Identifying Rules of Thumb . As you perform your market analysis, you should be on the lookout
for cost estimating rules of thumb that are commonly used in the product marketplace. For
example, when we compare the prices of houses, we typically do so in price per square foot.
Using this rule of thumb, we can compare the cost of different houses or the same house in
different parts of the country. There may be ways to develop more accurate estimates, but this
rule of thumb is widely accepted, relatively easy to calculate, and it provides reasonably accurate
results for many purposes. The same statement can probably be made about most rules of thumb.
You may be able to develop better cost estimating relationships, but given the time available and
the dollars involved, rules of thumb often provide useful tools for contract pricing.
Validate a Rule of Thumb Before Using . Like any CER, a rule of thumb can be based on another
cost, performance characteristic, or physical characteristic of the item being priced. Unlike other
CERs, rules of thumb typically have not been validated for use in specific estimating situations.
Validation has come from acceptable results produced in a variety of situations over a number of
years. Before you use a rule of thumb, consider the 6-step CER development process and ask the
following questions:
• Can the rule of thumb reasonably be used to estimate what you are trying to estimate
(e.g., cost, hours, or price)?
• Are there other rules of thumb that can be used to estimate the same cost, hours, or
price?
• Are the data required to use this rule of thumb readily available?
• Does the rule of thumb provide reasonably accurate results?
• If more than one rule of thumb is available, which one appears to produce the most
accurate estimate?
• Have technical experts or other buyers documented the results obtained from using the
rule of thumb?
Example of Rule of Thumb Validation. You just received two offers for 500 laboratory tables.
Each table is 4' x 6' (24 square feet of surface area), with an oak frame and legs. The work
surface is a unique composite material developed to meet Government requirements. The low
offer is $425; that offer is $175 less than the second low offer and $180 less than the
Government estimate. As a result, you are concerned that the price may be unreasonably low.
You have no acquisition history for this item and there are no similar items on the commercial
market. As a result, you have been looking for a CER that you can use in your pricing decision.
Another buyer, who has acquired similar tables, tells you that he has used a rule of thumb in
pricing similar tables -- $19 per square feet of surface area. You want to know the answers to the
following questions before you use it in making your own pricing decisions.
• Can the rule of thumb reasonably be used to estimate what you are trying to estimate
(e.g., cost dollars, hours, or product price)?
The answer appears to be yes. The buyer who recommended the CER has used it successfully.
Additional information shows that he learned of the CER from the scientists who developed the
table-top material.
• Are there other rules of thumb that can be used to estimate the same cost or price?
You have asked several "experts" and have been unable to identify any other rules of thumb for
estimating the price of these unique tables.
• Are the data required to use this rule of thumb readily available?
Yes, you already know the table surface area.
• Does the rule of thumb provide reasonably accurate results?
You have identified four recent acquisitions of similar tables and recorded the following
information comparing the estimates made using the rule of thumb and the actual prices paid:
Sq Ft Estimate Actual Price Percentage
Difference
15 $285 $310 + 8.8%
18 $342 $335 - 2.0%
32 $608 $580 - 4.6%
This sample size is too small to perform an effective statistical analysis, but you can still
subjectively evaluate rule of thumb estimate accuracy. All estimates are within 8.8 percent of the
actual price. For a rule of thumb, this appears reasonably accurate, especially since our
evaluation did not consider other acquisition situation differences (e.g., the number of tables on
each contract).
• If more than one rule of thumb is available, which one appears to produce the most
accurate estimates?
In this example, there is only one known rule of thumb to consider.
• Have technical experts or other buyers documented the results obtained from using the
rule of thumb?
In this case, the buyer documented every contract file when the rule of thumb was used. Such
documentation is not only valuable in supporting the contracting officer's decision on price
reasonableness; it provides valuable information to any contracting officer considering rule of
thumb use in the future.
Example of Using a Rule of Thumb in Estimate Development . Once you have determined that a
rule of thumb is acceptable for estimate development, you must apply it to the available data.
Using this rule of thumb, your estimate would be $456 (24 x $19). That estimate is about 7.3
percent higher than the low offer. Based on the rule of thumb, the price does not seem
unreasonable.
4.3 - Developing And Using Estimating Factors
Situations for Using Estimating Factors . An estimating rate or factor is a simple ratio, used to
estimate cost or price. The rule of thumb used to develop table price estimates in the previous
section is an example -- $19 per square foot. As the size of the table top increases, the price
estimate increases in direct proportion. Most rules of thumb are simple factors. Many CERs
developed by Government or industry are also simple factors. They are relatively easy to
develop, easy to understand, and in many cases quite accurate.
Development and use of estimating rates and factors involves two important implicit
assumptions.
• There is no significant element of the cost or price being estimated that is not related to
the independent variable (i.e., there is no "fixed cost" that is not associated with the
independent variable).
• The relationship between the independent variable and the cost being estimated is linear.
If you believe that there are significant costs that cannot be explained by the relationship or that
the relationship is not linear, you should either try to develop an equation that better tracks the
true relationship or limit your use of the estimating factor to the range of the data used in
developing the factor.
Example of Estimating Factor Development . Assume that you are negotiating a guard service
contract for your facility and you want to develop a CER to assist you in estimating a should-pay
contract price. Development should follow the 6-step CER process.
Step 1. Define the dependent variable. The objective is to develop an estimate of the price that
the Government should expect to pay for this contract.
Step 2. Select independent variables to be tested for developing estimates of the dependent
variable. Logically, the major driver of price in a guard service contract is the wages paid the
security guards manning the various posts identified in the contract.
Step 3. Collect data concerning the relationship between the dependent and independent
variables. You have collected information on prices, minimum manning requirements, and
service contract wage-rate determinations for the guard service contract at your facility for the
last three years. The minimum manning requirement for the current contract totals 75,000 (Guard
II) hours. The Service Contract Act (SCA) wage rate for the current year is $10.00 per hour. The
estimated direct labor cost for each year (Column D) is calculated by multiplying estimated
direct labor hours (Column B) by the Service Contract Act wage rate (Column C).
A B C D E
Year Estimated Direct SCA Minimum Estimated Direct Contract Price
Labor Hours Wage Rate Labor Cost
1 87,600 $9.15 $801,540 $1,346,585
2 78,840 $9.45 $745,038 $1,244,215
3 70,040 $9.50 $665,380 $1,124,490
Current 75,000 $10.00 $750,000
Step 4. Explore the relationship between the dependent and independent variables. The
following table demonstrates calculation of the Price to Direct Labor Cost Ratio. The ratio
(Column F) is calculated by dividing the contract price (Column E) by the estimated direct labor
cost (Column D). In Year 1 for example, price was 1.68 times the estimated direct labor cost.
A B C D E F
Year Estimated
Direct Labor
Hours
SCA Minimum
Labor Rate
Estimated
Direct
Labor Cost
Contract
Price
Price to Direct
Labor Cost
Ratio
1 87,600 $9.15 $801,540 $1,346,585 1.68
2 78,840 $9.45 $745,038 $1,244,215 1.67
3 70,040 $9.50 $665,380 $1,124,490 1.69
Current 75,000 $10.00 $750,000
Step 5. Select the relationship that best predicts the dependent variable. It appears from the
information above, that there is a relationship between contract price and the estimated direct
labor cost. The price is between 1.67 and 1.69 times the estimated direct labor cost. The average
ratio is 1.68.
Average Ratio = (1.68 + 1.67 + 1.69) / 3
= 1.68
You can now use this ratio to estimate the price of similar contracts.
Step 6. Document your findings. Your documentation of CER development should include the
information from the 6-step process above. Exact documentation requirements will vary with the
analysis involved.
Using an Estimating Factor in Estimated Development . Once you calculate an estimating factor,
you can use it to estimate should-pay price for similar products. Using the 1.68 factor from the
guard contract example, you can calculate a should-pay price for the current year. Using this
factor, the best estimate of a reasonable price would be $1,260,000, as shown in the table below:
A B C D E F
Year Estimated
Direct Labor
Hours
SCA Minimum
Labor Rate
Estimated
Direct
Labor Cost
Contract
Price
Price to Direct
Labor Cost
Ratio
Current 75,000 $10.00 $750,000 $1,260,000 1.68
Given the data above, you should be reasonably confident of your estimate, because the range of
ratios is only from 1.67 to 1.69. Even without statistical analysis, that range might be useful in
establishing a range of reasonable prices.
High side: 1.69 x $750,000 = $1,267,500
Mean: 1.68 x $750,000 = $1,260,000
Low side: 1.67 x $750,000 = $1,252,500
4.4 - Developing And Using Estimating Equations
Situations for Using Estimating Equations . Not all estimating relationships lend themselves to
the use of simple estimating factors. If there is a substantial element of the cost or price being
estimated that is not related to the independent variable (i.e., there is a "fixed cost" that is not
associated with the independent variable), you should consider using a linear estimating
equation. If the relationship is not linear, consider a nonlinear estimating equation.
Example of Estimating Equation Development . Assume that you are analyzing the costs
proposed for the construction of a new house and decide to develop a CER to support your
analysis. Development should follow the 6-step CER process described in the chapter
Introduction
Step 1. Define the dependent variable. The objective is to estimate the cost of a new base
housing model.
Step 2. Select independent variables to be tested for developing estimates of the dependent
variable. A variety of house characteristics could be used to estimate cost. These include such
characteristics as square feet of living area, exterior wall surface area, number of baths, and
others.
Step 3. Collect data concerning the relationship between the dependent and independent
variables.
Base Housing
Model
Unit Cost Baths Sq. Ft. Living
Area
Sq. Ft.
Exterior Wall
Surface
Burger $166,500 2.5 2,800 2,170
Metro $165,000 2.0 2,700 2,250
Suburban $168,000 3.0 2,860 2,190
Executive $160,500 2.0 2,440 1,990
Ambassador $157,000 2.0 1,600 1,400
New Home 2.5 2,600 2,100
Step 4. Explore the relationship between the dependent and independent variables.
Analysis of the relationship between the independent variable and house price could be
performed using many different techniques. In this situation most analysts would use regression
analysis. However, here we will use graphic analysis to demonstrate the thought process
involved. Three independent variables will be tested: number of baths, living area, and exterior
wall surface area.
• Price and the Number Of Baths. This graph appears to demonstrate that the number of
baths is not a good estimating tool, because three houses with a nearly $8,000 price
difference have the same number of baths.
• Price and Square Feet Of Living Area. This graph appears to depict a relationship.
• Price and Exterior Wall Surface Area. This graph also appears to depict a relationship.
Step 5. Select the relationship that best predicts the dependent variable. Based on the initial
analysis, it appears that square feet of living area and exterior wall surface have the most
potential for development of a CER. The two graphs below depict efforts to fit a straight line
through the observed values. Note that both graphs demonstrate efforts to fit a line with and
without using the data from the Ambassador model.
• Price and Living Area.
• Price and Exterior Wall Surface Area.
Consider Analysis Results and Other Data. Viewing both of these relationships, we might
question whether the Ambassador model data should be included in developing our CER. In
developing a CER, you need not use all available data if all data is not comparable. However,
you should not eliminate data just to get a better looking relationship. After further analysis, we
find that the Ambassador's size is substantially different from the other houses for which we have
data and the house for which we are estimating. This substantial difference in size might
logically affect relative construction cost. Based on this information, you might decide not to
consider the Ambassador data in CER development.
Final Analysis. If you exclude the Ambassador data, you find that the fit of a straight-line
relationship of cost to the exterior wall surface is improved. The relationship between cost and
square feet of living area is even closer, almost a straight line.
If you had to choose one relationship, you would probably select living area over exterior wall
surface because living area has so much less variance from the trend line. Since the relationship
between living area and price is so close, we can reasonably use it for our CER.
If the analysis of these relationships did not reveal a useful predictive relationship, you might
consider combining two or more of the relationships already explored or exploring new
relationships.
Step 6. Document your findings. In documenting our findings, we can relate the process
involved in selecting living area for price estimation. We may then present the following graph
developed as an estimating tool.
We might also convert the graphic relationship to an equation. The cost estimating relationship
(CER) would be:
Y = $117,750 + ($17.50 x Sq Ft of Living Area)
Using an Estimating Equation to Estimate Cost. Once developed, you can use an estimating
equation to contract cost or price in similar circumstances.
For example : Applying our new CER to the estimation of cost for our new 2,600 square-foot
house, we would estimate:
Y = $117,750 + ($17.50 x Sq Ft of Living Area)
= $117,750 + ($17.50 x 2,600)
= $117,750 + $45,500
= $163,250 estimated price
CERs, like most other tools of cost analysis, MUST be used with judgment. Judgment is required
to evaluate the historical relationships in the light of new technology, new design, and other
similar factors. Therefore, knowledge of the factors involved in CER development is essential to
proper application of the CER.
4.5 Identifying Issues And Concerns
Questions to Consider in Analysis . As you perform price or cost analysis, consider the issues
and concerns identified in this section as you consider use of a cost estimating relationship.
• Does the available information verify the existence and accuracy of the proposed
relationship?
Technical personnel can be helpful in analyzing the technical validity of the relationship. Audit
personnel can be helpful in verifying the accuracy of any contractor data and analysis.
• Is there a trend in the relationship?
For example, the cost of rework is commonly estimated as a factor of production labor. As
production continues, the production effort should become more efficient and produce fewer
defective units which require repair. The factor should decrease over time. You should also
consider the following related questions: Is the rate distorted by one bad run? What is being done
to control the rate? What else can be done?
• Is the CER used consistently?
If an offeror uses a CER to propose an element of cost, it should be used in all similar proposals.
Since the CER can be used to estimate the average value, some jobs should be expected to cost
more and others less. With a valid CER, you assume the variances will be minor and will
average out across all contracts. To use a CER in some cases and a discrete estimate in others
destroys it usefulness by over or understating costs across all proposals (e.g., using the average
unless a discrete estimate is lower/higher negates the averaging out of cost across all contracts
and is clearly unfair to one of the contracting parties).
• Has the CER been consistently accurate in the past?
No matter how extensive the price/cost information or how sophisticated the analysis technique,
if a CER does not do a good job of accurately projecting cost, then it is not a useful tool.
• How current is the CER?
Even the most accurate CER needs to be reviewed and updated. While the time interval between
updates will differ with CER sensitivity to change, in general a CER should be reviewed and
updated at least annually. A CER based on a moving average should be updated whenever new
data become available.
• Would another independent variable be better for developing and applying a CER?
If another independent variable would consistently provide a more accurate estimate, then it
should be considered. However, remember that the CER may be applicable to other proposals,
not just yours. It is possible that a relationship which works well on your contract would not
work well across the entire contract population. When assessing CER validity, you should
consider all affected contracts.
• Is the CER a self-fulfilling prophecy?
A CER is intended to project future cost. If the CER simply "backs into" a rate that will spread
the cost of the existing capacity across the affected contracts, then the CER is not fulfilling its
principle function. If you suspect that a CER is being misused as a method of carrying existing
resources, you should consider a should-cost type review on the functions represented by the
CER.
• Would use of a detailed estimate or direct comparison with actual cost from a prior
effort produce more accurate results?
Development of a detailed estimate can be time consuming and costly but the application of the
engineering principles required is particularly valuable in estimating cost of efficient and
effective contract performance.
• 5.0 - Chapter Introduction
• 5.1 - Identifying Situations For Use
• 5.2 - Developing And Using A Simple Regression Equation
• 5.3 - Analyzing Variation In The Regression Model
• 5.4 - Measuring How Well The Regression Equation Fits The Data
• 5.5 - Calculating And Using A Prediction Interval
• 5.6 - Identifying The Need For Advanced Regression Analysis
• 5.7 - Identifying Issues And Concerns
5.0 - Chapter Introduction
In this chapter, you will learn to use regression analysis in developing cost estimating
relationships and other analyses based on a straight-line relationship even when the data points
do not fall on a straight line.
Line-of-Best-Fit . The straight-line is one of the most commonly used and most valuable tools in
both price and cost analysis. It is primarily used to develop cost estimating relationships and to
project economic trends. Unfortunately, in contract pricing the data points that are used in
analysis do not usually fall exactly on a straight line. Much of the variation in a dependent
variable may be explained by a linear relationship with an independent variable, but there are
usually random variations that cannot be explained by the line. The goal in establishing a line-ofbest-fit is to develop a predictive relationship that minimizes the random variations. This can be
done visually with a graph and a ruler, but the visual line-of-best-fit is an inexact technique and
has limited value in cost or price analysis. Regression analysis is commonly used to analyze
more complex relationships and provide more accurate results.
This chapter will focus on simple regression (2-variable linear regression); in which a single
independent variable (X) is used to predict the value of a single dependent variable (Y). The
dependent variable will normally be either price or cost (e.g., dollars or labor hours), the
independent variable will be a measure related to the product (supply or service) being acquired.
It may be a physical characteristic of the product, a performance characteristic of the product, or
an element of cost to provide the product.
In some situations, you may need regression analysis tools that are more powerful than simple
regression. Multiple regression (multivariate linear regression) and curvilinear regression are
variations of simple regression that you may find useful. The general characteristics of both will
be addressed later in the chapter.
5.1 - Identifying Situations For Use
Cost Estimating Relationship Development and Analysis . Regression analysis is one of the
techniques most commonly used to establish cost estimating relationships (CERs) between
independent variables and cost or price. If you can use regression analysis to quantify a CER,
you can then use that CER to develop and analyze estimates of product cost or price.
Indirect Cost Rate Analysis ( FAR 31.203 ). Indirect costs are costs that are not directly
identified with a single final cost objective (e.g., a contract), i.e., indirect costs are identified with
two or more final cost objectives or an intermediate cost objective(s). Minor direct costs may be
treated as indirect costs if the treatment is consistently applied to all final cost objectives and the
allocation produces substantially the same results as treating the cost as a direct cost.
FAR 31.203 requires that indirect costs be accumulated into logical cost pools and allocated to
final cost objectives on the basis of the benefits accruing to the various cost objectives.
Regression analysis is commonly used to quantify the relationship between the indirect cost
allocation base and expense pool over time. If you can quantify the relationship, you can then use
that relationship to develop or analyze indirect cost rate estimates.
Time-Series Analysis . You can use regression analysis to analyze trends that appear to be related
to time. It is particularly useful when you can identify and adjust for other factors that affect
costs or prices (e.g., quantity changes) to isolate the effect of inflation/deflation for analysis. The
most common applications of this type are forecasting future wage rates, material costs, and
product prices.
In time-series analysis, cost or price data are collected over time for analysis. An estimating
equation is developed using time as the independent variable. The time periods are normally
weeks, months, quarters, or years. Each time period is assigned a number (e.g., the first month is
1, the fourth month is 4, etc.). All time periods during the analysis must be considered, whether
or not data were collected during that period.
Time does not cause costs or prices to change. Changes are caused by a variety of economic
factors. Do not use time-series analysis when you can identify and effectively measure the
factors that are driving costs or prices. If you can identify and measure one or more key factors,
you should be able to develop a better predictive model than by simply analyzing cost or price
changes over time. However, if you cannot practically identify or measure such factors, you can
often make useful predictions by using regression analysis to analyze cost or price trends over
time.
Just remember that regression analysis will not automatically identify changes in a trend (i.e., it
cannot predict a period of price deflation when the available data trace a trend of increasing
prices). As a result, regression analysis is particularly useful in short-term analysis. The further
you predict into the future, the greater the risk.
5.2 - Developing And Using A Simple Regression Equation
Simple Regression Model . The simple regression model is based on the equation for a straight
line:
Yc = A + BX
Where:
Yc = The calculated or estimated value for the dependent
(response) variable
A = The Y intercept, the theoretical value of Y when X = 0
X = The independent (explanatory) variable
B = The slope of the line (the change in Y divided by the change in X, i.e., the value by which Y
changes when X changes by one).
For a given data set, A and B are constants. They do not change as the value of the independent
variable changes. Yc is a function of X. Specifically, the functional relationship between Yc and
X is that Yc is equal to A plus the product of B times X.
The following figure graphically depicts the regression line:
Steps for Developing a 2-Variable Linear Regression Equation . To develop a regression
equation for a particular set of data, use the following 5-step least-squares-best-fit (LSBF)
process:
Step 1. Collect the historical data required for analysis. Identify the X and Y values for each
observation.
X = Independent variable
Y = Dependent variable
Step 2. Put the data in tabular form.
Step 3. Compute and .
Where:
= Sample mean for observations the independent
variable
= Sample mean for observations the dependent variable
= Summation of all the variables that follow the
symbol (e.g., X represents the sum of all X
values)
X = Observation value for the independent variable
Y = Observation value for the dependent variable
n = Total number of observations in the sample
Step 4. Compute the slope (B) and the Y intercept (A).
Step 5. Formulate the estimating equation.
2-Variable Linear Regression Equation Development Example . Assume a relationship between
a firm's direct labor hours and manufacturing overhead cost based on the use of direct labor
hours as the allocation base for manufacturing overhead. Develop an estimating equation using
direct labor hours as the independent variable and manufacturing overhead cost as the dependent
variable. Estimate the indirect cost pool assuming that 2,100 manufacturing direct labor hours
will be needed to meet 20X8 production requirements.
Step 1. Collect the Historical Data Required for Analysis.
Historical Data
Year
Manufacturing
Direct Labor Hours
Manufacturing
Overhead
20X2 1,200 $ 73,000
20X3 1,500 $ 97,000
20X4 2,300 $128,000
20X5 2,700 $155,000
20X6 3,300 $175,000
20X7 3,400 $218,000
20X8 2,100 (Est)
Step 2. Put The Data In Tabular Form.
X = Manufacturing direct labor hours in hundreds of
hours (00s)
Y = Manufacturing overhead in thousands of dollars ($000s)
Tabular Presentation
X Y XY X 2 Y 2
12 73 876 144 5,329
15 97 1,455 225 9,409
23 128 2,944 529 16,384
27 155 4,185 729 24,025
33 175 5,775 1,089 30,625
34 218 7,412 1,156 47,524
Column
Totals
144 846 22,647 3,872 133,296
Step 3. Compute and .
Step 4. Compute the slope (B) and the intercept (A).
Step 5. Formulate the estimating equation. Substitute the calculated values for A and B into
the equation:
Y C = A + BX
Y C = 5.8272 + 5.6322X
Where:
Yc = Manufacturing overhead ($000's)
X = Manufacturing direct labor hours (00's)
Example of Estimate Using Simple Regression Equation . Estimate manufacturing overhead
given an estimate for manufacturing direct labor hours of 2,100:
Y C = 5.8272 + 5.622X
= 5.8272 + 5.622(21)
= 5.8272 + 118.2762
= 124.1034 thousand dollars
Rounded to the nearest dollar, the estimate would be $124,103.
Sample Solution