Multiple Regression

[rhenne]

y(hat)=-10536+53.8x1+2.84x2, where x1=size of home in square feet and x2=lot size in square feet. For fixed lot size, how much is the house selling price predicted to increase for each square foot increase in home size

 

[JeffM]

y(hat)=-10536+53.8x1+2.84x2, where x1=size of home in square feet and x2=lot size in square feet. For fixed lot size, how much is the house selling price predicted to increase for each square foot increase in home size

Suppose I told you the size of a home was A sq. ft. and the lot size was B sq. ft. Could you tell me the predicted price of the house?

OK. Now what is the predicted price of a home of (A + 1) sq. ft. on a lot of B sq. ft.?

What is the difference between the predicted prices of the two houses?

 

[rhenne]

Hmmm, I am finding your example a bit confusing. But I am realizing (and perhaps this is what you are illustrating?) that I don't need to solve for anything.
But that the predicted selling price of the house has a mean of 53.8(in dollars) for every 1square foot increase in home size. Similarly, there is a 2.84 selling price (in mean dollars) for every 1 square foot increase in lot size. Is this what you are illustrating?

The next portion of the problem...(which I think I figured out) is worded in a way that is really difficult to interpret:
"For fixed home size, how much would lot size need to increase to have the same impact as 1 square foot increase in home size?

53.8=x/2.84
53.8/2.84=18.94

If you have any ideas of how to interpret the wording here that would be helpful for me to make sure I really understand the concept.

Thanks!

 

[Deleted member 4993]

y(hat)=-10536+53.8x1+2.84x2, where x1=size of home in square feet and x2=lot size in square feet. For fixed lot size, how much is the house selling price predicted to increase for each square foot increase in home size

If you have done calculus, then you are asked to find:

\(\displaystyle \frac{\partial {y}}{\partial x_1}\)

 

[JeffM]

What this problem is trying to do is to help you understand what the individual coefficients in a multiple regression mean.

Hmmm, I am finding your example a bit confusing. But I am realizing (and perhaps this is what you are illustrating?) that I don't need to solve for anything.
But that the predicted selling price of the house has a mean of 53.8(in dollars) for every 1square foot increase in home size. Exactly. Similarly, there is a 2.84 selling price (in mean dollars) for every 1 square foot increase in lot size. Is this what you are illustrating?

Yes. I was suggesting to you to USE your equation. So, if the size of the home is A and the lot size is B, the predicted price = -10,536 + 53.8A + 2.84B. If the size of the house is (A + 1) and the lot size is still B, the predicted price is -10,536 + 53.8(A + 1) + 2.84B. When you subtract the two prices, everything cancels except 53.8. But you got the concept without doing all that work. As S. Khan says, the answer is obvious if you remember your calculus, but I was guessing that you have not studied calculus yet. In any case, with LINEAR regression, the coefficient of a one independent variable tells you what the change in the number of units of the dependent variable (y hat) will be if you change the one independent variable by 1 unit and leave the other independent variables unchanged.

The next portion of the problem...(which I think I figured out) is worded in a way that is really difficult to interpret:
"For fixed home size, how much would lot size need to increase to have the same impact as 1 square foot increase in home size?

I agree that the question is not ideally worded. Perhaps it would be clearer if it just said: How much would lot size need to increase to have the same effect on predicted price as an increase of 1 sq foot in house size? Sadly, problems do not always come to us in the form easiest to understand.

53.8=x/2.84 I believe (based on your next line) that you understand conceptually what you are doing, but two comments. (I) You have defined house size and lot size as x[sub:1mak6c5j]1[/sub:1mak6c5j] and x[sub:1mak6c5j]2[/sub:1mak6c5j] respectively so your use of x is confusing. Let's use z. (II) More importantly, 53.8 = z / 2.84 is wrong. If you solve for z, you get z = 53.8 * 2.84.

53.8/2.84=18.94 There you go. z = (53.8 / 2.84).

If you have any ideas of how to interpret the wording here that would be helpful for me to make sure I really understand the concept. Just remember the concept that, in LINEAR regression, the coefficient of an independent variable shows the change in the dependent variable if you increase that independent variable by 1 unit and leave all other independent variables alone. Hope this helps.

Thanks! You're welcome.