Dankmar Böhning Southampton Statistical Sciences Research Institute University of Southampton, UK S 3 RI, 11-12 December 2014
An example with one random effect An example with two nested random effects
An example with one random effect An example: health awareness study three states in the US participated in a health awareness study each state independently devised a health awareness program three cities within each state were selected for participation and five households within each city were randomly selected to evaluate the effectiveness of the program a composite index (a count number) was formed (the large the index, the greater the awareness) the data have the following hierarchical structure:
An example with one random effect data: household state city 1 2 3 4 5 1 1 42 56 35 40 28 1 2 26 38 42 35 53 1 3 34 51 60 29 44 2 1 47 58 39 62 65 2 2 56 43 65 70 59 2 3 68 51 49 71 57 3 1 19 36 24 12 33 3 2 18 40 27 31 23 3 3 16 28 45 30 21
An example with one random effect Poisson model with random effect for state for the health awareness index Y ijk for household k, in city j, and state i: log E[Y ijk ] αi = log µ ijk = µ + α i with a state random effect α i N(0, σ 2 S ) and a Poisson error Y ijk Po(µ ijk )
An example with one random effect Poisson model with random effect for state let P(Y ijk = y) = Po(y µ ijk ) = Po(y µ + α i ) likelihood L = Po(y ijk µ + α i ) i,j,k (in the fixed effect case) but α i N(0, σs 2 ), e.g. normal random, so L = Po(y ijk µ + α i )φ(α i )dα i i α i j,k where φ(α i ) is a normal density with mean 0 and variance σ 2 S
An example with one random effect
An example with one random effect Mixed-effects Poisson regression Number of obs = 45 Group variable: state Number of groups = 3 Obs per group: min = 15 avg = 15.0 max = 15 Integration points = 1 Wald chi2(0) =. Log likelihood = -183.93181 Prob > chi2 =. index IRR Std. Err. z P> z [95% Conf. Interval] _cons 39.81456 7.124649 20.59 0.000 28.03645 56.54067 Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] state: Identity sd(_cons).307096.1278477.1358028.6944475 LR test vs. Poisson regression: chibar2(01) = 154.39 Prob>=chibar2 = 0.0000 Note: log-likelihood calculations are based on the Laplacian approximation.
An example with two nested random effects Poisson model with random effect for state and random effect for city nested within state let P(Y ijk = y) = Po(y µ ijk ) = Po(y µ + α i + β j(i) ) where β j(i) N(0, σt 2 ), e.g. normal random likelihood L = i α i β j j Po(y ijk µ + α i + β j(i) )φ(β j )dβ j φ(α i )dα i k where φ(α i ) is a normal density with mean 0 and variance σ 2 S where φ(β j ) is a normal density with mean 0 and variance σ 2 T
An example with two nested random effects
An example with two nested random effects Mixed-effects Poisson regression Number of obs = 45 No. of Observations per Group Integration Group Variable Groups Minimum Average Maximum Points state 3 15 15.0 15 1 city 9 5 5.0 5 1 Wald chi2(0) =. Log likelihood = -183.93181 Prob > chi2 =. index IRR Std. Err. z P> z [95% Conf. Interval] _cons 39.81457 7.124658 20.59 0.000 28.03644 56.54069 Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] state: Identity sd(_cons).3070964.127848.1358029.694449 city: Identity sd(_cons) 7.65e-12.0492277 0. LR test vs. Poisson regression: chi2(2) = 154.39 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference. Note: log-likelihood calculations are based on the Laplacian approximation.
Meta-Analysis on BCG vaccine against tuberculosis Colditz et al. 1974, JAMA provide a meta-analysis to examine the efficacy of BCG vaccine against tuberculosis
Data on the meta-analysis of BCG and TB the data contain the following details 13 studies each study contains: TB cases for BCG intervention number at risk for BCG intervention TB cases for control number at risk for control also two covariates are given: year of study and latitude expressed in degrees from equator
intervention control study year latitude TB cases total TB cases total 1 1933 55 6 306 29 303 2 1935 52 4 123 11 139 3 1935 52 180 1541 372 1451 4 1937 42 17 1716 65 1665 5 1941 42 3 231 11 220 6 1947 33 5 2498 3 2341 7 1949 18 186 50634 141 27338 8 1950 53 62 13598 248 12867 9 1950 13 33 5069 47 5808 10 1950 33 27 16913 29 17854 11 1965 18 8 2545 10 629 12 1965 27 29 7499 45 7277 13 1968 13 505 88391 499 88391
Data analysis on the meta-analysis of BCG and TB these kind of data can be analyzed by taking TB case as disease occurrence response intervention as exposure (fixed effect) study as random effect latitude and year as further fixed effects
Mixed Logistic Regression Model log p xij 1 p xij = µ + α i + β INTER INTER ij + β LAT LAT ij where α i N(0, σ 2 S ) each trial arm within each study contributes a binomial likelihood ( nij y ij ) p y ij x ij (1 p xij ) n ij y ij where p xij = exp(µ + α i + β INTER INTER ij + β LAT LAT ij ) 1 + exp(µ + α i + β INTER INTER ij + β LAT LAT ij )
Mixed Logistic Likelihood L = i ( nij α i y ij j ) p y ij x ij (1 p xij ) n ij y ij φ(α i )dα i where φ(α i ) is a normal density with mean 0 and variance σ 2 S
Integration points = 1 Wald chi2(1) = 134.12 Log likelihood = -196.3842 Prob > chi2 = 0.0000 cases Odds Ratio Std. Err. z P> z [95% Conf. Interval] intervention.6203579.0255761-11.58 0.000.5722016.672567 _cons.0149296.005892-10.65 0.000.0068885.0323576 Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] study: Identity sd(_cons) 1.410135.2813828.953694 2.085029 LR test vs. logistic regression: chibar2(01) = 3259.71 Prob>=chibar2 = 0.0000 Note: log-likelihood calculations are based on the Laplacian approximation..
Integration points = 1 Wald chi2(2) = 145.29 Log likelihood = -192.38326 Prob > chi2 = 0.0000 cases Odds Ratio Std. Err. z P> z [95% Conf. Interval] intervention.6206475.0255857-11.57 0.000.5724728.6728762 latitude 1.064961.0201333 3.33 0.001 1.026223 1.105162 _cons.001693.0012119-8.91 0.000.0004162.0068863 Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] study: Identity sd(_cons) 1.027901.2080747.6912662 1.528471 LR test vs. logistic regression: chibar2(01) = 2180.88 Prob>=chibar2 = 0.0000
Integration points = 1 Wald chi2(3) = 148.12 Log likelihood = -191.68717 Prob > chi2 = 0.0000 cases Odds Ratio Std. Err. z P> z [95% Conf. Interval] intervention.6207321.0255885-11.57 0.000.5725521.6729664 latitude 1.037569.0289224 1.32 0.186.9824025 1.095833 year.9560516.035339-1.22 0.224.8892379 1.027886 _cons.0363575.0948862-1.27 0.204.0002183 6.054397 Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] study: Identity sd(_cons).9696404.197926.6499213 1.44664 LR test vs. logistic regression: chibar2(01) = 2019.42 Prob>=chibar2 = 0.0000
Integration points = 1 Wald chi2(2) = 144.88 Log likelihood = -192.50774 Prob > chi2 = 0.0000 cases Odds Ratio Std. Err. z P> z [95% Conf. Interval] intervention.6206583.025586-11.57 0.000.5724831.6728877 year.9208262.0232455-3.27 0.001.8763747.9675324 _cons.7934227.9916217-0.19 0.853.0684969 9.190487 Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] study: Identity sd(_cons) 1.035618.2104227.6954207 1.542238 LR test vs. logistic regression: chibar2(01) = 2346.37 Prob>=chibar2 = 0.0000
model evaluation model log L AIC BIC intervention -196.3842 398.7684 402.5427 + latitude -192.3833 392.7665 397.7989 + year -191.6872 393.3743 399.6648 - latitude -192.5077 393.0155 398.0479