Multilevel analysis

Multilevel analysis can address the lack of independence of the observations when you are analyzing grouped data. See Stata Multilevel Mixed-Effects Reference Manual.

• groups of individuals
• panel data

Fixed-effects regression models

\[y_it = \beta_0 +\beta x_{it}+\mu_i+\eta_{it}\]

if \(\mu_i\) correlates with \(x_{it}\) -> Fixed-effects if \(\mu_i\) independent of \(x_{it}\) -> Random-effects models give consistent estimates

xtreg see Stata Longitudinal-Data/Panel-Data Reference Manual.

Random-effects regression models

\[y_it = \beta_0 +\beta x_{it}+\gamma z_i +\mu_i+\eta_{it}\]

assume \(\mu_i\) is independent of \(x_{it}\)

fixed component, \( \beta_0 +\beta x_{it}+\gamma z_i\) , describes the overall relationship between our dependent variable and our independent variable. The random component, \(\mu_i\) i represents the effects of the unobserved time-invariant variables.

score = fixed part + random effects + error

Going back and forth between wide and long formats : reshape wide and reshape long

reshape long drink, i(id) j(wave)

Random-intercept model

linear model

mixed drink c.wave || id:
estimates store linear
margins, at(wave=(0(2)10))
marginsplot

quadratic term

mixed drink c.wave##c.wave || id:
estimates store quadratic
margins, at(wave=(0(2)10))
marginsplot
lrtest linear quadratic

A proportional reduction in error (PRE) measuring how much the residual (error) variance is reduced by adding the quadratic term may be useful. We will call the random-intercept linear model “Model 1” and the random-intercept quadratic model “Model 2”.

PRE = (var(Residual)Model1-var(Residual)Model2)/var(Residual)Model1

Treating time as a categorical variable

mixed drink i.wave || id:
estimates store means
margins, at(wave=(0(2)10))
marginsplot
lrtest linear means
lrtest quadratic means

Random-coefficients model

mixed drink c.wave || id: wave, cov(unstructured)
predict yhat_drink, fitted

Including a time-invariant covariate

* Random coefficients model with time invariant covariate
* gender coded as male = 1, female = 0
mixed drink c.wave i.male || id: wave
margins male, at(wave=(0(2)8))
marginsplot

* Random coefficients, with wave interacting with the
* time invariant covariate--gender coded
mixed drink c.wave##i.male || id: wave
margins male, at(wave=(0(2)8))
marginsplot

mixed drink c.wave##c.wave##i.male || id: wave
margins male, at(wave=(0(2)8))
marginsplot