Background
Methodology
Study data
Variance-covariance structures and inference
Variance-covariance structures
Subject-specific random effect | Linear mixed model |
---|---|
M1: Random intercept |
\(Y_{ij} = \beta _{0} + \beta _{1}\,t_{ij} + \beta _{2}\,t_{ij}^{2} + \sum _{l=1}^{L} \theta _{l}\,x_{ijl} + b_{0_{i}} + \varepsilon _{ij}\)
|
M2: Linear random effects |
\(Y_{ij} = \beta _{0} + \beta _{1}\,t_{ij} + \beta _{2}\,t_{ij}^{2} + \sum _{l=1}^{L} \theta _{l}\,x_{ijl} + b_{0_{i}} + b_{1_{i}}\,t_{ij} + \varepsilon _{ij}\)
|
M3: Quadratic random effects |
\(Y_{ij} \,=\, \beta _{0} \,+\, \beta _{1}\,t_{ij} \,+\, \beta _{2}\,t_{ij}^{2} \,+\, \sum _{l=1}^{L} \theta _{l}\,x_{ijl} \,+\, b_{0_{i}} \,+\, b_{1_{i}}\,t_{ij} \,+\, b_{2_{i}}\,t_{ij}^{2} \,+\,\varepsilon _{ij}\)
|
Tests for zero variance components
Semi-parametric mixed effects model
Estimation of parameters
gamm
in mgcv
R package, for fitting the penalized spline model and the MIXED and GLIMMIX procedures in SAS [24]. This implementation of penalized smoothing in the linear mixed model framework also provides an automated approach to obtain a smoothing parameter and flexibility to extend the models [17].Model selection and inference
Rate of change over time and simultaneous confidence bands
gamm
function available in R
package mgcv
[29] and the linear mixed models using the lme
function available in R
package nlme
.Results
Patients baseline characteristics
Type of diabetes | |||||
---|---|---|---|---|---|
Characteristics | Type 1 | Type 2 | p-value | Overall | |
Gender | Male, N (%) | 87 (16.29%) | 255 (47.75) | 0.9935 | 342 (64.04%) |
Female, N (%) | 48 (8.99%) | 144 (26.97%) | 192 (35.96%) | ||
Family history | No, N (%) | 37 (6.93%) | 380 (71.16%) | <0.0001 | 417 (78.09%) |
Yes, N (%) | 98 (18.35%) | 19 (3.56%) | 117 (21.91%) | ||
Age, mean (SD) | 34.55 (11.92) | 48.63 (13.78) | <0.0001 | 45.4 (14.62) | |
Weight, mean (SD) | 58.83 (11.10) | 64.02 (13.74) | <0.0001 | 62.83 (13.36) | |
FBS, mean (SD) | 171.38 (102.39) | 162.73 (80.66) | 0.0139 | 164.72 (86.20) |
Parametric mixed models
Mean structure
Variance-covariance structure for random effects
Variance-components | ||||||
---|---|---|---|---|---|---|
Effects | Random intercept | Linear random effects | Quadratic random effects | |||
Estimate (s.e.) | p-value | Estimate (s.e.) | p-value | Estimate (s.e.) | p-value | |
Fixed effects
| ||||||
Intercept | 304.362 (14.616) | <0.0001 | 306.756 (15.743) | <0.0001 | 303.139 (15.678) | <0.0001 |
Age | 0.252 (0.183) | 0.1693 | 0.212 (0.179) | 0.2362 | 0.197 (0.179) | 0.2699 |
Gender, Male | -2.605 (5.487) | 0.6352 | -1.968 (5.983) | 0.7424 | -2.609 (5.933) | 0.6603 |
Diabetes type, Type 2 | -9.758 (8.697) | 0.2624 | -10.553 (8.814) | 0.2317 | -10.581 (8.852) | 0.2325 |
Family history, Yes | -12.763 (8.478) | 0.1328 | -12.335 (8.606) | 0.1523 | -12.593 (8.643) | 0.1457 |
Time | -4.462 (0.870) | <0.0001 | -5.614 (1.071) | <0.0001 | -5.549 (1.116) | <0.0001 |
Time2 | 0.123 (0.018) | <0.0001 | 0.135 (0.020) | <0.0001 | 0.153 (0.025) | <0.0001 |
Weight | -1.981 (0.196) | <0.0001 | -1.991 (0.216) | <0.0001 | -1.906 (0.215) | <0.0001 |
Time × Weight | 0.016 (0.013) | 0.2139 | 0.032 (0.016) | 0.0439 | 0.025 (0.016) | 0.1162 |
Gender, Male × Time | -0.412 (o.363) | 0.2563 | -0.482 (0.443) | 0.2761 | -0.425 (0.444) | 0.3390 |
Variance components
| ||||||
var(b0) | 2135.023 | 2797.766 | 3352.606 | |||
var(b1) | 4.575 | 40.343 | ||||
var(b2) | 0.048 | |||||
Residual | 5023.386 | 4873.227 | 4723.609 |
Effects | Estimate (s.e.) | p-value |
---|---|---|
Fixed effects
| ||
Intercept | 302.931 (13.330) | <0.0001 |
Time | -5.815 (1.061) | <0.0001 |
Weight | -1.968 (0.212) | <0.0001 |
Time × Weight | 0.031 (0.016) | 0.0509 |
Time2 | 0.134 (0.020) | <0.0001 |
Variance components
| ||
var(b0) | 2797.887 | |
var(b1) | 4.601 | |
Residual | 4877.259 |
Semi-parametric mixed model
Variance structures | ||||
---|---|---|---|---|
Effects | Random intercept | Linear random effects | ||
Estimate (s.e.) | p-value | Estimate (s.e.) | p-value | |
Fixed effects
| ||||
Weight | -1.908 (0.191) | <0.0001 | -1.899 (0.212) | <0.0001 |
Time | 28.264 (6.087) | <0.0001 | 26.742 (6.359) | <0.0001 |
Time × Weight | 0.017 (0.013) | 0.1837 | 0.031 (0.016) | 0.0536 |
Time2 | 0.408 (0.402) | 0.3095 | 0.448 (0.421) | 0.2875 |
s(Time)Fx1 | -2971.649 (551.992) | <0.0001 | -3014.737 (579.734) | <0.0001 |
Variance components
| ||||
Standard deviation | ||||
Intercept | 2104.479 | 2796.166 | ||
Linear | 4.814 | |||
Residual | 4919.429 | 4762.647 | ||
s(Time) | 13.287 | <0.0001 | 13.939 | <0.0001 |
Fit statistics | |||||
---|---|---|---|---|---|
Variance structure | −2 log(Lik) | AIC | BIC |
E
p
| AIC adj |
M5 | |||||
Random intercept | 50538.54 | 50554.54 | 50605.63 | 7.087 | 50545.627 |
Random linear | 50507.09 | 50527.09 | 50590.96 | 7.260 | 50514.350 |
M4 | 50583.51 | 50601.51 | 50658.98 |