Guide to Linear Regression (2015) | Hacker News Reader

Guide to Linear Regression (2015)

alexhwoods.com

84 pointsalexhwoods28 comments

Any summary of linear regression ought to at least point out that the "cost function", which depending on the circumstances is often viewed as modeling the variance in the data, is typically limited to one data coordinate. That is, the separations to the fit line to be minimized go entirely along the y-axis; which has the effect of assuming that there's perfect knowledge of the x value. And while that can be the case for some sampling protocols, it is also often not the case. So please consider including something like:

If there is uncertainty in both the x and y coordinates then one needs to pursue alternate approaches which admit variation in both, one of the most popular being "Deming regression"[1]

[1] https://en.wikipedia.org/wiki/Deming_regression

Can you give examples? It seems to me that in my cases, I never use X='temperature' as a predictor, but rather X='temperature reported by thermometer', and this is of course always known with 100% accuracy. In a predictive context, that's all we need. Are you referring to a social science context?

Here's an easy one to conceptualize: A good friend of mine is a state hydrologist. He crawls over streams with a ladder-like platform and at increments lowers a depth gauge to eventually estimate the stream's volume flow rate. For a number of reasons (including wild weather conditions) the cross stream increment is known only to an inch or so, that's the uncertainty in X, the depth gauge reading has another uncertainty in Y. These are applied to a function where the slope, via Deming regression, gives him an estimate of the flow rate. By the bye, most lab thermometers i've dealt with have an accuracy of about 0.1°C

Looks closer to support vector machine and its regression capabilities.

Although, linear regression, logistic regression, svm regression, are all almost the same thing, the loss function is different, all can be trained with gradient descent.

I believe Frank Harrell's Regression Modeling Strategies is one of the best references for performing regression analysis.

website: http://biostat.mc.vanderbilt.edu/wiki/Main/RmS Amazon book: http://amzn.to/1UdPWOv

Yep, there is really no other book like Harrell. There's a 2015 edition with some good improvements too. Definitely not an introduction, though – it assumes you already know how regression works and preferably have already used it quite a bit in practice. For an introduction, I might recommend Downey's Think Stats [1] for a gentle introduction, Kutner's Applied Linear Statistical Models for a more traditional, mathy introduction or Gelman and Hill's Data Analysis Using Regression and Multilevel/Hierarchical [2] for something in-depth but still very approachable.

[1] http://greenteapress.com/thinkstats2/

[2] http://www.stat.columbia.edu/~gelman/arm/

great points. it's not an intro to regression book. although Harrell's examples are very well laid out and easy to follow.

Yikes, no discussion of how one might interpret this model, what its assumptions are, and how to assess validity or fit? Look elsewhere--this is not the guide to linear regression anyone should be looking for.

I came here to say that, but in a slightly different way:

True:

> no discussion of how one might interpret this model, what its assumptions are, and how to assess validity or fit?

Quoting OP from the comments in his blog:

> Hey Victor, thanks for reading! You’re absolutely right, I should have taken some time to talk about the assumptions of linear regression. I tried to mention them in passing, but I’ll try to be more explicit next time. I should have brought up the pearson correlation coefficient. I tried to write this from a more machine learning point of view, not as much statistics.

Either way it was a decently written post and would read more in the future (though I would like to see more statistics side of things rather than just me thinking Alex here is waving his hands saying 'you see nothing!' when we look at the mathematical assumptions / 'pure' statistics).

In other thoughts:

I'm currently in the process of writing a blog series on the analytics of Firearm projectiles (What ammo is most accurate from a Ruger American 308 Win.). Seeing other statistical posts like this help me figure out how to make mine better. So thanks Alex!

would you mind sharing the blog post when you're finished? sounds very interesting!

Of course.

My biggest inspiration (other than taking a perfect shot at a whitetail deer and him just dropping dead) was a good PDF about shotGroups over on the CRAN[1]. It is a little dry and statistically heavy (meaning you need to have some statistical background to make sense of it), but well enough detailed for the info that triggered me to pursue this pet project.

I would love to write it at a level where if a person with no statistical background would be able to read through it and be able to understand it "enough". (Which requires a lot of rewriting on my part to make sure I'm also not boring a person with statistics background to death too explaining what Standard Deviation is :P)

[1] https://cran.r-project.org/web/packages/shotGroups/vignettes...

We wrote a couple docs about regression. I'd be curious to get folks' feedback.

Guide itself: http://docs.statwing.com/user-friendly-guide-to-regression/

Perhaps more interestingly to a lot of folks in this crowd, a guide to interpreting residuals: http://docs.statwing.com/interpreting-residual-plots-to-impr...

One valid critique is that the approach we describe is not super rigorous, there's more "add a variable and see if it sticks" and "explore a bunch of bivariate relationships to decide what to include" then you'd want to do if you were publishing a paper on your findings. To our minds, though, this approach is more realistic for the use cases we're more concerned about, like analyzing an ad hoc survey. You can always consider your results to be exploratory and then validate later. (Note that these docs occasionally refer to Statwing, our product, but really could be used for any tool).

Criticism is welcome.

Just as two quick comments, confidence intervals != prediction intervals. One gives a range around a true mean value, the other gives a range around where the variable may actually fall (prediction intervals are bigger).

You may also want to mention adjusted R^2 as a measure of quality of the model. I've never heard of AICR before, the standard metrics for quality of the model are AIC/BIC (and sometimes Mallus Cp).

Edit: fixed sloppy wording about confidence interval.

The prediction interval thing was sloppy, good catch.

The r^2 and AICR comments reflect the fact that these docs were built around Statwing's regression capabilities, which default to m-estimation instead of OLS. There's no adjusted r^2 for that, which I agree is a better measure when available, and it uses AICR, where the R stands for "robust". But still a good catch, since I didn't caveat ahead of time that we were only talking about robust methods (and we don't note it in the docs).

Much appreciated.

OLS is an M-estimator.

AIC depends on residual variation just like R^2, so I guess some people might be inclined to call it AIC_R to make that clear. Also, it's Mallows' C_p :-)

Thanks for the clarification about Mallow's C_p, I always get the name messed up for some reason.

Hey, just FYI, your posts are getting killed for some weird reason. See:

http://imgur.com/FmBkqgo

Thank you, I'm confused as to why this is happening.

Do you have a doc on whether the data are supposed to be Normally Distributed, and if they are then how to measure said Normality? I'd really appreciate such a doc if you do.

Sorry, we don't. My understanding, though, is that the typical statistical measures of normality (e.g., Kolmogorov–Smirnov) aren't really that effective, and the best way to assess normality is a visual inspection of a Q-Q plot. I don't have a specific source for this assertion, it's my memory from having researched this stuff a few years ago.

Multivariate statistics is something I really wish was covered better.

In my work as an engineer I use multiple linear regression from time to time. The best method I've found is forward selection stepwise approach. I've never really seen a simple explanation of how it works (it was introduced to me by a statistician at my work) but it is useful when you have many variables and you want to see how significant a single variable is relative to overall regression. The impression I get is pure stats people really dislike stepwise modelling.

More recently I've been looking into something called PCA (Principle Component Analysis) and PLS (projection to latent structure) after I came across it in a PHD thesis. I've yet to find a decent simple explanation about how it works though. Unfortunately my work no longer employees a statistician.

IMHO, a very confusing thing about Linear Regression is that there's widespread disagreement about whether the data are supposed to be Normally Distributed or not, and on how to measure said Normality.

You're right, there is lots of confusion in the topic and not all texts on regression get it right. Theoretically, linear regression depends only on a normally distributed outcome variable, the predictors don't have to be normally distributed. Practically speaking even that requirement can mostly be ignored: OLS is fairly insensitive to non-normal outcomes. Violations of the assumptions of regression usually don't affect the estimates much, they mainly affect the uncertainty around those parameter estimates -- the standard errors. This too is easily solved: use bootstrapping to calculate the standard errors.

In short: you can't just throw linear regression at anything and expect to get reasonable results, but it's pretty damn close.

Even that normality is not required; see, for instance, https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem.

Good catch. Outcome (or error, which boils down to the same thing) can't just be anything either, though, e.g. a common pattern is that the variance in an outcome increases when the predictor increases (e.g. a machine behaves more erratically when it spins faster) and that can mess with estimates.

Also some features of regression and hypothesis testing are actually fairly robust to non-normality, particularly with large sample sizes.

Look at normal qq-plot to measure normality

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