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MULTILEVEL ANALYSIS

MULTILEVEL ANALYSIS. Kate Pickett Senior Lecturer in Epidemiology. SUMBER: www-users.york.ac.uk/.../Multilevel%20 Analysis . ppt ‎University of York. Perspective. Health researchers: Are interested in answering research questions (not maths) Want to be able to apply statistical techniques

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MULTILEVEL ANALYSIS

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  1. MULTILEVEL ANALYSIS Kate Pickett Senior Lecturer in Epidemiology SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  2. Perspective • Health researchers: • Are interested in answering research questions (not maths) • Want to be able to apply statistical techniques • Want to be able to interpret results • Want to be able to communicate with consumers and statisticians SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  3. Aims for this session • Understand the rationale for multilevel analysis • Understand common terminology • Interpret output from multilevel models • Be able to read and critically appraise studies using multilevel models SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  4. Context and composition • Studying populations (groups) and individuals From Rose, G. Sick individuals and sick populations. Int J Epidemiol 1985;14:32-38 SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  5. Levels of analysis • Health researchers may collect and use data collected at the level of: • Individuals, patients • Families or other social groupings • Clinics or hospitals • Small areas, neighbourhoods • Large populations SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  6. Population A Population B How is Population A different from Population B? SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  7. Ecological studies • Data are aggregated and represent a group, rather than an individual • incidence rate of an illness • prevalence of a particular health service • We don’t know which particular individuals within the group were ill or received the service • These group-based outcome measures are analyzed by correlating them with determinants measured for the same groups SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  8. Source: Pickett KE, Kelly S, Brunner E, Lobstein T, Wilkinson RG. Wider income gaps, wider waistbands? An ecological study of obesity and income inequality. J Epidemiol Community Health 2005;59:670–674. SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  9. The ecological fallacy • Associations at the group level may not hold at an individual level • Eg, we might see that rates of obesity are correlated internationally with per capita calorie intake • But, we don’t know if it is the obese individuals who are eating all the calories • Many group-level variables are correlated so we may get spurious correlations • Eg, obesity rates may also be correlated with number of zoos per capita or some other completely unrelated factor SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  10. The atomistic fallacy • But the ecological fallacy has a flip side • Factors that affect outcomes in individuals may not operate in the same way at the population level • Eg, teenage births are more common among the poor, but teenage birth rates are very high in some very wealthy countries. SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  11. Example of teenage births Source: Pickett KE, Mookherjee S, Wilkinson RG. Adolescent Birth Rates,Total Homicides, and Income Inequality In Rich Countries, AJPH 2005;95:1181-1183. SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  12. Ecological variables • Sometimes ecological studies are done because it is quick and easy • Sometimes ecological studies are the best design for the research question BECAUSE • Some determinants are “ecological”: • Population density • Air quality/pollution • GNP • Income inequality • % unemployed • Ambient temperature

  13. Context and composition • But what if we are interested in both types of variables (individual and population) simultaneously? • Eg: we might want to know about the effect of population-level unemployment on health, above and beyond the health impact of being unemployed for any given individual

  14. Multilevel models

  15. Introduction to multilevel models • Hierarchical models • Mixed effects models • Random effects models

  16. Background • Developed in education research • Observations of students in a single class are not independent of one another • “Standard” statistical models assume that observations are independent • Two-level hierarchy • Students within classes • Three-level hierarchy • Students within classes within schools • Four-level hierarchy • Students within classes within schools within local authority areas

  17. Health research context • Patients within a medical practice • Residents within neighbourhoods • Subjects within trial clusters • Hospitals within PCTs….

  18. Examples for class • Some examples are drawn from Twisk JWR “Applied Multilevel Analysis” Cambridge University Press, 2006 • Example data are available at: http:\www.emgo.nl\researchtools • Research question: what is the relationship between total cholesterol and age? • Statistical software: Stata but note that MLwiN is free to UK academics: http://www.cmm.bristol.ac.uk/MLwiN/download/index.shtml)

  19. Simple linear regression Total cholesterol = β0 + β1 x age + ε

  20. Simple linear regression, adding a categorical variable Total cholesterol = β0 + β1 x age + β2 x gender +ε

  21. Simple linear regression, adding another variable (doctor) Total cholesterol = β0 + β1 x age + β2 x MD1 + β3 x MD2+ β4 x MD3+ β5 x MD4+…..+ βm x MDm-1+ ε

  22. Multilevel analysis • Instead of estimating all those separate intercepts, we estimate the variance of them • In our example that means estimating 1 additional parameter, rather than 11 • We are allowing the intercept to be random (random effects modelling) • An efficient way of correcting for a variable with many categories • Trade-off: • Assumes that the different intercepts are normally distributed SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  23. Example data Cholesterol Dataset • 441 patients • Age 44-86 years • Cholesterol 3.90-8.86 mmol/l • 12 doctors SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  24. Non-multilevel regression . regress cholesterol age Source | SS df MS Number of obs = 441 -------------+------------------------------ F( 1, 439) = 142.06 Model | 99.3395851 1 99.3395851 Prob > F = 0.0000 Residual | 306.984057 439 .699280312 R-squared = 0.2445 -------------+------------------------------ Adj R-squared = 0.2428 Total | 406.323642 440 .923462822 Root MSE = .83623 ------------------------------------------------------------------------------ cholesterol | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .0512619 .0043009 11.92 0.000 .042809 .0597148 _cons | 2.798691 .268571 10.42 0.000 2.270847 3.326536 ------------------------------------------------------------------------------ Example using Stata SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  25. . xtmixed cholesterol age ||doctor:, ml var Performing EM optimization: Performing gradient-based optimization: Iteration 0: log likelihood = -404.68939 Iteration 1: log likelihood = -404.68939 Computing standard errors: Mixed-effects ML regression Number of obs = 441 Group variable: doctor Number of groups = 12 Obs per group: min = 36 avg = 36.8 max = 39 Wald chi2(1) = 262.76 Log likelihood = -404.68939 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ cholesterol | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .0495866 .003059 16.21 0.000 .0435911 .0555822 _cons | 2.905812 .259134 11.21 0.000 2.397919 3.413705 ------------------------------------------------------------------------------ ------------------------------------------------------------------------------ Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval] -----------------------------+------------------------------------------------ doctor: Identity | var(_cons) | .3685781 .1541985 .1623381 .8368327 -----------------------------+------------------------------------------------ var(Residual) | .3314923 .0226341 .2899706 .3789597 ------------------------------------------------------------------------------ LR test vs. linear regression: chibar2(01) = 282.37 Prob >= chibar2 = 0.0000 Multilevel Model in Stata SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  26. Do we need the multilevel model? • Likelihood ratio test: • Compare -2 log likelihood of model with random intercept to -2 log likelihood of ordinary linear model • Difference has a Chi-square distribution with df = difference in number of parameters estimated • Difference = 284.73, highly significant SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  27. Model parameters • Effects of age in each model: • Coefficient in ordinary model = 0.0513 • Coefficient in multilevel model = 0.0496 • 95% CI in ordinary model (0.0428, 0.0597) • 95% CI in multilevel model (0.0435,0.0556) • Age is significant in both models

  28. Intraclass correlation coefficient • This measures how dependent the observations are within clusters • Eg, how correlated are the observations of patients belonging to the same doctor? • Defined as: • Variance between clusters/Total variance • The smaller the variance within clusters, the greater the ICC

  29. ICC (a) Distribution of an outcome variable Assume that the total variance = 10

  30. ICC (b) ICC is low because: Variance within groups is high (9) Variance between groups is low (1) Numerator is small, relative to denominator ICC = 1/10=0.1

  31. ICC (c) The groups are now more spread out, more different, and: ICC is bigger because: Variance within groups is lower (5) Variance between groups is higher (5) ICC=5/10 = 0.5

  32. ICC (d) The groups are now completely different, and: ICC is maximised because: Variance within groups is minimal (1) Variance between groups is maximal (9) Numerator is large, relative to denominator ICC=9/10 = 0.9 MUCH MORE DEPENDENCE WITHIN CLUSTER – each observation provides less unique information

  33. Impact on significance tests Table of alpha values under different conditions of sample size and ICC

  34. ICC in our example • ICC = between doctor variance/total variance • ICC = 0.3686/(0.3686+0.3315) = 0.3686/0.7001 = 0.526 52.6% of the total individual differences in cholesterol are at the doctor level SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  35. ICC • When ICC is high • Evidence of a contextual effect on the outcome • Evidence of differences in composition between the clusters • Explore by including explanatory variables at each level • When ICC is low • No need for a multilevel analysis SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  36. Back to unemployment example SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  37. Data Structure Population B Population A Red = unemployed SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  38. An ordinary regression model Health =b0 + b1 (unemployed) + b2 (% unemployed) + e e represents the effect of all omitted variables and measurement error and is assumed to have a random effect (so it gets ignored) SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  39. Data Structure Population B Population A Aside from unemployment, subjects in A are different from B in other ways: composition (shape, size), context (density) SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  40. A multi-level regression model i = individual, j=context: yij = bxij + BXi + Ej + eij Health = b (unemployedij) + B(% unemployedi) +Ej + eij SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  41. What does this mean for critical appraisal of the health literature? • When data are hierarchical or multi-level by nature, they should be analysed appropriately • The coefficients or odds ratios from the models can be interpreted as usual • The ICC shows how much variance in the outcome occurs between the higher-level contexts • If appropriate methods are not used, standard errors and significance tests may be wrong and coefficients biased SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

  42. A summary • Ecological studies • Appropriate when the research question concerns only ecological effects • Ecological fallacy may be a problem • Individual-level studies • Appropriate when the research question concerns only individual-level effects • Atomistic fallacy may be a problem • Multi-level studies • Appropriate when the research question concerns both context and composition of populations SUMBER: www-users.york.ac.uk/.../Multilevel%20Analysis.ppt‎University of York

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