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I am trying to wrap my head around the notation for this three-level model.

Level 1: Repeated observations Level 2: Client Level 3: Therapist

I am using a baseline intercept model to calculate intra-class correlation coefficients. In order to partition the variance at both the client and therapist level, I have random effects listed for both.

lme4 code:

mod01 <- lmer(var ~ 1 + (1 | client) + (1 | therapist), data = dat10)

And my notation thus far is:

()time,client,therapist = 00 + client + therapist + client,therapist

But is that error term correct? I'm trying to keep this as simple as possible for my audience by using the combined equation. Any feedback is greatly appreciated.

Robert Long
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b222
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2 Answers2

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I recognise that this is an old-ish question, but I think it's worth pointing out that the lmer formula in the OP is not for a three-level repeated measures random intercept model, unless the data are coded in a way that makes the nesting unambiguous.

The OP wrote:

mod01 <- lmer(var ~ 1 + (1 | client) + (1 | therapist), data = dat10)

This specifies a 2-level model, with crossed random effects for client and therapist. The correct 3-level formula, for repeated measures within client, where client is nested within therapist is:

mod01 <- lmer(var ~ 1 + (1 | therapist/client), data = dat10)

or equivalently:

mod01 <- lmer(var ~ 1 + (1 | therapist) + (1 | therapist:client), data = dat10)

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Robert Long
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  • thank you for the clarification and link to a well written summary of the issue. When I run the correct 3-level model code (therapist/client), I do end up with the same results across the board. If I understand the issue correctly, when the data are not nested, but in fact crossed, the results will differ. But when the data are truly nested, both formulas produce the same results... That being said, it will be good practice to know one's data and specify the correct formula going forward. Does that sound correct? – b222 Aug 12 '16 at 04:06
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    @bpace Yes it does ! – Robert Long Aug 12 '16 at 08:28
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Imo $\epsilon$client,therapist would imply a constant error over time. I guess that you don't make this assumption, so I would write $\epsilon$time,client,therapist. On this site you will find examples regarding your question.

Qaswed
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