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I got the following columns:

Level 1        240    12     7
Level 2        98     5      5
Level 3        46     4      6
Level 4        21     0      1

I try to prove that there is a correlation between the "Level" and the number of people (represented by each column) I used the following forumla (Table is the table..) :

RS2 = r2_score(Table.iloc[1:5,0], Table.iloc[1:5,1], Table.iloc[1:5,2])

print(RS2 )

The result is negative (-4....) which is wrong.

I am undecided, but maybe I should calculate the columns together as one group, finding mean and then the estimated values? if not, what should I do?

** note my data is small, concatenating the columns will probably give a large confidence interval..

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    $\begingroup$ I've taken the liberty of editing your title to be more informative. Take a look and see if that captures your main question, and feel free to improve on it by adding more specificity about your particular problem. $\endgroup$
    – Sycorax
    Dec 9, 2021 at 17:27
  • $\begingroup$ It does not make sense for there to be a negative correlation involving a nominal variable. Is there any order to the levels, or are they like "dog", "cat", "horse", "kangaroo"? // If you do want to find a negative correlation for variables where such a notion makes sense, $R^2$ is not a tool that will help you. That would be a correlation coefficient, which has a relationship to $R^2$ under certain circumstances. // Why do you want to show a negative correlation? $\endgroup$
    – Dave
    Dec 9, 2021 at 17:33
  • $\begingroup$ I edit my question. I want to show that as long as the level increases - > the number of people declining $\endgroup$ Dec 9, 2021 at 17:37
  • $\begingroup$ So there is an order to the levels? Do you know the difference between each of the levels? Is it constant in the sense that $L4 - L2 = L3 - L1$, etc? $\endgroup$
    – Dave
    Dec 9, 2021 at 17:39
  • $\begingroup$ The relations expressed here through "rating" from 1 -> 4 , if certain people "pass" L1 they go to L2 and so on. My theory: L1 is easiest stage, there are more people, as long as you keep increasing the levels, it's becoming harder so then less people capable to be on an advanced level. $\endgroup$ Dec 9, 2021 at 17:51

1 Answer 1

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$R^2$ has nothing to do with the sign of a correlation. While there are ways of getting $R^2<0$ in a regression model (an indication of a poor fit), the notation comes from the fact that $R^2 = r^2$, where $r$ is the sample correlation between two variables, when you fit a regression model $\hat y_i = \hat\beta_0 + \hat\beta_1x_i$ with the extremely common method of least squares.

However, it looks like you have one variable with the levels in it and one variable with the numerical observations>

$$ X = (1,2,3,4,1,2,3,4,1,2,3,4)\\ Y = (240, 98, 46, 21, 12, 5, 4, 0, 7, 5, 6, 1) $$

Since your levels appear to be ordinal---that is, ordered but with unclear differences between them---Spearman's rank correlation is appropriate here. In R, the line is cor(x, y, method = "spearman").

This gives me a result of about $- 0.5$. However, the plot is not so convincing. When I do a test of the Spearman correlation being nonzero via cor.test(x, y, method = "spearman"), I get a p-value that tends to be considered inconclusive, $p = 0.099$, along with a warning that the exact p-value cannot be computed, due to tied values. I am not sure how serious this is, but, combined with the graph, I am skeptical increasing the level decreases the $Y$ variable.

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  • $\begingroup$ Thanks, Probably due to low number of population (n)? $\endgroup$ Dec 9, 2021 at 18:01
  • $\begingroup$ "While there are ways of getting $R^2<0$"… Would you care to share what these ways are for a correlation between exactly two variables? $\endgroup$
    – Alexis
    Dec 9, 2021 at 18:21
  • $\begingroup$ @Alexis set.seed(2021); N <- 1000; x <- runif(1000); y <- 10*x + rnorm(N); preds <- 5 - y; plot(x, y); points(x, preds, col = 'red'); r2 <- 1 - (sum((y - preds)^2))/sum((y - mean(y))^2); r2 $\endgroup$
    – Dave
    Dec 9, 2021 at 18:33
  • $\begingroup$ Well, that's three variables, not 2. The correlation between $y$ and $preds$ is $-1$, and the $R^2$ is therefore 1: summary(lm(y~preds)) (look at R-squared) and cor(y,preds), so no: I do not think I am (yet) persuaded. (Alternately: summary(lm(preds~y)) gives the same thing: $R^2 = 1$.) $\endgroup$
    – Alexis
    Dec 9, 2021 at 20:00
  • $\begingroup$ @Alexis Those predictions are generated by $\hat y_i = 5 - 10x_i$, a model that gives $R^2<0$. $\endgroup$
    – Dave
    Dec 9, 2021 at 20:05

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