Combining metric and qualitative approach in a measure of similarity for ill-structured data sets

Sales, C. M. D., & Wakker, P. P. (2010). Combining metric and qualitative
approach in a measure of similarity for ill-structured data sets. Paper
presented at the XVII Meeting of the Portuguese Association of Data
Classification and Analysis JOCLAD, Lisbon.

Combining metric and qualitative approach
in a measure of similarity for ill-structured
data sets
Célia M. D. Sales & Peter P. Wakker
JOCLAD 2010, Lisboa

Metric-Frequency Similarity Measure (MF)

 Sales, C.M.D. & Wakker, P. P. (2009). The metric-
frequency measure of similarity for ill-structured data
sets, with an application to family therapy. British
Journal of Mathematical and Statistical Psychology, 62,
663-682.

 Software for calculation available:

http://people.few.eur.nl/wakker/miscella/mf.similarity.cal
culate/mfsimexplanation.htm

2 Sales & Wakker, JOCLAD 2010

The Team
 Célia M. D. Sales
 Dept. of Psychology and Sociology, Universidade Autónoma de
Lisboa (CIP/UAL)
 Center for Social Research and Intervention (CIS-ICSTE/IUL)
 Peter P. Wakker
 Dept. of Economics, H13-27, Erasmus University, Rotterdam, The
Netherlands

 Acknowledgements:
 Angela Fragoeiro
 Francisco Ortega Beviá
 Sónia Noronha


Outline

 Why developing MF
 MF rationale
 MF formula explained
 Illustration in a real case
 Next steps


The challenge
 The development of the MF arose from a problem in
psychotherapy research

 Individualized change measures
 Self-report measures
 Patient elaborates a list of problems and rates how much each
problem bothers him
 Each questionnaire is unique as it varies in the number of items and in
their content

 Example: Zarastro


Mother’s complaints

1. There were death threats to the mother.
2. I had to leave the house many times.
3. Family was very upset because of the constant threats to me.
4. He was 16 years without relating to the outside, broke up with
lifetime friends.
5. In the family we had continuous fights and arguments.
6. We called the police 5 times, in 2 years.
7. We went to a series of private psychiatrists.
8. He did very poorly in school.
9. We had short quiet times, going back to the threats, creating distress.
10. He was very strange and aggressive.


Zarastro’s Complaints

1. I’m very shy.
2. I don’t know how to keep a conversation going.
3. It’s hard to maintain my few friendships.
4. It bothers me to have eye-contact with people on the street.
5. I feel that people are watching me.
6. I’m not able to show my dislike or distress.
7. I’m worried about not having a job or schooling/studies/education.
8. I have a cold relationship with my younger brother.
9. It’s hard to talk and show my complaints at home.
10. I’m obsessed with the past.


His brother José
1. My brother Z. has a distorted perception of the relationships at home.
2. My brother Z. has difficulties relating to others.
3. My brother Z. has been losing his friends.
4. Lack of emotional communication.
5. My brother Amadeus is often impolite towards Zarastro
6. My brother Amadeus is very cold and reserved, doesn’t interact.
7. My brother Amadeus has lost interest in Zarastro’s problem.
8. My sister has a severe depression.
9. It affects my job because it decreases my attention.
10. It affects my relationships with others.
11. I feel down when I have to take care of Zarastro.


His brother Amadeus
1. I feel anxious sitting at the table between Zarastro and my mother.
2. My mother makes her children depend on her.
3. My mother wants her cubs surrounding her.
4. Zarastro is watching too much T.V.
5. We (extended family) share the same fate.


The challenge

 To what extent members have a similar perception of the
problems in the family?


ill-structured data set
 All family members have a different questionnaire
corresponding to their personal view of the existing
problems;
 There is no control over the content of the items that
each family member can raise;
 There is no limit to the number of items that are
conceivable;
 Each person is free to add new items or delete previous
items in subsequent questionnaire administration;
 Each item is weighted in a Likert scale.


MF: The rational
 Similarity must consider:
 numerical differences
 presence or absence of features (number of items raised)

 MF combines both metric and frequency components and
is targeted towards situations in which the number of
aspects is unpredictable.


MF: The rational
 Amos Tversky (1977)
 Assessment of similarity between stimuli described as a
comparison of features, rather than by the computation of
metric distance between points that represent objects
 Tversky's formula applies to similarity measurements where
items are either present or absent, but have no degrees of
intensity
 MF is an extension to the case where also intensities have
been measured
 The primary purpose of the MF is to be widely applicable
to handle situations that are ill structured and complex


 Imagine two members of a family (mother and
father)
 Each indicated a number of "problems" (items),
j = number of (“joint”)
and scored how serious they feel these
items raised by both (items
problems are (1-7 scale)
A,B,C,D= 4);
 Items not raised receive a score 0
f = number of items raised
by the father and not by the
mother (items E,F,G=3);
FATHER MOTHER
Items Scores Items Scores m = number of items raised
A 7 A 5 by the mother and not by
B 6 B 6 the father(item H=1)
C 1 C 2
D 1 D 1 The total number of items
E 3 H 1 is j + f + m (= 4 + 3 + 1 =
F 2 8).
G 2
1


MF: The formula explained

Score Frequency MF overall
Similarity Similarity Similarity


Score Frequency MF overall
Similarity Similarity Similarity

(1 | diff |) 0-1 scaled similarity measure based only on
the average differences of the scores of the

j f m father and the mother.


Stage 2: Frequency similarity

Frequency
similarity

Similarity based on the number of
items raised jointly by the father and
the mother

(Dis)similarity based on the difference of the
number of items raised by the father and the
mother


Step 2.1 – Similarity based on the number
of items raised jointly
 Reflected by a number j/N
 N is a normalization factor that ensures that j/N, f/N, and
m/N never exceed 1.
 N should be the same for all participants whose mutual
similarity weights are calculated
 Thus, it should exceed the maximum number of items raised
by any single participant in the group considered. For instance,
it can be the maximum number of conceivable items.
 In our example, N = 20 has been chosen, so that j/N = 4/20 =
0.2.


Step 2.1 – Similarity on the number of items
raised jointly
 Instead of the number j/N, we will use a transformation
j/N.
 The transformation is curved downwards (concave):
similarity increases less for high values of j (and j/N) than
for low values.
 Thus, an increase from j=0 to j=1 has more impact on
mother and father similarity than an increase from j=17
to j=18, which is plausible.
 In our example, the transformation yields
0.2 = 0.45


Step 2.2 – (Dis)similarity on the difference of
the number of items
FATHER MOTHER
f = number of items raised by
Items Scores Items Scores
the father and not by the
A 7 A 5
mother (items E,F,G=3)
B 6 B 6
C 1 C 2
D 1 D 1 m = number of items raised
E 3 H 1 by the mother and not by the
F 2 father (item H=1)
G 2
1
|fm|

| f/N- m/N|

1 - | f/N- m/N|


Frequency MF overall
Score Similarity
Similarity Similarity

(1 | diff |)
j f m

 ( j / N )  1 |  ( f / N )   (m / N ) |
2


Stage 3: MF overall similarity

(1 | diff |)
½ + ¼ + ¼(j/N)  ¼|(f/N)(m/N)|
j f m

The MF measure results as the half-half midpoint of the score-
similarity and the frequency-similarity


Pre-Treatment
Zarastro Mother José
Mother w: 0.45
stress formula 1 = 0.0328; stress formula 2 = 0.1054; r(monotonic)
f: 0.58
squared=0.9889; r-squared (p.v.a.f.)=0.9319)
0.51
José w: 0.33 w: 0.33 ---------------------- ZARASTRO
f: 0.55 f: 0.52 -------------------------|
0.44 0.43 | ------------ MOTHER
Amadeus w: 0.29 w: 0.37 w: |
f: 0.49 f: 0.49 0.33 |--------------------------------------------------- JOSÉ
0.39 0.43 f: 0.45 |
0.39 -------------------------------------------------------------------------
AMADEUS

Post-Treatment
Zarastro Mother José
Mother w: 0.81
stress formula 1 = 0.0000; stress formula 2 = 0.0000; r(monotonic)
f: 0.55
squared=1.0000; r-squared (p.v.a.f.)=0.9944)
0.68
(0.17)
José w: 0.81 w: 0.75
f: 0.54 f: 0.58 ---- ZARASTRO
0.68 0.67 ------------|
(0.24) (0.24) | -------------------- JOSÉ
|
Amadeus w: 0.71 w: 0.72 w:
|--- MOTHER
f: 0.49 f: 0.50 0.70 |
0.60 0.61 f: 0.44 -------------------------------------------------------------------------
(0.21) (0.18) 0.57 AMADEUS
(0.18)
1
w: the score-similarity; f: the frequency-similarity; printed in bold: the overall similarity; within
parentheses is the pre-post change, given by the difference in overall similarity of those two times.

Conclusion
 The MF is pragmatic and easily applicable to data sets
with little structure
 In particular it need not be anticipated which variables
will be observed, or how many variables, and they may be
metric or qualitative


Next Steps
 New software for MF calculation, available on-line

 Applying MF in psychotherapy research:

 Comparing alternative data entering (categorized vs. raw items)

 Implementation the MF in a software for patient progress
(Family Therapy and Group Therapy)

 Applying MF in other fields:

 Comparing agreements in open-ended judgments


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