# How do I account for opposite answers when calculating city block distances?

For a stats class, I have a question in which an occupational therapist uses a checklist about meal preparation. The checklist consisted of five statements to which persons responded using the following 1 to 7 scale:
1 = strongly disagree
2 = disagree
3 = slightly disagree
4 = neither agree nor disagree
5 = slightly agree
6 = agree
7 = strongly agree

Katie and Lorenz have been married for over 50 years. The occupational therapist asks you to calculate a city block distance in order to measure the extent to which the couple has the same view of meal preparation. What is the city block distance?

The following were Katie’s ratings:
6 1. I am responsible for preparing most of the meals in our household.
5 2. It takes a lot of time to prepare a nice meal.
6 3. We eat a variety of different foods over the course of a week
5 4. It takes a lot of energy to prepare a nice meal.
5 5. I enjoy eating a lot of different foods.

The following were Lorenz’s ratings.
1 1. I am responsible for preparing most of the meals in our household.
6 2. It takes a lot of time to prepare a nice meal.
6 3. We eat a variety of different foods over the course of a week
5 4. It takes a lot of energy to prepare a nice meal.
4 5. I enjoy eating a lot of different foods.

So to calculate the city block distance, you use Dcb = (y1-y2) + (x1-x2), etc. In this case, Katie is 1 and Lorenz is 2, and each letter corresponds to an answer on the questionnaire. So we want to figure out if these two people have the same perspective on their meal habits, so we go Dcb = (6-1)+(5-6)+(6-6)+(5-5)+(5-4).

Normally, the sum of the above equation would be our difference. BUT in this case, we have to read item 1 more closely, because both Katie and Lorenz are saying that Katie is responsible for preparing most of their meals. So they aren't actually disagreeing, they are mostly agreeing.

I'm not sure how to rectify this. Katie has said that she agrees (6) that she prepares most of the meals, and Lorenz said he strongly disagrees (1) that he prepares most of the meals. They are essentially saying the same thing here, but not quite; if Katie had put a 7 and Lorenz had put a 1, I would say they were absolutely in agreement in terms of who makes the meals, and would put that difference as a 0 for the first question on the scale. But 6 isn't the exact opposite of 1, so how do I account for that in my calculation?

To clarify: the question specifically wants us to determine the similarity of the two people's viewpoints on meal habits (this is for a health research statistics class). We are supposed to figure out the correct way to compensate for this in our calculation, but have been given no direction on how to do so.

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I was thinking that I should score the difference as a 1, since Lorenz strongly agrees that Katie cooks all the meals (which would really be a 7), and Katie agrees that she cooks all the meals (which is a 6). Would that be correct? –  Laura R Oct 1 '12 at 1:40
How is your question related to Mathematica programming ? –  Artes Oct 1 '12 at 1:43
I am not sure that you are actually measuring their distance with the calculation: $Dcb = (6-1)+(5-6)+(6-6)+(5-5)+(5-4)$. Notice that the first and last terms are positive, and the second term is negative. Despite both indicate a difference, they are reducing the total. I think you should calculate: $Dcb = |6-1|+|5-6|+|6-6|+|5-5|+|5-4|$ where $|\cdot|$ is the absolute value. –  Carl Morris Oct 1 '12 at 2:14

I think you can make the adjustment in the following way. Study each question in your survey to identify answers that refer to situations such that a high score for one person's answer is equivalent to a low score for the other person's answer. For those answers, re-score the answer according to the following procedure: if the score is $x$ for this question, set it to $8-x$. Make sure you only do this for one participants answers. This should correctly align the scores so that a high score on either answer (after re-scoring) means the same thing. The distance after re-scoring should make more sense.

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The city block distance of $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ is given by $$|x_1-y_1| + |x_2-y_2| + \ldots + |x_n-y_n|.$$

Think about it like this: if I live one block north of you or one block south of you, in either situation I would say I live one block away from you. There are no negative distances (hence the absolute values).

In this case the two vectors are $(6,5,6,5,5)$ and $(1,6,6,5,4)$ so their distance is $$|6-1| + |5-6| + |6-6| + |5-5| + |5-4|=5+1+0+0+1 = 7.$$

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I do not think you answered the OP's question. The actual question was about the fact that the first question has a certain symmetry. In other words, a 7 in the first questionnaire is equivalent to a 1 in the second questionnaire. I'll post an answer to explain what I think will fix it. –  Carl Morris Oct 1 '12 at 2:16
The actual content of the questions doesn't affect the city block distance of the results. OP is correct in that question 1 may not be measuring similarity of viewpoints, but this doesn't change the city block distance. The validity of the test is a separate issue from the value you compute from it. –  nullUser Oct 1 '12 at 2:35
I agree that the OP's calculation of the city block distance is incorrect, but the OP asks, "But 6 isn't the exact opposite of 1, so how do I account for that in my calculation?". So in this case I believe the question is not about how to calculate a city block distance but how to correct the calculation of the distance to compensate for the test's design. –  Carl Morris Oct 1 '12 at 2:51
I agree, @CarlMorris, you are correct. Hopefully OP will read these comments along with my answer. –  nullUser Oct 1 '12 at 3:42