Modeling time-to-good control of hypertension using Cox proportional hazard and frailty models at Bahir-Dar Felege Hiwot Referral Hospital

Figures & data

Table 1 Summary table for time-to-good control of hypertension

Covariates	Minimum	Maximum	Mean	Standard deviation	Median
Time (months)	9	60	31.99	14.195	30
Age (years)	18	95	54.2	17.548	55
Systolic blood pressure (mmHg)	145	240	169.8	17.426	170
Fasting blood sugar (mg/dL)	27	343	129.57	56.775	114.5
Blood urea nitrogen (mg/dL)	80	310	42.2	29.897	35
Creatinine (mg/dL)	0.1	10.6	1.356	0.8809	1.10

Table 2 Cross-tabulation of sex, age, and systolic blood pressure of hypertensive patients with status and time

Covariates	Group	Status		Time (months)		Total (%)
		Event(%)	Censor (%)	≤30 (%)	>30 (%)
Sex	Female	108 (52.7)	155 (52.5)	133 (50.6)	130 (54.9)	263 (52.6)
	Male	97 (47.3)	140 (47.5)	130 (49.4)	107 (45.1)	237 (47.4)
	Total	205 (100)	295 (100)	263 (100)	237 (100)	500 (100)
Age (years)	<45	75 (36.6)	84 (28.5)	93 (35.4)	66 (27.8)	159 (31.8)
	>45	130 (63.4)	211 (71.5)	170 (64.6)	171 (72.2)	341 (68.2)
	Total	205 (100)	295 (100)	263 (100)	237 (100)	500 (100)
Systolic blood pressure (mmHg)	<165	103 (50.2)	121 (41.0)	160 (60.8)	64 (27)	224 (44.8)
	>165	102 (49.8)	174 (59.0)	103 (39.2)	173 (73.0)	276 (55.2)
	Total	205 (100)	295 (100)	263 (100)	237 (100)	500 (100)

Figure 1 Kaplan–Meier estimates for hypertension data set by sex, age, SBP, FBS, BUN, and creatinine.

Abbreviations: BUN, blood urea nitrogen; FBS, fasting blood sugar; SBP, systolic blood pressure.

Table 3 Univariable and multivariable analyses of Cox proportional hazard

Covariates	Univariable analysis				Multivariable analysis
	β	P-value	HR	95% CI for HR	β	P-value	HR	95% CI for HR
Age (years)	−0.0121	0.0017	0.988	0.9805, 0.9955	−0.008050	0.0389	0.9920	0.6339, 1.1390
Systolic blood pressure (mmHg)	−0.0339	4.3e^−14	0.967	0.9581, 0.9752	−0.03652	2.59e^−13	0.9641	0.2691, 0.4948
Fasting blood sugar (mg/dL)	−0.00194	0.098	0.998	0.9958, 1	0.002912	0.0282	1.0029	1.0088, 1.7786
Blood urea nitrogen (mg/dL)	−0.00501	0.049	0.995	0.99, 1	−0.00128	0.6213	0.9987	0.6879, 1.2118
Creatinine (mg/dL)	0.15	0.036	1.16	1.01, 1.337	0.124024	0.0683	1.1320	0.5882, 1.3678

Note: β, coefficient for covariates.

Abbreviations: CI, confidence interval; HR, hazard ratio.

Table 4 Model selection for hypertension data set

Models	Covariates in the model	Number of parameters	Akaike Information Criterion	Likelihood ratio	P-value
1	Age, SBP, FBS, blood urea nitrogen, creatinine	5	2125.019	76.05	5.662e^−15
2	Age, SBP, FBS, creatinine	4	2123.271	75.8	1.332e^−15
3	Age, SBP, FBS	3	2124.25	72.82	1.11e^−15
4	Age, SBP, FBS, creatinine, agexSBP, agexFBS, agexcreatinine	7	2127.517	77.55	4.341e^−15
5	Age, SBP, creatinine, agexcreatinine	4	2126.033	73.03	5.218e^−15
6	Age, SBP, creatinine, agexSBP	4	2127.607	71.46	1.11e^−14
7	Age, SBP, creatinine, agexFBS	5	2125.162	75.91	5.995e^−15
8	Age, SBP, FBS, SBPxFBS	4	2124.215	74.85	2.109e^−15

Abbreviations: FBS, fasting blood sugar; SBP, systolic blood pressure.

Table 5 Multiple covariate analysis for the best model of Cox proportional hazard

Covariates	β	Standard error (β)	P-value	HR (95% CI)
Age (years)	−0.008161	0.003895	0.0361*	0.991873 (0.9843, 0.9995)
Systolic blood pressure (mmHg)	−0.036917	0.004932	7.13e^-14	0.963756 (0.9545, 0.9731)
Fasting blood sugar (mg/dL)	0.002905	0.001324	0.0282*	1.002910 (1.0003, 1.0055)
Creatinine (mg/dL)	0.124872	0.067934	0.0660	1.133004 (0.9918, 1.2944)

Note: *Statistically significant at P<0.05.

Abbreviations: CI, confidence interval; HR, hazard ratio.

Figure 2 Cox–Snell residual plot of the best-fitted model for hypertension data set.

Note: The red line indicates the linear trend, while the green line indicates the hypertension data set.

Figure 3 Martingale residuals plot of the best-fitted model for hypertension data set.

Abbreviations: FBS, fasting blood sugar; SBP, systolic blood pressure.

Figure 4 Schoenfeld residuals plot of the best-fitted model for hypertension data set.

Abbreviations: FBS, fasting blood sugar; SBP, systolic blood pressure.

Figure 5 Deviance residuals plot of the best-fitted model for hypertension data set.

Table 6 Statistical test for proportional hazard assumption of the covariates and their interaction with log of time and Schoenfeld residual

Covariates	P-value	Rho	χ2
Age	0.1626	0.10267	1.950281
Systolic blood pressure	0.2154	0.08689	1.534951
Fasting blood sugar	0.9847	0.000369	−0.00129
Creatinine	0.19292	5.202351	0.0226
Global	0.0531	NA	9.341522

Abbreviation: NA, not applicable.

Figure 6 Plots of scaled Schoenfeld residuals for each covariate in a best model.

Abbreviations: FBS, fasting blood sugar; SBP, systolic blood pressure.

Figure 7 Plots of log [−log S(t)] versus time for each covariate in a best model.

Abbreviations: FBS, fasting blood sugar; SBP, systolic blood pressure.

Table 7 Univariable and multivariable analyses of univariate frailty

Age	−0.0121	0.0039	9	1	0.0017	−0.0096	0.0044	4.77	1	2.9e^-02
Systolic blood pressure	−0.04	0.0052	58.7	1	1.9e^−14	−0.0405	0.0055	54.14	1	1.9e^−13
Fasting blood sugar	−0.0019	0.0012	2.74	1	0.098	0.0029	0.0015	4.13	1	4.2e^−02
Blood urea nitrogen	−0.005	0.0026	3.86	1	0.049	−0.0015	0.0028	0.27	1	6.0e^−01
Creatinine	0.15	0.072	4.41	1	0.036	0.1339	0.0791	2.86	1	9.1e^−02
Frailty (id)			0.01	0.01	0.71			59.39	52.9	2.5e^−01

Abbreviations: SE, standard error; DF, degree of freedom.

Table 8 Parameter estimates (SE) in the univariate gamma frailty model applied to hypertension data set

Covariates	β	SE (β)	c^2	DF	P-value
Age (years)	−0.00962	0.00436	4.87	1	2.7e^−02
Systolic blood pressure (mmHg)	−0.04082	0.00543	56.61	1	5.3e^−14
Fasting blood sugar (mg/dL)	0.00296	0.00145	2.91	1	4.1e^−02
Creatinine (mg/dL)	0.13419	0.07862	57.1	1	8.8e^−02
Frailty (id, dist=gamma)				57.1	2.6e^−01

Notes: Likelihood ratio test=179 on 54.5 DF, P-value=3.44e⁻¹⁵, n=500.

Abbreviation: SE, standard error.