combine means SDs

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StatsToDo : Combine Means and SDs Into One Group Program

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Introduction Javascript Program R Code

This page is a simple utility to combine multiple groups of n, mean, and SD into a single group. Two algorithms are offered, both producing the same results, but using different formulae.

Decomposition of means and Standard Deviation:

For each group :
- Σx = mean * n;
- Σx² = SD²(n-1)+((Σx)²/n)
The values are then added together
- tn = sum of all (n)
- tx = sum of all Σx
- txx = sum of all Σx²
The combine calculations are
- Combined n = tn
- Combined mean = tx / tn
- Combine SD = sqrt((txx-tx²/tn) / (tn-1))
When calculated the results from the example data should look like the following

n mean SD Σx Σx²

Grp1 10 11.8 2.4 118 1444.24

Grp2 20 15.3 3.2 306 4876.36

Grp3 15 8.4 4.1 126 1293.74

tn tx txx

Sum 45 550 7614.34

Combined 45 12.2222 4.5028

Algorithm described by Cochrane

Cochrane's formula combines two groups of n, mean, and SD (n1, m1, s1 and n2, m2, s2) with the following calculations

Combined n = n1 + n2
Combined mean = (n1*m1 + n2*m2) / (n1 + n2)
Combined Standard Deviation = sqrt(((n1-1)*s1*s1 + (n2-1)*s2*s2 + n1 * n2 / (n1 + n2) * (m1*m1 + m2*m2 - 2 * m1 * m2)) / (n1 + n2 -1));
When more tha 2 groups are to be combined, the first two groups are combined first, the results are then combined with the third group, then sequentiaaly with each subsequent group.

When calculated the results from the example data should look like the following

	Individual Groups			Combined with previous Groups
	n	Mean	Standard Deviation	n	Mean	Standard Deviation
Row 1	10	11.8	2.4	10	11.8	2.4
Row 2	20	15.3	3.2	30	14.1333	3.3634
Row 3	15	8.4	4.1	45	12.2222	4.5028

Please Note :

Combining n, mean, SD from different groups must be used with care, as the statistical assumption is that all the groups are merely sub-samples of a single group, and combining them merely restore them back into the original single group.

In many cases this assumption is faulty, as the groups may be from different populations, and sampled under different environments. It is much safer therefore to combine groups using the meta-analysis algorithm, using the Random Effect Model, available in the Meta-analysis for Comparing Two Unpaired Groups Program Page , using the mean and Standard Error of the mean for each group.

The Standard Error of the mean is calculated as SE = SD / sqrt(n) of each group.

After combining them using the Random Effect Model, the Standard Deviation can be recalculated as SD = SE * sqrt(tn), where tn is the sum of sample sizes from all the groups. The results should look like the following. I have made bold calculations of SE = SD / sqrt(n) before meta-analysis, and from SD = SE x sqrt(n) after meta-analysis

	n	mean	SD	SE
Grp1	10	11.8	2.4	0.7589
Grp2	20	15.3	3.2	0.7155
Grp3	15	8.4	4.1	1.0586
MetaAnalysis
Fixed Effect Model	45	12.6299	3.1341	0.4672
Random Effect Model	45	11.8975	12.6684	1.8885

References :

Altman DG, Machin D, Bryant TN and Gardner MJ. (2000) Statistics with Confidence Second Edition. BMJ Books ISBN 0 7279 1375 1. p. 28-31

Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect measures and computing estimates of effect. In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.0 (updated July 2019). Cochrane, 2019. Available from https://training.cochrane.org/handbook/current/chapter-06#section-6-5-2 (table 6.5.2a)

The two algorithms for combining means and Standard Deviations from 2 or more groups are presented as R codes, for those who are wish to check the validity of the calculations, incorporate the algorithm into their own applications, or merely interested.

The R code is designed to run from the source panel of R Studio. Those unfamiliar with R should read R Explained Page to learn how to set up and run R Studio.

The code is presented in maroon, and the results in navy. It is divided into 3 sections. Section 1 contains data entry, section 2 the first option, using the algorithm in StatsToDo, and section 3 the second option, using Cochrane's formulation.

The data. The same data as in the Javascript program is used, and a data frame is created

myDat = ("
n   mean   sd
10  11.8   2.4
20  15.3   3.2
15   8.4   4.1
") 
myDataFrame <- read.table(textConnection(myDat),header=TRUE)  # conversion to data frame    
#myDataFrame                                                  # optional check input

Algorithm 1: Decomposition of mean and SD to ex (Σx) and exx (Σx²)

nr = nrow(myDataFrame)   # number of rows
ex <- rep(0,nr)          # array to contain Σx
exx <- rep(0,nr)         # array to contain Σx²
tn = 0                   # total n
tx = 0                   # total Σx 
txx = 0                  # total Σx²
for(i in 1:nr)
{
  ex[i] = myDataFrame$n[i] * myDataFrame$mean[i]
  exx[i] = myDataFrame$sd[i]^2 * (myDataFrame$n[i]-1) + ex[i]^2 / myDataFrame$n[i]
  tn = tn + myDataFrame$n[i]
  tx = tx + ex[i]
  txx = txx + exx[i]
}
# concatenate Σx and Σx² to data frame
myDataFrame$ex <- ex
myDataFrame$exx <- exx
myDataFrame             # show data frame
# Calculate combined values
tMean = tx / tn
tSD = sqrt((txx-tx^2/tn)/(tn-1))
print("Combined n, mean, and SD")
print(c(tn,tMean,tSD))

The results are as follows

> myDataFrame
   n mean  sd  ex     exx
1 10 11.8 2.4 118 1444.24
2 20 15.3 3.2 306 4876.36
3 15  8.4 4.1 126 1293.74
> 
[1] "Combined n, mean, and SD"
[1] 45.000000 12.222222  4.502822

Algorithm 2: Cochrane's formula
Ref: https://handbook-5-1.cochrane.org/chapter_7/table_7_7_a_formulae_for_combining_groups.htm

Using the same data as in the Javascript program and algorithm 1

nr = nrow(myDataFrame)   # number of rows
newN <- rep(0,nr)        # array for combined n of this and previous group
newMean <- rep(0,nr)     # array for combined mean of this and previous group
newSD <- rep(0,nr)       # array for combined Standard Deviations of this and previous group
# Prime the first row by copying from data frame
newN[1] = myDataFrame$n[1]
newMean[1]  = myDataFrame$mean[1]
newSD[1] = myDataFrame$sd[1]
# designate values of current row as "old"
oldN = newN[1]
oldMean = newMean[1]
oldSD = newSD[1]
# Combining each pair of rows from row 2 onwards
for(i in 2:nr)
{
  # data from row
  n = myDataFrame$n[i]
  mean  = myDataFrame$mean[i]
  sd = myDataFrame$sd[i]
  #combining with old values (Cochrane's algorithm)
  newN[i] = oldN + n
  newMean[i] = (oldN * oldMean + n * mean) / (oldN + n)
  newSD[i] = sqrt(((oldN-1)*oldSD^2 + (n-1)*sd^2 + oldN * n / (oldN + n) * (oldMean^2 + mean^2
                  - 2 * oldMean * mean)) / (oldN + n -1))
  # designate values of current row as "old"
  oldN = newN[i]
  oldMean = newMean[i]
  oldSD = newSD[i]
}
# Concatenate columns of combined values to data frame
myDataFrame$newN <- newN
myDataFrame$newMean <- newMean
myDataFrame$newSD <- newSD
myDataFrame                # display dataframe containing original and combined results

The results are as follows

> myDataFrame
   n mean  sd newN  newMean    newSD
1 10 11.8 2.4   10 11.80000 2.400000
2 20 15.3 3.2   30 14.13333 3.363427
3 15  8.4 4.1   45 12.22222 4.502822

The final results from both algorithms are the same.