Study Results Obtained With Differing Models
| Description of the data | |||
|---|---|---|---|
| Variable | Description | Mean (SD) | Range |
| yij | Number of alcohol-free weeks in 1 year for patient i from clinic j | 14.61 (2.12) | 8.7–19.1 |
| xij | Total hours of physician advice per year for patient i from clinic j | 0.56 (0.30) | 0.002–1.23 |
| wj | Urbanicity: urban = 1; rural = 0 | 0.6 | 0–1 |
| Notation | |||
| i | Indexes patients within a clinic | 1–100 | |
| j | Indexes clinics | 1–5 | |
| HLM model 1: random-effects ANOVA model | |||||
|---|---|---|---|---|---|
| Fixed effects | Estimate* | SE | t | df | Pr > t |
| γ00 (grand mean) | 14.61 | 0.79 | 18.46 | 499 | .000 |
| Random effects | Estimate* | Pr(H0: τ= 0) | |||
| τ00 (between-clinic variance) | 1.76 | .000 | |||
| σ2 (residual variance) | 1.41 | ||||
| REG model 1: traditional linear regression model 1† | ||||
|---|---|---|---|---|
| Fixed effects | Estimate* | SE | t | Pr> t |
| β0 (γ10) – slope | 1.31 | 1.23 | 1.07 | .345 |
| β1 (γ00) – intercept | 13.87 | 1.19 | 11.69 | .000 |
| σ2 (residual variance) | 2.1 | |||
| HLM model 2: random-intercept model | |||||
|---|---|---|---|---|---|
| Fixed effects | Estimate* | SE | t | df | Pr> t |
| γ10 (slope) | 2.38 | 1.05 | 2.26 | 498 | .024 |
| γ00 (average intercept) | 13.27 | 1.30 | 10.24 | 4 | .000 |
| Random effects | Estimate* | Pr(H0: τ= 0) | |||
| τ00 (variability in clinic intercepts) | 3.47 | 0.000 | |||
| σ2 (residual variance) | 1.65 | ||||
| HLM model 3: random-coefficients model | |||||
|---|---|---|---|---|---|
| Fixed effects | Estimate* | SE | t | df | Pr> t |
| γ10 (average slope) | 2.96 | 0.89 | 3.31 | 4 | .040 |
| γ00 (average intercept) | 12.80 | 1.32 | 9.74 | 4 | .000 |
| Random effects | Estimate* | Pr(H0: τ= 0) | |||
| τ00 (variability in intercepts across clinics) | 10.71 | .000 | |||
| τ11 (variability in slopes across clinics) | 4.74 | .000 | |||
| τ01 (covariance between intercept and slope) | −7.10 | ||||
| σ2 (residual variance) | 1.18 | ||||
| HLM model 4: intercept as outcome model | |||||
|---|---|---|---|---|---|
| Fixed effects | Estimate* | SE | t | df | Pr> t |
| γ10 (slope) | 2.34 | 0.23 | 10.03 | 497 | .000 |
| γ01 (difference between urban and rural intercept) | −3.26 | 0.51 | −6.36 | 3 | .000 |
| γ00 (rural intercept) | 15.25 | 0.42 | 36.67 | 3 | .000 |
| τ00 (variability in clinic intercepts after adjusting for urban or rural location) | 0.55 | .000 | |||
| σ2 (residual variance) | 1.28 | ||||
| HLM model 5: intercept and slope as outcomes model | |||||
|---|---|---|---|---|---|
| Fixed effects | Estimate* | SE | t | df | Pr> t |
| γ11 (difference in slope between urban and rural areas) | 3.97 | 0.50 | 7.94 | 3 | .000 |
| γ10 (average slope in rural areas) | 0.67 | 0.44 | 1.54 | 3 | .220 |
| γ01 (difference in intercepts between urban and rural areas) | −5.53 | 0.83 | −6.67 | 3 | .000 |
| γ00 (average intercept in rural areas) | 16.15 | 0.60 | 26.77 | 3 | .000 |
| Random effects | Estimate* | Pr(H0: τ= 0) | |||
| τ00 (variability in intercepts after adjusting for urbanicity) | 1.51 | .000 | |||
| τ11 (variability in slopes after adjusting for Urbanicity) | 0.28 | .092 | |||
| τ01 (covariance between intercept and slope) | −0.44 | ||||
| σ2 (residual variance) | 1.39 | ||||
| REG model 2: traditional regression model 2 | ||||
|---|---|---|---|---|
| Fixed effects | Estimate* | SE | t | Pr> t |
| HLM = hierarchical linear model; H0 = null hypothesis; ANOVA = analysis of variance; REG = regression; Pr = probability. | ||||
| * Estimated number of alcohol-free weeks during the past year. | ||||
| † Residual variance = 2.0813. | ||||
| Slope | ||||
| γ11 (urban – rural) | 1.48 | 1.13 | 1.30 | .226 |
| γ10 (rural) | 0.83 | 0.49 | 1.71 | .163 |
| Intercept | ||||
| γ01 (urban – rural) | −4.04 | 1.01 | −4.01 | .016 |
| γ00 (rural) | 16.05 | 0.67 | 23.94 | .000 |