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I'm from a social science background and, like you, I often find myself hopelessly lost when it comes to what feels like very basic concepts in statistics. I think that's partly due to how statistics is taught in all non-mathematics disciplines - in theory we're taught how to use and evaluate quite complex statistical procedures, but with only 1-2 hours per week teaching, it's impossible for our lecturers to cover the fundamental building blocks that help us to understand what's actually going on.

Because of this, I've recently started a few MOOCs on Coursera, and I've found these massively helpful for covering research methods and statistics in far more depth than my undergraduate and postgraduate lecturers ever had time to delve into. In particular, a couple of courses I'd recommend are:

• Methods and Statistics in Social Sciences - This is particularly focused on quantitative methods in the social sciences (including quite a bit on behavioural and self-report research) so I'm not sure if it will be directly relevant with respect to neuroimaging and cognitive neuroscience, but this gives a great introduction to research methods in general. I've actually only done the first course in this series (Quantitative Research Methods), but they're very comprehensive and well made, so I'm confident that the whole series will be useful for any researcher.
  
• Probability and statistics: To p or not to p? - This one is a little bit more maths-heavy so might be a bit intimidating if you don't find that sort of material easy, but it's a good introduction to some of the core concepts in quantitative research, including some you mentioned (e.g. probability distributions). You don't really have to fully engage with or grasp the maths for it to be useful either.
  
  In terms of textbooks, I personally use Andy Field's Discovering Statistics Using R, and find that very helpful. Field is a psychologist who is very open about his difficulties with learning statistics, and I've found it quite useful and re-assuring to learn from someone with that mindset. He's also tried writing a statistics textbook in the form of a graphic novel, An Adventure in Statistics: The Reality Enigma, so if that sounds like something that might help you, check it out.
  
  I think a few people from a 'purer' statistics background are a bit more critical about Field's books because they're not as comprehensive as a book written by, for example, a statistics professor - and there might be some advice in there that's a little bit out-of-date or not quite correct. He also has a very hit-and-miss cheesy sense of humour, which you'll either love or find very annoying. But I think he takes the right sort of approach for helping people who aren't necessarily mathematically-inclined to dip their toes into the world of statistics.

Linear mixed effects models (also called mixed models or random effect models) are exactly what you want to base everything off of. Unfortunately, they are a tad bit advanced if you ever need to customize it. The basic idea is that in a linear model, you can treat some effects as random instead of fixed and it allows you to account for noise at multiple measurement levels. Let me give you some book recommendations that can teach you about this stuff while simultaneously building your stats background. (listed in order of approachability)

A Modern Approach to Regression with R is a fantastic book that starts with basic linear models and ends with a nice basic look at mixed models. Its also decently cheap for a textbook.

A similar book that covers more topics but assumes a slightly higher starting level of familiarity is An Introduction to Statistical Learning, which is probably the best text book I have ever used, plus its ultra cheap. Its coverage of mixed models is minimal, but you will learn a ton about modeling and how things like machine learning algorithms can be cast into a statistical framework.

There is an R package called lme4 that does mixed modeling. It isn't my favorite one to use (check out nlme) but the author wrote a pdf book about mixed models. It reads like an academic paper, but it is pretty comprehensive.

Profile analysis is related to mixed models, but it generally appears like a separate topic in most books. Now that I am thinking about it, it probably isn't the best idea for your application because it really only helps with one area of the problem whereas a mixed model can address the entire thing. If you are curious, look up books about multivariate analysis (not multiple analysis. very different). An example is Methods of Multivariate Analysis, but its pretty expensive and starts at a graduate level of understanding (that being said it is actually a lovely read if you are at that level).

There are other resources online that talk about this kind of stuff. Mixed models are especially useful in medical studies and pharmaceutical research, so searching for mixed models with those kinds of key words might bring up different perspectives based on different fields. Find whatever suits you best and be patient when learning this stuff. It takes time to learn and to settle in your mind.

Sorry, my post wasn’t very clear. Those were actually specific titles.

Practical Algebra:
https://www.barnesandnoble.com/w/practical-algebra-peter-h-selby/1114284979

Geometry and Trigonometry for Calculus:
https://www.barnesandnoble.com/p/geometry-and-trigonometry-for-calculus-peter-h-selby/1114965492/2676067143387

Those are both very good. My calculus recommendation is a little unconventional, so maybe it’s not for you, but I’d get Calculus: An Intuitive and Physical approach.
https://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics-ebook/dp/B00CB2MK6C

That book is far more wordy than your average calculus text, but I think that makes it great for self teaching. If you pick up something like 3000 Solved Calculus problems to go along with it you should be in great shape.

I know that’s not exactly cheap, but you should be able to pick up all of those for less than $100. Good luck!

Edit: all of the statistics texts in the last paragraph of my original post are available freely (legally, I believe) online.

You need Intuitive Biostatistics. It's written specifically for scientists and medical professionals without a math background to learn how to interpret data in scientific papers. I'm a PhD student in cell bio and it is invaluable. The only thing it might not cover super in-depth is probability, but it pretty much covers the gamut of everything else without delving into the mathematics behind everything. The guy who wrote it also makes the Graphpad Prism software, which a lot of bioscientists use for data analysis.

For the next step, I highly recommend JB Statistics videos. They are the best moderate math level explanations for the common concepts I have yet come across. Especially watch the sampling distribution playlist several times to fully comprehend the CLT.

Some other advice I wish I was told before I started learning statistics: 1) Statistics is the inverse of probability. 2) Statistics is unintuitive and hard to understand. You will have to read some things dozens of times and from different people's wording to fully understand a concept (looking at you, p-values). Best of luck.

Calculus by James Stewart is the best introductory Calculus book that I used in college - I definitely recommend it. It will get you through both single-variable calculus, as well as most of multi-variable calculus that you will need for for master's level probability and statistical theory. In particular, if you plan to use the book, you should focus on chapters 1-7 (for single variable calculus), chapter 11 (infinite sequences and series) and chapters 14 and 15 (partial derivatives and multiple integrals). These chapter numbers are based on the 7th edition.

If you have previously taken calculus, you might consider looking at Khan Academy for an overview instead.

If you have not previously taken linear algebra, or it has been awhile, you will definitely need to work through a linear algebra textbook (don't have any particular recommendations here) or visit Khan academy.

Finally, a book such as Stephen Abbott's Understanding Analysis is not necessary for master's level statistics, but could be helpful for getting into the mindset of calculus-based proofs.

I'm not sure what level of math you have previously completed, and what level of rigor the MS in Statistics program is, but you will likely need be very familiar with single- and multi-variable calculus as well as linear algebra to be successful in probability and statistical theory. It's certainly possible, just pointing out that there could be a lot of work! If you have any other questions, I'm happy to answer them.

"All models are wrong, but some are useful." - George Box

From my experience, it generally appears to be that it's always possible to have more information from which to source; however, barriers such as cost and time often prohibit you from being able to do so.

I think there will always be a better statistical model out there; I don't see anything wrong with updating your model over time as you come across more information or more effective techniques to do the analysis you would want.

I think these two sources should interest you. (1) (2)

Also Nate Silver's Signal and the Noise might be worth a read. It's not a technical manual, but it will give you some things to think about regarding the use of models in a wide variety of fields.

Hope that helps!

> I'm trying to create a forecasting model to predict ...was hoping someone could point me in the right direction with documentation, vignetts, papers that could help me figure this out, or ideas on how you would approach this problem.


Well you are in luck, there is an entire book that answers what you are trying to accomplish, and spells out the steps how to do it.


https://www.amazon.ca/Introductory-Time-Paul-S-P-Cowpertwait/dp/0387886974/ref=sr_1_1?keywords=introductory+time+series+r&qid=1574873740&sr=8-1

It may also be available for free through other means.

The specific chapters you want will probably be two and three (correlation, and forecasting strategies), though the rest of the book will likely be beneficial as well.

> I have some but I am not super well versed in the matter. (with regards to R and programming skills)

If that’s the case, it might be difficult to implement what is in the book. It assumes you know R at probably around a low/mid level. But if you power through you should be able to get some good out of it.

Also be aware that according to some of the reviews. the source code and data does not appear to be online any more. That makes direct copy and pasting the examples difficult, but if you are applying it to your own data set only, that might be less of an issue.

Its a pretty good book (its the one I read on forecasting originally), and it will get you going in the might direction but its not perfect and you might be able to find a more current one with available sample data. Read the reviews.

Get started and let me know how it goes :)

EDIT

All of the data, as well as functions and R code are indeed available online. Just had to look for it:

http://www.maths.adelaide.edu.au/emac2009/

Edit 2:

Here is are a few newer books as well. I skimmed it quickly and it may or may not be better:

http://db.ucsd.edu/static/TimeSeries.pdf

Again, you are looking at autocorrelation and forecasting. This one may lean a bit more on the math side.

/EDIT

I TFed an intro undergrad course that used Alan Agresti's Statistical Methods for the Social Sciences. I didn't read much of it, but the students seemed to like it. He also has another book that's probably also pretty good. The intro course for non-stats students at my graduate school is Applied Statistics for the Behavioral Sciences, which might also be worth a look. If those are too technical or hands-on, then the "for Dummies" book might also be a good choice - it's in very plain language and tries to keep things relevant to real-life examples.

Many of the bigger-picture "whys" become more apparent when you have a solid grounding in probability theory and the theory behind statistical inference, though. Some of them don't have very satisfying answers, either (Q: Why p = 0.05? A: Convention). In my opinion, the more you understand statistics, the more you realize it's less about finding exact answers than it is about quantifying imprecision. That can be hard for a layperson to wrap their head around!

It sounds like you want some kind of regression, especially to answer 2. In a GLM, you are not claiming that the data by itself has a Normal/Poisson/Negative Binomial/Binomial distribution, only that it has such a distribution when conditioned on a number of factors.

In a nutshell: you model the mean of the distribution as a linear combination of the inputs. Then you can read the weighting factors on each input to learn about the relationship.

In other words, it doesn't need to be that your data is Poisson or NB in order to do a Poisson or NB regression. It only has to be that the error, that is, the difference between the expected based on the mean function and the actual, follows such a distribution. In fact, there may be some simple transformations (like taking the log of the outcome) that lets you use a standard linear model, where you can reasonably assume that the error is Normal, even if the outcome is anything but.

If your variance is not dependent on any of your inputs, that's a great sign, since heteroskedasticity is a great annoyance when trying to do regressions.

If you have time, the modern classic in this area is http://www.amazon.com/Analysis-Regression-Multilevel-Hierarchical-Models/dp/052168689X. It starts with a pretty gentle introduction to regression and works its way into the cutting edge by the end.

Since you are still in college, why not take a statistics class? Perhaps it can count as an elective for your major. You might also want to consider a statistics minor if you really enjoy it. If these are not options, then how about asking the professor if you can sit in on the lectures?

It sounds like you will be able to grasp programming in R, may I suggest trying out SAS? This book by Ron Cody is a good introduction to statistics with SAS programming examples. It does not emphasize theory though. For theory, I would recommend Casella & Berger, many consider this book to be a foundation for statisticians and is usually taught at a grad level.

Good luck!

There is a bunch of engineering stats books out there. The one we teach out of at my uni is the one by Devore. I think it does a good job of teaching what it does. I know Ross has an engineering stats book out there, and so does Montgomery, and they are both people who have written good books in the past. The one by Ross seems to have some good topics in it from reading the table of contents.


Also, you probably want to pick up a regression book. I like the one by Kutner et al., but it is ungodly pricey. This one has a free pdf. I don't like a lot about it, but the first few chapters of every regression book are pretty much the same.

If you want to go deep into statistical theory, there is Casella and Berger as well.


For programs, I know MATLAB has a stats package that should be sufficient for the time being. If you want to go further in stats, you might want to consider R because it will have vastly more stats functions.

> the first half of my degree was heavy on theoretical statistics,

Really? Wow, I'm impressed. Actual coverage of even basic theoretical stats is extremely rare in psych programs. Usually it's a bunch of pronouncements from on high, stated without proof, along with lists of commandments to follow (many of dubious value) and a collection of bogus rules of thumb.

What book(s) did you use? Wasserman? Casella and Berger? Cox and Hinkley? or (since you say it was heavy on theory) something more theoretical than standard theory texts?

I'd note that reaction times (conditionally on the IVs) are unlikely to be close to normal (they'll be right skew), and likely heteroskedastic. I'd be inclined toward generalized linear models (perhaps a gamma model -probably with log-lnk if you have any continuous covariates- would suit reaction times?). And as COOLSerdash mentions, you may want a random effect on subject, which would then imply GLMMs

This book is a amazing:
Discovering Statistics Using R
by Andy Field

If you are doing self-study, it is easy to lose momentum. This book is hilarious, personal, and transcends the textbook genre.

Amazon Link

For gaining statistical intuition, I have yet to find a book as good as Statistics - Freedman

I will second this. I used this book for my year of undergrad foundations of probably and stats.
I also really like Casella and Berger's 'Statistical Inference.'

https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126

This is a classic. I took a grad level course with this textbook and every problem is nasty. But yes, it is really a classic.

Also, I just begun Data Analysis Using Regression and Multilevel/Hierarchical Models by Andrew Gelman and Jennifer Hill. Love his interpretation of linear regression. Linear regression might sound like basics, but it lays the foundation work for everything else and from time to time I feel compelled to review it. This book gave me a new way to look at a familiar topic.

If you are familiar with any statistical programming language/packages, I would highly suggest you implement the learnings from any books you have.