What I want to do is compare the ordering of variables determined by the ranking of each variable.
For example: Say, I have a rating system that is made up of 5 different ratings -
- Excellent
- Good
- Moderate
- Okay
- Bad
so the vector representing this list would be $$v = \{excellent, good, moderate, okay, bad\}$$
Furthermore the ordering and ranking of these variables is determined by the frequency of the variables. Let's say, $a$ is $$a = \{good = 156, excellent = 100, okay = 90, moderate = 50, bad = 10\}$$ $$a = \{2,1,4,3,5\}$$ So for $a$ good has rank, excellent has rank 2, etc.
For vector $b$ $$b = \{okay = 147,good = 100,exellent = 70,moderate = 60, bad = 20\}$$ $$b = \{3,2,4,1,5\}$$
My questions are: 1. Would a rank correlation be sufficient for this? 2. What about a case with vector $c$ where some of the variables have no frequency, so $$c = \{excellent = 100, good = 30, moderate = 1, okay = 0, bad = 0\}$$ $$c = \{1,2,3,0,0\}$$ or even a vector like $$d = \{excellent = 5, good = 0, moderate = 0, okay = 0, bad = 0\}$$ $$d = \{1,0,0,0,0\}$$ where zero would represent that variable not having any ranking?
I would want to compare $a$ with $b$, $a$ with $c$ and $a$ with $d$
$\bf{ADD}:$ The rankings are based on the order of the vector $v.$ $v = \{excellent, good, moderate, okay, bad\}$ so $v = \{1,2,3,4,5\}$ so $a$ becomes $a = \{2,1,4,3,5\}$, etc.