### Background

To increase the accuracy and specificity of the detection, we developed an assay over our Paired dCas9 Reporter (PC Reporter) System to get more sequence information from the target genome in the purpose of a more reliable result. We designed m pairs of gRNA specific target sites as m markers in the MTB genome. To make sure if the idea mention above actually work, here we used the target gene and the mismatched gene to have a test, respectively. In experimental group, the gRNAs were used to detect the target gene, while in control group, the gRNA were used to detect the mismatched gene. And to reduce the random error, both the experimental and the control group were repeated n times, the result would be shown as the optical power signals, which is generated by our Paired dCas9 Reporter System. Then by comparing the intensity of the optical power signal corresponding to the target gene and mismatched gene, the difference can be seen directly.

**Assumptions**

- There is no recognition site for gRNA in the mismatched gene.
- The measure values of control and experimental group is independent of each other.
- The measure values from n times repeated test compose the sample set, respectively, and the sample sets are both small.

### Model

**Wilcoxon Rank Sum Test of Block Design**

In view of the unknown distributions and different variances of the signals by our Paired dCas9 Reporter System, we chose a non-parametric statistics method called Wilcoxon Rank Sum Test of Block Design with the data Rank instead of ANOVA.

In the Block Design, we regarded the same gRNA detection of two treatment, i.e. target and mismatch DNA, as a block. To test the difference between two treatments, we test the null hypothesis that two treatment have no difference. The Wilcoxon Rank Sum statistics of each block is calculated first by

where indicates the serial number of in the population of both and . Note that Wilcoxon Rank Sum statistics are distribution free and its distribution is known as long as the sample number is known.

For example, if ,
,
,

so , which implies that

Under the null hypothesis, after calculate all the possible order of two sample sets, the distributions of the statistics are shown as below:

W_{j} |
6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|

f(W_{j}) |
0.05 | 0.05 | 0.10 | 0.15 | 0.15 | 0.15 | 0.15 | 0.10 | 0.05 | 0.05 |

Due to the small sample size, the minimal significance level is 0.05, which means only if leads to a rejection of the null hypothesis, in other words only when the minimum value of was greater than the maximum value of 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 as the statistics.

The Wilcoxon Rank Sum 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 converges in distribution to a standard normal distribution *N*(0,1)

So actually we use the statistics , also we can calculate the p-value , where is the distribution function of the standard normal distribution. If p-value is less than 0.01 or , then we accept the alternative hypothesis that the two treatment, i.e. target and mismatch DNA, is highly statistic significantly.

### Result

**Figure. 1 Results of high-throughput assay for MTB and control strain.** F denotes fragments obtained from MTB genome (a) or control strain (b); P denotes markers from each fragment.

In our experiment, , so

so target and mismatch DNA, are highly significantly different in signal.