Application of final optimism adjusted models to new individuals

The above findings show that the two prognostic modelling approaches gave similar results at each stage of model development and internal validation. In order to compare the predicted risk estimates produced by the final models, a fictitious patient (Patient Z) will be used for illustration. Patient Z is 25 years old and was diagnosed with pre-eclampsia 33.8 weeks into her pregnancy, she has no pre-existing medical conditions and has received treatment with magnesium sulphate. Their baseline test results are as follows: SBP= 159, platelet count= 226, ALT= 497, serum creatinine= 61. Patient Z’s baseline characteristics are displayed in Table 4.12.

In order to calculate Patient Z’s risk of experiencing an antenatal adverse event, the values above were incorporated into the final optimism adjusted prognostic model

models for Patient Z are given in Table 4.13. The difference between the linear predictor values from the two models for the hypothetical patient is 0.045.

Table 4.12: Patient Z’s baseline characteristics

Prognostic factors Patient Z

Maternal age (MA) 25

Gestational age (GA) 33.8

Medical history (MH) 0

Systolic blood pressure (SBP) 159

Platelet count (PC) 226

Serum Creatinine (SC) 61

Antihypertensive treatment (AH) 0

Magnesium sulphate treatment (MG) 1

Table 4.13: Linear predictor calculation for Patient Z

Cox proportional hazards model Royston-Parmar flexible parametric model 𝐿𝑃 = – 0.021 × ((𝑴𝑨 = 𝟐𝟓) − 30.244) + 0.015 × ((𝑮𝑨 = 𝟑𝟑. 𝟖) − 30.562) + 0.012 × ((𝑺𝑩𝑷 = 𝟏𝟓𝟗) − 158.634) – 0.003 × ((𝑷𝑪 = 𝟐𝟐𝟔) − 227.532) + 0.012 × ((𝑺𝑪 = 𝟔𝟏) − 61.615) + 1.437 × (𝑴𝑮 = 𝟏) = 1.599 𝐿𝑃 = – 0.032 × ((𝑴𝑨 = 𝟐𝟓) − 30.244) + 0.012 × ((𝑮𝑨 = 𝟑𝟑. 𝟖) − 30.562) + 0.011 × ((𝑺𝑩𝑷 = 𝟏𝟓𝟗) − 158.634) – 0.003 × ((𝑷𝑪 = 𝟐𝟐𝟔) − 227.532) + 0.012 × ((𝑺𝑪 = 𝟔𝟏) − 61.615) + 1.433 × (𝑴𝑮 = 𝟏) = 1.644

The values from the linear predictor calculations were then combined with the re- estimated baseline survival estimates to give the predicted cumulative risk of experiencing an antenatal adverse event at two days, one week, and four weeks for

Patient Z. The calculations for predicted risks are shown in Table 4.14.

The difference between the predicted risk estimates from the two models becomes larger as time progresses. A graphical display of Patient Z’s predicted cumulative risk of an antenatal event over time is depicted in Figure 4.5.

Table 4.14: Predicted cumulative risk of antenatal adverse events for Patient Z Cox proportional hazards model Royston-Parmar flexible parametric

model 𝑅𝑖𝑠𝑘 = 1 − 𝑆0(𝑡)𝑒𝑥𝑝(1.599) 1 − 0.980exp(1.599)= 9.5%, 𝑏𝑦 2 𝑑𝑎𝑦𝑠 1 − 0.956exp(1.599)= 20.0%, 𝑏𝑦 1 𝑤𝑒𝑒𝑘 1 − 0.878exp(1.599)= 47.5%, 𝑏𝑦 4 𝑤𝑒𝑒𝑘𝑠 1 − 0.868exp(1.599)= 50.4%, 𝑏𝑦 40 𝑑𝑎𝑦𝑠 1 − 0.847exp(1.599)= 56.0%, 𝑏𝑦 60 𝑑𝑎𝑦𝑠 1 − 0.847exp(1.599)= 56.0%, 𝑏𝑦 80 𝑑𝑎𝑦𝑠 𝑅𝑖𝑠𝑘 = 1 − 𝑆0(𝑡)𝑒𝑥𝑝(1.644) 1 − 0.979exp(1.644)= 10.4%, 𝑏𝑦 2 𝑑𝑎𝑦𝑠 1 − 0.955exp(1.644)= 21.2%, 𝑏𝑦 1 𝑤𝑒𝑒𝑘 1 − 0.893exp(1.644)= 44.3%, 𝑏𝑦 4 𝑤𝑒𝑒𝑘𝑠 1 − 0.870exp(1.644)= 51.4%, 𝑏𝑦 40 𝑑𝑎𝑦𝑠 1 − 0.840exp(1.644)= 59.4%, 𝑏𝑦 60 𝑑𝑎𝑦𝑠 1 − 0.816exp(1.644)= 65.1%, 𝑏𝑦 80 𝑑𝑎𝑦𝑠

Figure 4.5: Predicted cumulative risk for Patient Z

Both models appear to give quite similar predictions of cumulative risk up to around 40 days, however after this time the model predictions diverge slightly. For example, at 60 days the predicted risk is 56.0% from the Cox model but 59.4% from the Royston-Parmar model. Arguably, the estimates over time produced by the Royston-Parmar model provide a more realistic risk profile for a patient compared to those produced by the Cox model. It is much more likely that the real world risk of an event is smoothly increasing over time, rather than jumping at specific time points (when an event is observed in another participant) and remaining stable in between these time points.

In this chapter, both Cox and Royston-Parmar regression methods were applied to develop two new prognostic models which predict the risk of antenatal adverse events in women diagnosed with early-onset pre-eclampsia. The models were internally validated and adjusted for optimism; the included prognostic factors, predictive performance, and final optimism adjusted equations were compared. The aim of this chapter was to demonstrate the standard time-to-event methods to develop prognostic models and compare models developed using two popular time-to-event approaches.

Two commonly used time-to-event modelling methods, the Cox proportional hazards method (Cox, 1972a) and Royston-Parmar flexible parametric method (Royston and Parmar, 2002), were applied to develop prognostic models. Though the two modelling methods produced practically identical regression coefficient estimates, the difference between the estimation methods for the baseline survival function resulted in differences in predicted risks for individual patients over time, especially at later time points. The Royston-Parmar flexible parametric modelling approach directly estimates a smooth and flexible baseline cumulative hazard function, which can be utilised to produce smooth predicted risk estimates for study participants. This is advantageous over the Cox model, which requires additional modelling and estimation to estimate a non-parametric step function for the baseline cumulative hazard function. This stepped function becomes problematic at later time points as the number of participants and events decreases and the steps grow in size. At later time points the baseline hazard function from the Royston-Parmar approach extrapolates with little data, but the smooth function is more realistic. Hence, it is recommended the flexible parametric approach be used in prognostic model research and will be the focus of the remainder of this thesis.

This chapter adds to the work conducted by the PREP study team (Thangaratinam et al., 2017) in developing prognostic models which may be used to inform pre-eclampsia patients of their risks of adverse events. The alteration of the definition of the outcome to antenatal adverse events compliments the prognostic models developed by the PREP study team, and both models may be used to give the patient a broader picture of their risk of events during pregnancy and their risk of preterm delivery. However, changing the outcome does not completely eradicate the risk of treatment paradox bias (Cheong‐See et al., 2016), as patients were censored at the time of delivery of the baby. Therefore, the risk predictions calculated using the prognostic models developed in this chapter should be interpreted as the risk of experiencing an antenatal adverse event over time in a hypothetical scenario where delivery of the baby (even for treatment) is not possible.

Other methods have been proposed to reduce the risk of treatment paradox bias including: deleting the “trigger” prognostic factor from the model (Schuit et al., 2013), standardising treatment across predictor levels, using propensity scores (Cheong‐See et al., 2016), and modelling the probability of treatment (Groenwold et al., 2016). However, considering the definition of a competing event (an event which prevents or alters the risk of the event of interest from occurring (Koller et al., 2012)), a reasonable approach might be to model the treatment as a competing risk to the event of interest. By incorporating competing risks methodology into the development of prognostic models, one would be able to produce “real-world” risk prediction which would account for the presence and probability of the treatment happening. In the following chapter, the statistical methods to appropriately account for competing events will be utilised to develop prediction models for the risk of antenatal adverse events, accounting for the competing risk of delivery. Later, in Chapter 6, an additional dataset will be utilised to externally validate and compare the standard time-to-event and competing risks

5 A COMPARISON OF PROGNOSTIC MODELS