Math 217 Computer Lab –
Simple Linear Regression
Getting the Needed Files
Double
click on the My Computer icon on the desktop. Then double click on the campus_share on 'curtis' (U:) drive and then the Class_Share
folder. Finally, double click on the Math
folder and then the Math_217 folder. What you see in this folder are
(among others) the Minitab files we will use in today’s lab: AlcoholTobaccoSpending.MPJ, AlcoholMetabolism.MPJ,
and HousePricesAlbuquerque.MPJ.
As a
class, we cannot access these share files (only one person can assess them at a
time). Thus, you each need to copy the three files to your personal account.
You can do this by simply highlighting all the files (click on the first one,
then ctrl-click on the last one—this should highlight them all). Then press
Ctrl-C to copy the files. Now open the My Documents folder on the
desktop (this is the My Documents folder of your personal account). Once
you are in the My Documents folder, hit Ctrl-V to paste the four files
into your account.
Now open
the Minitab software (from the Start menu select Programs>Class
Programs and then Minitab>Minitab15). Then open the first file (HousePricesAlbuquerque.MPJ) in Minitab: go to the File menu and choose Open Project.
Description
of HousePricesAlbuquerque.MPJ
This data file contains information (selling price
and square feet of living space) on 115 homes sold in
Analysis
We already discussed this example “on paper” in the
classroom. Now we’ll investigate it with Minitab. Since we’re interested in the
relationship between selling price and square footage, the first step is to
create a scatterplot (Graph>Scatterplot>Simple). From
this graph, it’s clear that a positive linear relationship exists between these
variables.
There are multiple ways to perform regression
analysis in Minitab. First we’ll use the fitted line plot function. From
the Stat menu select Regression>Fitted Line Plot. Choose
selling price as the response variable and square feet as the predictor
variable. The default is linear regression, which is what we want. Now click on
the Graphs button. Note you can
select a histogram and normal plot of residuals (to check the normality
condition), and you can also select a plot of residuals versus the fitted
values (to check the constant-variance condition). But you cannot get a
prediction interval for selling price based on a specific square footage value.
(Also, this procedure doesn’t give as much regression output in the session
window.)
A more flexible procedure when doing regression is Stat>Regression>Regression. Select
selling price as the response and square feet as the predictor (note you can
choose more than one predictor—this procedure allows you to perform multiple
regression). Click on the Graphs
button and you’ll see all the diagnostic graphs from which you can
choose. Now click on the Options
button. Note you can ask for a prediction interval (or confidence interval)
for a new observation. Suppose we want a prediction interval for selling price
based on a house with 2000 square feet. Enter the value 2000 in the appropriate
box. By default the 95% level is used (but you can change this if desired).
Notice there is much output in the session window
(more than when you used the fitted line plot option). We will discuss this as
a class. At the last of the output, Minitab provides both a 95% confidence
interval for the average selling price of houses with 2000 square feet and a
95% prediction interval for the selling price of one house with 2000 square
feet.
The residual plot shows a slight fanning-out
pattern. This indicates that the constant-variance condition might not be met.
One possible remedy is to transform the response variable—often times a
natural log transformation works to stabilize the variability. Create a new
variable that is the natural log of selling price (Calc>Calculator). Then rerun the regression using log price as
the response variable. What do you think—did this transformation help? (Note
that you can easily transform predictions of log price back to price.)
Description
of AlcoholTobaccoSpending.MPJ
This
data file comes from the 1981 records of the Department of Employment (British
official statistics). It shows the average weekly household spending, in
British pounds, on tobacco products and alcoholic beverages for each of the 11
regions of
Analysis
Suppose
we’re interested in the relationship between alcohol and tobacco spending
(specifically, if tobacco spending helps predict alcohol spending). Create a scatterplot of these two variables. What is the most
obvious feature of this graph? Which region is the outlier? (Recall you can use
the Editor>Brush function to
identify the row of specific points. Furthermore, you can label all the points in a scatterplot based
on a third, categorical variable—this will be helpful for your homework.
Simply right-click on the scatterplot and choose Add>Data Labels. Then click on the
“Use labels from column” circle and select the Region variable.)
We can
now investigate the impact and influence
of this outlier on the regression analysis. Highlight the second and third
columns of the worksheet and copy them. Then paste them into the next two open
columns. Notice that Minitab gives them a default title. Change the titles
appropriately (e.g., “Alcohol Spending No N. Ireland”). Then delete the values
from
Now from
the Graph menu choose Scatterplot>With Regression. Enter two Y and two
X variables (your original data and the data with
Description
of AlcoholMetabolism.MPJ
Case study from The
Statistical Sleuth, Second Edition, by Ramsey and Schafer, Duxbury
Publishers:
“Women exhibit a lower tolerance for alcohol and
develop alcohol-related liver disease more readily than men. When men and women
of the same size and drinking history consume equal amounts of alcohol, the
women on average carry a higher concentration of alcohol in their bloodstream.
According to a team of Italian researchers, this occurs because
alcohol-degrading enzymes in the stomach (where alcohol is partially
metabolized before it enters the bloodstream and is eventually metabolized by
the liver) are more active in men than in women. The researchers studied the
extent to which the activity of the enzyme explained the first-pass alcohol
metabolism and the extent to which it explained the differences in first-pass
metabolism between women and men. This data file includes their data (M. Frezza, et al. (1990), “High Blood Alcohol Levels in
Women,” New England Journal of Medicine,
322, pp. 95-99).”
“The subjects were 18 women and 14 men, all volunteers
living in 
mol/min/g of tissue)
in mucus samples taken from the stomach linings of the subjects.”
Analysis
Does gastric AD activity help predict the first-pass
metabolism? And is this prediction different for males and females? First
create a scatterplot of first-pass metabolism versus
gastric AD activity. There seems to be a positive, linear relationship between
these two variables. Now go back to the scatterplot
option and choose With Groups (add Sex as the categorical variable). Do the
patterns seem to be different for males and females? Go back to the scatterplot option and choose With Regression and Groups. What do you notice in this graph? Just
based on this graph, what can we roughly say about the magnitude of the impact
of gastric AD activity on first-pass metabolism (think about the slopes)?
The scatterplot
option doesn’t allow you to do regression in detail. Suppose you want to analyze the data separately for males
and females. Then from the Data menu select Split Worksheet. In
the dialog box, select the Sex variable as the “By variable”. Minitab
will then create two new worksheets separating the data by sex (note that it
will also keep the original worksheet intact). The highlighted worksheet will
be the active worksheet and it’s the active worksheet that Minitab will work
with. It’s important that you label all
your graphs appropriately (males or females) so you can keep track of your
analysis (Minitab doesn’t do this for you).
Do a detailed regression analysis for both males and
females. What differences or similarities do you notice?