Statistical test

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