I was recently going through the concepts required to remove batch effects from gene expression data. In particular, I was going through [this tutorial][1] by Jeff Leek and trying out the codes mentioned there. From what I have understood from the analysis, SVA is similar to PCA and tries to find two components that try to explain any unknown variations in the data (or batch effects) in an unsupervised fashion. However, I wanted to clarify one plot from the tutorial.   First, they derive the sva components using the following code. 

    mod = model.matrix(~cancer,data=pheno)
    mod0 = model.matrix(~1, data=pheno)
    sva1 = sva(edata,mod,mod0,n.sv=2)
Then they see whether any of the sva components correlate with the batch effects

    summary(lm(sva1$sv ~ pheno$batch))

They get the following results

    ## Response Y1 :
    ## 
    ## Call:
    ## lm(formula = Y1 ~ pheno$batch)
    ## 
    ## Residuals:
    ##      Min       1Q   Median       3Q      Max 
    ## -0.26953 -0.11076  0.00787  0.10399  0.19069 
    ## 
    ## Coefficients:
    ##              Estimate Std. Error t value Pr(>|t|)
    ## (Intercept) -0.018470   0.038694  -0.477    0.635
    ## pheno$batch  0.006051   0.011253   0.538    0.593
    ## 
    ## Residual standard error: 0.1345 on 55 degrees of freedom
    ## Multiple R-squared:  0.00523,    Adjusted R-squared:  -0.01286 
    ## F-statistic: 0.2891 on 1 and 55 DF,  p-value: 0.5929
    ## 
    ## 
    ## Response Y2 :
    ## 
    ## Call:
    ## lm(formula = Y2 ~ pheno$batch)
    ## 
    ## Residuals:
    ##      Min       1Q   Median       3Q      Max 
    ## -0.23973 -0.07467 -0.02157  0.08116  0.25629 
    ## 
    ## Coefficients:
    ##              Estimate Std. Error t value Pr(>|t|)    
    ## (Intercept)  0.121112   0.034157   3.546 0.000808 ***
    ## pheno$batch -0.039675   0.009933  -3.994 0.000194 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ## 
    ## Residual standard error: 0.1187 on 55 degrees of freedom
    ## Multiple R-squared:  0.2248, Adjusted R-squared:  0.2107 
    ## F-statistic: 15.95 on 1 and 55 DF,  p-value: 0.0001945
It is quite clear that the second component is interesting because of the fit having a significant p value. Now they draw a boxplot with the second component as the y axis and the batch effects as the x axis. 

    boxplot(sva1$sv[,2] ~ pheno$batch)
This is where I am stuck. I am unable to interpret this plot. ![enter image description here][2]


  [1]: http://jtleek.com/genstats/inst/doc/02_13_batch-effects.html
  [2]: https://img.techpowerup.org/201029/boxplot.png
When you perform SVA, the method estimates surrogate variables which explain variance in your data (in the case of SVA, only that variance which is not explained by your main comparison). 

The surrogate variables (SVs) that are thus estimated, included in the `svobj` object, have 1 value for each sample. Thus, to check if the SVs are **associated** to the batch variable, a linear model is fit (`lm`). For the second SV, the coefficient tells you that, changing the batch, the value of the SV changes (by `-0.039675`), and this is significant, so it provides proof that the SV is associated to the batch. It is the equivalent of doing an ANOVA and checking if the values of the SV are different between the batches: i.e. if the means between the boxplots are different, which is what is plotted in the figure: the values of the SV, for each sample, segregated into the batch groups.

Additionally, I would add, because you mention that:

> It is quite clear that the second component is interesting because of the fit having a significant p value

Be careful with this: SVA estimates surrogate variables which explain your data. The importance of these surrogate variables particularly depends on how much variance they explain, not on whether they happen to correlate with one of the variables which you have measured in your experiment (such as the batch). What I mean is that, if you had not measured the batch effect, that SV would still be true and important, so I would not say that the importance on the SV relies on its correlation to other measured variables: SVA will find covariates explaining your data whether you happen to know them or not (and that's the nice thing about the method).