Decision Support Models

Transcription

1 Decision Support Models Measurement scales Carlos Bana e Costa João Lourenço Mónica Oliveira 2012/ MULTICRITERIA STEPS: Structuring vs. evaluation OPTIONS STRUCTURING EVALUATION Points of view Plausible descriptors of performance Options performance profile Partial value Sensitivity and robustness analysis Weights 1

2 3 Measurement When making measurements, we assign numbers to observations: The numbers allow an analysis by handling or making operations according to certain rules. We do not measure objects but specific characteristics of objects: Ex: weight or intelligence The relationship between objects and observed values, and the numbers assigned to the observations is so direct that by manipulating these numbers, new information is obtained about the objects! Measurement scales Context: It is a condition for a fundamental point of view (FPVj) being operational that it is associated with a descriptor and to a scale of local preference. Descriptors Scales of preference 2

3 5 Types of scales Nominal scales Ordinal scales Interval scales Ratio scales Scales used in multicriteria methodologies 6 Scales More information available Higher measurement level Higher statistical analysis allowed Nominal Ordinal Interval Ratio 3

4 Nominal scales 8 Nominal scale It corresponds to the lowest level of measurement. It uses numbers or other symbols to classify a person, object or characteristics. It has no implicit ordering: Operations such as addition / subtraction and multiplication / division operations are inadequate! 4

5 9 Examples Classification of a person regarding religion Classification of a person regarding marital status License plates Catholic, Jewish, Muslim, Hindu, Buddhist Single, Married, Divorced, Widow Nos. on the jerseys of football players 10 Nominal scale: Admissible operations The classification can be represented by any set of symbols: The nominal scale is single but can be subject to a biunovocal transformation. The only statistics that are permissible need to remain unchanged with that transformation: mode, frequency counts, etc. 5

6 11 Nominal scale: Admissible operations Example: In car plates there could be a prefix to indicate the district where the car was registered: A Lisbon car could have the prefix "L"; a Leiria could have the prefix "LA". It would be possible to exchange prefixes that represent the two districts, but this would only be preserved if this swap was done consistently and completed with the issuance of any new license plates. Ordinal scale 6

7 13 On an ordinal scale, the numbers of the scale (ordinal data) are obtained by sorting the elements by order of magnitude, according to a descriptor. There is only one order of preference, increasing or decreasing, among themselves, without being able to quantify how much a point in the range is more preferable than another. + preferable - preferable 14 Examples Military posts Sergeant > Cable > Soldier Knowledge of a student about Decision Analysis: Complete knowledge of matter > Good knowledge of the matter > Basic knowledge of the matter > Little knowledge of the matter > No knowledge of the matter Hotels classification: ***** > **** > *** > ** > * 7

8 15 Ordinal scale Elements in each category of a given scale are not only different, but keep some kind of relationship: Typical Relationships: taller, most preferred, easier, more disturbed, younger, etc. Relations with the symbol ">" in general mean "greater than." Particular scales: "preferable", "higher than", "more difficult than", etc. The specific meaning of this scale depends on the nature of the relationship that sets the scale. 16 Ordinal scale It requires full ordering information about the ">" for all pairs of classes. It respects a particular order but not a truly quantitative scale: Equal intervals on an ordinal scale do not imply equal intervals between the underlying classes! very weak weak normal strong very strong 8

9 17 Ordinal scale: Admissible operations Any transformation that preserves the order does not change the information contained in an ordinal scale: The scale is unique. All classes of numbers assigned to an ordinal scale can be changed in any way that does not alter the ordering of the elements of this scale. 18 Ordinal scale: Admissible operations Example General Brigadier This transformation does not change the order of the classes, because it involves no loss of information. 9

10 19 Ordinal scale: Admissible operations Statistic adequate to describe the central tendency: Median is not affected by modifications of any values below or above it, since the number of values below and above it remains the same. Interval scales 10

11 21 Interval scales All characteristics of the ordinal scale + It is known how great are the distances (ranges) between any two numbers on the scale Measurement considerably stronger than the ordinal Interval scale. (first scaling truly quantitative all ranges mentioned so far) 22 Interval scales Scale characterized by a constant and common unit of measurement that assigns a real number to every pair of objects in the ordered set. The ratio of any two intervals is independent of the measuring unit and zero point. The zero point and the unit of measure are arbitrary! 11

12 23 Examples Time measurement Temperature measurement Longitude measurement 24 Example temperature Celsius and Fahrenheit scales The unit of measure and the zero point are arbitrary and different for the two scales. Both scales contain the same amount and same type of information, being convertible through a positive linear transformation: F 9 5 C 32 F nº of degrees in a Fahrenheit scale C nº of degrees in a Celsius scale 12

13 25 Example temperature It is shown easily that: The ratios of the differences in temperature (intervals) do depend on the measurement unit and on the zero point. Ex: The water solidifies at 0 C and boils at 100 C. The water solidifies at 32 F and boils at 212 F. 26 Example temperature One can read from the scales: ºC ºF The ratio of the differences in a scale is equal to the ratio between differences in a different scale: In ºC 2 In ºF

14 27 Example temperature 80ºC = 2 temperature of 40ºC 80ºC 176ºF 40ºC 104ºF 176ºF 2 104ºF!!! 28 Example temperature The difference in temperature between 80ºC and 40ºC is double the difference in temperature between 60ºC and 40ºC. Confirmation 80ºC 40ºC = 2 (60ºC - 40ºC) 176ºF - 104ºF = 2 (140ºF 104ºF) 14

15 29 Interval scales: Admissible operations Any change in the numbers associated with the position of the measured object should preserve the order of the objects and the relative differences between the objects. The interval scale is unique unless a single linear transformation is done. 30 Interval scales: Admissible operations The information is not affected if each number is multiplied by a positive constant and it is then added a constant to the result: f(x) ax b Example: transforming values from an interval scale to another interval scale a b 100 a 0,1 500a b 0 b points converted into a new scale: f( 80 ) 0, f(x) 0,1x 50 15

16 31 Interval scales: Admissible operations Common parametric statistics apply. Mean, standard deviation, Pearson correlations, etc. Ratio scale 16

17 33 Ratio scale All the characteristics of an interval scale + True point in the origin Ratio scale It enables one to directly compare points (metric data) that compose it, and where the zero is fixed 34 Examples Lenght Weight Age Days Months Years There is the point zero grams and zero pounds. The ratio of two weights is independent of the measurement unit. If we determine the weights of two different objects in grams and in pounds, we observe that the ratio in grams is equal to the ratio in pounds. 17

18 35 Ratio scale: Admissible operations The numbers associated are true numbers associated with a true zero: Only the measurement unit is arbitrary. The ratio scale is "unique" unless it is subject to multiplication by a positive constant. In addition to the aforementioned statistical tests, statistics such as the geometric mean and the coefficient of variation can be used: These statistics require knowledge of the true zero point. 36 The values in the figure below express numerically a ratio scale for the attractiveness of the four alternatives (A, B, C and D) in the reliability criterion. You can read that: a) The alternative A is twice more attractive than alternative B in terms of Reliability b) Alternative D is worthless in terms of Reliability c) The difference in attractiveness in terms of Reliability between A and B is equal to twice the difference in attractiveness between B and C d) All the above e) None of the above 18

19 37 Bibliography Sidney Siegel; Nonparametric Statistics for the Behavioral Sciences, McGraw-Hill,