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− | <p>Due to the small sample size, the minimal significance level is 0.05, which means only if <img alt='Peking-Analysis-Wj%3D15.gif' src="https://static.igem.org/mediawiki/2015/e/e0/Peking-Analysis-Wj%3D15.gif" class='formula-inline' style="height:25px"> leads to a rejection of the null hypothesis, in other words only when the minimum value of <img alt='Peking-Analysis-X_j.gif' src="https://static.igem.org/mediawiki/2015/6/60/Peking-Analysis-X_j.gif" class='formula-inline' style="height:25px"> was greater than the maximum value of <img alt='Peking-Analysis-Y_j.gif' src="https://static.igem.org/mediawiki/2015/7/79/Peking-Analysis-Y_j.gif" class='formula-inline' style="width:30px"> to accept the alternative hypothesis instead of the null hypothesis, the two sets of data is significantly different. So the Wilcoxon Rank Sum Test may face challenge in single block test when the experimental and control group are slightly different.However, by using Block Design, we can integrate data from m blocks similar to the idea of ANOVA. We calculated the sum of <img alt='Peking-Analysis-W_j.gif' src="https://static.igem.org/mediawiki/2015/1/15/Peking-Analysis-W_j.gif" class='formula-inline' style="height:25px"> as the statistics. | + | <p>Due to the small sample size, the minimal significance level is 0.05, which means only if <img alt='Peking-Analysis-Wj%3D15.gif' src="https://static.igem.org/mediawiki/2015/e/e0/Peking-Analysis-Wj%3D15.gif" class='formula-inline' style="height:30px"> leads to a rejection of the null hypothesis, in other words only when the minimum value of <img alt='Peking-Analysis-X_j.gif' src="https://static.igem.org/mediawiki/2015/6/60/Peking-Analysis-X_j.gif" class='formula-inline' style="height:30px"> was greater than the maximum value of <img alt='Peking-Analysis-Y_j.gif' src="https://static.igem.org/mediawiki/2015/7/79/Peking-Analysis-Y_j.gif" class='formula-inline' style="height:30px"> to accept the alternative hypothesis instead of the null hypothesis, the two sets of data is significantly different.</p> |
| + | <p>So the Wilcoxon Rank Sum Test may face challenge in single block test when the experimental and control group are slightly different.However, by using Block Design, we can integrate data from m blocks similar to the idea of ANOVA. We calculated the sum of <img alt='Peking-Analysis-W_j.gif' src="https://static.igem.org/mediawiki/2015/1/15/Peking-Analysis-W_j.gif" class='formula-inline' style="height:30px"> as the statistics. |
| <div align='center'> | | <div align='center'> |
− | <img class='formula-line' alt="Peking-Analysis-W_sumWj.gif" src="https://static.igem.org/mediawiki/2015/2/29/Peking-Analysis-W_sumWj.gif" style='margin-top:0px;padding:0px' style="height:40px"> | + | <img class='formula-line' alt="Peking-Analysis-W_sumWj.gif" src="https://static.igem.org/mediawiki/2015/2/29/Peking-Analysis-W_sumWj.gif" style='margin-top:0px;padding:0px' style="height:35px"> |
| </div> | | </div> |
− | The Wilcoxon Rank Sum <img alt='Peking-Analysis-W_j.gif' src="https://static.igem.org/mediawiki/2015/1/15/Peking-Analysis-W_j.gif" class='formula-inline' style="height:25px"> from m blocks are independent and identically distributed (i.i.d), according to the central limit theorem (CLT), as m approaches infinity, the random variable <img class='big-formula-inline' alt="Peking-Analysis-W_BD_statistics.gif" src="https://static.igem.org/mediawiki/2015/6/68/Peking-Analysis-W_BD_statistics.gif" style="height:40px"> converges in distribution to a standard normal distribution <i>N</i>(0,1) | + | The Wilcoxon Rank Sum <img alt='Peking-Analysis-W_j.gif' src="https://static.igem.org/mediawiki/2015/1/15/Peking-Analysis-W_j.gif" class='formula-inline' style="height:30px"> from m blocks are independent and identically distributed (i.i.d), according to the central limit theorem (CLT), as m approaches infinity, the random variable <img class='big-formula-inline' alt="Peking-Analysis-W_BD_statistics.gif" src="https://static.igem.org/mediawiki/2015/6/68/Peking-Analysis-W_BD_statistics.gif" style="height:60px"> converges in distribution to a standard normal distribution <i>N</i>(0,1) |
| <div align="center"> | | <div align="center"> |
− | <img class='big-formula-line' alt="Peking-Analysis-W_BD_statistics_CLT.gif" src="https://static.igem.org/mediawiki/2015/7/73/Peking-Analysis-W_BD_statistics_CLT.gif "style="height:40px" style='margin:10px'> | + | <img class='big-formula-line' alt="Peking-Analysis-W_BD_statistics_CLT.gif" src="https://static.igem.org/mediawiki/2015/7/73/Peking-Analysis-W_BD_statistics_CLT.gif "style="height:60px" style='margin:10px'> |
| </div> | | </div> |
− | So actually we use the statistics <img class='big-formula-inline' alt="Peking-Analysis-W_BD_statistics.gif" src="https://static.igem.org/mediawiki/2015/6/68/Peking-Analysis-W_BD_statistics.gif" style="height:40px">, also we can calculate the p-value <img class='formula-inline' alt="Peking-Analysis-W_BD_p_value.gif" src="https://static.igem.org/mediawiki/2015/d/da/Peking-Analysis-W_BD_p_value.gif">, where <img class='formula-inline' alt="Peking-Analysis-Phi%28x%29.gif" src="https://static.igem.org/mediawiki/2015/1/19/Peking-Analysis-Phi%28x%29.gif" style="height:25px"> is the distribution function of the standard normal distribution. If p-value is less than 0.01 or <img class='formula-inline' alt="Peking-Analysis-W_BD_gt_2.33.gif" src="https://static.igem.org/mediawiki/2015/2/20/Peking-Analysis-W_BD_gt_2.33.gif" style="height:40px">, then we accept the alternative hypothesis that the two treatment, i.e. target and mismatch DNA, is highly statistic significantly.</p> | + | So actually we use the statistics <img class='big-formula-inline' alt="Peking-Analysis-W_BD_statistics.gif" src="https://static.igem.org/mediawiki/2015/6/68/Peking-Analysis-W_BD_statistics.gif" style="height:60px">, also we can calculate the p-value <img class='formula-inline' alt="Peking-Analysis-W_BD_p_value.gif" src="https://static.igem.org/mediawiki/2015/d/da/Peking-Analysis-W_BD_p_value.gif" style="height:30px">, where <img class='formula-inline' alt="Peking-Analysis-Phi%28x%29.gif" src="https://static.igem.org/mediawiki/2015/1/19/Peking-Analysis-Phi%28x%29.gif" style="height:30px"> is the distribution function of the standard normal distribution. If p-value is less than 0.01 or <img class='formula-inline' alt="Peking-Analysis-W_BD_gt_2.33.gif" src="https://static.igem.org/mediawiki/2015/2/20/Peking-Analysis-W_BD_gt_2.33.gif" style="height:30px">, then we accept the alternative hypothesis that the two treatment, i.e. target and mismatch DNA, is highly statistic significantly.</p> |
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