Analyzing Effects of Public Communication onto Player Behavior in Massively Multiplayer Online Games

  • Kiran LakkarajuEmail author
  • Jeremy Bernstein
  • Jon Whetzel
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)


In this preliminary work, we study how public forum communication reflects and shapes virtual world behavior. We find that in-game groups have differential public posting habits; that player behavior is reflected in public communication (in particular, players who attack more are mentioned more in the public forums), and finally that public and personal communications are linked, those who speak together publicly also speak together privately.


MMOsBehavioral predictionForum behaviorMMORPG 

1 Introduction

Massively multiplayer online games (MMOGs) are a fruitful domain to study a variety of social and behavioral issues. One interesting question is understanding how a players’ behavior in the game reflects, or is shaped by a players’ “real world” characteristics. In this work, we focus on how communicative behavior of players interacts with their behaviors within a game.

We cast this exploration through the idea of creating Richly communicative nonplayer characters (RC-NPCs). Socialization is an important motivation for playing massively multiplayer online games (MMOGs) [1]. Many games rely on socialization between players, however in some cases, there may be a lack of players to socialize with, which can be especially true in the initial stages of a game. Current technologies for nonplayer characters (NPC) do not provide rich interaction between players, often relying on prescripted dialog trees.

We suggest the development of agents that have capabilities for rich communication with other players. Unlike traditional NPCs, which focus on providing limited services (such as a shopkeeper) to a player, we envision agents that mimic the behavior of real players. These NPCs will act like a player, trading, fighting, and even communicating with other players and NPCs. We call these agents richly communicative nonplayer characters or RC-NPCs for short.

While there have been many studies on the behaviors of players, but less on understanding the communicative behaviors of players. Data have been collected for a few games, but these often focus on the personal communication between players [2]. Another aspect of communication are the “public forums”—places where all players in a game may post news and thoughts which can be commented on by other players1.

Forums often have lively conversations and are places to discuss opinions (often highly controversial) on other players, game modifications, and activity within the game. Forums are also a place for players to recruit others. Since forums are public and can be viewed by anyone, they are often the first measure of the social interaction in a game. Even though only a small fraction of players post, many people read the posts2.

We will use forum data as a representation of the “real world” behavior of players. The unconstrained nature of forums, where players can write nearly whatever they want, even if it is not relevant to the game, allows for the explicit and implicit expression of attitudes and sentiment. For instance, there is much work on inferring emotional states and other aspects from subject text [3, 4]. Forum data are also easily gathered, which is another benefit of using it.

In this preliminary chapter, we study the the relationship between communicating publicly and behavior within the games. We will discuss three patterns found within a 2-year data set of game behavior obtained by the authors:
  1. 1.

    In-game groups have differential public posting habits. In particular, the notion of “country” and “race” (which are somewhat tied together), impacts public communication frequency.

  2. 2.

    Public communication reflects in-game behavior. Specifically, we find a correlation between players who attack and players who are mentioned publicly.

  3. 3.

    Public and personal communications are linked. Specifically, those who talk to each other publicly also talk to each other in private.


By studying these relationships, we hope to increase our knowledge of real world/virtual world interaction.

2 Related Work

Learning behavior profiles is the closest work to ours. We can divide these into two categories: those that focus on player demographics and linking to in-game behavior (what we call the “demographic” aspect), and those who categorize players based solely on in-game, often kinetic behavior.

The main methodology in the demographic perspective is the use of surveys and other instruments to ascertain personality traits [5, 6], or motivations for play, [1, 7] and other factors. These are then linked to behavior in the game, including communication.

This study forms an important part of the puzzle by providing us clues on how people will play. However, they focus on the link between human and in-game. In creating RC-NPCs, we do not necessarily need to know the personality characteristics, but rather the characteristics of play in game. This leads us to behavioral profiling.

Many studies have attempted to identify roles and player types just from “telemetry data” [8]. Drachen and others [8, 9] used unsupervised clustering approaches to find evidence for different types of players with a variety of skills. For instance, players in BattleField 2: Bad Company 2 were divided into roles like “assault specialist,” “medical-engineer,” and “sniper” [8].

Other behavioral profiling approaches focused on churn prediction [10].

Most of these works have focused on what we call “kinetic” behaviors—actions that can directly change the state of the game world. This can include combat and trading activities, participating in raids, etc. Only a few have studied communicative behaviors, which is of interest for building bots for socialization. Ducheneaut and Moore [2] studied interaction patterns in Star Wars Galaxies.

Our work is different from these in terms of our end goal. Since we want to design RC-NPCs, we need to know how in-game action affects communication behavior.

3 Description of Game X

Game X3 is a browser-based exploration game which has players acting as adventurers owning a vehicle and traveling a fictional game world. There is no winning in Game X, rather players freely explore the game world and can mine resources, trade, and conduct war. There is the concept of money within Game X, which we refer to as marks. To buy vehicles and travel in the game world, players must gather marks. There is a vibrant market-based economy within Game X.

Players can communicate with each other through in-game personal messages, public forum posts, and in chat rooms. Players can also denote other players as friends or as hostiles. Players can take different actions, such as:
  1. 1.

    Move vehicle

  2. 2.

    Mine resources

  3. 3.

    Buy/sell resources

  4. 4.

    Build vehicles, products, factory outlets

  5. 5.

    Fight nonplayer characters (NPCs)

  6. 6.

    Fight other players.


Players can use resources to build factory outlets and create products that can be sold to other players.

Unlike other MMO’s like World of Warcraft (WoW) and Everquest (EV), Game X applies a “turn system.” Every day each player gets an allotment of “turns.” Every action (except communication) requires some number of turns to execute. For instance, if a player wants to move his vehicle by two tiles, this would cost, say, 10 turns. Turns can be considered a form of “energy” that players have.

The use of turns has two major impacts:
  1. 1.

    Players with varying time commitments can play together. Since everyone is limited to the same amount of actions per day, players with minimal time on their hands are not at a disadvantage. In contrast, in WoW, players leveling and experience can depend highly on the amount of time they play (e.g., “grinding”).

  2. 2.

    Players have to think about their moves ahead of time. Because there is a limit on turns, players must think and plan ahead before making their moves.

Fig. 1

Overview of the Game X world

Figure 1 is a schematic depicting the playing space of Game X. Players move from tile to tile in their vehicles. Tiles can contain resources and/or factory outlets and market centers. Only one factory outlet/market center may exist on a tile. The world is 2D, and does not wrap around.

Players can gather resources from tiles and sell them to market centers. Factory outlets allow the creation of new goods from resources—i.e., producing steel from iron ore. More advanced factory outlets exist which can create more advanced objects—i.e., taking steel and producing a sword. Gathering resources and selling to factory outlets is the main way of gaining marks in Game X.

Factory outlets and market centers can be built by players. Creating these structures is relatively straightforward and does not take much in terms of marks or experience. The difficulty lies in maintaining the structures. In order to prosper, the structures require certain resources. Once built, supplying your structures with the necessary resources can be time consuming. Joining a guild (see below) can be helpful as members of the guild can supply your structure.

Players can also engage in combat with NPCs, other players, and even market centers and factory outlets. Players can modify their vehicles to include new weaponry and defensive elements. Players have “skills” that can impact their ability to attack/defend.

3.1 Groups in Game X

There are four types of groups a player may belong to. Table 1 summarizes the properties of these.

3.2 Nations

There are three nations a player may join. We label them as A,B, and C. A player may choose not to join a nation as well.

Nations are fixed and defined by the game creators. Nation membership is open, players may join any nation they wish at any time and leave at any time.

Joining a nation provides several benefits:
  1. 1.

    Access to restricted, “nation controlled” areas

  2. 2.

    Access to special quests

  3. 3.

    Access to special vehicles and add-ons.


Nations have different strengths; one nation may be better suited for weaponry, and thus has more weaponry related add-ons. Another may be suited for trading.

Completing quests for a nation increases a player stature toward the nation which leads to access to special vehicles and add-ons.

Wars occur between nations.

3.3 Agency

An agency can be thought of as a social category. There are two agencies, X and Y. A player can only be a part of one agency at any time. To gain membership to an agency, certain requirements need to be met, but if those are met, anyone can join the agency.

Certain vehicles are open to particular agencies.

3.4 Race

Players may chose their race when they create a character. Different races have strengths in certain areas, implemented as different initial levels of skill. Race is fixed and cannot be changed once chosen. Race also determines starting location.

Race does not seem to play a strong role in the dynamics of the game.

3.5 Guild

Game X also allows the creation of player led guilds. These guilds allow members to cooperate to gain physical and economic control of the game world. Guilds comprise a leader and board who form policy and make decisions that impact the entire guild membership.

Guilds can be created by any player once they have met experience and financial requirements. Guilds have a minimum membership of one, and no upper limit on size.

Apart from the officers, there are the “privileged guild members”—a special set of guild players which is considered important. Finally there are the regular guild members.

Guilds are closed—players must submit an application and can be denied membership.

Guild members have access to private communication channels.

Guilds have a “guild account” which can store marks from players (taken in the form of taxes). These marks can be redistributed at the will of the CFO.
Table 1

Summary of properties of the groups in Game X







Fixed, 3

Fixed, 2

Dynamic, many


Membership type


Open (req’s)








3.5.1 Communication in Game X

Game X includes three methods by which players can communicate with each other:
  1. 1.

    Personal messages: An email like system for communicating with other players, or in some cases, groups of players.

  2. 2.

    Public forum: A Usenet like system in which players can post topics and replies (see below).

  3. 3.

    Chat roomt: An IM like system for players to chat with others in their guild.

The structures of the forums are shown in Fig. 2.
Fig. 2

Forum structure in Game X. Each forum can have multiple topics, and each topic can have multiple posts

Each forum post includes the name of the players who posted an image of their avatar in the game, and their guild affiliation.

3.5.2 Forum Based Measures

We have data on more than 700 days from the game. Included in this dataset are posts from seven different forums within the game. One of the forums is meant for role playing (RP) discussion, that is, all players must discuss in the role of their character. One forum is meant for Nonrole playing discussion (NRP). The rest of the forums are both RP and NRP.

Table 2 provides some high level statistics of the forums. We can see that forum 2, which was only RP, was the most popular in terms of posts. However, the number of topics was low—indicating higher average topic lengths.
Table 2

Overview of post/authors/and topics for each forum. Italic entries are the max values for the column. These are calculated over the entire 737-day period


# Posts

# Authors

# Topics


1. (NRP)





2. (RP)





3. (NRP/RP)





4. (NRP/RP)





5. (NRP/RP)





6. (NRP/RP)





7. (NRP/RP)





Fig. 3

New posts per day from day 5 to 739. Highlighted time periods indicate the two wars and the evaluation period

Figure 3 shows the posts per day over the entire time period of the dataset. We chose the 50-day time period starting from day 500 as our evaluation period. This time period had a relatively stable rate of posts per day, new players per day, and active players per day. Our goal was to reduce the impact of posting behavior from “newbies.” We mean the period from 500 to 550 in the following when we use the term “evaluation period.”

There are three ways of communicating with others on the forum:
  1. 1.

    Create a new topic.

  2. 2.

    Post on a topic created by another player.

  3. 3.

    Post on a topic and quote another player.


In this work we only consider the second type of public interaction, which we call “coposting.”

Definition: Coposters

The coposters of a player p as all players who have posted in a topic that player p has also posted in.

In Fig. 2, players p1 and p2 are coposters, as are p0 and p1; and p0 and p2 because they all have posted on the same topic (t0).

We choose to study “coposting” because it directly encodes participation in a conversation and it can be measured in many types of social media.

Players can reference others’ posts and players by either quoting them or mentioning them in a post. A players’ “mentions” is the number of times he was mentioned in another post.

4 Characteristics of Posters

What motivates players to publicly communicate? One part of the equation is the personality of the player [5]. We hypothesize that other, in-game factors can influence communication behavior as well. This could, for instance, be due to norms among subgroups.

In this section, we focus on identifying in-game characteristics of players that could influence public communication habits. These include groupings within the game (nation, guild, race, and agency), and the sex and experience of a player.

4.1 Evaluation

Table 3 describes some statistics about the forums during the evaluation period. Figure 3 indicates that this period was relatively stable in terms of number of posts. Even so, there were many active topics and newly created topics. The number of players (750) gives us a good sample of the population. The total number of active players during the evaluation time period was approximately 12,000, so the number of posters was about 6 % of the active player population during the evaluation time period.
Table 3

Context of posting evaluation period



# of posts


# of topics


# new topics


# of posters


Figure 4 shows, per day, several measures:
  1. 1.

    # of posters: The number of players who posted during the day (provided for context).

  2. 2.

    # of posts: The number of posts created that day (provided for context).

  3. 3.

    % Male: Fraction of the players who posted that day that were male.

  4. 4.

    % Guild: Fraction of the players who posted that day that were in a guild.

  5. 5.

    Entropy over race: The entropy over the proportions of players in the different races.

  6. 6.

    Entropy over nation: The entropy over the proportions of players in the different nations.

  7. 7.

    Entropy over agency: The entropy over the proportions of players in the different agencies.

Fig. 4

Per day measures. See text for details

We can see relatively stable behavior across all measures.

The “% male” measure varied between 0.8 and 0.9. This makes sense, as the overwhelming majority of characters in Game X are male.

Percentage of posters who were in a guild showed a 10 % range as well, from 0.875 to 0.975. The high correspondence could be because of a confound—those who have more experience participate in guilds more and they communicate more. To check this, we calculated two variables tracking the total amount of turns a player spent in his lifetime by the end of the evaluation period (“LifetimeTurns”) and whether the player posted during the evaluation period (“EvalPosted”). We then binned players based on LifetimeTurns, and calculated the percentage of players who were part of a guild and who posted during evaluation period.

Figure 5 shows the results. As expected, we see that most players with a high “LifetimeTurns” are in guilds, and in addition, they are more likely to post. Thus, we don’t know yet whether posting is due to being in a guild or because they are high in turns.
Fig. 5

Communication and guild membership as a function of experience

Figure 4 also indicated a relatively uniform distribution over Race and Nation. However, further observation indicated a statistically significant difference in mean proportion over days (Race: ANOVA with F = 144.7, d.f = 3.0, p value\(<2.2^{-16}\); Nation: ANOVA with F = 46.86, d.f. = 3.0, p value\(<2.2^{-16}\)). It turns out that Nation 1 and Race 4 are disproportionately involved in posting.

The reason for Nation 1 and Race 4 being more involved in public posting is most likely game specific. One explanation could be the aggressiveness of Nation 1 of the two large scale conflicts in our dataset, Nation 1 participated in both. Note that the evaluation period is between the two wars, and thus may reflect discussion of War 1 (by Nation 1), and may foreshadow War 2. Nation 1 may be making threats during this period that will escalate to war. Further work with the text of the public forum posting can help determine that.

The difference in race and posting may be tied to nation as well. While any race may join any nation, there are certain normative pairings. It turns out that Race 4 is one of two races that is usually aligned with Nation 1.

Most of the players (mean \(\approx 0.8\)) who posted did not have an agency affiliation.

4.2 Discussion

In general, we can say that public posting is not just a factor of player personality, but could also be influenced by groupings within the game. Future models of player behavior may not need models of personalities, but can assume certain traits based on in-game characteristics.

Further analysis, over other games, is needed to determine the factors that could cause a specific nation/race to be more prevalent. One thought is a selection bias, perhaps individuals who are more prone to communicating choose these specific nations/races. This could be true, however, players need to choose a race before they can even view the forums, thus they do not know the distribution of posters beforehand.

Another reason may be game specific. Territorial conditions, or the advent of specific leaders may encourage public communication. An interesting avenue of work may be to check these communication stats over other periods of time. The evaluation period we choose is between the two wars, so that may be one reason for the proportion of Nation 1 communications.

5 Attacks to Mentions

In this section we study the relationship between in-game kinetic behavior, such as attacking another player, and it’s relation to publicity.

As mentioned earlier, players can attack and destroy the vehicles and buildings of other players. These attacks are somewhat rare as they are difficult to execute and take a reasonable amount of resources. Since these attacks are rare, we hypothesized that attackers may be spoken about in the public forums. Personal experience with the forums indicate this is the case.

To test this hypothesis, we collected data from Game X on player combat behavior and player “mentions” in the forums. If the hypothesis is true, we would expect a correlation between the number of combat events a player makes, and their mentions within the forum.

Our data set consisted of 752 players from across the entire data set who had been active during the entire period. We tracked the number of times that each player had:
  • attacked another player

  • attacked a players’ market center

  • attacked a players’ factory.

We removed all players who did not participate in any combat actions at all.

We also tracked the number of mentions of the player in the public forums over the entire period of time. A confounder in the analysis is any aspect that may also influence a player being mentioned in the forums. Naturally, players who post more will be mentioned more often. In addition, players who have played more will also be mentioned more often. To address this, we segmented the population on turns used and postings according to the 33 and 66 % quantiles and labeled the players as either low/medium/high turns and low/medium/high posting.

To assess the relationship between these two variables, we calculate the Pearson product–moment correlation coefficient between combat activity and mentions. Due to the nonnormal distribution, a rankit4 transformation was first applied [11]. Table 4 shows the Pearson product–moment correlation coefficient (r) and 95 % confidence interval, segmented by the low/medium/high turns and posting values. Bold face entries are significant at 95 %.
Table 4

Correlations between combat activity and mentions


Low turns

Medium turns

High turns

Low posting





(-0.110, 0.230)

(-0.038, 0.385)

(-0.286, 0.329)

Medium posting





(0.014, 0.441)

(0.147, 0.516)

(0.056, 0.472)

High posting





(0.092, 0.632)

(0.116, 0.515)

(0.336, 0.597)

5.1 Discussion

The results indicate evidence for a correlation between combat behavior and mentions. This is mediated by the amount of posting one does, players who never post do not have significant correlations. For medium and high posters, however, significant and high correlations occur.

While only content analysis can tell us what is being said about players, we can clearly see a relationship between in-game behavior and posting on the public forum. Note that these are kinetic actions that translate to communicative actions by other players.

The impact of this future behavioral modeling is important to note. RC-NPCs may need to discuss current events and other players in the world.

6 Public and Private Overlap

Our final question is on the relationships between individuals. Will players who interact in the game world also communicate with each other in the public realm? This is an important question for other social media as well.

We addressed this question by comparing the coposting network (described in Sect. 3.5.2 and below) with other “private relationship” networks.

Usenet type discussion boards, due to their similarity to the forums in Game X, will have a notion of coposting. Facebook discussions or comment streams can also be analyzed for coposting behavior. Comment threads (for instance in the social bookmarking/commenting site Reddit) can also be analyzed for coposting behavior.

We constructed the “coposter” network, denoted by G cp by calculating the coposters for every player during the evaluation period of our data set. The nodes in the network are players, and an edge exists between nodes if either of the players is a coposter to the other. The edges are undirected. Each edge is weighted by the number of topics that both players posted on. So a value of 5 would indicate that the two players have both posted on 5 different topics during the evaluation time span. Self edges were removed—thus orphan topics (with no other posts except the original), were not counted.

The degree distribution of G cp (not shown here due to space constraints) indicates many people that only have a single coposter (indicating topics with only two posts). There are several people with an extremely high degree, which was most likely due to participation in topics with many posts that spanned many months.

6.1 Private Relationship Measures

For each edge in the G cp between players p i and p j we measured the following variables between the players:
  • Friendship Are either of the players friends of each other?

  • Hostility Are either of the players hostile to each other?

  • Personal Messaging] How many personal messages occurred between p i and p j ? (threshold of five)

  • Trades How many trades occurred between p i and p j ? (threshold of five)

  • Nation Are the two players part of the same nation?

  • Guild Are the two players part of the same guild?

We listed two players as communicating via personal messages if they sent and/or received more than five messages. We listed two players as traders if there have been more than five trades between the players. These thresholds were set in order to remove spurious relationships.

The link overlap on relationship R is the percentage of players who are coposters and have the relationship R. For instance, a link overlap of 0.7 on relationship “Friendship” means that 70 % of coposters also were friends.

Recall that several of the topics had hundreds of posts. In these cases, it may be that players were responding to long running topics (such as a feature proposal topic). Coposting on such a topic may not indicate a relationship between players. To address this, we calculate the link overlap by only considering pairs of players who had coposted on several topics—i.e., filtering edges based on edge weights.

More precisely, let N(x) be the set of all edges (pairs of nodes \((i,j)\)) in G cp such that the weight on the edge is greater than or equal to x. The link overlap for coposting threshold x is defined as:
$$L(x) = \frac{1}{|N(x)|} \sum_{i,j \in N(x)} M_{R}(i,j) $$
$$ M_{R}(i,j) =\left\{\begin{array}{ll} 1 & \text{If}\, \text{(i,j)}\, \text{satisfy the relation}\, R \\ 0 & \text{otherwise}\end{array}\right.$$
\(M_{R}(i,j)\) represents the private action measure R. For instance, M friendship is 1 if the two players i and j are friends. M communication is 1 if the two players had exchanged more than five messages.
Fig. 6

Link overlap between coposting and private interaction measures

7 Results and Discussion

Figure 6 shows the link overlap for different coposting thresholds.

We can see that nation affiliation had a steady overlap value of approximately 0.255 and peaking at 0.355. Nations do play a role in the large scale conflicts that take place in Game X. The evaluation time period started 40 days after the end of the first war, so it is possible that nation affiliation was still an important attribute. Another factor with nation affiliation: there are only three nations (plus “unaffiliated”). The high overlap value may be just a random effect. Note that we consider two people who are unaffiliated to be in the same nation. Further work will try to identify why this overlap is so high.

Surprisingly, guild overlap was quite low for all values of the coposting threshold. Guilds play an important role in the game and nearly every veteran player is part of a guild. Thus, guild member, intuitively, should have been high. One possible reason for this is the availability of other communication mechanisms. Guilds have a private chat room and have the ability to send personal messages to all other members of a guild. Thus, guild members may not need to communicate via public forum posting.

In contrast, however, results point to strong ties communicating by multiple means [12]. Assuming that guild relationships are strong, we would expect to see communication on multiple modalities. Further work is needed to explore this.

Trading overlap was quite low as well. A key component of Game X is the necessity to trade with other players. This is the primary method of getting marks. Thus, one would expect players to trade with many others, and thus have a high overlap. The lack of such is surprising. There could be two reasons for this:
  1. 1.

    Players may focus on trading with guild members. Guilds often “own” areas of the game world and setup their own economic systems. These often limit trade to members of the guild. Thus, while it is possible to trade with anyone, practically players may only trade with a limited number.

  2. 2.

    Geographical proximity may limit trading to a few locations. Since all movement takes some amount of turns, players may restrict themselves to small areas for trading, thus reducing the number of players they trade with.


Hostile overlap was quite minimal. This indicates players who were listed as hostile to each other did not copost publicly. This makes sense intuitively, if we consider coposting as a measure of the bond between players. However, in some cases, coposting can be used to “troll” others, that is, provide insulting or negative messages. These results show that while that may exist, it does not seem to have a large impact.

Friend overlap was higher than guild, trade and hostile overlap, but experienced change as a function of the coposting threshold, going from close to 0.025 to 0.15. Friendship relationships are of relatively low number, unlike other measures. Thus, they may be more affected by the spurious edges in the coposting network. There could also be a relationship between friendship and coposting, indicating that friends are more likely to copost.

The personal messaging overlap is the most interesting aspect. It is the largest by far, starting at close to 0.5 and peaking at a little higher than 0.7. It seems from this that public and private communications do interrelate.

8 Conclusions, Discussion and Future Work

The development of RC-NPCs can help increase socialization in MMOGs even when there are not many people playing the MMOGs yet. This will lead to a more rewarding experience for the players. To build RC-NPCs, we need an understanding of how in-game behavior relates to in-game communication. We focus on a somewhat neglected area, public communications via in-game forums and identify patterns of behavior and communication that can be used to develop RC-NPCs.

Through an analysis of 2 years worth of data from the MMOG Game X, we have identified the following general patterns:
  • Player communicative behavior is influenced by in-game groups.

  • Player behavior in-game is affected by his “popularity/notoriety” in the public communication sphere.

  • Player relationships in-game are reflected in relationships via public communication channels.

These three patterns can help progress toward agent-based models of communicative agents.

Our future work will focus on content analysis of the posts. An intriguing question is what we call “Expression to Action” (E2A). When does talk about action translate into action? Through our data set we can evaluate this relationship.


  1. 1.

    For instance, World of Warcraft offers a forum at

  2. 2.

    For instance, in the data set we investigate below, public forums were viewed more than 5 million times, even though the number of poster was a tiny fraction of the people who actually played.

  3. 3.

    To preserve the confidentiality of the game, we have anonymized some of the terms describing the game. No descriptions of gameplay dynamics have been changed.

  4. 4.

    Rankit is the Rank Inverse Log Transformation function: \(f(x) = \Phi^{-1}((x_{r}-.5)/n)\), where x r is the rank of x and \(\Phi^{-1}\) is the inverse normal cumulative distribution function.



Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the US Department of Energys National Nuclear Security Administration under contract DE-AC04-94AL85000.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kiran Lakkaraju
    • 1
    Email author
  • Jeremy Bernstein
    • 1
  • Jon Whetzel
    • 1
  1. 1.Sandia National LabsAlbuquerqueUSA

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