Interpretation of Regression Coefficient

[Trevor Andrew]

I have created a new independent variable based on the sum of the old variable. The old variable is a fraction that ranges from zero to one (e.g. 0; 0.1; 0.5; and so on). To create the new variable I sum up the old variable for each subgroup based on the number of the individuals in each subgroup. So if the first subgroup has 3 individuals I sum up the old variable for these three individuals, if the second subgroup has one individual I use the old variable for this individual, and so on. After I summed up the old variable I collapse the data set to be at the subgroup level which is also shown below. Below is an example at the individual level for 5 subgroups which contain 1 to 3 individuals and the collapsed data set.

My question is, if I run a regression where the new variable is the independent variable, how do I Interpret the coefficient of the new variable? Is it a unit increase? Is it a percent increase? I think it is something in between but I don't know what.

old data set				new data set
variable_old	subgroup	variable_new		subgroup	variable_new
0.31	1	0.80		1	0.80
0.28	1	0.80		2	0.35
0.20	1	0.80		3	0.18
0.17	2	0.35		4	1.21
0.18	2	0.35		5	1.13
0.18	3	0.18
0.46	4	1.21
0.75	4	1.21
0.65	5	1.13
0.15	5	1.13
0.33	5	1.13

[Bruce Weaver]

Trevor, your post reminded me of this line from the FAQ:

Asking about your real problem, not something else, may seem too obvious to mention, but do check http://xyproblem.info/.

In other words, what is it that you really want to do? You've said you want to estimate a regression model with variable_new as the explanatory variable. But you've not said what the Y-variable is. What is the unit of analysis? Is Y also a subgroup-level variable, or are there distinct values for the members of each subgroup? If you provide more information about what exactly you want to do, someone may be able to help. ;-)

HTH.

[Phil Bromiley]

The simple answer is that the parameter in a regression is the amount the predicted value increases for a one unit increase in the right-hand side variable.

What that means will depend on the scaling of the dependent variable and the scaling of the right-hand side variable. If we had a y in percentiles coded as 1 to 100, and coefficient of 3 would mean a one unit in the right-hand side variable increases the dependent variable by 3%. If it were coded .01 to 1, then a one unit increase in the right-hand side variable increases the dependent variable by 3 but this means 300%.

[Trevor Andrew]

Bruce Weaver you're right. I didn't want the question to be too long and that's why I left some information. The new variable is merged to individuals. So the data set is at the individual level and individuals in the same subgroup will have the same value of the new variable. I use educational attainment (measured in years) and probability of graduation (measured as a 0/1 dummy) as my dependent variables. I think I don't understand what a unit increase in my new variable means since it is a mix of fraction and integer. And I don't quite follow what Phil Bromiley says. Why is it 3% in the first case and 300 in the second? And how can you calculated it in general for any scale of the dependent variable?

[Bruce Weaver]

Good morning Trevor Andrew. Your variable_new is a level 2 explanatory variable, in that case, whereas subjects and scores on the outcome variable are at level 1. You need to use a multilevel model. For the graduation outcome, I suggest -melogit-. For years of education, it's less clear to me whether you should use -mixed- or one of the mixed effects count regression models (-mepoisson- or -menbreg-). You can find examples for all of these types of models in the Mixed Effects Reference Manual. There are also many tutorials and class notes online. HTH.