Beteta - Project Two MAT 303

In the context of the logistic regression model for predicting heart disease, the terms π and π 1 − π have specific meanings related to the probability of an individual having heart disease:  π: This represents the probability of an individual having heart disease. It is a value between 0 and 1, where 0 means no chance of heart disease and 1 means a certain presence of heart disease. In the context of our model, π is what we are trying to predict based on the input variables (age, resting blood pressure, exercise-induced angina, and maximum heart rate achieved).  π 1 − π : This is known as the odds ratio. It's a way of comparing the likelihood of having heart disease (π) to the likelihood of not having heart disease (1-π). For example, if π=0.5, the odds ratio π 1 − π is 1, meaning the odds of having heart disease are equal to the odds of not having it. If π > 0.5, the odds ratio is greater than 1, indicating a higher likelihood of heart disease. Conversely, if π <0.5, the odds ratio is less than 1, suggesting a lower likelihood of heart disease. In logistic regression, we use the natural logarithm of the odds ratio as the outcome variable. The model estimates these log odds as a linear combination of the predictor variables.  Prediction Model Equation: ln ( odds ) =− 1.0211 − 0.0175 age − 0.0149 trestbps − 1.625 exang 1 + 0.0311 thalach The estimated coefficient for the maximum heart rate achieved (thalach) in our logistic regression model is 0.031095. This indicates that in our model, for each one-unit increase in maximum heart rate (measured in beats per minute), the log odds of having heart disease increase by 0.031095. This positive coefficient suggests that higher maximum heart rates are associated with a greater likelihood of heart disease in our dataset. It's crucial to interpret this finding within the context of our model and other influencing factors; a higher heart rate is not a direct cause of heart disease but shows an association in the population we studied. This interpretation should be integrated with clinical insights and other relevant variables for a comprehensive understanding. Evaluating Model Significance The Hosmer-Lemeshow test statistic is based on a chi-square distribution, and the degrees of freedom are typically calculated as the number of groups minus 2. The test resulted in a chi-square value of 44.622 with 48 degrees of freedom, leading to a P-value of 0.612. Given that the P-value is significantly higher than the 0.05 (5%) significance level, we do not reject the null hypothesis. This implies that no substantial evidence suggests that our model fails to fit the data appropriately. According to the Hosmer-Lemeshow goodness of fit test, our model seems suitable for the data.

The odds of an individual with these characteristics having heart disease are approximately 0.3729. This means that such individuals are about 0.3729 times more likely to have heart disease than not to have it. While the odds are not extremely high, they still suggest an increased likelihood of heart disease for individuals with these specific attributes. This information can be valuable for medical assessments and risk evaluations. 2. The probability of an individual having heart disease at 50 years old, with a resting blood pressure of 130, no exercise-induced angina, and a maximum heart rate of 165, is 94.19%. With a probability value of 94.19%, the model indicates that individuals with these characteristics are highly likely to have heart disease. This probability suggests a strong risk of heart disease for individuals with this specific profile, emphasizing the importance of these factors in assessing heart disease risk. The odds of an individual with these characteristics having heart disease are approximately 16.41. This means that such individuals are about 16.41 times more likely to have heart disease than not to have it. These odds underscore the significant likelihood of heart disease for individuals with this particular combination of attributes, making it crucial information for medical evaluations and risk assessments. 4. Model #2 - Second Logistic Regression Model Reporting Results  General Form: E ( y ) = e ( β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + β 5 x 5 + β 6 x 6 + β 7 x 7 2 + β 8 x 2 x 6 ) 1 + e ( β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + β 5 x 5 + β 6 x 6 + β 7 x 7 2 + β 8 x 2 x 6 )  Prediction Equation: ln ( odds ) = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + β 5 x 5 + β 6 x 6 + β 7 x 7 2 + β 8 x 2 x 6  The prediction equation of this model in terms of the natural log of odds to express the beta terms in linear form: E ( y ) = e (− 15.56 + 0.1744 x 1 − 0.0196 x 2 + 1.913 x 3 + 2.037 x 4 + 1.777 x 5 + 0.1363 x 6 + 0.0008 x 7 2 − 0.0019 x 2 x 6 ) 1 + e (− 15.56 + 0.1744 x 1 − 0.0196 x 2 + 1.913 x 3 + 2.037 x 4 + 1.777 x 5 + 0.1363 x 6 + 0.0008 x 7 2 − 0.0019 x 2 x 6 )  Prediction Model Equation: ln ( odds ) =− 15.56 + 0.1744 age − 0.0196 trestbps + 1.913 c p 1 + 2.037 c p 2 + 1.777 c p 3 + 0.1363 thalach + 0.0008 age 2

 Recall (Sensitivity): Approximately 78.18%. This means that the model correctly identifies about 78.18% of the actual cases of heart disease. These metrics provide a comprehensive overview of the model's performance in predicting heart disease, considering its ability to identify true cases and its accuracy in ruling out the disease when it is not present. The ROC curve illustrates the True Positive Rate (TPR, or sensitivity) against the False Positive Rate (FPR, or 1 - specificity) at various threshold settings. The curve appears to rise quickly towards the upper left corner, which indicates an excellent true positive rate for the model at a low false positive rate. This shape suggests that the model has a strong ability to discriminate between the classes of interest, which in this context are individuals with and without heart disease. The AUC value is 0.8478, which is considered very good. This value represents the probability that the model will correctly rank a randomly chosen positive instance (a case with heart disease) higher than a randomly chosen negative instance (a case without heart disease) in terms of risk. An AUC value closer to 1.0 indicates a model with excellent predictive accuracy. The ROC curve is crucial for assessing a classification model's performance, mainly when the trade-off between sensitivity and specificity is vital. The high AUC value observed with this model suggests it effectively distinguishes between patients with and without heart disease.

These metrics indicate that the model's performance on the testing set is lower than on the training set, which is expected as the testing set represents new, unseen data. It shows room for improvement, particularly in reducing false positives and negatives to improve accuracy, precision, and recall. 6. Random Forest Regression Model Reporting Results The original dataset contains 303 rows. After splitting with an 80% training and 20% testing ratio using the specified seed, the training set contains 242 rows, and the testing set contains 61 rows. Based on the graphic, the Mean Squared Error (MSE) for the Random Forest Regression model stabilizes around the point where the model includes 20 trees. After reaching this number of trees, there is no significant error reduction with adding more trees. Hence, a Random Forest model with 20 trees is the optimal choice for this dataset, balancing the trade-off between model complexity and accuracy. Evaluating the Utility of the Random Forest Regression Model 1. Root Mean Squared Error for the Training Set (11.6282) : o This relatively low value indicates that the Random Forest model predicts the maximum heart rate quite accurately on the training data.