A quantitative dynamic systems model of health-related quality of life among older adults

Figures & data

Figure 1 The conceptual dynamic systems model of HRQOL showing the interactions between the physical, mental, and social domains over time (T1 … Tn).

Notes: Stable individual parameters are an indicator of age, comorbidity, and lifestyle, directly influencing the three main domains of HRQOL; The model shows four iterations (from T1 to T4, and it shows how the process continues until a generic Tn), each arrow is a connection between domains during time, representing the influences of one factor on another, causing the influenced factor to change over time (dynamic relationship).
Abbreviations: HRQOL, health-related quality of life; P, physical domain; M, mental domain; S, social domain; R_p, random factor in the physical domain; R_m, random factor in the mental domain; R_s, random factor in the social domain.

Figure 2 HRQOL developmental trajectory of subject X with low initial values in all domains.

Notes: Time (X-axis) is represented by means of the number of iterations computed by the model, assuming that one iteration represents 1 day; the Y-axis represents the level of HRQOL on a scale ranging from 0 to 1; in this particular simulation, the mental domain shows a slight increase, followed by a later decline whereas the physical and social domains start to decline relatively early.
Abbreviation: HRQOL, health-related quality of life.

Figure 3 HRQOL developmental trajectory of subject Y with moderate initial values in all three domains.

Notes: Time (X-axis) is represented by means of the number of iterations computed by the model, assuming that one iteration represents 1 day; the Y-axis represents the level of HRQOL on a scale ranging from 0 to 1; in this simulation, all the three domains show an increasing trend. The mental domain has more fluctuations than the other two, whereas the social domain presented the highest increase.
Abbreviation: HRQOL, health-related quality of life.

Figure 4 HRQOL developmental trajectory of the healthy and young subject Z with high initial values in all domains.

Notes: Time (X-axis) is represented by means of the number of iterations computed by the model, assuming that one iteration represents 1 day; the Y-axis represents the level of HRQOL on a scale ranging from 0 to 1; all the domains show a decreasing trend, more evident in the first iterations. In the final iterations, physical and social domains show an increase, whereas the mental domain remains more stable.
Abbreviation: HRQOL, health-related quality of life.

Table 1 Qualitative outcomes of the calibration procedure, analyzing trends and final results of simulated data starting from different sets of initial and parameter values

Initial and parameter values								Domain	Trend (%)		Output (%)
PI	PCR	MI	MCR	SI	SCR	Ini	Ad		Static	Explosion	Increase	Equal	Decrease
0.20	−0.20	0.20	−0.20	0.20	−0.20	0.50	0.80	P	0	0	18	2	80
								M	0	0	14	4	82
								S	0	0	20	8	72
0.40	0.20	0.40	0.20	0.40	0.20	0.65	0.60	P	0	0	70	2	28
								M	0	0	92	0	8
								S	0	0	76	0	24
0.60	−0.20	0.60	−0.20	0.60	−0.20	0.70	0.80	P	0	0	18	0	82
								M	0	0	20	0	80
								S	0	0	20	0	80
0.75	−0.30	0.45	−0.10	0.50	0.10	0.30	0.60	P	0	0	14	0	86
								M	0	0	28	1	71
								S	0	0	40	0	60
0.35	−0.10	0.08	0.50	0.63	−0.20	0.40	0.80	P	0	0	35	2	63
								M	0	0	0	0	100
								S	0	0	16	6	78
0.60	−0.15	0.15	0.40	0.65	−0.45	0.60	0.40	P	0	0	32	0	78
								M	0	0	4	0	96
								S	0	0	13	2	85

Abbreviations: P, physical domain; M, mental domain; S, social domain; P_I, initial value of the physical domain; P_CR, change rate of the physical domain; M_I, initial value of the mental domain; M_CR, change rate of the mental domain; S_I, initial value of the social domain; S_CR, change rate of the social domain; Ini, stable individual parameters; Ad, adaptability parameter.

Figure 5 Extremely low developmental trajectory.

Notes: Time (X-axis) is represented by means of the number of iterations computed by the model, assuming that one iteration represents one day; the Y-axis represents the level of HRQOL on a scale ranging from 0 to 1; with this set of parameters and initial data all the domains remain stable for the entire period showing very subtle fluctuations.
Abbreviation: HRQOL, health-related quality of life.

Figure 6 Extremely high developmental trajectory.

Notes: Time (X-axis) is represented by means of the number of iterations computed by the model, assuming that one iteration represents 1 day; the Y-axis represents the level of HRQOL on a scale ranging from 0 to 1; with this set of parameters and initial data all the domains rapidly increase to the maximum values and then remain stable.
Abbreviation: HRQOL, health-related quality of life.

Table 2 Baseline characteristics of the participants

Variable	N	Valid (%)
Sex (N)
Women	130	67
Men	64	33
Level of education (N)
Low	89	46.6
High	102	53.4
Family condition (N)
Never married	5	2.6
Married	107	55.4
Widow	71	36.8
Divorced	10	5.2
Work position (N)
Retired	163	98.8
Working	2	1.2
Past job (N)
Manual labor	65	34.2
Nonmanual labor	125	65.8
	Mean	Standard deviation
Age	73.5	6
Mini-Mental State Examination	28.4	2.5

Table 3 Cronbach’s α analysis of indexes

Wave	1	n
P	0.862	0.876
M	0.870	0.873
S	0.797	0.756

Abbreviations: Wave 1, baseline assessment; Wave n, final assessment; P, physical index; M, mental index; S, social index.

Table 4 Cluster centers

Cluster	1	2	3	4
N	39	27	88	40
PI	0.629	0.351	0.757	0.348
PCR	0.001	0.365	−0.003	0.051
MI	0.533	0.329	0.800	0.434
MCR	0.096	0.419	−0.058	0.011
SI	0.533	0.617	0.733	0.679
SCR	0.058	0.046	−0.019	−0.056

Abbreviations: P_I, initial value of the physical domain; P_CR, change rate of the physical domain; M_I, initial value of the mental domain; M_CR, change rate of the mental domain; S_I, initial value of the social domain; S_CR, change rate of the social domain.

Table 5 Monte Carlo analysis of the compared empirical and simulated outcomes, for different sets of initial change rates for each of the domains P, M, and S, and for the four different clusters

Cluster	1			2			3			4
Domain	P	M	S	P	M	S	P	M	S	P	M	S
Ini_CR	−0.1	−0.1	−0.1	−0.1	−0.1	−0.1	−0.1	−0.1	−0.1	−0.1	−0.1	−0.1
Average	0.644	0.568	0.562	0.605	0.687	0.643	0.766	0.782	0.738	0.375	0.451	0.668
Median	0.651	0.572	0.567	0.619	0.745	0.648	0.774	0.798	0.746	0.380	0.451	0.687
Minimum	0.358	0.385	0.351	0.090	0.071	0.346	0.472	0.316	0.421	0.191	0.197	0.238
Maximum	0.864	0.675	0.695	0.966	0.983	0.808	0.914	0.951	0.920	0.544	0.686	0.916
SD	0.084	0.052	0.059	0.195	0.213	0.061	0.070	0.094	0.082	0.060	0.085	0.120
P-value	0.589	0.078	0.314	0.327	0.494	0.386	0.604	0.632	0.650	0.360	0.531	0.688
Number of sim	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000
Ini_CR	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
Average	0.664	0.640	0.601	0.638	0.683	0.680	0.784	0.815	0.754	0.413	0.475	0.698
Median	0.673	0.656	0.608	0.679	0.725	0.688	0.793	0.820	0.762	0.413	0.478	0.704
Minimum	0.358	0.178	0.255	0.051	0.040	0.218	0.501	0.673	0.490	0.078	0.191	0.499
Maximum	0.879	0.946	0.916	0.973	0.990	0.930	0.919	0.895	0.904	0.762	0.746	0.817
SD	0.087	0.152	0.125	0.212	0.205	0.116	0.063	0.035	0.063	0.125	0.100	0.052
P-value	0.665	0.565	0.541	0.441	0.460	0.608	0.723	0.912	0.772	0.535	0.614	0.901
Number of sim	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000
Ini_CR	−0.1	0.1	0.1	−0.075	−0.1	0.1	−0.1	−0.1	−0.1	0.085	−0.1	−0.1
Average	0.647	0.642	0.604	0.656	0.711	0.660	0.766	0.782	0.738	0.403	0.454	0.654
Median	0.650	0.654	0.612	0.704	0.772	0.669	0.774	0.798	0.746	0.400	0.455	0.668
Minimum	0.345	0.116	0.185	0.042	0.045	0.318	0.472	0.316	0.421	0.118	0.174	0.164
Maximum	0.875	0.952	0.955	0.990	0.995	0.912	0.914	0.951	0.920	0.706	0.696	0.924
SD	0.086	0.147	0.129	0.219	0.219	0.112	0.070	0.094	0.082	0.114	0.085	0.118
P-value	0.601	0.555	0.559	0.483	0.544	0.535	0.604	0.632	0.650	0.506	0.550	0.636
Number of sim	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000	1,000

Notes: Average, the average outcome of 1,000 simulations; minimum, the lowest outcome of the 1,000 simulations; maximum, the highest outcome of the 1,000 simulations; P-value, chance that the empirical outcomes and the simulated outcomes stem from the same distribution of data.

Abbreviations: P, physical domain; M, mental domain; S, social domain; Ini_CR, change rate for the stable individual parameters; SD, standard deviation.

Table 6 Wilcoxon test for dependent samples, empirical vs simulated final data

Cluster	Domain	N	Z	P-value	CI (95%)	Empirical		Simulated
						Median	SD	Median	SD
1	P	39	−1.744	0.080	0.074–0.085	0.631	0.207	0.656	0.082
	M	39	−0.628	0.530	0.523–0.542	0.629	0.193	0.647	0.136
	S	39	−0.684	0.494	0.497–0.506	0.592	0.139	0.666	0.119
2	P	27	−0.336	0.737	0.739–0.756	0.717	0.225	0.720	0.227
	M	27	−0.216	0.829	0.833–0.847	0.748	0.166	0.786	0.231
	S	27	−0.505	0.614	0.611–0.630	0.663	0.168	0.686	0.126
3	P	88	−0.994	0.320	0.308–326	0.754	0.162	0.749	0.071
	M	88	−1.369	0.171	0.162–0.176	0.764	0.151	0.767	0.095
	S	88	−1.123	0.261	0.252–0.269	0.715	0.138	0.717	0.079
4	P	40	−0.739	0.460	0.456–0.475	0.399	0.162	0.374	0.134
	M	40	−1.505	0.132	0.127–140	0.445	0.159	0.467	0.072
	S	40	−0.672	0.502	0.498–0.518	0.623	0.116	0.648	0.127

Abbreviations: P, physical domain; M, mental domain; S, social domain; SD, standard deviation; CI, confidence interval; Z, scores based on the Wilcoxon test.

Table 7 Mann-Whitney U pairwise comparisons of simulated and empirical final data with Dunn-Sidak corrections

Domain	Cluster (I)	Cluster (J)	N (I)	N (J)	Empirical (I–J)		Simulated (I–J)		Simulated (I) vs Empirical (J)
					Z	P-value	Z	P-value	Z	P-value
P	1	2	39	27	−1.076	0.282	−0.789	0.430	−1.376	0.169
	1	3	39	88	−3.606	0.000*	−6.058	0.000*	−4.422	0.000*
	1	4	39	40	−3.555	0.000*	−6.658	0.000*	−4.158	0.000*
	2	3	27	88	−1.759	0.079	−1.149	0.136	−1.650	0.099
	2	4	27	40	−3.733	0.000*	−4.538	0.000*	−4.167	0.000*
	3	4	88	40	−7.279	0.000*	−8.739	0.000*	−7.860	0.000*
M	1	2	39	27	−2.719	0.007*	−2.119	0.034	−2.367	0.018
	1	3	39	88	−3.559	0.000*	−5.545	0.000*	−5.138	0.000*
	1	4	39	40	−3.942	0.000*	−5.354	0.000*	−3.716	0.000*
	2	3	27	88	−0.148	0.882	−0.046	0.963	−0.831	0.406
	2	4	27	40	−5.369	0.000*	−4.385	0.000*	−5.458	0.000*
	3	4	88	40	−7.323	0.000*	−8.811	0.000*	−7.516	0.000*
S	1	2	39	27	−1.369	0.163	−0.776	0.438	−1.337	0.181
	1	3	39	88	−2.993	0.003*	−5.533	0.000*	−4.429	0.000*
	1	4	39	40	−0.098	0.922	−0.157	0.875	−0.726	0.468
	2	3	27	88	−0.670	0.503	−1.759	0.073	−1.537	0.124
	2	4	27	40	−1.470	0.141	−0.793	0.428	−0.971	0.331
	3	4	88	40	−3.120	0.002*	−3.578	0.000*	−2.416	0.018

Notes:

* Significant after Dunn-Sidak correction. I and J refer to the number of the clusters analyzed.

Abbreviations: N, number of subjects in each cluster; P, physical domain; M, mental domain; S, social domain; Z, zeta score based on the Mann-Whitney U-test.