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Stat 13, Lab 11-12, Correlation and Regression Analysis
Part I: Before Class
Objective: This lab will give you practice exploring the relationship between two variables by using correlation, linear regression and graphical techniques.
Before starting this lab, you should…
- Be familiar with these terms:
- response “y” (or dependent) and explanatory “x”(or independent) variables;
- slope and intercept in a linear regression equation;
- positive and negative correlations.
Part II: In-class Activity
Suppose you were a Broadway producer. You would want your show to make as much money as possible, and one way of deciding whether or not to invest your time and money into a particular show would be to examine past shows to see how they did. We'll examine a simple question: how does the size of the theater affect box office receipts?
To begin, download the data:
use http://www.stat.ucla.edu/~dinov/courses_students.dir/STAT13_Fall01/STAT13_Fall01/data.dir/broadway
Some potentially useful Stata Commands:
use filename -> loads a Stata-format file dataset from the Web. If filename is specified without the Stata extension, “.dta” is assumed. edit -> opens Stata’s spreadsheet. label variable varname “ label ” -> enables you to attach an extra piece of information (up to 80 characters) about a variable. sort x -> arranges the observations of the current data in ascending order of the values of the variables. There is no limit to the number of the variables in the data. Missing values are interpreted as being larger than any other number and are thus placed last. graph y x -> for a scatterplot of y vs.x. graph y x, xlabel ylabel -> for scatterplots with improved axes (numerical lists). regress y x -> estimates a model from the list of variables using least-squares regression.
quietly regress y x -> suppresses the regression output for the duration of the command. predict newvar -> calculates the predicted values of a variable in a linear regression for each observation. The new values are stored under the name newvar. This command must follow a regress command. predict newvar , residuals -> calculates the residuals from a regression and places them in the variable named newvar. This command must follow a regress command. graph y newvar x , connect (.s) symbol (oi) -> displays a linear regression graph between two variables with fitted values ( newvar ) connected by a line. corr x y z w -> displays the correlation or covariance matrix for two or more continuous variables, or if they are not specified, for all variables in the data. Observations are excluded from the calculation when values are missing.
Does the size of a theater "predict" the average box-office receipts?
Your TA might ask the class the following questions, so you should jot down your observations and thoughts for class discussion.
- Make a scatterplot of the receipts against the capacity graph receipts capacity
If you want to reveal the name of the show for any unusual observations, issue this command: graph receipts capacity, symbol([show])
Which show had the highest box office receipts? Which show appeared in a theater with the most seats?
- Describe the trend: how are receipts and capacity related? Would you say this is a linear relationship?
- We can quantify the linear relationship with a least squares regression. (This works whether or not the relationship is really linear. If it is not linear, then our least squares regression will be a very poor description -- but we can still compute it.) Note that Stata gives us a lot more information than we are ready for right now. But you'll return to this later in your studies. Type:
regress receipt capacity
- Look in the column headed by "Coef." (Coefficient) to find the estimated intercept and slope. Write the equation of the line here:
Interpret the slope.
- To graph the line on top of the scatterplot, type:
Part III: Take-home Problem
You've probably been told, since the first day you complained about school, that education will help you get a better job. Certainly many jobs require a level of education, but does all that schoolwork pay off? Load this data set into Stata:
use http://www.stat.ucla.edu/projects/datasets/twins
This is data from a study of twins. You can learn details about the data, including how and why they were collected, at http://www.stat.ucla.edu/projects/datasets/twins-explanation.html.
Two variables of interest are hrwageh and hrwagel. These are the hourly wage of twin "1" and "2". (The twins were arbitrarily numbered.) You might want to focus your investigation on the difference in their hourly wage. To create this variable, type gen diffwage = hrwageh - hrwagel
Two other interesting variables are educh , the self-reported education level (in years) of the twin who reported earning hrwageh , and educl : the self-reported education level of the twin who reported earning hrwagel. For explanations of the other variables, see http://www.stat.ucla.edu/projects/datasets/twins-explanation.html
Are education and income related? Investigate this question with these data.
Report on your findings. Your report should include answers to these questions:
Do you expect the correlation between the twins' incomes to be positive or negative? High (close to positive or negative 1) or low (close to 0)? Check.
Find the correlation matrix for these variables: hrwageh, hrwagel, educl, educh, diffwage, diffeduc. What's the correlation between hrwageh and hrwagel? Interpret. Why does the correlation between hrwagel and diffeduc have a different sign than the correlation between hrwageh and diffeduc?
What's the typical difference in hourly wage between twins? Is it what you expected?
Describe the distribution of the difference in hourly wage. Are there any unusual features?
Make a scatterplot of the difference in income against the education level of either one of the twins. Interpret. Does it matter which twin's education level you chose?
Perform a regression of difference in income against a twin's education level.
What does the estimated slope say about the effect of education on income? (You might want to superimpose the regression line on the graph for a clearer picture.) Does your conclusion depend on which twin's education you used to predict the difference in income?
Create a new variable, diffeduc, that is the difference in education levels between twins. Perform a regression and use it to answer this question: is there evidence that the twin with more education makes more money?
Examine the residuals from this last regression. For what types of twins did the model have the largest error (that is, the greatest difference between the predicted value and the observed value)? Do you see any possible outliers?
This Lab was originally created by Prof. R. Gould A and Prof. V. Lew
