### Constant Sum Paired Comparison Procedure

In the Constant Sum Paired Comparison procedure, a study administrator will gather two objects to form a pair. This pair of objects gets presented, at the same time, to an observer. The observer must decide if there is a difference between the two objects. He or she does so by distributing 100 points between the objects. The value reported is the amount the left object obtains when 100 points gets divided between the two objects. If the observer sees no difference between the objects, the reported score is 50 for the left object. By subtraction, the right-hand object gets 50 points. But if the observer believes the object in the left position deserves a higher score, they may choose to assign a value of 60 points. This leaves 40 points for the other object on the right. The number given to the object on the left can vary from 1 to 99. It cannot be 0 or 100.

We call the method described the “Comrey Constant Sum Paired Comparison”. It’s named after the French Psychometrician Comrey.

### Constant Sum Paired Comparison Example

Let’s consider an example. Steel shafts, shown in the figure below, have visible scratches. In this case, we will assign fewer points to a shaft that is less severe and more points to a shaft that is more severe.

Typical practice would be to identify a boundary sample. We would use such a sample to judge the severity of scratches on production units. Productions units with scratches that are worst get scrapped while those that are less severe are not. This type of inspection is like a go-no-go gage. In this case, we can only render a binary decision; the production unit is either good or bad.

If we wanted to conduct an experiment to eliminate or reduce the non-conforming units we need continuous data. To collect such data we’d administer a random treatment and make a lot of samples for inspection. From that inspection we’d count the number of bad units and compute a fraction non-conforming. We would then administer the next treatment combination and compute another fraction non-conforming. Once the experiment is complete we would have 1 continuous number for each treatment combination. Now image how much production time and resources we’d consume to conduct this experiment. You’d have a hard time convincing management to run this experiment! There has got to be a better way of getting a continuous number from a visual attribute? The good news is there is!

### Constant Sum Paired Comparison Ratios

Let’s go back to our example. But this time we’ll use the Paired Comparison procedure to create a continuous scale for the shafts. In the table below, I show the random ordering of pairwise shafts. For each row, shown are the left and right scores and the left/right ratio. Remember a less severe shaft gets fewer points versus a more severe shaft will get more points.

### Constant Sum Paired Comparison Matrix

We will now arrange the ratio data into a 4 x 4 matrix as shown below. The columns and rows represent the Left and Right ratio scores.

Notice the diagonal values equal 1. These values represent 50:50 ratios when comparing the same object to itself. Also notice there’s several missing values in this matrix. We can compute these missing values. For example, cell (2, 1) is the inverse of cell (1, 2) which is 1/0.43. Likewise cell (3, 2) is the inverse of cell (2, 3) which is 1/1.50. I show the remaining values in the completed ratio matrix below.

### Computing the Geometric Average and Assigning a Quantitative Value to a Visual Attribute

We can now compute the geometric average of each column in our ratio matrix. Each value represents one of the four shafts in our example.

#### Shaft 1 = [ 1 x 0.43 x 0.67 x 0.25 ] ^1/4 = 0.52

Shaft 2 = [ 2.33 x 1 x 0.67 x 0.25 ] ^1/4 = 0.79

Shaft 3 = [ 1.49 x 1.49 x 1 x 0.67 ] ^1/4 = 1.11

Shaft 4 = [ 4 x 4 x 1.49 x 1 ] ^1/4 = 2.21

### Constant Sum Paired Comparison Ratio Scale

The following figure shows where each sample falls on the ratio scale.

As shown, the Pairs Comparison method assigns a quantitative value to each shaft. It also shows the severity between the shafts. Using this method you can build ratio scales, compute statistics, and conduct statistical analysis. Think about the experiment described earlier. Now you can make fewer samples per treatment combination and assign a number to it using a ratio scale. Assigning a number is easy! All you do is place the treatment sample between two suitable shafts say 3 and 4. You then judge the severity of the treatment sample by moving it closer to either shaft 3 or 4. Once you find a suitable position between shafts 3 and 4, just read the number on the scale and assign it to that treatment sample. Using this approach, you’ll consume less production time and resources, and solve problems sooner!

### Attribute Dimensions Improve Inspection Reproducibility

I have successfully used this procedure to assign a quantitative value to a visual attribute time and time again. This method also provides valuable insight into the evaluation of a visual attribute. For example, many observers that check for scratches may look for its length, width, depth, as well as the number of scratches. Knowing there are 4 attribute dimensions helps to improve inspection. I have often conducted a Gage Study and found differences in Operator Reproducibility using a ratio scale. This discrepancy is always due to the difference in attribute dimensions each Operator considers. One Operator may consider the number of scratches while another only considers the depth of scratches. Once Operators know what attribute dimensions to look for Operator Reproducibility is significantly reduced. Using a ratio scale and defining the attribute dimensions operators scrap fewer units. In one manufacturing plant we saved $350,000 by reducing the error in the interpretation of a visual attribute. In another manufacturing plant, we saved about $500,000 in customer returns. We gave the customer our ratio scale and told them which attribute dimensions to look for. This harmonized inspection between both plants. It reduced inspection disagreements and lead to fewer customer returns.

### Constant Sum Paired Comparison Applied In Industry

Many years ago I applied this method to the study of Automotive Seating Comfort. It was a significant contribution. In the past, the industry used subjective evaluations to assess automotive seating comfort. This was costly, time consuming, and led to long product development cycles. Using Paired Comparisons I discovered a strong statistical correlation to Body Pressure Distribution measurements. Using Body Pressure Distribution we shortened the product development cycle and improved the time to market. You can read the publication, just click on the article link. **Investigating Psychometric and Body Pressure Distribution Responses to Automotive Seating Comfort**.

### Now it’s your turn to tell me if you think this procedure will help you

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