1. Decision trees
part II

2. CHAID method : Chi-Squared Automatic Interaction Detection
LESSON TOPICS
CHAID method : Chi-Squared Automatic Interaction Detection
Chi-square test
Bonferroni correction factor
Examples

3. Principal features of CHAID method

4. CHAID merges categories of the predictor that are homogeneous with respect to the dependent variable , but keeps distinct all the categories which are heterogeneous

5. CHAID uses Bonferroni multiplier for doing the needed adjustments in order for making simultaneous statistical inferences

6. CHAID, a differenza di altri metodi di partizione iterativa, è limitato a caratteri di tipo ordinale e nominale

7. It uses chi-square test for veryfing indipendence between characters (together with Bonferroni factor) for assessing significativity of partition

8. Chi-square test of independence
  i j
( n ij - nij )2 *
nij
x2 =

9. where
nij
is the empirical frequency corresponding to the combination of modality i of the first character with modality j of the second character

10. nij = ninj *
Is the corresponding theoretical frequency according to the hypothesis of indipendence between the two characters

11. EXAMPLE
Families according to residence and personal computer ownership (empirical frequencies)

12. Geographic zone Ownership of personal computer North-Center South
Total
YES NO
150
500
650
100
250
350
750
1000

13. Families according to residence and personal computer ownership (theoretical frequencies)

14. Geographic zone Ownership of personal computer North-Center South
Total
YES NO
162,5
487,5
650,0
87,5
262,5
350,0
250,0
750,0
1000,0

15. Test calculations: (500-487,5)2/487,5+ (87,5-100)2/87,5+
(162,5-150)2/162,5+
( ,5)2/262,5=

16. Bonferroni adjustment factor
Let us consider the dependent variable R and the predictors B, with five modalities, and A, with two
Let us take that a is the first type error of the indipendence test in a two entry table with B e R (for example a =0,05)

17. There are 24 -1 = 15 different ways to make dichotomous variable B
If the 15 test of hypothesis were indipendent, the probability of making a first type error would be:
1-(1-a)15 > a

18. In the above example, 15 is called Bonferroni factor
If a è piccolo
1 - (1-a)M = Ma
For the predictor A the probability of making a first type error is simply a

19. In the CHAID method we compare the value of a associated with the test of indipendence for the variable A with the value of a for the variable B corrected with Bonferroni factor

20. Basic components of CHAID:

21. 1 2 3 A categorical dependent variable
A set of independent variables, categorical too, combinations of which are used for defining the partitions 3
A set of parameters

22. In each step of the analysis, each subgroup is analyzed and we get the best predictor, defined as that which has the smallest value of a corrected by the smallest Bonferroni factor

23. Kinds of predictive variables in CHAID
Monotonic 1
Free 2
Floating 3

24. The CHAID algorithm: STEP 1: Merging Step 2: Splitting
Step 3: Stopping

25. Merging

26. For each predictor

27. Construct the complete two ways table1

28. For each couple of categories that can be merged calculate chi-square test. For each couple which is not significative merge and go to step 3. If all the remaining couples are significative go to step 4 2

29. For each categories resulting from the merge of three or more categories originarie controlla con il test chi-quadrato se ogni categoria originaria può essere separata dalle altre. Torna al passo 2 3

30. Merge categories which have a too small number of observations, taking those which have the smallest value of chi-squared 4
Calculate the value of a corrected by Bonferroni factor on the table resulting by the merging process 5

31. Splitting
Take as the best predictor that which has the smallest value of a corrected by Bonferroni factor
If no predictor shows a significant value of a, do not split that subgroup

32. Stopping
Come back to step 1 and analyze the next subgroup. Stop when every subgroup has been analyzed or has too few observations

33. Example of chaid method
Dependent variable:
Response rate to a promotional offer of subscribing a magazine

34. Indipendent Variables

35. Head of the family age - 5 categories -floating (AGE)
gender - 2 categories -monotonic - (GENDER)
Presence of children - 2 categories - monotonic (KIDS)
Family income - 8 categories - monotonic (INCOME)

36. Credit card - 2 categories - monotonic (BANKCARD)
Number of components - 6 categories - floating - (HHSIZE)
Occupational status -4 categories - free (OCCUP)

37. Representation of the partition process by a dendrogram

38. Total ,040
HHSIZE
,384
,132
,198
? ,326
OCCUP
GENDER
-1-
-4-
W ,758
BO? ,374
M ,531
F ,795
-2-
-3-
-5-
-6-

39. Interpretation of results
Comparison of response accordin to the variable household size before and after merging

40. % of responses
HHSIZE
Frequency
Before merging
After merging 1 2 3 4 5
Missing value
25384
11240
4892
3187
3011
33326
1,09
1,49
1,59
1,79
2,06
0,87
1,52
1,92

41. Ranking of segments according to response rate

42. Rank Number Description Response rate 1 2 Segment 2 Segment 4 2,39
Household with two or tre components, head white collar
2,39
1,92
Households with four components and more

43. Household with one component
Rank
Number
Description
Response rate 3 4
Segment 3
Segment 1
Household with two or three components, head with occupational staus different from white collar
1,42
1,09
Household with one component

44. Household with missing number of components, head male
Rank
Number
Description
Response rate 5
Segment 6
Segment 5
Household with missing number of components, head female
1,o6
0,81
Household with missing number of components, head male