I'm embarrassed to say that I have a PhD and hold an asst professorship, but get tripped up when reading statistics research. I am in a field of Business that is similar to IO Psychology or Social Psych. I spend too much time reading applied stats books, but I find even with all the reading I don't have a firm grasp of what I'm actually doing. Everything is very 'seat of the pants.' (As sad as it seems, I think this is not a unique situation among the faculty in the social sciences...) The biggest problem comes when I need to apply a rarely used stat technique. I can find an article from a mathematical stats journal with the equations that would solve my problem, but I don't have the math to convert those into code. I am forever relying on other prof's R packages, and crossing my fingers hoping it will work (I can't even check to verify if it did or not). It's been over 15 years since I took Calculus and Algebra in undergrad, and I think I want to start at the beginning and truly understand probability and statistics.
I am starting with Gelfand's Algebra and Trigonometry books for a quick refresher of the basics -- I know it's hard to believe, but in an applied research field we rarely have use for sin or cos. I'm even trying to finally learn how to correctly do a proof, using the books from Velleman ("How to Prove It") and Houston ("How to Think Like a Mathematician") -- I'm serious about doing this right and understanding the subject. From there I want to move on to (correctly) learn the Calculus and Linear Algebra I need to tackle probability and statistics. I was thinking of using Strang's Calculus and Algebra books. But Apostol's Caculus comes highly recommended as well. After that I am completely at a loss. Further, I don't know how far to go into Calculus or Linear Algebra before I reach diminishing returns. (Apostle introduces Probability in the second half of Vol. 2 -- is it vital that I work through everything preceding it before tackling Probability?)
So my question is: if you had to do it over again with the goal of truly, deeply understanding statistics, where would you start? What books are the modern path to deep understanding? I would like to follow a modern path so that I can understand current research in statistics, including Bayesian approaches. But not in a machine learning context (which seems to be the all the rage at the moment), rather a social science / design and analysis of experiments / multilevel modeling context. Perhaps my goal would be the work of Andrew Gelman; his and Hill's book showed me how I should be looking at modeling and statistics (simulation, uncertainty estimates everywhere, bayesian inference, and so on). How should I go about relearning this material with that end goal in mind?
Update 1: Possible texts, starting from scratch with a focus on proofs and deep understanding. Not necessarily one after another.
Relearn the basics:
- How to Prove It by Velleman
- How to Think Like a Mathematician by Houston
- Algebra and Trigonometry by Gelfand (for understanding why and how instead of what)
- Precalculus in a Nutshell by Simmons (for reference)
- Measurement by Lockhart (for inspiration)
Calculus (which one(s), and how deep?):
- Calculus by Strang
- Calculus vol. 1 and Calculus vol. 2 by Apostol
- Calculus by Spivak (solutions)
- Introduction to Calculus and Analysis: Volume I by Curant (and II/1 II/2?)
Linear Algebra (which one(s) and how deep?):
- Intro to Linear Algebra by Strang
- Matrix Algebra Useful for Statistics by Searle
- Matrix Algebra: Theory, Computations, and Applications in Statistics by Gentle
Probability (which one(s)?):
- An Introduction to Probability Theory and Its Applications, Vol. 1 and Vol. 2 by Feller (for intuitive understanding)
- Introduction to Probability Theory by Hoel, Port, Stone
- A Probability Path by Resnick (for measure theoretic / modern approach?)
- Fifty Challenging Problems in Probability by Mosteller
Core Statistics (which one(s)?):
Other suggestions? Again with the goal of understanding and developing (or at least implementing) new methods in hierarchical modelling (generalized and linear).