More variables! For this one we are going to add all of the variables in their correct form in the data frame as qualitative or quantitative.
If you starting with this post, lets get the data loaded up, fix the column names, convert factors to a type of factor, and create a column for out non imperial friends so they can understand the mpg thing.
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In the last regression post we added more variables, but not all of them, I was holding back and not telling you why. So far we have been dealing with quantitative variables which ask how many or how much, the next is qualitative or categorical. Categorical usually asks which, and while it may be a number it would not make sense to perform math against it.
One item that might me a tiny bit helpful is to realize that as many moving parts as there are in regression it all boils down to a pretty simple formula for calculating a prediction. More of this will be covered piece by piece in the coming posts, but i wanted one post that will go through the formulas up to now.
The next few posts is just adding some more explanatory variables to see if we can get a better model from predicting mpg. We are going to keep it simple today and focus on just quantitative variables not categorical(qualitative), if that does not make any sense to you it will soon.
In the REAL world you would never predict a vehicle mpg by weight alone, there are dozens if not hundred of other variables to consider. Lucky for us the mtcars dataset only has 11 variables to consider. The grand finale of this linear regression will be a real dataset we can play with from the EPA with thousands of rows and dozens of columns. š
Lets get to it!
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Lets try and bring simple linear regression together before i move on to multiple. We started with a question, can we predict miles per gallon using weight of a vehicle? We looked at a scatter plot and saw a bit of linearity. We created a model and looked at the residuals and determined they are for he most part demonstrating constant variance and we looked at a histogram of the residuals and it is demonstrating enough normal distribution to move forward. I know, i’m not sounding very convincing am i? Its a small dataset and its for learning, having some values that are out in left and right field but are actually useful so i can demonstrate some other points later in this post.
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We need to talk about p-value. The calculation for p-value is a hot nightmare, not going to bother with it right now, if you need to know more about it you can find online calculators, but rarely an actual formula. Even the sites that will spill the beans on all the other formulas will resort to a t distribution calculator for p-value. Though i may fall back in a future post and spend some time on t-distribution, we shall see.
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My last linear regression post i mentioned that most of the numbers come form the residual errors, thats not entirely true. You have a basic understanding of lm you learned that R-square is the number to look at, that is based on residual error. You are also told to examine the p-value for each coefficient and for the entire model. P-value is a little bit harder to calculate, go search and find out for yourself. But in lieu of that i am going to provide the actual calculation for everything you may have seen reference in an lm.
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All the numbers, so many numbers, do we really need all of them…? If you want to make a an informed decision yes, you do. On the bright side they are not very hard to interpret, almost all of the numbers are related to just one number… The Error, more specifically the Residual Error. I’m going to blow your hair back again, its not an error, its nothing like and error, it should have never been called an error, its a difference. It is the difference between the line we ran through the middle of the scatter plot and the data points. Each point has a difference between the line and where the dot falls. Take a look at the visual below, the green line represents the distance between the line and the actual data point, thats our residual error. Its not hard to see that the larger the distance between the line and the data points the worse our model will perform. Not to mention the closer the data is to the the better.
I think itās difficult for a professor or teacher to know at exactly what point should linear regression be taught in a curriculum, it seems like it turns up everywhere calculus, algebra stats, modeling. It should be in all of them, but then the next question is do you need to know algebra, matrix algebra and linear algebra before knowing how to do a linear regression? I donāt know to be honest. Having worked with SQL for most of my adult life I have had to know and use all three and did not pay much attention to it or realize until I started formally beefing up my academics.
Regardless, the one thing I have heard from a few stats instructors is ādonāt worry about how its done or how it works, the software will take care of it for youā, to be fair, these were not stats professors at the local beauty college, these were ivy league educated (I checked) professors and teachers saying this. Which, my problem is if I donāt know how it works I will probably not truly understand it, ever. Depending on what you are doing a trivial knowledge may be sufficient, but what if itās not? If I am in an interview can I use the words āthe software will do it for meā as the answer to a hard question?
What?
In the next few posts, I will do my best to define Linear Regression in R using lm. Continue reading
