[R-meta] Model with intercept gives 0 heterogeneity but without intercept is ok

[Luke Martinez]

Thank you, Wolfgang. I visited the link you kindly shared. But that link
only discusses the effect of removing the intercept on the fixed parts, not
random parts.

Also, in that link the fixed parts only include either a categorical or
only a continuous moderator, but not both types of moderators together. For
example, if we have two categorical moderators and one
continuous moderator, as in:

data$gender <- sample(c("M","F"),nrow(data),replace = TRUE)
data$sector <- sample(c("Pr","Pv", "NGO"),nrow(data),replace = TRUE)

Then, removing the intercept is the matter of which categorical moderator
appears last in the formula! For example, in:

(A): rma.mv(yi ~  0 + gender + sector +  X , vi, random = ~ 1 |
study/outcome, data = data)

R removes the intercept for "sector" because it appears last. But, in:

(B): rma.mv(yi ~  0 + sector + gender +  X , vi, random = ~ 1 |
study/outcome, data = data)

R removes the intercept for "gender" because it appears last.

My question is that do these behaviors in the fixed-part, essentially,
change the meaning/nature (e.g., what average is varying across study
levels) of the random parts?

Apparently, the random part is not related to the fixed-part in rma.mv(),
and that's why both (A) and (B), with or without the intercepts (i.e., 4
specification) all give the exact same estimates of their two variance
components?

Thank you,
Luke



On Mon, Aug 30, 2021 at 2:43 PM Viechtbauer, Wolfgang (SP) <
wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:

> >-----Original Message-----
> >From: Luke Martinez [mailto:martinezlukerm using gmail.com]
> >Sent: Monday, 30 August, 2021 20:46
> >To: Viechtbauer, Wolfgang (SP)
> >Cc: R meta
> >Subject: Re: [R-meta] Model with intercept gives 0 heterogeneity but
> without
> >intercept is ok
> >
> >Dear Wolfgang,
> >
> >Thank you.
> >
> >1- To make sure I understand this correctly, you're saying that because I
> killed
> >the intercept, then the intercept for one or more continuous moderators
> equals 0
> >for all studies, thus, there is no intercept to vary across the levels of
> study,
> >hence no between-study variance component can be estimated (sigma^2.1 ==
> 0),
> >correct?
>
> This doesn't sound right. Removing the intercept in this model says that
> the average effect must be 0 when X = 0. One can still estimates the
> variance components whether there is an intercept or not. They have
> different interpretations though, since the variances are estimated as
> deviations from a line that has an intercept of 0 or not.
>
> >2- Under this circumstance (killing intercept with continuous moderators
> only),
> >the intercepts (or averages) for "study/outcome" combinations can still
> vary
> >across study-outcome combinations, and thus, in isolation from
> "sigma^2.1",  the
> >other "sigma^2.2" can be [correctly] estimated, correct?
>
> Again, both variance components can be estimated. It just happens to be
> the case that sigma^2.1 is estimated to be essentially 0 in Model1.
>
> >3- I have seen models where the intercept is killed in the fixed part,
> but present
> >in the random part. Based on what you said, in such models at least 1
> categorical
> >moderator must be present so the between-study variance component can be
> estimated
> >(e.g., below), correct?
>
> Again, all variance components can be estimated whether the fixed part
> includes an intercept or not.
>
> >data$gender <- sample(c("M","F"),nrow(data),replace = TRUE)
> >
> >rma.mv(yi ~ 0 + X + gender, vi, random = ~ 1 | study/outcome)
> >
> >                     estim    sqrt  nlvls  fixed         factor
> >sigma^2.1  0.0000  0.0001     60     no          study  <---   Still "0" ?
> >sigma^2.2  0.5932  0.7702    120     no  study/outcome
>
> In this case, the model is identical whether you use '0 + X + gender' or
> 'X + gender', just the parameterization of the fixed part is different.
> Please see the link I posted.
>
> >Thank you very much,
> >Luke
> >
> >On Mon, Aug 30, 2021 at 11:26 AM Viechtbauer, Wolfgang (SP)
> ><wolfgang.viechtbauer using maastrichtuniversity.nl> wrote:
> >Dear Luke,
> >
> >If X is a continuous moderator, removing the intercept forces the line to
> go
> >through the origin. That is very rarely a sensible thing to do. See also:
> >
> >
> https://www.metafor-project.org/doku.php/tips:models_with_or_without_intercept
> >
> >Best,
> >Wolfgang
> >
> >>-----Original Message-----
> >>From: R-sig-meta-analysis [mailto:
> r-sig-meta-analysis-bounces using r-project.org] On
> >>Behalf Of Luke Martinez
> >>Sent: Monday, 30 August, 2021 18:02
> >>To: R meta
> >>Subject: [R-meta] Model with intercept gives 0 heterogeneity but without
> >intercept
> >>is ok
> >>
> >>Dear Colleagues.
> >>
> >>I fitted two exact same models except that for one I included the
> intercept
> >>(Model 1) in the model, for the other, I didn't (Model 2).
> >>
> >>I wonder why for Model 1 the estimate of between-study heterogeneity is
> "0"
> >>but for Model 2 that estimate is not "0"?
> >>
> >>Thank you very much,
> >>Luke
> >>
> >>set.seed(132)
> >>data <- expand.grid(study = 1:60, outcome = rep(1:2,2))
> >>data$X <- rnorm(nrow(data))
> >>e <- rnorm(nrow(data))
> >>data$yi <- .8+.6*data$X + e
> >>data$vi <- runif(nrow(data))
> >>
> >>Model1 <- rma.mv(yi ~ 1 + X, vi, random = ~ 1 | study/outcome, data =
> dat)
> >>
> >>                       estim    sqrt  nlvls  fixed         factor
> >>sigma^2.1  0.0000  0.0001     60     no          study
> >>sigma^2.2  0.4707  0.6861    120     no  study/outcome
> >>
> >>
> >>Model2 <- rma.mv(yi ~ 0 + X, vi, random = ~ 1 | study/outcome, data =
> dat)
> >>
> >>                    estim    sqrt  nlvls  fixed         factor
> >>sigma^2.1  0.5634  0.7506     60     no          study
> >>sigma^2.2  0.4878  0.6984    120     no  study/outcome
>

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