2.1 - Examples of Linear Combinations

The Women's Health Survey data contains observations for the following variables:

• \(X_{1}\) calcium (mg)
• \(X_{2}\) iron (mg)
• \(X_{3}\) protein(g)
• \(X_{4}\) vitamin A(μg)
• \(X_{5}\) vitamin C(mg)

In addition to addressing questions about the individual nutritional component, we may wish to address questions about certain combinations of these components. For instance, we might want to ask what is the total intake of vitamins A and C (in mg). We note that in this case, Vitamin A is measured in micrograms while Vitamin C is measured in milligrams. There are a thousand micrograms per milligram so the total intake of the two vitamins, Y, can be expressed as the following:

\(Y = 0.001 X _ { 4 } + X _ { 5 }\)

In this case, our coefficients \(c_{1}\) , \(c_{2}\) and \(c_{3}\) are all equal to 0 since the variables \(X_{1}\), \(X_{2}\) and \(X_{3}\) do not appear in this expression. In addition, \(c_{4}\) is equal to 0.001 since each microgram of vitamin A is equal to 0.001 milligrams of vitamin A. In summary, we have

\(c _ { 1 } = c _ { 2 } = c _ { 3 } = 0 , c _ { 4 } = 0.001 , c _ { 5 } = 1\)

Another example where we might be interested in linear combinations is in the Monthly Employment Data. Here we have observations on 6 variables:

• \(X_{1}\) Number people laid off or fired
• \(X_{2}\) Number of people resigning
• \(X_{3}\) Number of people retiring
• \(X_{4}\) Number of jobs created
• \(X_{5}\) Number of people hired
• \(X_{6}\) Number of people entering the workforce

Net employment decrease:

In looking at the net job increase, which is equal to the number of jobs created, minus the number of jobs lost.

\(Y = X _ { 4 } - X _ { 1 } - X _ { 2 } - X _ { 3 }\)

In this case, we have the number of jobs created, (\(X_{4}\)), minus the number of people laid off or fired, (\(X_{1}\)), minus the number of people resigning, (\(X_{2}\)), minus the number of people retired, (\(X_{3}\)). These are all of the people that have left their jobs for whatever reason.

In this case

\(c _ { 1 } = c _ { 2 } = c _ { 3 } = - 1 \text { and } c _ { 4 } = 1\)

Because variables 5 and 6 are not included in this expression,

\(\mathrm {c } _ { 5 } = \mathrm { c } _ { 6 } = 0\)

Net employment increase:

In a similar fashion, net employment increase is equal to the number of people hired, (\(X_{5}\)), minus the number of people laid off or fired, (\(X_{1}\)), minus the number of people resigning, (\(X_{2}\)), minus the number of people retired, (\(X_{3}\)).

\(Y = X _ { 5 } - X _ { 1 } - X _ { 2 } - X _ { 3 }\)

In this case

\(c _ { 1 } = c _ { 2 } = c _ { 3 } = - 1 , c _ { 4 } = c _ { 6 } = 0 , \text { and } c _ { 5 } = 1\)

Net unemployment increase:

Net unemployment increase is going to be equal to the number of people laid off or fired, (\(X_{1}\)), plus the number of people resigning, (\(X_{2}\)), plus the number of people entering the workforce, (\(X_{6}\)), minus the number of people hired, (\(X_{5}\)).

\(Y = X _ { 1 } + X _ { 2 } + X _ { 6 } - X _ { 5 }\)

Unfilled jobs:

Finally, if we wanted to ask about the number of jobs that went unfilled, this is simply equal to the number of jobs created, (\(X_{4}\)), minus the number of people hired, (\(X_{5}\)).

\(Y = X _ { 4 } - X _ { 5 }\)

In other applications, of course, other linear combinations would be of interest.