0 - SAS

LOGISTIC REGRESSION
ЛОГИСТИЧЕСКАЯ РЕГРЕССИЯ
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OVERVIEW
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ПОМНИТЕ? APPLICATIONS: ПРЕДСКАЗАНИЕ VS. ИССЛЕДОВАНИЕ
•
•
Предикторы, их знаки и
статистическая значимость
представляют вторичный
интерес.
Фокусируемся на построении
модели, лучшей с точки зрения
предсказания будущих значений Y,
т.е. более точной модели.
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•
•
Фокусируемся на понимании
взаимосвязи между целевой
(зависимой) переменной и
предикторами (независимыми)
переменными.
Поэтому, важна статистическая
значимость предикторов, а также
значения и знаки коэффициентов в
модели.
LOGISTIC
ПРИМЕРЫ ЗАДАЧ
REGRESSION
Target Marketing
Attrition Prediction
Credit Scoring, Collection Scoring
http://www.sas.com/ru_ru/academic/konkurs.html
Fraud Detection
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LOGISTIC
REGRESSION AND OTHER MODELS
REGRESSION
Type of
Predictors
Categorical
Continuous
Continuous and
Categorical
Continuous
Analysis of
Variance (ANOVA)
Ordinary Least
Squares (OLS)
Regression
Analysis of Covariance
(ANCOVA)
Categorical
Contingency Table
Analysis or Logistic
Regression
Logistic Regression
Logistic Regression
Type of
Response
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LOGISTIC
TYPES OF LOGISTIC REGRESSION
REGRESSION

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LOGISTIC
SUPERVISED (BINARY) CLASSIFICATION
REGRESSION
(Binary) Target
Input Variables
y
Cases
1
...
2
...
3
...
4
...
5
..
.
...
n
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x1 x2 x3 x4 x5 x6 ... xk
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
...
LOGISTIC
ЗАДАЧА И ДАННЫЕ
REGRESSION
1= yes
0= no
~32’000 obs
47 vars
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Did customer
purchase variable
annuity product?
Other product
usage in a three
month period
Demographics
Sav
Saving Account
32264
MMCred
0
Money Market Credits
32264
0
SavBal
Saving Balance
32264
MTG 0
Mortgage
32264
0
32264
MTGBal
0
Mortgage Balance
32264
0
32264
CC
Credit Card
28131
4133
Number Point of Sale
28131
CCBal
4133
Credit Card Balance
28131
4133
POSAmt Amount Point of Sale
N Miss
CD
Certificate of Deposit
2070
CDBal
CD Balance
28131
CCPurc
4133
Credit Card Purchases
28131
4133
32264
SDB 0
Safety Deposit Box
32264
0
32264
Income
0
Income
26482
5782
IRA 0
IRABal0
Retirement Account
32264
HMOwn
0
Owns Home
26731
5533
IRA Balance
32264
LORes
0
Length of Residence
26482
5782
Line of Credit
32264
HMVal
0
Home Value
26482
5782
Line of Credit Balance
32264
Age 0
Age
25907
6357
Investment
28131
CRScore
Credit Score
4133
31557
707
LOGISTIC
ATM
ATM
ЗАДАЧА
ИWithdrawal
ДАННЫЕ
ATMAmt ATM
Amount
REGRESSION
POS
Variable Label
N
0
AcctAge
Age of Oldest Account
30194
DDA
Checking Account
32264
DDABal
Checking Balance
32264
Dep
Checking Deposits
32264
DepAmt
Amount Deposited
32264
LOC 0
0
LOCBal
CashBk
Number Cash Back
32264
Inv
Checks
Number of Checks
32264
Investment Balance
28131
Moved
4133
Recent Address Change
32264
0
DirDep
Direct Deposit
32264
InvBal0
ILS 0
Installment Loan
32264
InArea0
Local Address
32264
0
NSF
Number Insufficient Fund
32264
Loan Balance
32264
0
NSFAmt
Amount NSF
32264
ILSBal0
MM 0
Money Market
32264
0
Phone
28131
4133
MMBal
Money Market Balance
32264
0
32264
0
Mortgage
32264
Sav
Saving Account
32264
Mortgage Balance
32264
SavBal
Saving Balance
32264
Credit Card
28131
4133
ATM
ATM
32264
Credit Card Balance
28131
4133
ATMAmt
ATM Withdrawal Amount
32264
Credit Card Purchases
28131
4133
POS
Number Point of Sale
28131
Safety Deposit Box
32264
0
POSAmt Amount Point of Sale
28131
Income
26482
5782
CD
Certificate of Deposit
32264
Owns Home
26731
5533
CDBal
CD Balance
32264
MMCred
0
MTG
0
MTGBal
0
CC
0
CCBal
0
CCPurc
4133
SDB
4133
Income
0
HMOwn
0
LORes
0
HMVal
Money Market Credits
Teller
Number Telephone
Banking
Teller Visits
Length of Residence
26482
5782
Home Value
26482
5782
IRA
Retirement Account
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32264
32264
0
Frequen
cy
0 Ins
0
21089
0
1
11175
Cumulati Cumulati
ve
ve
Frequen
cy Percent
Percent
65.36
21089
65.36
34.64
32264
100.00
ANALYTICAL CHALLENGES
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ANALYTICAL
OPPORTUNISTIC DATA
CHALLENGES
Operational / Observational
Massive
Подготовка данных для
моделирования:
Errors and Outliers
Missing Values
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• BENCHMARK: 80/20
• [MY] LIFE: 99/1
ANALYTICAL
MIXED MEASUREMENT SCALES
CHALLENGES
sales, executive, homemaker, ...
88.60, 3.92, 34890.50, 45.01, ...
F, D, C, B, A
0, 1, 2, 3, 4, 5, 6, ...
M, F
27513, 21737, 92614, 10043, ...
12
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ANALYTICAL
RARE TARGET EVENT
CHALLENGES
Event
respond
churn
default
fraud
No Event
not respond
stay
pay off
legitimate
13
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ANALYTICAL
NONLINEARITIES AND INTERACTIONS
CHALLENGES
E(y)
E(y)
x1
x2
Linear
Additive
x1
x2
Nonlinear
Nonadditive
14
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ANALYTICAL
HIGH DIMENSIONALITY
CHALLENGES
15
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ANALYTICAL
MODEL SELECTION
CHALLENGES
I
II I I I II II III II II I I IIIIII IIII I III II IIIIIIII IIIIIIIIII IIIIIIIII I
Underfitting
Overfitting
Just Right
I III II I III I I III II I II IIIIIIII IIIIII IIIIII I III I IIII I
I
16
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THE MODEL & ITS INTERPRETATION
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LOGISTIC
ПОЧЕМУ НЕ ВЗЯТЬ ЛИНЕЙНУЮ?
REGRESSION
OLS Reg: Yi=0+1X1i+i
•
•
•
Если целевая переменная
категориальная, как представить ее в
виде числовой?
Если целевая закодирована (1=Yes and
0=No) а результат модели 0.5 или 1.1
или -0.4, что это означает?
Если переменная имеет только два
значения (или несколько), имеет ли
смысл требовать постоянства
дисперсии или нормальности ошибок?
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Linear Prob. Model: pi=0+1X1i
•
•
•
•
Вероятность ограничена, а линейная
функция принимает любые значения. (Once
again, how do you interpret a predicted value
of -0.4 or 1.1?)
Примая во внимание ограниченность
вероятности, вы можете предполагать
линейную связь между X и p?
Можно ли предполагать ошибку с
постоянной дисперсией?
Что такое наблюденная вероятность для
конкретного наблюдения?
LOGISTIC
FUNCTIONAL FORM
REGRESSION
posterior probability
logit(pi )  0  1 x1i    k xki
parameter
input
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LOGISTIC
THE LOGIT LINK FUNCTION
REGRESSION
 pi 
1
    pi 
logit(pi )  ln

1

p
1

e
i 

pi = 1
pi = 0
smaller    larger
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LOGISTIC
THE FITTED SURFACE
REGRESSION
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LOGISTIC
LOGISTIC PROCEDURE
REGRESSION
proc logistic data=develop plots(only)=(effect(clband
x=(ddabal depamt checks res))
oddsratio (type=horizontalstat));
class res (param=ref ref='S');
model ins(event='1') =
dda ddabal dep depamt cashbk checks res / stb clodds=pl;
units ddabal=1000 depamt=1000 / default=1;
oddsratio 'Comparisons of Residential Classification'
res / diff=all cl=pl;
run;
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LOGISTIC
PROPERTIES OF THE ODDS RATIO
REGRESSION
No Association
Группа в числителе
имеет более высокие
шансы
0
1

Группа в знаменателе
имеет более высокие
шансы наступления
события
𝑝 𝑒𝑣𝑒𝑛𝑡 (𝐴)
𝑂𝑑𝑑𝑠 =
𝑝𝑒𝑣𝑒𝑛𝑡 (𝐵)
Estimated logistic regression model:
logit(p) = .7567 + .4373*(gender)
where females are coded 1 and males are coded 0
Estimated odds ratio (Females to Males):
odds ratio = (e-.7567+.4373)/(e-.7567) = 1.55
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LOGISTIC
RESULTS FROM ODDSRATIO
REGRESSION
oddsratio 'Comparisons of Residential Classification' res / diff=all cl=pl;
Odds Ratio Estimates and Profile-Likelihood Confidence Intervals
95% Confidence
Label
Estimate
Limits
Comparisons of Residential Classification
Res R vs S
0.954
0.897
1.015
Comparisons of Residential Classification 2
Res R vs U
0.991
0.933
1.053
Comparisons of Residential Classification 3
Res U vs S
0.963
0.911
1.017
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LOGISTIC
RESULTS FROM PLOTS = (EFFECT(…
REGRESSION
plots(only)=(effect(clband x=(ddabal depamt checks res))
Odds Ratio Estimates and Profile-Likelihood
Confidence Intervals
95% Confidence
Effect
Unit
Estimate
Limits
DDABal
1000.0
1.074
1.067
1.082
DepAmt
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1000.0
1.018
1.012
1.025
LOGISTIC
LOGISTIC DISCRIMINATION
REGRESSION
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OVERSAMPLING
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OVERSAMPLING SAMPLING DESIGNS
Joint
(x,y),(x,y),(x,y),
(x,y),(x,y),(x,y),
(x,y),(x,y),(x,y),
(x,y),(x,y),...
{(x,y),(x,y),(x,y),(x,y)}
Separate
x,x,x,
x,x,x,
x,x,x,
x,x,...
y=0
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{(x,0),(x,0),(x,1),(x,1)}
x,x,x,
x,x,x,
x,x,x,
x,x,...
y=1
OVERSAMPLING THE EFFECT OF OVERSAMPLING
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OVERSAMPLING OFFSET
Два способа корректировки
1. Включить параметр «сдвига» в
модель
model … / offset=X
  0 1 

ln 
  1 0 
𝜋1 , 𝜋0 - в действительности
𝜌1 , 𝜌0 - в выборке
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2. Скорректировать вероятности на
выходе модели
Adjusted Probability:
𝑎𝑑𝑗
𝑝1
𝑝1 𝜋1 𝜌0
=
𝑝1 𝜋1 𝜌0 + 1 − 𝑝1 𝜋0 𝜌1
OVERSAMPLING КОРРЕКТИРОВКА ВЕРОЯТНОСТЕЙ
/* Specify the prior probability
/* to correct for oversampling
%let pi1=.02;
*/
*/
/* Correct predicted probabilities */
proc logistic data=develop;
model ins(event='1')=dda ddabal dep depamt cashbk checks;
score data=new out=scored priorevent=&pi1;
run;
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PREPARING THE INPUT VARIABLES
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MISSING VALUES
DOES PR(MISSING) DEPEND ON THE DATA?
• No
o MCAR (missing completely at random)
• Yes
o that unobserved value
o other unobserved values
o other observed values (including the target)
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14
2
2
67
1
4
?
3
1
33
1
7
18
2
1
6
0
1
31
3
8
51
1
8
MISSING VALUES COMPLETE CASE ANALYSIS
Cases
Input Variables
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...
MISSING VALUES COMPLETE CASE ANALYSIS
Cases
Input Variables
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MISSING VALUES NEW MISSING VALUES
Fitted Model:
ˆ)  2.1  .07 2x1  .89x2  1.4 x3
logit(p
New Case:
x1 ,
x2 , x3   2, ?,  .5
Predicted Value:
ˆ)  2.1  .1 44 .89( )  .7
logit(p
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MISSING VALUES MISSING VALUE IMPUTATION
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6
12
03
04
2.6
1.8
0
0
8.3 42
0.5 86
66
65
C03
C14
6.5
8
6
3
2
10
7
6.5
01
01
01
01
02
03
01
01
2.3 .33 4.8 37
2.1 1 4.8 37
2.8 1 9.6 22
2.7 0 1.1 28
2.1 1 5.9 21
2.0 0 0.8 0
2.5 0 5.5 62
2.4 0 0.9 29
66
64
66
64
63
63
67
63
C00
C08
C99
C00
C03
C99
C12
C05
MISSING VALUES IMPUTATION + INDICATORS
Incomplete
Data
Completed
Data
Missing
Indicator
34
63
.
22
26
54
18
.
47
20
34
63
30
22
26
54
18
30
49
20
0
0
1
0
0
0
0
1
0
0
Median = 30
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MISSING VALUES IMPUTATION + INDICATORS
data develop1; /* Create missing indicators */
set develop;
/* name the missing indicator variables */
array mi{*} MIAcctAg MIPhone … MICRScor;
/* select variables with missing values */
array x{*} acctage phone
… crscore;
do i=1 to dim(mi);
mi{i}=(x{i}=.);
end;
run;
proc stdize data=develop1
reponly
method=median /* Impute missing values with the median */
out=imputed;
var &inputs;
run;
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MISSING VALUES CLUSTER IMPUTATION [AT LATER LECTURES]
X1 =
X2 = ?
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CATEGORICAL INPUTS
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CATEGORICAL
DUMMY VARIABLES
INPUTS
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X
DA
DB
DC
DD
D
B
C
C
A
A
D
C
A
.
.
.
0
0
0
0
1
1
0
0
1
.
.
.
0
1
0
0
0
0
0
0
0
.
.
.
0
0
1
1
0
0
0
1
0
.
.
.
1
0
0
0
0
0
1
0
0
.
.
.
CATEGORICAL
SMARTER VARIABLES
INPUTS
ZIP
99801
99622
99523
99523
99737
99937
99533
99523
99622
.
.
.
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HomeVal
75
100
150
150
150
75
100
150
100
.
.
.
Urbanicity Local ...
1
2
1
1
3
3
2
1
3
.
.
.
1
1
1
0
1
1
1
0
1
.
.
.
CATEGORICAL
QUASI-COMPLETE SEPARATION
INPUTS
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0
1
DA
DB
Dc
DD
A
28
7
1
0
0
0
B
16
0
0
1
0
0
C
94
11
0
0
1
0
D
23
21
0
0
0
1
CATEGORICAL
CLUSTERING LEVELS
INPUTS
0
1
A
28
7
B
16
0
C
94
11
D
23
21
Merged:
2 =
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31.7
100%
...
CATEGORICAL
CLUSTERING LEVELS
INPUTS
0
A
B
28
16
1
0
1
28
7
7
0
110 11
C
94
11
23
D
23
21
Merged:
2 =
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21
31.7
100%
B&C
30.7
97%
...
CATEGORICAL
CLUSTERING LEVELS
INPUTS
0
1
0
A
28
7
0
28
B
16
1
1
7
0
138 18
110 11
C
94
11
23
23
D
23
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21
21
Merged:
2 =
21
31.7
100%
B&C
A & BC
30.7
97%
28.6
90%
...
CATEGORICAL
CLUSTERING LEVELS
INPUTS
0
1
0
A
28
7
0
28
B
16
1
1
7
0
0
138 18
110 11
C
94
11
23
21
21
21
Merged:
2 =
161 39
23
23
D
1
31.7
100%
B&C
A & BC
30.7
97%
ABC & D
28.6
90%
0
0%
Greenacre (1988, 1993) PROC MEANS – PROC CLUSTER – PROC TREE -… HOME WORK
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VARIABLE CLUSTERING
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VARIABLE
REDUNDANCY
CLUSTERING
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VARIABLE
CLUSTERING
PROC VARCLASS [LATER LECTURE]
Mortgage
Balance
Checking
Deposits
Number of
Checks
Teller Visits
Credit Card
Balance
Age
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VARIABLE
UNIVARIATE SCREENING
SCREENING
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VARIABLE
UNIVARIATE SMOOTHING
SCREENING
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EMPIRICAL LOGITS

Mi
 mi 
2
ln 

Mi
 M i  mi 
2







where
mi= number of events
Mi = number of cases
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EMPIRICAL LOGIT
PLOTS
1. Hand-Crafted New Input Variables
2. Polynomial Models
3. Flexible Multivariate Function Estimators
4. Do Nothing
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SUBSET SELECTION
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SUBSET SELECTION SCALABILITY IN PROC LOGISTIC
Stepwise
Time
All
Subsets
25
50
75
100
150
Number of Variables
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200
MEASURING CLASSIFIER PERFORMANCE
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HONEST
THE OPTIMISM PRINCIPLE
ASSESSMENT
Training
Test
Accuracy = 70%
Accuracy = 47%
x1
 gray black 
x2
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 gray black 
x2
HONEST
ASSESSMENT
DATA SPLITTING
Validation
Training
Test
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HONEST
OTHER APPROACHES – CROSS VALIDATION
ASSESSMENT
A
1)
2)
3)
4)
5)
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B
C
Train
BCDE
ACDE
ABDE
ABCE
ABCD
D
Validate
A
B
C
D
E
E
CONFUSION MATRIX & ROC CURVE
0
1
0
True
Negative
False
Positive
Actual
Negative
1
False
Negative
True
Positive
Actual
Positive
Predicted
Negative
Predicted
Positive
SENSITIVITY (true positive rate (TPR), hit rate, recall)
TPR = TP / (TP+FN)
Sensitivity
Actual Class
Predicted Class
SPECIFICITY (SPC) (true negative rate (TNR))
SPC = TN / (FP + TN)
http://en.wikipedia.org/wiki/Receiver_operating_characteristic
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ALLOCATION RULES CUTOFFS
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ALLOCATION RULES PROFIT MATRIX
Bayes Rule:
Actual Class
Decision
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0
1
0
1
 TN
 FP
 FN
 TP
Decision 1 if
P
1
  TP   FN 
1




 TN
FP 
ALLOCATION RULES CLASSIFIER PERFORMANCE
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OVERSAMPLED TEST SET
Actual
Predicted
0
1
0
29
21
50
56
41
97
1
17
33
50
1
2
3
46
54
57
43
Sample
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Predicted
0
1
Population
ADJUSTMENTS FOR OVERSAMPLING
Actual Class
Predicted Class
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0
1
0
0·Sp
0(1—Sp)
0
1
1(1—Se)
1·Se
1
ALLOCATION RULES PROFIT MATRIX
Total Profit
Actual
Predicted
0
1
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0
1
$0
$0
-$1
$99
70
5
9
16
66
9
4
21
57
18
1
24
16*99 - 5 = $1579
21*99 - 9 = $2070
24*99 - 18 = $2358
ALLOCATION RULES USING PROFIT TO ASSESS FIT
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OVERALL
CLASS SEPARATION
PREDICTIVE POWER
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OVERALL
K-S STATISTIC
PREDICTIVE POWER
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OVERALL
AREA UNDER THE ROC CURVE
PREDICTIVE POWER
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ROC AND ROCCONTRAST STATEMENTS
ROC <'label'> <specification> </ options>;
ROCCONTRAST <'label'><contrast></ options>;
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