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| | ==Comparing Proportions== | | ==Comparing Proportions== |
2023年9月10日 (日) 13:19時点における版
| Navigation Menu
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| General issues of Vaccine
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| General issues of Tropical med.
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| General issues of Travel med.
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| Trematode (fluke, distoma)
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Comparing Proportions
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Independent samples
(Unpaired in case of two)
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Dependent samples
(Paired in case of two)
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| 2 proportions
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- [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]
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| ≥ 3 proportions
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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
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- 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
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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
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| ≥1 cell expected value <5
Fisher's exact test
- very rare in real researches
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Comparing Means
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Parametric
i.e., normally distributed
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Non-parametric
i.e., not normally distributed
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Independent samples
(Unpaired in case of two)
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Dependent samples
(Paired in case of two)
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Independent samples
(Unpaired in case of two)
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Dependent samples
(Paired in case of two)
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| 2 means
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Enough large sample
- [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]
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- [math]\displaystyle{ H_0 }[/math] is mean of paired differences in the population is zero.
- [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
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- Wilcoxon rank sum test
=Mann-Whitney test
- [math]\displaystyle{ H_0 }[/math] is medians or means of ranks in the two populations 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
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- Wilcoxon signed rank test
- [math]\displaystyle{ 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 ('signed rank')
- To look up critical value relevant to numbers of pairs with non-0 differences and whether the smaller sum of rank (=statistics) is smaller than the critical value
- if smaller than the critical value, p-value is smaller than designated
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Small sample size <30 in a group
- [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]
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Large discrepancy in SDs between groups
- Bootstrap
- Non-parametric
- Fisher-Behrens
- Welch
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| ≥ 3 means
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- [math]\displaystyle{
\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]\displaystyle{ n }[/math] is sample size (whole combined number of observations)
- [math]\displaystyle{ k }[/math] is number of groups
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- Linear regression model
- Repeated measures ANOVA
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- [math]\displaystyle{ H_0 }[/math] is medians or means of ranks in the all populations 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]\displaystyle{ H = \frac{n-1}{n} \sum_{i=1}^k \frac{n_i(\bar{R}-E_R)}{s^2} }[/math]
- [math]\displaystyle{ H }[/math] is Kruskal-Wallis statistics
- [math]\displaystyle{ n_i }[/math] is number of observations in group [math]\displaystyle{ i }[/math]
- [math]\displaystyle{ \bar{R} }[/math] is the mean of rank sum in group [math]\displaystyle{ i }[/math]
- [math]\displaystyle{ E_R }[/math] is expected value of the rankings
- [math]\displaystyle{ 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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- *needs try to transform data into parametric (e.g., logarithmic), or other considerations
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Comparing Survival time
| Life table
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Kaplan-Meyer
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- Log rank test
= Mantel-Cox [math]\displaystyle{ \chi^2 }[/math] test
- [math]\displaystyle{ H_0 }[/math] is event (survival) rates in each interval are all the same in two groups
- [math]\displaystyle{ Log\ rank\ statistics = \frac{}{} }[/math]
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