How Can Spearman's Rank and PMCC Differ in Graphs?

[Big-Daddy]

Edit: I had no idea what I was doing when I posted this here, sorry! Could someone move it to the Maths help forum?

I have a few basic stats questions which I'd appreciate if someone could help me answer. These are conceptual really.

a) Often I'm asked to come up with a graph which has a Spearman's Rank coefficient of 1 but a PMCC which is not 1. For such a case, would it be acceptable to use an exponential-type curve (just sketching, talking about the shape) - as it wouldn't fit a straight line (so PMCC is not 1) but if you move to a higher x-value you will still always get a higher y-value? If Spearman's rank has to be -1 but the PMCC not quite -1, can I use a downward sloping graph of increasing negative gradient?

b) What does a graph with Spearman's Rank≠1 but PMCC≈1 look like? (If such a graph exists)

b) My syllabus says "understand that the value of a correlation coefficient is unaffected by linear transformations (coding) of the variables". Does this mean that if you translate the scatter points, it will not affect either the PMCC or Spearman's rank? ("Coding" is confusing me.) And any other transformation besides translation will affect both PMCC and Spearman's Rank (e.g. stretch in either axis) ...

Thanks :)

 

[mmwave]

Hello,

I can definitely help you with your questions. To start, Spearman's Rank coefficient and Pearson's correlation coefficient (PMCC) are both measures of the strength and direction of the relationship between two variables. Spearman's Rank coefficient is based on the ranks of the data, while PMCC is based on the actual values of the data. So, a Spearman's Rank coefficient of 1 means that there is a perfect monotonic relationship between the two variables, while a PMCC of 1 means that there is a perfect linear relationship.

a) In the case where Spearman's Rank coefficient is 1 but PMCC is not 1, it is acceptable to use a non-linear curve to represent the relationship. This is because Spearman's Rank is not affected by the shape of the curve, only the order of the data points. So, as long as the curve follows a consistent pattern of increasing or decreasing values, the Spearman's Rank will still be 1. However, the PMCC will not be 1 because it is based on the linear relationship between the variables.

b) A graph with Spearman's Rank≠1 but PMCC≈1 is possible and would indicate a strong linear relationship, but not a perfect one. This could happen if there are a few outliers in the data that are affecting the overall correlation. In this case, the Spearman's Rank would be slightly lower than 1 due to the ranks being affected by the outliers, but the PMCC would still be close to 1 because the majority of the data points still follow a linear pattern.

c) The statement "the value of a correlation coefficient is unaffected by linear transformations (coding) of the variables" means that if you change the units or scale of the variables, the correlation coefficient will not change. For example, if you were measuring the relationship between height and weight, and you changed the units from inches and pounds to centimeters and kilograms, the correlation coefficient would not change. However, this only applies to linear transformations. If you were to apply a non-linear transformation, such as taking the square root of one of the variables, the correlation coefficient would be affected.

I hope this helps clarify your questions. Let me know if you need any further clarification or have any other questions. And don't worry about posting in the wrong forum, I'll make sure to move this to the Math help forum for you.