Relationships, Correlation

Agent Smith

Agent Smith

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Relationships 2.PNG
These are scatter plots of 2 variables.
My question is regarding Fig. C. Visually we can discern the absence of either a positive/negative trend. In this case the correlation coefficient r = 0.
I drew the green horizontal line M to represent the fact that correlation is 0. Change in the independent variable (x - axis) is not accompanied by corresponding trend changes in the dependent variable (y axis). The line M itself is quite arbitrary I think. My aim was to show that the 2 variables in question are not correlated. My hunch is that the green line shows that as the variable on the x - axis varies, the variable on the y - axis doesn't respond "appropriately" (per the green line B, it remains constant). Is this a valid way to represent absence of correlation?

For comparison is the image below, where the blue line N is a good fit for the scatter plot. This too will compute to 0 correlation.
capture 2.PNG
 
Agent Smith said:
My hunch is that the green line shows that as the variable on the x - axis varies, the variable on the y - axis doesn't respond "appropriately" (per the green line B, it remains constant). Is this a valid way to represent absence of correlation?
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As a non-statistician, I haven't wanted to answer, but I can state my impression.

Your first Fig. C represents the usual case, where 0 correlation means that ANY line through the middle would represent the data equally well; it is a sort of "indeterminate" slope.

Your second Fig. C seems to me like a special case, there is clearly a best fit line, but it is horizontal, and by the nature of the definition (and the formula), correlation has to be (near) 0 rather than 1.

On the other hand, if the data exactly fit the line, then r would not be 0! The formula yields 0/0. In fact, your two C's essentially differ only in the vertical scale, which doesn't affect correlation.

I found a reasonable discussion of this case here, including this comment:

correlation quantifies the degree to which two variables increase or decrease together, not strictly the degree to which a line "fits" the data. A correlation of zero indicates that one variable does not tend to increase (or decrease) with the second variable.​

My two cases are two ways in which this can happen: by not fitting, or by fitting a lack of increase.

Checking Wikipedia, I see that they do sort of touch on this:

1725216829942.png​
In the usual case of 0 correlation, it represents 0 strength (no effect); in the horizontal case, it represents "no direction (neither up nor down)".

There is probably at least one error in what I've said here.
 
@blamocur , si that's a scatter plot that's not exhibiting either correlation or association between the variables x and y. Gracias.

@Dr.Peterson gracias for the detailed explanation

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It seems as though correlation is reserved for linear relationships. I read in a logic course that although correlation doesn't imply causation, it is the only kind of (cor)relation that can be considered in establishing causality. I don't understand why though and I can't quite accept the implication here that all causal relationships are linear relationships; couldn't a causal relationship be a parabolic relationship (any nonlinear relationship is deemed an association, defined as some kind of dependence, not a correlation).
 
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Sounds like a question about terminology. I'd try searching the net and see if there is unanimity in definitions.
 
Agent Smith said:
It seems as though correlation is reserved for linear relationships. I read in a logic course that although correlation doesn't imply causation, it is the only kind of (cor)relation that can be considered in establishing causality. I don't understand why though and I can't quite accept the implication here that all causal relationships are linear relationships; couldn't a causal relationship be a parabolic relationship (any nonlinear relationship is deemed an association, defined as some kind of dependence, not a correlation).
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Correlation in this particular sense (Pearson correlation coefficient, or its relatives) is only about linear relationships; whatever they mean by correlation in logic, it is obviously broader than that. Your inference (which you call an implication) is, of course, nonsense, and if you did accept it, I would be worried about you.

As Wikipedia says, "In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data." Clearly that isn't the only kind of correlation!

Never assume that a given word is only used in one way.
 
@blamocur I've checked multiple sources and the received wisdom is that causality is limited to linear relationships, to which correlation coefficient applies. The better the linear model (regression line) the stronger the case for causation (cogito).

Why restrict causality to straight lines?
 
@Dr.Peterson , you have a point, but read my reply to blamocur, the usage of "correlation" is consistent across at least 2 domains (logic &statistics): straight lines.

Say I'm wrong, any examples where causality is demonstrated based on a nonlinear correlation (?)?
 
Agent Smith said:
@blamocur I've checked multiple sources and the received wisdom is that causality is limited to linear relationships, to which correlation coefficient applies. The better the linear model (regression line) the stronger the case for causation (cogito).

Why restrict causality to straight lines?
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Please identify and quote the source(s). It's nonsense.

And if we disagree with the source, then you have to ask those people to defend their claim, not us!

And if what they say is, as you say here, "The better the linear model (regression line) the stronger the case for causation", I hope you know enough logic to see that that does not say "causality is limited to linear relationships"! It says "IF there's strong linear correlation, then there's reason to expect causation", not "ONLY IF ...". And I don't think that's true anyway!

Agent Smith said:
Say I'm wrong, any examples where causality is demonstrated based on a nonlinear correlation (?)?
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You did that yourself:
Agent Smith said:
couldn't a causal relationship be a parabolic relationship
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How about the correlation of gravity and distance?
 
@blamocur I understand.

@Dr.Peterson , so gravity is the cause of planetary orbits and falling apples. The greater the mass the greater the gravitational force. Isn't that a linear relationship?
 
Agent Smith said:
@blamocur I understand.

@Dr.Peterson , so gravity is the cause of planetary orbits and falling apples. The greater the mass the greater the gravitational force. Isn't that a linear relationship?
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Maybe distance causes a decrease in gravitational force ... which is not linear?

Why haven't you shown us the sources you claim insist causality must be linear? I'd like to see what they are really saying.
 
A follow up question.

Capture.PNG
The above is a graph of Boyle's law. I think the formula is [imath]P_1 V_1 = P_2 V_2[/imath]

It's not a straight line and so per what I learned (causal relations are linear), pressure (y axis) and volume (x axis) are NOT causally connected.
 
Agent Smith said:
A follow up question.

View attachment 38706
The above is a graph of Boyle's law. I think the formula is [imath]P_1 V_1 = P_2 V_2[/imath]

It's not a straight line and so per what I learned (causal relations are linear), pressure (y axis) and volume (x axis) are NOT causally connected.
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Be careful about mixing terminologies. It is true that Boyle's law is [imath]P_1V_1 = P_2V_2[/imath], but the graph you are showing is for the relationship between pressure and volume which has this formula: [imath]\displaystyle P = \frac{k}{V}[/imath] or [imath]\displaystyle PV = k[/imath].
 
@mario99 gracias for the warning. I would like to know if the relationship between P and V is causal. I learned that causal relationships have to be linear and as you can see the relationship between P and V (T being constant) is nonlinear.
 
Agent Smith said:
@mario99 gracias for the warning. I would like to know if the relationship between P and V is causal. I learned that causal relationships have to be linear and as you can see the relationship between P and V (T being constant) is nonlinear.
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Where did you learn that?

Two professors have already told you that it makes no sense. And professor Dave asked for the source that taught you that, but you claimed it was lost. First, let us read the source (sources) that you have read, then we can say something. Maybe you understood them in the wrong way!
 
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