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?


 A: 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.  
A: 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".
