# How would you explain generalized linear models to people with no statistical background?

I always have a hard time explaining statistical techniques to audience with no statistical background. If I wanted to explain what GLM is to such audience (without throwing out statistical jargon), what would be the best or most effective way?

I usually explain GLM with three parts -- (1) the random component which is response variable, (2) the systematic component which is linear predictors, and (3) the link function which is the "key" to connecting (1) and (2). Then I would give an example of linear or logistic regression and explain how the link function is selected based on the response variable. Hence it acts as the key connecting two components.

• What kind of background do the audience have? Explaining GLM to a mathematician or a biologist is very different. – user10525 May 3 '12 at 16:17
• There will be few mathematicians with no statistical background, @Procrastinator. But your point is a good one: having a clearer idea of the intended audience will help keep the replies consistent and focused. Would you mind editing the question to expand on this, Ken? – whuber May 3 '12 at 17:44
• I see your point, @Procrastinator but I was hoping to get an easy-to-understand answer for everyone (mathematicians and/or biologists), in general because if I don't have math or biology background (which is the case), I wouldn't know how to explain GLM to them with respect to their background anyway. – Ken May 3 '12 at 18:12
• I think it's important to keep in mind that you can get a bachelor's, a master's, or even a doctorate in biology without ever having taken a statistics class, even at many tier one universities. My degree in biochemistry required two semesters of introductory calculus and one semester of differential equations. The substance of these classes is quickly forgotten because many students will never use these skills again! So I really do think it's necessary to dumb down the explanation for typical non-statisticians. – Alexander May 3 '12 at 19:11
• A comment to add to the answers below; if you can get across fitting a line (i.e. the link function and linear predictors) then the connection to efficient inverse-variance weighting is not so hard to communicate; we simply want to upweight precise contributions and downweight the rest. This lets you avoid saying anything too technical about the randomness of the outcomes. NB GLMs were devised as (just) the models where IWLS can be used to give the MLE, so the way of thinking about them described above captures most of why they're actually useful. – guest May 4 '12 at 6:38

If the audience really has no statistical background, I think I would try to simplify the explanation quite a bit more. First, I would draw a coordinate plane on the board with a line on it, like so:

Everyone at your talk will be familiar with the equation for a simple line, $\ y = mx + b$, because that's something that is learned in grade school. So I would display that alongside the drawing. However, I would write it backwards, like so:

$\ mx + b = y$

I would say that this equation is an example of a simple linear regression. I would then explain how you (or a computer) could fit such an equation to a scatter plot of data points, like the one shown in this image:

I would say that here, we are using the age of the organism that we are studying to predict how big it is, and that the resultant linear regression equation that we get (shown on the image) can be used to predict how big an organism is if we know its age.

Returning to our general equation $\ mx + b = y$, I would say that x's are variables that can predict the y's, so we call them predictors. The y's are commonly called responses.

Then I would explain again that this was an example of a simple linear regression equation, and that there are actually more complicated varieties. For example, in a variety called logistic regression, the y's are only allowed to be 1's or 0's. One might want to use this type of model if you are trying to predict a "yes" or "no" answer, like whether or not someone has a disease. Another special variety is something called Poisson regression, which is used to analyse "count" or "event" data (I wouldn't delve further into this unless really necessary).

I would then explain that linear regression, logistic regression, and Poisson regression are really all special examples of a more general method, something called a "generalized linear model". The great thing about "generalized linear models" is that they allow us to use "response" data that can take any value (like how big an organism is in linear regression), take only 1's or 0's (like whether or not someone has a disease in logistic regression), or take discrete counts (like number of events in Poisson regression).

I would then say that in these types of equations, the x's (predictors) are connected to the y's (responses) via something that statisticians call a "link function". We use these "link functions" in the instances in which the x's are not related to the y's in a linear manner.

Anyway, those are my two cents on the issue! Maybe my proposed explanation sounds a bit hokey and dumb, but if the purpose of this exercise is just to get the "gist" across to the audience, perhaps an explanation like this isn't too bad. I think it's important that the concept be explained in an intuitive way and that you avoid throwing around words like "random component", "systematic component", "link function", "deterministic", "logit function", etc. If you're talking to people who truly have no statistical background, like a typical biologist or physician, their eyes are just going to glaze over at hearing those words. They don't know what a probability distribution is, they've never heard of a link function, and they don't know what a "logit" function is, etc.

In your explanation to a non-statistical audience, I would also focus on when to use what variety of model. I might talk about how many predictors you are allowed to include on the left hand side of the equation (I've heard rules of thumb like no more than your sample size divided by ten). It would also be nice to include an example spread sheet with data and explain to the audience how to use a statistical software package to generate a model. I would then go through the output of that model step by step and try to explain what all the different letters and numbers mean. Biologists are clueless about this stuff and are more interested in learning what test to use when rather than actually gaining an understanding of the math behind the GUI of SPSS!

I would appreciate any comments or suggestions regarding my proposed explanation, particularly if anyone notes errors or thinks of a better way to explain it!

• Not everyone is familiar with the equation for a line; not even all graduate students are, nor all people with PhDs. – Peter Flom - Reinstate Monica May 3 '12 at 20:20
• I mean, I'm sure a graduate student exists out there in the world who doesn't know the equation for a line, but presumably an audience to which you would want to explain generalized linear models would at least have half a clue about high school level algebra! :-o – Alexander May 3 '12 at 22:35
• I agree with you Alexander and your approach seems very natural to me. I wouldn't focus on the "g" of the glm too much (or too early) and would also not go into distinctions on random vs fixed. Of course it depends on the amount of time you have to explain all this. – Dominic Comtois May 4 '12 at 2:41
• You could also explain the $Y = \alpha X + \beta$ form as I have: "For every 1 this(X) goes up by, this(Y) goes up by $\alpha$". – Eoin Jul 7 '14 at 14:47

I wouldn't call the response a random component. It is a combination of a deterministic and a random component.

I think I would describe generalized linear models this way. We have a response variable and a set of related variables that can aid in predicting the response. However the response and the predictors are not linearly related. The link function provides a transformation of the response so that the transformed response is linearly related to the predictors. For example in logistic regression the predictor could be continuous variables that can take on values over the entire real line. But the response is a probability (the probability of a successful outcome in a clinical trial for example). So the response is constrained to fall between 0 and 1. The link function in logistic regression is called the logit function. It equals $\log(p/(1-p))$. You can see that the logit function transforms a variable constrained to $[0,1]$ to a variable that can take values over the entire real line. In this case the link function makes the response compatible with the predictor variables and hence it is possible to make it a linear function of the predictors plus a random component.

• I wonder about this use of "response." Our intended audience would likely understand that to mean the observed response: yes or no, 0 or 1, etc. In logistic regression we model something unobserved (and never directly observable); namely, the hypothetical chance of the response. The "link" merely is a matter of expressing those chances as log odds rather than as probabilities. Logistic regression assumes the log odds vary linearly with the IVs. (My use of "model," "assume," and "hypothetical," rather than "is" and "predict," indicates a different cognitive and ontological viewpoint, too.) – whuber May 3 '12 at 17:41
• Good point whuber. – Michael R. Chernick May 13 '12 at 21:54

I would explain it saying that sometimes I need things predicted. For instance, the price of a house given some information about it. Say, its size, location, how old the construction is, etc. I want to factor that into a model that takes into account the influence of these factors to predict the price.

Now taking a sub-example, lets say, I consider only the size of the house. That would imply that nothing else affects the price. It could be a case where I am comparing houses which are in the same locality, were constructed around the same time etc. Or it could be that I don't want to complicate matters for myself and hence want the real life to conform to how far I can think. Moving on, I make a model where I have a list of sizes and corresponding prices of similar properties (say, from sales that have been happening recently... but that would have serious bias from houses that are not for sale and hence affect price of houses that are. but lets ignore that).

Now I see that a 100 sq feet house costs \$1m(get over yourself, this is a simplified example). So, naturally you would expect a 200sq feet house to cost double. And that is what we would call a "linear pattern". Of course when we collect the data and plot size vs price, we see that it is not exactly double. But there is definitely an increasing trend.

So I try to quantify the trend. How much increase for every increased sq foot? That is linear regression.

INSERT terminology map and continue with statistical concepts. One way of explaining random and systematic component could be that whatever you forgot to model, or couldn't possibly gauge, is random. Whatever you could is systematic. (For instance, say it is 2008 and you want to sell a house.)

Assumptions that underlie this model are that the scatterplot should look like a rod. Which is that Both X and Y are "Normal". and all have similar variance.

If that is not the case, enter GLM. and now explain link function n all that.

It is simplified, but it should work as an introduction.

You can put in history of GLMs and factorial models. Where Fisher required things to start varying together and this framework was suitable for that kind of complexity.

Hope this helps...

• We appreciate your efforts but there's no need to post your material until you have actually finished writing it. In its present form, the way it decays into sparse cryptic notes at the end will disappoint readers. – whuber Feb 14 '14 at 23:25