher et al. (2006). Results are given in terms of estimated hazard ratios (HRs). HRs should be given with confidence intervals, but we have omitted them for brevity. With BE(0.05), we select with both approaches a model containing the three markers: tumour grade, number of involved lymph nodes, and progesterone receptor. Using AIC and BIC for model assessment, the two models with categorised covariates fit better than the corresponding models based on the assumption of a linear effect of continuous covariates. For both approaches, the model selected by BE(0.05) would be preferred to the corresponding full model due to smaller values of AIC and BIC.

BE(0.01) yields the same model as BE(0.05) when a linear effect is assumed, whereas only two markers (number of lymph nodes, progesterone receptor) were selected with BE(0.01) for categorised data (Table 2.2). Applying BE(0.157) in the latter situation, age and menopausal status were included in the selected model, but both markers were eliminated when assuming a linear effect. However, tumour size is now included. Assessing models by BIC, the smallest model would be preferred due to the large penalty term for additional markers; the smallest AIC value was obtained with BE(0.05) for categorised covariates. However, in spite of the better fit, the categorisation can always be criticised because of some degree of arbitrariness and subjectivity concerning the number of cutpoints, the specific cutpoints chosen, and a loss of information [see Royston, Altman, and Sauerbrei (2006)].

25 35 45 55 65 75

25 35 45 55 65 75

Age (years)

Figure 2.1: Different risk functions for the effect of age obtained in the GBSG-2 study: categorised risk function (solid line), linear risk function (dashed line), and fractional polynomial (dotted line)

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