Government%20Financial%20Accounting

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Regression Analysis
Defense Resources Management Institute
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Unscheduled Maintenance Issue

• 36 flight squadrons
• Each experiences unscheduled maintenance actions
  (UMAs)
• UMAs costs 1000 to repair, on average.

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Youve got the Data Now What?
Unscheduled Maintenance Actions (UMAs)
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What do you want to know?

• How many UMAs will there be next month?
• What is the average number of UMAs ?

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Sample Mean
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Sample Standard Deviation
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UMA Sample Statistics
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UMAs Next Month
95 Confidence Interval
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Average UMAs
95 Confidence Interval
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Model Cost of UMAs for one squadron

• If the cost per UMA 1000, the
• Expected cost for one squadron 60,000

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Model Total Cost of UMAs

• Expected Cost for all squadrons
• 60 1000 36 2,160,000

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Model Total Cost of UMAs

• Expected Cost for all squadrons
• 60 1000 36 2,160,000
• How confident are we about this estimate?

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95
mean (60) standard error 12/?36 2
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56 58 60 62 64 (1
standard unit 2)
95
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95 Confidence Interval on our estimate of UMAs
and costs

• 60 2(2) 56, 64
• low cost 56 1000 36 2,016,000
• high cost 64 1000 36 2,304,000

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What do you want to know?

• How many UMAs will there be next month?
• What is the average number of UMAs ?
• Is there a relationship between UMAs and and some
  other variable that may be used to predict UMAs?
• What is that relationship?

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Relationships

• What might be related to UMAs?
• Pilot Experience ?
• Flight hours ?
• Sorties flown ?
• Mean time to failure (for specific parts) ?
• Number of landings / takeoffs ?

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Regression

• To estimate the expected or mean value of UMAs
  for next month
• look for a linear relationship between UMAs and a
  predictive variable
• If a linear relationship exists, use regression
  analysis

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Regression analysis

• describes and evaluates
• relationships between one variable
• (dependent or explained variable), and
• one or more other variables (called the
  independent or explanatory variables).

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What is a good estimating variable for UMAs?

• quantifiable
• predictable
• logical relationship with dependent variable
• must be a linear relationship
• Y a bX

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Sorties
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Pilot Experience
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Sample Statistics
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Describing the Relationship

• Is there a relationship?
• Do the two variables (UMAs and sorties or
  experience) move together?
• Do they move in the same direction or in opposite
  directions?
• How strong is the relationship?
• How closely do they move together?

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Positive Relationship
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Strong Positive Relationship
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Negative Relationship
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Strong Negative Relationship
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No Relationship
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Relationship?
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Correlation Coefficient

• Statistical measure of how closely two variables
  are moving together in a coordinated fashion
• Measures strength and direction
• Value ranges from -1.0 to 1.0
• 1.0 indicates perfect positive linear relation
• -1.0 indicates perfect negative linear relation
• 0 indicates no relation between the two variables

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Correlation Coefficient
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Sorties vs. UMAs
r .9788
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Experience vs. UMAs
r .1896
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Correlation Matrix
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A Word of Caution...

• Correlation does NOT imply causation
• It simply measures the coordinated movement of
  two variables
• Variation in two variables may be due to a third
  common variable
• The observed relationship may be due to chance
  alone

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What is the Relationship?

• In order to use the correlation information to
  help describe the relationship between two
  variables we need a model
• The simplest one is a linear model

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Fitting a Line to the Data
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One Possibility
Sum of errors 0
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Another Possibility
Sum of errors 0
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Which is Better?

• Both have sum of errors 0
• Compare sum of absolute errors

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Fitting a Line to the Data
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One Possibility
Sum of absolute errors 6
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Another Possibility
Sum of absolute errors 6
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Which is Better?

• Sum of the absolute errors are equal
• Compare sum of errors squared

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The Correct Relationship Y a bX U
Y
systematic random
100
90
80
70
60
50
X
100
110
120
130
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The correct relationship
Y a bX U
Y
systematic random
100
90
80
70
60
50
X
100
110
120
130
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Least-Squares Method

• Penalizes large absolute errors
• Y- intercept
• Slope

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Assumptions

• Linear relationship
• Errors are random and normally distributed with
  mean 0 and variance
• Supported by Central Limit Theorem

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Least Squares Regression for Sorties and UMAs
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Regression Calculations
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Sorties vs. UMAs
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Regression Calculations Confidence in the
predictions
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Confidence Interval for Estimate
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95 Confidence Interval for the model (b)
Y
X
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Testing Model Parameters

• How well does the model explain the variation in
  the dependent variable?
• Does the independent variable really seem to
  matter?
• Is the intercept constant statistically
  significant?

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Variation
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Coefficient of Determination

• Values between 0 and 1
• R2 1 when all data on line (r1)
• R2 0 when no correlation (r0)

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Regression Calculations How well does the model
explain the variation?
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Does the IndependentVariable Matter?

• If sorties do not help predict UMAs we expect b
  0
• If b is not 0, is it statistically significant?

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Regression Calculations Does the Independent
Variable Matter?
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95 Confidence Interval for the slope (a)
Y
Mean of Y
Mean of X
X
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Confidence Interval for Slope
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Is the InterceptStatistically Significant?
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Confidence Intervalfor Y-intercept
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Basic Steps ofRegression Analysis

• Formulate the model
• Plot scatter diagram for visual inspection
• Compute correlation coefficient
• Fit the regression line
• Test the model

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Factors affecting estimation accuracy

• Sample size (larger is better)
• Range of X values (wider is better)
• Standard deviation of U (smaller is better)

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Uses and Limitationsof Regression Analysis

• Identifying relationships
• Not necessarily cause
• May be due to chance only
• Forecasting future outcomes
• Only valid over the range of the data
• Past may not be good predictor of future

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Common pitfalls in regression

• Failure to draw scatter diagrams
• Omitting important variables from the model
• The two point phenomenon
• Unfounded claims of model sophistication
• Insufficient attention to interval estimates and
  predictions
• Predicting too far outside of known range

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Lines can be deceiving...
R2 .6662
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Nonlinear Relationship
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Best fit?
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Misleading data
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Summary

• Regression Analysis is a useful tool
• Helps quantify relationships
• But be careful
• Does not imply cause and effect
• Dont go outside range of data
• Check linearity assumptions
• Use common sense!

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Non-linear relationship between output and cost