Can a Correlation Be Not Significant While a Regression Is Significant for the Same Two Variables?

Much depends on what 'a regression being significant' means and what kind of correlation test we are using. This blog post will explore the nuances between these two statistical measures and provide examples to help clarify the concepts.

Introduction to Correlation and Regression

Correlation and regression are key statistical tools used to assess the relationship between two variables. Correlation measures the strength and direction of a linear relationship between two variables, while regression helps to predict the value of one variable based on the value of another. Both are crucial in understanding and analyzing data, but they are often interpreted differently.

Simple Linear Regression

Consider a simple linear regression model defined as:

y ax b with normally distributed residuals.

The hypothesis that a 0 is essentially identical to the hypothesis that the Pearson correlation between y and x is 0. This means that a being significantly different from 0 is equivalent to the Pearson correlation being significantly different from 0. If one test yields a p-value of 0.049 and the other yields a p-value of 0.051, it#39;s simply because the software uses different approximations for the two tests. However, an exception occurs when y is constant. In such a scenario, a is 0 while R is undefined. This might seem like a technicality, but it makes sense when we consider what a and R are meant to convey:

1. If y is constant, it doesn#39;t depend on x; thus, a is zero, even if y is actually not constant if measured with greater accuracy.

2. Conversely, if y could be slightly non-constant, R could range from -1 to 1.

Interpreting the Coefficients

Another consideration is that a 'significant' regression could refer to either coefficient being significantly different from 0. For instance:

a 0 but b is positive.

In contrast, if a is the only coefficient:

y ax with normally distributed residuals.

It is completely possible that the correlation is zero while a is non-zero. However, in this case, the model without an intercept does not fit the data very well. This is illustrated in the figure below:

Conclusion

While correlation and regression often yield similar results, differences can arise due to the nature of the relationship between the variables and the specific tests being used. Understanding these nuances is crucial for accurate analysis and interpretation of data. As Accident rightly points out, the terminology and context play a significant role in determining the significance of the results.

References:

Athreya, K., Pennington, T. (2018). Introduction to Correlation and Regression Analysis. Johnston, J. (1984). Econometric Methods. New York: McGraw-Hill.