[R-meta] "Categorical" moderator varying within and between studies
Simon Harmel
@|m@h@rme| @end|ng |rom gm@||@com
Thu Dec 3 03:35:39 CET 2020
Hi James,
I keep coming back to our informative discussion in this thread. So a quick
follow-up. Last time, we did the following to obtain the contextual effect
for males. Given this model (i.e., mg_b_w), can we obtain the contextual
effect for females OR we need to fit a new model, this time with males as
the reference group?
Thank you, Simon
library(dplyr)
library(fastDummies)
library(lme4)
hsb <- read.csv("
https://raw.githubusercontent.com/rnorouzian/e/master/hsb.csv")
hsb2 <- hsb %>%
mutate(gender = ifelse(female==0,"M","F")) %>% # create 'gender'
from variable ‘female’
dummy_columns(select_columns = "gender") %>% # create dummies
for 'gender’ (creates 2 but we need 1)
group_by(sch.id) %>% # group by cluster
id 'sch.id'
mutate(across(starts_with("gender_"), list(wthn = ~ . - mean(.), btw = ~
mean(.))))
mg_b_w <- lmer(math ~ gender_M_wthn + gender_M_btw + (1|sch.id), data =
hsb2)
fixef(mg_b_w)[["gender_M_btw"]] - fixef(mg_b_w)[["gender_M_wthn"]] # gives
1.92 as the contextual effect for males
On Thu, Oct 29, 2020 at 2:09 PM James Pustejovsky <jepusto using gmail.com> wrote:
> My apologies! I had this backwards in my head. Revised explanation below:
>
> With gender, if you include the group-mean-centered dummy variables and
> the cluster-level means, then the contextual effect will be as you
> described (gender_M_btw - gender_M_wthn). However, another approach would
> be to leave the dummy variables uncentered. If you do it this way, then the
> coefficient on gender_M_btw corresponds exactly to the contextual effect,
> with no need to subtract out the coefficient on gender_M_wthn.
>
> R code verifying the equivalence of these approaches:
>
> library(dplyr)
> library(fastDummies)
> library(lme4)
>
> hsb <- read.csv("
> https://raw.githubusercontent.com/rnorouzian/e/master/hsb.csv")
>
> hsb2 <- hsb %>%
> mutate(gender = ifelse(female==0,"M","F")) %>% # create 'gender’ from
> variable ‘female’
> dummy_columns(select_columns = "gender") %>% # create dummies for
> 'gender’ (creates 2 but we need 1)
> group_by(sch.id) %>% # group by cluster id '
> sch.id'
> mutate(across(starts_with("gender_"), list(wthn = ~ . - mean(.), btw = ~
> mean(.))))
>
> mg_b_w <- lmer(math ~ gender_M_wthn + gender_M_btw + (1|sch.id), data =
> hsb2)
>
> mg_b_d <- lmer(math ~ gender_M + gender_M_btw + (1|sch.id), data = hsb2)
>
> fixef(mg_b_w)[["gender_M_btw"]] - fixef(mg_b_w)[["gender_M_wthn"]]
> fixef(mg_b_d)[["gender_M_btw"]]
>
> On Thu, Oct 29, 2020 at 1:57 PM Simon Harmel <sim.harmel using gmail.com> wrote:
>
>> Thank you, James. For uniformity, I always (i.e., for both categorical &
>> numeric predictors) use the following method (using a dataset I found on
>> Stack Overflow).
>>
>> So, in the case below, you're saying gender_M_btw is the contextual
>> effect itself?
>>
>> Simon
>>
>> library(dplyr)
>> library(fastDummies)
>> library(lme4)
>>
>> hsb <- read.csv("
>> https://raw.githubusercontent.com/rnorouzian/e/master/hsb.csv")
>>
>> hsb2 <- hsb %>%
>> mutate(gender = ifelse(female==0,"M","F")) %>% # create 'gender’ from
>> variable ‘female’
>> dummy_columns(select_columns = "gender") %>% # create dummies for
>> 'gender’ (creates 2 but we need 1)
>> group_by(sch.id) %>% # group by cluster id '
>> sch.id'
>> mutate(across(starts_with("gender_"), list(wthn = ~ . - mean(.), btw = ~
>> mean(.))))
>>
>> mg_b_w <- lmer(math ~ gender_M_wthn + gender_M_btw + (1|sch.id), data =
>> hsb2)
>>
>> On Thu, Oct 29, 2020 at 1:31 PM James Pustejovsky <jepusto using gmail.com>
>> wrote:
>>
>>> Hi Simon,
>>>
>>> There are different ways to parameterize contextual effects. With
>>> gender, if you include the regular dummy variables (without
>>> group-mean-centering) plus the cluster-level means, then the contextual
>>> effect will be as you described (gender_M_btw - gender_M_wthn). However,
>>> another approach would be to first group-mean-center the dummy variables.
>>> In this approach, for a male student, gender_M_wthn would be equal to 1
>>> minus the proportion of male students in the cluster, and for a female
>>> student, gender_M_wthn would be equal to the negative of the proportion of
>>> male students in the cluster. If you do it this way, then the coefficient
>>> on gender_M_btw corresponds exactly to the contextual effect, with no need
>>> to subtract out the coefficient on gender_M_wthn.
>>>
>>> All that said, if you have more than two categories you will have more
>>> than one contextual effect. In your example, you have a contextual effect
>>> for M, which would be the average difference in the DV between two units
>>> who are both male, but belong to clusters that differ by 1 percentage point
>>> in the composition of males *and have the same proportion of
>>> other-gender students *(i.e., clusters that have 1 percentage point
>>> difference in males, and a -1 percentage point difference in females). And
>>> then you have a contextual effect for other, corresponding the average
>>> difference in the DV between two units who are both other-gender, but
>>> belong to clusters that differ by 1 percentage point in the composition of
>>> other *and have the same proportion of male-gender students *(i.e.,
>>> clusters that have 1 percentage point difference in other, and a -1
>>> percentage point difference in females).
>>>
>>> James
>>>
>>> On Thu, Oct 29, 2020 at 12:24 PM Simon Harmel <sim.harmel using gmail.com>
>>> wrote:
>>>
>>>> Dear James,
>>>>
>>>> This makes perfect sense, many thanks. However, one thing remains. I
>>>> know the contextual effect coefficient is "b_btw - b_wthn". If we have two
>>>> categories (as in the case of "gender") and take females as the
>>>> reference category, then the contextual effect coefficient will be:
>>>>
>>>> gender_M_btw - gender_M_wthn
>>>>
>>>> But if we have more than two categories (say we add a third "gender"
>>>> category called OTHER), then will the contextual effect coefficient be (sum
>>>> of the betweens) - (sum of the withins)?
>>>>
>>>> (gender_M_btw + gender_OTHER_btw) - (gender_M_wthn +
>>>> gender_OTHER_wthn)
>>>>
>>>>
>>>>
>>>> On Thu, Oct 29, 2020 at 9:44 AM James Pustejovsky <jepusto using gmail.com>
>>>> wrote:
>>>>
>>>>> Hi Simon,
>>>>>
>>>>> With a binary or categorical predictor, one could operationalize the
>>>>> contextual effect in terms of proportions (0-1 scale) or percentages (0-100
>>>>> scale). If proportions, like say proportion of vegetarians, then the
>>>>> contextual effect would be the average difference in the DV between two
>>>>> units who are both vegetarian (i.e., have the same value of the predictor),
>>>>> but belong to clusters that are all vegetarian versus all omnivorous (i.e.,
>>>>> that differ by one unit in the proportion for that predictor). That will
>>>>> make the contextual effects look quite large because it's an extreme
>>>>> comparison--absurdly so, in this case, since there can't be a vegetarian in
>>>>> a cluster of all omnivores.
>>>>>
>>>>> If you operationalize the contextual effect in terms of percentages
>>>>> (e.g., % vegetarians) then you get the average difference in the DV
>>>>> between two units who are both vegetarian, but belong to clusters that
>>>>> differ by 1 percentage point in the proportion of vegetarians.
>>>>>
>>>>> All of this works for multi-category predictors also. Say that you had
>>>>> vegetarians, pescatarians, and omnivores, with omnivores as the reference
>>>>> category, then the model would include group-mean-centered dummy variables
>>>>> for vegetarians and pescatarians, plus group-mean predictors representing
>>>>> the proportion/percentage of vegetarians and proportion/percentage of
>>>>> pescatarians. You have to omit one category at each level to avoid
>>>>> collinearity with the intercept.
>>>>>
>>>>> James
>>>>>
>>>>> On Thu, Oct 29, 2020 at 1:32 AM Simon Harmel <sim.harmel using gmail.com>
>>>>> wrote:
>>>>>
>>>>>> Dear James,
>>>>>>
>>>>>> I'm returning to this after a while, a quick question. In your gender
>>>>>> example, you used the term "%female" in your interpretation of the
>>>>>> contextual effect. If the categorical predictor had more than 2 categories,
>>>>>> then would you still use the term % in your interpretation?
>>>>>>
>>>>>> My understanding of contextual effect is below:
>>>>>>
>>>>>> Contextual effect is the average difference in the DV between two
>>>>>> units (e.g., subjects) which have the same value on an IV (e.g., same
>>>>>> gender), but belong to clusters (e.g., schools) whose mean/percentage on
>>>>>> that IV differs by one unit (is unit percentage if IV is categorical?).
>>>>>>
>>>>>> Thank you, Simon
>>>>>>
>>>>>>
>>>>>>
>>>>>> On Sun, Jun 7, 2020 at 7:30 AM James Pustejovsky <jepusto using gmail.com>
>>>>>> wrote:
>>>>>>
>>>>>>> Yes, it’s general and also applies outside the context of
>>>>>>> meta-analysis. See for example Raudenbush & Bryk (2002) for a good
>>>>>>> discussion on centering and contextual effects in hierarchical linear
>>>>>>> models.
>>>>>>>
>>>>>>> On Jun 6, 2020, at 11:07 PM, Simon Harmel <sim.harmel using gmail.com>
>>>>>>> wrote:
>>>>>>>
>>>>>>> Many thanks James. A quick follow-up. The strategy that you
>>>>>>> described is a general, regression modeling strategy, right? I mean even if
>>>>>>> we were fitting a multi-level model, the fixed-effects part of the formula
>>>>>>> had to include the same construction of (i.e., *b1 (%
>>>>>>> female-within)_ij + b2 (% female-between)_j*) in it?
>>>>>>>
>>>>>>> Thanks,
>>>>>>> Simon
>>>>>>>
>>>>>>> On Thu, Jun 4, 2020 at 9:42 AM James Pustejovsky <jepusto using gmail.com>
>>>>>>> wrote:
>>>>>>>
>>>>>>>> Hi Simon,
>>>>>>>>
>>>>>>>> Please keep the listserv cc'd so that others can benefit from these
>>>>>>>> discussions.
>>>>>>>>
>>>>>>>> Unfortunately, I don't think there is any single answer to your
>>>>>>>> question---analytic strategies just depend too much on what your research
>>>>>>>> questions are and the substantive context that you're working in.
>>>>>>>>
>>>>>>>> But speaking generally, the advantages of splitting predictors into
>>>>>>>> within- and between-study versions are two-fold. First is that doing this
>>>>>>>> provides an understanding of the structure of the data you're working with,
>>>>>>>> in that it forces one to consider *which* predictors have
>>>>>>>> within-study variation and *how much *variation there is (e.g.,
>>>>>>>> perhaps many studies have looked at internalizing symptoms, many studies
>>>>>>>> have looked at externalizing symptoms, but only a few have looked at both
>>>>>>>> types of outcomes in the same sample). The second advantage is that
>>>>>>>> within-study predictors have a distinct interpretation from between-study
>>>>>>>> predictors, and the within-study version is often theoretically more
>>>>>>>> interesting/salient. That's because comparisons of effect sizes based on
>>>>>>>> within-study variation hold constant other aspects of the studies that
>>>>>>>> could influence effect size (and that could muddy the interpretation of the
>>>>>>>> moderator).
>>>>>>>>
>>>>>>>> Here is an example that comes up often in research synthesis
>>>>>>>> projects. Suppose that you're interested in whether participant sex
>>>>>>>> moderates the effect of some intervention. Most of the studies in the
>>>>>>>> sample are of type A, such that only aggregated effect sizes can be
>>>>>>>> calculated. For these type A studies, we are able to determine a) the
>>>>>>>> average effect size across the full sample (pooling across sex) and b) the
>>>>>>>> sex composition of the sample (e.g., % female). For a smaller number of
>>>>>>>> studies of type B, we are able to obtain dis-aggregated results for
>>>>>>>> subgroups of male and female participants. For these studies, we are able
>>>>>>>> to determine a) the average effect size for males and b) the average effect
>>>>>>>> size for females, plus c) the sex composition of each of the sub-samples
>>>>>>>> (respectively 0% and 100% female).
>>>>>>>>
>>>>>>>> Without considering within/between variation in the predictor, a
>>>>>>>> meta-regression testing for whether sex is a moderator is:
>>>>>>>>
>>>>>>>> Y_ij = b0 + b1 (% female)_ij + e_ij
>>>>>>>>
>>>>>>>> The coefficient b1 describes how effect size magnitude varies
>>>>>>>> across samples that differ by 1% in the percent of females. But the
>>>>>>>> estimate of this coefficient pools information across studies of type A and
>>>>>>>> studies of type B, essentially assuming that the contextual effects
>>>>>>>> (variance explained by sample composition) are the same as the
>>>>>>>> individual-level moderator effects (how the intervention effect varies
>>>>>>>> between males and females).
>>>>>>>>
>>>>>>>> Now, if we use the within/between decomposition, the
>>>>>>>> meta-regression becomes:
>>>>>>>>
>>>>>>>> Y_ij = b0 + b1 (% female-within)_ij + b2 (% female-between)_j + e_ij
>>>>>>>>
>>>>>>>> In this model, b1 will be estimated *using only the studies of
>>>>>>>> type B*, as an average of the moderator effects for the studies
>>>>>>>> that provide dis-aggregated data. And b2 will be estimated using studies of
>>>>>>>> type A and the study-level average % female in studies of type B. Thus b2
>>>>>>>> can be interpreted as a pure contextual effect (variance explained by
>>>>>>>> sample composition). Why does this matter? It's because contextual effects
>>>>>>>> usually have a much murkier interpretation than individual-level moderator
>>>>>>>> effects. Maybe this particular intervention has been tested for several
>>>>>>>> different professions (e.g., education, nursing, dentistry, construction),
>>>>>>>> and professions that tend to have higher proportions of females are also
>>>>>>>> those that tend to be lower-status. If there is a positive contextual
>>>>>>>> effect for % female, then it might be that a) the intervention really is
>>>>>>>> more effective for females than for males or b) the intervention is equally
>>>>>>>> effective for males and females but tends to work better when used with
>>>>>>>> lower-status professions. Looking at between/within study variance in the
>>>>>>>> predictor lets us disentangle those possibilities, at least partially.
>>>>>>>>
>>>>>>>> James
>>>>>>>>
>>>>>>>> On Wed, Jun 3, 2020 at 9:27 AM Simon Harmel <sim.harmel using gmail.com>
>>>>>>>> wrote:
>>>>>>>>
>>>>>>>>> Indeed that was the problem, Greta, Thanks.
>>>>>>>>>
>>>>>>>>> But James, in meta-analysis having multiple categorical variables
>>>>>>>>> each with several levels is very pervasive and they often vary both
>>>>>>>>> within and between studies.
>>>>>>>>>
>>>>>>>>> So, if for each level of each of such categorical variables we
>>>>>>>>> need to do this, this would certainly become a daunting task in addition to
>>>>>>>>> making the model extremely big.
>>>>>>>>>
>>>>>>>>> My follow-up question is what is your strategy after you create
>>>>>>>>> within and between dummies for each of such categorical variables? What are
>>>>>>>>> the next steps?
>>>>>>>>>
>>>>>>>>> Thank you very much, Simon
>>>>>>>>>
>>>>>>>>> p.s. After your `robu()` call I get: `Warning message: In
>>>>>>>>> sqrt(eigenval) : NaNs produced`
>>>>>>>>>
>>>>>>>>> On Wed, Jun 3, 2020 at 8:45 AM Gerta Ruecker <
>>>>>>>>> ruecker using imbi.uni-freiburg.de> wrote:
>>>>>>>>>
>>>>>>>>>> Simon
>>>>>>>>>>
>>>>>>>>>> Maybe there should not be a line break between "Relative and
>>>>>>>>>> Rating"?
>>>>>>>>>>
>>>>>>>>>> For characters, for example if they are used as legends, line
>>>>>>>>>> breaks
>>>>>>>>>> sometimes matter.
>>>>>>>>>>
>>>>>>>>>> Best,
>>>>>>>>>>
>>>>>>>>>> Gerta
>>>>>>>>>>
>>>>>>>>>> Am 03.06.2020 um 15:32 schrieb James Pustejovsky:
>>>>>>>>>> > I'm not sure what produced that error and I cannot reproduce
>>>>>>>>>> it. It may
>>>>>>>>>> > have to do something with the version of dplyr. Here's an
>>>>>>>>>> alternative way
>>>>>>>>>> > to recode the Scoring variable, which might be less prone to
>>>>>>>>>> versioning
>>>>>>>>>> > differences:
>>>>>>>>>> >
>>>>>>>>>> > library(dplyr)
>>>>>>>>>> > library(fastDummies)
>>>>>>>>>> > library(robumeta)
>>>>>>>>>> >
>>>>>>>>>> > data("oswald2013")
>>>>>>>>>> >
>>>>>>>>>> > oswald_centered <-
>>>>>>>>>> > oswald2013 %>%
>>>>>>>>>> >
>>>>>>>>>> > # make dummy variables
>>>>>>>>>> > mutate(
>>>>>>>>>> > Scoring = factor(Scoring,
>>>>>>>>>> > levels = c("Absolute", "Difference
>>>>>>>>>> Score", "Relative
>>>>>>>>>> > Rating"),
>>>>>>>>>> > labels = c("Absolute", "Difference",
>>>>>>>>>> "Relative"))
>>>>>>>>>> > ) %>%
>>>>>>>>>> > dummy_columns(select_columns = "Scoring") %>%
>>>>>>>>>> >
>>>>>>>>>> > # centering by study
>>>>>>>>>> > group_by(Study) %>%
>>>>>>>>>> > mutate_at(vars(starts_with("Scoring_")),
>>>>>>>>>> > list(wthn = ~ . - mean(.), btw = ~ mean(.))) %>%
>>>>>>>>>> >
>>>>>>>>>> > # calculate Fisher Z and variance
>>>>>>>>>> > mutate(
>>>>>>>>>> > Z = atanh(R),
>>>>>>>>>> > V = 1 / (N - 3)
>>>>>>>>>> > )
>>>>>>>>>> >
>>>>>>>>>> >
>>>>>>>>>> > # Use the predictors in a meta-regression model
>>>>>>>>>> > # with Scoring = Absolute as the omitted category
>>>>>>>>>> >
>>>>>>>>>> > robu(Z ~ Scoring_Difference_wthn + Scoring_Relative_wthn +
>>>>>>>>>> > Scoring_Difference_btw + Scoring_Relative_btw,
>>>>>>>>>> > data = oswald_centered, studynum = Study, var.eff.size =
>>>>>>>>>> V)
>>>>>>>>>> >
>>>>>>>>>> > On Tue, Jun 2, 2020 at 10:20 PM Simon Harmel <
>>>>>>>>>> sim.harmel using gmail.com> wrote:
>>>>>>>>>> >
>>>>>>>>>> >> Many thanks, James! I keep getting the following error when I
>>>>>>>>>> run your
>>>>>>>>>> >> code:
>>>>>>>>>> >>
>>>>>>>>>> >> Error: unexpected symbol in:
>>>>>>>>>> >> "Rating" = "Relative")
>>>>>>>>>> >> oswald_centered"
>>>>>>>>>> >>
>>>>>>>>>> >> On Tue, Jun 2, 2020 at 10:00 PM James Pustejovsky <
>>>>>>>>>> jepusto using gmail.com>
>>>>>>>>>> >> wrote:
>>>>>>>>>> >>
>>>>>>>>>> >>> Hi Simon,
>>>>>>>>>> >>>
>>>>>>>>>> >>> The same strategy can be followed by using dummy variables
>>>>>>>>>> for each
>>>>>>>>>> >>> unique level of a categorical moderator. The idea would be to
>>>>>>>>>> 1) create
>>>>>>>>>> >>> dummy variables for each category, 2) calculate the
>>>>>>>>>> study-level means of
>>>>>>>>>> >>> the dummy variables (between-cluster predictors), and 3)
>>>>>>>>>> calculate the
>>>>>>>>>> >>> group-mean centered dummy variables (within-cluster
>>>>>>>>>> predictors). Just like
>>>>>>>>>> >>> if you're working with regular categorical predictors, you'll
>>>>>>>>>> have to pick
>>>>>>>>>> >>> one reference level to omit when using these sets of
>>>>>>>>>> predictors.
>>>>>>>>>> >>>
>>>>>>>>>> >>> Here is an example of how to carry out such calculations in
>>>>>>>>>> R, using the
>>>>>>>>>> >>> fastDummies package along with a bit of dplyr:
>>>>>>>>>> >>>
>>>>>>>>>> >>> library(dplyr)
>>>>>>>>>> >>> library(fastDummies)
>>>>>>>>>> >>> library(robumeta)
>>>>>>>>>> >>>
>>>>>>>>>> >>> data("oswald2013")
>>>>>>>>>> >>>
>>>>>>>>>> >>> oswald_centered <-
>>>>>>>>>> >>> oswald2013 %>%
>>>>>>>>>> >>>
>>>>>>>>>> >>> # make dummy variables
>>>>>>>>>> >>> mutate(
>>>>>>>>>> >>> Scoring = recode(Scoring, "Difference Score" =
>>>>>>>>>> "Difference",
>>>>>>>>>> >>> "Relative Rating" = "Relative")
>>>>>>>>>> >>> ) %>%
>>>>>>>>>> >>> dummy_columns(select_columns = "Scoring") %>%
>>>>>>>>>> >>>
>>>>>>>>>> >>> # centering by study
>>>>>>>>>> >>> group_by(Study) %>%
>>>>>>>>>> >>> mutate_at(vars(starts_with("Scoring_")),
>>>>>>>>>> >>> list(wthn = ~ . - mean(.), btw = ~ mean(.))) %>%
>>>>>>>>>> >>>
>>>>>>>>>> >>> # calculate Fisher Z and variance
>>>>>>>>>> >>> mutate(
>>>>>>>>>> >>> Z = atanh(R),
>>>>>>>>>> >>> V = 1 / (N - 3)
>>>>>>>>>> >>> )
>>>>>>>>>> >>>
>>>>>>>>>> >>>
>>>>>>>>>> >>> # Use the predictors in a meta-regression model
>>>>>>>>>> >>> # with Scoring = Absolute as the omitted category
>>>>>>>>>> >>>
>>>>>>>>>> >>> robu(Z ~ Scoring_Difference_wthn + Scoring_Relative_wthn +
>>>>>>>>>> >>> Scoring_Difference_btw + Scoring_Relative_btw, data =
>>>>>>>>>> oswald_centered,
>>>>>>>>>> >>> studynum = Study, var.eff.size = V)
>>>>>>>>>> >>>
>>>>>>>>>> >>>
>>>>>>>>>> >>> Kind Regards,
>>>>>>>>>> >>> James
>>>>>>>>>> >>>
>>>>>>>>>> >>> On Tue, Jun 2, 2020 at 6:49 PM Simon Harmel <
>>>>>>>>>> sim.harmel using gmail.com> wrote:
>>>>>>>>>> >>>
>>>>>>>>>> >>>> Hi All,
>>>>>>>>>> >>>>
>>>>>>>>>> >>>> Page 13 of *THIS ARTICLE
>>>>>>>>>> >>>> <
>>>>>>>>>> >>>>
>>>>>>>>>> https://cran.r-project.org/web/packages/robumeta/vignettes/robumetaVignette.pdf
>>>>>>>>>> >>>>> *
>>>>>>>>>> >>>> (*top of the page*) recommends that if a *continuous
>>>>>>>>>> moderator *varies
>>>>>>>>>> >>>> both within and across studies in a meta-analysis, a
>>>>>>>>>> strategy is to break
>>>>>>>>>> >>>> that moderator down into two moderators by:
>>>>>>>>>> >>>>
>>>>>>>>>> >>>> *(a)* taking the mean of each study (between-cluster effect),
>>>>>>>>>> >>>>
>>>>>>>>>> >>>> *(b)* centering the predictor within each study
>>>>>>>>>> (within-cluster effect).
>>>>>>>>>> >>>>
>>>>>>>>>> >>>> BUT what if my original moderator that varies both within
>>>>>>>>>> and across
>>>>>>>>>> >>>> studies is a *"categorical" *moderator?
>>>>>>>>>> >>>>
>>>>>>>>>> >>>> I appreciate an R demonstration of the strategy recommended.
>>>>>>>>>> >>>> Thanks,
>>>>>>>>>> >>>> Simon
>>>>>>>>>> >>>>
>>>>>>>>>> >>>> [[alternative HTML version deleted]]
>>>>>>>>>> >>>>
>>>>>>>>>> >>>> _______________________________________________
>>>>>>>>>> >>>> R-sig-meta-analysis mailing list
>>>>>>>>>> >>>> R-sig-meta-analysis using r-project.org
>>>>>>>>>> >>>> https://stat.ethz.ch/mailman/listinfo/r-sig-meta-analysis
>>>>>>>>>> >>>>
>>>>>>>>>> > [[alternative HTML version deleted]]
>>>>>>>>>> >
>>>>>>>>>> > _______________________________________________
>>>>>>>>>> > R-sig-meta-analysis mailing list
>>>>>>>>>> > R-sig-meta-analysis using r-project.org
>>>>>>>>>> > https://stat.ethz.ch/mailman/listinfo/r-sig-meta-analysis
>>>>>>>>>>
>>>>>>>>>> --
>>>>>>>>>>
>>>>>>>>>> Dr. rer. nat. Gerta Rücker, Dipl.-Math.
>>>>>>>>>>
>>>>>>>>>> Institute of Medical Biometry and Statistics,
>>>>>>>>>> Faculty of Medicine and Medical Center - University of Freiburg
>>>>>>>>>>
>>>>>>>>>> Stefan-Meier-Str. 26, D-79104 Freiburg, Germany
>>>>>>>>>>
>>>>>>>>>> Phone: +49/761/203-6673
>>>>>>>>>> Fax: +49/761/203-6680
>>>>>>>>>> Mail: ruecker using imbi.uni-freiburg.de
>>>>>>>>>> Homepage: https://www.uniklinik-freiburg.de/imbi.html
>>>>>>>>>>
>>>>>>>>>>
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