Modelo para la evaluación institucional

We examined the amount of data available for learning each parameter in the BN. Data were too small, or not available, to learn the parameters for some relations. In this section, we describe the techniques used for learning the parameters when they have 1) small amount of data, 2) no data at all (latent variables) and 3) adequate amount of data.

Variables with Small Data

We learned the parameters that had less than 20 data instances from a combination of meta-analysis and data using the technique described in Section 6.2.2. Table 6.10 shows the data available for learning the NPT of the ‘repair failure’ variable. There was small amount of data to learn some of these parameters (shown by bold in Table 6.10) therefore we learned these parameters by combining the meta-analysis results with the data. We used the mean and variance of the predictive distributions from the meta-analysis (see Table 6.7) as observations to 𝜇_{𝑃𝑛𝑒𝑤} and 𝜎_{𝑃𝑛𝑒𝑤}2 variables in the Bayesian learning model (see Figure 6.6).

The meta-analysis provides us the pooled probabilities of an unsuccessful outcome conditioned on each individual clinical factor (see Table 6.7). The variable equivalent to an unsuccessful outcome is ‘nonviable extremity’ in the LEVT BN. However, the meta-analysis results could also be used for the NPT of the ‘repair failure’ variable as 1) our model assumes a deterministic relation between an unsuccessful outcome and repair failure 2) the parents of ‘repair failure’ can influence ‘nonviable extremity’ through only one pathway. For example, the ‘arterial repair’ variable can affect ‘nonviable extremity’ through the following pathway in our model:

𝐴𝑟𝑡𝑒𝑟𝑖𝑎𝑙 𝑅𝑒𝑝𝑎𝑖𝑟 → 𝑅𝑒𝑝𝑎𝑖𝑟 𝐹𝑎𝑖𝑙𝑢𝑟𝑒 → 𝐵𝑙𝑜𝑜𝑑 𝑆𝑢𝑝𝑝𝑙𝑦 → 𝑁𝑜𝑛𝑣𝑖𝑎𝑏𝑙𝑒 𝐸𝑥𝑡𝑟𝑒𝑚𝑖𝑡𝑦 (𝐴𝑅) (𝑅𝐹) (𝐵𝑆) (𝑁𝐸)

The probability provided by the meta-analysis, 𝑃(𝑁𝐸 = 𝑇𝑟𝑢𝑒|𝐴𝑅 = 𝐺𝑟𝑎𝑓𝑡), is equivalent to marginalisation of ‘repair failure’ and ‘blood supply’ from this pathway:

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𝑃(𝑁𝐸 = 𝑇𝑟𝑢𝑒|𝐴𝑅 = 𝐺𝑟𝑎𝑓𝑡) = ∑ 𝑃(𝑁𝐸 = 𝑇𝑟𝑢𝑒|𝐵𝑆)𝑃(𝐵𝑆|𝑅𝐹)𝑃(𝑅𝐹|𝐴𝑅 = 𝐺𝑟𝑎𝑓𝑡)

𝐵𝑆,𝑅𝐹

In our model, a repair failure always leads to inadequate blood supply, and inadequate blood supply always leads to a nonviable extremity:

𝑃(𝐵𝑆 = 𝐿𝑜𝑤|𝑅𝐹 = 𝑇𝑟𝑢𝑒) = 1, 𝑃(𝑁𝐸 = 𝑇𝑟𝑢𝑒|𝐵𝑆 = 𝐿𝑜𝑤) = 1 𝑃(𝐵𝑆 = 𝐿𝑜𝑤|𝑅𝐹 = ¬𝑇𝑟𝑢𝑒) = 0, 𝑃(𝑁𝐸 = 𝑇𝑟𝑢𝑒|𝐵𝑆 = ¬𝐿𝑜𝑤) = 0

By using these values in the marginalisation equation above, we get: 𝑃(𝑁𝐸 = 𝑇𝑟𝑢𝑒|𝐴𝑅 = 𝐺𝑟𝑎𝑓𝑡) = 𝑃(𝑅𝐹 = 𝑇𝑟𝑢𝑒|𝐴𝑅 = 𝐺𝑟𝑎𝑓𝑡). Consequently, we can use the probabilities from the meta-analysis for learning the relation between ‘repair failure’ and its parents.

The meta-analysis does not provide any information about the multiplicity of tibial arteries (see Section 6.3.2 how the multiplicity of tibial arteries is modelled in the BN). Therefore, we model the number of injured tibial arteries as a risk modifier. Our model assumes that a repair failure leads to a non-viable extremity if all 3 tibial arteries are injured. However, there is a chance of a successful outcome, which is learned from data, if only 1 or 2 tibial arteries are injured.

Table 6.10 Amount of Data Available for Learning Parameters of Repair Failure Variable

AR* Graft True Femoral Graft True Popliteal Graft True Tibial Graft False Femoral Graft False Popliteal Graft False Tibial Primary True Femoral Primary True Popliteal … MAI* AS* RF* Data 14 6 2 71 115 38 1 3 …

*AR: Arterial Repair, MAI: Multiple Levels, AS: Anatomical Site, RF: Repair Failure We used the OpenBUGS software (Lunn et al., 2009) to calculate the posteriors of the auxiliary learning model for this case study. Since OpenBUGS uses MCMC sampling, it is necessary to assess the convergence of the Markov chain to ensure that sampled values cover the entire distribution. We used Gelman and Rubin (1992) diagnostic technique, sample plots and autocorrelation plots to assess the convergence. None of the diagnostic techniques can prove convergence but they can assist detecting the lack of convergence. We discarded the first 10,000 samples in

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MCMC as the burn-in samples, and calculated the posterior distributions based on the next 70,000 samples.

Table 6.11 shows extracts of the NPTs of the ‘Repair Failure’ and the amount of data available for learning its parameters. The values in bold and italic fonts are the parameters learned by combining the results of the meta-analysis with the data, and the values in normal fonts are the parameters learned purely from the data. The parameters learned from two approaches differ substantially for smaller amounts of data. The effects of this difference to the model performance are discussed in Section 6.4.1.

Table 6.11 Learning from Data, and from Combining Data and Meta-Analysis Results

AR*: Graft True Femoral Graft True Popliteal Graft True Tibial … Primary True Femoral Primary True Popliteal … MAI*: AS*: RF*: …. True 0.17 0.17 0.39 0.41 0.99 0.49 0.01 0.14 0.01 0.29 … False 0.83 0.83 0.61 0.59 0.01 0.51 0.99 0.86 0.99 0.71 Data: 14 6 2 1 3

*AR: Arterial Repair, MAI: Multiple Levels, AS: Anatomical Site, RF: Repair Failure

Latent Variables

The BN contained several latent variables as described in Section 6.3.2 (see Table 6.8 for a list of these variables). Ranked nodes were used to model the NPT of these variables (Fenton et al., 2007). A ranked node is an approximation of the truncated normal distribution to the multinomial distribution with ordinal scale (see Section 2.6.1 for a more detailed description of ranked nodes). We used the framework proposed by Fenton et al. (2007) to elicit the parameters of ranked nodes.

For each of the latent variables we first asked the domain experts to describe the relation between the variable and its parents. Afterwards, we selected a suitable ranked node function and elicited initial weights that imitate the described relation. We presented the behaviour of the ranked node under various combinations of observations to the domain experts, and refined the weights based on their comments.

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Adequate Amount of Data

After the parameters with small or no data were defined, the remainder of the parameters were learned purely from the data. The EM algorithm was used to learn the parameters as the dataset contained missing values. The parameters that were already defined from the experts or meta-analysis were kept fixed while EM was applied.