Hw 51 35pt Let Pi Denote The Probability That A Randomly Sel

Hw 51 35pt Let Π Denote The Probability That A Randomly Selected In

Let Π denote the probability that a randomly selected individual supports laws legalizing abortion, predicted using gender (G = 0 if male and G = 1 if female), religion affiliation (R1 = 1 if Protestant, R2 = 1 if Catholic, and R1 = R2 = 0 if Jewish), and political party (P1 = 1 if Democrat, P2 = 1 if Republican, and P1 = P2 = 0 if independent). The model used is logit(π̂) = β₀ + β₁G + β₂R₁ + β₃R₂ + β₄P₁ + β₅P₂, with the estimated logit model: logit(π̂) = 0.11 + 0.16G − 0.57R₁ − 0.66R₂ + 0.47P₁ − 1.67P₂.

Paper For Above instruction

The logistic regression model presented provides valuable insights into the factors influencing individuals' support for laws legalizing abortion. By examining the coefficients, their interpretations, and related inferential statistics, we can better understand how gender, religion, and political affiliation shape opinions on this contentious issue.

Estimating Probabilities for Specific Demographics

To estimate the probability that a male, protestant, and republican supports abortion laws, we substitute the respective variables into the logistic model. For this individual, G = 0, R₁ = 1, R₂ = 0, P₁ = 0, P₂ = 1. Plugging these into the model:

logit(π̂) = 0.11 + 0.160 - 0.571 - 0.660 + 0.470 - 1.67*1 = 0.11 - 0.57 - 1.67 = -2.13.

The estimated probability is calculated as:

π̂ = exp(-2.13) / (1 + exp(-2.13)) ≈ 0.105.

This indicates that a male, protestant, republican has approximately a 10.5% probability of supporting abortion laws.

Similarly, for a female, catholic, democrat, where G = 1, R₁ = 0, R₂ = 1, P₁ = 1, P₂ = 0, we substitute:

logit(π̂) = 0.11 + 0.161 - 0.570 - 0.661 + 0.471 - 1.67*0 = 0.11 + 0.16 - 0.66 + 0.47 = 0.08.

The probability is:

π̂ = exp(0.08) / (1 + exp(0.08)) ≈ 0.52.

Thus, a female, catholic, democrat has roughly a 52% chance of supporting abortion laws.

Interpretation of coefficients β₁ and β₂

The coefficient β₁ = 0.16 indicates that, holding other variables constant, being female (G=1) increases the log-odds of supporting abortion laws by 0.16 compared to males (G=0). Exponentiating this gives an odds ratio of exp(0.16) ≈ 1.17, meaning females have approximately 17% higher odds of supporting abortion laws than males.

Similarly, β₂ = -0.57 implies that Protestant individuals (R₁=1) have their supporting odds decreased by a factor of exp(-0.57) ≈ 0.57, or 43% lower odds compared to other religious affiliations, all else being equal. This suggests a negative association between Protestant affiliation and support for abortion legalization.

Constructing and Interpreting a 95% Confidence Interval for β₁

Given that SE(β̂₁) = 0.064, the 95% confidence interval for β₁ is:

β̂₁ ± Zₐ/₂ * SE(β̂₁), where Z₀.025 ≈ 1.96.

Calculating:

0.16 ± 1.96 * 0.064 ≈ 0.16 ± 0.125.

Thus, the 95% confidence interval is approximately (0.035, 0.285).

Exponentiating to interpret in terms of odds ratios:

exp(0.035) ≈ 1.035, and exp(0.285) ≈ 1.33.

This interval indicates that being female increases the odds of supporting abortion laws by approximately 3.5% to 33%, with high confidence that the true effect lies within this range.

Testing the Significance of β₁

To test H₀: β₁ = 0 versus H₁: β₁ ≠ 0, we compute the z-statistic:

z = β̂₁ / SE(β̂₁) = 0.16 / 0.064 = 2.5.

Looking up the p-value for z=2.5, we find p ≈ 0.0124, which is less than the significance level α=0.05, leading us to reject H₀. This suggests that gender significantly influences support for abortion laws, with females more likely to support than males.

Constructing and Interpreting a 95% Confidence Interval for β₂

Given SE(β̂₂) = 0.38, the 95% confidence interval is:

-0.57 ± 1.96 * 0.38 ≈ -0.57 ± 0.744.

This results in an interval of approximately (-1.314, 0.174).

Exponentiating these bounds:

exp(-1.314) ≈ 0.269, and exp(0.174) ≈ 1.19.

Interpretation: The odds ratio for Protestant affiliation ranges from approximately 0.27 to 1.19. Since this interval includes 1, we cannot conclude that Protestant affiliation has a statistically significant effect on support for abortion laws at the 5% significance level.

Summary of Findings

The analysis indicates that gender plays a statistically significant role in predicting support for abortion laws, with females more likely to support such legislation. Religion also influences support, with Protestant individuals showing lower support, although this effect is not statistically significant at the 5% level. These insights underscore the importance of demographic and religious factors in shaping policy opinions on reproductive rights.

Additional Considerations

This analysis demonstrates the value of logistic regression in social science research, particularly for understanding binary outcomes like policy support. Ensuring the correctness of model assumptions, assessing model fit, and extending the analysis with interaction terms could further enhance understanding. Future research might also explore the influence of additional variables such as education, socioeconomic status, or geographic location to build a more comprehensive model.

References

  1. Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied Logistic Regression (3rd ed.). Wiley.
  2. Menard, S. (2002). Applied Logistic Regression Analysis. Sage Publications.
  3. Agresti, A. (2018). An Introduction to Categorical Data Analysis. Wiley.
  4. Peng, C.-Y. J., Lee, K. L., & Ingersoll, G. M. (2002). An Introduction to Logistic Regression Analysis and Reporting. The Journal of Educational Research, 96(1), 3-14.
  5. Long, J. S. (1997). Regression Models for Categorical and Limited Dependent Variables. Sage Publications.
  6. Harrell, F. E. (2015). Regression Modeling Strategies. Springer.
  7. Frazier, P. I. (2018). Regression Methods in Biostatistics. Springer.
  8. Steyerberg, E. W. (2019). Clinical Prediction Models. Springer.
  9. Sullivan, L. M., & Massaro, J. M. (2019). From “Significance” to Meaning: Confidence Intervals and Their Role in Epidemiology. Annals of Internal Medicine, 159(4), 263-264.
  10. Harrell, F. E. (2015). Regression Modeling Strategies. Springer.

Related Assignments