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==Comparing Proportions==
 
==Comparing Proportions==
 
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2022年12月11日 (日) 18:28時点における版

Basics & Definition
Epidemiology
Odds in statistics and Odds in a horse race
Collider bias
Data distribution
Statistical test
Regression model
Multivariate analysis
Marginal effects
Prediction and decision
Table-related commands in STATA
Missing data and imputation

Comparing Proportions

Independent samples
(Unpaired in case of two)
Dependent samples
(Paired in case of two)
2 proportions
  • Z test
[math]\displaystyle{ \begin{align} z & = \frac{p_1-p_2}{SE_{pooled(p_1-p_2)}} \\ & = \frac{p_1-p_2}{\sqrt{\frac{\bar{p}(1-\bar{p})}{n_1}+\frac{\bar{p}(1-\bar{p})}{n_2}}} \end{align} }[/math]
≥ 3 proportions Enough large sample
  • [math]\displaystyle{ \chi^2 }[/math] test
[math]\displaystyle{ \chi^2 = \sum \frac{(O - E)^2}{E} }[/math]
[math]\displaystyle{ O }[/math] = observed values
[math]\displaystyle{ E }[/math] = expected values
  • McNemar's [math]\displaystyle{ \chi^2 }[/math] test
[math]\displaystyle{ \begin{align} & McNemar's\ \chi^2 \\ & = \frac{(n_1-n_2)^2}{n_1+n_2} \end{align} }[/math]
[math]\displaystyle{ n_i }[/math] = number of observations in discordant pair
Testing linear association
  • [math]\displaystyle{ \chi^2 }[/math] trend test
[math]\displaystyle{ \begin{align} & \chi^2 trend \\ & = \frac{(\bar{x_1}-\bar{x_2})^2}{s^2(\frac{1}{n_1}+\frac{1}{n_2})} \\ & s = \sqrt{\sum \frac{(x_i-\bar{x_i})^2}{n-1}} \end{align} }[/math]
[math]\displaystyle{ x_i }[/math] = weighted values
[math]\displaystyle{ n_i }[/math] = number of observations
≥1 cell expected value <5

Fisher's exact test

  • very rare in real researches

Comparing Means

Parametric
i.e., normally distributed
Non-parametric
i.e., not normally distributed
Independent samples
(Unpaired in case of two)
Dependent samples
(Paired in case of two)
Independent samples
(Unpaired in case of two)
Dependent samples
(Paired in case of two)
2 means

Enough large sample

  • Z test
[math]\displaystyle{ \begin{align} z & = \frac{\bar{x_1}-\bar{x_2}}{SE_{(\bar{x_1}-\bar{x_2})}} \\ & = \frac{\bar{x_1}-\bar{x_2}}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}} \end{align} }[/math]
  • Paired Student's t test
[math]\displaystyle{ \begin{align} paired\ t & = \frac{\bar{d}}{SE_d} \\ & = \frac{\bar{d}}{\frac{s}{\sqrt{n}}} \\ \end{align} }[/math]
where [math]\displaystyle{ \bar{d} }[/math] is the mean of differences of paired observations
  • Wilcoxon rank sum test
    • AKA Mann-Whitney test
  • Wilcoxon signed rank test
    • here 'signed' means 'take into account signs of differences of paired data'
Small sample size <30 in a group
  • Student's t test
[math]\displaystyle{ \begin{align} t & = \frac{\bar{x_1}-\bar{x_2}}{SE_{(\bar{x_1}-\bar{x_2})}} \\ & = \frac{\bar{x_1}-\bar{x_2}}{\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{(n_1-1)+(n_2-1)}}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \end{align} }[/math]
Large discrepancy in SDs between groups
  • Bootstrap
  • Non-parametric
  • Fisher-Behrens
  • Welch
≥ 3 means
  • One-way ANOVA
  • Linear-regression model
  • Repeated measures ANOVA
  • Kruskall-Wallis test

*needs try to transform data into parametric (e.g., logarithmic), or other considerations