Modelling admission lengths within psychiatric intensive care units

Results

From the four units involved, data were acquired from 368 patients (table 2).

Data from 40 type 1 (typical) patients were easily collected from each unit. Of the other types, amount collected from units over the maximum 4-year time period varied.

Survival plots for patient types are shown in figure 1A. These show lengthier admissions for the longer secure care group (patient type 2). Assuming an exponential distribution, hazard of discharge for this group is significantly lower (at the 5% level) than the other two groups. A log-rank test for differences (using patient-type 1 as the reference) gave a    value of 97.8 on two degrees of freedom (p≤2×10⁻⁶). Survival plots of patients from the different units are shown in figure 1B. These plots are closer together than the patient-type survival distributions but do indicate differences between unit distributions. Indeed, a log-rank test for differences (using unit 1 as reference) gave a    value of 20.6 on three degrees of freedom (p=0.001). As survival plots are initial inspection tools, to make them easier to view and to avoid doubts concerning the power of log-rank tests, only group subsets containing 20 or more patients were considered for other variables (figure 1C–E). The log-rank test for differences in survival distributions gave p=0.0007 for ICD chapter differences, p=0.1 for cluster and p=0.03 for gender differences.

Intra-class correlation (ICC) measures how similar outcomes of individuals within a group are likely to be, relative to those from other groups. Measurement is based on ANOVA, assuming normal distributions. Using    (admission length_(days)+ 1) as the outcome variable, the ICC for patient type was 0.37, unit=0.0585, cluster=0.0425, ICD-10 chapter=0.0259 and gender=0.014. None can be regarded as good, however, in the context of this data set, patient type was an order of magnitude better than any other.

ICC does not account for effect of interactions between groups, for example, the effect that patient type has may be confounded by the effect that unit has on admission length. Figure 1F shows how the patient-type variable varies with unit. This indicated that any model should include an interaction term between them.

From the three models described above, both elpd-loo and LOO-IC favoured model 3. Compared with model 1, the LOO-IC difference for model 2 was 25.5 and for model 3, 42.4. Differences in elpd-loo were 12.8 and 21.2, respectively. In models analogous to model 1, patient type was favoured above using either ICD_10 chapters or clustering measurements. There was an elpd-loo difference of 49.3 for patient type versus clustering and 50.5 versus ICD_10 chapter. LOO-IC differences were 98.5 and 100.9, respectively.

Adding further parameters to the model did not improve model comparison measures and there was some suggestion of model overfitting (with some models having a higher R² value but negative differences in elpd-loo and LOO-IC compared with model 3).

The middle plot in the last row of figure 2 displays autocorrelation from chain 1 of model 3 (other chains are similar). It shows that correlation settled after 3–4 lags (ideally, it should be around zero from lag 1 onwards). It was decided that this was satisfactory and that no thinning or increase in draws was needed.27

Posterior predictive checks give the model’s predictive distribution for a replication of y, denoted y_rep. The first plot in figure 2’s final row shows posterior predictive distributions of 1000 draws from model 3. It shows a reasonable fit.

One method of plotting residuals, using the posterior predictive distribution, is to use what Kay has termed a probability residual.28 Here, for each observation, the predicted probability of generating a value less than or equal to the actual observation is calculated:   . If the predictive distribution is well calibrated, these probabilities should be uniform and if the inverse cumulative distribution function of the standard Normal distribution is applied to these probability residuals, the result should be approximately standard normal. These are quantile residuals (z_residual):    and the final plot in figure 2 shows a Q–Q plot of these residuals using model 3 to be acceptable.

Actual mean values can be calculated (noting the model gives values of log_e(days admitted+1)). Table 3 displays means, 0.025 and 0.975 quantiles of actual admission lengths for types of patients from different units.

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