# [R-meta] Estimate variance from time series data

Viechtbauer, Wolfgang (SP) wolfg@ng@viechtb@uer @ending from m@@@trichtuniver@ity@nl
Thu Aug 16 09:57:49 CEST 2018

Dear Arne,

"Calculating raw mean differences is no problem, as you will realize."

Do you mean the mean difference based on the 'aggregated' means over time within the two groups? Sure, that is easy to compute. But computing the sampling variance thereof is difficult.

"Because different studies involve different numbers of time steps, and because the effect sizes are expected to vary with time, I do not want to calculate the effect size per time step, but an overall effect size, based on the entire time series."

Could you explain why you do not want to calculate effects per time step?

"As an alternative approach, I was thinking of calculating the effect size per time step and then averaging over time, but the question then remains how to estimate the sampling variance."

I think calculating the effect size per time step is exactly what you should do. However, I would not recommend averaging over time. You say that you expect effect sizes to vary with time, so unless you take extra precautions, averaging over time would incorrectly assume that they do not vary over time.

Instead, I would analyze the effects as they are, using an appropriate mixed-effects model that accounts for the dependency in the estimates. For one, sampling errors of multiple effects over time are correlated. If you do not know the correlation between the measurements over time, you could guestimate them, compute the covariances between the effects within the same study, and then do sensitivity analyses. In addition, the underlying true effects are likely to be correlated and for time series data, autoregressive structures like AR(1) and continuous-time AR(1) are often appropriate. See help(rma.mv) and take a look at the paragraph starting with: "For meta-analyses of studies reporting outcomes at multiple time points ...".

These two papers are also highly relevant:

Ishak, K. J., Platt, R. W., Joseph, L., Hanley, J. A., & Caro, J. J. (2007). Meta-analysis of longitudinal studies. Clinical Trials, 4, 525-539.

Trikalinos, T. A., & Olkin, I. (2012). Meta-analysis of effect sizes reported at multiple time points: A multivariate approach. Clinical Trials, 9, 610-620.

There is also this tutorial-type paper:

Musekiwa, A., Manda, S. O., Mwambi, H. G., & Chen, D. G. (2016). Meta-analysis of effect sizes reported at multiple time points using general linear mixed model. PLOS ONE, 11(10), e0164898.

If you are only interested in the fixed effects, instead of guestimating the correlations (and then computing the covariances), you could start with a working model that assumes that the covariances are zero, and then use cluster-robust inference methods, using robust() from metafor or, even better, the clubSandwich package, which also works nicely together with metafor.

Best,
Wolfgang

-----Original Message-----
From: Arne Janssen [mailto:arne.janssen using uva.nl]
Sent: Wednesday, 15 August, 2018 16:37
To: Viechtbauer, Wolfgang (SP)
Cc: r-sig-meta-analysis using r-project.org
Subject: Re: [R-meta] Estimate variance from time series data

Dear Wolfgang,

I would like to calculate the standardized mean difference between time
series of different treatments, each replicated. Calculating raw mean
differences is no problem, as you will realize.

Some background: I want to do a meta-analysis of population-dynamical
data, so time series. I am interested in the effect of one treatment
compared to a control, as usual. Each treatment and control will consist
of several replicate time series, of which the averages and s.d. are
usually given per time step. Because different studies involve different
numbers of time steps, and because the effect sizes are expected to vary
with time, I do not want to calculate the effect size per time step, but
an overall effect size, based on the entire time series.

Hope this clarifies things a bit. In any case, it seems that I need to
make some assumptions on the correlations between the time series within
each treatment, which is indeed not trivial.

As an alternative approach, I was thinking of calculating the effect
size per time step and then averaging over time, but the question then
remains how to estimate the sampling variance.

Thanks and best wishes,
Arne

On 15-Aug-18 14:54, Viechtbauer, Wolfgang (SP) wrote:
> If you do not know the correlations, then you cannot compute the sampling variances correctly. You could 'guestimate' the correlations and then do sensitivity analyses. I do not know what you actually want to compute based on the combined means and SDs of the two groups -- do you want to compute a mean difference or standardized mean difference or some other effect size measure? One would have to work out the correct equation for the sampling variance that takes the correlations into consideration. That part alone may not be trivial.
>
> Best,
> Wolfgang
>
> -----Original Message-----
> From: Arne Janssen [mailto:arne.janssen using uva.nl]
> Sent: Wednesday, 15 August, 2018 14:35
> To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis using r-project.org
> Subject: Re: [R-meta] Estimate variance from time series data
>
> Dear Wolfgang,
>
> Exactly, and there's the problem, because the correlations are never
> reported. So what do do in this case?
>
> Best wishes,
> Arne
>
>
> On 15-Aug-18 14:26, Viechtbauer, Wolfgang (SP) wrote:
>
>> Dear Arne,
>>
>> In this example, there are 15 observations in total. The code I provided shows how to obtain the mean and standard deviation of these 15 observations. However, these 15 observations are not independent and hence any sampling variance you compute for the combined mean (or some function thereof) would need to take the degree of correlation into consideration.
>>
>> Best,
>> Wolfgang
>>
>> -----Original Message-----
>> From: Arne Janssen [mailto:arne.janssen using uva.nl]
>> Sent: Wednesday, 15 August, 2018 14:10
>> To: Viechtbauer, Wolfgang (SP); r-sig-meta-analysis using r-project.org
>> Subject: Re: [R-meta] Estimate variance from time series data
>>
>> Dear Wolfgang,
>>
>> Thanks for your quick reply. The question really is what is the sample
>> size. Suppose there are 3 time series of 5 data points through time each
>> that I want to combine. Given are the average and s.d. of the 3 series
>> per time (so 5 averages and s.d.).
>>
>> I would like to obtain an overall average  and s.d. of these 3 time
>> series. If we consider that N = 15, I can use the standard method to
>> combine the 3 series. If we consider N to be 3, because there are only 3
>> time series, I would indeed need to know the correlation among the time
>> series to estimate the s.d., but this correlation is unknown. Please advise.
>>
>> Thanks and best wishes,
>> Arne
>>
>> On 14-Aug-18 22:50, Viechtbauer, Wolfgang (SP) wrote:
>>
>>
>>> Hi Arne,
>>>
>>> It is not entirely clear to me what you are trying to do. Do you want to know the mean and SD when throwing together the N1 measurements from timepoint 1 and the N1 measurements from timepoint 2 from the same group, such that there are 2*N1 measurements in total now for the group? (or 3*N1 if there were three timepoints and so on).
>>>
>>>
>> Reply: This is indeed what I want to do.
>>
>>
>>>     Then the same equation could be used as if there are independent subgroups.
>>>
>>> For example:
>>>
>>> ### Suppose we have the mean, SD, and size of several subgroups, but we
>>> ### need the mean and SD of the total/combined groups. Code below shows
>>> ### what we need to compute to obtain this.
>>>
>>> ### simulate some data
>>> n.total<- 100
>>> grp<- sample(1:4, size=n.total, replace=TRUE)
>>> 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))
>>>
>>>
>>>
>> Here is my doubt: The sum(ni) is now larger than the number of
>> replicates (4 time series, so 4 replicates, n should be 4), am I correct?
>>
>>
>>> ### check that we get the right values
>>> m.total
>>> sd.total
>>>
>>> This would be the case for independent subgroups. Now let's simulate data for 50 individuals measured twice:
>>>
>>> library(MASS)
>>>
>>> Y<- mvrnorm(50, mu=c(0,0), Sigma=matrix(c(1, .8, .8, 1), nrow=2))
>>> y<- c(t(Y))
>>> grp<- c(1:50, 1:50)
>>>
>>> ### 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
>>>
>>> Still works. However, when it comes to computing the sampling variance for m.total (or some function thereof), one cannot treat these two cases as the same. In the first case, we really have sum(ni) independent measurements, so var(y) / sum(ni) would be the correct sampling variance of m.total, but not so for the second case. You would need to know the correlation between the measurements over time to compute an appropriate sampling variance of m.total in the second case.
>>>
>>> Best,
>>> Wolfgang
>>>
>>> -----Original Message-----
>>> From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On Behalf Of Arne Janssen
>>> Sent: Monday, 13 August, 2018 19:23
>>> To: r-sig-meta-analysis using r-project.org
>>> Subject: [R-meta] Estimate variance from time series data
>>>
>>> Dear list members,
>>>
>>> I am doing a meta-analysis with data that are often presented as
>>> repeated measures of population densities, but authors sometimes also
>>> give overall averages and s.d. or s.e.. Because I want to combine these
>>> data into one analysis, I am interested in the overall effect size of
>>> the repeated measures, so would like to combine all data of the time
>>> series into one average and s.d. The time series are repeated several
>>> times, yielding data of the following form:
>>> Time                Treatment 1                Treatment 2
>>>                             N    Ave    s.d.                N    Ave    s.d.
>>> 1                      N1    x1,1    sd1,1          N2    x2,1    sd2,1
>>> 2                      N1    x1,2    sd1,2           n2    x2,2    sd2,2
>>> ...
>>> ...
>>> ...
>>>
>>> What I want to obtain is one average and s.d. per treatment through time.
>>> The average is straightforward, but I cannot come up with a calculation
>>> for the s.d.
>>>
>>> The formula normally used for calculating the combined variance of two
>>> series of measurements:
>>>
>>> Var = (s1^2(n1 -- 1) + s2^2(n2 -- 1) + n1(X-x1)^22 + n2(X-x2)^22)/( (n1
>>> + n2 -- 1)
>>>
>>> does not seem to apply when combining the measurements through time,
>>> because this increases the number of replicates, which in my opinion,
>>> should be the number of time series and not the number of observations.
>>>
>>> I hope I made myself clear, and would be very grateful if you could
>>> advise me on this matter.
>>>
>>> Thanks very much in advance.
>>> Arne Janssen