1 Introduction
A sentiment lexicon is a dictionary that assigns each term a polarity score
representing the strength of the positive or negative affect associated with the term. In general, word polarity strength depends on the context, and its representation by a single number can therefore only be a crude approximation. Nevertheless, such sentiment lexica are an important tool for opinion mining and have been proven to be very useful. Examples for recent use cases are the sentiment analysis of tweets and SMS
[1, 2] or the political classification of newspapers [3].There are two approaches to building a sentiment lexicon: corpus based automatic assignment or manual annotation. Corpus based approaches start with a set of seed words of known polarity and extend this set with other words occurring in a text corpus or a synonym lexicon. One possible approach is to compute the “Pointwise Mutual Information” (PMI) [4] from cooccurrences of seed words and other words. The German sentiment lexicon SentiWS [5] was built in this way. The English sentiment lexicon SentiWordNet [6] is based on the propagation of seed words, too, but by means of semisupervised classification and random walks. Yet another sophisticated corpusbased method was implemented by Cambria et al. for SenticNet [7, 8, 9].
Corpus based methods have the advantage of building large lexica in an automated way without time consuming experiments with human annotators. They have two drawbacks, however: due to peculiarities in the corpus, some words can obtain strange scores. In SentiWS 1.8, e.g., “gelungen” (successful) has the highest positive score (1.0) while the more positive word “fantastisch” (fantastic) only has a score of 0.332. In SenticNet 3.0, “inconsequent” has a strong positive polarity (0.948). Moreover, it is not possible to assign a score value to words that are absent from the corpus.
Assigning polarity scores by manual annotations can be done in two different ways. One is by direct assignment of an ordinal score to each word on a Likerttype scale. In this way, Wilson et al. have created a subjectivity lexicon with English words [10], which has also been used by means of automated translations for sentiment analysis of German texts [11]. The other method is to present words in pairs and let the observer decide which word is more positive or more negative. Comparative studies for other use cases have shown that scores from paired comparisons are more accurate than direct assignments of scores [12]
. The main advantage is their invariance to scale variances between different test persons. This is especially important when words are added at some later point when the original test persons are no longer available. Unfortunately, paired comparisons are much more expensive than direct assignments: for
words, direct assignments only require judgments, while a complete comparison of all pairs requires judgments. For large , this becomes prohibitive and must be replaced by incomplete comparisons, i.e. by omitting pairs. Incomplete paired comparisons are widely deployed in the estimation of chess players’ strength [13, 14].In the present article, we propose a method for building a sentiment lexicon from paired comparisons in two steps. At first, an initial lexicon is built from a limited set of 199 words by comparison of all pairs. This lexicon is then subsequently extended with new words, which are only compared to a limited number of words from the initial set, which are determined by a modified version of Silverstein & Farrell’s sorting method [15].
As we plan to use the lexicon ourselves for the sentiment analysis of German news reporting and thus were in need of a reliable German sentiment lexicon, and as our test persons were native German speakers, the scope of our investigation was restricted to German words. In order to use this lexicon for an evaluation of English lexica, we utilized the “averaged translation” technique described in Sec. 4.4. Automatic translation is a common method in multilingual sentiment analysis [16, 17]. It should be noted, however, that the ground truth scores of our new sentiment lexicon can only be used to compare sentiment lexica with respect to their accuracy in the sentiment analysis of German texts.
This article is an extended and corrected version of a presentation made at the ICDM workshop “Sentire” in December 2016 [18]. Compared to that presentation, the following changes and additions have been made:

The scores in [18] had been computed with an approximation by Elo that turned out to be grossly inaccurate in this use case [19]
. The formula has been replaced by a numeric optimization and all scores have been recomputed. This correction also fixes the odd result of the leaveoneout experiment in
[18]. 
The validity of the probabilistic model is checked and validated with a goodnessoffit test.

An error inherited from SentiWS has been corrected that had lead to a duplicate word in two different spellings, which has been corrected by merging the duplicates.
Our article is organized as follows: Sec. 2 provides an overview over the mathematical model that underpins the method of paired comparisons and presents formulas for computing scores from paired comparison results, Sec. 3 describes the criteria for choosing the initial set of words and our experimental setup, and Sec. 4 presents the results for the initial lexicon, evaluates the method for adding new words, and presents an evaluation of the other lexica Sentiws, SenticNet, and SentiWordNet by means of our ground truth initial lexicon.
2 Method of paired comparison
The method of paired comparison goes back to the early 20th century [20]. See [14] for a comprehensive presentation of the model and its estimation problems, and [21] for a review of recent extensions. Applied to word polarity, it makes the assumption that each word has a hidden score (or rating)
. The probability that
is more positive than (symbolically: ) in a randomly chosen context depends on the difference between the hidden scores:(1a)  
(1b)  
(1c) 
where is approximately proportional to the drawing probability of equal strength players and can thus be considered as a draw width, and
is the cumulative distribution function of a zerosymmetric random variable. Thurstone’s model
[20] uses anbased on the normal distribution, a model that can be derived from the assumption that the polarity of a word
is normally distributed around its mean inherent score . Although this is the only model with a sound statistical justification, simpler distribution functions have also been used for convenience, e.g. the logistic distribution (BradleyTerry model) [22]or the uniform distribution
[23]. The shapes of these functions are quite similar (see Fig. 1), and they typically provide similar fits to data [24], although the asymptotic error of the score estimates behaves differently [25]^{1}^{1}1Beware, however, that both Stern and Vojnovic & Yun only considered models without a draw possibility, i.e., they only considered the case .. The standard deviation of the distribution function is a scale parameter that determines the range of the ratings . We have set it to so that the estimated scores approximately fell into the range for our data.As the probabilities in Eq. (1) only depend on rating differences, the origin cannot be determined from the model, but must be defined by an external constraint. Typical choices are the average rating constraint , or the reference object constraint, i.e. for some
. For sentiment lexica, a natural constraint can be obtained by separately classifying words into positive and negative words and choosing the origin in such a way that the scores from the paired comparison model coincide with these classifications.
The ratings and the drawwidth must be estimated from the observed comparisons. During our two steps of building a sentiment lexicon, two different estimation problems occur:

Estimation of one unknown of a new word from comparisons with old words with known ratings .

Estimation of and all unknown from arbitrary pair comparisons.
In [18], analytic formulas were given for the estimators, but these were based on an approximation by Elo [13, paragraph 1.66] which turned out to be grossly inaccurate in this use case [19]. The formulas given in [18] should therefore not be used, and it is necessary instead to estimate the parameters via numeric optimization either with the maximum likelihood principle or with nonlinear least squares.
2.1 Case 1: one unknown rating
Let us first consider this simpler case. The loglikelihood function ( is considered as given) is
(2)  
As this is a function of only one variable , it can be easily maximized numerically, e.g., with the R function optimize.
Batchelder & Bershed [14]
suggested an alternative methodofmoments estimation by setting a combination of the observed number of wins
and draws equal to its expectation value and solve the resulting equation for the parameters:(3)  
This combination is constructed in such a way that a Taylor expansion of the right hand side around cancels out the term linear in , so that we obtain for small
(4) 
Even this approximation must be solved numerically for , but it makes the score estimation independent of the estimate for , and it shows that the draw width has only little effect on the score estimator.
It should be noted that both the solution of (3) or (4) and the maximization of (2) yield (or ) when is chosen to be the normal or logistic distribution and the word wins (or looses) all comparisons. For the uniform distribution, the solution or minimum is not unique in this case, but this ambiguity could be resolved by choosing the solution with .
2.2 Case 2: all ratings and unknown
The loglikelihood function in this case is
(5)  
Due to the large number of parameters, numerical methods for maximizing the loglikelihood function (5) might be very slow or fail to converge. In this situation, an alternative solution obtained via the leastsquares method can be helpful.
For a least squares fit of the parameters, let us consider the same observable as in case 1, but now for each word :
(6) 
When we set this “scoring” equal to its expectation value and make again a Taylor series expansion at , we obtain
(7) 
where are the indices of all words that have been compared with ^{2}^{2}2Note that indices may occur more than once in because might have been compared with some other word more than once. Joint estimates for all ratings can then be obtained by minimizing the sum of the squared deviations
(8) 
Minimizing (8) is a nonlinear least squares problem, which can be solved efficiently with the LevenbergMarquardt algorithm as it is provided, e.g., by the R package minpack.lm [26].
To obtain an approximate estimator for the draw width , let us consider the total number of draws of each word as an observable and set it equal to its expectation value from its comparisons:
(9) 
Keeping only the first nonzero term in a Taylor expansion around of the sum on the right hand side yields
(10) 
Again, we can determine by minimizing the sum of the squared deviations
(11) 
The minimum of expression (11) can be found analytically by solving for the zero of , which yields
(12) 
The leastsquares solution (8) and (12) can either be used as an estimator for the parameters, or it can be used as a starting point for maximizing the loglikelihood function.
3 Experimental design
To select 200 words for building the initial lexicon from round robin pair comparisons, we have started with all adjectives from SentiWS [5]. We have chosen adjectives because sentiments are more often carried by adjectives and adverbs than by verbs and nouns [27]. To build an intersection of these words with SenticNet [7], we translated all words into English with both of the GermanEnglish dictionaries from www.dict.cc and www.freedict.org, and removed all words without a match in SenticNet. From the remaining words, we selected manually 10 words that appeared strongly positive to us, and 10 strongly negative words. This was to make sure that the polarity range is sufficiently wide in the initial lexicon. The remaining words were ranked by their SentiWS score and selected with equidistant ranks, such that we obtained 200 words, with an equal number of positive and negative words according to SentiWS. As we noticed after finishing the experiments, there are duplicate words in different spellings in SentiWS, and we eventually only had 199 different words by an unhappy coincidence. In the computations of section 4 in the present paper, we have taken care of this by correcting one of the spellings, i.e. replaced the word “phantastisch” with “fantastisch” (fantastic).
We let ten different test persons assign polarity scores to these words in two different experiments. The first one consisted of direct assignment of scores on a five degree scale (see Fig. 2(a)), which resulted in ten evaluations for each word. An average score was computed for each word by replacing the ordinal scale with a metric value ( = strong negative, = weak negative, = neutral, = weak positive, = strong positive).
The second experiment consisted of twofold round robin paired comparisons between the original 200 words, with all pairs evenly distributed among the ten test persons, such that each person evaluated pairs. Due to the duplicate word, this actually was not a round robin experiment, but one word (“fantastisch”) was compared more often than the others. See Fig. 2(b) for the graphical user interface presented to the test persons.
The scores were computed both with the maximum likelihood (ML) method and the nonlinear least squares (LSQ) method as described in section 2.2. The standard deviation of the distribution function was set to , which corresponds to the distribution functions in Fig. 1. Maximization of the loglikelihood function (5) took about 12 minutes on an i74770 3.40 GHz with the “BFGS” method of the Rfunction optim. As this method uses numerically computed gradients, it was not applicable in the case of the uniform distribution, because its distribution function is not differentiable. We therefore resorted to the much slower optimization by Nelder & Mead in this case with the Elo approximate solution [18] as a starting point, which took 45 minutes. The LSQ solution, on the contrary, only took 12 seconds in all cases because we could use the more efficient LevenbergMarquardt algorithm from the R package minpack.lm [26] for minimizing (8). When the LSQ solution was used as a starting point for the ML optimization, the runtime of the BFGS algorithm reduced to 6 minutes. This was not applicable in the case of the uniform distribution, though, because the LSQ solution corresponds to a loglikelihood function that is minus infinity.
For a reasonable choice for the origin , we shifted all scores such that they best fitted to the discrimination between positive and negative words from the direct comparison experiment. To be precise: when is the score from the direct assignment and the score from the paired comparisons with an arbitrarily set origin, we chose the shift value that minimized the squared error
(13) 
For adding new words, we implemented the method by Silverstein & Farrell, which uses comparison results to sort the new word into a binary sort tree built from the initial words [15]. For initial words, this only leads to comparisons, which generally are too few for computing a reliable score. We therefore extended this method by adding comparisons with words from the initial set which have the closest rank to the rank obtained from the sort tree process. Algorithm 1 lists the resulting algorithm in detail. In practice, the selection of words in the binary sort process (lines 2 & 18 in Algorithm 1) might be randomized by adding a small random index offset. This would avoid that the test persons permanently have to compare with the same pivot elements. This algorithm can be applied sequentially to more than one test person by estimating the resulting rating from all scores obtained from all test persons. We have evaluated this method with a leaveoneout experiment using the comparisons from our twofold roundrobin comparison experiment.
4 Results



4.1 Goodness of fit
Both the ML method and the LSQ method yield estimators for the sentiment scores even in cases where the model assumptions in Eq. (1) do not hold. It is thus necessary to verify that the model explains the experimental observations. Moreover, it is also interesting to check which of the six solutions (three different distributions functions in combination with ML or LSQ) provides the best fit to the experimental data.
A visual way to verify the model consists in cutting the possible score differences into bins^{3}^{3}3To achieve an equal distribution of positive and negative , the pair order has been shuffled randomly for the computation of Fig. 3. and to compare the observed frequencies for and with the probabilities (1) predicted by the model. The resulting plots for the LSQ estimates are shown in Fig. 3. The figures show that the model indeed predicts the observed frequencies, albeit better for wins or losses than for draws. This is no surprise because only one fourth of the comparisons were draws and consequently the ML or LSQ estimation step fits the scores rather to the winning probability than to the draw probability.
LSQ  ML  LSQ  ML  

normal  
logistic  
uniform  
1487 
Goodnessoffit test statistic
and valuesfor the different score estimators. The last row gives the corresponding threshold for rejecting the null hypothesis that the model produced the result at a significance level
.For a more formal goodnessoffit test, we computed the statistic which is the sum of the squared differences between the expected and observed frequencies. For our paired comparison model, it reads [28]
(14)  
where is the number of bins into which the are grouped, and , , and are the number of wins, draws, or losses in each group. The expected number of wins, draws, or losses are computed as the sum of the respective probabilities according to the model (1), e.g.
(15)  
(16) 
The statistic (14) is approximately
distributed when the number of members in each group is not too small. The number of degrees of freedom is
, because out of the three possible outcomes in each group only two are a free choice (when the outcome is neither win nor draw, it must be a loss) and there are fitted parameters in the model: the ratings and the draw width .Douglas suggested in [28] to use each possible score difference as its own group, but this would mean that we would only have two items per group and the test statistic would not be
distributed. We therefore formed the groups as quantiles among the
values^{4}^{4}4We have computed the quantiles on basis of instead of because otherwise the value of the test statistic would depend on the pair order.such that each group had approximately the same number of samples, a method commonly deployed in testing logistic regressions
[29].The resulting values are listed in Table 1. Surprisingly, the LSQ estimators lead to a clearly better goodnessoffit than the ML estimators, with the ML estimators even yielding such a high value for that the model would be rejected as unlikely. As usual for tests, the result varies with the number of groups however, and the values are different for different numbers of groups . Nevertheless, the relative values of for fixed allow to compare different estimation methods, and in all cases the LSQ estimator yielded a clearly better fit. With respect to the distribution function , the uniform distribution provides an impossible LSQ fit because it leads to wins with zero probability in some groups. is smallest for the logistic choice for , which is therefore the model which conforms best to the observed test person answers.
4.2 Score values
normal  logistic  uniform  
unpraktisch (unpractical)  0.378  0.396  0.344 
rüde (uncouth)  0.628  0.629  0.631 
t (draw width)  0.126  0.122  0.132 
0.644  0.647  0.635  
0.226  0.227  0.228 
For the 199 different words, we estimated the polarity scores with the LSQ method with the three distribution functions of Fig. 1. The draw width turned out to be 0.126 for the normal distribution, 0.122 for the logistic distribution, and 0.132 for the uniform distribution. Fig. 4 shows a kernel density plot [30] for the resulting score distributions. The valley around zero (neutrality) is due to the fact that the words were drawn from the SentiWS data, which only contains positive or negative words. The comparative shapes are as expected from Fig. 1: the slightly steeper slope of the logistic distribution at leads to a slightly stronger separation of the positive and negative scores. The range of the score values depends on the distribution , too: for the normal distribution it is , for the logistic distribution , and for the uniform distribution . The example in Table 2 shows that the varying difference in ratings for the different distribution functions has only little effect on the “winning probabilities”, because the greater rating difference for the uniform distribution, e.g., is compensated by its greater draw width .
adjective  

paradiesisch (paradisaical)  1.00  0.966  0.039 
wunderbar (wonderful)  1.00  0.915  0.045 
perfekt (perfect)  0.95  1.083  0.057 
traumhaft (dreamlike)  0.95  1.041  0.050 
prima (great)  0.75  0.793  0.041 
zufrieden (contented)  0.75  0.613  0.034 
kinderleicht (easypeasy)  0.50  0.462  0.034 
lebensfähig (viable)  0.50  0.351  0.032 
ausgeweitet (expanded)  0.05  0.012  0.027 
verbindlich (binding)  0.00  0.143  0.023 
kontrovers (controversial)  0.05  0.268  0.029 
unpraktisch (unpractical)  0.50  0.396  0.026 
rüde (uncouth)  0.50  0.629  0.026 
falsch (wrong)  0.75  0.627  0.025 
unbarmherzig (merciless)  0.75  0.772  0.027 
erbärmlich (wretched)  1.00  0.804  0.025 
tödlich (deadly)  1.00  1.042  0.031 
It is interesting to compare the scores from paired comparisons for words which have obtained the same score from direct assignment on the five grade scale. The examples in table 3 show that the paired comparisons indeed lead to a different and finer rating scheme than averaging over coarse polarity scores from direct assignments, and that they even can lead to a reversed rank order (see, e.g., “traumhaft” and “wunderbar”). We have also estimated the variances of the polarity score estimates as the bootstrap variance from 200 bootstrap replications of all paired results [31]. These can be used to test whether, for , the score difference is significant by computing the value , where is the distribution function of the standard normal distribution. For the words “unpraktisch” and “rüde”, e.g., the value is much less than 5% and the difference is therefore statistically significant, although they obtained identical scores on the five grade scale from direct estimation.
4.3 Adding new words
To obtain a lower bound for the error in estimating scores for unknown words, we have first computed the scores for all words with the estimators for one unknown rating as described in section 2.1, where each word was compared with all other words and the scores for other words were considered to be known from the results in the preceding section. For the logistic distribution, the mean absolute error with respect to the nonlinear LSQ score was much higher for the maximumlikelihood estimator () than for methodofmoments estimator () computed with Eq. (4). This was to be expected because the “ground truth score” was also based on the difference between observed and expected wins and draws. With respect to the ML score, the ML estimator was better ( versus ), but, as we have seen in section 4.1, the ML scores are a much poorer model fit and therefore should not be used as a point of reference. We therefore have used the methodofmoments estimator (4) in our evaluations.
For a reasonable recommendation for the number of incomplete comparisons, we have varied the number of comparisons in a leaveoneout twofold application of Algorithm 1 on basis of the twofold round robin experiment. The results are shown in Fig. 5. As a point of reference, the median of the bootstrap standard deviation of the ground truth scores is given too (). The median of the absolute score errors reaches this reference value first for , which corresponds to picking neighbors after Silverstein & Farrel’s method.
To check whether our word selection method based on Silverstein & Farrel’s method actually is an improvement over picking words for comparison at random, we did a 100fold MonteCarlo experiment with choosing words at random and computing from all corresponding twofold comparisons the methodofmoments score estimate. This yielded on average a mean error of and a median error , which are considerably greater than the errors of Algorithm 1. We therefore conclude that incomplete comparisons with only 18 out of 199 words yield a reasonably accurate score estimate, provided the words are selected with our method.
4.4 Comparison to corpusbased lexica
The polarity scores computed in our experiments provide nice ground truth data for the evaluation of corpusbased polarity scores. We therefore compared the scores from SentiWS 1.8, SenticNet 3.0, SenticNet 4.0, and SentiWordNet 3.0 with the scores computed from test persons’ paired comparisons. SenticNet and SentiWordNet only contain English words, from which we have computed scores for the German words by translating each German word with both of the GermanEnglish dictionaries from www.dict.cc and www.freedict.org and by averaging the corresponding scores, a method that we call “averaged translation”.
A natural measure for the closeness between lists of polarity scores is Pearson’s correlation coefficient , which has the advantage that it is invariant both under scale and translation of the variables. This is crucial in our case, because score values from paired comparisons allow for arbitrary shift and scale as explained in section 2. is highest for a linear relationship and smaller for other monotonous relationships. As can be seen in Table 4, this means that its value depends on the shape of the model distribution function , albeit only slightly. Whatever function is used, the correlation between the scores from direct assignment and paired comparison is very strong. This was to be expected, because both values stem from the same test persons.
choice for  

normal  logistic  uniform  
direct  
SentiWS  
SenticNet 3.0  
SenticNet 4.0  
SentiWordNet 3.0 
From the corpus based lexica, the correlation with the paired scores is highest for SenticNet 4.0. According to the significance tests in the R package cocor [32], the difference to the correlation of SentiWordNet is not statistically significant at a 5% significance level, but it is significant with respect to the other two corpus based lexica, including SenticNet 3.0.
Another comparison criterion that is in favor of SenticNet 4.0 can be seen in Fig. 7: the bimodal shape of the score distribution is only reflected in the SenticNet scores. SentiWordNet has an unimodal score distribution centered around zero, and SentiWS has many identical scores with values and , which show up as horizontal lines in Fig. 6(a). This peculiar distribution of the SentiWS scores was also reported in the original paper presenting the SentiWS data set by Remus et al. (see Fig. 1 in [5]). The identical scores show up in Fig. 7 as a peak around neutrality, which corresponds to a valley (sic!) in the score distribution from paired comparisons.
As an alternative to our “averaged translation” approach, the third party Python module senticnetapi^{5}^{5}5https://github.com/yurimalheiros/senticnetapi also provides sentiment scores for German words, which cannot be recommended, however. These have been obtained from SenticNet 4.0 by automatic translation from English to German, which is the reverse direction than our approach. This has the effect that only 35% of the words in our lexicon have matches in senticnetapi, and these only have a correlation of , which is even less than the value for SentiWS.
Based on these observations, we consider the polarity scores from SenticNet 4.0 the most reliable among the investigated corpus based lexica. For foreign languages, we recommend to translate from the foreign language to English and compute the mean score among the matches, rather than to translate from English into the foreign language.
5 Conclusions
The new sentiment lexicon from paired comparison is a useful resource that can be used for different aims. It can be used, e.g., as ground truth data for testing and comparing automatic corpusbased methods for building sentiment lexica, as we did in section 4.4. Our results suggest that the most reliable corpusbased sentiment lexicon, among the tested ones, is SenticNet 4.0. When it is used with nonEnglish languages, the “averaged translation” method should be used rather than a translation from English into the foreign language.
Our lexicon can also be used as a starting point for building specialized lexica for polarity studies. The method for adding new words makes the method of paired comparison applicable to studies with an arbitrary vocabulary because it yields accurate polarity scores even for rare words. We make our new lexicon, a GUI for adding words via test users, and the software for computing scores freely available on our website^{6}^{6}6http://informatik.hsnr.de/~dalitz/data/sentimentlexicon/.
Although our results suggest that only twofold comparisons are sufficient for estimating the score of a new, yet unknown word, this is still very timeconsuming in practice. It would thus be interesting to combine the pairedcomparison method with a corpusbased lexicon, such that only missing words or words estimated with a low confidence are added by experiments with test persons. Such a hybrid approach would be an interesting subject for further research.
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