Simplifying stats

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- Regression models can be used to predict outcomes and understand which factors influence them. Examples given include predicting India's energy consumption based on GDP growth and the probability of a customer defaulting on a loan. - Simple and multiple regression models define the dependent variable Y and identify independent variables X to estimate relationships and interpret results. - Non-linear probability models like logistic (logit) and probit models are better suited when the dependent variable is dichotomous like default/no default. These transform the probability in a non-linear way compared to the linear probability model (LPM).

Analytics using SAS© Beacon Learning
Regression Models

Analytics using SAS© Beacon Learning
Regression
 Predictive Modeling
 Which Factors Explain?
 Regressive vs. Correlation
Examples:
 What will be India’s Energy Consumption as GDP grows by 6.5%?
 What is the probability that a customer will default on housing loan
 How many fatal road accidents will you have in Delhi if the traffic
volume increases by 10%?

Analytics using SAS© Beacon Learning
Simple vs multiple Regression
Define Y
Identify X
Estimate
Interpret
uxxxy kk   ....22110

Analytics using SAS© Beacon Learning
Non Linear Probability Models
 ktiit Xfp  ,

Analytics using SAS© Beacon Learning
Linear Probability Model (LPM)
uxxxp kk   ....22110 …….
where,
p
kxxx ,...,, 10
is the probability of default
kare the explanatory variables
.
yRegress an indicator variable on kxxx ,...,, 10
y is a dichotomous variable with possible values




defaultednothasfirmtheif0
defaultedhasfirmtheif1
y

Analytics using SAS© Beacon Learning
Problems with LPM
 Goodness of Fit
 Improbable Probability Estimates
 Linear Incremental Effect of variables on Default Probability

Analytics using SAS© Beacon Learning
Goodness of Fit

Analytics using SAS© Beacon Learning
 Improbable Probability Estimates
 Linear Incremental Effect of variables on Default
Probability
Other Problems with LPM

Analytics using SAS© Beacon Learning
How should it look like?

Analytics using SAS© Beacon Learning
Non Linear Probability Models
 Linear vs Non Linear Regression
 Logit Model
 Probit Model

Analytics using SAS© Beacon Learning
Logistic Model (Logit Models)





 k
j
ij
k
j
ij
e
e
P
1
0
1
0
1



Analytics using SAS© Beacon Learning
Linear Transformation




k
j
ij
i
i
i
P
P
L
1
0
1
ln 
 Log-Odds Ratio

Analytics using SAS© Beacon Learning
How does the probability change?
)1( PP
dx
dP
j
j
 

Analytics using SAS© Beacon Learning
Estimation and Interpretation
  

01
1
ii y
i
y
i PP
Maximum Likelihood Technique
Likelihood function
.Choose j to maximize

Analytics using SAS© Beacon Learning
Goodness of Fit: Concordant Analysis/Specificity vs Sensitivity
Estimated
Equation
Actual  Won Lost Total
Predicted
Won 160 61 221
Lost 133 1345 1478
Total 293 1406 1699
Correct 160 1345 1505
% Correct 54.6 95.7 88.6
% Incorrect 45.4 4.3 11.4
Constant
Probability
Won Lost Total
0 0 0
293 1406 1699
293 1406 1699
0 1406 1406
0.0 100.0 82.8
100.0 0.0 17.2
Sensitivity 54.61%
Specificity 95.66%
Positive predictive value 72.40%
Negative predictive value 91.00%

Analytics using SAS© Beacon Learning
Probit Model

Analytics using SAS© Beacon Learning
Logit versus Probit

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2. Analytics using SAS© Beacon Learning Regression  Predictive Modeling  Which Factors Explain?  Regressive vs. Correlation Examples:  What will be India’s Energy Consumption as GDP grows by 6.5%?  What is the probability that a customer will default on housing loan  How many fatal road accidents will you have in Delhi if the traffic volume increases by 10%?

3. Analytics using SAS© Beacon Learning Simple vs multiple Regression Define Y Identify X Estimate Interpret uxxxy kk   ....22110

5. Analytics using SAS© Beacon Learning Linear Probability Model (LPM) uxxxp kk   ....22110 ……. where, p kxxx ,...,, 10 is the probability of default kare the explanatory variables . yRegress an indicator variable on kxxx ,...,, 10 y is a dichotomous variable with possible values     defaultednothasfirmtheif0 defaultedhasfirmtheif1 y

6. Analytics using SAS© Beacon Learning Problems with LPM  Goodness of Fit  Improbable Probability Estimates  Linear Incremental Effect of variables on Default Probability

8. Analytics using SAS© Beacon Learning  Improbable Probability Estimates  Linear Incremental Effect of variables on Default Probability Other Problems with LPM

10. Analytics using SAS© Beacon Learning Non Linear Probability Models  Linear vs Non Linear Regression  Logit Model  Probit Model

14. Analytics using SAS© Beacon Learning Estimation and Interpretation     01 1 ii y i y i PP Maximum Likelihood Technique Likelihood function .Choose j to maximize

15. Analytics using SAS© Beacon Learning Goodness of Fit: Concordant Analysis/Specificity vs Sensitivity Estimated Equation Actual  Won Lost Total Predicted Won 160 61 221 Lost 133 1345 1478 Total 293 1406 1699 Correct 160 1345 1505 % Correct 54.6 95.7 88.6 % Incorrect 45.4 4.3 11.4 Constant Probability Won Lost Total 0 0 0 293 1406 1699 293 1406 1699 0 1406 1406 0.0 100.0 82.8 100.0 0.0 17.2 Sensitivity 54.61% Specificity 95.66% Positive predictive value 72.40% Negative predictive value 91.00%

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