Multilevel models for analyzing people s daily moving behaviour

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1 Multilevel models for analyzing people s daily moving behaviour Matteo BOTTAI 1 Nicola SALVATI 2 Nicola ORSINI 3 13th European Colloquium on Theoretical and Quantitative Geography Lucca 5th - 9th September, Institute of Information Science and Technology National Research Council of Italy and Dep. Epidemiology ad Biostatistics, University of South Carolina, Columbia, USA m.bottai@isti.cnr.it 2 Department of Statistics G.Parenti University of Florence salvati@ds.unifi.it 3 Institute of Information Science and Technology National Research Council of Italy n.orsini@isti.cnr.it

2 OUTLINE Data - hierarchical data structure Statistical model - multilevel generalized linear model Application - A survey on daily moving behaviours of the people residing in the area of Pisa 2

3 HIERARCHICAL DATA STRUCTURE LEVEL UNIT SUBSCRIPT RANGE TOTAL 1 Individual j = 1,, n i 1 n i 5 N=806 2 Family i = 1,, m k 1 m k 76 M=373 3 Area k = 1,, K K=35 N K = k = 1 m k i= 1 n i = 806 M K = k = 1 m k = 373 3

4 FEATURES OF THE DATA Dependence - Correlation between individual observations in the same level. Ignoring hierarchical data structure - Misleading statistical inference Traditional statistical models should be adapted - Using random variables and relative assumptions to describe this correlation 4

5 MULTILEVEL GENERALIZED LINEAR MODEL Probability distribution Y ~ Exponential Family Conditional expectation Linear predictor Link function η = V v= 1 E[ Y x, z, u] = µ x v β v + L l= 2 g ( µ ) =η R l r= 1 u ( l ) r z ( l ) r 5

6 2-LEVEL RANDOM INTERCEPT MODEL NORMAL L = 2 R l = 1 V = 2 j = 1,, n i i = 1,, m k 2 Distribution Y ~ N(, σ ) µ Expectation Linear predictor Link E ( Y x, z, u ) = µ η = x β + x β + µ = η (2) 1 i < µ < + u (2) (2) 1 1i 2 2 1i 1 z (2) x 1 = 1 z 1 = 1 6

7 3-LEVEL RANDOM INTERCEPT MODEL NORMAL L = 3 R l = 1 V = 2 j = 1,, n i i = 1,, m k k = 1,, K 2 Distribution Y ~ N(, σ ) k µ k Expectation Linear predictor Link E ( Y x, z, u, u ) = µ k k k k (2) (3) 1 i 1k k < µ k < + η = x β + x β + u z + µ = η k (2) (2) (3) (3) 1 k 1ik 2 k 2 1i 1 k 1k 1 k k u z (2) (3) x 1 k = 1 z 1 k = 1 z 1 k = 1 7

8 2-LEVEL RANDOM INTERCEPT MODEL POISSON L = 2 R l = 1 V = 2 j = 1,, n i i = 1,, m k Distribution Y ~ µ ) P( Expectation Linear predictor Link E ( Y x, z, u ) = µ η = x β + x β + log( µ ) = η (2) 1 i 0 < µ < + u (2) (2) 1 1i 2 2 1i 1 z (2) x 1 = 1 z 1 = 1 8

9 RANDOM EFFECTS DISTRIBUTION ( l) 2 u ~ N(0, σ ) l 1,..., L r lr = r = 1,..., Rl Covariance between two random effects at the same level COV = σ with l = l and r r ( l) ( l ) 2 ( ur,ur ) lrr Covariance between two random effects at different level COV ( l) ( l ) ( ur,ur ) = 0 with l l and r r 9

10 MAXIMUM LIKELIHOOD ESTIMATORS Define the likelihood for the data L ( Y w) = f ( Y u, w) p( u w) du - w denote a vector containing all parameters to be estimated - f ( ) is the probability distribution of the outcome - p ( ) is the probability distribution of the random effects Maximize the likelihood respect to w in order to make inferences about w Optimization Algorithms Differences between linear and non linear multilevel model 10

11 Fixed effects - Wald test Random effects TESTING HYPOTHESIS H0 : β v = W = ˆ βv ˆ σ ˆ β 2 H 0 : σ lr = - Likelihood Ratio test LRT = 2 abs(l1-l0) where L1 and L0 are the log-likelihoods evaluated in a model with and 2 (l) without the variance component σ lr, that is, the random effect u r. v

12 STATISTICAL SOFTWARE Several packages are available - STATA (command GLLAMM and XT) - SPLUS or R (command LME and NLME) - SAS (command PROC MIXED) - MLwiN - HLM Differences in computational algorithms - Newton-Raphson - Expectation-Maximization or Fisher Scoring - Iterative Generalized Least Squares 12

13 MODEL 1. xi: gllamm distance gender i.agec, f(gaussian) l(identity) i(family) number of level 1 units = 806 number of level 2 units = 373 log likelihood = (some STATA output is omitted) distance Coef. Std. Err. z P> z [95% Conf. Interval] gender _Iagec_ _Iagec_ _Iagec_ _Iagec_ _Iagec_ >74 _cons Variance at level ( ) Variances and covariances of random effects level 2 (family) var(1): ( ) INTRA-FAMILY CORRELATION = 1562/( ) =

14 MODEL 2. xi: gllamm distance gender i.agec, f(gaussian) l(identity) i(family area) number of level 1 units = 806 number of level 2 units = 373 number of level 3 units = 35 log likelihood = (some STATA output is omitted) distance Coef. Std. Err. z P> z [95% Conf. Interval] gender _Iagec_ _Iagec_ _Iagec_ _Iagec_ _Iagec_ >74 _cons Variance at level ( ) Variances and covariances of random effects level 2 (family) var(1): ( ) scalar LRT = 2*abs( ) level 3 (area).di chiprob(1, LRT) var(1): ( )

15 MODEL 1 (adaptive quadrature). xi: gllamm distance gender i.agec, f(gaussian) l(identity) i(family) adapt number of level 1 units = 806 number of level 2 units = 373 log likelihood = (some STATA output is omitted) distance Coef. Std. Err. z P> z [95% Conf. Interval] gender _Iagec_ _Iagec_ _Iagec_ _Iagec_ _Iagec_ >74 _cons Variance at level ( ) Variances and covariances of random effects level 2 (family) var(1): ( ) INTRA-FAMILY CORRELATION = 2233/( ) =

16 MODEL 2 (adaptive quadrature). xi: gllamm distance gender i.agec, f(gaussian) l(identity) i(family area) adapt number of level 1 units = 806 number of level 2 units = 373 number of level 3 units = 35 log likelihood = (some STATA output is omitted) distance Coef. Std. Err. z P> z [95% Conf. Interval] gender _Iagec_ _Iagec_ _Iagec_ _Iagec_ _Iagec_ >74 _cons Variance at level ( ) Variances and covariances of random effects level 2 (family) var(1): ( ) scalar LRT = 2*abs( ) level 3 (area).di chiprob(1, LRT) var(1): ( )

17 MODEL 3. xi: gllamm ntrip gender i.agec, f(poisson) l(log) i(family) number of level 1 units = 806 number of level 2 units = 373 log likelihood = (some STATA output is omitted) ntrip Coef. Std. Err. z P> z [95% Conf. Interval] gender _Iagec_ _Iagec_ _Iagec_ _Iagec_ _Iagec_ >74 _cons Variances and covariances of random effects level 2 (family) var(1): ( ). display 2*abs( ) ( from poisson regression) display chiprob(1, )

18 MODEL 3 (adaptive quadrature). xi: gllamm ntrip gender i.agec, f(poisson) l(log) i(family) adapt number of level 1 units = 806 number of level 2 units = 373 log likelihood = ntrip Coef. Std. Err. z P> z [95% Conf. Interval] gender _Iagec_ _Iagec_ _Iagec_ _Iagec_ _Iagec_ >74 _cons Variances and covariances of random effects level 2 (family) var(1): ( ). display 2*abs( ) ( from poisson regression) display chiprob(1, )

19 SUMMARY Close relation between features of the data and statistical techniques - Multilevel Statistical models take into account a hierarchical data structure Define a multilevel generalized linear model and obtained MLEs using the STATA command GLLAMM - Normal and Poisson probability distributions as special case in the exponential family 19

20 POTENTIAL FUTURE RESEARCH Testing variance components and confidence intervals - Score test Assessing the adequacy of multilevel models - Assumptions of the random variables Multi-stage sampling design 20

21 REFERENCES Raudenbush S.W., Bryk A.S. (2002) Hierarchical Linear Models, Sage Publications. McCulloch C.E., Searle S.R. (2001) Generalized, Linear and Mixed Models, Wiley Series in Probability and Statistics. Rabe-Hesketh, S., Pickles, A. and Skrondal, A. (2001b). GLLAMM Manual. Technical Report 2001/01, Department of Biostatistics and Computing, Institute of Psychiatry, King's College, London. 21

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