1

SCREENING PROCEDURES IN HUMAN MEDICINE

EVALUATION OF RESULTS BY MULTIPLE CORRESPONDENCE ANALYSIS

Jože Rovan1, Vilma Urbančič-Rovan2, Mira Slak2

1Faculty of Economics, Dept. of Statistics, University of Ljubljana2University Medical Centre, Dept. of Endocrinology

Ljubljana, Slovenia

2

Screening

mass examination of the population to detect the existence of a particular disease.

Dorland’s illustrated Medical Dictionary,

25th Ed, WB Saunders, 1974

3

Diabetes mellitus

• chronic metabolic disorder

• elevated blood sugar levels

• incidence: 4 - 7 % of the population

• Slovenia:

~ 80.000 diabetics (4% of the population)

4

Late complications of diabetes

develop after 7-10 years of high blood sugar:

• blood vessel disease - diabetic angiopathy

• nerve disease - diabetic neuropathy:– impaired sensation (pain, temperature,

vibration, light touch)– decreased muscle strength– disturbed function of the autonomic nerves

5

The diabetic foot

a group of disorders of the foot, caused by late complications of diabetes:

• poor blood supply (ischaemia)

• disturbed nerve function

• infection

6

Gangrene and amputation

are among the most feared complications of diabetes mellitus:

• 50% of all non-traumatic amputations are performed on diabetics

• foot ulcer develops in 15% of the diabetics

• very often, gangrene and amputation of one leg are followed by gangrene and amputation of the other.

7

Foot screening protocol

In order to prevent gangrene and amputation, the patients at risk for such complications must be detected on time and treated properly.

Foot screening protocol is a world-wide adopted set of simple and cheap diagnostic procedures that helps us identify the patients at risk.

8

Data acquisition

• Out-Patient Diabetes Clinic, University Medical Centre Ljubljana

• observation period: Nov. 96 - Nov. 98

• foot screening procedure:– 1275 patients– 50.8% women, 49.2% men– average age 63.63 years

9

Data acquisition

• demographic data (ID, age, sex) • medical history (previous foot ulcer, amputation,

various symptoms) • foot examination (various deformities, hard skin,

ulcer, dry skin, redness, arterial pulses)• risk status classification (groups 1 - 4)• therapeutic measures (education, footwear

prescription, referral to: foot clinic, angiologist, surgeon, pedicurist)

10

Data acquisition

• 56 variables were analysed

• all, except for age, were nominal, mostly dichotomous

• age has been recoded to 3 age groups:– 1: under 51 years– 2: 51 - 70– 3: more than 70

• altogether 117 categories

11

Multiple correspondence analysis (MCA)

Correspondence analysis is a multivariate method for exploring categorical data.

The primary goal of MCA is to transform numerical information into graphical displays (“maps”) and related numerical statistics.

The position of the category-points in MCA maps is the basis for revealing the relationship among the investigated variables.

12

Burt table

11 12 1

21 22 2

1 2

F ( 1,2,..., ) diagonal matrices

F ( , 1,2,..., , ) contingency tables

qq

qq

Q

Q

Q Q QQ

q Q

q q Q q q

F F F

F F FB

F F F

13

Dimensionality of MCA solution

Based on the Burt table we can form the nonsimetric matrix

- the number of variables (i.e. 56 in our example) and

- a diagonal matrix with the frequencies of the categories on t

1 112

(1)

f

Q Q

J

f fQ

D

D BD B

2

1 11

he main diagonal ( 117 ).

Spectral decomposition of the matrix results in:

- a diagonal matrix of principal inertias and

- the matrix of standard coordinates.

f fQ

J

D BD B

D

Y

14

MCA includes the fitting of the diagonal submatrices ( =1,2,..., )

of the Burt table. As a result, the total inertia is inflated and thus the

proportions of the first few principal inertias as the p

qq q QF

2

arts of the total

inertia are reduced.

To eliminate this problem Benzécri calculates modified inertias :

11,2, (2)

1

and suggests to consider only those principal axes tha

k

k k

Qk

Q Q

t fullfill

the condition 1/ .

The number of inertias has been reduced from 61 ( 61) nontrivial

principal inertias to only 12 modified inertias , with strongly

dominating values of the first t

k

k

Q

J Q

hree principal inertias.

15

T h e v a l u e s o f p r i n c i p a l i n e r t i a s k , t h e v a l u e s o f m o d i f i e d i n e r t i a s k ,t h e p e r c e n t a g e s o f m o d i f i e d i n e r t i a a n d t h e c u m u l a t i v e p e r c e n t a g e s o fm o d i f i e d i n e r t i a

P r i n c i p a li n e r t i a

k

M o d i f i e di n e r t i a

k

P e r c e n t a g eo f m o d i f i e d

i n e r t i a

C u m u l a t i v ep e r c e n t a g e

o f m o d i f i e di n e r t i a

1 0 2 0 3 0 4 0 % - - - - + - - - - + - - - - + - - - - + - - - -

1 0 . 0 2 2 2 9 1 0 . 0 1 7 9 1 1 4 7 . 6 8 4 7 . 6 8 * * * * * * * * * * * * * * * * * * * * * * * *2 0 . 0 0 6 4 8 0 0 . 0 0 4 0 6 8 1 0 . 8 3 5 8 . 5 1 * * * * *3 0 . 0 0 5 6 1 9 0 . 0 0 3 3 8 0 9 . 0 0 6 7 . 5 0 * * * *4 0 . 0 0 2 9 2 8 0 . 0 0 1 3 6 3 3 . 6 3 7 1 . 1 3 * *5 0 . 0 0 1 8 8 3 0 . 0 0 0 6 7 6 1 . 8 0 7 2 . 9 3 *6 0 . 0 0 1 7 9 0 0 . 0 0 0 6 2 0 1 . 6 5 7 4 . 5 8 *7 0 . 0 0 1 5 3 1 0 . 0 0 0 4 6 9 1 . 2 5 7 5 . 8 3 *8 0 . 0 0 1 4 2 6 0 . 0 0 0 4 1 1 1 . 0 9 7 6 . 9 2 *9 0 . 0 0 1 3 1 0 0 . 0 0 0 3 4 8 0 . 9 3 7 7 . 8 5

1 0 0 . 0 0 1 1 1 0 0 . 0 0 0 2 4 8 0 . 6 6 7 8 . 5 11 1 0 . 0 0 1 0 8 9 0 . 0 0 0 2 3 8 0 . 6 3 7 9 . 1 41 2 0 . 0 0 1 0 2 2 0 . 0 0 0 2 0 6 0 . 5 5 7 9 . 6 9

2 0 . 0 3 7 5 6 8

16

T h e v a l u e s o f p r i n c i p a l i n e r t i a s k , t h e v a l u e s o f m o d i f i e d i n e r t i a s k ,t h e p e r c e n t a g e s o f m o d i f i e d i n e r t i a a n d t h e c u m u l a t i v e p e r c e n t a g e s o fm o d i f i e d i n e r t i a

P r i n c i p a li n e r t i a

k

M o d i f i e di n e r t i a

k

P e r c e n t a g eo f m o d i f i e d

i n e r t i a

C u m u l a t i v ep e r c e n t a g e

o f m o d i f i e di n e r t i a

1 0 2 0 3 0 4 0 % - - - - + - - - - + - - - - + - - - - + - - - -

1 0 . 0 2 2 2 9 1 0 . 0 1 7 9 1 1 4 7 . 6 8 4 7 . 6 8 * * * * * * * * * * * * * * * * * * * * * * * *2 0 . 0 0 6 4 8 0 0 . 0 0 4 0 6 8 1 0 . 8 3 5 8 . 5 1 * * * * *3 0 . 0 0 5 6 1 9 0 . 0 0 3 3 8 0 9 . 0 0 6 7 . 5 0 * * * *4 0 . 0 0 2 9 2 8 0 . 0 0 1 3 6 3 3 . 6 3 7 1 . 1 3 * *5 0 . 0 0 1 8 8 3 0 . 0 0 0 6 7 6 1 . 8 0 7 2 . 9 3 *6 0 . 0 0 1 7 9 0 0 . 0 0 0 6 2 0 1 . 6 5 7 4 . 5 8 *7 0 . 0 0 1 5 3 1 0 . 0 0 0 4 6 9 1 . 2 5 7 5 . 8 3 *8 0 . 0 0 1 4 2 6 0 . 0 0 0 4 1 1 1 . 0 9 7 6 . 9 2 *9 0 . 0 0 1 3 1 0 0 . 0 0 0 3 4 8 0 . 9 3 7 7 . 8 5

1 0 0 . 0 0 1 1 1 0 0 . 0 0 0 2 4 8 0 . 6 6 7 8 . 5 11 1 0 . 0 0 1 0 8 9 0 . 0 0 0 2 3 8 0 . 6 3 7 9 . 1 41 2 0 . 0 0 1 0 2 2 0 . 0 0 0 2 0 6 0 . 5 5 7 9 . 6 9

2 0 . 0 3 7 5 6 8

17

The next question is the quality of the presentation of the

position of the profiles based on a few first principal

coordinates. M. Greenacre calculates the percentage of inertia

as follows:

% 100 kk

2

2 2'

2 2' 2

1 ' 1 1'

1,2, (3)

where is an average of the off-diagonal inertias , i.e.

1(4)

( 1) 1

qq

Q Q J Q

qq jq q k

q q

k

Q J Q

Q Q Q Q

18

T h e v a l u e s o f p r i n c i p a l i n e r t i a s k , t h e v a l u e s o f m o d i f i e d i n e r t i a s k ,t h e p e r c e n t a g e s o f m o d i f i e d i n e r t i a a n d t h e c u m u l a t i v e p e r c e n t a g e s o fm o d i f i e d i n e r t i a

P r i n c i p a li n e r t i a

k

M o d i f i e di n e r t i a

k

P e r c e n t a g eo f m o d i f i e d

i n e r t i a

C u m u l a t i v ep e r c e n t a g e

o f m o d i f i e di n e r t i a

1 0 2 0 3 0 4 0 % - - - - + - - - - + - - - - + - - - - + - - - -

1 0 . 0 2 2 2 9 1 0 . 0 1 7 9 1 1 4 7 . 6 8 4 7 . 6 8 * * * * * * * * * * * * * * * * * * * * * * * *2 0 . 0 0 6 4 8 0 0 . 0 0 4 0 6 8 1 0 . 8 3 5 8 . 5 1 * * * * *3 0 . 0 0 5 6 1 9 0 . 0 0 3 3 8 0 9 . 0 0 6 7 . 5 0 * * * *4 0 . 0 0 2 9 2 8 0 . 0 0 1 3 6 3 3 . 6 3 7 1 . 1 3 * *5 0 . 0 0 1 8 8 3 0 . 0 0 0 6 7 6 1 . 8 0 7 2 . 9 3 *6 0 . 0 0 1 7 9 0 0 . 0 0 0 6 2 0 1 . 6 5 7 4 . 5 8 *7 0 . 0 0 1 5 3 1 0 . 0 0 0 4 6 9 1 . 2 5 7 5 . 8 3 *8 0 . 0 0 1 4 2 6 0 . 0 0 0 4 1 1 1 . 0 9 7 6 . 9 2 *9 0 . 0 0 1 3 1 0 0 . 0 0 0 3 4 8 0 . 9 3 7 7 . 8 5

1 0 0 . 0 0 1 1 1 0 0 . 0 0 0 2 4 8 0 . 6 6 7 8 . 5 11 1 0 . 0 0 1 0 8 9 0 . 0 0 0 2 3 8 0 . 6 3 7 9 . 1 41 2 0 . 0 0 1 0 2 2 0 . 0 0 0 2 0 6 0 . 5 5 7 9 . 6 9

2 0 . 0 3 7 5 6 8

19

T h e v a l u e s o f p r i n c i p a l i n e r t i a s k , t h e v a l u e s o f m o d i f i e d i n e r t i a s k ,t h e p e r c e n t a g e s o f m o d i f i e d i n e r t i a a n d t h e c u m u l a t i v e p e r c e n t a g e s o fm o d i f i e d i n e r t i a

P r i n c i p a li n e r t i a

k

M o d i f i e di n e r t i a

k

P e r c e n t a g eo f m o d i f i e d

i n e r t i a

C u m u l a t i v ep e r c e n t a g e

o f m o d i f i e di n e r t i a

1 0 2 0 3 0 4 0 % - - - - + - - - - + - - - - + - - - - + - - - -

1 0 . 0 2 2 2 9 1 0 . 0 1 7 9 1 1 4 7 . 6 8 4 7 . 6 8 * * * * * * * * * * * * * * * * * * * * * * * *2 0 . 0 0 6 4 8 0 0 . 0 0 4 0 6 8 1 0 . 8 3 5 8 . 5 1 * * * * *3 0 . 0 0 5 6 1 9 0 . 0 0 3 3 8 0 9 . 0 0 6 7 . 5 0 * * * *4 0 . 0 0 2 9 2 8 0 . 0 0 1 3 6 3 3 . 6 3 7 1 . 1 3 * *5 0 . 0 0 1 8 8 3 0 . 0 0 0 6 7 6 1 . 8 0 7 2 . 9 3 *6 0 . 0 0 1 7 9 0 0 . 0 0 0 6 2 0 1 . 6 5 7 4 . 5 8 *7 0 . 0 0 1 5 3 1 0 . 0 0 0 4 6 9 1 . 2 5 7 5 . 8 3 *8 0 . 0 0 1 4 2 6 0 . 0 0 0 4 1 1 1 . 0 9 7 6 . 9 2 *9 0 . 0 0 1 3 1 0 0 . 0 0 0 3 4 8 0 . 9 3 7 7 . 8 5

1 0 0 . 0 0 1 1 1 0 0 . 0 0 0 2 4 8 0 . 6 6 7 8 . 5 11 1 0 . 0 0 1 0 8 9 0 . 0 0 0 2 3 8 0 . 6 3 7 9 . 1 41 2 0 . 0 0 1 0 2 2 0 . 0 0 0 2 0 6 0 . 5 5 7 9 . 6 9

2 0 . 0 3 7 5 6 8

20

T h e v a l u e s o f p r i n c i p a l i n e r t i a s k , t h e v a l u e s o f m o d i f i e d i n e r t i a s k ,t h e p e r c e n t a g e s o f m o d i f i e d i n e r t i a a n d t h e c u m u l a t i v e p e r c e n t a g e s o fm o d i f i e d i n e r t i a

P r i n c i p a li n e r t i a

k

M o d i f i e di n e r t i a

k

P e r c e n t a g eo f m o d i f i e d

i n e r t i a

C u m u l a t i v ep e r c e n t a g e

o f m o d i f i e di n e r t i a

1 0 2 0 3 0 4 0 % - - - - + - - - - + - - - - + - - - - + - - - -

1 0 . 0 2 2 2 9 1 0 . 0 1 7 9 1 1 4 7 . 6 8 4 7 . 6 8 * * * * * * * * * * * * * * * * * * * * * * * *2 0 . 0 0 6 4 8 0 0 . 0 0 4 0 6 8 1 0 . 8 3 5 8 . 5 1 * * * * *3 0 . 0 0 5 6 1 9 0 . 0 0 3 3 8 0 9 . 0 0 6 7 . 5 0 * * * *4 0 . 0 0 2 9 2 8 0 . 0 0 1 3 6 3 3 . 6 3 7 1 . 1 3 * *5 0 . 0 0 1 8 8 3 0 . 0 0 0 6 7 6 1 . 8 0 7 2 . 9 3 *6 0 . 0 0 1 7 9 0 0 . 0 0 0 6 2 0 1 . 6 5 7 4 . 5 8 *7 0 . 0 0 1 5 3 1 0 . 0 0 0 4 6 9 1 . 2 5 7 5 . 8 3 *8 0 . 0 0 1 4 2 6 0 . 0 0 0 4 1 1 1 . 0 9 7 6 . 9 2 *9 0 . 0 0 1 3 1 0 0 . 0 0 0 3 4 8 0 . 9 3 7 7 . 8 5

1 0 0 . 0 0 1 1 1 0 0 . 0 0 0 2 4 8 0 . 6 6 7 8 . 5 11 1 0 . 0 0 1 0 8 9 0 . 0 0 0 2 3 8 0 . 6 3 7 9 . 1 41 2 0 . 0 0 1 0 2 2 0 . 0 0 0 2 0 6 0 . 5 5 7 9 . 6 9

2 0 . 0 3 7 5 6 8

21

T h e v a l u e s o f p r i n c i p a l i n e r t i a s k , t h e v a l u e s o f m o d i f i e d i n e r t i a s k ,t h e p e r c e n t a g e s o f m o d i f i e d i n e r t i a a n d t h e c u m u l a t i v e p e r c e n t a g e s o fm o d i f i e d i n e r t i a

P r i n c i p a li n e r t i a

k

M o d i f i e di n e r t i a

k

P e r c e n t a g eo f m o d i f i e d

i n e r t i a

C u m u l a t i v ep e r c e n t a g e

o f m o d i f i e di n e r t i a

1 0 2 0 3 0 4 0 % - - - - + - - - - + - - - - + - - - - + - - - -

1 0 . 0 2 2 2 9 1 0 . 0 1 7 9 1 1 4 7 . 6 8 4 7 . 6 8 * * * * * * * * * * * * * * * * * * * * * * * *2 0 . 0 0 6 4 8 0 0 . 0 0 4 0 6 8 1 0 . 8 3 5 8 . 5 1 * * * * *3 0 . 0 0 5 6 1 9 0 . 0 0 3 3 8 0 9 . 0 0 6 7 . 5 0 * * * *4 0 . 0 0 2 9 2 8 0 . 0 0 1 3 6 3 3 . 6 3 7 1 . 1 3 * *5 0 . 0 0 1 8 8 3 0 . 0 0 0 6 7 6 1 . 8 0 7 2 . 9 3 *6 0 . 0 0 1 7 9 0 0 . 0 0 0 6 2 0 1 . 6 5 7 4 . 5 8 *7 0 . 0 0 1 5 3 1 0 . 0 0 0 4 6 9 1 . 2 5 7 5 . 8 3 *8 0 . 0 0 1 4 2 6 0 . 0 0 0 4 1 1 1 . 0 9 7 6 . 9 2 *9 0 . 0 0 1 3 1 0 0 . 0 0 0 3 4 8 0 . 9 3 7 7 . 8 5

1 0 0 . 0 0 1 1 1 0 0 . 0 0 0 2 4 8 0 . 6 6 7 8 . 5 11 1 0 . 0 0 1 0 8 9 0 . 0 0 0 2 3 8 0 . 6 3 7 9 . 1 41 2 0 . 0 0 1 0 2 2 0 . 0 0 0 2 0 6 0 . 5 5 7 9 . 6 9

2 0 . 0 3 7 5 6 8

22

* *

We have already calculated the matrix of standard coordinates.

Next, we need to transform the first 12 columns of , denoted

by into principal coordinates using the modified inertias

given by

Y

Y

Y G

* * 1 2

formula (2)

(5)

where is the diagonal matrix of the 12 modified inertias. The

position of the projections of profilepoints in the optimal subspace

of chosen dimensionality is defined by t

G Y D

D

he principal coordinates.

23

Maps and analysis

When the cumulative percentage of inertia of the first two dimensions is relatively high (i.e. 58.51% in our case), then most of the profiles are well represented in a two-dimensional map (by their projections onto a plane).

24

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

25

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

26

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

27

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

28

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5

Referral to angiologist

Absent pulses of distal arteries

Risk group 3

29

0.1

0.2

0.3

0.4

0.5

0.0 0.1 0.2 0.3 0.4 0.5

Referral to angiologist

Absent pulses of distal arteries

Risk group 3

Referral to foot clinic

Acute foot ulceration

30

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4

31

-0.10

-0.05

0.00

0.05

0.10

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20

Female

Male

Foot deformity

Nail changes

Abundant callus

Referral to pedicurist

32

Conclusions (1)

• As we have expected, for most of the variables under consideration, there was not much difference in the characteristics of the left and the right foot.

• The patients with poor blood supply form a special group.

33

Conclusions (2)

• Abundant callus, hallux valgus and toe nail deformities are more frequent in women than in men - possibly due to improper footwear.

• The category-points representing the groups with an acute foot ulcer, loss of protective sensation, absent foot pulses, foot deformity, abundant callus and history of previous foot ulcer were close together, confirming the influence of the known risk factors on ulcer development.

34

Conclusions (3)

• By MCA, we have confirmed most of the causal relationships in the development of foot pathology that are known from the literature.

• In human medicine, we are often faced with the situations where categorical (nominal and ordinal) variables are predominant. Even more, some laboratory results, although physical readings, are essentially of ordinal nature. For that reason, we believe MCA can be a fruitful approach in the analysis of medical data.