Linear regression models


Notes on linear regression analysis (pdf file)

Introduction to linear regression analysis

Mathematics of simple regression

Regression examples

·         Baseball batting averages

·         Beer sales vs. price, part 1: descriptive analysis

·         Beer sales vs. price, part 2: fitting a simple model

·         Beer sales vs. price, part 3: transformations of variables

·         Beer sales vs. price, part 4: additional predictors

·         NC natural gas consumption vs. temperature

What to look for in regression output

What’s a good value for R-squared?
What's the bottom line? How to compare models
Testing the assumptions of linear regression
Additional notes on regression analysis
Stepwise and all-possible-regressions
Excel file with simple regression formulas

Excel file with regression formulas in matrix form

If you are a PC Excel user, you must check this out:
RegressIt: free Excel add-in for linear regression and multivariate data analysis


Regression example, part 1:  descriptive analysis

Any regression analysis (or any sort of statistical analysis, for that matter) ought to begin with a careful look at the raw material:  the data.  Where did it come from, how was it measured, is it clean or dirty, how many observations are available, what are the units, what are typical magnitudes and ranges of the values, and very importantly, what do the variables look like?  Much of your brain is devoted to the processing of visual information, and failure to engage that part of your brain is like shooting in the dark. Visual analysis helps you to identify systematic patterns as well as unusual events and data errors.

The objective of this analysis will be to explain and predict how the quantity of weekly sales of a popular brand of beer depends on its price at a small chain of supermarkets.  The data file contains 52 weeks of average-price and total-sales records for three different carton sizes:  12-packs, 18-packs, and 30-packs.  (This is real data, apart from some very minor adjustments for the 30-packs.)  One of the first things to consider in assembling a data set for regression analysis is the choice of units (i.e., scaling) for the variables.  At the end of the day you will be looking at error measures that are expressed in the units of the dependent variable, and the model coefficients will be measured in units of predicted change in the dependent variable per unit of change in the independent variable.  Ideally these numbers should scaled in a way that makes them easy to read and easy to interpret and compare.  In this analysis, the price and sales variables have already been converted to a per-case (i.e., per-24-can) basis, so that relative sales volumes for different carton sizes are directly comparable and so that regression coefficients are directly comparable for models fitted to data for different carton sizes.  The first few rows of the data set (in an Excel file) look like this:

The column headings were chosen to be suitable as descriptive variable names for the analysis.   The value of 19.98 for PRICE_12PK in week 1 means that 24 cans of beer cost $19.98 when purchased in 12 packs that week (i.e., the price of a single 12-pack was $9.99), and the value of 223.5 for CASES_12PK means that 447 12-packs were sold (because a case is two 12-packs).

Let’s begin with a look at the descriptive statistics, which show typical magnitudes and the ranges of the variables:

Here it is seen that sales volume (measured in comparable units of cases) was greater for the smaller carton sizes  (399 cases’ worth of 12-packs vs. 165 for 30-packs, with 18-packs in the middle), while the average price-per-case was significantly smaller for the larger carton sizes ($14.38 per case on average for 30-packs, vs. $19.09 per case for 12-packs, with 18-packs again in the middle).  However, there was considerable variation in prices of each carton size, as shown by the minimum and maximum values.

Because these are time series variables, it is vitally important to look at their time series plots, as shown below.  (Actually, you should look at plots of your variables versus row number even if they are not time series  You never know what you might see.  For non-time series data, you would not want to draw connecting lines between the dots, however.)  What stands out clearly in these plots is that (as beer buyers will attest) the prices of different carton sizes are systematically manipulated from week to week over a wide range, and there are spikes in sales in weeks where there are price cuts.  For example, there was deep cut in the price of 18-packs in weeks 13 and 14, and a corresponding large increase in sales in those two weeks.  In fact, if you look at all the cases-sold plots, you can see that sales volume for every carton size is rather low unless its price is cut in a given week.  (High-volume beer drinkers are very price-sensitive.)  Another thing that stands out is the pattern of price manipulation was not the same for all the carton sizes:  prices of 12-packs were not manipulated very often, whereas prices of 30-packs were manipulated on an almost week-to-week basis in the first half of the year, and prices of 18-packs were more frequently manipulated in the second half of the year.  So, at this point we have a pretty good idea of what the qualitative patterns are in weekly prices and sales.

If our goal is to measure the price-demand relationships by fitting regression models, we are also very interested in the correlations among the variables and in the appearance of their scatterplots.  Here is the correlation matrix, i.e., the table of all pairwise correlations among the variables.  (Remember that the correlation between two variables is a statistic that measures the relative strength of the linear relationship between them, on a scale of -1 to +1.)  What stands out clearly here is that (as we already knew from looking at the time series plots) there are very strong, negative correlations between price and sales for all three carton sizes (greater than 0.8 in magnitude, as it turns out), which are measures of “the price elasticity of demand.”  There are also some weaker positive correlations between the price of one carton size and sales of another--for example, a correlation of 0.521 between price of 18-packs and sales of 30-packs.  These are measures of “cross-price elasticities”, i.e., substitution effects.  Consumers tend to buy fewer 30-packs when the price of 18-packs is reduced, presumably because they buy 18-packs instead.

Last but not least, we should look at the scatterplot matrix of the variables, i.e., the matrix of all 2-way scatterplots.  The scatterplot matrix is the visual counterpart of the correlation matrix, and it should always be studied as a prelude to regression analysis if there are many variables.  (Virtually all commercial regression software offers this feature, although the results vary a lot in terms of graphical quality.  The ones produced by RegressIt, which are shown here, optionally include the regression line, center-of-mass point, correlation, and squared correlation, and each separate chart can be edited.)   The full scatterplot matrix for these variables is a 6x6 array, but we are especially interested in the 3x3 submatrix of scatterplots in which sales volume is plotted vs. price for different combinations of carton sizes:

Each of these plots shows not only the price-demand relationship for sales of one carton size vs. price of another, but it also gives a preview of the results that will be obtained if a simple regression model is fitted.   On the pages that follow, regression models will be fitted to the sales data for 18-packs.  From the scatterplot in the center of the matrix, we already know a lot about the results we will get if we regress sales of 18-packs on price of 18-packs.  Some “red flags” are already waving at this point, though.  The price-demand relationships are quite strong, but the variance of sales is not consistent over the full range of prices in any of these plots.

Go on to next step:  interpreting simple regression output

By the way, all of the output shown above was generated at one time on a single Excel worksheet with a few keystrokes using the Data Analysis procedure in RegressIt, as shown below.  Hopefully your software will make this relatively easy too.