Statistical test for significance in ranking positions I do a user test and the user haves to rank lists. The user will answer around 14 question. For each question they get 4 answers and they haves to order them from good to bad answers. The 4 answers come from two categories (categorie A and B). There are always two answers from A and two answers from B. So for example an output could be:


*

*A A B B

*B A B A

*A B A B

*B B A A


Around 200 users will participate in this test. All users will answer the exact same questions. So I have the output of the 14 questions around 200 times. How can I statistically test if answers from categorie A are significantly better then answers from categorie B? The crux for me is how to deal with those two categories. 
 A: As was pointed out in the comments, my previous approach didn't address the entire question. 
Suppose there is no difference between $A$ answers and $B$ answers. Then there are $6$ different ranking sequences. Let us score these sequences by summing the position of the $A$'s.


*

*$AABB$ - 3

*$ABAB$ - 4

*$ABBA$ - 5

*$BBAA$ - 7

*$BABA$ - 6

*$BAAB$ - 5


Thus, assuming no difference between the answers, we have a simple pmf for the score of $A$ for each question under the null hypothesis.
$$P(A_i = x) = \begin{cases}\frac{1}{6}, & x \in \{3,4,6,7\} \\
\frac{1}{3}, & x=5
\end{cases}$$
Quick calculations for the mean and variance yeild:
$$\mu_o = E(A_i) = 5$$
$$\sigma_o^2 = Var(A_i) = \frac{5}{3}$$
Since your samples size is large, the normal approximation should work reasonably well. Let $N$ be the total sample size (i.e. if $n$ users each answer $m$ questions then $N = nm$). Finally, denote $$\bar{A} = \frac{1}{N}\sum_{i=1}^N A_i$$. By the CLT,
$$Z = \frac{\bar A - \mu_o}{\sigma_o/\sqrt{N}}$$
will be approximately standard normal under the null hypothesis. Via a quick simulation, here is a plot which shows that the CLT does provide a good approximation here.

If answers from category $A$ are better than $B$, we might expect $A$ to be less than $5$ on average. We know the distribution of $A$ under the null hypothesis test, and we can define the P-value to be
$$p = P(Z <z|H_o) \approx \Phi\left(\frac{\sqrt{N}(z-5)}{\sqrt {5/3}}\right)$$
For example, suppose $200$ users each answer $14$ questions and we calculate $\bar A = 4.9$, we can obtain an approximate P-value for this test as follows (using R).
N <- 200*14
Abar <- 4.9
mu0 <- 5
sigma0 <- sqrt(5/3)
Z <- (Abar - mu0)/(sigma0/sqrt(N))
pval <- pnorm(Z, lower.tail=TRUE) 

In this case, we get $Z = -4.10$ and a P-value of $2.07e^{-5}$. Thus in this hypothetical example, it appears that we have evidence to conclude that $A$ is getting ranked more favorably than it would under the null hypothesis.
A: Suppose every respondent answers every question. If not you have a missing data problem you must address.
There are 24 rankings of the 4 answers. There are 6 arrangements of A and B and they can be evaluated as A better, worse, or unclear.
ID  Seq     Evaluation                          Score1  Score2
1   AABB    A is clearly better                   1       1
2   ABAB    A is probably better                  2       1
3   ABBA    unclear which is better (type 1)      3       2
4   BAAB    unclear which is better (type 2)      3       2
5   BABA    B is probably better                  4       3
6   BBAA    B is clearly better                   5       3
If a respondent answers randomly then each pattern will occur an average of 14/6 times or 2  1/3. 
If respondents have preferences for A then AABB and ABAB will occur more frequently than randomly. Conversely the categories 5 and 6 will occur less often. If A is preferred then it is unclear what the effect on the frequencies of categories 3 and 4 will be except that they should be preferred more frequently than categories 5 and 6 and less frequently than categories 1 and 2.
You can summarize the results using a 14 by 6 table of questions by responses. It would be useful to look at the total for this table. It might suggest that some questions (or responses) are not easily answered by the respondents, if its' frequencies appear to be nearly random. This might be worth some discussion in your report. Examine the marginal frequencies of the six response categories for the expected pattern of preferences for A. 
For each respondent and each question you can compute a score that reflects the preference for A. There are two suggested scorings shown above. The analysis of these scores will be helpful. You can do an ANOVA on these scores with the 14 questions as a factor (repeated measures). A lower average score will indicate a preference for A over all 14 questions. Examine the averages of the questions for possible outliers to mention in your report.
Also examine the distribution of average scores of the 14 questions by respondent. This might show that some respondents are not able to distinguish A from B or else have no strong preference. Some, of course, might be able to distinguish and have a strong preference.
As a practical matter I think you could stop there.
However, these analyses don’t give you some estimates that might be desirable such as the probability of preferring A to B. In any analysis of the probabilities of preference I would suggest starting with the three categories in Score2. I believe the ordering is clear among those three and reducing the number to three should make any estimation procedure better behaved, numerically. If that works well you might consider analyzing the classification into six categories but I think that will prove problematic. This is a lot of categories and it is difficult to decide how to interpret categories 3 and 4 as a preference for A or B or even a lack of such preference.
It would be desirable if categories 3 and 4 were quite rare. This would help show that people have preferences and can express them as your questions and answers are designed.
ADDED INFORMATION
You might want to evaluate the consistency of the subjects in the scores you use for analysis. You could use intraclass correlation for an overall measure. 
