Testing for differences between pooled biological samples I am trying to test for differences in absorbance between a control and treatment group.  Due to cost, each measurement is derived from the mixture of 5 independent samples.  Overall there are 5 sample measurements each derived from 5 individual samples.  The data would look like:
Control:
Measurement 1 (from 5 combined samples)
Measurement 2 (from 5 combined samples)
Measurement 3 (from 5 combined samples)
etc.

Treatment:
Measurement 1 (from 5 combined samples)
Measurement 2 (from 5 combined samples)
Measurement 3 (from 5 combined samples)
etc.

Here my issue is that each measurement would be close to the mean value of the 5 combined samples and we would not know much about the actual variance that would be observed across individual samples.  Because of this I'm thinking that I can't apply a simple test like a t test to test for a difference between groups.
I appreciate any suggestions about how to properly test between the groups using data gathered in this fashion.  Also, would your suggestion work if each measurement was collected from varying numbers of samples?  For example: 3 sample for one measurement, 5 samples for another measurement, 2 samples for another?
 A: One possibility is that your 50 original samples are Poisson counts, in which case pooling
would give you 10 pooled samples with somewhat larger Poisson counts---five Control and five Treatment.
Poisson data with reasonably large counts are approximately normal. A difficulty might
be that different population means between the Treatment and Control groups would imply different population variances also. There are two ways to handle potentially different variances in this situation. A traditional approach is to use a square root transformation to 'stabilize' variances. Another would be to use a Welch 2-sample t test, which does not require equal variances. I would prefer the latter.
Maybe your data are something like the simulated data below:
set.seed(1126)
x1 = rpois(5, 20)
x2 = rpois(5, 35)
summary(x1); length(x1);  sd(x1)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
     16      23      24      23      26      26 
[1] 5            # sample size
[1] 4.123106     # sample standard deviation

summary(x2); length(x2);  sd(x2)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   25.0    30.0    35.0    33.2    36.0    40.0 
[1] 5
[1] 5.80517

stripchart(list(x1,x2), ylim=c(.5, 2.5), meth="stack", pch=20)


For illustration, here are results from a Welch t test on the counts and a pooled 2-sample t test on square
roots of counts. Both tests find significant differences at the 2% level.
[You should choose one test.]
t.test(x1, x2)

        Welch Two Sample t-test

data:  x1 and x2
t = -3.2032, df = 7.217, p-value = 0.01439
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -17.68411  -2.71589
sample estimates:
mean of x mean of y 
     23.0      33.2 

t.test(sqrt(x1), sqrt(x2), var.eq=T)

         Two Sample t-test

data:  sqrt(x1) and sqrt(x2)
t = -3.1448, df = 8, p-value = 0.01371
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -1.6726184 -0.2573859
sample estimates:
mean of x mean of y 
 4.778570  5.743572 

Notes: (1) For the test on square roots of counts, the confidence interval may be difficult to
interpret. (2) If any counts are $0,$ it is customary to add $0.5$ to all
counts before taking square roots.
