# Different definitions of Pearson residuals (Logistic Regression)

I needed help from friends about Pearson residuals in Logistic Regression:

a) The Pearson residual is the difference between the observed and estimated probabilities divided by the binomial standard deviation of the estimated probability. Therefore standardizing the residuals.

From Menard, Scott (2002). Applied logistic regression analysis, 2nd Edition. Thousand Oaks, CA: Sage Publications. Series: Quantitative Applications in the Social Sciences, No. 106. First ed., 1995. See Chapter 4.4

But

b) Pearson residuals are given by:

From Kutner (2013). Applied Linear Statistical Models.1 McGraw Hill India, 5th edition. See page 591

In my understanding the two formulas are different.

• What is the formula given in (a)? It sounds like these are the same formulas. Possibly in (a) it is referring to a binomial case while in (b) it is Bernoulli, in which case (b) is a special case of (a). May 10 '20 at 21:31
• @user0, I included the formula in the first example. Thanks May 10 '20 at 22:38
• The first definition seems odd because we cannot observe $P(Y_i=1)$. We can observe the proportion of Ys that are 1, so maybe that’s what’s meant by $P(Y_i=1)$. May 10 '20 at 22:51

My library has the first book. The notation isn't all that clear even given the book, but it appears to be $$P(Y_j=1)$$ for the empirical proportion of 1s in case j and $$\hat P(Y_j=1)$$ for the fitted probability from the model.
For binary (Bernoulli) data, $$P(Y_j=1)$$ is just $$Y_j$$, but for Binomial data $$P(Y_j=1)$$ will be number of successes/number of trials for that covariate pattern.