Details the changes made to limma’s diffSplice method.

Here we describe the improvements to *limma*’s
`diffSplice`

method. `diffSplice`

works on the bin-wise coefficient of the
linear model which corresponds to the log2 fold changes between conditions. It
compares the log2(fold change) \(\hat{\beta}_{k,g}\) of a bin \(k\) belonging to
gene \(g\), to a weighted average of log2(fold change) of all the other bins of
the same gene combined \(\hat{B}_{k,g}\) (the subscript \(g\) will be henceforth
omitted for ease of reading). The weighted average of all the other bins in the
same gene is calculated by
\[\begin{equation}
\hat{B}_{k}= \frac{\sum_{i, i\neq k}^{N}{w_{i} \hat{\beta}_{i}}}{\sum_{i, i\neq k}^{N}w_{i}}
\end{equation}\]
where \(w_{i}=\frac{1}{u_{i}^2}\) and \(u_{i}\) refers to the diagonal elements of
the unscaled covariance matrix \((X^TVX)^{-1}\). \(X\) is the design matrix and \(V\)
corresponds to the weight matrix estimated by `voom`

. The difference of log2
fold changes, which is also the coefficient returned by `diffSplice()`

is then
calculated by \(\hat{C}_{k} = \hat{\beta}_{k} - \hat{B}_{k}\). Instead of
calculating the t-statistic with \(\hat{C}_{k}\), this value is scaled again in
the original code:

\[\begin{equation} \hat{D}_{k} =\hat{C}_{k}\sqrt{1-\frac{{w}_{k}}{\sum_{i}^{N}w_{i}}} \end{equation}\]

and the *t*-statistic is calculated as:

\[\begin{equation} t_{k} =\frac{\hat{D}_{k}}{u_{k}s_{g}} \end{equation}\]

\(s_{g}^2\) refers to the posterior residual variance of gene \(g\), which is calculated by averaging the sample values of the residual variances of all the bins in the gene, and then squeezing these residual variances of all genes using empirical Bayes method. This assumes that the residual variance is constant across all bins of the same gene.

In , we applied three changes to the above method. First,
the residual variances are not assumed to be constant across all bins of the
same gene. This results in the sample values of the residual variances of
every bin now being squeezed using empirical Bayes method, resulting in
posterior variances \(s_{i}^2\) for every individual bin \(i\). Second, the
weights \(w_{i}\), used to calculate \(\hat{B}_{k}\), now incorporate the
individual variances by \(w_{i}=\frac{1}{s_{i}^2u_{i}^2}\). Third, the
\(\hat{C}_{k}\) value is directly used to calculate the *t*-statistic, which
after all these changes now corresponds to

\[\begin{equation} t_{k} =\frac{\hat{C}_{k}}{u_{k}s_{i}}. \end{equation}\]

Using a simulation from a previous DEU benchmark
Soneson et al. 2016, we show that
our modifications improve the accuracy of the original
*limma* method:

Of note, however, in most settings the accuracy of `diffSplice2`

is still
inferior to that of *DEXSeq*, which therefore remains
the method of choice for DEU. However, as the computationally-heavy
*DEXSeq* does not scale well and is not applicable in
all situations, we recommend `diffSplice2`

as the next best method.