Can linear regression or any other models be used to judge if Y increases as X increases?

Question

I have a database like:

Site    X    Y
S1      1    1.5
S1      1    1.3
S1      2    1.7
S1      1    1.1
S1      4    5.9
S2      3    4.0
S2      2    2.5
S2      4    9.1
S2      4    9.2
S2      1    2.3

I need to find if $Y$ increases as $X$ increases for every site. In other words, Bigger $X$ corresponds to bigger $Y$.

I know linear regression might suit this problem. But please take a look at the following graph:

Figure 1 is not what I want because small $X$ corresponds to big $Y$. However, Figure 2 is what I want. When I use a linear regression model and RMSE as a measure, it cannot tell the difference between Figure 1 and Figure 2.

Another thing about my database is: $X$ are like levels, which are the same to all the sites. But the $Y\text{'s}$ of every sites are different. For example, for one site, $X=1$ and $Y=20$ means $20$ is a small value because it corresponds the lowest level of $X$. But for another site, $Y=15$ and $X=6$ means $15$ is the highest value because $X$ is the highest level.

So, my problem is: for every site, I need to use a linear model or any other algorithms to judge if $Y$ increases as $X$ does. Then, I need to use a measure to select some sites.

Answer 1

Linear Regression will help you decide whether Y tends to increase with X, but it is not a good tool to prove that Y always increases with X. For this you need to design an algorithm.

To prove that Y always increases with X at each site, first create a table of unique X in ascending order with the corresponding min and max Y for each site:

S1:
X    min(Y)    max(Y)
1    1.1       1.5
2    1.7       1.7    
4    5.9       5.9

S2:
X    min(Y)    max(Y)
1    2.3       2.3
2    2.5       2.5    
3    4.0       4.0
4    9.1       9.2

Now for each site verify the following: For each X check that max(Y) is less than min(Y) for X+1. If this condition ever fails then you have shown that Y does not always increase with X at every site, otherwise you can say that it does.

Answer 2

I don’t think linear model is a good idea for your problem, because it can capture only linear pattern. Furthermore, if I understand you correctly the results for different sites will not be comparable due to different scales/ slopes.

Instead I suggest using the Spearman rank correlation coefficient.