Hello,
For research I am doing three replicate measurements of a property. Each of the three data set consists of several data points from which I compute a mean and standard deviation. The data set are close to normal distributed.
However, given that each data set represents a replicate measurement, I want to combine the three distributions into one with its own mean standard deviation in order for me to calculate the standard error of the combined data.
Normally, I would simply calculate mean and standard deviation from the three mean values, however, I feel this is far from representative because the computed standard deviation is much less than the corresponding value for each replicate measurement. Also, would I not lose information if I were to discard the individual standard deviations?
While surfing on the internet it seems to me, that the best solution is to make a weighted mixture of the distributions?
You wont loose YouTube information if you discard. You can mail me and İ can help you with regration analyses if you want. İf you mail, please note point bu point what you need, ill try to help. My mail adress: [email protected]
Dear Kristian,
If I understood U right (3 samples of the same variable/es distributed almost normally with obs>30.).
Then
1. No need to calculate mean and standart deviation (!) between these samples because they are ALMOST representative (repeat the distribution of the analyzed attribute in population etc).
2. You can try to improve the representativity of your samples for example by combining 3 samples in one sample. After this procedure your variable/es probably will be distributed more normally than before.
3. If 2nd step is impossible due to study design or smthng else than "from 3 evils you need to choose the less" >:-)
HOW? To compute descriptive statistics (Mean, Mo, Me, 95%CI, SD, SE, percentiles, don't forget about 'Box-and-Whiskers' plots) for your variable/es and choose the most representative sample from your three ones. Also it would be great to conduct Kolmogorova-Smirnova, Lilliephorce, Shapiro-Wilks tests to determine p-value (normal probability) for your samles' variables. After these procedures you will get the general vision of your variables, data & dataset.
Good Luck!
You have two ways: the approximate and the rigorous.
In the approximate you can assume that each one of your replicates is the result of a random variable normally distributed with the same media and variance. Under this assumption you can estimate the media and variance from the arithmetic average of the whole responses and from standard deviation of the whole response. You can verify your assumption by means and variance comparison between replicates.
In the rigorous you must use probabilistic calculations. That is, First, you must fit (adjusting the media and variance) each replicate to a normal cumulative density function. You can use a normal paper or more elaborated software. Then you have Y1, Y2 and Y3 as random variables normally distributed for replicates 1, 2 and 3 and you have their parameters (media and variance). The combination that you desire is,
Y=(Y1+Y2+Y3)/3
You can simulate Y by MonteCarlo simulation. That is you must perform >100 random assays:
Y=(N1^(-1)(U(0,1))+ N2^(-1)(U(0,1))+ N2^(-1)(U(0,1)))/3
Where Nm^(-1)(U(0,1)) is the inverse of cumulative density function of Ym as function of a random number uniformly distributed between 0 to 1.
Finally you must fit (adjusting the media and variance) the assay results to a normal cumulative density function.
Weighting: This is a good practice when the weighting is on something like sample size. However, weighting could be abused. I think that this replicate went better than the others so I will give it greater weight. Now the results simply conform to preconceived ideas about how the experiment should turn out.
Sampling: beware pseudoreplication, but technical replicates are fine so long as they are properly incorporated into the analysis. Say you have an experiment with two treatments (A and B). You will apply these to 10-hectare blocks of land and you have enough product to treat three such blocks with each treatment. The treatments are applied consistently, but you know that the application is not uniform. You also expect that each block is not uniform. To account for this you take four samples from each 10-hectare block.
Analysis: You could take an average of the four samples from each block, and then use a t-test for the difference between the three values of treatment A and three values of treatment B. You have discarded some information about the within-plot variability, but do you really care? If so, you could discuss the within plot variability as a separate analysis. Alternatively you could analyze all the data. Then the model will need another variable to account for where each sample came from. They would be subplots.
If I understamd your problem cOrrectlY, you have three replicates of single property and all these three replicates are normlly distributed. . And you want to convert these three different replicates(samples) into one sample. So if you have equal sample size for all three replicates then you may proceed as: sum the first values of all three replicates, it gives you firat value of new generated sample, then sum the second values of all three variables it will gives you second value of new sample, and continue tha process to complete the new sample.
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