Pairwise comparison
A paired comparison is a comparison method, are compared in pairs in which individual objects. In contrast, is considered when scaling or " ranking " of each object individually and sorted on a scale. The pair comparison is often used when subjective criteria are to be recorded, such as " beauty " or "good tasting food ."
The advantage of the pairwise comparison is the accuracy and the ability to show small differences.
The pair-wise comparison is used in many fields, for example, used in empirical social research or medical statistics.
- 2.1 Example
- 2.2 Full paired comparison
- 2.3 Pairwise comparison
Empirical Social Research
The pair comparison as a basis for measures of association in empirical social research
In the descriptive statistics of empirical social research, the pair-wise comparison is often used to measure a relationship between at least ordinal variables. There are a number of measures of association based on the pair comparison and include the possible pairings in different ways or calculate. The choice of a particular measure of association is dependent on the structure of the data.
Procedure
When paired comparison case pairs are examined by analyzing the characteristics of two variables of the two cases are compared with each other. What really interested is the comparison of the two variables. So when comparing the variables education and media literacy considering each individual case ( the respondent ) and compares its manifestations in the two variables with the characteristics of any other case in the record. Three respondents (A, B, C) so arise three pairs (A with B, A with C and B with C), interviewed at N persons arising N ( N-1 ): 2 pairings. When paired comparison so each case is compared with another case. This pair (these two cases ) examined the relationship of their values (or attributes). There are five ratios that are possible: the pair, or the values are concordant or discordant bound in x or y or x and y.
Possible pairings
The concordant values of a pair of (that is two cases) are different for the two variables, the direction of the relationship is the same for both variables. That is, the values will change the direction of the relationship is when two variables the same.
Example 1
Case 1: x = 1, y = 3 Example 2: x = 3, y = 4
X2 > x1 and y2 > y1
Example 2
Case 1: x = 4, y = 3 Example 2: x = 2, y = 1
X2 < x1 and y2 < y1
Discordant The values of a pair (that is two cases) are different for the two variables, the direction of the relationship also varies. That is, the values change, the direction of the relationship is different for the two 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 ( ie two cases) are equal to x, y for variables different for variable.
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 ( ie two cases) are x different, same for variable y variable.
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 the x and y variables, the values of both of a pair of (that is two cases) are equal.
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 Example 2: x = 3, y = 3
X2 = x1 and y2 = y1
When comparing it comes to the ratio of occurrences of two variables in the calculation of measures of association it comes to the question of the frequencies, that is, to the question of how many pairings which character ( concordant, discordant, or tied ). The number of pairings is best calculated using a frequency table. If one uses a statistical program that calculates the pairings, helps a frequency table in the assessment of pairings; this is important for the choice of the connection metric.
In our fictional example of the comparison of education (x) and political interest (y) could the frequencies look like this (see table): 33 cases ( people) have basic education, the expression 1 (no education ) and political interest, the expression 1 (very low interest ), 20 cases ( people) have basic education, the expression 1 (no education ) and political interest, the expression 2 ( some interest ), 6 cases ( people) have basic education, the expression 1 ( no schooling ) and political interest in the expression 3 ( great interest ), etc.
The pairings are calculated for each cell the number of cases with this cell ( or with all cases in the cell) concordant, discordant and tied, or one calculates all concordant pairs, all discordant couples, all in x, y and all all pairs bound in x and y, and they are added in each case.
The concordant pairings for the first cell ( x = 1 and y = 1 ), all cells whose values are larger ( than 1) in this case the cells in the big red bow. The number of pairs that are concordant with the cases in the first cell (33 cases), calculated by multiplying the sum of the first cell.
33 * (41 18 29 30 18 23 )
[ Sum of: ( frequencies of the output cell * sum of the frequencies of concordant cells) ]
The sum of all concordant combinations is the sum of all calculated in this way pairs, ie, the concordant pairs to the output cell 1 /1 ( for x = 1 and y = 1), ½, 2/1, 2/2, 3/1 ( however 0 ), 3 / second The cells 1/3, 2/ 3, 3/ 3, 4 /3, 4/1 and 4/2 are not concordant combinations.
The sum of all concordant pairs is NC and is:
= 33 * (41 18 29 30 18 23 ) 20 * (18 30 23 ) 11 * (29 30 18 23 ) 41 * (30 23) 29 * 23
Similarly, the discordant and in the x and y in the bound pairs are calculated.
For example, the following cells for the cell half 20 discordant cases: 2/1 (11 cases ), 3/ 1 (0 cases) and 4/1 (2 cases). The number of discordant pairs with cell ½ is therefore 20 * (11 0 2).
For example, the following cells are responsible for cell 1 /1 with 33 cases bound in x 1 /2 and 1/3. The number of bound cells with 1/1 in x pairs is therefore 33 * ( 20 6).
For example, the following cells are responsible for cell 1 /1 with 33 cases bound to y 2/ 1, 3/1 and 4 / first The number of bound cells with 1/1 in the y pairs is thus 33 * (11 0 2).
The bonded pairs in x and y are calculated.
This takes place as follows:
For example there is for the cell 1/1 with 33 cases, the following number of pairings: 33 * ( 33-1 ) / 2 = 528
Notes:
1 Always just " count in one direction "! While it is also, for example, 2/2 with 1/1 concordant, this pairing but was already taken into account in the sum of the initial cell 1/1. Although ½ is also bound with 1/ 1 in x, this pairing but was already taken into account in the sum of output cell 1/1. 2 discordant means of connection is in opposite (negative), i.e., x is less more Y or vice versa.
One can consider having considered all pairings by all concordant, discordant and tied pairs added. This result must be the same as the quotient of N ( N -1): 2 That is to say: ( ie the total of concordant pairs ) ( ie the total number of pairings ) = N (N -1): 2
Formula: = N (N -1): 2
Other Applications
Example
Christmas tree candles are to be sorted by " beauty " so then the best can 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, so each candle is a number (between 1 and 10 = beautiful = ugly) give, get first all pink candles a 10, every 0.5 cm thin a 10 and so on, but keep it " quantized " and indistinguishable red, white all over with 8 and 10 cm length with the "Note" 1. This sorting is fast because each candle has to be considered only once and evaluated.
The pair-wise comparison is more complex, since each candle is compared with everyone. Here also, for example, the red 8 cm long like with the white 8 cm long and then you have a decision; the result is a unique sequence for all the candles. The exact sequence of this comparison is described in the full pair comparison.
Full paired comparison
Fully a pair of comparison is called when each subject has evaluated each pair. The evaluation is carried out in a matrix of all ratings. In the matrix, for each rating, is added in the pleasant noise and the line of the comparison - noise column one. The example shows a matrix for the comparison of five candles and filled with reviews of four subjects.
In this example, K3 is against K1 of three subjects preferred K1 and K3 against of one. The completeness is easy to check, because it must be the sum of (row, column ) ( column, row) always correspond to the number of subjects. By transposing a matrix would be composed of the matrix for " beautiful candles " are for " uglier candles ".
The ranking of candles can be calculated with the complete pair comparison with the column sum, where simply the largest sum gets the first rank, and is counted down to the smallest sum.
Pairwise comparison
Using the method Pairwise comparison one can perform difficult quality reviews. Standing at a decision, several alternatives to choose from, so they can be systematically compared by the paired comparison. In order to decide the various properties of the product are compared. This method turns out to personal preference, so it can an objective decision to be made.
Example:
An electronics company plans to manufacture of portable MP3 players. What qualities will probably rate the highest of customer? For this purpose, a customer profile must be created. The customer profile indicates which group of customers is to be achieved, so that the property valuation is precisely related to the customer.
Customer Profile:
- Age 16-25 years
- Earnings below the average
- The client listens to music a lot and
- Sets value on quality
After creating the customer profile then the properties of the product are defined according to the customer profile.
Properties of the product:
The properties are numbered and entered into the matrix of pairwise comparison or in the form.