Advanced Techniques in Regression Analysis
Structural Variations
•
Interactions
–
When the effect of one explanatory variable on the
dependent variable depends on the value of another
explanatory variable
•
The “trick”: Introduce the product of the two as a new
artificial explanatory variable.
(Session 2)Example
•
Nonlinearities
–
When the impact of an explanatory variable on the
dependent variable “bends”
•
The “trick”: Introduce the square of that variable as a new
artificial explanatory variable.
(Session 3)Example
Interactions: Summary
•
When the effect (i.e., the coefficient) of one explanatory
variable on the dependent variable depends on the value
of another explanatory variable
–
Signaled only by judgment
–
The “trick”: Introduce the product of the two as a new artificial
explanatory variable. After the regression, interpret in the
original “conceptual” model.
–
For example, Cost = a + (b
1
+b
2
Age)
Mileage + … (rest of model)
–
The latter explanatory variable (in the example, Age) might or
might not remain in the model
–
Cost: We lose a meaningful interpretation of the beta-weights
Nonlinearity: Summary
•
When the direct relationship between an explanatory variable
and the dependent variable “bends”
–
Signaled by a “U” in a plot of the residuals against an explanatory
variable
–
The “trick”: Introduce the square of that variable as a new artificial
explanatory variable: Y = a + bX + cX
2
+ … (rest of model)
–
One trick can capture 6 different nonlinear “
shapes
”
–
Always keep the original variable (the linear term, with coefficient
“b”, allows the parabola to take any horizontal position)
–
c (positive = upward-bending parabola, negative = downward-
bending)
–
-b/(2c) indicates where the vertex (either maximum or minimum) of
the parabola occurs
–
Cost: We lose a meaningful interpretation of the beta-weights
Examples from the Sample Exams
Revenue
pred
= -1224.82 + 62.37
Age – 0.5201
Age
2
– 121.9
Sex + 1.99
Direct + (0.8527+1.4377
Sex)
Indirect
Caligula’s Castle:
The Age effect on Revenue is greatest at
Age = -(62.37)/(2(-0.5201)) = 59.96 years
Give direct incentives (house chips, etc.)
to men
Give indirect incentives (flowers, meals)
to women
Examples from the Sample Exams
CustSat
pred
= 84.40 – 0.8667
Wait – 0.0556
Wait
2
– 5.602
Size + (-40.0845+8.7747
Size)
Franz?
Hans and Franz:
Set Franz? = 0 (assign Hans) when the
party size is < 40.0845/8.7747 = 4.568
Customers’ anger grows more quickly
the longer they wait: c < 0, and
b = -(-0.8667)/(2
(-0.0556)) = -7.794
(i.e., in negative territory).
Explore various advanced techniques in regression analysis, including structural variations, interactions, and nonlinearities. Learn how to handle situations where the effect of one explanatory variable depends on another, or when the relationship between variables bends non-linearly. Discover tricks to introduce artificial explanatory variables and interpret regression results effectively, with practical examples.
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Presentation Transcript
Structural Variations Interactions When the effect of one explanatory variable on the dependent variable depends on the value of another explanatory variable The trick : Introduce the product of the two as a new artificial explanatory variable. Example (Session 2) Nonlinearities When the impact of an explanatory variable on the dependent variable bends The trick : Introduce the square of that variable as a new artificial explanatory variable. Example (Session 3)
Interactions: Summary When the effect (i.e., the coefficient) of one explanatory variable on the dependent variable depends on the value of another explanatory variable Signaled only by judgment The trick : Introduce the product of the two as a new artificial explanatory variable. After the regression, interpret in the original conceptual model. For example, Cost = a + (b1+b2 Age) Mileage + (rest of model) The latter explanatory variable (in the example, Age) might or might not remain in the model Cost: We lose a meaningful interpretation of the beta-weights
Nonlinearity: Summary When the direct relationship between an explanatory variable and the dependent variable bends Signaled by a U in a plot of the residuals against an explanatory variable The trick : Introduce the square of that variable as a new artificial explanatory variable: Y = a + bX + cX2+ (rest of model) One trick can capture 6 different nonlinear shapes Always keep the original variable (the linear term, with coefficient b , allows the parabola to take any horizontal position) c (positive = upward-bending parabola, negative = downward- bending) -b/(2c) indicates where the vertex (either maximum or minimum) of the parabola occurs Cost: We lose a meaningful interpretation of the beta-weights
Examples from the Sample Exams Regression: Revenue Caligula s Castle: Sex Ind 1.43767 constant -1224.84 62.37502 Age Age2 -0.5201 -121.899 1.992615 Sex Direct Indirect 0.85276 coefficient Revenuepred = -1224.82 + 62.37 Age 0.5201 Age2 121.9 Sex + 1.99 Direct + (0.8527+1.4377 Sex) Indirect revenue / $ incentive direct indirect Men (Sex=0) $1.99 $0.85 Women (Sex=1) $1.99 $2.29 Give direct incentives (house chips, etc.) to men Give indirect incentives (flowers, meals) to women The Age effect on Revenue is greatest at Age = -(62.37)/(2(-0.5201)) = 59.96 years
Examples from the Sample Exams Regression: CustSat Hans and Franz: Size Franz? constant 84.4016931 -0.8666595 -0.0556165 -5.6022949 -40.084506 8.77474654 Wait Wait2 Size Franz? coefficient CustSatpred = 84.40 0.8667 Wait 0.0556 Wait2 5.602 Size + (-40.0845+8.7747 Size) Franz? Set Franz? = 0 (assign Hans) when the party size is < 40.0845/8.7747 = 4.568 Customers anger grows more quickly the longer they wait: c < 0, and b = -(-0.8667)/(2 (-0.0556)) = -7.794 (i.e., in negative territory).
