I am trying to perform a variance component analysis (nested ANOVA) to quantify the contribution of several factors to the variation in my data. Those data are measured on samples taken from several batches of products that were prepared at two temperatures. Hence I wish to evaluate 4 factors here: 1). Temperature (T1 and T2) 2). Inter-batch variability (batch B1, B2, B3, etc., prepared at each temperature) 3). Intra-batch variability (sample S1, S2, S3, etc., for each batch) 4). Measurement error (measurement result R1, R2 for each sample)
Among those factors, the production temperatures are defined (only two temperatures) and the number of measurements is also defined (only 2 measurements for each sample). However, the number of batches and the number of samples of each batch are not defined. Apparently the more batches and more samples are measuremented, the more representative the analytical results. However, increasing the number of batch/sample dramatically increases the cost of measurements (see below a summary of total samples and total measurements at sample size of 2, 3, and 4).
| Temperature | Batch | Sample | measurments | total samples | total measurements |
|---|---|---|---|---|---|
| 2 | 2 | 2 | 2 | 8 | 16 |
| 2 | 3 | 3 | 2 | 18 | 36 |
| 2 | 4 | 4 | 2 | 32 | 64 |
Therefore I need an optimal sample size that balances the cost and statistical representativeness. Would a sample size of 3 be the choice? My thought is that, from 2 to 3, there is a considerable decrease in multiplicative factor for confidence interval calculation (see below the link):
Is it meaningful to calculate standard deviation of two numbers?
For variance component analysis, the way of calculating confidence interval is different from the example described in the above linked webpage, but still depends on the sample size. Does it make sense here too for variance component analysis to choose a sample size of 3? By the way, I am going to use the VCA pacakage of R for doing this and have found below how confidence interval is caculated for variance component below:
