# Multinomial Logit Model Building via TreeNet and Association Rules Analysis: An Application via a Thyroid Dataset

## Abstract

**:**

## 1. Introduction

## 2. Generating Variables and Interactions with TreeNet and Association Rules Analysis for the Multinomial Logit Model

#### 2.1. TreeNet

#### 2.2. Association Rules Analysis (ASA)

_{1}, X

_{2}, X

_{3}, …, X

_{k}, and a binary response, Y. Each variable has two levels denoted by 0 and 1. Many rules can be generated. As an example, the first rule could be “If X

_{1}= 1, X

_{2}= 0, then Y = 1” and the second rule could be “If X

_{1}= 0, X

_{2}= 1, X

_{3}= 1, then Y = 1”.

_{1}= 1, X

_{2}= 0 and Y = 1), and support (s) for the second rule is equal to P (X

_{1}= 0, X

_{2}= 1, X

_{3}= 1 and Y = 1). The second measurement is confidence (c), which is the probability of the left-hand side item(s) and the right-hand side item(s) divided by the probability of the left-hand side item(s). From the example, confidence (c) for the first rule is equal to P (X

_{1}= 1, X

_{2}= 0, and Y = 1)/P (X

_{1}= 1 and X

_{2}= 0) and confidence (c) for the second rule is equal to P (X

_{1}= 0, X

_{2}= 1, X

_{3}= 1, and Y = 1)/P (X

_{1}= 0, X

_{2}= 1, X

_{3}= 1). Similar to ASA, CRM finds all the rules that meet two key thresholds: minimum support and minimum confidence [22]. These selected rules will be used to form a classifier [23,24].

## 3. Proposed Method

- Step 1:
- Discretization

- Step 2:
- Rules Generation

_{i}’s = x

_{i}’s, then Y = y,” where x

_{i}is the level of predictor X

_{i}and y the level of response Y.

- Step 3:
- Rules Selection

- Step 4:
- Variable Generation

_{i}= x

_{i}, X

_{j}= x

_{j}, and X

_{k}= x

_{k}, then Y = y”. The interaction between the predictors X

_{i}, X

_{j}, and X

_{k}from this rule is generated by setting this interaction as 1 if X

_{i}= x

_{i}, X

_{j}= x

_{j}, and X

_{k}= x

_{k}, and as 0 otherwise. This interaction is denoted by X

_{i}(x

_{i})X

_{j}(x

_{j})X

_{k}(x

_{k}). For example, for the rule “If X

_{1}= 1, X

_{2}= 1, and X

_{3}= 1, then Y = 1”, we create an interaction between X

_{1}, X

_{2}, and X

_{3}denoted by X

_{1}(1)X

_{2}(1)X

_{3}(1). In this case, X

_{1}(1)X

_{2}(1)X

_{3}(1) = 1, if X

_{1}= 1, X

_{2}= 1, and X

_{3}= 1, and 0 otherwise. Note that the level of Y in each rule does not have any role in generating the interactions.

- Step 5:
- Model Selection

## 4. Illustrated Example: Thyroid Dataset

#### 4.1. Results from Model 1

_{1}/p

_{0})], where p

_{1}is the probability that the variable is in class 1 and where p

_{0}is the probability that the variable is in class 0. For simplicity, we refer to 0.5[log(p

_{1}/p

_{0})] as the logit or log-odds. The interpretation from TreeNet is based on comparing the relative values of the log-odds; i.e., the higher the value of the log odds, the higher the probability that a variable is in class 1.

#### 4.2. Results from Model 2

_{2}/p

_{0})], where p

_{2}is the probability that a variable is in class 2 and where p

_{0}is the probability that a variable is in class 0. The higher the value of the log-odds, the higher the probability that a variable belongs in class 2.

- Rule 6: If X20L1 = 1, X18LL1 = 0, and X21LL4 = 1, then Y = 0 with s = 8.537%, c = 100%.
- Rule 8: If X11 = 0, X19LL3 = 1, and X18LL1 = 0, then Y = 0 with s = 3.075%, c = 100%.
- Rule 22: If X8 = 0, X17L1 = 0, and X19L3 = 0, then Y = 1 with s = 1.935%, c = 97.26%.
- Rule 26: If X3 = 0, X17LL1 = 0, and X19LL2 = 1, then Y = 2 with s = 5.276%, c = 88.945%.

#### 4.3. Performance Comparison Using the Training Set

^{2}(McFadden), and accuracy. There are 21 candidate predictors for Model 1, 231 for Model 2, and 90 for Model 3. The comparison shows that for all four criteria, the proposed model (Model 3) outperformed each of the other two models. The misclassification error from our proposed model is only 0.32%, as shown in Table 9.

#### 4.4. Performance Comparison Using the Test Set

#### 4.5. Performance Comparison with Other Methodologies

## 5. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Attribute | Description | Variable |
---|---|---|

Age | Age in years | X1 |

Sex | Gender | X2 = 1 if male and X2 = 0 if female |

On Thyroxine | Patient on Thyroxine | X3 = 1 if yes and X3 = 0 if no |

Query Thyroxine | Maybe on Thyroxine | X4 = 1 if yes and X4 = 0 if no |

On antithyroid | On antithyroid medication | X5 = 1 if yes and X5 = 0 if no |

Sick | Patient reports malaise | X6 = 1 if yes and X6 = 0 if no |

Pregnant | Patient pregnant | X7 = 1 if yes and X7 = 0 if no |

Thyroid surgery | History of thyroid surgery | X8 = 1 if yes and X8 = 0 if no |

I131 treatment | Patient on I131 treatment | X9 = 1 if yes and X9 = 0 if no |

Query hypothyroid | Maybe hypothyroid | X10 = 1 if yes and X10 = 0 if no |

Query hyperthyroid | Maybe hyperthyroid | X11 = 1 if yes and X11 = 0 if no |

Lithium | Patient on lithium | X12 = 1 if yes and X12 = 0 if no |

Goiter | Patient has goiter | X13 = 1 if yes and X13 = 0 if no |

Tumor | Patient has tumor | X14 = 1 if yes and X14 = 0 if no |

Hypopituitary | Patient hypopituitary | X15 = 1 if yes and X15 = 0 if no |

Psych | Psychological symptoms | X16 = 1 if yes and X16 = 0 if no |

Thyroid Stimulating Hormone (TSH) | TSH value, if measured | X17 |

Triiodothyronine (T3) | T3 value, if measured | X18 |

Total Thyroxine (TT4) | TT4 value, if measured | X19 |

Thyroxine Uptake (T4U) | T4U value, if measured | X20 |

Free Thyroxine Index (FTI) | FTI—calculated from TT4 and T4U | X21 |

Variable | Score | |
---|---|---|

X17 | 100.00 | |||||||||||||||||||||||||||||||||||||||||| |

X21 | 63.88 | |||||||||||||||||||||||||| |

X8 | 25.08 | |||||||||| |

X3 | 17.38 | |||||| |

X19 | 12.16 | |||| |

X18 | 6.67 | || |

X20 | 6.56 | || |

X2 | 6.42 | || |

X1 | 3.11 | |

X10 | 2.71 | |

X9 | 2.00 | |

X11 | 1.93 |

Original Variables | Generated Binary Variables |
---|---|

TSH (X17) | X17L1 = 1 if X17 < 0.025 and X17L1 = 0 otherwise |

X17L2 = 1 if 0.025 ≤ X17 and X17L2 = 0 otherwise | |

FTI (X21) | X21L1 = 1 if X21 < 0.055 and X21L1 = 0 otherwise |

X21L2 = 1 if 0.055 ≤ X21 < 0.07 and X21L2 = 0 otherwise | |

X21L3 = 1 if 0.07 ≤ X21 and X21L3 = 0 otherwise | |

TT4 (X19) | X19L1 = 1 if X19 < 0.042 and X19L1 = 0 otherwise |

X19L2 = 1 if 0.042 ≤ X19 < 0.065 and X19L2 = 0 otherwise | |

X19L3 = 1 if 0.065 ≤ X19 and X19L3 = 0 otherwise | |

T3 (X18) | X18L1 = 1 if X18 < 0.006 and X18L1 = 0 otherwise |

X18L2 = 1 if 0.006 ≤ X18 and X18L2 = 0 otherwise | |

T4U (X20) | X20L1 = 1 if X20 < 0.097 and X20L1 = 0 otherwise |

X20L2 = 1 if 0.097 ≤ X20 and X20L2 = 0 otherwise | |

Age (X1) | X1L1 = 1 if X1 < 0.15 and X1L1 = 0 otherwise |

X1L2 = 1 if 0.15 ≤ X1 and X1L2 = 0 otherwise |

Original Variables | Generated Variables with Linear Trend |
---|---|

FT1 (X21) | X21Q2 = X21 if 0.055 ≤ X21 < 0.07 and X21Q2 = 0 otherwise |

TT4 (X19) | X19Q2 = X19 if 0.042 ≤ X19 < 0.065 and X19Q2 = 0 otherwise |

Variable | Score | |
---|---|---|

X17 | 100.00 | |||||||||||||||||||||||||||||||||||||||||| |

X3 | 44.88 | |||||||||||||||||| |

X8 | 25.20 | |||||||||| |

X19 | 22.47 | ||||||||| |

X18 | 17.51 | ||||||| |

X21 | 10.87 | |||| |

X20 | 8.04 | || |

X1 | 6.68 | || |

X10 | 5.15 | | |

X5 | 4.21 | | |

X12 | 4.02 | | |

X11 | 3.20 | |

X2 | 0.84 |

Original Variables | Generated Binary Variables |
---|---|

TSH (X17) | X17LL1 = 1 if X17 < 0.006 and X17LL1 = 0 otherwise |

X17LL2 = 1 if 0.006 ≤ X17 and X17LL2 = 0 otherwise | |

FTI (X21) | X21LL1 = 1 if X21 < 0.057 and X21LL1 = 0 otherwise |

X21LL2 = 1 if 0.057 ≤ X21 < 0.071 and X21LL2 = 0 otherwise | |

X21LL3 = 1 if 0.071 ≤ X21 < 0.115 and X21LL3 = 0 otherwise | |

X21LL4 = 1 if 0.115 ≤ X21 < 0.217 and X21LL4 = 0 otherwise | |

X21LL5 = 1 if 0.217 ≤ X21 and X21LL5 = 0 otherwise | |

TT4 (X19) | X19LL1 = 1 if X19 < 0.065 and X19LL1 = 0 otherwise |

X19LL2 = 1 if 0.065 ≤ X19 < 0.145 and X19LL2 = 0 otherwise | |

X19LL3 = 1 if 0.145 ≤ X19 < 0.161 and X19LL3 = 0 otherwise | |

X19LL4 = 1 if 0.161 ≤ X19 and X19LL4 = 0 otherwise | |

T3 (X18) | X18LL1 = 1 if X18 < 0.02 and X18LL1 = 0 otherwise |

X18LL2 = 1 if 0.02 ≤ X18 < 0.045 and X18LL2 = 0 otherwise | |

X18LL3 = 1 if 0.045 ≤ X18 and X18LL3 = 0 otherwise | |

T4U (X20) | X20LL1 = 1 if X20 < 0.07 and X20LL1 = 0 otherwise |

X20LL2 = 1 if 0.07 ≤ X20 < 0.15 and X20LL2 = 0 otherwise | |

X20LL3 = 1 if 0.15 ≤ X20 and X20LL3 = 0 otherwise | |

Age (X1) | X1LL1 = 1 if X1 < 0.75 and X1LL1 = 0 otherwise |

X1LL2 = 1 if 0.75 ≤ X1< 0.85 and X1LL2 = 0 otherwise | |

X1LL3 = 1 if 0.85 ≤ X1 and X1LL3 = 0 otherwise |

Original Variables | Generated Variables with Linear Trend |
---|---|

FT1 (X21) | X21QQ2 = X21 if 0.057 ≤ X21 < 0.071 and X21QQ2 = 0 otherwise |

X21QQ4 = X21 if 0.115 ≤ X21 < 0.217 and X21QQ4 = 0 otherwise | |

TT4 (X19) | X19QQ3 = X19 if 0.145 ≤ X19 < 0.161 and X19QQ3 = 0 otherwise |

T3 (X18) | X18QQ2 = X18 if 0.02 ≤ X18 < 0.045 and X18QQ2 = 0 otherwise |

Age (X1) | X1QQ2 = X1 if 0.75 ≤ X1< 0.85 and X1QQ2 = 0 otherwise |

Model | Candidate Predictors | BIC | AIC | R^{2}(McFadden) | Accuracy |
---|---|---|---|---|---|

Model 1 | Main effects (X1–X21) | 916.05 | 841.23 | 65.58% | 97.03% |

Model 2 | Main effects (X1–X21) All two-way interactions | 561.56 | 449.32 | 82.59% | 98.36% |

Model 3 | Main effects (X1–X21) Generated variables from step 2 Generated interactions from step 4 | 233.41 | 121.17 | 96.41% | 99.68% |

Actual Class | ||||
---|---|---|---|---|

0 | 1 | 2 | ||

Predict class | 0 | 3479 | 2 | 1 |

1 | 3 | 91 | 0 | |

2 | 6 | 0 | 190 |

Actual Class | ||||
---|---|---|---|---|

0 | 1 | 2 | ||

Predict class | 0 | 3165 | 6 | 8 |

1 | 4 | 67 | 0 | |

2 | 9 | 0 | 169 |

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Changpetch, P.
Multinomial Logit Model Building via TreeNet and Association Rules Analysis: An Application via a Thyroid Dataset. *Symmetry* **2021**, *13*, 287.
https://doi.org/10.3390/sym13020287

**AMA Style**

Changpetch P.
Multinomial Logit Model Building via TreeNet and Association Rules Analysis: An Application via a Thyroid Dataset. *Symmetry*. 2021; 13(2):287.
https://doi.org/10.3390/sym13020287

**Chicago/Turabian Style**

Changpetch, Pannapa.
2021. "Multinomial Logit Model Building via TreeNet and Association Rules Analysis: An Application via a Thyroid Dataset" *Symmetry* 13, no. 2: 287.
https://doi.org/10.3390/sym13020287