Hi, I am unclear on your question. Could you be more specific and perhaps precise in your question? I'd be glad to help so.

In Pearson correlation the p value is above 0.5 indicates these variables strongly correlated and below 0.5 shows weekly correlated meanwhile negative values also will come

If the relation between two variables is not linear but, for instance, has the shape of an inverted U, then the Pearson correlation will be near zero, yet the one variable is dependent on the other one.

If correlation coefficient between two variables is zero then

(1) the relationship between the two variables are not linear ,

& (2) they are either independent or they are related by some non-linear relationship.

if there is zero correlation it means there is no linear relationship between the two variables

Dear Kabo Lupe , Is it possible that other type of relationship between the two variables can exist in this situation if there is no linear relationship between them ? If so, what can be that relationship ?

Dear researcher,

To my knowledge, you should check the null hypothesis that no relationship exists Then with some comparison of the null hypothesis parameters, you can reject the hypothesis that there is no relationship or not.

There are many videos concerning this method if it is suitable for your case study.

@ Belgacem Mohamed El Ghazali ,

Details of the following two questions are requested:

(1) How to check the null hypothesis that no relationship exists between the two variables or equivalently to check the null hypothesis that the two variables are independent.

(2) What are the the null hypothesis parameters which are to be compared for rejecting the hypothesis that there is no relationship?

Assuming that you do not know the origin of the sample, there is no test showing independence given zero correlation.

Assuming data from a well-designed randomized experiment - the designed experiment will assure independence given zero correlation.

For details: see "Statistics for experimenters", Box Hunter Hunter, Chap 1 - 3, 2005

@ Jan Mikael Åkerström,

I agree with the opinion of Jan Mikael Åkerström

that (1) assuming the origin of the sample to be unknown, there is no test showing independence given zero correlation.

and that (2) assuming data from a well-designed randomized experiment - the designed experiment will assure independence given zero correlation.

However, if the two variables are not independent in the first case then what will be the relationship between the two variables. This is the thrust of my study.

One suggestion is a sample from a square function eg

x y

1 1

2 4

-1 1

-2 4

You can check the correlation to be zero.

@ Jan Mikael Åkerström,

The example namely

"A sample from a square function eg

x y

1 1

2 4

-1 1

-2 4

has zero correlation "

establishes that If the value of Kearl Pearson's correlation between two variables is found to be zero then one possibility is that the dependent variable is a quadratic function of the independent variable.

My thrust is to know if there is(are) any more possibility(possibilities).

The relationship might not be linerar and most of the experts have already added this comment Dhritikesh Chakrabarty quite interesting question and it is good to see the subject related questions

@ Srinivas Kasulla ,

Certainly, the relationship will not be linear in this situation. But what can be the relationship between them ? This is the main thrust of the question problem..

Zero correlation coefficient implies either absence of linear association or presence of non-linear correlation between the two variables.

It means there is no linear association between variables and they are not influenting to each other directly.

Correlation coefficient measures degree and direction of association between two variables. It follows principle of additivity and Linearity. Therefore, the result (0) here simply implies independence of linear association between variables.

In plant science, we know that any trait is interaction of several traits that certainly will lead to no linear association between two variables/traits. But we measure it for several trait combination.

So, my supplimentry querry is that even in case two variables have any other relationship and not a completely linear relationship, is not this a situation that any other relationship is just a deviation from linearity [just as SCA (nonlinear) is measures as deviation from GCA (linear)] and it should show relationship in terms of significant correlation coefficient. Why we should not get a zero value only in case of Independence of any relationship?

@ Mithilesh Kumar Singh

I do agree with the view that the value "0" of Kearl Pearson's correlation coefficient here simply implies independence of linear association between variables.

However, the independence of linear association between the variables does not imply that there is no relationship between them.

It is to be noted that non-linear relationship is not just a deviation from linearity but a set of infinite specific mathematical relationships.

Then no relation between them

@ Muhammad Ihsan ,

There may exists relationship between the variables.

In the instant situation, either the variables are independent or related by some by some relationship which is not linear but some other mathematical relationship.

@ Prakash Singh ,

It does not mean, as I know, that it is not the only option there is no linear association between variables and they are not influencing to each other directly.

One counter example is as follows:

Take the following pairs of values on (X , Y)

(-3 , 9) , (-2 , 4) , (-1 , 1) , (1 , 1) , (2 , 4) , (3, 9)

Here,the correlation coefficient is 0.

But the variables X and Y are not independent, They have quadratic relationship.

@ Arbind K. Choudhary ,

I do agree with the view of Arbind K. Choudhary that

Zero correlation coefficient implies either absence of linear association or presence of non-linear correlation between the two variables.

Accordingly, my question is what is the specific form of that non-linear correlation.

@Dhritikesh Chakrabarty

Then if any relationship exists and it is further decomposed it should have have an element of linearity and all deviations from it will comprise real association. Under such situation it should contribute some value attributed to linearity and value should not come to zero (though of a miniature value).

@ Mithilesh Kumar Singh ,

Suppose, some relationship exists between the variables.

Then, the relationship may be any mathematical type other than linear (since Kearl Pearson's coefficient of correlation is 0).

However, it is not necessary that the relationship will always contain an element / term of linearity . For example, consider the relationship

Y = a. Square(X)

where "a" is constant.

This relationship does not contain a linear term.

@Dhritikesh Chakrabarty

Y=a.square(x)

Y=8, 18, 32, 50, 72 (a=2, x=2, 3, 4, 5, 6)

If it was purely linear, trend should have been 4,6,8,10,12. The squared trend (8, 18,32,50,72) has already 4,6,8,10,12 (linear) trend within it and the deviations (4,12,24,40,60) makes it a squared relationship going beyond linearity. Then is not there a measure to partition the squared trend as linear and deviation from linearity and then based on linear component give the Carl pearson correlation. How far am I correct?

No relation. Independent. But there may be some unknown and unseen factors. As in real life if one relation descends, there may be some other that ascends