Index
Mixed Model Analysis – continuous variables .............................................................................................. 3
Random intercept (p.8) ............................................................................................................................. 3
Random slopes (p.16, 19, 32) .................................................................................................................... 3
Fixed part of the model ............................................................................................................................. 4
How to report ............................................................................................................................................ 4
Random part of the model ........................................................................................................................ 4
Likelihood ratio test (p.12) ........................................................................................................................ 4
Intraclass Correlation Coefficient (ICC) (p.14) ........................................................................................... 4
Centering independent variable (p.22) ..................................................................................................... 5
Steps to conduct a mixed model analysis – 3 levels model ...................................................................... 5
Confounding .............................................................................................................................................. 5
Effect modification .................................................................................................................................... 5
Application of Mixed Models .................................................................................................................... 6
Explain variance ..................................................................................................................................... 6
Explanatory factors................................................................................................................................ 6
Aggregate analysis ................................................................................................................................. 6
Disaggregate / Naïve analysis ................................................................................................................ 6
Mixed Model Analysis ........................................................................................................................... 6
Sample size ................................................................................................................................................ 7
Correction for dependency ................................................................................................................... 7
Mixed Model Analysis – dichotomous and count outcomes (p.46) .............................................................. 8
Dichotomous ............................................................................................................................................. 8
Regression coefficient (p. 47) ................................................................................................................ 8
Odds Ratio (OR) ..................................................................................................................................... 8
Random intercept.................................................................................................................................. 8
Intraclass correlation coefficient (ICC) .................................................................................................. 8
Likelihood ratio test............................................................................................................................... 8
Confounding and effect modification ................................................................................................... 8
Residual variance (p. 47) ....................................................................................................................... 9
Count outcome variables (p. 65) ............................................................................................................... 9
Output ................................................................................................................................................... 9
,Longitudinal data ........................................................................................................................................... 9
GLM for repeated measures ......................................................................................................................... 9
Multivariate ............................................................................................................................................... 9
Univariate test ........................................................................................................................................... 9
Sums of squares ...................................................................................................................................... 10
Post-hoc procedures ............................................................................................................................... 11
One within, one between design ............................................................................................................ 11
Tests of between subject effects......................................................................................................... 11
Multivariate test .................................................................................................................................. 11
Tests of within-subjects effects (univariate approach) ....................................................................... 11
Disadvantages of GLM ............................................................................................................................. 12
Mixed Models with Longitudinal continuous data...................................................................................... 12
Random intercept variance ..................................................................................................................... 12
Random slope .......................................................................................................................................... 12
Regression coefficient ............................................................................................................................. 13
Interpretation of effect estimate ............................................................................................................ 13
Mixed Model vs GEE ................................................................................................................................ 13
GEE analysis for longitudinal data ............................................................................................................... 13
Correlation structures p.59 ..................................................................................................................... 13
Independent structure ........................................................................................................................ 13
Exchangeable structure ....................................................................................................................... 14
Stationary m-dependent structure ..................................................................................................... 14
Autoregressive structure ..................................................................................................................... 14
Unstructured ....................................................................................................................................... 14
Output ..................................................................................................................................................... 14
Regression coefficient ......................................................................................................................... 14
Scale parameter................................................................................................................................... 15
Alternative Models ...................................................................................................................................... 15
Time Lag Models...................................................................................................................................... 15
Hybrid models ......................................................................................................................................... 16
Modelling of changes .............................................................................................................................. 16
Longitudinal Data Analysis with dichotomous and count outcomes .......................................................... 16
Dichotomous outcome ............................................................................................................................ 16
, Count outcomes ...................................................................................................................................... 17
Modelling of time ........................................................................................................................................ 18
Growth curve analysis ............................................................................................................................. 18
Adjustment for time ................................................................................................................................ 18
Interaction with time............................................................................................................................... 18
Missing data ................................................................................................................................................ 19
Cross sectional data and missing data .................................................................................................... 19
Longitudinal data and missing data......................................................................................................... 19
Analysis of RCT ............................................................................................................................................ 20
One-follow up measurement .................................................................................................................. 20
Analysis of covariance ......................................................................................................................... 20
More than 1 follow-up measure (continuous outcome)......................................................................... 20
Interaction with time............................................................................................................................... 21
RCT Analysis – Dichotomous outcome .................................................................................................... 21
Classic analysis ..................................................................................................................................... 21
Alternative analysis ............................................................................................................................. 21
Mixed Model Analysis – continuous variables
B0 = the value of the outcome when independent variable = 0
B1 = reflects the difference in outcome variable for subjects who differ 1 unit in independent variable
Random intercept (p.8)
• Correction for cluster (e.g. neighborhood, region) is carried out by estimating the variance of
intercepts for different clusters
• Assumption: normal distribution
Random slopes (p.16, 19, 32)
• Effect modification with cluster is carried out by estimating variance of the slopes
• It checks if relationship between variables is area dependent → if yes! Random slope!
• It is the variance over the regression coefficients for the independent variable for the different
areas
• Assumption: normal distribution
• Likelihood test → evaluate whether random slope is needed !
• When adding random intercept and random slope in the model this must be UNSTRUCTURED
and add covariance between intercept and slope → it models dependency (not model
dependency does not make sense! P 21)
, • Important: random slope can only be added at a level higher from the level it was measured (p
32)
• Check covariance sign between variance of random slope and random intercept variance
o Positive sign: areas (e.g. neighborhoods) with high intercept, have high slope
▪ Example: Neighborhood with a high level of depression have a higher regression
coefficient for age
o Negative sign: areas with high intercept, low slope
o For example:
▪ The regression coefficient for age is higher in the neighborhoods with a lower
intercept, which means a lower average value of depression
▪ In neighborhood with high average value of depression we have a lower
regression coefficient for age
Fixed part of the model
• Regression coefficient (B1)
o Reflects the difference in outcome variable when there is 1 unit difference in the
independent variable
• Intercept (_cons) (B0)
o the value of the outcome when independent variable = 0
o sometimes not informative when independent variable does not have value 0
• Standard error (SE)
• Z-value = regression coefficient / SE
• P-value
• 95% CI = regression coefficient +/- 1.96*SE
• the value of the outcome when independent variable = 0
How to report
• a difference in 1 unit in activity (independent variable) between subjects is associated with a
difference of 0.59 units in health (outcome variable)
Random part of the model
• var(residual) → residual variance → error variance / unexplained variance
• var(_cons) → random intercept variance → the variance in intercepts for the different areas
around the intercept value given in the fixed part of the model (_cons)
Likelihood ratio test (p.12)
• evaluates whether we need random coefficients in the model
• The difference between – 2 * log likelihood model 1 and -2 * log likelihood model 2
• Check with chi-square distribution with n degrees of freedom (p.12)
• Check last line of STATA output LR test vs linear model
Intraclass Correlation Coefficient (ICC) (p.14)
• Measure of dependency
• ICC = between group variance / total variance = variance between neighborhoods / total
variance = var(_cons) / var(_cons) + var(Residual)
, • Gives the magnitude of the average correlation that you can find within the grouping variable
• ICC and correlation are related
• Note: calculate ICC with an intercept only model, as more independent variables that are related
to the outcome will explain some of the variance and lower the true ICC estimate
Centering independent variable (p.22)
• Makes intercept value and random variance of intercept more interpretable
• Everything else in the model stays the same
• We just move the Y-axis
• for each subject, you have to subtract your independent variable value by the average the entire
set of the independent variable
Steps to conduct a mixed model analysis – 3 levels model
1. Naïve model
2. Add random intercept on neighborhood More important !
3. Add random intercept on region
4. Add random slope for “age” on neighborhood More difficult to
5. Add random slope for “age” on region
estimate !
New interpretation of the intercept
• The neighborhood variance over the intercept is the difference between neighborhoods within
regions
Mixed model analysis with continuous outcome variable is identical to linear regression analysis
with an additional correction
Confounding
• Add confound variable to model
• Check regression coefficient of variable of interest
• More than 10% difference → confounding !
Effect modification
• Add interaction term
• Significant p-value → effect modification !
• Results should be presented for different groups
• Positive regression coefficient of the interaction means that relation between variables X and Y is
stronger for cities for covariate = 1 compared to covariate = 0
• Regression coefficient for variable X is always for the subgroup of the population where
covariate = 0
Mixed Model Analysis – continuous variables .............................................................................................. 3
Random intercept (p.8) ............................................................................................................................. 3
Random slopes (p.16, 19, 32) .................................................................................................................... 3
Fixed part of the model ............................................................................................................................. 4
How to report ............................................................................................................................................ 4
Random part of the model ........................................................................................................................ 4
Likelihood ratio test (p.12) ........................................................................................................................ 4
Intraclass Correlation Coefficient (ICC) (p.14) ........................................................................................... 4
Centering independent variable (p.22) ..................................................................................................... 5
Steps to conduct a mixed model analysis – 3 levels model ...................................................................... 5
Confounding .............................................................................................................................................. 5
Effect modification .................................................................................................................................... 5
Application of Mixed Models .................................................................................................................... 6
Explain variance ..................................................................................................................................... 6
Explanatory factors................................................................................................................................ 6
Aggregate analysis ................................................................................................................................. 6
Disaggregate / Naïve analysis ................................................................................................................ 6
Mixed Model Analysis ........................................................................................................................... 6
Sample size ................................................................................................................................................ 7
Correction for dependency ................................................................................................................... 7
Mixed Model Analysis – dichotomous and count outcomes (p.46) .............................................................. 8
Dichotomous ............................................................................................................................................. 8
Regression coefficient (p. 47) ................................................................................................................ 8
Odds Ratio (OR) ..................................................................................................................................... 8
Random intercept.................................................................................................................................. 8
Intraclass correlation coefficient (ICC) .................................................................................................. 8
Likelihood ratio test............................................................................................................................... 8
Confounding and effect modification ................................................................................................... 8
Residual variance (p. 47) ....................................................................................................................... 9
Count outcome variables (p. 65) ............................................................................................................... 9
Output ................................................................................................................................................... 9
,Longitudinal data ........................................................................................................................................... 9
GLM for repeated measures ......................................................................................................................... 9
Multivariate ............................................................................................................................................... 9
Univariate test ........................................................................................................................................... 9
Sums of squares ...................................................................................................................................... 10
Post-hoc procedures ............................................................................................................................... 11
One within, one between design ............................................................................................................ 11
Tests of between subject effects......................................................................................................... 11
Multivariate test .................................................................................................................................. 11
Tests of within-subjects effects (univariate approach) ....................................................................... 11
Disadvantages of GLM ............................................................................................................................. 12
Mixed Models with Longitudinal continuous data...................................................................................... 12
Random intercept variance ..................................................................................................................... 12
Random slope .......................................................................................................................................... 12
Regression coefficient ............................................................................................................................. 13
Interpretation of effect estimate ............................................................................................................ 13
Mixed Model vs GEE ................................................................................................................................ 13
GEE analysis for longitudinal data ............................................................................................................... 13
Correlation structures p.59 ..................................................................................................................... 13
Independent structure ........................................................................................................................ 13
Exchangeable structure ....................................................................................................................... 14
Stationary m-dependent structure ..................................................................................................... 14
Autoregressive structure ..................................................................................................................... 14
Unstructured ....................................................................................................................................... 14
Output ..................................................................................................................................................... 14
Regression coefficient ......................................................................................................................... 14
Scale parameter................................................................................................................................... 15
Alternative Models ...................................................................................................................................... 15
Time Lag Models...................................................................................................................................... 15
Hybrid models ......................................................................................................................................... 16
Modelling of changes .............................................................................................................................. 16
Longitudinal Data Analysis with dichotomous and count outcomes .......................................................... 16
Dichotomous outcome ............................................................................................................................ 16
, Count outcomes ...................................................................................................................................... 17
Modelling of time ........................................................................................................................................ 18
Growth curve analysis ............................................................................................................................. 18
Adjustment for time ................................................................................................................................ 18
Interaction with time............................................................................................................................... 18
Missing data ................................................................................................................................................ 19
Cross sectional data and missing data .................................................................................................... 19
Longitudinal data and missing data......................................................................................................... 19
Analysis of RCT ............................................................................................................................................ 20
One-follow up measurement .................................................................................................................. 20
Analysis of covariance ......................................................................................................................... 20
More than 1 follow-up measure (continuous outcome)......................................................................... 20
Interaction with time............................................................................................................................... 21
RCT Analysis – Dichotomous outcome .................................................................................................... 21
Classic analysis ..................................................................................................................................... 21
Alternative analysis ............................................................................................................................. 21
Mixed Model Analysis – continuous variables
B0 = the value of the outcome when independent variable = 0
B1 = reflects the difference in outcome variable for subjects who differ 1 unit in independent variable
Random intercept (p.8)
• Correction for cluster (e.g. neighborhood, region) is carried out by estimating the variance of
intercepts for different clusters
• Assumption: normal distribution
Random slopes (p.16, 19, 32)
• Effect modification with cluster is carried out by estimating variance of the slopes
• It checks if relationship between variables is area dependent → if yes! Random slope!
• It is the variance over the regression coefficients for the independent variable for the different
areas
• Assumption: normal distribution
• Likelihood test → evaluate whether random slope is needed !
• When adding random intercept and random slope in the model this must be UNSTRUCTURED
and add covariance between intercept and slope → it models dependency (not model
dependency does not make sense! P 21)
, • Important: random slope can only be added at a level higher from the level it was measured (p
32)
• Check covariance sign between variance of random slope and random intercept variance
o Positive sign: areas (e.g. neighborhoods) with high intercept, have high slope
▪ Example: Neighborhood with a high level of depression have a higher regression
coefficient for age
o Negative sign: areas with high intercept, low slope
o For example:
▪ The regression coefficient for age is higher in the neighborhoods with a lower
intercept, which means a lower average value of depression
▪ In neighborhood with high average value of depression we have a lower
regression coefficient for age
Fixed part of the model
• Regression coefficient (B1)
o Reflects the difference in outcome variable when there is 1 unit difference in the
independent variable
• Intercept (_cons) (B0)
o the value of the outcome when independent variable = 0
o sometimes not informative when independent variable does not have value 0
• Standard error (SE)
• Z-value = regression coefficient / SE
• P-value
• 95% CI = regression coefficient +/- 1.96*SE
• the value of the outcome when independent variable = 0
How to report
• a difference in 1 unit in activity (independent variable) between subjects is associated with a
difference of 0.59 units in health (outcome variable)
Random part of the model
• var(residual) → residual variance → error variance / unexplained variance
• var(_cons) → random intercept variance → the variance in intercepts for the different areas
around the intercept value given in the fixed part of the model (_cons)
Likelihood ratio test (p.12)
• evaluates whether we need random coefficients in the model
• The difference between – 2 * log likelihood model 1 and -2 * log likelihood model 2
• Check with chi-square distribution with n degrees of freedom (p.12)
• Check last line of STATA output LR test vs linear model
Intraclass Correlation Coefficient (ICC) (p.14)
• Measure of dependency
• ICC = between group variance / total variance = variance between neighborhoods / total
variance = var(_cons) / var(_cons) + var(Residual)
, • Gives the magnitude of the average correlation that you can find within the grouping variable
• ICC and correlation are related
• Note: calculate ICC with an intercept only model, as more independent variables that are related
to the outcome will explain some of the variance and lower the true ICC estimate
Centering independent variable (p.22)
• Makes intercept value and random variance of intercept more interpretable
• Everything else in the model stays the same
• We just move the Y-axis
• for each subject, you have to subtract your independent variable value by the average the entire
set of the independent variable
Steps to conduct a mixed model analysis – 3 levels model
1. Naïve model
2. Add random intercept on neighborhood More important !
3. Add random intercept on region
4. Add random slope for “age” on neighborhood More difficult to
5. Add random slope for “age” on region
estimate !
New interpretation of the intercept
• The neighborhood variance over the intercept is the difference between neighborhoods within
regions
Mixed model analysis with continuous outcome variable is identical to linear regression analysis
with an additional correction
Confounding
• Add confound variable to model
• Check regression coefficient of variable of interest
• More than 10% difference → confounding !
Effect modification
• Add interaction term
• Significant p-value → effect modification !
• Results should be presented for different groups
• Positive regression coefficient of the interaction means that relation between variables X and Y is
stronger for cities for covariate = 1 compared to covariate = 0
• Regression coefficient for variable X is always for the subgroup of the population where
covariate = 0
