Other Forms of Regression . In 2-variable regression analysis, you use a single independent
variable (X) to estimate the dependent variable (Y), and the relationship is assumed to form a
straight line. This is the most common form of regression analysis used in contract pricing.
However, when you need more than one independent variable to estimate cost or price, you
should consider multiple regression (or multivariate linear regression). When you expect that a
trend line will be a curve instead of a straight line, you should consider curvilinear regression.
A detailed presentation on how to use multiple regression or curvilinear regression is beyond the
scope of this text. However, you should have a general understanding of when and how these
techniques can be applied to contract pricing. When you identify a situation that seems to call for
the use of one of these techniques, consult an expert for the actual analysis. You can obtain more
details on the actual use of these techniques from advanced forecasting texts.
Multiple Regression Situation . Multiple regression analysis assumes that the change in Y can be
better explained by using more than one independent variable. For example, suppose you want to
determine the relationship between main-frame computer hours, field-audit hours expended in
audit analysis, and the cost reduction recommendations sustained during contract negotiations.
Computer
Hours
Field Audit
Hours
Sustained
Reduction
1.4 45 $290,000
1.1 37 $240,000
1.4 44 $270,000
1.1 45 $250,000
1.3 40 $260,000
1.5 46 $280,000
1.5 47 $300,000
It is beyond the purpose of this text to demonstrate how a multivariate equation would be
developed using this data. However, we will describe the elements of the multivariate equation
and the results of a regression analysis.
Three-Variable Linear Equation . Multiple regression can involve any number of independent
variables. To solve the audit example above, we would use a three-variable linear equation — two
independent variables and one dependent variable.
Y C = A + B1 X 1 + B2 X 2
Where:
Yc = The calculated or estimated value for the dependent
(response) variable
A = The Y intercept, the value of Y when X 1 = 0 and X 2 = 0
X 2 = The first independent (explanatory) variable
B2 = The slope of the line related to the change in X 1 , the value by
which Y changes when X 1 changes by one.
X 2 = The second independent (explanatory) variable
B2 = The slope of the line related to the change in X 2 , the value by
which Y changes when X 2 changes by one.
Results of Audit Data Three-Variable Linear Regression AnalysisI. Using the above data on audit
analysis and negotiated reductions, an analyst identified the following three variables:
X 2 = Computer Hours
X 2 = Field Audit Hours
Y = Cost Reductions Sustained
The results of analysts analysis are shown in the following table:
Regression Results
Predictor Variable Equation r 2
Computer Hours Y = A + BX 1 .82
Field Audit Hours Y = A + B X 2 .60
Comp Hrs and Field Audit Hrs Y = A + B1 X 1 + B2 X 2 .88
You can see from the r 2 values in the above table that computer hours explains more of the
variation in cost reduction recommendations sustained than is explained by field audit hours. If
you had to select one independent variable, you would likely select Computer Hours. However,
the combination of the two independent variables in multiple regression explains more of the
variation in cost reduction recommendations sustained than the use of computer hours alone. The
combination produces a stronger estimating tool.
Curvilinear Regression Analysis . In some cases, the relationship between the independent
variable(s) may not be linear. Instead, a graph of the relationship on ordinary graph paper would
depict a curve. You cannot directly fit a line to a curve using regression analysis. However, if
you can identify a quantitative function that transforms a graph of the data to a linear
relationship, you can then use regression analysis to calculate a line of best fit for the
transformed data.
Common
Transformation
Functions
Examples
Reciprocal
Square Root
Log-Log logX
Power X 2
For example, improvement curve analysis (presented later in this text) uses a special form of
curvilinear regression. While it can be used in price analysis and material cost analysis, the
primary use of the improvement curve is to estimate labor hours. The curve assumes that less
cost is required to produce each unit as the total units produced increases. In other words, the
firm becomes more efficient as the total units produced increases.
There are many improvement curve formulations but one of the most frequently used is:
Y = AX B
Where:
Y = Unit cost (in hours or dollars of the Xth unit)
X = Unit number
A = Theoretical cost of the first unit
B = Constant value related to the rate of efficiency improvement
Obviously, this equation does not describe a straight line. However, using the logarithmic values
of X and Y (log-log transformation), we can transform this curvilinear relationship into a linear
relationship for regression analysis. The result will be an equation in the form:
logY = logA + BlogX
Where:
logY = The logarithmic value of Y
logA = The logarithmic value of A
logX = The logarithmic value of X
We can then use the linear equation to estimate the logarithmic value of Y, and from that Y.
5.7 – Identifying Issues And Concerns
Questions to Consider in AnalysisI. As you perform price/cost analysis, consider the issues and
concerns identified in this section, whenever you use regression analysis.
• Does the r 2 value indicate a strong relationship between the independent variable and
the dependent variable?
The value of r 2 indicates the percentage of variation in the dependent variable that is explained
by the independent variable. Obviously, you would prefer an r 2 of .96 over an r 2 of .10, but there
is no magic cutoff for r 2 that indicates that an equation is or is not acceptable for estimating
purposes. However, as the r 2 becomes smaller, you should consider your reliance on any
prediction accordingly.
• Does the T-test for significance indicate that the relationship is statistically significant?
Remember that with a small data set, you can get a relatively high r 2 when there is no statistical
significance in the relationship. The T-test provides a baseline to determine the significance of
the relationship.
• Have you considered the prediction interval as well as the point estimate?
Many estimators believe that the point estimate produced by the regression equation is the only
estimate with which they need to be concerned. The point estimate is only the most likely
estimate. It is part of a range of reasonable estimates represented by the prediction interval. The
prediction interval is particularly useful in examining risk related to the estimate. A wide interval
represents more risk than a narrow interval. This can be quite valuable in making decisions such
as contract type selection. The prediction interval can also be useful in establishing positions for
negotiation. The point estimate could be your objective, the lower limit of the interval your
minimum position, and the upper limit your maximum position.
• Are you within the relevant range of data?
The size of the prediction interval increases as the distance from increases. You should put the
greatest reliance on forecasts made within the relevant range of existing data. For example, 12 is
within the relevant range when you know the value of Y for several values of X around 12 (e.g.,
10, 11, 14, and 19).
• Are time series forecasts reasonable given other available information?
Time series forecasts are all outside the relevant range of known data. The further you estimate
into the future, the greater the risk. It is easy to extend a line several years into the future, but
remember that conditions change. For example, the low inflation rates of the 1960s did not
predict the hyper-inflation of the 1970s. Similarly, inflation rates of the 1970s did not predict
inflation rates of the 1980s and 90s.
• Is there a run of points in the data?
A run consisting of a long series of points which are all above or all below the regression line
may occur when historical data are arranged chronologically or in order of increasing values of
the independent variable. The existence of such runs may be a symptom of one or more of the
following problems: