Conjoint analysis, or the multiattribute compositional model, is a statistical technique that was developed in the field of mathematical psychology. Conjoint analysis involves the measurement of psychological judgments (such as preferences, or acceptabilities) or perceived similarities or differences between choice alternatives¹. In many studies, users are asked to rank combinations of attributes in order to determine the relative importance assigned to individual attributes. By isolating the attributes, users can more effectively see which ones are affecting their decision-making process espcially in more complex scenarios.

In order to demonstrate the basics of how conjoint analysis works, we will perform a Two Factor at a Time Tradeoff Analysis using football as the data for our system.

For the sake of the example, imagine that you have just been hired as the defensive coordinator for a small college football program. Because of limited resources such as time and staff size, your players can only excel at one aspect of their relative positions. Before you rank your preferences, the following steps will provide you with the details of the trade offs you'll face at each position.

STEP 1
First, let us consider the choices at defensive line. Your team's defensive ends and tackles can either specialize in pass rushing or at stopping the run.

Run Stop -- strength and size are important factors for players of this type. Strength will allow defenders to break through blocks while a larger frame will aid in plugging up holes that running backs may try to escape through.

Pass Rush -- speed and agility are the keys of success for players of this type. Defensive ends require speed and a good set of moves in order to get around the opponent's tackles while defensive tackles must be able to accelerate quickly once lanes to the quarterback open up.

STEP 2
Next, we'll take a look at the team's second line of defense. At the linebacker position, the choice is to either train your players in block recognition or at dropping back into proper zone coverage.

Block Recognition -- the ability to read plays before the snap is an important skill for linebackers to possess. By studying blocking assignments, linebackers will improve in the area of pursuit as well as increase the effectiveness of larger blitz packages.

Zone Drop -- in order to execute an effective zone, it is crucial that every player on the defense drops into their correct position. Training linebackers to efficiently maintain their shallow zones while simultaneously spying the quarterback is a critical part when attempting to mix up defensive coverage.

STEP 3
Now, we'll move to the secondary and the team's primary defense against the pass. Cornerbacks may be proficient at either upper body moves or they can rely more on lower body techniques.

Upper Body -- defensive backs who focus on honing upper body moves are typically more physical players who depend on strength and timing to make up for their slower speed. This type of back will incorporate the bump-and-run to slow down faster recievers as well as deliver big hits to break up plays underneath.

Lower Body -- defensive backs who work at refining lower body techniques can be depended on to keep up with the opponent's fastest reciever. By practicing skills like the back pedal or the quick step, this type of back will make most quarterbacks reluctant to throw to their side of the field and can be counted on to come down with a majority of the jump balls.

STEP 4
Lastly, we need to think about the role of the safeties. This position will be the team's last line of defense when the rest of the defenders get beat.

Support -- one of the main jobs of the safeties is to come up and provide assistance in stopping the run or screen pass. In order to perform this task well, safeties should practice tackling and blitzing in order to avoid giving up long ground plays and also to help apply pressure to the quarterback.

Protection -- stopping the long pass is also one of the safety's main duties. Players looking to accomplish this task must gameplan and study a number of tapes on opposing players and passing formations if they are to make the right reads during the game.

STEP 5
Keeping the previous choices in mind, the next step is to rank the following combinations of attributes in the order of your most preferred to your least preferred. Each row in the list represents one of the possible combinations while each column refers to one of the attributes, or positions in our case, mentioned in steps 1-4.

Instructions -- press a bar to activate it, then drag and drop it to its desired position. Once list has been completed, 'click' the calculate button to view results.

Relative Importance
The first table in the Results section shows you the relative importance of each of the attributes as determined by your selections. In a conjoint analysis study, relative importance scores should always add up to 100% except in certain cases where rounding will lead to a loss in precision. The larger a percentage is for an attribute, the more weight that attribute has in your overall decision-making process. With these percentages, you can also calculate how much influence an attribute has over another by dividing an attribute's score by a target's score.

For example, say that in our study you find that your relative importance for the Linebackers attribute is 30%, and likewise the Safeties attribute gets a score of 15%. This scenerio would demonstrate that you place twice the amount of importance in your player's performance at Linebacker than you do at the position of Safety. This can be calculated by dividing the Linebackers score of 30(%) by the Safeties score of 15(%).

The exact formula used in this study to calculate relative importance can be found here:
http://www.sma.org.sg/smj/4610/4610bs1.pdf ²

Part Worth
The second table in the Results section lists the part worth values for each of the levels (or choices). These values are scored on a scale of 0 to 1 and can be calculated by subtracting the highest average rank from the target's average and then dividing by the difference between the lowest and highest average ranks.

(average score - highest average score) / (lowest average score - highest average score).

For example, say the average rank for the SUPPORT level is 6.0, and the highest average rank score for any of the other levels is equal to 9.0 while the lowest average rank score is equal to 3.0. The part worth value for SUPPORT would be .5 or (6.0 - 9.0) / (3.0 - 9.0).

While part worth values cannot be compared directly to each other, you can compare the differences of part worths between two attributes. For instance, say the levels for the Defensive Line attribute scored .8 and .2 while the levels for the Cornerbacks attribute scored .6 and .4. The difference in the ranges gives you a net value of 3 (or .6 / .2). This means you are willing invest three times the amount resources in your defensive line than you would in your cornerbacks. Although we didn't assign time or money increments to any of the attributes, in other studies with such units you could take the net value of the ranges and multiply it by the range of a target to get a perceived value for that attribute.

Overall Utility
Finally, the last number in the Results section rates how useful the study was in discovering your preferences. A higher value shows that the user made more of a distinction between attribute level importances, while a lower value tells the opposite. This value is calculated by summing the ranges of the attributes' part worth utilities which is explained in greater detail by the formula found at the previous address.