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Thread: How to Determine if 2 variables are dependent?

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    Re: How to Determine if 2 variables are dependent?




    Quote Originally Posted by GretaGarbo View Post
    No, R^2 is not by definition a measure of covariation.
    Sure, bad choice of words. My point still stands even if it was possible to missunderstand.

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    Re: How to Determine if 2 variables are dependent?

    Quote Originally Posted by Dason View Post
    I was just pointing out that you can't use the fact that R^2 = 0 to imply that there is no dependence.
    I think we've discussed different things. Correct me if I'm wrong, but you've discussed the observed R^2 while I've had the true unknown R^2 in mind.

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    Re: How to Determine if 2 variables are dependent?

    Either or. Even if the "true" R^2 is 0 (which is the case for X ~ Unif(-a, a), Y = X^2) there can be dependence.
    I don't have emotions and sometimes that makes me very sad.

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    Re: How to Determine if 2 variables are dependent?

    Quote Originally Posted by Dason View Post
    Either or. Even if the "true" R^2 is 0 (which is the case for X ~ Unif(-a, a), Y = X^2) there can be dependence.
    But that isn't true. The R^2 is per definition not zero in that case. It can be estimated to be zero, but in that case the model used to predict Y is seriously flawed.

    I agree on this: If R^2 is estimated to be 0, then we cannot draw the conclusion that there are no dependence. But if the true unknown R^2 is 0, then there are no dependence. I would say that R^2 is 1 in the example you refer to, because there is a perfect relationship between the variables described. But if you try to fit a linear model, then the observed R^2 will be far from 1.

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    Re: How to Determine if 2 variables are dependent?

    Consider the following distribution:

    \begin{tabular}{|c|c|c|}
\hline
x & y & Prob(Y = y $|$ X = x) \\
\hline
0 & 0  & 1 \\
1 & 1  & .5 \\
1 & -1 & .5 \\
\hline
\end{tabular}

    In this case the best "regression" we could come up with is predicting y = 0 regardless of x. But there is still dependence here. For if we know x = 0 then we KNOW y = 0. If we know x = 1 then y could be either -1 or 1. So the distribution of Y depends on the value of X. So R^2 is 0 but there is still dependence.
    I don't have emotions and sometimes that makes me very sad.

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    Re: How to Determine if 2 variables are dependent?

    A friend of mine surprised me when he said:
    “You can choose to get as large R^2 as you want.“

    “What?!” I said.

    “If there is a linear relationship between x and y and you can design where to put the x-values, then just by stretching out the x-values far enough you will get a large enough R^2 value“, he said.


    Another aspect is that if you have an observational study and there has not been very much variation in the x-values – the x-values have been roughly constant (as often happens in observational studies) – then the R^2 will be low. That does not mean that the model is bad. It can be a good description of reality. A good model is a model that fits to the data. Not if R^2 is high or low. Lack of fit measures are far more important than R^2.

    The residual variance has an influence on R^2 (by increasing the residual sums of squares). So you can make a two-by-two “table” or graph with high and low variation in the x-values and with high and low residual variation. I think that is more important to think of than the R^2.

    I would be primary concerned by the parameter estimates and if they are significant, the standard deviation in the residuals and lack-of-fit-measures.

    Quote Originally Posted by noetsi View Post
    but there are no perfect tools least of all those that can be understood by the 99.9999 percent that are not statisticians by trade
    I don’t think that R^2 is understood by 99 percent. I think it is overemphasized and misused.

    Besides, I think it gives increased confidence if someone talks both about a models strengths AND weaknesses. This is valid for statistical investigations and used car sellers.

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    Re: How to Determine if 2 variables are dependent?

    Greta! Go get one more post and then you'll have a surprise on the TalkStats homepage for you.
    I don't have emotions and sometimes that makes me very sad.

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    Re: How to Determine if 2 variables are dependent?


    Quote Originally Posted by Englund View Post
    But that isn't true. The R^2 is per definition not zero in that case. It can be estimated to be zero, but in that case the model used to predict Y is seriously flawed.
    The usual is to think of regression parameters like “beta” and “sigma” to have a population value that can be estimated from a sample.

    But does R^2 have a population value? I have never heard of that.

    Think of a linear regression model with a nonzero slope (beta).

    Imagine that a first experiment is having the x-values in a narrow range. That will give one R^2 value.

    Imagine a second experiment with exactly the same parameters but with the x-values in a wider range. That will give a higher R^2 values for exactly the same parameter values, that is, for the same population beta and sigma values.

    No, I don’t think it is meaningful to think of R^2 as population parameters.

    I think of R^2 as a simple descriptive of the data at hand.

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