[R-sig-ME] Standard Error of a coef. in a 2-level model vs. 2 OLS models

[Phillip Alday]

The answer is no.

Consider the case where the between-cluster variation goes to zero (i.e.
a singular fit), but the residual variation is not zero. The
between-cluster variation is clearly not the same as the "total
variation" / residual variation in a non mixed OLS regression on the
data aggregated within clusters.

The other way to see this is that lme4 actually uses the scaled random
effects, i.e. the variance relative to the residual variance, for
fitting. (This what the entries in the theta vector reflect: the lower
Cholesky factor of the variance-covariance matrix of the scaled random
effects.)

Best,

Phillip

On 17/09/2020 19:15, Simon Harmel wrote:
> Thanks Harold! I may have been insufficiently clear.
>
> At its core, the question is asking: "Is the "between-cluster" variation
> (i.e., τ00) in a two-level nested mixed-effects model (i.e., `m1`), the
> same as the amount of "total variation" that we would otherwise get if we
> only use the cluster-level data in a corresponding, NON-HLM regression
> model (i.e., `ols2`)?"
>
>
> library(lme4)
> library(tidyverse)
>
> hsb <- read.csv('
> https://raw.githubusercontent.com/rnorouzian/e/master/hsb.csv')
> hsb_ave <- hsb %>% group_by(sch.id) %>% mutate(math_ave = mean(math)) %>%
> slice(1)
>
> ols2 <- lm(math_ave ~ sector, data = hsb_ave)
>
> m1 <- lmer(math ~ sector + (1|sch.id), data = hsb)
>
> On Thu, Sep 17, 2020 at 11:16 AM Harold Doran <
> harold.doran using cambiumassessment.com> wrote:
>
>> Simon
>>
>>
>>
>> Crossposting like this is frowned on a bit, but I’ve been in your shoes
>> trying to get an answer before. I think you might be a bit confused. I saw
>> your questions in both places and you’re asking how to get the standard
>> errors of the OLS model using by fitting a mixed model and using the
>> variance components from that mixed model to get the standard errors from
>> an OLS model.
>>
>>
>>
>> This is what we refer to in statistics as “bass-ackwards”. 😊
>>
>>
>>
>> If you want the standard errors of the fixed effects from an OLS model,
>> compute them as follows:
>>
>>
>>
>> ctl <- c(4.17,5.58,5.18,6.11,4.50,4.61,5.17,4.53,5.33,5.14)
>>
>> trt <- c(4.81,4.17,4.41,3.59,5.87,3.83,6.03,4.89,4.32,4.69)
>>
>> group <- gl(2, 10, 20, labels = c("Ctl","Trt"))
>>
>> weight <- c(ctl, trt)
>>
>> lm.D9 <- lm(weight ~ group)
>>
>> X <- model.matrix(lm.D9)
>>
>>
>>
>> ### Compute Standard errors of fixed effects
>>
>> sqrt(diag(solve(crossprod(X)) * .6964^2))
>>
>>
>>
>> ### use built in extractor function to get them instead
>>
>> sqrt(diag(vcov(lm.D9)))
>>
>>
>>
>> Similarly, get the standard errors of the mixed model from its own
>> variance/covariance matrix.
>>
>>
>>
>>
>>
>> *From:* Simon Harmel <sim.harmel using gmail.com>
>> *Sent:* Wednesday, September 16, 2020 5:54 PM
>> *To:* Harold Doran <harold.doran using cambiumassessment.com>
>> *Cc:* r-sig-mixed-models <r-sig-mixed-models using r-project.org>
>> *Subject:* Re: [R-sig-ME] Standard Error of a coef. in a 2-level model
>> vs. 2 OLS models
>>
>>
>>
>> Hi Harold,
>>
>>
>>
>> I improved my question, and asked it on CrossValidated: (
>> https://stats.stackexchange.com/questions/487363/using-variance-components-of-a-mixed-model-to-obtain-std-error-of-a-coef-from-a
>> )
>>
>>
>>
>> I would appreciate your answer, either here or on CrossValidated.
>>
>>
>>
>> Many thanks, Simon
>>
>>
>>
>> On Wed, Sep 16, 2020 at 4:45 PM Harold Doran <
>> harold.doran using cambiumassessment.com> wrote:
>>
>> This is not how standard errors are computed for linear or mixed linear
>> models. I'm not sure what you're goal is, but the SEs are the square roots
>> of the diagonal elements of the variance/covariance matrix of the fixed
>> effects.
>>
>> See ?vcov on how to extract that matrix.
>>
>> -----Original Message-----
>> From: R-sig-mixed-models <r-sig-mixed-models-bounces using r-project.org> On
>> Behalf Of Simon Harmel
>> Sent: Sunday, September 13, 2020 7:51 PM
>> To: r-sig-mixed-models <r-sig-mixed-models using r-project.org>
>> Subject: Re: [R-sig-ME] Standard Error of a coef. in a 2-level model vs. 2
>> OLS models
>>
>> External email alert: Be wary of links & attachments.
>>
>>
>> Just a clarification.
>>
>> For `ols1` model, I can approximate its SE of the sector coefficient by
>> using the within and between variance components from the HLM model:
>> sqrt(( 6.68  + 39.15  )/45)/(160*.25))
>>
>> BUT  For `ols2` model, how can I approximate its SE of the sector
>> coefficient by using the within and between variance components from the
>> HLM model?
>>
>> On Sun, Sep 13, 2020 at 6:37 PM Simon Harmel <sim.harmel using gmail.com> wrote:
>>
>>> Dear All,
>>>
>>> I have fit two ols models (ols1 & ols2) and an mixed-effects model (m1).
>>> ols1 is a simple lm() model that ignores the second-level. ols2 is a
>>> simple
>>> lm() model that ignores the first-level.
>>>
>>> For `ols1` model,  `sigma(ols1)^2` almost equals sum of variance
>>> components in the `m1` model: 6.68 (bet.) + 39.15 (with.) For `ols2`
>>> model, I wonder what does `sigma(ols2)^2` represents when compared to
>>> the `m1` model?
>>>
>>> Here is the fully reproducible code:
>>>
>>> library(lme4)
>>> library(tidyverse)
>>>
>>> hsb <- read.csv('
>>> https://raw.githubusercontent.com/rnorouzian/e/master/hsb.csv')
>>> hsb_ave <- hsb %>% group_by(sch.id) %>% mutate(math_ave = mean(math))
>>> %>%
>>> slice(1) # data that only considers grouping but ignores lower level
>>>
>>> ols1 <- lm(math ~ sector, data = hsb)
>>> summary(ols1)
>>>
>>> m1 <- lmer(math ~ sector + (1|sch.id), data = hsb)
>>> summary(m1)
>>>
>>> # `sigma(ols1)^2` almost equals 6.68 (bet.) + 39.15 (with.) from lmer
>>>
>>> But if I fit another ols model that only considers the grouping
>>> structure (ignoring lower level):
>>>
>>> ols2 <- lm(math_ave ~ sector, data = hsb_ave)
>>> summary(ols2)
>>>
>>> Then what does `sigma(ols2)^2` should amount to when compared to the
>>> `m1` model?
>>>
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