Statistical Inference for Proportions Summary

Statistical Inferences for Proportions

• Key Concepts:

  • Estimation and inference for one proportion (population parameter $p$)
  • Comparison and inference between two population proportions

• Statistical Methods:

  • Hypothesis testing and confidence intervals for one and two populations
  • Differences in proportions, relative rates, odds ratios

One Population Proportion

• Estimation:
  • Point estimator: $p$
  • Standard error: $SE(p) = \frac{ ext{sqrt}( ext{p}<em>0(1 - ext{p}</em>0))}{n}$
  • Z-test for hypothesis testing against a constant $p_0$.
  • 95% Confidence Interval: $CI = p ext{ ± } 1.96 imes SE(p)$

Confidence Intervals for One Proportion

• Data example: 837 out of 1810 respondents favored legalized abortion.
• Point estimate: 46.24%.
• R function: binom.confint(x, n, conf.level) from the binom package, providing multiple CI methods (11 types).

Two Population Proportions

• Comparison Methods:

  • Difference: $p1 - p2$
  • Relative risk: $RR = \frac{p<em>1}{p</em>2}$
  • Odds ratio: $OR = \frac{p<em>1(1 - p</em>2)}{p<em>2(1 - p</em>1)}$

• Data Structure:

  • Independent groups in rows, binary results in columns.

Chi-Squared Tests

• Use:
  • Confirms independence of variables (e.g., gender and political party).
  • Standard output: Chi-squared statistic and p-value.

R Functions for Proportions

• One population: prop.test() for CI and hypothesis testing.
• Two populations: prop.test() for statistical inference.
• Relative Risk CI: Function from Proxya package.
• CMH test for ordinal data: Using CMH.test() function, requires ordinal data scoring.