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384 Chapter 6 Association Analysis

Table 6.19. A two-way contingency table between the sale of high-definition television and exercise machine.

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Buv Exercise Machine Yes No


No 99 o4

81 66

180 t20

153 L47 300

Table 6,20. Example of a three-way contingency table.

Customer Group


Buy Exercise Machine Total YCS No

College Students Yes No


9 30

10 34

Working Adult YES

No 98 50

72 36

770 86

6.7.3 Simpson’s Paradox

It is important to exercise caution when interpreting the association between variables because the observed relationship may be influenced by the presence of other confounding factors, i.e., hidden variables that are not included in the analysis. In some cases, the hidden variables may cause the observed relationship between a pair of variables to disappear or reverse its direction, a phenomenon that is known as Simpson’s paradox. We illustrate the nature of this paradox with the following example.

Consider the relationship between the sale of high-definition television (HDTV) and exercise machine, as shown in Table 6.19. The rule {HDTV:Yes} ——+ {Exercise machine:Yes} has a confidence of 99/180:557o and the rule

{HDTV:No} -_.-+ {Exercise machine:Yes} has a confidence of 541720 : 45%o. Together, these rules suggest that customers who buy high-definition televi- sions are more likely to buy exercise machines than’those who do not buy high-defi nition televisions.

However, a deeper analysis reveals that the sales of these items depend on whether the customer is a college student or a working adult. Table 6.20 summarizes the relationship between the sale of HDTVs and exercise machines among college students and working adults. Notice that the support counts given in the table for college students and working adults sum up to the fre- quencies shown in Table 6.19. Furthermore, there are more working adults



6.7 Evaluation of Association Patterns 385

than college students who buy these items. For college students:

c({UOfV:Yes} —‘ {Exercise machine:Yes})

c({HOtV:No} —–* {Exercise machine:Yes})

I l l 0 :10To ,

4134: 7I.8To,

while for working adults:

c({HnfV:Yes} —–* {Exercise machine:Yes}) : 981170 : 57.7Vo,

c({UOfV:No} —– {Exercise machine:Yes}) : 50/86 :58.IVo.

The rules suggest that, for each group, customers who do not buy high-

definition televisions are more likely to buy exercise machines, which contradict the previous conclusion when data from the two customer groups are pooled

together. Even if alternative measures such as correlation, odds ratio’ or

interest are applied, we still find that the sale of HDTV and exercise machine

is positively correlated in the combined data but is negatively correlated in

the stratified data (see Exercise 20 on page 414). The reversal in the direction

of association is known as Simpson’s paradox. The paradox can be explained in the following way. Notice that most

customers who buy HDTVs are working adults. Working adults are also the

largest group of customers who buy exercise machines. Because nearly 85% of

the customers are working adults, the observed relationship between HDTV

and exercise machine turns out to be stronger in the combined data than what it would have been if the data is stratified. This can also be illustrated

mathematically as follows. Suppose

o,lb < cf d and plq < rls,

where afb andplqmay represent the confidence of the rule A —, B in two

different strata, while cld and rf s may represent the confidence of the rule

A ——+ B in the two strata. When the data is pooled together’ the confidence values of the rules in the combined data are ([email protected]+q) and (c+r)[email protected]+s),

respectively. Simpson’s paradox occurs when

a * P c l r

b+q> d+r ‘

thus leading to the wrong conclusion about the relationship between the vari-

ables. The lesson here is that proper stratification is needed to avoid generat-

ing spurious patterns resulting from Simpson’s paradox. For example, market



386 Chapter 6 Association Analysis

1000 1500 Items sorted by support

Figure 6.29. Support distribution of items in the census data set.

basket data from a major supermarket chain should be stratified according to store locations, while medical records from various patients should be stratified according to confounding factors such as age and gender.

6.8 Effect of Skewed Support Distribution

The performances of many association analysis algorithms are influenced by properties of their input data. For example, the computational complexity of the Apri,ori algorithm depends on properties such as the number of items in the data and average transaction width. This section examines another impor- tant property that has significant influence on the performance of association analysis algorithms as well as the quality of extracted patterns. More specifi- cally, we focus on data sets with skewed support distributions, where most of the items have relatively low to moderate frequencies, but a small number of them have very high frequencies.

An example of a real data set that exhibits such a distribution is shown in Figure 6.29. The data, taken from the PUMS (Public Use Microdata Sample) census data, contains 49;046 records and 2113 asymmetric binary variables. We shall treat the asymmetric binary variables as items and records as trans- actions in the remainder of this section. While more than 80% of the items have support less than 1%, a handfuI of them have support greater than 90%.



6.8 Effect of Skewed Support Distribution 387

Table 6.21. Grouping the items in the census data set based on their support values.

Group G 1 G2 Gs Support < t% r% -90% > gUYa

Number of Items 1 – D r I ‘ d U 358 20

To illustrate the effect of skewed support distribution on frequent itemset min- ing, we divide the items into three groups, Gt, Gz, and G3, according to their support levels. The number of items that belong to each group is shown in Table 6.21.

Choosing the right support threshold for mining this data set can be quite

tricky. If we set the threshold too high (e.g., 20%), then we may miss many interesting patterns involving the low support items from Gr. In market bas- ket analysis, such low support items may correspond to expensive products (such as jewelry) that are seldom bought by customers, but whose patterns are still interesting to retailers. Conversely, when the threshold is set too low, it becomes difficult to find the association patterns due to the following reasons. First, the computational and memory requirements of existing asso- ciation analysis algorithms increase considerably with low support thresholds. Second, the number of extracted patterns also increases substantially with low support thresholds. Third, we may extract many spurious patterns that relate a high-frequency item such as milk to a low-frequency item such as caviar. Such patterns, which are called cross-support patterns, are likely to be spu- rious because their correlations tend to be weak. For example, at a support threshold equal to 0.05yo, there are 18,847 frequent pairs involving items from Gr and G3. Out of these, 93% of them are cross-support patterns; i.e., the pat-

terns contain items from both Gr and G3. The maximum correlation obtained from the cross-support patterns is 0.029, which is much lower than the max- imum correlation obtained from frequent patterns involving items from the same group (which is as high as 1.0). Similar statement can be made about many other interestingness measures discussed in the previous section. This example shows that a large number of weakly correlated cross-support pat-

terns can be generated when the support threshold is sufficiently low. Before presenting a methodology for eliminating such patterns, we formally define the concept of cross-support patterns.



388 Chapter 6 Association Analysis

Definition 6.9 (Cross-Support Pattern). A cross-support pattern is an itemset X : {ir,,i2,. . . ,i6} whose support ratio

(6 .13)

is less than a user-specified threshold h“.

Example 6.4. Suppose the support for milk is 70To, while the support for sugar is 10% and caviar is 0.04%. Given h” : 0.01, the frequent itemset

{milk, sugar) caviar} is a cross-support pattern because its support ratio is

min [0.7,0.1,0.0004]0.0004: 0.00058 < 0.01. max10 .7 ,0 .1 ,0 .00041 0 .7


Existing measures such as support and confidence may not be suffi.cient to eliminate cross-support patterns, as illustrated by the data set shown in Figure 6.30. Assuming that h. :0.3, the i temsets {p,q}, {p,r} , and {p,q,r} are cross-support patterns because their support ratios, which are equal to 0.2, are less than the threshold h”. Although we can apply a high support threshold, say, 20Vo, to eliminate the cross-support patterns, this may come at the expense of discarding other interesting patterns such as the strongly correlated itemset, {q, r} that has support equal to L6.7To.

Confidence pruning also does not help because the confidence of the rules extracted from cross-support patterns can be very high. For example, the confidence for {q} – {p} is 80% even though {p,S} is a cross-support pat- tern. The fact that the cross-support pattern can produce a high-confidence rule should not come as a surprise because one of its items (p) appears very frequently in the data. Therefore, p is expected to appear in many of the transactions that contain q. Meanwhile, the rule {q} – {r} also has high confidence even though {q,r} is not a cross-support pattern. This example demonstrates the difficulty of using the confidence measure to distinguish be- tween rules extracted from cross-support and non-cross-support patterns.

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