# Transition from using statistical software to understanding mathematical equations?

### Context:

I'm a Psychology PhD student. As with many psychology PhD students, I know how to perform various statistical analyses using statistical software, up to techniques such as PCA, classification trees, and cluster analysis. But it's not really satisfying because though I can explain why I did an analysis and what the indicators mean, I can't explain how the technique works.

The real problem is that mastering statistical software is easy, but it is limited. To learn new techniques in articles requires that I understand how to read mathematical equations. At present I couldn't calculate eigenvalues or K-means. Equations are like a foreign language to me.

### Question:

• Is there a comprehensive guide that helps with understanding equations in journal articles?

### Edit:

I thought the question would be more self explanatory: above a certain complexity, statistical notation becomes gibberish for me; let's say I would like to code my own functions in R or C++ to understand a technique but there's a barrier. I can't transform an equation into a program. And really: I don't know the situation in US doctoral schools, but in mine (France), the only courses I can follow is about some 16th century litterary movement...

• @Coronier Sorry, I doubt there is a comprehensive guide for understanding psychology articles that use statistical modeling. But the required background should all be at the level of a master's degree in statistics. If your program will pay for it, consider getting an M.A. in stats. The next best option for your purposes might be to re-take the stats department's version of multivariate stats -- usually they provide notes with the mathematical background for PCA, clustering, trees, etc. You will need a background in linear algebra and basic mathematical statistics regardless. – lockedoff Mar 21 '11 at 18:56
• Please ask more specific questions. – user88 Mar 21 '11 at 19:05
• I'm a Psychology PhD student as well, and I made the choice to take a significant amount of math in my undergraduate years because there were so many Psychology PhDs who have no idea how a PCA (for example) was computed. The very first thing you need to do is work your way through any decent linear algebra textbook. What's a decent linear algebra textbook? Gilbert Strang's is the bomb, and he has video lectures of his linear algebra course on MIT's website to boot. You can even get them on iTunes. – Phillip Cloud Mar 21 '11 at 19:07
• The question is so broad that it will not get a satisfying answer in a few paragraphs. Statistics are like questions: it gets easier if you break it down to several manageable components. – Fr. Mar 21 '11 at 22:11
• I can only agree with the above comments. Either you'll have to focus on a particular issue, or you just need to work through some textbooks or online handouts first. A decent textbook that covers basic concepts for multivariate statistics with illustrations is Mathematical Tools for Applied Multivariate Analysis, by Carroll and Green (AP, 1997, Rev. Ed.). Another one is Applied Multivariate Statistics and Mathematical Modeling, by Tinsley and Brown (AP, 2000). – chl Mar 21 '11 at 22:23

### Overview:

• My impression is that your experience is common to a lot of students in the social sciences.
• The starting point is a motivation to learn.
• You can go down either self-taught or formal instruction routes.

### Formal instruction:

There are many options in this regard. You might consider a masters in statistics or just taking a few subjects in a statistics department. However, you'd probably want to check that you have the necessary mathematical background. Depending on the course, you may find that you need to revisit pre-calculus mathematics, and perhaps some material such as calculus and linear algebra before tackling university-level mathematically rigorous statistics subjects.

### Self-taught

Alternatively, you could go down the self-taught route. There are heaps of good resources on the internet. In particular, reading and doing exercises in mathematics text books is important, but probably not sufficient. It's important to listen to instructors talking about mathematics and watch them solve problems.

It's also important to think about your mathematical goals and the mathematical prerequisites required to achieve those goals. If equations are like a foreign language to you, then you may find that you need to first study elementary mathematics.

I've prepared a few resources aimed at assisting people who are making the transition from using statistical software to understanding the underlying mathematics.

• Thanks, the ressources you provide are great. Btw, your blog is totally absorbing (I'm an I/OP student and useR, it's like a revelation for me). – Coronier Mar 22 '11 at 9:57
• @Coronier It's great to meet another person combining R with I/O Psych. – Jeromy Anglim Mar 22 '11 at 10:46

I get the impression that you think that you can get insight into a statistical equation by programming it into either R or C++; you can't. To understand a statistical equation, find an "undergraduate" textbook with lots of homework problems at the end of each chapter that contains the equation, and then do the homework at the end of the chapter containing the equation.

For example, to understand PCA you do need a good understanding of linear algebra and in particular singular value decomposition. While learning quantum computing through Michael Nielsen's book, it became apparent to me that I needed to review linear algebra. I came across Gilbert Strang's videos, they were extremely helpful in establishing a foundational understanding of concepts. However, the nuance of the material did not get through until I found a linear algebra book containing alot of homework problems, and then I needed to do them.

• @ schenectady while i sympathesise with your viewpoint, for me at least, the R code provides a bridge that i can use to greater my understanding of the equations and maths concerned. That being said, i wholeheartedly agree with the need for problems, statistics and maths in general is something that can only be learned by doing. – richiemorrisroe Mar 22 '11 at 19:13

I understand your difficulty as I have a similar problem when I try to do something new in statistics (I'm also a grad student, but in a different field). I have found examining the R code quite useful to get an idea how something is calculated. For example, I have been recently learning how to use kmeans clustering and have many basic questions, both conceptual and how it is implemented. Using an R installation (I recommend R Studio, http://www.rstudio.org/, but any installation works), just type kmeans in to the command line. Here is an example of part of the output:

x <- as.matrix(x)
m <- nrow(x)
if (missing(centers))
stop("'centers' must be a number or a matrix")
nmeth <- switch(match.arg(algorithm), Hartigan-Wong = 1,
Lloyd = 2, Forgy = 2, MacQueen = 3)
if (length(centers) == 1L) {
if (centers == 1)
nmeth <- 3
k <- centers
if (nstart == 1)
centers <- x[sample.int(m, k), , drop = FALSE]
if (nstart >= 2 || any(duplicated(centers))) {
cn <- unique(x)
mm <- nrow(cn)
if (mm < k)
stop("more cluster centers than distinct data points.")
centers <- cn[sample.int(mm, k), , drop = FALSE]
}
}


I'm not sure how practical it is to examine the source every time, but it really helps me get an idea what is going on, assuming you have some familiarity with the syntax.

A previous question I asked on stackoverflow pointed me in this direction, but also helpfully told me that the comments about the code are sometimes included here.

More generally, the Journal of Statistical Software illustrates this link between theory and implementation, but it is frequently about advanced topics (that I personally have difficulty understanding), but is useful as an example.