Capturing simple and complex time-dependent effects using flexible parametric survival models: A simulation study

Figures & data

Figure 1. Upper-left panel (a) shows hazard ratios from a flexible parametric proportional hazards model with 5 degrees of freedom (solid line) and a flexible parametric non-proportional hazards model with 5,3 degrees of freedom (dashed line) on the log cumulative hazard scale. Upper-right panel (b) shows the hazard ratios from flexible parametric non-proportional hazards models with varying degrees of freedom on the log cumulative hazard scale. Lower-left panel (c) shows the same as panel (a) but for models on the log hazard scale. Lower-right panel (d) shows the same as panel (b) but for models on the log hazard scale.

Table 1. Agreement of AIC and BIC with SKL for different models on the log cumulative hazard scale.

		AIC										BIC
		3,1	3,3	5,1	5,3	5,5	7,1	7,3	7,5	7,7	Total	3,1	3,3	5,1	5,3	5,5	7,1	7,3	7,5	7,7	Total
Scenario 1; SKL	3,1	341	45	43	4	9	32	2	2	3	481	480	1	0	0	0	0	0	0	0	481
	3,3	4	0	1	0	1	1	0	0	0	7	7	0	0	0	0	0	0	0	0	7
	5,1	11	1	0	0	0	0	0	0	0	12	12	0	0	0	0	0	0	0	0	12
	5,3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	5,5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,7	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	Total	356	46	44	4	10	33	2	2	3	500	499	1	0	0	0	0	0	0	0	500
Scenario 2; SKL	3,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	3,3	0	0	0	0	0	0	2	0	0	2	0	0	1	0	0	0	1	0	0	2
	5,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	5,3	0	0	0	1	4	0	45	74	13	137	0	2	1	57	9	1	64	3	0	137
	5,5	0	0	0	3	4	1	147	70	15	240	0	2	11	114	0	6	106	1	0	240
	7,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,3	0	0	0	2	3	0	6	12	4	27	0	0	1	16	4	0	6	0	0	27
	7,5	0	0	0	10	3	0	45	19	7	84	0	1	3	66	0	0	13	1	0	84
	7,7	0	0	0	2	0	0	4	4	0	10	0	0	1	7	0	0	2	0	0	10
	Total	0	0	0	18	14	1	249	179	39	500	0	5	18	260	13	7	192	5	0	500
Scenario 3; SKL	3,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	3,3	0	1	0	0	5	0	0	1	3	10	2	1	0	0	7	0	0	0	0	10
	5,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	5,3	0	0	0	0	1	0	0	0	0	1	0	0	0	0	1	0	0	0	0	1
	5,5	1	54	0	11	284	0	5	46	20	421	109	282	1	1	28	0	0	0	0	421
	7,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,5	0	9	0	2	51	0	0	0	3	65	18	46	0	0	1	0	0	0	0	65
	7,7	0	1	0	0	2	0	0	0	0	3	0	3	0	0	0	0	0	0	0	3
	Total	1	65	0	13	343	0	5	47	26	500	129	332	1	1	37	0	0	0	0	500

Figure 2. Survival proportions, hazard rates and log hazard ratios of the three simulated scenarios.

Table 2. Agreement of AIC and BIC with SKL for different models on the log hazard scale.

		AIC										BIC
		3,1	3,3	5,1	5,3	5,5	7,1	7,3	7,5	7,7	Total	3,1	3,3	5,1	5,3	5,5	7,1	7,3	7,5	7,7	Total
Scenario 1; SKL	3,1	338	46	50	8	8	18	4	2	3	477	476	1	0	0	0	0	0	0	0	477
	3,3	6	0	2	0	1	1	0	0	0	10	10	0	0	0	0	0	0	0	0	10
	5,1	10	3	0	0	0	0	0	0	0	13	13	0	0	0	0	0	0	0	0	13
	5,3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	5,5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,5	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,7	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	Total	354	49	52	8	9	19	4	2	3	500	499	1	0	0	0	0	0	0	0	500
Scenario 2; SKL	3,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	3,3	0	0	0	3	1	0	8	7	57	76	0	0	5	21	4	5	36	1	4	76
	5,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	5,3	0	0	0	0	6	0	30	54	150	240	1	10	8	80	23	15	94	1	8	240
	5,5	0	0	0	4	3	0	68	3	41	119	1	9	8	62	0	7	32	0	0	119
	7,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,3	0	0	0	2	16	0	10	5	15	48	1	6	5	33	3	0	0	0	0	48
	7,5	0	0	0	3	1	0	7	0	4	15	0	2	2	11	0	0	0	0	0	15
	7,7	0	0	0	0	0	0	2	0	0	2	0	0	1	1	0	0	0	0	0	2
	Total	0	0	0	12	27	0	125	69	267	500	3	27	29	208	30	27	162	2	12	500
Scenario 3; SKL	3,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	3,3	0	2	0	0	15	0	0	6	4	27	7	15	0	1	4	0	0	0	0	27
	5,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	5,3	0	0	0	0	0	0	0	0	1	1	0	1	0	0	0	0	0	0	0	1
	5,5	1	60	0	1	133	0	6	45	53	299	91	189	0	3	16	0	0	0	0	299
	7,1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
	7,3	0	1	0	0	0	0	0	0	0	1	1	0	0	0	0	0	0	0	0	1
	7,5	0	7	0	1	42	0	0	3	12	65	18	43	0	1	3	0	0	0	0	65
	7,7	0	28	0	5	63	0	0	6	5	107	35	69	0	2	1	0	0	0	0	107
	Total	1	98	0	7	253	0	6	60	75	500	152	317	0	7	24	0	0	0	0	500

Figure 3. Bias in the survival proportions, from models on the log cumulative hazard, log(H), and log hazard, log(h), scales with different degrees of freedom. Sample size 1 000. Mean values based on 500 simulations.

Figure 4. Bias in the hazard rates from models on the log cumulative hazard, log(H), and log hazard, log(h), scales with different degrees of freedom. Sample size 1 000. Mean values based on 500 simulations.

Figure 5. Bias in the survival proportions log hazard ratios from models on the log cumulative hazard, log(H), and log hazard, log(h), scales with different degrees of freedom. Sample size 1 000. Mean values based on 500 simulations.