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I have the following data:

data <- data.frame(id_pers=c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11),
                       Income=c(10000, 12000, 1000, 9000, 10000, 
                       16000, 100000, 120000, 119000, 10000, 11000),
                       family=c(1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5),
                       BIRTH_ORDER=c("firstborn",  "notfirstborn", 
                       "firstborn", "notfirstborn", "firstborn", 
                       "notfirstborn", "firstborn", "notfirstborn", 
                       "notfirstborn", "firstborn", "notfirstborn"),
                       income_father=c(12000, 12000, 20000, 20000, 
                       15000, 15000, 92000, 92000, 92000, 9000, 9000),
                       income_father_cat=c("low", "low", "middle", 
                         "middle", "middle", "middle", "high", "high", 
                        "high", "low", "low"),
                       region=c("south", "south", "north","north", 
                          "west", "west", "south", 
                          "south","south","north","north"))

I want to find the effect of being part of a certain family. What does that mean? How much of the variation in Income of a certain person can be explained with being part of that family? In lme4, we would write: baseline_model <- lmer(Income ~ (1 | family), data= data)

At the end I want to test, whether the Interclass-correlation-coefficient (ICC) is getting smaller after controlling for a certain family-internal or family-external factors. For most of the variables it's pretty clear what to do. For example in testing the family-external effect of the region or the family-internal (categorized) income of the father

region_model <- Income ~ (1 | family) + (1 | region) , data= data)

father_model <- Income ~ (1 | family) + (1 | income_father_cat) , 
    data= data)

All of these effects I will treat as crossed-random-effects, because the "sub-classes" belonging to every family. Nice explanation I read here: Crossed vs nested random effects: how do they differ and how are they specified correctly in lme4?

As far as good!

But now it's the question how to handle to family-internal factor of being first-born or not firstborn sibling within a family. There will always be at least one first-born individual but eventually more than one not first-born siblings (as in family "4"). How do I handle that problem with using either treating BIRTH_ORDER as a crossed random effect or as a nested random effect?

crossed_model <- Income ~ (1 | family) + (1 | BIRTH_ORDER) , 
    data= data)
or
nested_model <- Income ~ (1 | family) +  (1 | family:BIRTH_ORDER) , 
    data= data)

What does these different models tell us about the family background?

What do you think about it?

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1 Answer 1

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It doesn't really make sense (IMO) to treat birth order on its own as a random effect, as the levels are not exchangeable: that is, it wouldn't really make sense given birth order "first"/"not first" to switch the labels (whereas if you have families "Jones", "Smith", "Woodward", relabeling the families won't change anything in the interpretation). Therefore, I would go with (1|family:BIRTH_ORDER), which says that birth order within family matters. In fact, you might want (BIRTH_ORDER|family) (or (BIRTH_ORDER-1|family)), which would allow for a correlation in the effects of different birth orders within families.

(Furthermore, since BIRTH_ORDER only has two levels, it wouldn't really be practical to treat it as a random effect - unless you have strong Bayesian priors it doesn't really work to fit factors with fewer than 5-6 levels as random effects.)

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