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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.

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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:

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

In my understanding the two formulas are different.

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  • $\begingroup$ 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). $\endgroup$
    – user0
    May 10 '20 at 21:31
  • $\begingroup$ @user0, I included the formula in the first example. Thanks $\endgroup$ May 10 '20 at 22:38
  • $\begingroup$ 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 $P(Y_i=1)$. $\endgroup$
    – user0
    May 10 '20 at 22:51
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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.

So, the two formulas are the same for binary data; the first formula is more general.

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