**3.3 Prediction equation of score in relation to dose allowing tail dependence**

In this subsection, we focus on 7 of the 10 selected patients reported in **Table 8**. Specifically, two patients with both lower and upper tails (P32 and P35), three

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


#### **Table 9.**

*Upper and/or lower tail dose relationships, equations, and associated change in adjusted R<sup>2</sup> for simple LR* vs *our novel approach. Shaded rows indicate the poor trackers.*

patients with only upper tails (P4, P23, P37), and two patients with only lower tails (P2 and P15). For each of these seven patients, our novel equation relating each patient's nurses' agitation severity rank score versus the patient's infusion dose with parameter estimates and p-values is reported below. In addition, interpretation of the equations in regard to regions where the nurses'score either under or overestimates the patient's agitation with respect to so-called ground truth, this being the patient's automated infusion dose is reported per patient.

Patient 32 poor tracker, 2 tails, R<sup>2</sup> squared (non adj, adjusted) = (0.60, 0.67).

Score = 3.23 + 0.43Dose-0.75Dose\*LT + 0.33Dose\*UT (**Table 10**): The intercept of 3.23 indicates that the patient is experiencing severe "chronic" background agitation. The slope of 0.43 indicates that in the moderate agitation zone nurses tend to strongly underestimate the patient's agitation severity. In mild agitation zone, nurses tend to still underestimate the agitation severity. In severe agitation zone, nurses tend to overestimate the patient's agitation severity on average 0.33 points higher (**Figures 7** and **11**). For P32 LT *tau* = 0.73, UT *tau* = 0.40, and its bivariate (LT, UT) *dose* thresholds are (3.90, 7.02), with (LT, UT) *nurses' score* thresholds of (3.62, 7.63).

Patient 25 – good tracker, 2 tails, R<sup>2</sup> squared (non adj, adjusted) = (0.60, 0.70).

Score = 1.55 + 0.46Dose-0.35Dose\*LT + 0.34Dose\*UT (**Table 10**): The intercept of 1.55 indicates that the patient is experiencing "chronic" background agitation. The slope of 0.46 indicates that in the moderate agitation zone nurses tend to strongly underestimate the patient's agitation severity. In severe agitation zone, nurses tend to overestimate the agitation severity. In the mild agitation zone, nurses tend to assign a rating that is, on average, 0.35 points lower than expected for the patient's given agitation severity. In the severe agitation zone, nurses tend to overestimate the agitation severity on average, 0.34 points higher than expected for the patient's given agitation severity (**Figure 9**, see also **Figure 11**). For P25 LT tau = 0.06, UT tau = 0.06, and its (LT, UT) dose threshold = (2.68, 4.43), (LT, UT) *score* threshold of (2.41, 3.94).

#### **Figure 11.**

*Line plots of nurses' score (observed, red) and dose (black line), with 95% WPB bands for P32, P25 (both tails); P4, P23 (upper tail); P2, P15 (lower tail).*

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

**Figure 12.**

*Line plots of nurses' score (observed, red) vs. dose (black line), with 90% WPB bands for P37 good tracker.*


#### **Table 10.**

*P32 equation, poor tracker with 2 tails and LT* tau *= 0.73, UT* tau *= 0.40. P25 equation, poor tracker with 2 tails and LT* tau *= 0.06, UT* tau *= 0.06.*

Score = 0.79 + 0.52Dose+0.45Dose\*UT (**Table 11**): The slope of 0.52 indicates that in the moderate agitation zone nurses tend to strongly underestimate the patient'sagitation severity. In the severe agitation zone, nurses tend to overestimate the agitation severity. There are 14 (out of a total of 63) such occurrences indicating that approximately only one in every four ratings, nurses tend to overestimate patients' agitation severity (**Figure 13**, see also **Figure 11**). For P4 UT *tau* = 0.69, and its UT dose threshold = 2.82 and UT nurses'score threshold = 2.75.

Patient 23 good tracker, UT R<sup>2</sup> squared (non adj, adjusted) = (0.64, 0.68).

Score = 0.34 + 0.58Dose+0.52Dose\*UT (**Table 11**): The slope of 0.58 indicates that in the moderate agitation zone nurses tend to strongly underestimate the patient's agitation severity. In periods of severe agitation, nurses tend to overestimate the agitation severity. There are 10 (out of a total of 57) such occurrences indicating that approximately only one in every five ratings, nurses tend to overestimate patient 23's agitation (**Figure 14**, see also **Figure 11**). For P23 UT *tau* = 0.68, and its UT dose threshold = 1.77 and UT nurses'score threshold = 1.97.

Patient 37 good tracker, UT R<sup>2</sup> squared (non adj, adjusted) = (0.74, 0.75). Patient 4 - poor, Upper Tail Score = 0.17+ 0.78Dose+0.32Dose\*UT (**Table 11**): The slope of 0.78 indicates that in the moderate agitation zone nurses tend to moderately


#### **Table 11.**

*P4 equation, poor tracker with upper tail and UT* tau *= 0.69. P23 equation, poor tracker with upper tail and UT* tau *= 0.68.*

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

#### **Figure 13.**

*P4 poor tracker with upper tail - copula plot, main* tau, *K- plot and tail* tau *(upper panel); best fit line(s) (lower panel) relating P4 nurses' A-S score with dose. UT* tau *= 0.69, UT dose threshold = 2.82 and UT nurse score threshold = 2.75.*

#### **Figure 14.**

*P23 good tracker with upper tail - copula plot, main* tau, *K- plot and tail* tau *(upper panel); best fit line(s) (lower panel) relating P23 nurses' A-S score with dose. UT* tau *= 0.68, UT dose threshold = 1.77 and UT nurse score threshold = 1.97.*

underestimate the patient's agitation severity. In the patient's severe agitation zone, nurses tend to overestimate agitation severity. There are 25 (out of a total of 123) such occurrences indicating that approximately only in one in every five ratings, nurses tend to overestimate patients' agitation severity accurately. (**Figure 10**, see also **Figure 12**). For P37 UT tau = 0.77, and its UT dose threshold = 2.53 and UT nurses'score threshold = 2.99.

P37 equation, good tracker with upper tail and UT *tau* = 0.77.

Patient 2 poor tracker, LT R2 squared (non adj, adjusted) = (0.42, 0.44).

Score = 0.27 + 0.85Dose - 0.37Dose\*LT (**Table 12**): The slope of 0.85 indicates that in the moderate agitation zone nurses tend to underestimate the patient's agitation severity. In the mild agitation zone, nurses tend to assign a rating that is, on average, 0.37 points lower than expected for the patient's given agitation severity. There are 28 out of a total of 63 such occurrences indicating that in around one in every two ratings, nurses tend to underestimate agitation severity (**Figure 15**, also **Figure 11**).

Patient 15 good tracker, LT R2 squared (non adj, adjusted) = (0.45, 0.47).

Score = 0.86 + 0.83Dose - 0.34Dose\*LT (**Table 12**): The slope of 0.83 indicates that in the moderate agitation zone nurses tend to underestimate the patient's agitation severity accurately. In the mild agitation zone, nurses tend to underestimate the patient's agitation severity even more. There are 22 (out of a total of 43) such occurrences indicating that approximately one in every three ratings, nurses tend to *strongly* underestimate the patients' agitation severity (**Figure 16**, see also **Figure 11**).

Patient 28 poor tracker, LT R<sup>2</sup> squared (non adj, adjusted) = (0.57, 0.59)

Score = 0.01 + 0.97Dose-0.35Dose\*LT (**Table 13**).

The slope of 0.97 indicates that in the moderate agitation zone nurses tend to estimate the patient's agitation severity fairly accurately. In the mild agitation zone, however,


#### **Table 12.**

*P2 equation, poor tracker with lower tail and LT* tau *= 0.64. P15 equation, good tracker with lower tail and LT* tau *= 0.74.*

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

#### **Figure 15.**

*P2 poor tracker with lower tail - copula plot, main* tau, *K- plot and tail* tau *(upper panel); best fit line(s) (lower panel) relating P2 nurses' A-S score with dose. LT* tau *= 0.64, with LT dose threshold = 0.96 and LT nurses' score threshold = 0.83.*

#### **Figure 16.**

*P15 good tracker with lower tail - copula plot, main* tau, *K- plot and tail* tau *(upper panel); best fit line(s) (lower panel) relating P15 nurses' A-S score with dose. LT tail* tau *= 0.74, LT dose threshold = 3.47 and LT score threshold = 3.05.*


#### **Table 13.**

*P28 equation, poor tracker with lower tail LT* tau *= 0.79. P35 equation, poor tracker with lower tail LT* tau *= 0.74. P27 equation, poor tracker no tails.,* tau *= 0.78 (Figures 18 and 19).*

#### **Figure 17.**

*Line plots of nurses' score (observed, red) vs. dose (black line), with 95% WPB bands for P27 and P35 poor trackers with and P27 poor tracker with no tails.*

nurses tend to underestimate patient 28's agitation severity more. There are 29 (out of a total of 203) such occurrences indicating that one in every seven ratings, nurses tend to strongly underestimate patients' agitation severity (**Figure 17**, see also **Figure 18).**

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

#### **Figure 18.**

*Line plots of nurses' score (observed, red) vs. dose (black line), with 95% WPB bands for P28 and P35 both poor trackers with lower tails.*

Patient 35 poor tracker, LT R<sup>2</sup> squared (non adj, adjusted) = (0.63, 0.64) Score = 0.23 + 0.91Dose-0.79Dose\*LT (**Table 13**): The intercept of 0.23 indicates that the patient is experiencing mild "chronic" background agitation. The slope of 0.91 indicates that in the moderate agitation zone nurses tend to slightly underestimate the patient's agitation severity. In the mild agitation zone, nurses tend to *strongly* underestimate the patient's agitation severity even more. There are 42 (out of a total of 211) such occurrences indicating that approximately one in every five ratings, nurses tend to underestimate patients' agitation severity (**Figures 8** and **18**).
