# How to start building a regression model when the most strongly associated predictor is binary

I have data set containing 365 observation of three variables namely pm, temp and rain. Now I want to check be behavior of pm in response to changes in other two variables. My variables are:

• pm10 = Response (dependent)
• temp = predictor (independent)
• rain = predictor(independent)

The following is the correlation matrix for my data:

> cor(air.pollution)
pm        temp       rainy
pm     1.00000000 -0.03745229 -0.15264258
temp  -0.03745229  1.00000000  0.04406743
rainy -0.15264258  0.04406743  1.00000000


The problem is when I was studying the construction of regression models, it was written that the additive method is to start with the variable that is most highly related to response variable. In my data set rain is highly correlated with pm (as compared to temp), but same time it is a dummy variable (rain=1, no rain=0), so I have now clue from where should I start. I have attached two images with the question: The first is a scatterplot of data, and the second image is a scatterplot of pm10 vs. rain, I am also unable to interpret scatterplot of pm10 vs. rain. Can some one help me how to start?

• This is a perfectly viable question, IMO, even if it proceeds from a misunderstanding. – gung - Reinstate Monica Nov 23 '16 at 15:26
• One thing to remember with regression is that the $y_i$ dependent variable is assumed to be a random variate, while the predictors $x_{i1}, x_{i2},\ldots,x_{ip}$ are assumed to be fixed experimentally-controlled variates. (thus, be careful when you turn something on its head). There doesn't appear to be any strong positive or negative correlation in your matrix, since none of the off-diagonal $|r_{jk}|>0.8$? – user32398 Nov 23 '16 at 15:32

Many people believe that you should use some strategy like starting with the most highly associated variable, and then adding additional variables in turn until one is not significant. However, there is no logic that compels this approach. Moreover, this is a kind of 'greedy' variable selection / search strategy (cf., my answer here: Algorithms for automatic model selection). You do not have to do this, and really, you shouldn't. If you want to know the relationship between pm, and temp and rain, just fit a multiple regression model with all three variables. You will still need to assess the model to determine if it is reasonable and the assumptions are met, but that's it. If you want to test some a-priori hypothesis, you can do so with the model. If you want to assess the model's out of sample predictive accuracy, you can do that with cross-validation.

You needn't really worry about multicollinearity either. The correlation between temp and rain is listed as 0.044 in your correlation matrix. That is a very low correlation and shouldn't cause any problems.

• thank you very much for your kind suggestions . i am new to this site dont know how to use it, can you please provide some extra suggestions or studying materials – Syed Asif Ali Shah Nov 23 '16 at 17:38
• @SyedAsifAliShah, other than that English doesn't seem to be your native language, I don't see any problems w/ how you're using the site. Regarding study materials, you might look at this or this, or just browse our threads with the references tag. – gung - Reinstate Monica Nov 23 '16 at 17:50
• should i try linear model or GLM for my data ?? – Syed Asif Ali Shah Nov 24 '16 at 13:43
• @SyedAsifAliShah, presumably a linear model is fine for your data. – gung - Reinstate Monica Nov 24 '16 at 13:43
• bro i need your help – Syed Asif Ali Shah Jan 12 '17 at 16:47

While this doesn't directly address your already gathered data set, another thing you could try the next time you are gathering data like this is to avoid recording "rain" as a binary. Your data would probably be more informative if you had instead measured rain rate (cm/hour), which would give you a variable distributed continuously (up to your precision of measurement) from 0...max_rainfall.

This would let you correlate not just "is it raining" to the other variables, but also "how much is it raining".

• hi bro i did the same according to your suggestion i gathered full data of rain and construct model – Syed Asif Ali Shah Jan 12 '17 at 16:47
• can i ask for your email please??? i just want to ask a few question – Syed Asif Ali Shah Jan 12 '17 at 17:05
• If you have additional questions about setting up your model, perhaps a new StackExchange question would be the way to go. That way you can get feedback from more folks, many of whom are more expert than I. – JKreft Jan 16 '17 at 18:56
• stats.stackexchange.com/questions/255959/… – Syed Asif Ali Shah Jan 16 '17 at 19:34