Decision Tree Problem, Part 2

Reference: P. Winston, 1992.

Factors Affecting Sunburn (continued)

Phase 2: From Tree to Rules

We may now establish rules from the decision tree.

Rule	If	then
1	the person's hair color is blonde the person uses no lotion	the person gets sunburned.
2	the person's hair color is blonde the person uses lotion	nothing happens.
3	the person's hair color is red	the person gets sunburned.
4	the person's hair color is brown	nothing happens.

Phase 3: Simplify Rules

Once a rule set has been generated, simplify it.

[The training data has been multiplied by a factor of 4 to permit the use of the chi-square test below]

Assume a significance level of = 0.05

a) Eliminate Unnecessary Rule Antecedents

• Consider the two antecedents in Rule #1: blonde hair and lotion.

1. Blonde

   Actual:

   	No Change	Sunburned	Marginal Sum
   Blonde	16	16	32
   Not Blonde	20	12	32
   Marginal Sum	36	28	64

   Expected:

   Sample expected value calculation for contingency cell X₁₁:

   

   	No Change	Sunburned
   Blonde	18	14
   Not Blonde	18	14

   Sample calculation, where o_ij and e_ij are the observed (actual) and expected values for cell x_ij, respectively, and 1 i 2, 1 j 2:

   

   Sample degrees of freedom calculation:

   df = (r - 1)(c - 1) = (2 - 1)(2 - 1) = 1

   From the chi-square table, = 3.84

   Since , we accept the null hypothesis of independence, H₀.

   We thus conclude, according to the training examples, that sunburn is independent from blonde hair, and thus we may eliminate this antecedent from Rule #1 and Rule #2.

    

2. Lotion

   For argument's sake we will also examine the lotion antecedent for independence.

   Actual:

   	No Change	Sunburned	Marginal Sum
   Lotion	12	0	12
   No Lotion	8	12	20
   Marginal Sum	20	12	32

   Expected:

   	No Change	Sunburned
   Lotion	7.5	4.5
   No Lotion	12.5	7.5

   = 11.52

   df = 1

   From the chi-square table, = 3.84

   Since , we reject the null hypothesis of independence, H₀, and accept the alternate hypothesis of dependence, H_a.

   Therefore, according to the training examples, sunburn is clearly dependent upon the use of lotion, so we cannot eliminate this antecedent.

    

b) Eliminate Unnecessary Rules (under construction)

The tentative rule set is as follows:

Rule	If	then
1	the person uses no lotion	the person gets sunburned.
2	the person uses lotion	nothing happens.
3	the person's hair color is red	the person gets sunburned.
4	the person's hair color is brown	nothing happens.