The 3rd edition of Hox's book (with Moerbeek & Schoot) says a bit more about

the effects of group size. The following excerpt is from pp. 215-216. Look

for the sentence starting with, "Theall et al. (2011) studied the effects of

small group sizes."

--- start of excerpt ---

It is clear that with increasing sample sizes at all levels, estimates and

their standard errors become more accurate. Kreft (1996) suggests a rule of

thumb, which she calls the ‘30/30 rule.’ To be on the safe side, researchers

should strive for a sample of at least 30 groups with at least 30

individuals per group. From various simulations, this seems sound advice if

the interest is mostly in the fixed parameters. However, it seems that this

rule will likely not yield high power levels for fixed effects at both

levels (Bell et al. 2014). For certain applications, one may modify this

rule of thumb. Specifically, if there is strong interest in cross-level

interactions, the number of groups should be larger, which leads to a 50/20

rule: about 50 groups with about 20 individuals per group. If there is

strong interest in the random part, the variance and covariance components

and their standard errors, the number of groups should be considerably

larger, which leads to a 100/10 rule: about 100 groups with at least about

10 individuals per group. Theall et al. (2011) studied the effects of small

group sizes (i.e. less than five). They found that when the number of groups

was as large as 459, the fixed and random effects were not affected by group

size. When the number of groups decreased, inflated standard errors of fixed

and random effects were found. Group-level variance estimates were more

inflated than fixed effects. Raudenbush (2008) also treats the case of many

small groups. Many small groups always arise when the object of study is

twins, married couples, families, or short time series (Raudenbush, 2008, p.

215). The general advice is to keep the model simple, with few random

components at the second level. The exception are short time series, where

often relatively much and reliable variation is found at the subject level

(Raudenbush, 2008, p. 218).

When the number of groups is smaller than 20, fixed parameter estimates and

their standard errors become inaccurate. When the interest is in variance

components, as in structural equation modeling (SEM), the minimum number of

groups is 50 (Meuleman & Billiet, 2009). Hox, van de Schoot and Mattijsse

(2012) show that with Bayesian estimation, SEM with as few as 20 groups is

feasible. We refer to McNeish and Stapleton (2016) for a general review of

the problems associated with having a small number of groups.

These rules of thumb take into account that there are costs attached to data

collection, so if the number of groups is increased, the number of

individuals per group decreases. In some cases, this may not be a realistic

reflection of costs. For instance, in school research an extra cost will be

incurred when an extra class is included. Testing only part of the class

instead of all pupils will usually not make much difference in the data

collection cost. Given a limited budget, an optimal design should reflect

the various costs of data collection. Snijders and Bosker (1993), Cohen

(1998), Raudenbush and Liu (2000) and Moerbeek, van Breukelen and Berger

(2000) all discuss the problem of choosing sample sizes at two levels while

considering costs. Moerbeek, van Breukelen and Berger (2001) discuss the

problem of optimal design for multilevel logistic models. Essentially,

optimal design is a question of balancing statistical power against data

collection costs. Data collection costs depend on the details of the data

collection method. The problem of estimating power in multilevel designs is

treated later in this chapter.

--- end of excerpt ---

Hox, J. J., Moerbeek, M., & Van de Schoot, R. (2017). Multilevel analysis:

Techniques and applications. Routledge.

Bruce Weaver wrote

> I had not heard of that 30-30 rule of thumb, but discovered that Joop Hox

> mentions it in the second edition (2010) of his book, Multilevel Analysis.

> Here's what he says on p. 235.

>

> --- snip

-----

--

Bruce Weaver

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http://sites.google.com/a/lakeheadu.ca/bweaver/"When all else fails, RTFM."

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