Please Show Stata Commands Using Data From 82 Subjects

Please Show Stata Commands Using Data From 82 Subjects A Can

Please show Stata commands using data from 82 subjects. A cancer epidemiologist studied the relationship between a “lung cancer risk score” (y), ranging from 0 to 150, and a specific biomarker (x). The output from a simple linear regression analysis includes the following: estimated intercept (α̂) = 10, standard error (SE(α̂)) = 2; estimated slope (β̂) = 2.5, standard error (SE(β̂)) = 0.75; R-squared (R²) = 0.5; and a 95% confidence interval for α being (6, 14), and for β being (1, 4).

Write down the estimated regression line. Obtain a point estimate of the difference in average risk scores for two individuals whose biomarker values differ by 5 units. Determine whether there is a statistically significant association between the biomarker and the risk score, explaining your reasoning. Finally, find the biomarker value associated with a risk score of 100.

Paper For Above instruction

In analyzing the relationship between the lung cancer risk score and a particular biomarker, the first step involves estimating the regression model based on the provided data. The simple linear regression model is expressed as:

\[ y = \alpha + \beta x + \varepsilon \]

Using the provided estimates, the regression equation becomes:

\[ \hat{y} = 10 + 2.5 x \]

This equation signifies that for every one-unit increase in the biomarker value, the estimated lung cancer risk score increases by 2.5 units, with an intercept of 10 indicating the expected risk score when the biomarker is zero.

In Stata, such analysis can be conducted through the following commands, assuming the dataset contains variables named "risk_score" for \( y \) and "biomarker" for \( x \):

. regress risk_score biomarker

This command initiates a linear regression of the risk score on the biomarker and outputs the estimated coefficients, standard errors, R-squared, and confidence intervals.

To specifically obtain the confidence intervals for the intercept and slope, the following command is used:

. regress risk_score biomarker, level(95)

The output will include the confidence intervals matching those provided: (6, 14) for the intercept and (1, 4) for the slope.

Next, to estimate the difference in the average risk score between two individuals with biomarker values differing by 5 units, we leverage the estimated slope:

Difference in risk scores \( = \betâ \times \Delta x = 2.5 \times 5 = 12.5 \)

This signifies that, on average, an increase of 5 units in the biomarker is associated with a 12.5 point increase in the risk score.

To assess whether the association is statistically significant, we examine the p-value for the slope coefficient from the regression output. Alternatively, the t-statistic can be calculated as:

t = \frac{\hat{\beta} - 0}{SE(\hat{\beta})} = \frac{2.5}{0.75} \approx 3.33

The degrees of freedom are \( n - 2 = 80 \). Consulting a t-distribution table or using Stata:

. display ttail(80, abs(3.33))*2

we find the p-value is less than 0.01, indicating the association is statistically significant at the 5% significance level.

Finally, to find the biomarker value corresponding to a risk score of 100, we rearrange the regression equation:

\[ x = \frac{y - \alpha}{\beta} \]

Plugging in the estimated values:

x = \frac{100 - 10}{2.5} = \frac{90}{2.5} = 36

Hence, a biomarker value of approximately 36 is associated with a predicted risk score of 100.

References

  • Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied Logistic Regression (3rd ed.). Wiley.
  • LaMorte, W. W. (2019). Introduction to Linear Regression. Boston University School of Public Health.
  • StataCorp. (2021). Stata Statistical Software: Release 17. College Station, TX: StataCorp LLC.
  • Wooldridge, J. M. (2016). Introductory Econometrics: A Modern Approach (6th ed.). Cengage Learning.
  • Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2004). Applied Linear Statistical Models (5th ed.). McGraw-Hill.
  • Selvin, S. (1992). Statistical Analysis of Epidemiologic Data. Oxford University Press.
  • Agresti, A. (2007). An Introduction to Categorical Data Analysis. Wiley.
  • Faraway, J. J. (2014). Linear Models with R (2nd ed.). Chapman and Hall/CRC.
  • Kleinbaum, D. G., Kupper, L. L., & Muller, K. E. (1988). Applied Regression Analysis and Other Multivariable Methods. PWS-Kent Publishing Company.
  • Harrell, F. E. (2015). Regression Modeling Strategies. Springer.

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