[R-meta] Reproducing results using regtest in metafor

Thu Aug 15 15:43:29 CEST 2019

Thanks James

Again, your response is super helpful.

(I did have a dig into the code of regtest but wasn't confident I knew enough about it to reliably unpick it). The point you make about coherence is important. It is interesting to note that the multiplicative error of the test as specified (originally by Egger) was due to it being conceived with respect to a Galbraith plot (which has standardised effect on one axis so regression is "divided through" by the standard error) not any real belief about the error structure - but it has stuck and does deal with the scenario of underdispersion (less variability than expected by chance even under fixed effect) which could occur if the most extreme study results under a fixed effect assumption have been suppressed.

So while the test as regtest conducts it does not mix errors, there is still the issue that the analysis strategy does (i.e. why use different error structures for baseline risk on its own and when adding in the standard error for asymmetry testing?). One thing in favour of the "2-stage" analysis is that it allows the construction of a funnel based on the residuals to see how its appearance changes as a result of adjusting for the initial covariate.

Thanks again for all your help - I now understand what is going on and thus you have comprehensively answered my question.

Very best wishes

Alex
________________________________
From: James Pustejovsky <jepusto using gmail.com>
Sent: 14 August 2019 19:50
To: Sutton, Alex (Prof.) <ajs22 using leicester.ac.uk>
Cc: Michael Dewey <lists using dewey.myzen.co.uk>; r-sig-meta-analysis using r-project.org <r-sig-meta-analysis using r-project.org>; DOLEMAN, Brett (UNIVERSITY HOSPITALS OF DERBY AND BURTON NHS FOUNDATION TRUST) <brett.doleman using nhs.net>; Freeman, Suzanne C. (Dr.) <suzanne.freeman using leicester.ac.uk>
Subject: Re: [R-meta] Reproducing results using regtest in metafor

Alex,

Looking at the source code for regtest.rma, it looks like my previous syntax can be simplified even further, to simply:
lm_fit <- lm(yi ~ m.c + sqrt(vi), data = ma.dataset, weights = 1 / vi)
summary(lm_fit)

The random effects structure / weighting scheme used to fit the initial metaregression is simply disregarded when re-fitting the model. I think that's reasonable because it forces one to make a coherent assumption about the structure of the errors in the regression. In contrast, the other approach involves fitting the first model based on random effects weighting, and then fitting the Egger regression on the residuals, but using a different weighting scheme. I can't think of any data generating process where this would be an efficient way to fit the model or a valid approach for inference. Thus, it would seem preferable to fit the model all at once, using assumptions that can at least be clearly stated.

James

On Mon, Aug 12, 2019 at 7:34 AM Sutton, Alex (Prof.) <ajs22 using leicester.ac.uk<mailto:ajs22 using leicester.ac.uk>> wrote:
Dear James

Thank you for taking the time to think about this and write. I think your approach of writing the model out is a good one for discussing exactly what is going on "under the hood".

Your justification of the degrees of freedom seems sound - much appreciated.

I am still a little uncertain about the model specification because an additive random effect is specified in the initial meta-regression using the rma command. But then a multiplicative error is specified in the regtest command. These probably dont have any bearing on the degrees of freedom, but I would greatly appreciate it if someone could explain how these are jointly implemented.

With very best wishes

Alex

________________________________
From: James Pustejovsky <jepusto using gmail.com<mailto:jepusto using gmail.com>>
Sent: 09 August 2019 19:04
To: Sutton, Alex (Prof.) <ajs22 using leicester.ac.uk<mailto:ajs22 using leicester.ac.uk>>
Cc: Michael Dewey <lists using dewey.myzen.co.uk<mailto:lists using dewey.myzen.co.uk>>; r-sig-meta-analysis using r-project.org<mailto:r-sig-meta-analysis using r-project.org> <r-sig-meta-analysis using r-project.org<mailto:r-sig-meta-analysis using r-project.org>>; DOLEMAN, Brett (UNIVERSITY HOSPITALS OF DERBY AND BURTON NHS FOUNDATION TRUST) <brett.doleman using nhs.net<mailto:brett.doleman using nhs.net>>; Freeman, Suzanne C. (Dr.) <suzanne.freeman using leicester.ac.uk<mailto:suzanne.freeman using leicester.ac.uk>>
Subject: Re: [R-meta] Reproducing results using regtest in metafor

Option 1 amounts to fitting the model

Y_i = b0 + b1 m.c_i + b2 se_i + e_i

with the assumption that Var(e_i) = sigma^2 v_i for unknown sigma^2.  It can be fit using the nlme package using the following syntax (building off of the previous code):

library(nlme)
gls_fit <- gls(yi ~ m.c + sqrt(vi), data = ma.dataset, weights = varFixed(~ vi))
summary(gls_fit)

I think the correct df would be 27 in this case.

With option 2, the residuals from the first stage have lost 2 degrees of freedom (down to 28), and then they lose two more with the second stage fit (because the intercept is re-estimated when it should be constrained to zero).

James

On Fri, Aug 9, 2019 at 8:24 AM Sutton, Alex (Prof.) <ajs22 using leicester.ac.uk<mailto:ajs22 using leicester.ac.uk>> wrote:
Dear Michael

Many thanks for taking the time to write. (And let me thank you for all your hard work on the R meta-analysis web page - this is a fantastic resource I use all the time)

I think your suggestion is very plausible and I was thinking along related lines. Then I convinced myself that the df used in first regression was reflected in the se of the residuals (in a very "hand wavy" sort of way!) .

As you say, the issue then focuses on which is "better" - my concern with option 1 is that I don't know how to reproduce it outside of the package - but as you say it may well be theoretically superior.

I am also hoping Wolfgang can supply some definitive insight!

Thanks again

Alex

Alex Sutton
Professor of Medical Statistics

Department of Health Sciences
College of Life Sciences
University of Leicester
George Davies Centre
University Road
LEICESTER LE1 7RH UK

Please use Lancaster Road, Leicester, LE1 7HA for SatNav

Member of the Complex Reviews Support Unit
http://www.nihrcrsu.org/<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.nihrcrsu.org%2F&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229669644&sdata=mqqxssr3NWoR1U8dDRfjkdIf6bslU%2Fc8uQKWAR%2Fj0sY%3D&reserved=0>

t: +44 (0)116 229 7268
e:  ajs22 using le.ac.uk<mailto:ajs22 using le.ac.uk>
w: www.le.ac.uk<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.le.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229669644&sdata=a3yGqJ%2F%2Fv7ktwLIj8%2BU5oIuADcb%2BVhVnDFDrUToRbJY%3D&reserved=0>

Follow us on Twitter or visit our Facebook page

-----Original Message-----
From: Michael Dewey [mailto:lists using dewey.myzen.co.uk<mailto:lists using dewey.myzen.co.uk>]
Sent: 09 August 2019 13:19
To: Sutton, Alex (Prof.) <ajs22 using leicester.ac.uk<mailto:ajs22 using leicester.ac.uk>>; r-sig-meta-analysis using r-project.org<mailto:r-sig-meta-analysis using r-project.org>
Cc: DOLEMAN, Brett (UNIVERSITY HOSPITALS OF DERBY AND BURTON NHS FOUNDATION TRUST) <brett.doleman using nhs.net<mailto:brett.doleman using nhs.net>>; Freeman, Suzanne C. (Dr.) <suzanne.freeman using leicester.ac.uk<mailto:suzanne.freeman using leicester.ac.uk>>
Subject: Re: [R-meta] Reproducing results using regtest in metafor

Dear Alex

I suspect the answer is that regtest.rma refits the model and so "knows"
you have a moderator and hence loses you a degree of freedom whereas regtest.default does not "know" that and so proceeds as though you had just the single predictor (sei). That probably explains the change in t too although we may need to wait for Wolfgang to give us the definitive answer.

Of course that leaves open the question which is better. I think your option 1 is what I would prefer but that is just based on instainct rather than any higher mathematics.

Michael

On 08/08/2019 17:42, Sutton, Alex (Prof.) wrote:
> Hi All list members
>
> I wish to adjust a meta-analytic dataset by the control arm event rate
> using meta-regression (ignoring any regression to the mean issues) and
> then run Egger's funnel asymmetry test after the adjustment.
>
> (I paste complete data, R code and output  below my query.)
>
> The regtest command in metafor can do this and an example is given in
> the manual (option 1 in the code).
>
> But a colleague tried to do the same analysis in another package -
> STATA
> - but did not get the same result.
>
> I have tried to reconcile the 2, and have succeeded getting the STATA
> result in R using the code in option 2. But I don't understand why
> option 1 and option 2 code below give different results - I thought
> they should be equivalent? (The degrees of freedom are different as
> well as the test p-values)
>
> The key lines extracted from the below are:
>
> model2 <- rma(yi, vi, data = ma.dataset, mods = cbind(m.c),
> control=list(maxiter=200, stepadj=0.5))
>
> print(summary(model2))
>
> # Option 1: Put model 2 directly into regtest
>
> option1 <- regtest(model2, model = "lm", predictor="sei")
>
> print(option1)
>
> # Option 2: Extract residuals and SE from model 2 and then put these
> into regtest manually
>
> a <- rstandard.rma.uni(model2)
>
> option2 <- regtest(x=a$resid, sei=a$se, model = "lm", predictor="sei")
>
> print(option2)
>
> Hoping someone can offer me some insight.
>
> Many thanks in advance.
>
> Alex
>
> DATA BELOW:
>
> n.c
>
>
>
> n.t
>
>
>
> m.c
>
>
>
> m.t
>
>
>
> sd.c
>
>
>
> sd.t
>
> 15
>
>
>
> 15
>
>
>
> 43.34736
>
>
>
> 22.6778
>
>
>
> 20.14634
>
>
>
> 13.2187
>
> 15
>
>
>
> 15
>
>
>
> 36.66456
>
>
>
> 26.95877
>
>
>
> 14.24828
>
>
>
> 14.89395
>
> 15
>
>
>
> 15
>
>
>
> 51.74782
>
>
>
> 28.44658
>
>
>
> 23.83604
>
>
>
> 26.70313
>
> 15
>
>
>
> 15
>
>
>
> 49.80753
>
>
>
> 19.04398
>
>
>
> 24.16307
>
>
>
> 11.40324
>
> 15
>
>
>
> 15
>
>
>
> 60.67109
>
>
>
> 44.62121
>
>
>
> 29.01649
>
>
>
> 23.60531
>
> 15
>
>
>
> 15
>
>
>
> 31.34018
>
>
>
> 14.65917
>
>
>
> 18.27904
>
>
>
> 13.30416
>
> 25
>
>
>
> 25
>
>
>
> 44.48585
>
>
>
> 22.18729
>
>
>
> 24.98121
>
>
>
> 11.94804
>
> 25
>
>
>
> 25
>
>
>
> 42.19917
>
>
>
> 27.97304
>
>
>
> 27.69334
>
>
>
> 19.4292
>
> 25
>
>
>
> 25
>
>
>
> 44.99957
>
>
>
> 26.77364
>
>
>
> 33.80657
>
>
>
> 18.46828
>
> 25
>
>
>
> 25
>
>
>
> 62.6953
>
>
>
> 36.15569
>
>
>
> 32.1401
>
>
>
> 28.75357
>
> 25
>
>
>
> 25
>
>
>
> 35.38889
>
>
>
> 19.77613
>
>
>
> 16.96604
>
>
>
> 15.69643
>
> 25
>
>
>
> 25
>
>
>
> 36.46263
>
>
>
> 27.69418
>
>
>
> 22.18271
>
>
>
> 22.04955
>
> 25
>
>
>
> 25
>
>
>
> 42.63761
>
>
>
> 26.83291
>
>
>
> 22.60301
>
>
>
> 23.47935
>
> 50
>
>
>
> 50
>
>
>
> 57.92533
>
>
>
> 30.96353
>
>
>
> 28.90083
>
>
>
> 17.40233
>
> 50
>
>
>
> 50
>
>
>
> 26.15427
>
>
>
> 21.06599
>
>
>
> 17.39962
>
>
>
> 14.29371
>
> 50
>
>
>
> 50
>
>
>
> 39.98736
>
>
>
> 29.13268
>
>
>
> 19.99763
>
>
>
> 18.44498
>
> 50
>
>
>
> 50
>
>
>
> 46.11657
>
>
>
> 27.34703
>
>
>
> 24.23706
>
>
>
> 15.55458
>
> 50
>
>
>
> 50
>
>
>
> 48.0687
>
>
>
> 34.85292
>
>
>
> 25.19593
>
>
>
> 21.33037
>
> 50
>
>
>
> 50
>
>
>
> 53.43902
>
>
>
> 31.4177
>
>
>
> 31.33719
>
>
>
> 19.16536
>
> 50
>
>
>
> 50
>
>
>
> 56.2192
>
>
>
> 39.338
>
>
>
> 28.10021
>
>
>
> 24.66106
>
> 50
>
>
>
> 50
>
>
>
> 29.25929
>
>
>
> 15.60332
>
>
>
> 17.00783
>
>
>
> 12.57914
>
> 50
>
>
>
> 50
>
>
>
> 30.72385
>
>
>
> 20.22988
>
>
>
> 18.93253
>
>
>
> 15.11565
>
> 15
>
>
>
> 15
>
>
>
> 55.86973
>
>
>
> 27.38925
>
>
>
> 33.32485
>
>
>
> 22.29828
>
> 15
>
>
>
> 15
>
>
>
> 47.10929
>
>
>
> 28.39797
>
>
>
> 21.93013
>
>
>
> 25.12405
>
> 15
>
>
>
> 15
>
>
>
> 31.87787
>
>
>
> 15.31057
>
>
>
> 17.05947
>
>
>
> 10.99065
>
> 15
>
>
>
> 15
>
>
>
> 36.91457
>
>
>
> 26.14574
>
>
>
> 18.46971
>
>
>
> 18.57387
>
> 15
>
>
>
> 15
>
>
>
> 49.84609
>
>
>
> 34.04816
>
>
>
> 24.64221
>
>
>
> 21.22332
>
> 25
>
>
>
> 25
>
>
>
> 57.06665
>
>
>
> 32.91816
>
>
>
> 27.6288
>
>
>
> 19.84755
>
> 25
>
>
>
> 25
>
>
>
> 50.57364
>
>
>
> 36.87816
>
>
>
> 33.87053
>
>
>
> 21.38415
>
> 25
>
>
>
> 25
>
>
>
> 28.80046
>
>
>
> 21.97428
>
>
>
> 16.45233
>
>
>
> 15.75454
>
> CODE BELOW:
>
> library(metafor)
>
> rm(list=ls())
>
> ##simdat <- read.csv("s4_data.csv")
>
> ma.dataset <- escalc(n1i = n.t, n2i = n.c, m1i = m.t, m2i = m.c,
>
>                       sd1i = sd.t, sd2i = sd.c, data = simdat, measure
> = "MD",
>
>                       append = TRUE)
>
> model2 <- rma(yi, vi, data = ma.dataset, mods = cbind(m.c),
> control=list(maxiter=200, stepadj=0.5))
>
> print(summary(model2))
>
> # Option 1: Put model 2 directly into regtest
>
> option1 <- regtest(model2, model = "lm", predictor="sei")
>
> print(option1)
>
> # Option 2: Extract residuals and SE from model 2 and then put these
> directly into regtest
>
> a <- rstandard.rma.uni(model2)
>
> option2 <- regtest(x=a$resid, sei=a$se, model = "lm", predictor="sei")
>
> print(option2)
>
> OUTPUT BELOW:
>
> Mixed-Effects Model (k = 30; tau^2 estimator: REML)
>
>    logLik  deviance       AIC       BIC      AICc
>
> -84.1061  168.2122  174.2122  178.2088  175.2122
>
> tau^2 (estimated amount of residual heterogeneity):     0 (SE =
> 6.3793)
>
> tau (square root of estimated tau^2 value):             0
>
> I^2 (residual heterogeneity / unaccounted variability): 0.00%
>
> H^2 (unaccounted variability / sampling variability):   1.00
>
> R^2 (amount of heterogeneity accounted for):            100.00%
>
> Test for Residual Heterogeneity:
>
> QE(df = 28) = 17.1497, p-val = 0.9456
>
> Test of Moderators (coefficient(s) 2):
>
> QM(df = 1) = 24.9726, p-val < .0001
>
> Model Results:
>
>           estimate      se     zval    pval    ci.lb<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fci.lb&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229679522&sdata=hvM6FGU78Nu9Jonyo2H8x1FrTwQiTErlraP7AmYD9mw%3D&reserved=0>    ci.ub
>
> intrcpt    3.5641  3.8211   0.9327  0.3510  -3.9251  11.0534
>
> m.c       -0.4633  0.0927  -4.9973  <.0001  -0.6450  -0.2816  ***
>
> ---
>
> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
>
>>
>
>>
>
>> # Option 1: Put model 2 directly into regtest
>
>> option1 <- regtest(model2, model = "lm", predictor="sei")
>
>> print(option1)
>
> Regression Test for Funnel Plot Asymmetry
>
> model:     weighted regression with multiplicative dispersion
>
> predictor: standard error
>
> test for funnel plot asymmetry: t = -0.2787, df = 27, p = 0.7826
>
>> # Option 2: Extract residuals and SE from model 2 and then put these
>> directly into regtest
>
>> a <- rstandard.rma.uni(model2)
>
>> residuals <- a$resid
>
>> SEresiduals <- a$se
>
>>
>
>> option2 <- regtest(x=residuals, sei=SEresiduals, model = "lm",
>> predictor="sei")
>
>> print(option2)
>
> Regression Test for Funnel Plot Asymmetry
>
> model:     weighted regression with multiplicative dispersion
>
> predictor: standard error
>
> test for funnel plot asymmetry: t = -0.2170, df = 28, p = 0.8297
>
> *Alex Sutton
> Professor of Medical Statistics
>
> ***
>
> Department of Health Sciences
>
> College of Life Sciences
>
> University of Leicester
>
> George Davies Centre
>
> University Road
>
> LEICESTER LE1 7RH UK
>
> *Please use Lancaster Road, Leicester, LE1 7HA for SatNav*
>
> Member of the Complex Reviews Support Unit
>
> https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.n<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.n&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229679522&sdata=PekJV8vVxkcS%2B%2BdzXNOMVe0C4VjLR1tP%2BTl5jo7A7dA%3D&reserved=0>
> ihrcrsu.org<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fihrcrsu.org&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229689515&sdata=ziFRHVLy7Blao6Bk5BUuy%2FfNRVpaPIhYLc95qQ7iyMM%3D&reserved=0>%2F&data=02%7C01%7Cajs22%40leicester.ac.uk<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2F40leicester.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229689515&sdata=Lx2akKH3oGeTGN88Q0Eb0AAYE82Lg3LSUL3KS8gQ4eo%3D&reserved=0>%7Ce117781ca2
> a44ca568b608d71cc3bd96%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C1%7C63
> 7009499341114715&sdata=0iAykupucHNp6qBZHrBdJiw9BLww5gTGlTsWDIFg2Zo
> %3D&reserved=0
>
> *t:*+44 (0)116 229 7268
> *e:*ajs22 using le.ac.uk<mailto:ajs22 using le.ac.uk> <mailto:ajs22 using le.ac.uk<mailto:ajs22 using le.ac.uk>>
> *w:*
> https://eur03.safelinks.protection.outlook.com/?url=www.le.ac.uk&d<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.le.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229699510&sdata=wi6Crc1N4yszpUN2aHEJ5muVCfk13Agv3amAqPnYiEE%3D&reserved=0>
> ata=02%7C01%7Cajs22%40leicester.ac.uk<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2F40leicester.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229699510&sdata=EEymDZDqS6wetqsvdMOE9qCugBOccZsdmbyfRcFjpls%3D&reserved=0>%7Ce117781ca2a44ca568b608d71cc3bd
> 96%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C1%7C637009499341114715&amp
> ;sdata=i4CJas8E39vy%2BcRWETeeFLVYCndvYsojfoKxwVwLT3c%3D&reserved=0
> <https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww<http://www>.
> le.ac.uk<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fle.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229699510&sdata=KCPiRzaSMixZxTU9M8eFBPqrOiSzYmxQnlrq3E2Cg7g%3D&reserved=0>%2F&data=02%7C01%7Cajs22%40leicester.ac.uk<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2F40leicester.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229709500&sdata=Ak0uBMOfscN0wCQnO8J2%2FksDxvN5L1eELUBMyVcHSec%3D&reserved=0>%7Ce117781ca2a44
> ca568b608d71cc3bd96%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C1%7C63700
> 9499341114715&sdata=D02HtRTW%2BRjXayhrM6tkUlvvOY21cPt2FT4Yo5qfzFE%
> 3D&reserved=0>_
> _cid:image001.gif using 01D0F6E3.BA906A20
> Follow us on Twitter
> <https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Ftwi<https://twi>
> tter.com<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Ftter.com&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229709500&sdata=UMNeS8B1MmeQMBGX%2BMtR4AMtdkbf7ApDMWAAhDe64SU%3D&reserved=0>%2Funiofleicester&data=02%7C01%7Cajs22%40leicester.ac.uk<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2F40leicester.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229719494&sdata=ybmiJgTScrUVz%2FPjKl90PKleSqsk8EnXVcrpTsM4pPg%3D&reserved=0>%7
> Ce117781ca2a44ca568b608d71cc3bd96%7Caebecd6a31d44b0195ce8274afe853d9%7
> C0%7C0%7C637009499341114715&sdata=mxn3i9TylT7fLGKv44ULs3xQjYpxULu2
> 1X6guJqxiuY%3D&reserved=0> or visit our Facebook
> <https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww<http://www>.
> facebook.com<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Ffacebook.com&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229719494&sdata=%2BZzH5iuS2J5B5EjFiDTOhkIdYIds0Kj20euqKu6%2BQUs%3D&reserved=0>%2Funiofleicester&data=02%7C01%7Cajs22%40leicester.ac<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2F40leicester.ac&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229729490&sdata=LrGrs%2F4bZor7cymEQ1o2a%2FF5Tw3awaW%2F3dZKIDckpbg%3D&reserved=0>.
> uk%7Ce117781ca2a44ca568b608d71cc3bd96%7Caebecd6a31d44b0195ce8274afe853
> d9%7C0%7C0%7C637009499341114715&sdata=JakhmIjpCe2VXycfx9ZYxdMT1z4r
> Kx6wv1fyrfzdJpg%3D&reserved=0> page__
>
>
>
> cid:image002.png using 01D0F6E3.BA906A20
>
> **
>
>
> <https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.avg.com%2Femail-signature%3Futm_medium%3Demail%26utm_source%3Dlink%26utm_campaign%3Dsig-email%26utm_content%3Demailclient%26utm_term%3Doa-4885-b&data=02%7C01%7Cajs22%40leicester.ac.uk%7Ce117781ca2a44ca568b608d71cc3bd96%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C1%7C637009499341114715&sdata=G94bbFUIO8t%2FguFLoO1HFHodjMF3mVGLFiM7ejuFIko%3D&reserved=0<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.avg.com%2Femail-signature%3Futm_medium%3Demail%26utm_source%3Dlink%26utm_campaign%3Dsig-email%26utm_content%3Demailclient%26utm_term%3Doa-4885-b&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229729490&sdata=fwBaN7gCcTMAUckWvlO4oz0Yh5LmPDkVuJfjmAJGSOA%3D&reserved=0>>
>       Virus-free.
> https://eur03.safelinks.protection.outlook.com/?url=www.avg.com&da<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.avg.com&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229739486&sdata=63O2TN%2Ft76UYBHhhnYpueED58DIYfu1Q49hLHyWTGUI%3D&reserved=0>
> ta=02%7C01%7Cajs22%40leicester.ac.uk<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2F40leicester.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229739486&sdata=RB4l9TlhOgYrTdq244ILAXfdXnNcy2r82s8X3UhapYY%3D&reserved=0>%7Ce117781ca2a44ca568b608d71cc3bd9
> 6%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C1%7C637009499341124710&
> sdata=aYlIcQAyezpanV387TaeLwlIqki0jpJ0%2FDspYgWNvfk%3D&reserved=0
> <https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww<http://www>.
> avg.com<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Favg.com&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229749477&sdata=SrlUhyXXA2cOrugkByVkAnjhwOnQXMC%2Bfi9YED3j30c%3D&reserved=0>%2Femail-signature%3Futm_medium%3Demail%26utm_source%3Dlink%26u
> tm_campaign%3Dsig-email%26utm_content%3Demailclient%26utm_term%3Doa-48
> 85-b&data=02%7C01%7Cajs22%40leicester.ac.uk<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2F40leicester.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229749477&sdata=xSgH2W48LTJvUpkgEkVR3o5QPLvOHavORSg9Zt1OViM%3D&reserved=0>%7Ce117781ca2a44ca568b6
> 08d71cc3bd96%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C1%7C637009499341
> 124710&sdata=cnQ7G%2Fd9EBWdWuKkagE9enGenkla%2FCpUF4X6T3IaXxU%3D&am
> p;reserved=0>
>
>
> <#DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2>
>
> _______________________________________________
> R-sig-meta-analysis mailing list
> R-sig-meta-analysis using r-project.org<mailto:R-sig-meta-analysis using r-project.org>
> https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fstat<https://stat>
> .ethz.ch<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fethz.ch&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229759473&sdata=hZDQ0d0%2BWLIx1TCZyCYs29h1l4rHuMsvURgHVTP6yQ0%3D&reserved=0>%2Fmailman%2Flistinfo%2Fr-sig-meta-analysis&data=02%7C01%7
> Cajs22%40leicester.ac.uk<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2F40leicester.ac.uk&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229759473&sdata=MQyuv3CV6cn%2FLDhmZeTqqpLYPK4pdyhmkBtACNdahW8%3D&reserved=0>%7Ce117781ca2a44ca568b608d71cc3bd96%7Caebecd6a
> 31d44b0195ce8274afe853d9%7C0%7C1%7C637009499341124710&sdata=uZQWN8
> g5lR%2Flox9rs2YExt4Czj%2B8lRl9Ch0j563nAyE%3D&reserved=0
>

--
Michael
https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.dewey.myzen.co.uk%2Fhome.html&data=02%7C01%7Cajs22%40leicester.ac.uk%7Ce117781ca2a44ca568b608d71cc3bd96%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C1%7C637009499341124710&sdata=eIw%2FeMVcYsc8gQBrJkDF77GUTZMZxehWneBmAPG%2FhpQ%3D&reserved=0<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.dewey.myzen.co.uk%2Fhome.html&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229769465&sdata=uMmKAGjssbh2qm2bKbPrLupYY1%2FBIUd%2BkLJgP1urTX8%3D&reserved=0>

_______________________________________________
R-sig-meta-analysis mailing list
R-sig-meta-analysis using r-project.org<mailto:R-sig-meta-analysis using r-project.org>
https://stat.ethz.ch/mailman/listinfo/r-sig-meta-analysis<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fstat.ethz.ch%2Fmailman%2Flistinfo%2Fr-sig-meta-analysis&data=02%7C01%7Cajs22%40leicester.ac.uk%7Cdb52331b2e8e49371a6508d720e84168%7Caebecd6a31d44b0195ce8274afe853d9%7C0%7C0%7C637014054229769465&sdata=4q5Wyf1%2BoNhsIiqeLnZL67B%2F9nP87z87csWlzcMORuQ%3D&reserved=0>

	[[alternative HTML version deleted]]