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**here ::''<math>H_0</math> is '''median of paired differences''' in the population is zero'':#To calculate differences between pairs and discard 0 differences:#To rank the absolute values of differences (ignoring 0):#To make the sum of ranks of '''positive difference''' and the sum of ranks of '''negative differences''' ('signedrank'):#To look up '''critical value' means 'take into account signs ' relevant to numbers of pairs with non-0 differences and whether '''the smaller sum of paired datarank'''(=statistics) is smaller than the critical value::if smaller than the critical value, p-value is smaller than designated
→Basic concept of statistical tests
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==Basic concept of statistical tests==
*First remind that
**Statistical test is to test
==Comparing Proportions==
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|}
==Comparing Means==
{|class="wikitable"
!rowspan="2" style="width:80px"|
|-
!rowspan="3"|2 means
|''Enough large sample''
*'''Z test'''
::<math>
\end{align}
</math>
|rowspan="3" style="vertical-align:top"|
*'''Paired Student's t test'''
::''<math>H_0</math> is '''mean of paired differences''' in the population is '''zero'''.''
::<math>
\begin{align}
</math>
::where <math>\bar{d}</math> is the mean of differences of paired observations
|rowspan="3" style="vertical-align:top"|
*'''Wilcoxon rank sum test'''**AKA <br>='''Mann-Whitney test'''::''<math>H_0</math> is '''medians or means of ranks''' in the two population'''s''' are the same'':#To rank whole combined observations of two groups:#To separate back the ranks into two groups:#To look up ''critical range'' relevant to both numbers of observations and whether '''the sum of ranks''' in '''the group of smaller number of observation''' (=statistics) is outside the range or not::if outside the range, p-value is smaller than designated
|rowspan="3" style="vertical-align:top"|
*'''Wilcoxon signed rank test'''
|-
|''Small sample size <30 in a group''
\end{align}
</math>
|-
|''Large discrepancy in SDs between groups''
*Fisher-Behrens
*Welch
|-
!≥ 3 means
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*'''One-way ANOVA'''
::<math>
\begin{align}
F & = \frac{ \sum_{j=1}^k \sum_{j=1}^{n_j} (x_{ij}-\bar{x_j})^2 }{ k-1 } \\
& \div \frac{ \sum_{j=1}^k (\bar{x_j}-\bar{x})^2 }{ n-k }
\end{align}
</math>
::<math>n</math> is sample size (whole combined number of observations)
::<math>k</math> is number of groups
<!--
::The variance of whole combined observations is
::<math>s^2=\frac{\sum_{i=1}^n (x_i-\bar{x})^2}{n-1}</math>
::The numarator is ''sum of square''
::<math>\sum_{i=1}^n (x_i-\bar{x})^2</math>
::<math>= \sum_{i=1}^n x_i^2 - 2\bar{x} \sum_{i=1}^n x_i + \bar{x}^2 \sum_{i=1}^n 1</math>
-->
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*'''Linear-regression model'''
*Repeated measures ANOVA
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*'''Kruskall-Wallis test'''
::''<math>H_0</math> is '''medians or means of ranks''' in the all population'''s''' are the same''
:#To rank whole combined observations of all groups
:#To separate back the ranks into original groups
:#To make sum of ranks in each group
::<math>H = \frac{n-1}{n} \sum_{i=1}^k \frac{n_i(\bar{R}-E_R)}{s^2}</math>
::<math>H</math> is Kruskal-Wallis statistics
::<math>n_i</math> is number of observations in group <math>i</math>
::<math>\bar{R}</math> is the mean of rank sum in group <math>i</math>
::<math>E_R</math> is expected value of the rankings
::<math>s^2</math> is the variance of rank
::To look up ''critical values'' relevant to '''sum of ranks''' in '''the group of smaller number of observation'''
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*<nowiki>*</nowiki>''needs try to transform data into parametric (e.g., logarithmic), or other considerations''|} ==Comparing Survival time=={|class="wikitable"|-!style="width:50%"|Life table!style="width:50%"|Kaplan-Meyer|-|colspan="2"|*'''Log rank test<br>= Mantel-Cox <math>\chi^2</math> test'''::<math>H_0</math> is event (survival) rates in each interval are all the same in two groups::<math>Log\ rank\ statistics = \frac{}{}</math>
|}
