Notebook

Nov 15, 2024, 3:50 PM

The purpose of a study was to characterize acute hepatitis A in patients more than 40 years old. They performed a retrospective chart review of 20 subjects who were diagnosed with acute hepatitis A, but were not hospitalized. Of interest was the use predict bilirubin levels (mg/dl). The following data were collected.

Age (Years)- 78 , 72 ,81, 59, 64,48,46,44,42,45,78,47,50,57,42,58,52,52,58,45Bilirubin (mg/dl)-7.5,12.9,14.3,8.0,14.1,10.9,12.3,7,1.8,.8,3.8,3.5,5.1,16.5,1,5.2,5.1,3.5,5.6,1.9
b) Calculate the correlation coefficient between age and bilirubin. Interpret your results. Evaluate the significance of the associations at the 0.05 level.
c) Determine the linear regression equation between age and bilirubin. Interpret the estimated regression coefficients. Construct a 95% confidence interval for β. For age 93, what is the estimated bilirubin value? Is the slope obtained for the regression equation statistically significant (at the 0.05 level)
The correlation coefficient between age and bilirubin levels is approximately 0.466. This indicates a moderate positive correlation, suggesting that as age increases, bilirubin levels tend to increase as well.
The p-value for this correlation is approximately 0.038, which is less than the significance level of 0.05. This means that the correlation is statistically significant at the 0.05 level, indicating that there is a significant association between age and bilirubin levels in this sample.
Next, let's determine the linear regression equation between age and bilirubin, interpret the coefficients, and construct a 95% confidence interval for the slope (β). We will also estimate the bilirubin level for an age of 93 and evaluate the significance of the slope.
The error occurred because the shape of the input data for prediction was not aligned correctly with the model's expected input shape. The model.predict() function expects the input to have the same number of columns as the training data, including the constant term added for the intercept.
To fix this, we need to ensure that the input for prediction (age_93) is a 2D array with the same structure as the training data, including the constant term. Let's correct this and try again.
The linear regression equation between age and bilirubin levels is given by:
Bilirubin=−2.98+0.18×Age\text{Bilirubin} = -2.98 + 0.18 \times \text{Age}

Interpretation of the Regression Coefficients:

  • Intercept (-2.98): This is the estimated bilirubin level when age is 0. However, since age 0 is not within the range of the data, this value is not practically meaningful.
  • Slope (0.18): For each additional year of age, the bilirubin level is expected to increase by approximately 0.18 mg/dl.

Confidence Interval for the Slope:

The 95% confidence interval for the slope (β) is approximately (0.01, 0.35). This interval does not include 0, suggesting that the slope is statistically significant.

Estimated Bilirubin for Age 93:

The estimated bilirubin level for a person aged 93 is approximately 13.69 mg/dl.

Significance of the Slope:

The p-value for the slope is approximately 0.038, which is less than the significance level of 0.05. Therefore, the slope is statistically significant at the 0.05 level, indicating a significant linear relationship between age and bilirubin levels.
If you have any further questions or need additional analysis, feel free to ask!
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