「Statistical test」の版間の差分

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==Means==
 
==Means==
 
{|class="wikitable"
 
{|class="wikitable"
!colspan="2" rowspan="2"|comparisons
+
!rowspan="2"|comparisons
 
!colspan="2"|Parametric<br>i.e., normally distributed
 
!colspan="2"|Parametric<br>i.e., normally distributed
 
!colspan="2"|Non-parametric<br>i.e., not normally distributed
 
!colspan="2"|Non-parametric<br>i.e., not normally distributed
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!style="width:200px"|Dependent samples<br>(Paired in case of two)
 
!style="width:200px"|Dependent samples<br>(Paired in case of two)
 
|-
 
|-
!rowspan="2" style="width:80px"|Means
+
!rowspan="3" style="width:40px"|2 means
!style="width:40px"|2
 
 
|
 
|
 
''Enough large sample''
 
''Enough large sample''
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\end{align}
 
\end{align}
 
</math>
 
</math>
''Small sample size <30 in a group''
+
|rowspan="3" style="vertical-align:top"|
*'''Student's t test'''
 
::<math>
 
\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'''
 
|style="vertical-align:top"|
 
 
*'''Paired Student's t test'''
 
*'''Paired Student's t test'''
 
::<math>
 
::<math>
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</math>
 
</math>
 
::where <math>\bar{d}</math> is the mean of differences of paired observations
 
::where <math>\bar{d}</math> is the mean of differences of paired observations
|style="vertical-align:top"|
+
|rowspan="3" style="vertical-align:top"|
 
*'''Wilcoxon rank sum test'''
 
*'''Wilcoxon rank sum test'''
 
**AKA '''Mann-Whitney test'''
 
**AKA '''Mann-Whitney test'''
|style="vertical-align:top"|
+
|rowspan="3" style="vertical-align:top"|
 
*'''Wilcoxon signed rank test'''
 
*'''Wilcoxon signed rank test'''
 
**here 'signed' means 'take into account signs of differences of paired data'
 
**here 'signed' means 'take into account signs of differences of paired data'
 
|-
 
|-
!3 or more
+
|''Small sample size <30 in a group''
 +
*'''Student's t test'''
 +
::<math>
 +
\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
 +
|-
 +
!&ge; 3 means
 
|style="vertical-align:top"|
 
|style="vertical-align:top"|
 
*'''ANOVA'''
 
*'''ANOVA'''

2022年12月11日 (日) 18:20時点における版

Proportions

comparisons Independent samples
(Unpaired in case of two)
Dependent samples
(Paired in case of two)
Propotions 2
  • 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 or more 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

Means

comparisons 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
  • ANOVA
  • Linear-regression model
  • Repeated measures ANOVA
  • Kruskall-Wallis test

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