Solved – What does “dependency between variables” mean (graphically)

correlationnon-independent

For an exercise I'm asked to resolve the "dependency" of two variables (data sets) using x-y plots. So I plot to the x-axis the variable that I infer as being the predictor and to the y-axis the one that I infer as being the response.

However,

how am I supposed to infer whether and what kind of dependency there is by just looking at the following plot?

To me it seems like there's some correlation, that increasing mom_iq also increases kid_score, but that this correlation is not "clean", but rather there's a lot of variation in as well. I.e. in some cases kid_score is high (or higher than it should be) even if the mom_iq wouldn't be.

So in this case, where do I strike a line between "there's strong correlation" and "there's weak correlation"?

Best Answer

Dependence between two variables in general is whenever the following equation does not hold:

$P(x,y)\neq p(x)p(y) $

There are quite a few tests that test this hypothesis HHG, Hoffdings Test and more.

If you are interested in the different types of dependence, you can use Pearson correlation which evaluates the linear relationship between two variables, or Spearman correlation which evaluates the monotonic relationship between two variables.

In the case you presented it seems that you want to say something on the nature of relationship. In order to do so we use Regression. If the assumptions of the regression hold, you can test the Pearson correlation and see if you reject the null that the Pearson correlation equals zero. You can also estimate the coefficients and test them.

In the plot you presented it seems that there is dependence, but is not linear. You could try to fit a polynomial regression.

$Y = \beta_0 + \beta_1 x_1 + \beta_2 x_1^2$

$y$ being the kids score and $x_1$ the mom iq. Then you can evaluate your adjusted $R^2$ which in the one dimensional case coincides with the Pearson correlation. Check if the assumption hold and then see the size and significance of the coefficients.

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Solved – A way to show Simpson’s paradox is happening in the data

You can call it "confounding" or "omitted variable bias":

Confounding: The statistical relationship between $X$ and $Y$ is confounded by the grouping factor "color". [Maybe you remember that a confounder is a variable which is both related to $X$ and $Y$, which is the case in your picture.]
Omitted variable bias: The estimated statistical relationship between $X$ and $Y$ suffers from an omitted variable bias if the grouping factor "color" is omitted.

Both are describing the same problem in different words. Simpson's paradoxon can be viewed as an extreme case of those phenomena.

When I reread your post I was somehow unconfortable about my answer because usually, we want to avoid confounding/omitted variable bias but it seems that you are preferring the "crude", unadjusted relationship between $X$ and $Y$.

Usually, we first describe the crude relationship and then revise it by confounder adjusting it. But I am sure also the converse can be useful in some cases.

Now, how can you "quantify" confounding? In a regression setting, you would compare the estimated slopes between two regressions both run on the pooled data set: One by regressing $Y$ on $X$ and a more complex one by regressing $Y$ on $X$, $\text{Color}$ and eventually on the interaction of $X$ $\text{Color}$ (then you have two comparisons to make). It is a very broad topic, so it is difficult to provide a complete answer.

Solved – How to interpret a vertical/horizontal scatter plot

I agree that the plots suggest no obvious relationship between the two variables. But I want to mention some things.

Firstly, the correlation coefficient r for data like this could still be quite high and significant so I would say it is always a good idea to visualize the data as you did. Consider the following

# generate data
set.seed(1)
a <- c(rep(1, 100), 1:100) + rnorm(200, 0, 10)
b <- c(1:100, rep(1, 100)) + rnorm(200, 0, 1)
plot(a, b)

plot

cor.test(a, b)
....
p-value < 2.2e-16
cor= - .56481

But it would be clearly inappropriate to assume a linear trend here.

Secondly, I don't think this is a big problem in your case judging by your plot but you still maybe want to read a little bit about overplotting. I suggest this because your can't really see much in the "tall cloud" in the range of latency_normalized 0 to 0.5.

Thirdly, the count-standardized variable is clearly right-skewed and zero-inflated. Maybe this is an issue but it depends on what you know about the data.

In the end I would say it depends on the meaning of the data how to interpret this plots. I am absolutely not sure about this interpretation but maybe there is some threshold going on here. The threshhold would be that there can't be any high count_standardized values once latency reached >=.5 (with a few exceptions, obviously). And maybe if you would apply transparency to the plot you would then see some relationship for the latency_standardized <.5 values with the count_standardized variable. But I can't tell whether this makes sense in your case to assume such a threshold.

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