Reddit reviews Modern Applied Statistics with S (Statistics and Computing)

This is a huge brain dump but hopefully some of it is useful. Mostly personal opinion so take it with a grain of salt.

Statistical Culture

Go and read a copy of The Lady Tasting Tea. Now.

The typical Stats101 course is like "Wheee!!! A seemingly arbitrary collection of formulas to cookbook your way through!!" Do not be discouraged. Although there is no winner, there are a series of 'philosophies' of statistics which each present a cogent, unified perspective on how to proceed (Fisherian, Neyman-Pearson, Bayesian). The messiness comes from trying to give the engineers a cookbook of results to follow. [Resources for (too?) advanced extra credit: I haven't yet found a good intro to this but maybe look at the personal appendix in "Principles of Statistical Inference" by D Cox. I've been reading this paper recently.]

Mathematical Background

The core mathematical background is probably: Linear Algebra, Multivariable Calculus, Real Analysis. Eventually measure theory too when you get to probability for grown ups.

Applied Stats

The best introduction to applied statistics I have discovered is the Statistical Sleuth.

The most useful activity I can recommend is to do little projects where you get some real, raw data, analyse it and then write up a report. So many issues crop up which you can't really understand from your textbooks. What data is available to me? Is this the data I really want? (Eg, observational vs randomised experiment.) What am I going to do about errors in it? (Eg, missing entries, outliers,...) How do I get it into my statistical analysis program? Do I actually have enough? Are my models at all good? It is a huge lie to say this but in a sense, once you have enough of the right data and it's all scrubbed up and nicely formatted, all the rest is easy.

The standard tools of the trade for applied work are the statistical analysis program/language R and the typesetting program Latex. This is the standard text for data analysis using R (R is a free version of the S language developed at Bell Labs.) There are online tutorials for Latex.

Learn to program. The language is less important than learning how programming works. The generic programming advice is to learn Python or Scheme. The former is probably more useful practically, the latter will give you more street cred with computer scientists. Within mathematics people tend to use R/Matlab/Mathematica or Fortran/C++ if they are doing heavy duty simulation. I'd recommend just putzing around with Python and R, and learn anything else as needed. The big BUT is if you want to go into finance, then you'll definitely want to dabble with C++. There are probably people who can give better advice but most seem to recommend Accelerate/Effective C++.

Studying Mathematics
Mathematics develops linearly, each step up builds on the preceding material. Even after you have finished a course, go back a couple of times over the next few years to refresh the material.

When studying mathematics I like to work top down and then bottom up. That is, start with a broad understanding of what you are doing and then go fill in the details within that framework.

For getting the high level view I like making mind maps or dependency trees. This isn't legible but hopefully it gives you the idea. I've summarised 24 pages of notes so that I can see the main branches covered, definitions of the basic objects and can quickly find the four super important theorems (Written as 'Theorem 1, p14: Rough idea of what it says'). With a bit more time I'd go through with another colour and draw dependency arrows to show which theorems/lemmas are used to prove which other ones. Having this big map somehow compresses the 'intellectual content' of the course down: making it easier to see interrelationships and not panic.

As an aside. Pure mathematics can be broken into four parts: definitions, important/key theorems, lemmas/propositions needed to prove important/key theorems, applicationy examples/results proved using the important/key theorems.

Then to actually learn the material we fill in details:

0. Find somewhere quiet with no friends/technology. Take your notes, some paper and a pen, and for the sake of cliche a cup of coffee.

1. Pick one of the early key theorems to work on.
   
   
2. Do you know from memory the definitions of the terms used in the statement of the key theorem? If not, look them up and play around with the definition and some examples until you have it memorised. Eg: an integer n is even if there exists an integer m such that n = 2m. The number 6 is even because 6 = 2(3). However, the number 7 is not even: 2(3) = 6 < 7 < 8 = 2(4).
   
   
3. What is the theorem 'saying'? Get a simple, concrete example down on paper. Eg, for Theorem: if n and m are even integers then so is n+m, take 4 + 6 = 10.
   
   
4. Spend a bit of time trying to prove the theorem is true. If you get stuck sometimes it can help to try drawing a picture, considering a special case or constructing a counter example (figuring out why you can't get your counter example to work can help you see why the theorem must be true). If you succeed, great.
   
   
5. If you didn't come up with a proof start working through the proof line by line. It can be helpful to keep your concrete simple example in mind as you do this. Inevitably there will be gaps which you should try to fill in ('He says that this implies that but it isn't obvious how. Can I prove that it works?') If the proof involves an earlier lemma, you have two choices: go back and repeat this process on that lemma, or push on regarding the lemma as a 'black box'. My advice is to do the latter but take a bit of time to think about what it is exactly that your lemma is accomplishing within the proof. Something like: "To show f(x) has property Y given condition A, we first need to know that f(x) has property Z and our proof uses that to show property Y. We need Lemma 1 to show that condition A implies f(x) has property Z." One of the problem with mathematics lectures is that they present the material in a logical way and so sometimes lemmas crop up unmotivated because 'we use this later'. By doing this process you put the motivation front and center. If you get totally stuck make a note and talk to your teacher.
   
   
6. You now in principle understand how to prove the result, perhaps conditionally on assuming some lemmas. Spend some time really making the proof your own. Knowing the result can you see an easier way to prove it? Can you put the steps in a more sensible order? Can you fill in the gaps in the proof? Can you add in some helpful comments which explain what you are doing? Can you cut anything out? At some point in the future (a few hours+ later) you should sit down and state the theorem from memory and then try to construct either the whole proof or a sketch of the proof also from memory. The 100% absolute best way to cement something into your mind is to teach it. If you can find a classmate to work with great, but often I will just find an empty room and lecture the material to the wall. There have been sooo many times when I've said something like "And this is true because...um...actually I'm not sure." The most efficient way to study is to test yourself, find out what you don't know, and then focus on filling those gaps. Explaining seems to cause me to think of questions I wouldn't have thought of if I was learning.
   
   
7. Having figured out your tool you now want to use it to make sure you understand it. Solve examples from your problem sheets which use the theorem. Try to use it to prove corollaries with it.
   
   
8. Now go and back fill any skipped lemmas which were used in the build up to the proof of the theorem.
   
   Career
   
   As Mark notes, it's worthwhile spending some time to learn a bit about a particular discipline. People want you to solve problems. The fact that you're doing it with statistics is irrelevant, if you could get correct answers by divining in chicken guts they'd be quite happy to accept that methodology too. Having a domain in which you can apply your knowledge gives you in idea of what the problems are and gives you the language to talk to the people who have the problems. I think you can sometimes pick this stuff up on the fly, but it's nice to just have it.
   
   Definitely try to get internships over your vacations. Ideally with a company you want to go on and work for.
   
   I don't know much about Actuarial work. Apparently there are a series of industry exams you need to pass. Look into that.

With a degree in maths, you are already ahead of most scientists

It depends on exactly what you want it for. Asking for a comprehensive statistical reference is akin to asking for a comprehensive compendium of pure maths or computer science or physics. It's not going to happen and any book which purports to be a complete reference is going to be so vague and high-level it is useless.

For instance if you are interested in vanilla statistical modelling, there are a couple of books which may be of interest. Davidson's Statistical Models and Pawitan's In All Likelihood are great technical introductions (with examples). Vennable and Ripley's Modern Applied Statistics with S is much more applied and focuses on the practice of modelling rather than the theory. Although it is import to understand some theory because violating model assumptions can result in erroneous inferences, without theory you are really operating in the dark!

EDIT: perhaps seek a book specific to your academic discipline (bioinformatics etc.)

I see "Discovering Statistics using R" suggested often too, but I borrowed it from the library and the first page I opened to had a glaring error - and one that was really easy to check, pretty much simply by typing what was being discussed into R (it said something didn't work in R, but it did)

A bit later I went to another page. Another error.

I opened to another page. A couple of errors. I put the book down.

Next day, tried another page ... another error.

Another ... same result

I let it sit a few days. Tried once more ... and while there wasn't an actual error this time, something was so misleadingly explained I don't know how someone who didn't already know the material would end up understanding it.

I left it a couple more days planning to try some more, but it got recalled. I returned the book.

Maybe it was bad luck and I just hit the only bad places in the whole book, but it looks to me like there are at least some problems with editing/checking.

Nevertheless a lot of people in certain application areas seem to like it. I don't know if they just can't see the errors or they don't care about them. If it helps you, use it, but try to not take everything it says too seriously.

Oh, and when I tried to report the first problem I struck ... it ended up taking me about 40 minutes to figure out exactly how (I kept running into links that didn't work or pages saying, basically "here's why I don't respond to people"), and for which no feedback is offered whatever (not even a 'I got your message'). For all I know it disappeared down a black hole. So I didn't try that again. [If you're going to write a book that you don't want to be full of errors, you have to make it easier for people to let you know, and you have to be prepared to actually communicate with them a little even if that's uncomfortable for you. If you can't deal with that, you either have to be a hell of a lot better at checking stuff, or you need to give up any hope of writing a book that's not full of mistakes.]

A couple of years ago I managed to get hold of The R book for a few hours but it didn't especially grab me; that may just be lack of time spent with the book, I don't know. I hear that the more recent edition (2103) is substantially better, though.


For myself, for learning R I found Braun and Murdoch's A First Course in Statistical Programming with R quite useful (unfortunately, someone borrowed it from me and I don't have it any more) and after that, R for Dummies and Matloff's Art of R programming book were reasonably good as well. For stats in R, I got a lot of value out of Venables and Ripley's [Modern Applied Statistics with S](http://www.amazon.com/Modern-Applied-Statistics-Computing/dp/0387954570
) (R is an implementation of the S language), but your mileage may vary.

You have an ordered categorical (Likert) response variable and one quantitative predictor variable? You need to read up on ordered categorical data analysis. There are discussions of this in Agresti and in Venables and Ripley and, of course, lots of other places.

This one is fairly standard: http://www.amazon.com/dp/0387954570. After all, it's where the MASS library comes from.