# Notation for Pearson residual vs Pearson correlation coefficient

From my book, the Pearson residual is defined as:

$$R = \frac{Observed - Expected}{\sqrt{Expected}}$$

And from wikipedia, the Pearson correlation coefficient is defined as:

$$r_{xy}={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}}}}$$

So, are these two measures completely distinct unrelated things that just happen to use lowercase and capital R?

One of the reasons I ask is because, wikipedia also says that the Coefficient of determination is denoted either $$r^2$$ or $$R^2$$, and it equals, "the square of the Pearson correlation coefficient."

The fact that there seems to be both Pearson $$r$$ and Pearson $$R$$ values, and then the coefficient of determination is denoted either $$r^2$$ or $$R^2$$, is confusing me, notationally speaking.

• What they have in common is an intention to honour (Karl) Pearson, although there is a difference: he certainly discussed correlation, but Pearson residuals are one remove from chi-square contribution computationally and a even bigger remove conceptually -- unless someone can document his using them! – Nick Cox Oct 15 '19 at 16:58

## 1 Answer

Pearson residual and Pearson correlation are totally different concepts from different contexts.

Generally there are far more concepts in statistics than letters (even if you are prepared to use four different alphabets), so inevitably, at least if you go through a number of books and papers, the same notation will be used for different things.

The standard attitude should be that all notation needs to be defined specifically for the book or paper you are reading, and that you cannot rely on knowing what notation means unless it is explicitly defined. If you want to know what notation means, look up the definition in the text you're reading - if it's not defined, it's bad writing.

That said, particularly in statistics there are a number of conventions that are used fairly generally. $$r_{xy}$$ is often used for the Pearson (sample) correlation and $$R^2$$ is often used for the coefficient of determination (which in fact is the square of a specific correlation in standard regression), whereas I haven't seen $$R$$ for what you call Pearson residual that often (and even the term Pearson residual itself is not universally used). Using $$R$$ for the Pearson residual and $$R^2$$ for the coefficient of determination in the same book is actually misleading; such things should be avoided, and even $$R$$ and $$r$$ are better used for two things that are clearly related (such as random variables and their realisations), but such things happen fairly often.

Some authors think that they can use standard conventional notation without definition, but I wouldn't agree. The base line is that things are what they are defined to be, and there is no guarantee whatsoever that notation is consistent between different texts. (Within the same text I can of course not guarantee it either, but the author should make an effort.)

• The principle of it being good to explain notation is right, but numbers and elementary operators are notation too: what would you think of a book that defined + - and / ? Also if a book uses calculus notation, exp() and ln() without definition and that is a puzzle to the reader, it's arguable that the reader is reading the wrong book: typically a preface or introduction should explain prerequisites. Worse than unexplained notation is inconsistent notation! – Nick Cox Oct 15 '19 at 16:56
• Ok, fair enough... I admit that for practical reasons authors need to take some notation for granted and should only start defining at a certain level. That level may be lower than you think though, some people use log() for what others write as ln() and occasionally there's danger of misunderstanding from that. And wherever such a danger exists, explicit definition is safer. As author ask yourself whether you are sure that every (serious) reader will understand your symbol, otherwise define it! – Lewian Oct 15 '19 at 17:19
• I agree about log(). That's why ln() is good notation and was in my example, despite what some mathematicians say. And more generally: jstor.org/stable/20001459 is me saying this to geographers in 1979... – Nick Cox Oct 15 '19 at 17:26