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Latent class models for cross-national comparisons: the association between individual and national-level fertility and partnership characteristics

citly model micro-macro level interactions, without the loss of information associated with tradition-
al multilevel models, and countries clustered within classes which afford a substantive interpretation.
2. Methodological Discussion

Random effects models are a form of regression, which in their most basic form function by captur-
ing deviations in clusters from the overall regression equation by partitioning error terms. Deviations
from the overall population line due to country/cluster level variation are captured by the cluster lev-
el random effect, which is a draw from a normal (or similar) distribution. Deviation is assumed to be
due to a sampling process with an approximately normal sampling distribution — the stochastic na-
ture of this model leads to the term random effects models. Residuals (cluster level deviations) will
typically be shrunk or precision weighted, to take account of the fact that j level units with few indi-
viduals (i level observations) will be unreliably estimated (Efron and Morris, 1973).
This model is attractive to social scientists wishing to investigate the interaction between individ-
ual and higher order clusters. The random effects model will correct for standard error overestima-
tion due to clustering, provides estimates of the relative size of individual and cluster level correla-
tions and allow for more complex models such as random slopes models or cross level interaction
which allow effects of interest to vary between clusters.
That said, random effects models have some limitations which means that they may not be me-
thodologically germane to research questions which specify certain types of cluster, for example,
countries. Many analyses will typically be limited in the number of countries analysed (for example
Neels et al. (2013) analysed only 14 countries, Billingsley and Farrini (2014) used 21). This is prob-
lematic when trying to obtain estimates for country level random effects, due to a lack of precision
and since many iterative methods (such as IGLS) will assume normality, which is difficult to verify
with such a small sample. Small sample sizes can typically result in underpowered analysis; Hox et
al. (2012) find that at least 20 higher order units is required for the accurate interpretation of regres-
sion coefficients, and a sample size of at least 50 higher order units is required for sufficiently po-
wered analysis of variance parameters. The use of Bayesian estimation techniques can produce more
reliable estimates for a far lower number of higher level units (Hox, van de Schoot, and Matthijsse,
2012). However, the small number of j units can mean that reliable estimates require exceptionally
long model runs (when using MCMC; Browne, 2014) or the use of informative priors.
Interpretation can also be difficult when trying to establish cluster-specific effects. Interpreting the
deviation for a particular country requires interpretation of posterior or empirical Bayes residuals,
and is not intuitive. Where the model is more complicated, for example through the addition of ran-
dom slopes, country specific estimates can become increasingly obtuse.
Finally, it is questionable whether this model is conceptually valid. The fundamental assumption
of the random effects model is that the random effect approximates variation that is characteristic of
a sampling process, where the countries in the observed dataset were drawn at random from a larger
population. For many researchers, the selection of countries within a dataset may often be purposive.
At best, it is unclear what the population of countries to which inference is being made actually is,
since enumeration of higher order units will tend to be complete or approaching the finite population
from which they are drawn (Stegmueller, 2013). Further, it is assumed that the cluster level devia-
tions from the population line can be well approximated by a draw from an i.i.d. normal distribution.
Evidence in demographic literature of the existence of country typologies (Korpi, Ferrarini, and En-
glund, 2013; Korpi, 2000; Kalwij, 2010) or ‘clumpy’ groupings of higher order units within multile-
vel models (Billingsley and Farrini, 2014) would indicate that this assumption is shaky when coun-
tries are specified as the level 2 unit.
2.1 Fixed Effects

In fixed effects models deviation from the overall population line, rather than being captured by a

46 International Journal of Population Studies | 2016, Volume 2, Issue 2