# Should I draw the conclusions from average or individual data? Is this approach using Pearson CC correct?

I have made some research and I acquired the following data:

• 40 "packs" of data
• each "pack" contains 10 scores of variable A and 10 scores of variable B
• A is always 1,2,3,4,5,6,7,8,9,10
• B is a measurement result and scores in 0-100%

My thesis is: the higher A, the lower B.

Now, having all this info, I have used Shapiro-Wilk test on every "pack" to deremine normality and p value was always > 0.05. Based on the data I have and the results of SW test, I concluded that Pearson Correlation Coefficient is the right choice here. I calculated the correlation and that's where I am at this moment.

Question 1: Can I calculate the average of B for every A (i.e. avarege of B where A = 1 from all "packs", repeat for all rows 1-10) and then use Pearson to calculate the correlation of the averages? Is this valid approach?

Question 2: Are there any other statistics that may be important for my thesis and this set of data apart from correlations?

Question 3: Should I draw a conclusion from all the individual results? Or from the average? Or maybe something else?

I am not a statistician in my daily work, so I would be very grateful for any help. Thank you.

• You want to test the assumptions of the Pearson correlation coefficient first. If you have normality, then plot the variable A versus variable B for some of the packs i.e. one plot should only contain data from a pack. Your plots should show a linear relationship. Then you should be fine, at least at a cursory level. – Heteroskedastic Jim Jun 17 '17 at 15:45
• Yeah, thats exactly what I am doing. Every pack has normality and I calculated Pearson CC for each pack. Thanks for comment :) – Sikor Jun 17 '17 at 15:56

Be sure to draw your conclusion from individual analysis not averages. Unless you would be answering a different question that includes packs in the hypothesis and not just A and B. Averaging then performing the analysis results in threats to ecological validity.

The data within the packs could possibly be related, and this could be biasing your standard errors, making them smaller than they should be, resulting in small p-values.

• Calculate your correlation for all your data, thus obtaining a single value. I assume -0.5. Transform it to Fisher's z (davidmlane.com/hyperstat/A50760.html) = -0.549. Calculate its standard error (SE) (davidmlane.com/hyperstat/B8544.html). Assuming 400 pairs = 0.050. Multiply the SE by the DEFT obtained from the tool above to account for clustering. If DEFT = 2, then $0.050 * 2 = 0.10$. Your 95% CI would then be $[-0.5 - 1.96 (.1), -0.5 + 1.96 * (.1)]$ = [-0.696, -0.304]. This interval does not include 0, hence your correlation of -0.5 has a p-value less than .05. – Heteroskedastic Jim Jun 17 '17 at 15:35