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.

  • $\begingroup$ 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. $\endgroup$ – Heteroskedastic Jim Jun 17 '17 at 15:45
  • $\begingroup$ Yeah, thats exactly what I am doing. Every pack has normality and I calculated Pearson CC for each pack. Thanks for comment :) $\endgroup$ – 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.

You could go about this in two ways:

  • Calculate something called the design effect, and if it is large enough (greater than 2), you might need to account for the bias in your standard errors. One way to do this would be to multiply your standard errors by the root of the design effect (DEFT), and conduct the statistical test of significance for the correlation again.
  • Another thing you could do would be to model the clustered nature of your data. One way to do this would be a mixed effects model. This is possible in many statistical packages. And your analysis would be quite easy given the simple nature of your problem.

As a starting point, you can use this tool to calculate the design effect under the Intraclass correlation option. If you can upload your datafile in CSV format. Use the pack ID as the cluster variable, and B as the outcome (I'm assuming it is). I'd be glad to answer any questions you have in the comments.

This article briefly introduces the problem with clustered data, and has several references. This other article goes into some detail of how one can use the DEFT to reduce the bias in the standard errors.

  • $\begingroup$ Thanks a lot for your input. I have few additional questions if you don't mind. If p value is < 0.05 (few packs have that), does it mean I should just ignore them during the conclusion or do they act as the opposite of my thorem? Also, Pearson is the correct choice to calculate correlation given this data, correct? $\endgroup$ – Sikor Jun 17 '17 at 14:41
  • $\begingroup$ 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. $\endgroup$ – Heteroskedastic Jim Jun 17 '17 at 15:35
  • $\begingroup$ In my comment above, I tried to outline how you would go about this if you wanted to use the Pearson correlation, which is acceptable. $\endgroup$ – Heteroskedastic Jim Jun 17 '17 at 15:39
  • $\begingroup$ Thanks a lot. Appreciate it. I will write if I have any further questions. $\endgroup$ – Sikor Jun 17 '17 at 16:03

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