Introduction This research paper delves into the theoretical aspects of association rules for empirical analysis. We discuss the key components to consider when using association rules for empirical analysis, such as support, confidence, and lift, and present R markdown code examples to demonstrate the application of these methods. By understanding the theoretical underpinnings of association rules, researchers can more effectively utilize these techniques in empirical studies. Association rules are a popular method for discovering relationships between variables in large datasets. They are commonly employed in market basket analysis, where the goal is to uncover associations between items that customers purchase together. In empirical research, association rules can be used to explore relationships among variables, thus providing valuable insights for decision-making processes. In this paper, we discuss the analytical elements one should consider when using association rules for empirical analysis.
Support, Confidence, and Lift When applying association rules to empirical research, three key measures should be considered: support, confidence, and lift. These measures assess the strength and significance of the discovered associations.
Support Support refers to the proportion of transactions in the dataset that contain a particular itemset. In empirical analysis, a higher support value indicates that the association is more common and therefore more relevant.
# example for calculating support
library(arules)## Loading required package: Matrix##
## Attaching package: 'arules'## The following objects are masked from 'package:base':
##
## abbreviate, writedata("Adult")
# Generate association rules
rules <- apriori(Adult, parameter = list(supp = 0.2, conf = 0.8, target = "rules"))## Apriori
##
## Parameter specification:
## confidence minval smax arem aval originalSupport maxtime support minlen
## 0.8 0.1 1 none FALSE TRUE 5 0.2 1
## maxlen target ext
## 10 rules TRUE
##
## Algorithmic control:
## filter tree heap memopt load sort verbose
## 0.1 TRUE TRUE FALSE TRUE 2 TRUE
##
## Absolute minimum support count: 9768
##
## set item appearances ...[0 item(s)] done [0.00s].
## set transactions ...[115 item(s), 48842 transaction(s)] done [0.02s].
## sorting and recoding items ... [18 item(s)] done [0.00s].
## creating transaction tree ... done [0.01s].
## checking subsets of size 1 2 3 4 5 6 7 done [0.00s].
## writing ... [1306 rule(s)] done [0.00s].
## creating S4 object ... done [0.00s].# Inspect support values
inspect(head(sort(rules, by = "support")))## lhs rhs support confidence
## [1] {} => {capital-loss=None} 0.9532779 0.9532779
## [2] {} => {capital-gain=None} 0.9173867 0.9173867
## [3] {} => {native-country=United-States} 0.8974243 0.8974243
## [4] {capital-gain=None} => {capital-loss=None} 0.8706646 0.9490705
## [5] {capital-loss=None} => {capital-gain=None} 0.8706646 0.9133376
## [6] {} => {race=White} 0.8550428 0.8550428
## coverage lift count
## [1] 1.0000000 1.0000000 46560
## [2] 1.0000000 1.0000000 44807
## [3] 1.0000000 1.0000000 43832
## [4] 0.9173867 0.9955863 42525
## [5] 0.9532779 0.9955863 42525
## [6] 1.0000000 1.0000000 41762Confidence Confidence measures the likelihood that a certain association occurs, given the presence of a particular itemset. In other words, it is the conditional probability of finding the consequent itemset when the antecedent itemset is present.
# example for calculating confidence
library(arules)
data("Adult")
# Generate association rules
rules <- apriori(Adult, parameter = list(supp = 0.2, conf = 0.8, target = "rules"))## Apriori
##
## Parameter specification:
## confidence minval smax arem aval originalSupport maxtime support minlen
## 0.8 0.1 1 none FALSE TRUE 5 0.2 1
## maxlen target ext
## 10 rules TRUE
##
## Algorithmic control:
## filter tree heap memopt load sort verbose
## 0.1 TRUE TRUE FALSE TRUE 2 TRUE
##
## Absolute minimum support count: 9768
##
## set item appearances ...[0 item(s)] done [0.00s].
## set transactions ...[115 item(s), 48842 transaction(s)] done [0.02s].
## sorting and recoding items ... [18 item(s)] done [0.00s].
## creating transaction tree ... done [0.01s].
## checking subsets of size 1 2 3 4 5 6 7 done [0.00s].
## writing ... [1306 rule(s)] done [0.00s].
## creating S4 object ... done [0.00s].# Inspect confidence values
inspect(head(sort(rules, by = "confidence")))## lhs rhs support confidence coverage lift count
## [1] {relationship=Husband} => {sex=Male} 0.4036485 0.9999493 0.4036690 1.495851 19715
## [2] {marital-status=Married-civ-spouse,
## relationship=Husband} => {sex=Male} 0.4034028 0.9999492 0.4034233 1.495851 19703
## [3] {relationship=Husband,
## capital-loss=None} => {sex=Male} 0.3778715 0.9999458 0.3778920 1.495845 18456
## [4] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-loss=None} => {sex=Male} 0.3776258 0.9999458 0.3776463 1.495845 18444
## [5] {relationship=Husband,
## race=White} => {sex=Male} 0.3656075 0.9999440 0.3656279 1.495843 17857
## [6] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White} => {sex=Male} 0.3654027 0.9999440 0.3654232 1.495843 17847Lift
Lift evaluates the strength of an association rule by comparing the confidence of the rule to the support of the consequent itemset. A lift value greater than one indicates that the antecedent and consequent itemsets are positively correlated.
# example for calculating lift
library(arules)
data("Adult")
# Generate association rules
rules <- apriori(Adult, parameter = list(supp = 0.2, conf = 0.8, target = "rules"))## Apriori
##
## Parameter specification:
## confidence minval smax arem aval originalSupport maxtime support minlen
## 0.8 0.1 1 none FALSE TRUE 5 0.2 1
## maxlen target ext
## 10 rules TRUE
##
## Algorithmic control:
## filter tree heap memopt load sort verbose
## 0.1 TRUE TRUE FALSE TRUE 2 TRUE
##
## Absolute minimum support count: 9768
##
## set item appearances ...[0 item(s)] done [0.00s].
## set transactions ...[115 item(s), 48842 transaction(s)] done [0.02s].
## sorting and recoding items ... [18 item(s)] done [0.00s].
## creating transaction tree ... done [0.01s].
## checking subsets of size 1 2 3 4 5 6 7 done [0.00s].
## writing ... [1306 rule(s)] done [0.00s].
## creating S4 object ... done [0.00s].# Inspect lift values
inspect(head(sort(rules, by = "lift")))## lhs rhs support confidence coverage lift count
## [1] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## native-country=United-States} => {relationship=Husband} 0.3382335 0.9931466 0.3405675 2.460299 16520
## [2] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## capital-gain=None,
## native-country=United-States} => {relationship=Husband} 0.2967528 0.9929438 0.2988616 2.459797 14494
## [3] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {relationship=Husband} 0.3159371 0.9927942 0.3182302 2.459427 15431
## [4] {workclass=Private,
## marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## native-country=United-States} => {relationship=Husband} 0.2164735 0.9926767 0.2180705 2.459136 10573
## [5] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## capital-gain=None,
## capital-loss=None,
## native-country=United-States} => {relationship=Husband} 0.2744564 0.9925218 0.2765243 2.458752 13405
## [6] {marital-status=Married-civ-spouse,
## sex=Male,
## native-country=United-States} => {relationship=Husband} 0.3628025 0.9923279 0.3656075 2.458272 17720Pruning and Rule Selection
In empirical analysis, it is crucial to filter out irrelevant or redundant association rules to focus on the most significant ones. Pruning techniques, such as setting minimum thresholds for support and confidence, can help achieve this goal.
# example for pruning association rules
library(arules)
data("Adult")
# Generate association rules with minimum support and confidence thresholds
rules <- apriori(Adult, parameter = list(supp = 0.3, conf = 0.9, target = "rules"))## Apriori
##
## Parameter specification:
## confidence minval smax arem aval originalSupport maxtime support minlen
## 0.9 0.1 1 none FALSE TRUE 5 0.3 1
## maxlen target ext
## 10 rules TRUE
##
## Algorithmic control:
## filter tree heap memopt load sort verbose
## 0.1 TRUE TRUE FALSE TRUE 2 TRUE
##
## Absolute minimum support count: 14652
##
## set item appearances ...[0 item(s)] done [0.00s].
## set transactions ...[115 item(s), 48842 transaction(s)] done [0.02s].
## sorting and recoding items ... [14 item(s)] done [0.00s].
## creating transaction tree ... done [0.01s].
## checking subsets of size 1 2 3 4 5 6 done [0.00s].
## writing ... [326 rule(s)] done [0.00s].
## creating S4 object ... done [0.00s].# Inspect pruned rules
inspect(rules)## lhs rhs support confidence coverage lift count
## [1] {} => {capital-gain=None} 0.9173867 0.9173867 1.0000000 1.0000000 44807
## [2] {} => {capital-loss=None} 0.9532779 0.9532779 1.0000000 1.0000000 46560
## [3] {education=HS-grad} => {capital-gain=None} 0.3020966 0.9348074 0.3231645 1.0189895 14755
## [4] {education=HS-grad} => {capital-loss=None} 0.3111052 0.9626837 0.3231645 1.0098668 15195
## [5] {marital-status=Never-married} => {capital-gain=None} 0.3161214 0.9579947 0.3299824 1.0442649 15440
## [6] {marital-status=Never-married} => {capital-loss=None} 0.3199910 0.9697214 0.3299824 1.0172494 15629
## [7] {sex=Female} => {capital-gain=None} 0.3123132 0.9420702 0.3315180 1.0269063 15254
## [8] {sex=Female} => {capital-loss=None} 0.3201753 0.9657856 0.3315180 1.0131207 15638
## [9] {relationship=Husband} => {marital-status=Married-civ-spouse} 0.4034233 0.9993914 0.4036690 2.1811642 19704
## [10] {relationship=Husband} => {sex=Male} 0.4036485 0.9999493 0.4036690 1.4958506 19715
## [11] {relationship=Husband} => {race=White} 0.3656279 0.9057618 0.4036690 1.0593175 17858
## [12] {relationship=Husband} => {capital-loss=None} 0.3778920 0.9361432 0.4036690 0.9820255 18457
## [13] {marital-status=Married-civ-spouse} => {capital-loss=None} 0.4293026 0.9369498 0.4581917 0.9828716 20968
## [14] {age=Middle-aged} => {capital-gain=None} 0.4632079 0.9170281 0.5051185 0.9996091 22624
## [15] {age=Middle-aged} => {capital-loss=None} 0.4800786 0.9504276 0.5051185 0.9970100 23448
## [16] {income=small} => {capital-gain=None} 0.4849310 0.9581311 0.5061218 1.0444135 23685
## [17] {income=small} => {capital-loss=None} 0.4908480 0.9698220 0.5061218 1.0173549 23974
## [18] {hours-per-week=Full-time} => {capital-gain=None} 0.5435895 0.9290688 0.5850907 1.0127342 26550
## [19] {hours-per-week=Full-time} => {capital-loss=None} 0.5606650 0.9582531 0.5850907 1.0052191 27384
## [20] {sex=Male} => {capital-gain=None} 0.6050735 0.9051455 0.6684820 0.9866565 29553
## [21] {sex=Male} => {capital-loss=None} 0.6331027 0.9470750 0.6684820 0.9934931 30922
## [22] {workclass=Private} => {capital-gain=None} 0.6413742 0.9239073 0.6941976 1.0071078 31326
## [23] {workclass=Private} => {capital-loss=None} 0.6639982 0.9564974 0.6941976 1.0033773 32431
## [24] {race=White} => {native-country=United-States} 0.7881127 0.9217231 0.8550428 1.0270761 38493
## [25] {race=White} => {capital-gain=None} 0.7817862 0.9143240 0.8550428 0.9966616 38184
## [26] {race=White} => {capital-loss=None} 0.8136849 0.9516307 0.8550428 0.9982720 39742
## [27] {native-country=United-States} => {capital-gain=None} 0.8219565 0.9159062 0.8974243 0.9983862 40146
## [28] {native-country=United-States} => {capital-loss=None} 0.8548380 0.9525461 0.8974243 0.9992323 41752
## [29] {capital-gain=None} => {capital-loss=None} 0.8706646 0.9490705 0.9173867 0.9955863 42525
## [30] {capital-loss=None} => {capital-gain=None} 0.8706646 0.9133376 0.9532779 0.9955863 42525
## [31] {marital-status=Never-married,
## capital-gain=None} => {capital-loss=None} 0.3061300 0.9683938 0.3161214 1.0158567 14952
## [32] {marital-status=Never-married,
## capital-loss=None} => {capital-gain=None} 0.3061300 0.9566831 0.3199910 1.0428352 14952
## [33] {sex=Female,
## capital-gain=None} => {capital-loss=None} 0.3009705 0.9636817 0.3123132 1.0109136 14700
## [34] {sex=Female,
## capital-loss=None} => {capital-gain=None} 0.3009705 0.9400179 0.3201753 1.0246692 14700
## [35] {marital-status=Married-civ-spouse,
## relationship=Husband} => {sex=Male} 0.4034028 0.9999492 0.4034233 1.4958506 19703
## [36] {relationship=Husband,
## sex=Male} => {marital-status=Married-civ-spouse} 0.4034028 0.9993913 0.4036485 2.1811641 19703
## [37] {marital-status=Married-civ-spouse,
## sex=Male} => {relationship=Husband} 0.4034028 0.9901503 0.4074157 2.4528768 19703
## [38] {marital-status=Married-civ-spouse,
## relationship=Husband} => {race=White} 0.3654232 0.9058059 0.4034233 1.0593691 17848
## [39] {relationship=Husband,
## race=White} => {marital-status=Married-civ-spouse} 0.3654232 0.9994400 0.3656279 2.1812704 17848
## [40] {relationship=Husband,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3628230 0.9993233 0.3630687 2.1810156 17721
## [41] {relationship=Husband,
## capital-gain=None} => {marital-status=Married-civ-spouse} 0.3550018 0.9993660 0.3552271 2.1811088 17339
## [42] {marital-status=Married-civ-spouse,
## relationship=Husband} => {capital-loss=None} 0.3776463 0.9361043 0.4034233 0.9819847 18445
## [43] {relationship=Husband,
## capital-loss=None} => {marital-status=Married-civ-spouse} 0.3776463 0.9993498 0.3778920 2.1810735 18445
## [44] {relationship=Husband,
## sex=Male} => {race=White} 0.3656075 0.9057570 0.4036485 1.0593119 17857
## [45] {relationship=Husband,
## race=White} => {sex=Male} 0.3656075 0.9999440 0.3656279 1.4958427 17857
## [46] {relationship=Husband,
## native-country=United-States} => {sex=Male} 0.3630482 0.9999436 0.3630687 1.4958421 17732
## [47] {relationship=Husband,
## capital-gain=None} => {sex=Male} 0.3552066 0.9999424 0.3552271 1.4958403 17349
## [48] {relationship=Husband,
## sex=Male} => {capital-loss=None} 0.3778715 0.9361400 0.4036485 0.9820221 18456
## [49] {relationship=Husband,
## capital-loss=None} => {sex=Male} 0.3778715 0.9999458 0.3778920 1.4958454 18456
## [50] {relationship=Husband,
## race=White} => {native-country=United-States} 0.3384587 0.9256916 0.3656279 1.0314982 16531
## [51] {relationship=Husband,
## native-country=United-States} => {race=White} 0.3384587 0.9322168 0.3630687 1.0902574 16531
## [52] {relationship=Husband,
## capital-gain=None} => {race=White} 0.3212604 0.9043804 0.3552271 1.0577019 15691
## [53] {relationship=Husband,
## race=White} => {capital-loss=None} 0.3421236 0.9357151 0.3656279 0.9815764 16710
## [54] {relationship=Husband,
## capital-loss=None} => {race=White} 0.3421236 0.9053476 0.3778920 1.0588330 16710
## [55] {relationship=Husband,
## native-country=United-States} => {capital-loss=None} 0.3396053 0.9353747 0.3630687 0.9812193 16587
## [56] {relationship=Husband,
## capital-gain=None} => {capital-loss=None} 0.3294501 0.9274352 0.3552271 0.9728906 16091
## [57] {marital-status=Married-civ-spouse,
## sex=Male} => {race=White} 0.3684739 0.9044173 0.4074157 1.0577451 17997
## [58] {marital-status=Married-civ-spouse,
## sex=Male} => {capital-loss=None} 0.3815569 0.9365295 0.4074157 0.9824307 18636
## [59] {marital-status=Married-civ-spouse,
## race=White} => {native-country=United-States} 0.3788133 0.9226090 0.4105892 1.0280632 18502
## [60] {marital-status=Married-civ-spouse,
## native-country=United-States} => {race=White} 0.3788133 0.9249613 0.4095451 1.0817719 18502
## [61] {marital-status=Married-civ-spouse,
## race=White} => {capital-loss=None} 0.3844232 0.9362721 0.4105892 0.9821607 18776
## [62] {marital-status=Married-civ-spouse,
## native-country=United-States} => {capital-loss=None} 0.3833176 0.9359596 0.4095451 0.9818329 18722
## [63] {marital-status=Married-civ-spouse,
## capital-gain=None} => {capital-loss=None} 0.3743704 0.9283611 0.4032595 0.9738619 18285
## [64] {age=Middle-aged,
## sex=Male} => {capital-gain=None} 0.3177798 0.9085109 0.3497809 0.9903249 15521
## [65] {age=Middle-aged,
## sex=Male} => {capital-loss=None} 0.3304738 0.9448022 0.3497809 0.9911088 16141
## [66] {age=Middle-aged,
## workclass=Private} => {capital-gain=None} 0.3361451 0.9192609 0.3656689 1.0020430 16418
## [67] {age=Middle-aged,
## workclass=Private} => {capital-loss=None} 0.3481225 0.9520157 0.3656689 0.9986759 17003
## [68] {age=Middle-aged,
## race=White} => {native-country=United-States} 0.3902584 0.9174970 0.4253511 1.0223669 19061
## [69] {age=Middle-aged,
## race=White} => {capital-gain=None} 0.3884976 0.9133574 0.4253511 0.9956079 18975
## [70] {age=Middle-aged,
## race=White} => {capital-loss=None} 0.4034233 0.9484477 0.4253511 0.9949330 19704
## [71] {age=Middle-aged,
## native-country=United-States} => {capital-gain=None} 0.4106711 0.9148878 0.4488760 0.9972761 20058
## [72] {age=Middle-aged,
## native-country=United-States} => {capital-loss=None} 0.4262929 0.9496898 0.4488760 0.9962361 20821
## [73] {age=Middle-aged,
## capital-gain=None} => {capital-loss=None} 0.4381680 0.9459424 0.4632079 0.9923049 21401
## [74] {age=Middle-aged,
## capital-loss=None} => {capital-gain=None} 0.4381680 0.9127004 0.4800786 0.9948918 21401
## [75] {hours-per-week=Full-time,
## income=small} => {capital-gain=None} 0.3010114 0.9602247 0.3134802 1.0466957 14702
## [76] {hours-per-week=Full-time,
## income=small} => {capital-loss=None} 0.3041644 0.9702828 0.3134802 1.0178383 14856
## [77] {workclass=Private,
## income=small} => {capital-gain=None} 0.3488186 0.9607511 0.3630687 1.0472696 17037
## [78] {workclass=Private,
## income=small} => {capital-loss=None} 0.3526678 0.9713528 0.3630687 1.0189608 17225
## [79] {race=White,
## income=small} => {native-country=United-States} 0.3873101 0.9139089 0.4237951 1.0183687 18917
## [80] {race=White,
## income=small} => {capital-gain=None} 0.4054093 0.9566163 0.4237951 1.0427623 19801
## [81] {race=White,
## income=small} => {capital-loss=None} 0.4106916 0.9690806 0.4237951 1.0165772 20059
## [82] {native-country=United-States,
## income=small} => {capital-gain=None} 0.4313501 0.9576799 0.4504115 1.0439217 21068
## [83] {native-country=United-States,
## income=small} => {capital-loss=None} 0.4366119 0.9693622 0.4504115 1.0168727 21325
## [84] {capital-gain=None,
## income=small} => {capital-loss=None} 0.4696573 0.9685033 0.4849310 1.0159716 22939
## [85] {capital-loss=None,
## income=small} => {capital-gain=None} 0.4696573 0.9568282 0.4908480 1.0429934 22939
## [86] {sex=Male,
## hours-per-week=Full-time} => {capital-gain=None} 0.3421441 0.9188431 0.3723639 1.0015876 16711
## [87] {sex=Male,
## hours-per-week=Full-time} => {capital-loss=None} 0.3551042 0.9536482 0.3723639 1.0003884 17344
## [88] {workclass=Private,
## hours-per-week=Full-time} => {capital-gain=None} 0.3952131 0.9354495 0.4224847 1.0196894 19303
## [89] {workclass=Private,
## hours-per-week=Full-time} => {capital-loss=None} 0.4064330 0.9620063 0.4224847 1.0091562 19851
## [90] {race=White,
## hours-per-week=Full-time} => {native-country=United-States} 0.4394988 0.9109272 0.4824741 1.0150463 21466
## [91] {race=White,
## hours-per-week=Full-time} => {capital-gain=None} 0.4468081 0.9260768 0.4824741 1.0094727 21823
## [92] {race=White,
## hours-per-week=Full-time} => {capital-loss=None} 0.4618566 0.9572671 0.4824741 1.0041847 22558
## [93] {hours-per-week=Full-time,
## native-country=United-States} => {capital-gain=None} 0.4801196 0.9269507 0.5179559 1.0104253 23450
## [94] {hours-per-week=Full-time,
## native-country=United-States} => {capital-loss=None} 0.4959052 0.9574275 0.5179559 1.0043529 24221
## [95] {capital-gain=None,
## hours-per-week=Full-time} => {capital-loss=None} 0.5191638 0.9550659 0.5435895 1.0018756 25357
## [96] {capital-loss=None,
## hours-per-week=Full-time} => {capital-gain=None} 0.5191638 0.9259787 0.5606650 1.0093657 25357
## [97] {workclass=Private,
## sex=Male} => {capital-gain=None} 0.4160763 0.9110145 0.4567176 0.9930540 20322
## [98] {workclass=Private,
## sex=Male} => {capital-loss=None} 0.4341346 0.9505536 0.4567176 0.9971422 21204
## [99] {race=White,
## sex=Male} => {native-country=United-States} 0.5415421 0.9204803 0.5883256 1.0256912 26450
## [100] {sex=Male,
## native-country=United-States} => {race=White} 0.5415421 0.9051090 0.5983170 1.0585540 26450
## [101] {race=White,
## sex=Male} => {capital-gain=None} 0.5313050 0.9030799 0.5883256 0.9844048 25950
## [102] {race=White,
## sex=Male} => {capital-loss=None} 0.5564268 0.9457804 0.5883256 0.9921350 27177
## [103] {sex=Male,
## native-country=United-States} => {capital-gain=None} 0.5406003 0.9035349 0.5983170 0.9849008 26404
## [104] {sex=Male,
## native-country=United-States} => {capital-loss=None} 0.5661316 0.9462068 0.5983170 0.9925823 27651
## [105] {sex=Male,
## capital-gain=None} => {capital-loss=None} 0.5696941 0.9415288 0.6050735 0.9876750 27825
## [106] {workclass=Private,
## race=White} => {native-country=United-States} 0.5433848 0.9144157 0.5942427 1.0189334 26540
## [107] {workclass=Private,
## race=White} => {capital-gain=None} 0.5472339 0.9208931 0.5942427 1.0038221 26728
## [108] {workclass=Private,
## race=White} => {capital-loss=None} 0.5674829 0.9549683 0.5942427 1.0017732 27717
## [109] {workclass=Private,
## native-country=United-States} => {capital-gain=None} 0.5689570 0.9218444 0.6171942 1.0048592 27789
## [110] {workclass=Private,
## native-country=United-States} => {capital-loss=None} 0.5897179 0.9554818 0.6171942 1.0023119 28803
## [111] {workclass=Private,
## capital-gain=None} => {capital-loss=None} 0.6111748 0.9529145 0.6413742 0.9996188 29851
## [112] {workclass=Private,
## capital-loss=None} => {capital-gain=None} 0.6111748 0.9204465 0.6639982 1.0033354 29851
## [113] {race=White,
## native-country=United-States} => {capital-gain=None} 0.7194628 0.9128933 0.7881127 0.9951019 35140
## [114] {race=White,
## capital-gain=None} => {native-country=United-States} 0.7194628 0.9202807 0.7817862 1.0254689 35140
## [115] {race=White,
## native-country=United-States} => {capital-loss=None} 0.7490480 0.9504325 0.7881127 0.9970152 36585
## [116] {race=White,
## capital-loss=None} => {native-country=United-States} 0.7490480 0.9205626 0.8136849 1.0257830 36585
## [117] {race=White,
## capital-gain=None} => {capital-loss=None} 0.7404283 0.9470983 0.7817862 0.9935175 36164
## [118] {race=White,
## capital-loss=None} => {capital-gain=None} 0.7404283 0.9099693 0.8136849 0.9919147 36164
## [119] {capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.7793702 0.9481891 0.8219565 0.9946618 38066
## [120] {capital-loss=None,
## native-country=United-States} => {capital-gain=None} 0.7793702 0.9117168 0.8548380 0.9938195 38066
## [121] {marital-status=Married-civ-spouse,
## relationship=Husband,
## sex=Male} => {race=White} 0.3654027 0.9058011 0.4034028 1.0593635 17847
## [122] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White} => {sex=Male} 0.3654027 0.9999440 0.3654232 1.4958427 17847
## [123] {relationship=Husband,
## race=White,
## sex=Male} => {marital-status=Married-civ-spouse} 0.3654027 0.9994400 0.3656075 2.1812703 17847
## [124] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male} => {relationship=Husband} 0.3654027 0.9916653 0.3684739 2.4566299 17847
## [125] {marital-status=Married-civ-spouse,
## relationship=Husband,
## native-country=United-States} => {sex=Male} 0.3628025 0.9999436 0.3628230 1.4958421 17720
## [126] {relationship=Husband,
## sex=Male,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3628025 0.9993233 0.3630482 2.1810155 17720
## [127] {marital-status=Married-civ-spouse,
## sex=Male,
## native-country=United-States} => {relationship=Husband} 0.3628025 0.9923279 0.3656075 2.4582715 17720
## [128] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-gain=None} => {sex=Male} 0.3549814 0.9999423 0.3550018 1.4958402 17338
## [129] {relationship=Husband,
## sex=Male,
## capital-gain=None} => {marital-status=Married-civ-spouse} 0.3549814 0.9993660 0.3552066 2.1811087 17338
## [130] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-gain=None} => {relationship=Husband} 0.3549814 0.9896684 0.3586872 2.4516830 17338
## [131] {marital-status=Married-civ-spouse,
## relationship=Husband,
## sex=Male} => {capital-loss=None} 0.3776258 0.9361011 0.4034028 0.9819813 18444
## [132] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-loss=None} => {sex=Male} 0.3776258 0.9999458 0.3776463 1.4958454 18444
## [133] {relationship=Husband,
## sex=Male,
## capital-loss=None} => {marital-status=Married-civ-spouse} 0.3776258 0.9993498 0.3778715 2.1810735 18444
## [134] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-loss=None} => {relationship=Husband} 0.3776258 0.9896974 0.3815569 2.4517548 18444
## [135] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White} => {native-country=United-States} 0.3382540 0.9256499 0.3654232 1.0314518 16521
## [136] {marital-status=Married-civ-spouse,
## relationship=Husband,
## native-country=United-States} => {race=White} 0.3382540 0.9322837 0.3628230 1.0903358 16521
## [137] {relationship=Husband,
## race=White,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3382540 0.9993951 0.3384587 2.1811723 16521
## [138] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-gain=None} => {race=White} 0.3210761 0.9044351 0.3550018 1.0577659 15682
## [139] {relationship=Husband,
## race=White,
## capital-gain=None} => {marital-status=Married-civ-spouse} 0.3210761 0.9994264 0.3212604 2.1812407 15682
## [140] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White} => {capital-loss=None} 0.3419188 0.9356791 0.3654232 0.9815386 16700
## [141] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-loss=None} => {race=White} 0.3419188 0.9053944 0.3776463 1.0588878 16700
## [142] {relationship=Husband,
## race=White,
## capital-loss=None} => {marital-status=Married-civ-spouse} 0.3419188 0.9994016 0.3421236 2.1811864 16700
## [143] {relationship=Husband,
## capital-gain=None,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3188035 0.9992941 0.3190287 2.1809518 15571
## [144] {marital-status=Married-civ-spouse,
## relationship=Husband,
## native-country=United-States} => {capital-loss=None} 0.3393596 0.9353310 0.3628230 0.9811734 16575
## [145] {relationship=Husband,
## capital-loss=None,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3393596 0.9992765 0.3396053 2.1809136 16575
## [146] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-gain=None} => {capital-loss=None} 0.3292248 0.9273891 0.3550018 0.9728423 16080
## [147] {relationship=Husband,
## capital-gain=None,
## capital-loss=None} => {marital-status=Married-civ-spouse} 0.3292248 0.9993164 0.3294501 2.1810005 16080
## [148] {relationship=Husband,
## race=White,
## sex=Male} => {native-country=United-States} 0.3384382 0.9256874 0.3656075 1.0314935 16530
## [149] {relationship=Husband,
## sex=Male,
## native-country=United-States} => {race=White} 0.3384382 0.9322129 0.3630482 1.0902530 16530
## [150] {relationship=Husband,
## race=White,
## native-country=United-States} => {sex=Male} 0.3384382 0.9999395 0.3384587 1.4958360 16530
## [151] {relationship=Husband,
## sex=Male,
## capital-gain=None} => {race=White} 0.3212399 0.9043749 0.3552066 1.0576955 15690
## [152] {relationship=Husband,
## race=White,
## capital-gain=None} => {sex=Male} 0.3212399 0.9999363 0.3212604 1.4958312 15690
## [153] {relationship=Husband,
## race=White,
## sex=Male} => {capital-loss=None} 0.3421031 0.9357115 0.3656075 0.9815726 16709
## [154] {relationship=Husband,
## sex=Male,
## capital-loss=None} => {race=White} 0.3421031 0.9053424 0.3778715 1.0588271 16709
## [155] {relationship=Husband,
## race=White,
## capital-loss=None} => {sex=Male} 0.3421031 0.9999402 0.3421236 1.4958370 16709
## [156] {relationship=Husband,
## capital-gain=None,
## native-country=United-States} => {sex=Male} 0.3190082 0.9999358 0.3190287 1.4958305 15581
## [157] {relationship=Husband,
## sex=Male,
## native-country=United-States} => {capital-loss=None} 0.3395848 0.9353711 0.3630482 0.9812155 16586
## [158] {relationship=Husband,
## capital-loss=None,
## native-country=United-States} => {sex=Male} 0.3395848 0.9999397 0.3396053 1.4958363 16586
## [159] {relationship=Husband,
## sex=Male,
## capital-gain=None} => {capital-loss=None} 0.3294296 0.9274310 0.3552066 0.9728862 16090
## [160] {relationship=Husband,
## capital-gain=None,
## capital-loss=None} => {sex=Male} 0.3294296 0.9999379 0.3294501 1.4958335 16090
## [161] {relationship=Husband,
## race=White,
## native-country=United-States} => {capital-loss=None} 0.3161623 0.9341238 0.3384587 0.9799071 15442
## [162] {relationship=Husband,
## race=White,
## capital-loss=None} => {native-country=United-States} 0.3161623 0.9241173 0.3421236 1.0297440 15442
## [163] {relationship=Husband,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.3161623 0.9309700 0.3396053 1.0887994 15442
## [164] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male} => {native-country=United-States} 0.3405675 0.9242652 0.3684739 1.0299087 16634
## [165] {marital-status=Married-civ-spouse,
## sex=Male,
## native-country=United-States} => {race=White} 0.3405675 0.9315115 0.3656075 1.0894326 16634
## [166] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-gain=None} => {race=White} 0.3238606 0.9029054 0.3586872 1.0559769 15818
## [167] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male} => {capital-loss=None} 0.3449081 0.9360449 0.3684739 0.9819224 16846
## [168] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-loss=None} => {race=White} 0.3449081 0.9039493 0.3815569 1.0571978 16846
## [169] {marital-status=Married-civ-spouse,
## sex=Male,
## native-country=United-States} => {capital-loss=None} 0.3421031 0.9357115 0.3656075 0.9815726 16709
## [170] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-gain=None} => {capital-loss=None} 0.3328283 0.9279068 0.3586872 0.9733854 16256
## [171] {marital-status=Married-civ-spouse,
## race=White,
## capital-gain=None} => {native-country=United-States} 0.3321936 0.9211423 0.3606322 1.0264289 16225
## [172] {marital-status=Married-civ-spouse,
## capital-gain=None,
## native-country=United-States} => {race=White} 0.3321936 0.9233965 0.3597519 1.0799419 16225
## [173] {marital-status=Married-civ-spouse,
## race=White,
## native-country=United-States} => {capital-loss=None} 0.3540600 0.9346557 0.3788133 0.9804651 17293
## [174] {marital-status=Married-civ-spouse,
## race=White,
## capital-loss=None} => {native-country=United-States} 0.3540600 0.9210162 0.3844232 1.0262884 17293
## [175] {marital-status=Married-civ-spouse,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.3540600 0.9236727 0.3833176 1.0802649 17293
## [176] {marital-status=Married-civ-spouse,
## race=White,
## capital-gain=None} => {capital-loss=None} 0.3344662 0.9274441 0.3606322 0.9729000 16336
## [177] {marital-status=Married-civ-spouse,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.3335244 0.9270958 0.3597519 0.9725346 16290
## [178] {age=Middle-aged,
## workclass=Private,
## native-country=United-States} => {capital-loss=None} 0.3064985 0.9508988 0.3223250 0.9975043 14970
## [179] {age=Middle-aged,
## workclass=Private,
## capital-gain=None} => {capital-loss=None} 0.3185987 0.9478012 0.3361451 0.9942549 15561
## [180] {age=Middle-aged,
## workclass=Private,
## capital-loss=None} => {capital-gain=None} 0.3185987 0.9151914 0.3481225 0.9976071 15561
## [181] {age=Middle-aged,
## race=White,
## native-country=United-States} => {capital-gain=None} 0.3557799 0.9116521 0.3902584 0.9937490 17377
## [182] {age=Middle-aged,
## race=White,
## capital-gain=None} => {native-country=United-States} 0.3557799 0.9157839 0.3884976 1.0204581 17377
## [183] {age=Middle-aged,
## race=White,
## native-country=United-States} => {capital-loss=None} 0.3695385 0.9469073 0.3902584 0.9933171 18049
## [184] {age=Middle-aged,
## race=White,
## capital-loss=None} => {native-country=United-States} 0.3695385 0.9160069 0.4034233 1.0207065 18049
## [185] {age=Middle-aged,
## race=White,
## capital-gain=None} => {capital-loss=None} 0.3665698 0.9435573 0.3884976 0.9898030 17904
## [186] {age=Middle-aged,
## race=White,
## capital-loss=None} => {capital-gain=None} 0.3665698 0.9086480 0.4034233 0.9904744 17904
## [187] {age=Middle-aged,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.3880881 0.9450095 0.4106711 0.9913263 18955
## [188] {age=Middle-aged,
## capital-loss=None,
## native-country=United-States} => {capital-gain=None} 0.3880881 0.9103789 0.4262929 0.9923612 18955
## [189] {workclass=Private,
## native-country=United-States,
## income=small} => {capital-gain=None} 0.3066214 0.9603694 0.3192744 1.0468534 14976
## [190] {workclass=Private,
## native-country=United-States,
## income=small} => {capital-loss=None} 0.3098153 0.9703732 0.3192744 1.0179332 15132
## [191] {workclass=Private,
## capital-gain=None,
## income=small} => {capital-loss=None} 0.3384178 0.9701825 0.3488186 1.0177332 16529
## [192] {workclass=Private,
## capital-loss=None,
## income=small} => {capital-gain=None} 0.3384178 0.9595936 0.3526678 1.0460078 16529
## [193] {race=White,
## native-country=United-States,
## income=small} => {capital-gain=None} 0.3705213 0.9566527 0.3873101 1.0428021 18097
## [194] {race=White,
## capital-gain=None,
## income=small} => {native-country=United-States} 0.3705213 0.9139437 0.4054093 1.0184076 18097
## [195] {race=White,
## native-country=United-States,
## income=small} => {capital-loss=None} 0.3749437 0.9680710 0.3873101 1.0155182 18313
## [196] {race=White,
## capital-loss=None,
## income=small} => {native-country=United-States} 0.3749437 0.9129568 0.4106916 1.0173078 18313
## [197] {race=White,
## capital-gain=None,
## income=small} => {capital-loss=None} 0.3923058 0.9676784 0.4054093 1.0151063 19161
## [198] {race=White,
## capital-loss=None,
## income=small} => {capital-gain=None} 0.3923058 0.9552321 0.4106916 1.0412535 19161
## [199] {capital-gain=None,
## native-country=United-States,
## income=small} => {capital-loss=None} 0.4175505 0.9680084 0.4313501 1.0154524 20394
## [200] {capital-loss=None,
## native-country=United-States,
## income=small} => {capital-gain=None} 0.4175505 0.9563423 0.4366119 1.0424637 20394
## [201] {race=White,
## sex=Male,
## hours-per-week=Full-time} => {capital-loss=None} 0.3028132 0.9530253 0.3177388 0.9997350 14790
## [202] {sex=Male,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-loss=None} 0.3117808 0.9527623 0.3272389 0.9994591 15228
## [203] {sex=Male,
## capital-gain=None,
## hours-per-week=Full-time} => {capital-loss=None} 0.3248843 0.9495542 0.3421441 0.9960938 15868
## [204] {sex=Male,
## capital-loss=None,
## hours-per-week=Full-time} => {capital-gain=None} 0.3248843 0.9148985 0.3551042 0.9972878 15868
## [205] {workclass=Private,
## race=White,
## hours-per-week=Full-time} => {native-country=United-States} 0.3155890 0.9012981 0.3501495 1.0043165 15414
## [206] {workclass=Private,
## race=White,
## hours-per-week=Full-time} => {capital-gain=None} 0.3265223 0.9325225 0.3501495 1.0164989 15948
## [207] {workclass=Private,
## race=White,
## hours-per-week=Full-time} => {capital-loss=None} 0.3365751 0.9612326 0.3501495 1.0083446 16439
## [208] {workclass=Private,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-gain=None} 0.3440686 0.9331964 0.3686991 1.0172334 16805
## [209] {workclass=Private,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-loss=None} 0.3543057 0.9609618 0.3686991 1.0080605 17305
## [210] {workclass=Private,
## capital-gain=None,
## hours-per-week=Full-time} => {capital-loss=None} 0.3791614 0.9593846 0.3952131 1.0064059 18519
## [211] {workclass=Private,
## capital-loss=None,
## hours-per-week=Full-time} => {capital-gain=None} 0.3791614 0.9329001 0.4064330 1.0169105 18519
## [212] {race=White,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-gain=None} 0.4059826 0.9237399 0.4394988 1.0069253 19829
## [213] {race=White,
## capital-gain=None,
## hours-per-week=Full-time} => {native-country=United-States} 0.4059826 0.9086285 0.4468081 1.0124848 19829
## [214] {race=White,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-loss=None} 0.4200278 0.9556974 0.4394988 1.0025380 20515
## [215] {race=White,
## capital-loss=None,
## hours-per-week=Full-time} => {native-country=United-States} 0.4200278 0.9094335 0.4618566 1.0133818 20515
## [216] {race=White,
## capital-gain=None,
## hours-per-week=Full-time} => {capital-loss=None} 0.4261906 0.9538560 0.4468081 1.0006064 20816
## [217] {race=White,
## capital-loss=None,
## hours-per-week=Full-time} => {capital-gain=None} 0.4261906 0.9227768 0.4618566 1.0058756 20816
## [218] {capital-gain=None,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-loss=None} 0.4580689 0.9540725 0.4801196 1.0008335 22373
## [219] {capital-loss=None,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-gain=None} 0.4580689 0.9237026 0.4959052 1.0068847 22373
## [220] {workclass=Private,
## race=White,
## sex=Male} => {native-country=United-States} 0.3656894 0.9111825 0.4013349 1.0153307 17861
## [221] {workclass=Private,
## sex=Male,
## native-country=United-States} => {race=White} 0.3656894 0.9042171 0.4044265 1.0575109 17861
## [222] {workclass=Private,
## race=White,
## sex=Male} => {capital-gain=None} 0.3647066 0.9087338 0.4013349 0.9905679 17813
## [223] {workclass=Private,
## race=White,
## sex=Male} => {capital-loss=None} 0.3809631 0.9492399 0.4013349 0.9957640 18607
## [224] {workclass=Private,
## sex=Male,
## native-country=United-States} => {capital-gain=None} 0.3674297 0.9085202 0.4044265 0.9903351 17946
## [225] {workclass=Private,
## sex=Male,
## native-country=United-States} => {capital-loss=None} 0.3838500 0.9491217 0.4044265 0.9956400 18748
## [226] {workclass=Private,
## sex=Male,
## capital-gain=None} => {capital-loss=None} 0.3934933 0.9457238 0.4160763 0.9920757 19219
## [227] {workclass=Private,
## sex=Male,
## capital-loss=None} => {capital-gain=None} 0.3934933 0.9063856 0.4341346 0.9880082 19219
## [228] {race=White,
## sex=Male,
## native-country=United-States} => {capital-gain=None} 0.4881045 0.9013233 0.5415421 0.9824900 23840
## [229] {race=White,
## sex=Male,
## capital-gain=None} => {native-country=United-States} 0.4881045 0.9186898 0.5313050 1.0236961 23840
## [230] {sex=Male,
## capital-gain=None,
## native-country=United-States} => {race=White} 0.4881045 0.9028935 0.5406003 1.0559629 23840
## [231] {race=White,
## sex=Male,
## native-country=United-States} => {capital-loss=None} 0.5113632 0.9442722 0.5415421 0.9905529 24976
## [232] {race=White,
## sex=Male,
## capital-loss=None} => {native-country=United-States} 0.5113632 0.9190124 0.5564268 1.0240556 24976
## [233] {sex=Male,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.5113632 0.9032585 0.5661316 1.0563898 24976
## [234] {race=White,
## sex=Male,
## capital-gain=None} => {capital-loss=None} 0.4994062 0.9399615 0.5313050 0.9860309 24392
## [235] {sex=Male,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.5084149 0.9404636 0.5406003 0.9865576 24832
## [236] {workclass=Private,
## race=White,
## native-country=United-States} => {capital-gain=None} 0.4994062 0.9190656 0.5433848 1.0018301 24392
## [237] {workclass=Private,
## race=White,
## capital-gain=None} => {native-country=United-States} 0.4994062 0.9126010 0.5472339 1.0169114 24392
## [238] {workclass=Private,
## race=White,
## native-country=United-States} => {capital-loss=None} 0.5181401 0.9535418 0.5433848 1.0002768 25307
## [239] {workclass=Private,
## race=White,
## capital-loss=None} => {native-country=United-States} 0.5181401 0.9130498 0.5674829 1.0174114 25307
## [240] {workclass=Private,
## race=White,
## capital-gain=None} => {capital-loss=None} 0.5204742 0.9511000 0.5472339 0.9977153 25421
## [241] {workclass=Private,
## race=White,
## capital-loss=None} => {capital-gain=None} 0.5204742 0.9171628 0.5674829 0.9997559 25421
## [242] {workclass=Private,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.5414807 0.9517075 0.5689570 0.9983526 26447
## [243] {workclass=Private,
## capital-loss=None,
## native-country=United-States} => {capital-gain=None} 0.5414807 0.9182030 0.5897179 1.0008898 26447
## [244] {race=White,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.6803980 0.9457029 0.7194628 0.9920537 33232
## [245] {race=White,
## capital-loss=None,
## native-country=United-States} => {capital-gain=None} 0.6803980 0.9083504 0.7490480 0.9901500 33232
## [246] {race=White,
## capital-gain=None,
## capital-loss=None} => {native-country=United-States} 0.6803980 0.9189249 0.7404283 1.0239581 33232
## [247] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## sex=Male} => {native-country=United-States} 0.3382335 0.9256458 0.3654027 1.0314471 16520
## [248] {marital-status=Married-civ-spouse,
## relationship=Husband,
## sex=Male,
## native-country=United-States} => {race=White} 0.3382335 0.9322799 0.3628025 1.0903313 16520
## [249] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## native-country=United-States} => {sex=Male} 0.3382335 0.9999395 0.3382540 1.4958359 16520
## [250] {relationship=Husband,
## race=White,
## sex=Male,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3382335 0.9993950 0.3384382 2.1811722 16520
## [251] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## native-country=United-States} => {relationship=Husband} 0.3382335 0.9931466 0.3405675 2.4602995 16520
## [252] {marital-status=Married-civ-spouse,
## relationship=Husband,
## sex=Male,
## capital-gain=None} => {race=White} 0.3210556 0.9044296 0.3549814 1.0577594 15681
## [253] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## capital-gain=None} => {sex=Male} 0.3210556 0.9999362 0.3210761 1.4958311 15681
## [254] {relationship=Husband,
## race=White,
## sex=Male,
## capital-gain=None} => {marital-status=Married-civ-spouse} 0.3210556 0.9994264 0.3212399 2.1812406 15681
## [255] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## capital-gain=None} => {relationship=Husband} 0.3210556 0.9913390 0.3238606 2.4558216 15681
## [256] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## sex=Male} => {capital-loss=None} 0.3418984 0.9356755 0.3654027 0.9815348 16699
## [257] {marital-status=Married-civ-spouse,
## relationship=Husband,
## sex=Male,
## capital-loss=None} => {race=White} 0.3418984 0.9053893 0.3776258 1.0588818 16699
## [258] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## capital-loss=None} => {sex=Male} 0.3418984 0.9999401 0.3419188 1.4958369 16699
## [259] {relationship=Husband,
## race=White,
## sex=Male,
## capital-loss=None} => {marital-status=Married-civ-spouse} 0.3418984 0.9994015 0.3421031 2.1811863 16699
## [260] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## capital-loss=None} => {relationship=Husband} 0.3418984 0.9912739 0.3449081 2.4556604 16699
## [261] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-gain=None,
## native-country=United-States} => {sex=Male} 0.3187830 0.9999358 0.3188035 1.4958304 15570
## [262] {relationship=Husband,
## sex=Male,
## capital-gain=None,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3187830 0.9992940 0.3190082 2.1809517 15570
## [263] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-gain=None,
## native-country=United-States} => {relationship=Husband} 0.3187830 0.9920357 0.3213423 2.4575475 15570
## [264] {marital-status=Married-civ-spouse,
## relationship=Husband,
## sex=Male,
## native-country=United-States} => {capital-loss=None} 0.3393391 0.9353273 0.3628025 0.9811696 16574
## [265] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-loss=None,
## native-country=United-States} => {sex=Male} 0.3393391 0.9999397 0.3393596 1.4958362 16574
## [266] {relationship=Husband,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3393391 0.9992765 0.3395848 2.1809135 16574
## [267] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {relationship=Husband} 0.3393391 0.9919205 0.3421031 2.4572622 16574
## [268] {marital-status=Married-civ-spouse,
## relationship=Husband,
## sex=Male,
## capital-gain=None} => {capital-loss=None} 0.3292044 0.9273849 0.3549814 0.9728380 16079
## [269] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-gain=None,
## capital-loss=None} => {sex=Male} 0.3292044 0.9999378 0.3292248 1.4958335 16079
## [270] {relationship=Husband,
## sex=Male,
## capital-gain=None,
## capital-loss=None} => {marital-status=Married-civ-spouse} 0.3292044 0.9993163 0.3294296 2.1810004 16079
## [271] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-gain=None,
## capital-loss=None} => {relationship=Husband} 0.3292044 0.9891117 0.3328283 2.4503040 16079
## [272] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## native-country=United-States} => {capital-loss=None} 0.3159576 0.9340839 0.3382540 0.9798652 15432
## [273] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## capital-loss=None} => {native-country=United-States} 0.3159576 0.9240719 0.3419188 1.0296933 15432
## [274] {marital-status=Married-civ-spouse,
## relationship=Husband,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.3159576 0.9310407 0.3393596 1.0888820 15432
## [275] {relationship=Husband,
## race=White,
## capital-loss=None,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3159576 0.9993524 0.3161623 2.1810792 15432
## [276] {relationship=Husband,
## race=White,
## sex=Male,
## native-country=United-States} => {capital-loss=None} 0.3161418 0.9341198 0.3384382 0.9799029 15441
## [277] {relationship=Husband,
## race=White,
## sex=Male,
## capital-loss=None} => {native-country=United-States} 0.3161418 0.9241128 0.3421031 1.0297389 15441
## [278] {relationship=Husband,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.3161418 0.9309659 0.3395848 1.0887945 15441
## [279] {relationship=Husband,
## race=White,
## capital-loss=None,
## native-country=United-States} => {sex=Male} 0.3161418 0.9999352 0.3161623 1.4958296 15441
## [280] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## native-country=United-States} => {capital-loss=None} 0.3182302 0.9344114 0.3405675 0.9802088 15543
## [281] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## capital-loss=None} => {native-country=United-States} 0.3182302 0.9226523 0.3449081 1.0281115 15543
## [282] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.3182302 0.9302172 0.3421031 1.0879189 15543
## [283] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## capital-gain=None} => {capital-loss=None} 0.3002948 0.9272348 0.3238606 0.9726805 14667
## [284] {marital-status=Married-civ-spouse,
## sex=Male,
## capital-gain=None,
## capital-loss=None} => {race=White} 0.3002948 0.9022515 0.3328283 1.0552121 14667
## [285] {marital-status=Married-civ-spouse,
## race=White,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.3074403 0.9254854 0.3321936 0.9708453 15016
## [286] {marital-status=Married-civ-spouse,
## race=White,
## capital-gain=None,
## capital-loss=None} => {native-country=United-States} 0.3074403 0.9191969 0.3344662 1.0242611 15016
## [287] {marital-status=Married-civ-spouse,
## capital-gain=None,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.3074403 0.9217925 0.3335244 1.0780659 15016
## [288] {age=Middle-aged,
## race=White,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.3350600 0.9417621 0.3557799 0.9879198 16365
## [289] {age=Middle-aged,
## race=White,
## capital-loss=None,
## native-country=United-States} => {capital-gain=None} 0.3350600 0.9066984 0.3695385 0.9883492 16365
## [290] {age=Middle-aged,
## race=White,
## capital-gain=None,
## capital-loss=None} => {native-country=United-States} 0.3350600 0.9140416 0.3665698 1.0185166 16365
## [291] {race=White,
## capital-gain=None,
## native-country=United-States,
## income=small} => {capital-loss=None} 0.3581549 0.9666243 0.3705213 1.0140005 17493
## [292] {race=White,
## capital-loss=None,
## native-country=United-States,
## income=small} => {capital-gain=None} 0.3581549 0.9552231 0.3749437 1.0412437 17493
## [293] {race=White,
## capital-gain=None,
## capital-loss=None,
## income=small} => {native-country=United-States} 0.3581549 0.9129482 0.3923058 1.0172982 17493
## [294] {workclass=Private,
## race=White,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-loss=None} 0.3028950 0.9597768 0.3155890 1.0068174 14794
## [295] {workclass=Private,
## race=White,
## capital-gain=None,
## hours-per-week=Full-time} => {capital-loss=None} 0.3129479 0.9584274 0.3265223 1.0054019 15285
## [296] {workclass=Private,
## race=White,
## capital-loss=None,
## hours-per-week=Full-time} => {capital-gain=None} 0.3129479 0.9298011 0.3365751 1.0135324 15285
## [297] {workclass=Private,
## capital-gain=None,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-loss=None} 0.3296753 0.9581672 0.3440686 1.0051289 16102
## [298] {workclass=Private,
## capital-loss=None,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-gain=None} 0.3296753 0.9304825 0.3543057 1.0142752 16102
## [299] {race=White,
## capital-gain=None,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-loss=None} 0.3865116 0.9520399 0.4059826 0.9987013 18878
## [300] {race=White,
## capital-loss=None,
## hours-per-week=Full-time,
## native-country=United-States} => {capital-gain=None} 0.3865116 0.9202047 0.4200278 1.0030718 18878
## [301] {race=White,
## capital-gain=None,
## capital-loss=None,
## hours-per-week=Full-time} => {native-country=United-States} 0.3865116 0.9068985 0.4261906 1.0105571 18878
## [302] {workclass=Private,
## race=White,
## sex=Male,
## native-country=United-States} => {capital-gain=None} 0.3313951 0.9062203 0.3656894 0.9878280 16186
## [303] {workclass=Private,
## race=White,
## sex=Male,
## capital-gain=None} => {native-country=United-States} 0.3313951 0.9086622 0.3647066 1.0125224 16186
## [304] {workclass=Private,
## sex=Male,
## capital-gain=None,
## native-country=United-States} => {race=White} 0.3313951 0.9019280 0.3674297 1.0548338 16186
## [305] {workclass=Private,
## race=White,
## sex=Male,
## native-country=United-States} => {capital-loss=None} 0.3464027 0.9472594 0.3656894 0.9936865 16919
## [306] {workclass=Private,
## race=White,
## sex=Male,
## capital-loss=None} => {native-country=United-States} 0.3464027 0.9092815 0.3809631 1.0132124 16919
## [307] {workclass=Private,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.3464027 0.9024429 0.3838500 1.0554360 16919
## [308] {workclass=Private,
## race=White,
## sex=Male,
## capital-gain=None} => {capital-loss=None} 0.3443348 0.9441419 0.3647066 0.9904162 16818
## [309] {workclass=Private,
## race=White,
## sex=Male,
## capital-loss=None} => {capital-gain=None} 0.3443348 0.9038534 0.3809631 0.9852480 16818
## [310] {workclass=Private,
## sex=Male,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.3468531 0.9439987 0.3674297 0.9902660 16941
## [311] {workclass=Private,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {capital-gain=None} 0.3468531 0.9036164 0.3838500 0.9849897 16941
## [312] {race=White,
## sex=Male,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.4579256 0.9381711 0.4881045 0.9841528 22366
## [313] {race=White,
## sex=Male,
## capital-gain=None,
## capital-loss=None} => {native-country=United-States} 0.4579256 0.9169400 0.4994062 1.0217463 22366
## [314] {sex=Male,
## capital-gain=None,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.4579256 0.9006927 0.5084149 1.0533890 22366
## [315] {workclass=Private,
## race=White,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.4741616 0.9494506 0.4994062 0.9959851 23159
## [316] {workclass=Private,
## race=White,
## capital-loss=None,
## native-country=United-States} => {capital-gain=None} 0.4741616 0.9151223 0.5181401 0.9975317 23159
## [317] {workclass=Private,
## race=White,
## capital-gain=None,
## capital-loss=None} => {native-country=United-States} 0.4741616 0.9110184 0.5204742 1.0151479 23159
## [318] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## sex=Male,
## native-country=United-States} => {capital-loss=None} 0.3159371 0.9340799 0.3382335 0.9798611 15431
## [319] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## sex=Male,
## capital-loss=None} => {native-country=United-States} 0.3159371 0.9240673 0.3418984 1.0296883 15431
## [320] {marital-status=Married-civ-spouse,
## relationship=Husband,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {race=White} 0.3159371 0.9310366 0.3393391 1.0888772 15431
## [321] {marital-status=Married-civ-spouse,
## relationship=Husband,
## race=White,
## capital-loss=None,
## native-country=United-States} => {sex=Male} 0.3159371 0.9999352 0.3159576 1.4958296 15431
## [322] {relationship=Husband,
## race=White,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {marital-status=Married-civ-spouse} 0.3159371 0.9993524 0.3161418 2.1810791 15431
## [323] {marital-status=Married-civ-spouse,
## race=White,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {relationship=Husband} 0.3159371 0.9927942 0.3182302 2.4594265 15431
## [324] {workclass=Private,
## race=White,
## sex=Male,
## capital-gain=None,
## native-country=United-States} => {capital-loss=None} 0.3121084 0.9418016 0.3313951 0.9879612 15244
## [325] {workclass=Private,
## race=White,
## sex=Male,
## capital-loss=None,
## native-country=United-States} => {capital-gain=None} 0.3121084 0.9009989 0.3464027 0.9821364 15244
## [326] {workclass=Private,
## race=White,
## sex=Male,
## capital-gain=None,
## capital-loss=None} => {native-country=United-States} 0.3121084 0.9064098 0.3443348 1.0100125 15244Visualization
Visualizing association rules can provide a more intuitive understanding of the relationships among variables. Tools such as scatter plots, parallel coordinate plots, and graph-based visualizations can be used to represent the strength and
#example for visualizing association rules library(arules) library(arulesViz) data(“Adult”)
#Generate association rules rules <- apriori(Adult, parameter = list(supp = 0.2, conf = 0.8, target = “rules”))
#Scatter plot plot(rules, method = “scatter”, measure = c(“support”, “confidence”), shading = “lift”)
plot(rules, method = “paracoord”, control = list(reorder = TRUE))
#Graph-based visualization plot(rules, method = “graph”, measure = c(“support”, “confidence”), shading = “lift”)
Conclusion:
Association rules are valuable tools for empirical analysis, as they allow researchers to uncover meaningful relationships among variables. By understanding the theoretical features of association rules, such as support, confidence, and lift, and applying appropriate pruning and visualization techniques, researchers can more effectively analyze their datasets and derive actionable insights.
References: 1. Agrawal, R., Imieliński, T., & Swami, A. (1993). Mining Association Rules between Sets of Items in Large Databases. Proceedings of the 1993 ACM SIGMOD International Conference on Management of Data, 207–216. 2. Hahsler, M., Grün, B., & Hornik, K. (2005). arules - A Computational Environment for Mining Association Rules and Frequent Itemsets. Journal of Statistical Software, 14(15), 1–25. 3. Hahsler, M. (2017). arulesViz: Interactive Visualization of Association Rules with R. R Journal, 9(2), 163–175.