Stat 280 Lab #5: Understanding Regression Output

Preliminaries

Data – As usual, the data set is located at: http://www.stat.rice.edu/~hflores/stat280_gmdavis/

The data is the same used for lab 4: stat280L4.xls.

Purpose

Our goal for this lab will be to help you understand what you have learned about simple linear regression in class and apply it to data.  Specifically, I want you to know how to interpret the output that you get out of most Regression/ANOVA software.  Please realize that regression is one of the most important and easy to use (and abuse) statistical tools in existence, so this material is extremely important to understand.

Analysis

1. Select 2 variables from the dataset from our previous lab, "Vice City" (the dataset referred to above), with the exception of City and Gambling.  As you will recall, City is merely a "counter" for our cities and Gambling is a binary variable.  

2. Our goal is to fit a regression line to these two variables.  In other words, we are going to try to apply a linear model (y = a + b*x, where a,b are predicted constants, x is the independent (predictor) variable and y is the dependent (response) variable) to x to get y.  Since we have a several observations of (x,y) [the variables you just chose], the question is, what would be the best choices for a and b?  THE FIRST question you should ask yourself is, "Well, if I'm going to fit a line to this data then these points should have some sort of linear association...so do they?"  Generate a scatter plot and compute a correlation coefficient (a scatter plot is often sufficient) to answer this question.  If you notice a linear association (either positive or negative), move on to (3).  If not, try a different combination of variables.  Could you fit a regression line to variables that don't show a linear association?  Explain.

3. Fit the regression line (i.e. let the computer make the necessary computations to tell you what the best choices (in the least-squares sense) for a & b are). 

To do this: 
•	click on Tools->Data Analysis (make sure the "analysis tool-pak" is installed in the Add-ins), and select Regression.
•	Select your independent (x) variable column and your dependent (y) variable column (Click and drag).  Usually, you want the variable placement to make sense.  For example, if we were dealing with continuous variables like age and weight, it would make sense to regress weight on age (y = weight, x = age), but it does not make sense (at least to me) how age could be dependent on weight (although it could be argued that weight might be a reasonable predictor for age).  In the mathematical sense, however, it doesn't matter which one you choose as the predictor and response if linear association is present.
•	Make sure to "check" all options under Residuals and Normal Probability Plots.

This should cause another spreadsheet to be created containing the results.

4. Now comes the hard part.  In your report, try to figure out what it all means.  Use your book and notes to help you.  Specifically, I need you to interpret all of the following output values:

Regression Statistics
•	r^2 (excellent discussion in your book)

ANOVA (ignore if not discussed in class, otherwise interpret all values)

Un-named table containing coefficients
•	coefficients
•	standard error (how does this play a role in computing confidence intervals?)
•	p-value 
•	confidence intervals (they may be reproduced twice for some reason in your output)

Residual Output
•	predicted Y
•	Residuals
•	Standardized Residuals (what they mean, not how to calculate them)

Probability Output (ignore)

Plots
You should get 3 plots.  Interpret

In your interpretation, tell me what the general value means, and then what the actual value means.  For example, "r^2 in general means {blah blah}, and in this case it is approximately 0.897, which implies that {blah blah}".  Make sure to ask your TA for clarification on any of these values.

5 As part of your final report, you should report your results in the following order:

Fit of {variable A} and {variable B}
•	coefficients, standard error, p-values, and confidence intervals (the actual values of the least-squares fit), including your interpretation
•	the value of r^2, and what it implies about the linear fit
•	information on residuals (including interpretations of residual and normal quantile plots) and what they imply about the assumptions behind the fit
•	any other relevant information you wish to include