How should I analyze my data? I am new here, so I hope this question is on topic. I understand a little about statistics, but my knowledge of what methods to use in real case scenarios is a little limited.
I want to statistically analyse the accuracy of my developed system. This system is used to obtain a specific measure of an user, let's call it Method A. Only the user can use the system to obtain his/her own measure. To compare the results I can take that measure of an user using two other methods, but these are operated by myself (Method B and Method C).
So what I have is a sample of 40 subjects and three measures of that distance, for each subject. Two that come from already existing systems and the third one comes from my developed system.
Below there is a table that reflects the possible results of this test. Note that due to budget limitations I can't really acquire more data than this. Not even take more measures of a subject with my system.
Subject | Method A  | Method B  | Method C    
1       | 122.6     | 123.6     | 125.6     
2       | 102.6     | 104.4     | 105.2    
(...)   | (...)     | (...)     | (...)     
40      | 152.6     | 151.0     | 149.7

The two measures from the already existing system are also not considered correct, that is. They are the methods in which the community most relies to obtain that particular measure, but are not considered as "correct". Note that each subject has a different measure and the measure is not related inter-subjects. That is, any two subjects can have entirely different measures. 
How can I statistically analyse my data in order to prove that my system is accurate or not. Or at least how to show that the measures that I obtain from my method are close to the establish methods. Which kind of statistical analysis should I perform with my data to obtain either conclusion?
Is there any modifications I can add to the design of my experiment in order to help prove the accuracy of my system?
Thanks in advance.a
 A: If Method A is considered the established "gold standard" (it's unclear from your question whether this is so), and if there are important values of the measure that are used to make decisions (e.g. underperforming, in high-risk zone, diagnostic category, etc.) you might use some form of receiver operator characteristic to see what values of Method B and Method C correspond to the specific values of Method A for a given sensitivity, specificity, and total correct classification.
However, if you are simply interested in the agreement between, say Measure A and Measure B, or Measure A and Measure C, you might consider performing an equivalence test. You do not say how your values of Measure A (and B and C) are distributed, but if they are distributed normally, then you could perform a t test for equivalence, and if they are not distributed normally, you could perform a nonparametric rank sum test for equivalence or approximate z test for equivalence (the latter might be a bit easier). See the description of the two one-sided tests for equivalence to get a basic idea about these tests. There's software available for R and for Stata that performs these tests. Probably for SAS also.
A: Essentially, this seems to be a question about estimation. Does Method A provide similar scores as Method B & C do (The second point made by Alexis above)?
I would approach this by looking at a graph of Method means (if the distributions seem normal) plotted with confidence intervals. This is a very simple approach, and here's how to do that in R. Note that you would also be interested in the variances of each method: Perhaps your method provides scores that are much more/less variable than other methods. Looking at means will answer the simple question of similarity in the average score between methods.
library(dplyr)
library(reshape2)
library(ggplot)

Melt data into long form, then get summaries (using reshape2 and dplyr).  
data_long = melt(data, id="Subject")
methods = group_by(data_long, variable)
d_summary = summarise(methods, mean = mean(value), sd = sd(value), n = n(), se = sd/sqrt(n), ci = qt(.975, n-1) * se)

Then we just plot these values (with ggplot2).
ggplot(d_summary, aes(x = variable, y = mean)) + geom_pointrange(aes(ymin = mean - ci / 2, ymax = mean + ci / 2))

A: If Method B is valid in absolute terms, then you probably want to compare the mean for A with the mean for B (such as with a paired t-test).  If the absolute value is of no interest to you, you might want to see if A is correlated with B (as with a Pearson product-moment correlation - but plot the data first to see if the relationship appears linear).  (Of course, it is possible to have a high correlation but still have very different means.)  Do you think either of these approaches will answer your research question?
