Quantifying the difference between micro and macro averaging in cross-validation

Question

I am studying the effect of micro vs macro averaging the results of a cross-validation run. Using true positive rate as a running example, given a cross-validation run of $K$ folds, the final $TPR$ can be computed by:

• micro-averaging: $$TPR_\mu = \dfrac{\sum_{i=1}^K TP_i}{\sum_{i=1}^K N_i^+}=\dfrac{\sum_{i=1}^K TP_i}{N^+}$$
• macro-averaging: $$TPR_M=\dfrac1K\sum_{i=1}^K \dfrac{TP_i}{N_i^+}$$ where $N_i^+$ is the number of positive examples in fold $i$ and $N^+$ is the total number of positive examples.

Other answers and literature [1,2] argue that $TPR_\mu$ is the most unbiased estimator for the $TPR$. Here, I am interested in quantifying the difference between $TPR_\mu$ and $TPR_M$, particularly with respect to the number of positive examples in each fold, to better understand the sources of a potential bias in $TPR_M$.

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My attempt is the following:

Let $f_i=\dfrac{N_i^+}{N_i}$ be the fraction of positives in fold $i$, and $f = \dfrac{N^+}{N}$ be the fraction of positives in the entire dataset. We have that: $$ \begin{align} N_i^+ &= f_i \cdot N_i = f_i \cdot \dfrac{N}{K} \\ N_i^+ &\sim \mathsf{Binomial} \left( \frac{N}{K}, \ f \right) \\ f_i &\sim \mathcal{N} \left( f, \ \sqrt{\frac{K}{N} \cdot f \cdot (1-f)} \right) \end{align} $$

Finally, $$ \begin{align} TPR_\mu &= \dfrac{\sum_{i=1}^K TP_i}{\sum_{i=1}^K N_i^+} = \dfrac{1}{N}\sum_{i=1}^K \color{blue}{\dfrac{TP_i}{f}} \\ TPR_M &= \dfrac1K\sum_{i=1}^K \dfrac{TP_i}{N_i^+}=\dfrac1K\sum_{i=1}^K \dfrac{TP_i}{f_i\cdot\frac{N}{K}} = \dfrac{1}{N}\sum_{i=1}^K\color{magenta}{\dfrac{TP_i}{f_i}} \end{align} $$

So, essentially, the difference between micro and macro averaging is solely due to the fraction of positives in each split. If each split contains the same fraction of positives (i.e. $f_i = f, \ \forall i$), then $TPR_\mu = TPR_M$.

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My questions:

1. Can I quantify some sort of difference between $TPR_\mu$ and $TPR_M$ given the variance of $f_i$? For example, if the variance is high, I expect $TPR_M$ to greatly differ from $TPR_\mu$, whereas if all $f_i$ are close to the mean, there wouldn't be much difference.
2. Are there any interesting insights I can extract from this calculations?

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[1] https://www.kdd.org/exploration_files/v12-1-p49-forman-sigkdd.pdf

[2] Mean(scores) vs Score(concatenation) in cross validation