这个问题其实在10年有过一篇文章讲过这个事情，从统计学上说就是如何平衡组内和组间的效应。R语言中的sva包是可以之间处理的。除此之外，推荐给大家一个在线网站。http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a3.htm。

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Batch is a Nuisance Factor | The two nuisance factors in this experiment are the batch number and the lab. There are two batches and eight labs. Ideally, these factors will have minimal effect on the response variable. We will investigate the batch factor first. | |

Bihistogram | This bihistogram shows the following. - There does appear to be a batch effect.
- The batch 1 responses are centered at 700 while the batch 2 responses are centered at 625. That is, the batch effect is approximately 75 units.
- The variability is comparable for the 2 batches.
- Batch 1 has some skewness in the lower tail. Batch 2 has some skewness in the center of the distribution, but not as much in the tails compared to batch 1.
- Both batches have a few low-lying points.
Although we could stop with the bihistogram, we will show a few other commonly used two-sample graphical techniques for comparison. | |

Quantile-Quantile Plot | This q-q plot shows the following. - Except for a few points in the right tail, the batch 1 values have higher quantiles than the batch 2 values. This implies that batch 1 has a greater location value than batch 2.
- The q-q plot is not linear. This implies that the difference between the batches is not explained simply by a shift in location. That is, the variation and/or skewness varies as well. From the bihistogram, it appears that the skewness in batch 2 is the most likely explanation for the non-linearity in the q-q plot.
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Box Plot | This box plot shows the following. - The median for batch 1 is approximately 700 while the median for batch 2 is approximately 600.
- The spread is reasonably similar for both batches, maybe slightly larger for batch 1.
- Both batches have a number of outliers on the low side. Batch 2 also has a few outliers on the high side. Box plots are a particularly effective method for identifying the presence of outliers.
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Block Plots | A block plot is generated for each of the eight labs, with "1" and "2" denoting the batch numbers. In the first plot, we do not include any of the primary factors. The next 3 block plots include one of the primary factors. Note that each of the 3 primary factors (table speed = X1, down feed rate = X2, wheel grit size = X3) has 2 levels. With 8 labs and 2 levels for the primary factor, we would expect 16 separate blocks on these plots. The fact that some of these blocks are missing indicates that some of the combinations of lab and primary factor are empty. These block plots show the following. - The mean for batch 1 is greater than the mean for batch 2 in
**all**of the cases above. This is strong evidence that the batch effect is real and consistent across labs and primary factors.
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Quantitative Techniques | We can confirm some of the conclusions drawn from the above graphics by using quantitative techniques. The F-test can be used to test whether or not the variances from the two batches are equal and thetwo sample t-test can be used to test whether or not the means from the two batches are equal. Summary statistics for each batch are shown below.Batch 1: NUMBER OF OBSERVATIONS = 240 MEAN = 688.9987 STANDARD DEVIATION = 65.5491 VARIANCE = 4296.6845 Batch 2: NUMBER OF OBSERVATIONS = 240 MEAN = 611.1559 STANDARD DEVIATION = 61.8543 VARIANCE = 3825.9544 | |

F-Test | The two-sided F-test indicates that the variances for the two batches are not significantly different at the 5 % level.H0: σ12 = σ22 Ha: σ12 ≠ σ22 Test statistic: | |

Two Sample t-Test | Since the F-test indicates that the two batch variances are equal, we can pool the variances for the two-sided, two-sample t-test to compare batch means.H0: μ1 = μ2 Ha: μ1 ≠ μ2 Test statistic:The t-test indicates that the mean for batch 1 is larger than the mean for batch 2 at the 5 % significance level. | |

Conclusions | We can draw the following conclusions from the above analysis.- There is in fact a significant batch effect. This batch effect is consistent across labs and primary factors.
- The magnitude of the difference is on the order of 75 to 100 (with batch 2 being smaller than batch 1). The standard deviations do not appear to be significantly different.
- There is some skewness in the batches.
This batch effect was completely unexpected by the scientific investigators in this study. Note that although the quantitative techniques support the conclusions of unequal means and equal standard deviations, they do not show the more subtle features of the data such as the presence of outliers and the skewness of the batch 2 data. |