**3. Results**

#### **3.1 Copula types across WPB status and tails status, and copula** *taus*

**Table 4** Gives the summary of copulas distributions selected as optimal for the total 36 patients stratified by WPB tracking status, copula type, number and type of tail(s), as gleaned from the more detailed **Table 5**, which gives each patient's WPB status, number of time points captured over ICU stay, patient's copula type, in

#### **Figure 6.**

*P8 bivariate plots (LHS), copula type and main region tau (upper panel) and K-plot and lower tail (LT)* tau *= 0.70, with fit lines (bottom panel) relating P8's score with dose. P8 is a good tracker with lower tail thresholds for dose and nurses' score: LT dose threshold = 3.02 and LT nurses' score threshold = 2.57.*



*Copula Modelling of Agitation-Sedation (A-S) in ICU: Threshold Analysis of Nurses'… DOI: http://dx.doi.org/10.5772/intechopen.105753*

#### **Table 7.**

*List of optimal copulas for the 36 patients with each patient's WPB status, number of time points over ICU stay, copula type, values for copula tau (τ) for the main region, the lower or upper regions taus for the patient's (score, dose) bivariate profiles.*

addition to the values for copula tau (τ) for the main region, the lower or upper regions, as applicable for the patients' (nurses'score, dose) bivariate profiles.

In regards to the distribution of copula types by WPB (good versus poor) tracking status, there are four Frank copula types of the 13 poor trackers, in comparison to 1 good tracker being a Frank copula type. Of the 23 good trackers, the majority are either Survival Gumbel (5 of these), Survival BB8 (5 such), or Gaussian (4) copula types (**Table 4**).

Furthermore, of the 23 good trackers, eight patients (35%) have bivariate dependence with a lower tail, 12 patients (52%) with no tails, 1 patient P32 (4.3%) with both upper and lower tails, and 2 (8.7%) (P23 and 27) with an upper tail. In comparison of

the 13 poor trackers, there are four patients (31%) with a lower tail, seven (54%) with no tails, one patient P25 (7.7%) with both tails, and one patient 4 (7.7%) with an upper tail (**Table 4**).

Copulas that are unique to the poor trackers are the Survival BB7 (P32 with upper and lower tails, a poor tracker, displayed in **Figure 7** and the Rotated Tawn type 1, 1800 (P35 with an upper tail, a poor tracker displayed in **Figure 8**). Unique copula types found only for the good trackers are the Clayton (P3 and P19, each have lower tails), the BB8 (P30 and P36 both with no tails), the t copula (P25 with upper and lower tails) shown in **Figure 9** and the Joe copula (P37, upper tail) displayed in **Figure 10**.
