Hypothetical multilevel regression analysis of systolic blood pressure (SBP) (mm Hg) in 25 000 people aged 35 to 64 from 39 neighbourhoods of a city
| Empty model | Model with individual variables | Model with individual variables and random slopes | Model with individual variables, random slopes, and non-constant individual variance | |
|---|---|---|---|---|
| Model 1 | Model 2 | Model 3 | Model 4 | |
| “Fixed effects” and “random effects” are expressions that are often used in MLRA. In very simple terms, fixed effects are used to model means, whereas random effects are used to model variances. BMI, body mass index; ICC, intraclass correlation; VPC, variance partition coefficient. *As the variance is a function of individual BMI, the values in the table are for the intercepts (people with BMI equal to 25 kg/m2). †When performing statistical modelling, you try to find a function that predicts SBP as well as possible and, therefore, decrease unexplained variance. The goodness of this fit is measured by different statistical techniques. One very common technique that we used in our study model is the reduction in “deviance”. This technique is used to evaluate the fit of consecutive models with additional terms.6–8 Table 1 shows that in our study model, as compared with the empty model, every consecutive model significantly decreases the deviance and improves the goodness of the fit of the model. | ||||
| Fixed effects | ||||
| Mean SBP of the city (intercept) | 130.2 | 128.1 | 128.0 | 127.9 |
| Regression coefficients (β), 95% confidence intervals (CI) | β (95% CI) | β (95% CI) | β (95% CI) | β (95% CI) |
| Age in years | 0.81 (0.78, 0.87) | 0.80 (0.77, 0.86) | 0.80 (0.77, 0.86) | |
| BMI (1 unit kg/m2) | 0.88 (0.86, 0.92) | 0.89 (0.83, 1.01) | 0.91 (0.85, 1.03) | |
| Antihypertensive medication | 14.7 (14.3, 15.3) | 14.7 (14.3, 15.3) | 15.5 (14.8, 15.8) | |
| Random effects | ||||
| Components of variance | Variance (SE) | Variance (SE) | Variance (SE)* | Variance (SE)* |
| Variance between neighbourhood intercepts | 36.2 (8.1) | 27.6 (6.4) | 27.6 (6.4) | 27.5 (6.4) |
| Variance between individuals | 433.4 (3.9) | 307.6 (2.8) | 305.1 (2.8) | 298.5 (3.5) |
| BMI–SBP slope variance between neighbourhoods | 0.11 (0.03) | 0.11 (0.03) | ||
| BMI related individual SBP variance | 0.16 (0.08) | |||
| Neighbourhood covariance (intercept slope) | 0.93 (0.34) | 0.92 (0.34) | ||
| Individual level covariance | 5.16 (0.31) | |||
| Proportional change in variance (PCV) by the new model | PCV | PCV | PCV* | PCV* |
| Between neighbourhoods | Reference | 24% | 24% | 24% |
| Between individuals | Reference | 29% | 30% | 30% |
| ICC–VPC | ICC = 0.08 | ICC = 0.08 | VPC = 0.08* | VPC = 0.08* |
| Deviance† | 222 764 (Reference) | 211 585 | 211 443 | 211 058 |