Discriminant analysis (a type of regression)
1. Step: Preprocessing: Transform the nonlinear world of benefits for illness
to a linear framework

2. Step: Discriminant analysis• Build clusters of values (groups) of the random varable Y (cost of "linearized" illnes),that is to be predicted• The random variables Xi (representing the verious diseases, i=1,...,15,000) will predict Y• Modelling the total (linearized) cost of illnes using "discriminant parameters bi":
YD = b0 + b1X1 + b2X2 + + bmXm

(Discriminant function)• determine bi so, such that:the distance of the mean values of the groups is maximal
the variance within the groups is minimal• Select adequate predictors Xi such that the loss of information is minimal.

3. Step: Postprocessing Return to the nonlinear world of cost of illnes by adjusting functions© 2005 RISK-CONSULTING Prof. Dr. Weyer GmbHGoals of the selection of coefficientsfrequency

 large distance between the "centroids" (mean value of the considered group) small variance witin groups large variance between groups metric is the Mahalanobis distanceGroup 1 (no benefit)

Group 2 (benefit of amount a)

independent variable (e.a. age, anamnesis hypertension)

Group 2 (benefit of amount a)frequency

Group 1 (no benefit)

independent variable (e.a. age, anamnesis hypertension)© 2005 RISK-CONSULTING Prof. Dr. Weyer GmbH1

DECISIONS by DISCRIMINANT FUNCTION•

YD = b 0 + b1X1 + b2X2 + … + bmXm•

evaluate dicriminant function for known customers with. special values of the predictors Xi•

the discriminant functions ' values are assigned to points on the discriminant axis .

•compare the values with the group -centroids.

(Discriminant function )Group A_YAGroup B

Y*YD_YB__YA , YB =Group-centroids = mean value of discriminant function YD for thegroup A or B respectively.
Y*•= criterion of separation

evaluate YD for new customers with special values of the predictors Xi according to the value ofseparation Y* of the. group A or B

© 2005 RISK-CONSULTING Prof. Dr. Weyer GmbHSELECTION of SIGNIFICANT PREDICTORS
for DISCRIMINANT ANALYSIS•selection of significant predictors is realized step by step.

select an additional variable in every step,, such that discrimination is optimized between groups
with the smallest (Mahalanobis-) distance (F to enter)•if variables contribute only insignificantly to discrimination in the current multivariate system,.
they can be removed
in a later step
Wilks Lambda λ:λ=non explained scattering
total scattering•

measure the quality of dicriminance by•

the level of significance of the discriminant analysis results from λ as a χ²-distributed variable.
χ ² = - N J + G 
λ2 - 1 ln( )N = number of customers in the sample, J = number of predictors•G = number of groups

the level of significance is better (lower) than α= 0.001 for the present study© 2005 RISK-CONSULTING Prof. Dr. Weyer GmbH2DISCRIMINANCE BETWEEN 2 GROUPS
group 1 = no benefit; payed(i)Xkl=group 2 = benefit; payedi. e. G = 2 ( 2 groups )random variable, describing the predictor number k ,of the person number l,
who belongs to the with benefits (i), l=1...ni ; n = n1 + n2(i)Yl=random variable, describing the "probability to benefits" of person l belonging to group i
(1)

Yl(2)

Yl(1)

(1)

(1)

:=

b1X 1l + b2X 2l + … + bmX ml

group (1)

:=

(2)
(2)
(2)
b1X 1l + b2X 2l + … + bmX ml

group (2)group centroids:n1_ (1)1(1)Y:= n ∑ Y l
1_ (i)
Xk =n_ (2)1 2 (2)
Y:= n ∑ Y l2

l=1

l=1

mean value of the predictor k in group i© 2005 RISK-CONSULTING Prof. Dr. Weyer GmbHCRITERIONof DISCRIMINANCE__(1)

S = S(b 1, …, bm) = Y-Y(2)= distance of the centre of the group = max!

n1n2(1) _ (1)(2) _ (2)
T= T(b 1, …, bm) = ∑ (Y l - Y )2 + ∑ (Y l - Y )2

l=1
S2
Q = T = max!

= variance between groups = min!

l=1
⇒

∂Q
∂bk = 0under minor condition .

ST=1

∧
∧
Solutionb 1, …, bm is inserted in Sm_ (1) _ (2) _ (1) _ (2)∧∧S(b 1, …, b m) = (n - g) ∑ wij* (X i - X i )(X j - X j )i,j = 1wij* = elements of inversion of the co variance-matrix (within-groups)

D2 = S

is called Mahalanobis - distance© 2005 RISK-CONSULTING Prof. Dr. Weyer GmbH3

Criterion of selection of significant predictors•

not all predictors

Xi are considered immediately.

search for predictors, that .separate the groups optimally•

for that purpose: calculate S depending. on only some of the bi•

first select such Xi (or bi respectively) , maximizing the distance S between groups, that are nearest•(n - 1 - m) n n
F = m (n - 2) (n +1 n2 ) S
12is F-distributed.

© 2005 RISK-CONSULTING Prof. Dr. Weyer GmbH4