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There are a few problems with your first paragraph which may make your question difficult to answer.
We know that a polynomial can approximate any function.
Can it? If you're referring to a Taylor polynomial, then the function must be smooth. Not every function is a smooth function.
In binary logistic regression we're trying to fit a decision boundary ...
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EDIT: I see you mentioned the Fieller method in your original post. Perhaps you were referring to the solution I provided below.
Here is a great paper on the topic. Using a logistic regression with a logit link function you can model the proportion of fish as a function of length, with $\lambda:=$LD50. Based on the asymptotic normality of
$$ \frac{(\hat{\...
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There is no reason to stop at the normal! What you propose, is called the Cauchit link function, for an example at this site, see GARMA models for counts.
Here, at arXiv is a paper looking at and comparing various link functions, cauchit included, for binary regression.
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Comments to the question suggest the following interpretation:
Given any two non-overlapping finite collections of points $A$ and $B$ in a Euclidean space $E^n,$ does there always exist a polynomial function $f_{A,B}:E^n\to\mathbb R$ that perfectly separates the collections? That is, $f_{A,B}$ has positive values on all points of $A$ and negative values on ...
1
Correlation of the variables can give this, and I would expect correlation when you start including lots of polynomial terms. Check out the simulation I give in the appendix here.
To give intuition, when variables are correlated, it is hard to say which variable is contributing to the model (testing individual factors). Is it $X_1?$ Is it $X_2$? Who knows!? ...
1
The key question is what to predict / or what is the dependent variable.
If the study is about investigating the "correlation" between gender and care setting, or how to use care setting to predict gender, then the original formulation can be used.
If we want to use gender and other variable to "predict" care setting, then your ...
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You can set up your data as a contingency table, as with the following R code:
yourtable <- as.table( matrix(c(5, 5, 16,
367-5, 408-5, 420 -16),
byrow=TRUE, nrow=2, ncol=3))
rownames(yourtable) <- c("cases", "nocases")
colnames(yourtable) <- c("2018", "2019", "...
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This can happen when you have multi-colinearity between your variables.
Imagine $X_1=X_2+X_3$ and $Y=2X_1 + 0.2X_2 + 0.1X_3$. So most of the predictive power is contained in $X_1$ and we can do a pretty good one variable prediction with just $X_1$. However, if we're allowed to have two variables we can do a better job with $X_2$ and $X_3$ since there's ...
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