Pair comparison

A pair comparison is a comparison method in which individual objects are compared in pairs. In contrast to this, with the scaling or “ranking”, each object is viewed individually and sorted on a scale. Pair comparison is often used when subjective criteria are to be captured, e.g. B. “Beauty” or “Good Tasting Food”.

The advantage of the pair comparison lies in the accuracy or in the ability to show small differences.

The pair comparison is used in many fields, e.g. B. in empirical social research or medical statistics .

Empirical Social Research

The pair comparison as the basis for measures of correlation in empirical social research

In the descriptive statistics of empirical social research, the pair comparison is often used to measure a relationship between at least ordinally scaled variables . There are a number of measures of association which are based on the pair comparison and which include or calculate the possible pairings in different ways. The decision for a certain measure of correlation depends on the structure of the data.

Action

In the pair comparison, pairs of cases are checked by comparing the characteristics of two variables in the two cases. What is really interesting is the comparison of the two variables. When comparing the variables school education and media literacy, one looks at each individual case (the person questioned) and compares its characteristics in the two variables with the characteristics of every other case in the data set. With three interviewed persons (A, B, C) there are three pairings (A with B, A with C and B with C), with N interviewed persons there are N (N – 1): 2 pairings. When comparing pairs, each case is compared with another case. This pair (these two cases) is checked for the relationship between their values (or characteristics). There are five possible relations: the pair or the values are concordant, discordant or bound in x or y or in x and y.

Possible pairings

concordant The values of a pair (i.e. two cases) are different for both variables, the direction of the relationship is the same for both variables. That is, the values change, the direction of the relationship is the same for both variables.

example 1

Case 1:

{\ displaystyle x = 1, y = 3}

Case 2:

{\ displaystyle x = 3, y = 4}

${\ displaystyle \ Rightarrow x_ {2}> x_ {1}}$ and ${\ displaystyle y_ {2}> y_ {1}}$

Example 2

Case 1:

{\ displaystyle x = 4, y = 3}

Case 2:

{\ displaystyle x = 2, y = 1}

${\ displaystyle \ Rightarrow x_ {2} <x_ {1}}$ and ${\ displaystyle y_ {2} <y_ {1}}$

discordant The values of a pair (i.e. two cases) are different for both variables, the direction of the relationship is also different. That is, the values change, the direction of the relationship is different for both variables.

example 1

Case 1: x = 1, y = 2 Case 2: x = 2, y = 1

x2> x1 and y2 <y1

Example 2

Case 1: x = 2, y = 1 Case 2: x = 1, y = 2

x2 <x1 and y2> y1

bound in x The values of a pair (i.e. two cases) are the same for variable x and different for variable y.

example 1

Case 1: x = 1, y = 2 Case 2: x = 1, y = 1

x2 = x1 and y2 <y1

Example 2

Case 1: x = 1, y = 1 Case 2: x = 1, y = 2

x2 = x1 and y2> y1

bound in y The values of a pair (i.e. two cases) are different for variable x and the same for variable y.

example 1

Case 1: x = 2, y = 2 Case 2: x = 3, y = 2

x2> x1 and y2 = y1

Example 2

Case 1: x = 3, y = 2 Case 2: x = 2, y = 2

x2 <x1 and y2 = y1

bound in x and y The values of both variables of a pair (i.e. two cases) are the same.

example 1

Case 1: x = 2, y = 4 Case 2: x = 2, y = 4

x2 = x1 and y2 = y1

Example 2

Case 1: x = 3, y = 3 Case 2: x = 3, y = 3

x2 = x1 and y2 = y1

The comparison is about the ratio of the characteristics of two variables, the calculation of the correlation measures is about the frequency, that is, the question of how many pairings have which character (concordant, discordant or tied). The number of pairings is best calculated using a frequency table. If a statistics program is used that calculates the pairings, a frequency table helps to assess the pairings; this is important for the choice of the measure of correlation.

In our fictitious example of comparing school education (x) and political interest (y), the frequencies could be as follows (see table): 33 cases (people) have value 1 (no schooling) for school education and value 1 for political interest 1 (very little interest), 20 cases (people) have value 1 (no schooling) for school education and value 2 (some interest) for political interest, 6 cases (people) have value 1 (no schooling) and for school education in the case of political interest, value 3 (great interest), etc.

Using the example of an output cell, the table shows how concordant pairings are determined in a frequency table

When forming pairs, one calculates for each cell how many cases with this cell (or with all cases in the cell) are concordant, discordant and bound, or one calculates all concordant pairs, all discordant pairs, all in x, all in y as well as all pairs bound in x and y and add them up.

The concordant pairings for the first cell (x = 1 and y = 1) are all cells whose values are greater (than 1), in this case the cells in the large red ribbon. The number of pairings concordant with the cases in the first cell (33 cases) is calculated by multiplying their sum by the first cell.

${\ displaystyle 33 \ cdot (41 + 18 + 29 + 30 + 18 + 23)}$

${\ displaystyle N_ {C}: \ sum (h_ {AZ} \ sum h_ {c})}$ [Sum of all: (frequencies of the starting cell sum of the frequencies of the concordant cells)] ${\ displaystyle \ cdot}$

The sum of all concordant pairings is the sum of all pairs calculated in this way, i.e. the concordant pairs for the output cell 1/1 (for x = 1 and y = 1), ½, 2/1, 2/2, 3/1 (but is 0 ), 3/2. The cells 1/3, 2/3, 3/3, 4/3, 4/1 and 4/2 have no concordant pairings.

The sum of all concordant pairs is called and is: ${\ displaystyle N_ {C}}$

${\ displaystyle N_ {C} = 33 \ cdot (41 + 18 + 29 + 30 + 18 + 23) +20 \ cdot (18 + 30 + 23) +11 \ cdot (29 + 30 + 18 + 23) +41 \ cdot (30 + 23) +29 \ cdot 23}$

In the same way, the discordant and the pairs bound in x and in y are calculated.

For example, for cell 1/2 with 20 cases, the following cells are discordant: 2/1 (11 cases), 3/1 (0 cases) and 4/1 (2 cases). So the number of pairs discordant with cell ½ is . ${\ displaystyle 20 \ cdot (11 + 0 + 2)}$

For example, for cell 1/1 with 33 cases, the following cells are bound in x: 1/2 and 1/3. So the number of pairs bound with cell 1/1 in x is . ${\ displaystyle 33 \ cdot (20 + 6)}$

For example, for cell 1/1 with 33 cases, the following cells are bound in y: 2/1, 3/1 and 4/1. So the number of pairs bound with cell 1/1 in y is . ${\ displaystyle 33 \ cdot (11 + 0 + 2)}$

The pairings bound in x and y are also calculated.

This is done as follows:

For example there is for the cell 1/1 with 33 cases, the following number of pairings: . ${\ displaystyle 33 \ cdot {\ tfrac {33-1} {2}} = 528}$

Hints:

1. Always “calculate in one direction”! For example, 2/2 is also concordant with 1/1, but this pairing has already been taken into account in the sum with the output cell 1/1. Although ½ is also tied to 1/1 in x, this pairing was already taken into account in the sum of the output cell 1/1. 2. Discordant means that the connection is opposite (negative), that is, more x means less y or vice versa.

One can check whether one has accounted for all pairings by adding up all concordant, discordant, and tied pairings. This result must be identical to the quotient from . That means: (i.e. the sum of all concordant pairs) + (i.e. the total number of pairings) ${\ displaystyle {\ tfrac {N (N-1)} {2}}}$ ${\ displaystyle N_ {C}}$ ${\ displaystyle N_ {D} + T_ {X} + T_ {Y} + T_ {XY} = N_ {T}}$ ${\ displaystyle = {\ tfrac {N (N-1)} {2}}}$

Formula:

{\ displaystyle N_ {C} + N_ {D} + T_ {X} + T_ {Y} + T_ {XY} = N_ {T} = {\ frac {N (N-1)} {2}}}

Other uses

example

Christmas tree candles should be sorted according to "beauty" so that the most beautiful can then be offered for sale.

We have red, yellow, green, blue, pink and white, in lengths of 5, 8, 10 and 15 cm and thicknesses of 0.5, 1, 1.5 and 2 cm in diameter.

If we now apply the scaling , i.e. give each candle a number (between 1 = beautiful and 10 = ugly), first all pink candles get a 10, every 0.5 cm thin a 10 and so on, but it remains "quantized" and indistinguishable red, white with 8 and 10 cm length all with the “grade” 1 left. This sorting is quick, because each candle only has to be viewed and rated once.

The pair comparison is more complex, because every candle is compared with everyone. Also z. As the 8 cm long long with the white 8 cm and then red must be a decision is made; this results in a clear sequence for all candles. The exact process of this comparison is described in the complete pair comparison.

Complete pair comparison

A pair comparison is called complete when each subject has assessed each pair. The evaluation takes place in a matrix of all evaluations. One is added to the matrix for each evaluation in the column for the more pleasant noise and in the row for the comparison noise. The example shows a matrix for the comparison of five candles and is filled with ratings from four test subjects.

Comparison candle					nicer candle
	K1	K2	K3	K4	K5
K1	.	3	3	4th	1
K2	1	.	3	3	1
K3	1	1	.	2	1
K4	0	1	2	.	1
K5	3	3	3	3	.

In this example, K3 is preferred over K1 by three subjects and K1 over K3 by one. The completeness can be easily checked because the sum (row, column) + (column, row) must always correspond to the number of test subjects. By transposing, the matrix for “nicer candles” would become a matrix for “uglier candles”.

The ranking of the candles can be calculated with the complete pair comparison with the column sum, whereby the largest sum simply gets the first rank and is counted down to the smallest sum.

Pairwise comparison

Difficult quality assessments can be carried out with the pairwise comparison method . If there are several alternatives to choose from when making a decision, they can be systematically compared using the pairwise comparison. To make a decision, the various properties of the product are compared with one another. This method switches off personal preferences, so an objective decision can be made ( see also preference matrix ).

Example:

An electronics company is planning to manufacture portable MP3 players. Which properties will the customer rate the most? A customer profile must be created for this. The customer profile states which group of customers is to be reached so that the property evaluation is more precisely related to the customer.

Customer profile:

Age 16-25 years
Earnings below average
The customer likes to listen to music a lot
Quality is important

After the creation of the customer profile, the properties of the product are then determined according to the customer profile.

Features of the product:

Display
Battery with charging cable
1 gigabyte of memory
Good audio quality
Inexpensive
Modern design

The properties are numbered consecutively and entered in the matrix of the pairwise comparison or in the form.

literature

Behnke, Joachim and Nathalie Behnke 2006. Basics of statistical data analysis. An introduction for political scientists. Wiesbaden. VS. Pages 169-181.

Müller-Benedict, Volker 2007. Basic course in statistics in the social sciences. Wiesbaden. VS. 4th edition. Chapter 11: Ordinally scaled measures of association 208–238, including pair comparison 208–216.

Web links

Method sheet for pair comparison (PDF; 57 kB)