Interval estimation for minimal clinically important difference and its classification error via a bootstrap scheme

Figures & data

Figure 1. The illustration of the 0–1 loss $L_{01}$, its surrogate non-smooth ramp loss $L_{\delta }$, and the DC decomposition $L_{\delta }=L_1-L_2$.

Table 1. Confidence interval for MCID in simulation studies.

		MCID
Sample size	Kernel	Lower	Upper	Length	CP
Scenario 1
n=500	Linear (Correct)	$-4.3040$	$-3.3198$	0.9842	0.934
	Kernel	$-4.6326$	$-2.3341$	2.2985	0.928
n=1000	Linear (Correct)	$-4.2669$	$-3.6130$	0.6539	0.952
	Kernel	$-4.5890$	$-1.5640$	3.0250	0.982
Scenario 2
n=500	Linear (Incorrect)	$-7.5186$	$-4.1525$	3.3661	0.004
	Kernel	$-10.7414$	$-7.4439$	3.2974	0.990
n=1000	Linear (Incorrect)	$-7.5371$	$-4.8906$	2.6465	0.000
	Kernel	$-10.5402$	$-4.7963$	5.7439	0.998

Note: CP: coverage probability.

Table 2. Confidence interval for the classification error in simulation studies.

		Classification Error
Sample Size	Kernel	Lower	Upper	Length	CP
Scenario 1
n=500	Linear (Correct)	0.1951	0.3068	0.1117	1.000
	Kernel	0.1985	0.3198	0.1213	0.996
n=1000	Linear (Correct)	0.2080	0.2871	0.0791	0.992
	Kernel	0.2063	0.3428	0.1365	1.000
Scenario 2
n=500	Linear (Incorrect)	0.3595	0.4861	0.1266	0.000
	Kernel	0.2041	0.3335	0.1294	0.996
n=1000	Linear (Incorrect)	0.3762	0.4687	0.0924	0.000
	Kernel	0.2086	0.3883	0.1797	1.000

Note: CP: coverage probability.

Table 3. Interval estimation for the ChAMP trial analysis.

	MCID				Classification error
		Interval estimation			Interval estimation
	Point Estimation	Lower	Upper	Length	Lower	Upper	Length
No model: $c\left(\mathbf{z}\right)=c$
MCID_all	1.4199	$-0.9868$	2.3224	3.3092	0.3050	0.4421	0.1371
MCID_trt=1	2.0216	$-0.9944$	3.8266	4.8210
MCID_trt=0	$-0.0843$	$-0.9868$	1.7207	2.7076
MCID_diff	2.1059	$-1.5042$	3.9109	5.4151
Model 1: $c\left(\mathbf{z}\right)=\alpha +{\text{β}}_1\text{treatment}$
α	$-0.0009$	$-2.3885$	1.6182	4.0067	0.2842	0.5263	0.2421
${\text{β}}_1$	1.9008	$-1.4109$	3.4050	4.8159
MCID_all	0.9676	$-1.0497$	1.8041	2.8538
MCID_trt=1	1.8999	$-0.8843$	2.5207	3.4050
MCID_trt=0	$-0.0009$	$-2.3885$	1.6182	4.0067
MCID_diff	1.9008	$-1.4109$	3.4050	4.8159
Model 2: $c\left(\mathbf{z}\right)=\alpha +{\text{β}}_1\text{treatment}+{\text{β}}_2\text{age}+{\text{β}}_3\text{sex}+{\text{β}}_4\text{damage}$
α	$-6.7261$	$-21.4810$	11.3416	32.8226	0.2737	0.5158	0.2421
${\text{β}}_1$	2.4068	$-1.9171$	3.9081	5.8252
${\text{β}}_2$	0.1680	$-0.1918$	0.3768	0.5686
${\text{β}}_3$	$-0.9713$	$-3.4509$	2.2291	5.6800
${\text{β}}_4$	$-1.1430$	$-1.8663$	1.3246	3.1909
MCID_all	0.2467	$-1.6964$	2.0215	3.7178
MCID_trt=1	1.6050	$-1.6592$	3.3212	4.9804
MCID_trt=0	$-1.1645$	$-3.1225$	1.7468	4.8693
MCID_diff	2.7694	$-1.6298$	4.0097	5.6395
Model 3: $c\left(\mathbf{z}\right)$ is nonparametric
MCID_all	$-0.3465$	$-1.5073$	1.5785	3.0858	0.0526	0.4424	0.3897
MCID_trt=1	$-0.2987$	$-1.4469$	1.8478	3.2947
MCID_trt=0	$-0.3962$	$-1.7511$	1.5378	3.2890
MCID_diff	0.0975	$-0.5132$	1.2298	1.7430

Table 4. Sensitivity analysis for the ChAMP trial.

		Bootstrap resampling size (B)
		600	900	1200	1500
No model: $c\left(\mathbf{z}\right)=c$
MCID_all	Lower	$-0.9868$	$-0.9868$	$-0.9868$	$-0.9868$
	Upper	2.3600	2.3224	2.3224	2.3224
MCID_trt=1	Lower	$-0.9868$	$-0.9868$	$-1.2877$	$-1.2877$
	Upper	3.8266	3.8266	3.8266	3.8266
MCID_trt=0	Lower	$-0.9868$	$-0.9868$	$-0.9868$	$-0.9868$
	Upper	1.7207	1.7207	1.7207	1.7207
MCID_diff	Lower	$-1.5042$	$-1.5042$	$-1.5042$	$-1.5042$
	Upper	3.9109	3.9109	3.9109	3.9109
Classification Error	Lower	0.3053	0.2997	0.3053	0.2947
	Upper	0.4526	0.4476	0.4526	0.4526
Model 1: $c\left(\mathbf{z}\right)=\alpha +{\text{β}}_1\text{treatment}$
α	Lower	$-2.2125$	$-2.3885$	$-2.3885$	$-2.3885$
	Upper	1.6182	1.6182	1.6182	1.6182
${\text{β}}_1$	Lower	$-1.2034$	$-1.3565$	$-1.4084$	$-1.2992$
	Upper	3.4050	3.4050	3.4050	3.3092
MCID_all	Lower	$-0.8760$	$-1.0214$	$-1.0473$	$-1.0458$
	Upper	1.7648	1.7624	1.8041	1.9027
MCID_trt=1	Lower	$-0.7909$	$-0.8843$	$-0.8843$	$-0.8843$
	Upper	2.5223	2.5207	2.5208	2.5208
MCID_trt=0	Lower	$-2.2125$	$-2.3885$	$-2.3885$	$-2.3885$
	Upper	1.6182	1.6182	1.6182	1.6182
MCID_diff	Lower	$-1.2034$	$-1.3565$	$-1.4084$	$-1.2992$
	Upper	3.4050	3.4050	3.4050	3.3092
Classification Error	Lower	0.2842	0.2842	0.2842	0.2842
	Upper	0.5161	0.5263	0.5263	0.5263
Model 2: $c\left(\mathbf{z}\right)=\alpha +{\text{β}}_1\text{treatment}+{\text{β}}_2\text{age}+{\text{β}}_3\text{sex}+{\text{β}}_4\text{damage}$
α	Lower	$-22.5595$	$-21.4412$	$-21.3968$	$-21.1874$
	Upper	9.7443	10.3888	10.7200	11.2746
${\text{β}}_1$	Lower	$-1.9235$	$-1.8822$	$-1.9171$	$-1.9289$
	Upper	4.0760	3.8862	3.9378	3.9518
${\text{β}}_2$	Lower	$-0.1830$	$-0.1842$	$-0.1830$	$-0.2019$
	Upper	0.4176	0.3760	0.3751	0.3715
${\text{β}}_3$	Lower	$-3.5843$	$-3.4407$	$-3.5837$	$-3.4926$
	Upper	2.3425	2.2252	2.2467	2.3068
${\text{β}}_4$	Lower	$-1.7888$	$-1.8370$	$-1.8925$	$-1.8103$
	Upper	1.3664	1.3070	1.3238	1.3833
MCID_all	Lower	$-1.7405$	$-1.7091$	$-1.6945$	$-1.7689$
	Upper	2.0215	2.0263	1.9981	2.0001
MCID_trt=1	Lower	$-1.7345$	$-1.6240$	$-1.5910$	$-1.5944$
	Upper	3.3077	3.3142	3.3077	3.2833
MCID_trt=0	Lower	$-3.2409$	$-3.1269$	$-3.0683$	$-3.1346$
	Upper	1.8563	1.7076	1.6739	1.6121
MCID_diff	Lower	$-1.6303$	$-1.5644$	$-1.6227$	$-1.6262$
	Upper	4.2119	3.9789	4.0669	4.0933
Classification Error	Lower	0.2734	0.2737	0.2737	0.2737
	Upper	0.5055	0.5158	0.5053	0.5158
Model 3: $c\left(\mathbf{z}\right)$ is nonparametric
MCID_all	Lower	$-1.4763$	$-1.4921$	$-1.5270$	$-1.5201$
	Upper	1.5442	1.5951	1.5785	1.6139
MCID_trt=1	Lower	$-1.4830$	$-1.4605$	$-1.5064$	$-1.4834$
	Upper	1.7903	1.8483	1.8195	1.8338
MCID_trt=0	Lower	$-1.7379$	$-1.7465$	$-1.7623$	$-1.7518$
	Upper	1.5213	1.5583	1.5378	1.6119
MCID_diff	Lower	$-0.4841$	$-0.5003$	$-0.5468$	$-0.5488$
	Upper	1.2007	1.2294	1.2000	1.1966
Classification Error	Lower	0.0526	0.0526	0.0526	0.0526
	Upper	0.4526	0.4476	0.4421	0.4421