how to understand an unit of analysis

Question

I came a cross a definition which I cannot understand the logic. is there a way to explain what is the problem ?

a longitudinal design for an intervention study in 10 participants where the researchers are interested in evaluating whether there is a correlation between their main measure and a clinical condition using a simple regression analysis. Their unit of analysis should be the number of data points (1 per participant, 10 in total), resulting in 8 df. For df = 8, the critical R value (with an alpha level of. 05) for achieving significance is 0.63. That is, any correlation above the critical value will be significant (p≤0.05). If the researchers combine the pre and post measures across participants, they will end up with df = 18, the critical R value is now 0.44, rendering it easier to observe a statistically significant effect. This is inappropriate because they are mixing within- and between- analysis units, resulting in dependencies between their measures – the pre-score of a given subject cannot be varied without impacting their post-score, meaning they only truly have 8 independent df. This often results in interpretation of the results as significant when in fact the evidence is insufficient to reject the possibility that there is no effect.

My questions are these

1- if the data point is 10 then how comes that the degree of freedom is 8? 2- how to calculate the critical value ? 3- how is it 18 when one does pre and post measures calculation?

Answer 1

1) Degree of freedoms is defined as the number of observations minus the number of parameters. In their case they have 10 observations and a linear regression has two parameters (slope and intercept) so you get 10 - 2 = 8. See reference here: https://blog.minitab.com/blog/statistics-and-quality-data-analysis/what-are-degrees-of-freedom-in-statistics

2) Critical values are usually looked up in a table so in this case you would look for 8 degrees of freedom and an alpha value of 0.05 (it looks like it's a two tailed test here) and you get 0.632. For an exampler of such a table please see here: http://turner.faculty.swau.edu/mathematics/math241/materials/rctable/rc.php or you can google "critical R value table". Actually calculating the critical value is hard because it doesn't have a closed form expression and here is starting place for reading about that: https://www.researchgate.net/post/What_is_the_formula_to_calculate_the_critical_value_of_correlation

3) The reason that the researchers got 18 degrees of freedom is because they now had two measurements for each of their 10 participants so they have 20 observations and their regression still has 2 parameters. So 20 - 2 is 18 degrees of freedom. So now they need to go look for a different entry in their critical values table.