Article Figures & Data
Figures
- Figure 1.
(a) Scatter plot of patient number of alcohol-free weeks during 1 year vs physician number of hours of advice per year. (b) Fitted line based on predictions for the whole data set using REG model 1 (traditional linear regression model 1).
- Figure 2.
Fitted lines based on predictions for individual clinics using REG model 1 (traditional linear regression model 1). (a) Lines for 2 clinics. (b) Lines for 3 clinics.
Tables
Multilevel/hierarchical/clustered/nested data: Data that have some inherent group membership (eg, students within schools, patients with clinics) or hierarchical structure Multilevel models/hierarchical linear models (HLMs): A type of statistical procedure belonging to the class of general linear models, adapted for analysis of clustered data Analysis of variance (ANOVA): A statistical procedure used to compare means of a continuous outcome variable for more than 2 groups, classified by 1 or more categorical variables; for example, comparison of patient scores on a functional health survey for 4 non-overlapping diagnostic groups by 2 sex categories Analysis of covariance (ANCOVA): An extension of ANOVA in which the means of a continuous outcome variable are compared across groups, as described above, adjusting for 1 or more continuous covariates; for example, comparison of patient scores on a functional health survey for 4 nonoverlapping diagnostic groups, adjusted for age Fixed effects: A condition in which the levels of a factor include all levels of interest to the researcher (eg, sex: male or female) Intraclass correlation coefficient (ICC): A measure that describes the extent to which individuals within the same group are more similar to each other than individuals in different groups Random effects: A condition in which the levels of a factor represent a random sample of all possible levels (eg, clinics) Linear regression analysis: Simple linear regression analysis assesses how a continuous outcome variable (or dependent variable) changes per unit change in a predictor variable (or independent variable). Multiple linear regression analysis assesses the relationship between 1 dependent variable and more than 1 independent variables. Residual variance: The remaining variance in the outcome variable (dependent variable) after accounting for all predictors (independent variables) and random effects of interest 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 Original Sample Size* No. of Clinics No. of Patients per Clinic (m) ICC VIF Adjusted Sample Size* m = designator for number of patients per cluster to be used in calculations; ICC = intraclass correlation coefficient; VIF = variance inflation factor. *Number of patients per treatment group. 100 25 4 .00 1.0 100 100 29 4 .05 1.15 115 100 33 4 .10 1.30 130 100 10 10 .00 1.0 100 100 15 10 .05 1.45 145 100 19 10 .10 1.90 190 100 5 20 .00 1.0 100 100 10 20 .05 1.95 195 100 15 20 .10 2.9 290 100 4 25 .00 1.0 100 100 9 25 .05 2.2 220 100 14 25 .10 3.4 340
Supplemental Appendix
Supplemental Appendix. Descriptions of Statistical Models
Files in this Data Supplement:
- Supplemental data: Appendix - PDF file, 4 pages, 175 KB
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