## Power Analysis

Power can be defined as the probability of finding a real difference, if it exists. 80% or 0.8 is considered an acceptable value for power. The power analysis is performed independently for each protein. Given the sample size, the variance of abundance values and the size of the difference we want to detect, we can calculate the power. Also, for a given power of 80% we can determine how many samples are required to ensure we find a difference if it actually exists.

### Power calculation

For each protein, we know the abundance variance, the sample size and the difference between the group means. Using this data, we can calculate the power. If the observed difference between group means is a true difference, then the power tells us the probability of finding it to be significant. In other words, if two groups means are truly different and we have a power of 0.4, then we are only 40% likely to find that the difference exists. Clearly, we would like to have a much higher power; 0.8 for example. We are more likely to find as significant those proteins with a large difference between groups. Thus, these proteins will have a high power. Also, these proteins will have a lower p-value, as the greater the difference between proteins (all other factors being equal), then the lower the p-value.

### Sample size calculation

For each protein, we know the abundance variance and difference between the group means. We choose a power level of 80%. With this data, we calculate the sample size that would be required to find a significant difference between groups. There are some points to note. Clearly, if group means will be relatively close together, then a larger number of replicates will be required to find this difference significant. If group means are relatively far apart, then less replicates will be required to find a significant difference. Thus, proteins with a high p value will need a high number of replicates to achieve an 80% power while proteins with a low p-value will require far fewer replicates.