[R-sig-ME] Fwd: syntax equation of random intercepts and slopes model

Viechtbauer, Wolfgang (SP) wolfg@ng@viechtb@uer @ending from m@@@trichtuniver@ity@nl
Fri May 18 15:27:07 CEST 2018

It should be:

u_0i ~ N(0, τ^2_0)
u_1i ~ N(0, τ^2_1)
e_ij ~ N(0, sigma^2)

and it is also worth mentioning that the model allows for correlation between u_0i and u_1i. So, technically, the assumption is:

[u_0i] ~ MVN([0], [τ^2_0  rho*τ_0*τ_1])
[u_1i]      ([0]  [       τ^2_1      ])

And if one wants to be really explicit, we assume that u_0i and e_ij are independent and u_1i and e_ij are independent.


-----Original Message-----
From: R-sig-mixed-models [mailto:r-sig-mixed-models-bounces at r-project.org] On Behalf Of Juan Pablo Edwards Molina
Sent: Friday, 18 May, 2018 1:34
To: Ben Bolker
Cc: R SIG Mixed Models
Subject: Re: [R-sig-ME] Fwd: syntax equation of random intercepts and slopes model

Thanks prof. Bolker,
Do you mean this?

u_i∼N(0,τ^2)      e_ij∼N(0,v_i)


2018-05-17 16:57 GMT-03:00 Ben Bolker <bbolker at gmail.com>:
> That looks about right.  You didn't specify the variance of e_ij in
> your description, and you didn't say explicitly that the u_ and e_
> values are Normally distributed ...
> On Thu, May 17, 2018 at 2:27 PM, Juan Pablo Edwards Molina
> <edwardsmolina at gmail.com> wrote:
>> Sorry, I edited the lmer function...
>> ============================================
>> Dear list,
>> I fitted a linear mixed effects models to a set of 41 field trials
>> with plot-level assessments of x,y, for estimating the linear
>> regression coefficients β_0 and β_1
>> res1 <- lmer(y ~ x+ (x|trial), data=mydata, REML=F)
>> I wish to write the model equation for its publication, so this is my first try:
>> W_ij= (β_0 + u_0i)+ (β_1+ u_1i) x_ij + e_ij
>> where j subscript represents the j-plot within i-trial, both for y or
>> x. β0 and β1 are the population average intercept and slope; u0i and
>> u1i are the effect of the i-trial on the intercept and the slope,
>> respectively, considered as random variables (with mean 0 and
>> variances  τ_u0 and  τ_u1 a )
>> I´m not sure if I´m in the right path... I would really appreciate any guidance.
>> Juan Edwards
>> National Institute of Agriculture Technology - Argentina

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