The latent indicator variable 1

Well it does seem a while since we last posted and there is still much work to do as we continue from the previous post. The situation in which you either have chlorine or hydrogen at C4 of the phenyl should be easy to spot using any of a number of substituent parameters and comparing average pIC50 values for the two groups of compounds will give you a good idea of whether or not substitution with chloro is good for activity. If substitution with chloro at C4 leads to a consistent increase in potency, you’ll get model that is both predictive and that can be validated. So exactly what is your point, we hear you cry.

OK let’s be a bit more specific. We’ll use the Wikipedia as our source of Hammett sigma constants. The Hammett sigma constant for meta-chloro is +0.37 and (by definition) that for hydrogen is zero. If chloro substitution leads to a significant increase in potency you should get a reasonable model by fitting pIC50 to sigma. It will satisfy validation criteria and Senior Pharma Fellow (SPF) will be able to rattle off an impressive array of quality control metrics in his next presentation. Aren’t we clever! Surely it’s time to use the model to do some predicting.

Our chemists want to know what happens if we introduce methoxy or fluoro at C4. Actually they don’t like Senior Pharma Fellow (SPF) any more than we do but there is a directive from the Project Management Politburo that these models are to be used even if they are not believed. Furthermore you need to run the model so that you can tick the relevant boxes on the Authorisation For Synthesis form that the tiresome Black-Belted Half-Wits have set up for the gathering of Base-line Productivity Indicators. At least we know that we won’t be extrapolating because the Hammett sigma values for meta-methoxy and meta-fluoro are +0.11 and +0.34 respectively so both lie within the space spanned by the training set. We’d predict that replacing chloro at C4 with fluoro would to lead to a small drop in potency because the relevant Hammett sigma values are so similar. We’d be particularly confident in our predictions for the methoxy-substituted analogs because this represents interpolation to a greater extent than if we were doing predictions for the compounds with which the model was built.

Now for the sake of argument, let’s suppose we’d decided to use the Hammett constants for these substituents at the para position. The value for chlorine is now +0.23 and that for hydrogen is still zero (by definition) as before so the quality of the model. However fluoro (sigma-para = +0.06) looks much more like hydrogen than chloro while methoxy (sigma-para = -0.27) now lies well outside the space spanned by the training set. Needless to say this is a very different picture to what we saw using sigma-meta values.

What does this all mean? This is obviously a toy example that we’ve created to illustrate a point. However it is clear that if we’re building models using pIC50s for compounds that are either unsubstituted or have chloro at C4 then sigma-para will work just as well as sigma-meta. The sigma values function as indicator variables and any parameter which has different values for chloro and hydrogen substituents will do the job just as well. The problem is that for these models having anything other than hydrogen or chloro at C4 represents an extrapolation while the continuous nature of sigma constants suggests that we might be interpolating. Real models are typically a lot more complex than this toy example and it is often not clear when linear combinations of continuous variables are actually functioning as indicator variables. We’ll pick up in the next post since it is getting late and there is cider to be drunk. It should be fun and hopefully we will not encounter a latent indicator variable (LIV).

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