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This has been answered in part in many places but here I am asking again.

Background: I did calculus and vectors and advanced functions in high school. Then I did a bachelor's degree in social work and a master's in social work and now I'm looking to do a PhD. I consider myself statistically literate, more so than average social work students, and can navigate SPSS. But to conduct rigorous research that is firmly grounded in theory in the social sciences and in the math behind statistics I feel a need for a thorough understanding of mathematics so I can say why I actually did the analyses, why I use certain alpha scores, what I did, and what do the results really mean rather than relying solely on convention and audience ignorance.

What I've gathered so far is I should start with topics in linear algebra, and real analysis... and maybe avoid Discovering Statistics (although it is a favorite among many profs and students)... but otherwise I am totally lost.

Ultimately, I want to be able to run and understand multi-group confirmatory factor analysis to look at scalar invariance but later in the future also have the flexibility to do some SEM, IRT, Bayesian statistics, and natural language processing.

So for someone looking to self study with no undergraduate math experience....what is my trajectory (e.g. MOOCs, books, get another bachelor's degree)?

BTW I'm going to focus on learning R - for flexibility in the future.

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    $\begingroup$ Since you mentioned linear algebra, MIT has a free video series from Gilbert Strang that covers the basics pretty well. $\endgroup$ – Frans Rodenburg Oct 26 '18 at 1:54
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    $\begingroup$ This is a very broad and basic question. The exact path will highly depend on the particular strengths of your PhD program and/or the availability of courses at universities that you can visit. I'd say that a typical study plan formula would be along the lines: 1) Follow some basic courses in statistics (some doctorate programs include a part of studying/lectures where you can keep developing your knowledge in specific fields and your research tools) 2) Then follow some more advanced courses (or just read a couple of good books/key articles) depending on what topic is your interest/focus/need $\endgroup$ – Martijn Weterings Oct 26 '18 at 7:45
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    $\begingroup$ I have removed the self-study tag. While this question is about the topic of self-study, the self-study tag is reserved for specific exercises/questions about solving statistical problems/issues. $\endgroup$ – Martijn Weterings Oct 26 '18 at 7:49
  • $\begingroup$ There are several links like the following. Some are math/stats focus. staff.science.uu.nl/~gadda001/goodtheorist/… $\endgroup$ – EngrStudent Oct 26 '18 at 12:52
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    $\begingroup$ I think the title would benefit from becoming more specific, but the information in the body of the question is quite detailed about what skills are desired by the end, and on that basis I do not think it is too broad or opinion-based to ask how these might best be developed. $\endgroup$ – Silverfish Oct 26 '18 at 13:19
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I see various areas you should have a look into:

  • Basics of probability

Here you should understand the most common continuous probability distributions (e.g. normal distribution, t-distribution) and the most common discrete distributions (e.g. binomial distribution and geometric distribution). You should also understand how they are related to each other, e.g. a t-distribution converges to a normal distribution if n goes to infinity. You should also understand concepts like conditional probability and Bayes' theorem and you should have a look into random processes, e.g. random walk.

  • Basics of inferential statistics

You should understand the basics of inferential statistics and statistical testing. In statistical testing p-values and power of tests is important.

  • Linear algebra

Linear algebra is one of the most important mathematical concepts for statistics. Important concepts are e.g. the inverse and the transpose of a matrix. You should also be able to calculate with matrices, e.g. multiplication.

  • Regression and econometrics

There are three different areas of regression analysis: Cross-sectional regressions, panel data and time series analysis. You should go through all of the three areas. Time series analysis might be the most important area of this three areas for practitioners as it is used for forecasting.

  • Machine learning algorithms

After having an overview of the different areas of machine learning you should have look in some of the most common supervised machine learning algorithms (e.g. regression and classification) and the most common unsupervised machine learning algorithms (e.g. clustering, cimensionality reduction and anomaly detection)

  • Coding with statistical software

R and Python are the most widespread languages for statistical computing. If I were you I would choose R as you need less pre-knowledge in object-oriented computing for using it.

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    $\begingroup$ (+1) given the OP's interests (CFA, SEM0 I think your comments on linear algebra are well taken but s/he could go a long way in social science without hitting a time series. $\endgroup$ – mdewey Oct 26 '18 at 13:28
  • $\begingroup$ You are right. Especially in political science and economics time series analysis is very important. Therefore I mentioned it in the bullet point "regression". $\endgroup$ – Ferdi Oct 26 '18 at 13:29
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Since you mentioned Bayesian statistics, let me recommend Data Analysis: A Baysesian Tutorial by Sivia and Skilling to you. I am reading it myself at the moment and find it fantastic.

The book really helps in understanding the big picture of (Bayesian) probability theory. Also it finally links together all the divergent pieces of information that I had accumulated but never really understood in statistics classes. It has just the right amount of mathematical rigor and practical application. I could go on, give it a try!

Here is what the book says about itself:

Statistics lectures have been a source of much bewilderment and frustration for generations of students. This book attempts to remedy the situation by expounding a logical and unified approach to the whole subject of data analysis.

This text is intended as a tutorial guide for senior undergraduates and research students in science and engineering. After explaining the basic principles of Bayesian probability theory, their use is illustrated with a variety of examples ranging from elementary parameter estimation to image processing. Other topics covered include reliability analysis, multivariate optimization, least-squares and maximum likelihood, error-propagation, hypothesis testing, maximum entropy and experimental design.

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