[R-meta] Collapsing a between subject factor

Tue, Jan 30, 2018 2:54 AM

This still isn't quite right. You can compute the mean and SD for the combined sample exactly:

### simulate some data
n.total <- 100
grp <- sample(1:2, size=n.total, replace=TRUE, prob=c(.2,.8))
y   <- rnorm(n.total, mean=grp, sd=2)

### means and SDs of the subgroups
ni  <- c(by(y, grp, length))
mi  <- c(by(y, grp, mean))
sdi <- c(by(y, grp, sd))

### want to get mean and SD of the total group
mean(y)
sd(y)

### mean = weighted mean (weights = group sizes)
m.total <- sum(ni*mi)/sum(ni)

### SD = sqrt((within-group sum-of-squares plus between-group sum-of-squares) / (n.total - 1))
sd.total <- sqrt((sum((ni-1) * sdi^2) + sum(ni*(mi - m.total)^2)) / (sum(ni) - 1))

### check that we get the right values
m.total
sd.total

This also generalizes to any number of groups. Try with:

grp <- sample(1:3, size=n.total, replace=TRUE, prob=c(.2,.6,.3))

Best,
Wolfgang

-----Original Message-----
From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces at r-
project.org] On Behalf Of Oliver Clark
Sent: Monday, 29 January, 2018 12:14
To: Michael Dewey
Cc: r-sig-meta-analysis at r-project.org; Oliver Clark
Subject: Re: [R-meta] Collapsing a between subject factor

Dear Michael,

Many thanks for your response.  Indeed, the sample sizes are unequal
which is apparently why it was treated as two analyses.

I?ve been playing with this example others and your example below
overestimates the variance.  I think this is because the means are being
squared rather than the delta from the combined:

S_c <- (n_1 * (v_1 + (m_1 - m_c) ^2 ) + n_2*( v_2 + ( m_2 - m_c) ^2) ) /
( n_1 + n_2)

This still overestimates the known population variance of 4, so applying
the bessel correction:

S_c_2 <- ( (n_1 - 1 )*( v_1 +( m_1 - m_c ) ^2 ) + ( n_2 - 1)*( v_2 + (
m_2 - m_c)^2) ) / ( ( n_1 + n_2) -1 )

leads to a good estimate of the combined variance.  Code:

M <- rnorm(34,5,3)
F <- rnorm(57,5,3)

comb <- c(M,F)
n_1 = 34
n_2 = 57
m_1 = mean(M)
m_2 = mean(F)

v_1 = sd(M)^2
v_2 = sd(F)^2

m_c = (n_1 * m_1 + n_2 * m_2) / (n_1 + n_2)

S_c_2 <- ( (n_1 - 1 )*( v_1 +( m_1 - m_c ) ^2 ) + ( n_2 - 1)*( v_2 + (

m_2 - m_c)^2) ) / ( ( n_1 + n_2) -1 )

sd(comb) - sqrt(S_c_2)

[1] 0.001710072

Many thanks for your advice - I?d have been stuck without your input!

Best wishes,

Oliver

On 29 Jan 2018, at 10:02, Michael Dewey <lists at dewey.myzen.co.uk>

wrote:

Dear Oliver

You do not say whether the sample sizes are equal or not so I give the

procedure for unequal.

For the means you need to weight by sample size

(n_1 * m_1 + n_2 * m_2) / (n_1 + n_2)

where n are sample sizes and m means

For variance you need

(n_1 * (m_1^2 + v_1) + n_2 * (m_2^2 + v_2) / (n_1 + n_2)) - m_c

where v are variances and m_c is the combined mean you got above.

I suggest double checking this with a few examples in case of

transcription errors at my end or yours.

Michael

On 28/01/2018 21:49, Oliver Clark wrote:

Hi all,
I am currently coding studies for a meta-analysis and have come across

a case in which I have a set of studies in which all but one do not
include sex as a between subject factor.  The reason given was unequal
cell sizes, differences in visual stimuli (it is not clear what these
differences are so they are unlikely to be systematic, rather an
artefact)  and strength differences between men and women.

With my limited experience, I don?t see the benefit in treating these

both as separate cases and was wondering whether it would make sense to
merge the means and SDs for both groups and use that with the total N to
calculate an effect size?

Combining the means seems relatively straightforward but I am not sure

how to do the standard deviations.  I have tried averaging the variance
in the following simulation to get there but must admit that I am
stabbing in the dark!:

M <- rnorm(10,5,2)
F <- rnorm(10,5,2)

comb <- c(M,F)

(mean(M) + mean(F)) / 2 == mean(comb)

[1] TRUE

sqrt((sd(M)^2 + sd(F)^2)/2) == sd(comb)

[1] FALSE
Can anyone offer any advice on the best path for this? Should I treat

them as different studies, attempt to merge the means and SDs, use a
different aggregation method or omit this study?

Many thanks,
Oliver Clark
PhD Student
Manchester Metropolitan University

[R-meta] Collapsing a between subject factor

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