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 oij and eij are the observed (actual) and expected values for cell xij, 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, Ha.

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.