---
title: "ClusterSignificance Vignette"
author: "Jesper Robert Gådin and Jason T. Serviss"
date: '`r Sys.Date()`'
output:
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---

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```{r style, echo = FALSE, results = 'asis'}
BiocStyle::markdown()
library(ggplot2)
```

# Introduction
The ClusterSignificance package provides tools to assess if 
"class clusters" in dimensionality reduced data 
representations have a separation different from permuted 
data. The term "class clusters" here refers to, clusters of 
points representing known classes in the data. This is particularly
useful to determine if a subset of the variables, e.g. genes in a
specific pathway, alone can separate samples into these established 
classes. Evaluation of this is accomplished in a three stage process 
*Projection*, *Serparation classification*, and *Score permutation*. 
To be able to compare class cluster separations, we give them a score
based on this separation. First, all data points are projected 
onto a line (*Projection*), after which the best separation for 
two groups at a time is identified and scored 
(*Separation classification*). Finally, to get a p-value for the 
separation, we compare the separation score for our real data to 
the separation score for permuted data (*Score permutation*).

The package includes 2 different methods for accomplishing the 
Projection step outlined above, *Mean line projection* (**Mlp**) 
and *Principal curve projection* (**Pcp**). Here we will first underline 
the assumptions made by the ClusterSignificance method, followed by a 
description of the similarities of the three stages of ClusterSignificance 
independent of which Projection method is used. This is followed by an 
example of the Mlp and Pcp methods and a description of the unique 
features of each. Furthermore, we provide an example where 
ClusterSignificance is used to examine the separation between 2 class 
clusters downsteam of a principal component analysis (PCA). 

# Assumptions
The ClusterSignificance package operates based on three simple assupmtions
shown below:

1. The dimensionality reduction method was sufficiently capable of detecting
the dissimilarities, characterized by features in high-dimensional space, that 
correspond to the class separations under investigation.

2. The obtained principal curve accurately depicts the data with respect to 
the class separations under consideration.

3. Under the null hypothesis, the joint distribution remains invariant under 
all rearrangements of the subscripts.

# Methods
## Projection

The inputs to the Projection methods are a matrix of the data representation 
after dimensionality reduction and the class labels. (Dimensionality reduction 
is not strictly necessary which is discussed further in 
<a href="#common-questions">common questions</a>) The input matrix should be 
constructed with data points (or samples) on the rows, and dimensions 
(or principal components) in the columns. For the method to know which rows 
are in the same class, the class argument must be specified. The class argument 
should be a character vector without NA's. The projection step utilizes either 
the Mlp or Pcp method to project all data points onto a line, after which they 
are moved into a single dimension.

## Seperation classification

Now that all of the points are in one dimension, it allows us to easily
define a perpendicular line that best separates the two classes. This is
accomplished by utilizing the user defined classes to calculate the 
sensivitity and specificity at each possible seperation of the projected 
data. The score for each possible separation is then calculated based on 
the formula below which measures the complement of the Euclidean distance 
from each seperation to the ROC curves operating point:

#### $$score = 1 - \sqrt{(1-specificity)^2 + (1-sensitivity)^2}$$

In addition to the seperation scores, the Seperation classification stage
also outputs the area under the curve (AUC) for each class seperation under
investigation. AUC serves as an intuitave measure of the class seperation in 
the projected space.

## Score permutation

The null-hypothesis of the permutation stage is that the class labels are 
independent of the features and, thus rejection of the null indicates a 
dependence of these variables and consequently, a separation between 
classes. Permutation is performed by randomly sampling the input matrix, with
replacment, and assigning the data points to the groups. The projection and 
classification steps are then run for the sampled matrix. The max separation 
score is recorded and the next permutation is performed. The p-value can 
subsequently be calculated with the following formula:

####$$p.value=\frac{count\ of\ permuted\ scores >= real\ score}{iterations}$$

If none of the permuted scores are greater that the real score the p-value
is instead calculated as:

#### $$p.value < 10^{-log10(iterations)}$$

Due to the fact that the score permutation stage will typically be a 
monte carlo test, rather that a permutation test, the p-value is 
actually a p-value estimate with a confidence interval. A discussion 
concerning this and the number of permutations necessary can be 
found <a href="#common-questions">here</a>. 

# Choosing a projection method
As previously mentioned, there are two projection methods provided with the 
ClusterSignificance package; *Principal curve projection* (**Pcp**)
and *Mean line projection* (**Mlp**) . These are outlined
below together with the situations where each can be used.

## Pcp
The Pcp method is suitable in situations where more than 2 dimensions need
to be considered simultaneously (although Pcp works with only 2 dimensions)
and/or more than 2 class separations should
be analyzed. It is limited by the fact that, the Pcp method will not work
for < 5 data points per class and must have > 5 unique values per dimension.
The Pcp method utilizes the principal_curve method from the princurve package
to fit a principal curve to the data points utilizing all the dimensions input
by the user. A principal curve can be described as a "smooth curve that
passes through the middle of the data in an orthogonal sense". All data
points are then projected onto the principal curve and transferred into
one dimension for scoring.

## Mlp
The Mlp method is suitable for comparing clusters in a maximum of 2 dimensions
when the separation between a maximum of 2 classes will be evaluated at
a time. Briefly, Mlp functions by drawing a regression line through the
means of the two classes and then projects all points onto that line. To
project the points into one dimension, the points are then rotated onto
the x-axis. A detailed description and graphical representation of the
individual steps can be seen in the Mlp example below. It should be noted that,
despite the fact that the Mlp method will work with as low as 2 data points
per class, it may not be advisable to perform an analysis with such a small
amount of data.

# Examples

```{r echo=TRUE, eval=TRUE, message=FALSE}
library(ClusterSignificance)
```

## Demonstration data

The ClusterSignificance package includes 2 small example data matrices, 
one used for demonstrating the Mlp method and one used for demonstrating 
the Pcp method. The Mlp demo matrix has 2 dimensions whereas, the Pcp demo 
matrix has 4. In addition, the class variable has been added as the rownames 
of each matrix, with the Mlp demo matrix having 2 classes and the Pcp demo 
matrix having 3 classes. Finally, colnames, indicating the dimension number, 
were also added to each matrix. Neither colnames or rownames are needed for 
the input matrix to ClusterSignificance functions and we have only added 
them here only for clarity.

```{r, demoDataMlp}
data(mlpMatrix)
```

```{r headMat, echo=FALSE, eval=TRUE}
##Mlp
printMat <- mlpMatrix
classes <- rownames(mlpMatrix)
dimnames(printMat) <- list(classes, paste("dim", 1:2, sep = ""))
knitr::kable(
    printMat[1:3,], 
    digits = 2, 
    caption = "Head; Mlp matrix.", 
    align = c(rep('l', 2))
)

gTable <- as.data.frame(table(classes))
knitr::kable(
    gTable, 
    digits = 2, 
    align = c(rep('l', 2)), 
    caption = "Rownames table; Mlp matrix."
)
```

```{r, demoDataPcp}
data(pcpMatrix)
```

```{r headPcp, echo=FALSE}
##Pcp
printMat <- pcpMatrix
classes <- rownames(pcpMatrix)
dimnames(printMat) <- list(classes, paste("dim", 1:3, sep=""))
knitr::kable(
    printMat[1:3,], 
    digits = 2, 
    caption = "Head; Pcp matrix.", 
    align = c(rep('l', 2))
)

gTable <- as.data.frame(table(classes))
knitr::kable(
    gTable, 
    digits = 2, 
    align = c(rep('l', 2)), 
    caption = "Rownames table; Pcp matrix."
)
```

## Pcp method
### Projection

To provide the readers with an easy to follow example, 
we generated in silico data representative of output from
a dimensionality reduction algorithm. The demonstration data was simulated 
so that one cluster has a visible separation from the two other clusters, 
whereas two of the clusters are largely overlapping. Therefore, we can imagine
that ClusterSignificance will find 2 separations to be significant where as 
the other will be insignificant.

```{r PcpProjection, fig.align='center', fig.width=10, fig.height=10}
## Create the group variable.
classes <- rownames(pcpMatrix)

## Run Pcp and plot.
prj <- pcp(pcpMatrix, classes)
plot(prj)
```

The following steps take place during Pcp projection:

1. The input matrix.
2. Drawing the principal curve.
3. Projection of points onto the principal curve.
4. The points after projection onto the principal curve.
5. Projection of points into one dimension.
6. The final projection of points.

### Seperation classification
The classification function works in the same manner for both the Mlp and
Pcp methods and is previously described in the [Classification][] section.
The lines perpendicular to the x-axis in the classification plot represent 
the separation scores for all possible separations. The highest scoring 
separation of the two classes is represented by the highest perpendicular line.
Due to the fact that the Pcp method can perform several class comparisons
at once, when > 2 classes are used all possible separations are automatically
analyzed and plotted.

```{r classifyPcp, message=FALSE, fig.width=10, fig.height=7}
cl <- classify(prj)
plot(cl)
```

Furthermore, as a measure of effect size we can examine the AUC for each seperation.

```{r}
getData(cl, "AUC")
```

### Score permuation

```{r permutePcp, message=FALSE, fig.width=10, fig.height=7}
#permute matrix
iterations <- 100 ## Set the number of iterations.

## Permute and plot.
pe <- permute(
    mat = pcpMatrix, 
    iter = iterations, 
    classes = classes, 
    projmethod = "pcp"
)

plot(pe)
```

```{r pValuePcp, echo=TRUE, eval=TRUE, message=FALSE}
pvalue(pe)
```

The permutation plot shows a histogram of the max separation scores from
all permutations with the vertical red line representing the max score
from the real data. The p-values can be obtained using `pvalue(pe)`. 
The results from the permutation indicate that 
ClusterSignificance performed as we would expect. The separation for 
class1 vs class3 is not significant at alpha = 0.05, whereas the separations 
for both class1 vs class2 and class2 vs class3 are significant.

We can also calculate a confidence interval for the p-value estimates using the 
following command:

```{r, confIntPcp, echo=TRUE, eval=TRUE, message=FALSE}
conf.int(pe, conf.level = .99)
```

If it is unclear why these are p-value estimates or why we would want 
a confidence interval for them, please refer to 
<a href="#common-questions">Common questions</a>.

## Mlp method
### Projection

The Mlp method is suitable for comparing clusters in a maximum of 2 dimensions
when the separation between a maximum of 2 classes will be evaluated at
a time. Briefly, Mlp functions by drawing a regression line through the
means of the two classes and then projects all points onto that line. To
project the points into one dimension, the points are then rotated onto
the x-axis. A detailed description and graphical representation of the
individual steps can be seen in the example below. It should be noted that,
despite the fact that the Mlp method will work with as low as 2 data points
per group, it may not be advisable to perform an analysis with such a small
amount of data.

We use the demonstration data included with the package to demonstrate 
the Mlp function.

```{r, fig.align='center', fig.width=10, fig.height=8}
## Create the group variable.
classes <- rownames(mlpMatrix)

## Run Mlp and plot.
prj <- mlp(mlpMatrix, classes)
plot(prj)
```

The plot above shows the 6 steps Mlp takes to project the points into one
dimension.

1. The input matrix.
2. Finding the mean of each class.
3. Regression line and move to origo.
    a. The regression line is drawn through the mean of each class.
    b. Adjustment of the line and points so the line passes through origo.
        i. Arrows show the move from the original position to new position.
4. Orthogonal projection onto the regression line.
5. Points shown after projection.
6. The final rotation of points into one dimension.

### Spereation classification

```{r classifyMlp, message=FALSE, fig.align='center', fig.width=8, fig.height=6}
## Classify and plot.
cl <- classify(prj)
plot(cl)
```

The classification function works in the same manner for both the Mlp and
Pcp methods and is previously described in the [Classification][] section.

### Score permuation

```{r permuteMlp, message=FALSE, fig.align='center', message = FALSE}
## Set the seed and number of iterations.
iterations <- 500 

## Permute and plot.
pe <- permute(
    mat = mlpMatrix, 
    iter = iterations, 
    classes = classes, 
    projmethod = "mlp"
)

plot(pe)
```

```{r pValueMlp, echo=FALSE, eval=TRUE, message=FALSE}
## p-value
pvalue(pe)
conf.int(pe)
```

From the permutation plot can see that when comparing the seperation
of class1 vs. class2 the permuted data, using 100 iterations, never scores higher
than the real data. Thus, the separation thus has a p-value < 0.01. 

# Principal Component Analysis

One probable application for ClusterSignificance would be the analysis
of cluster separation post-PCA. Here we use the leukemia dataset from the
plsgenomics package to demonstrate how this may be done. The dataset includes
"gene expression data (3051 genes and 38 leukemia samples) from the leukemia
microarray study of Golub et al. (1999)". The tumor samples are from both
acute lymphoblastic leukemia (ALL) and acute myeloid leukemia (AML) patients.
Due to the inherent differences in the transcriptome of these two cancer
types, we hypothesize that they would form seperate clusters post-PCA and
that we can use ClusterSignificance to examine if this separation is 
significant.


```{r echo=TRUE, eval=TRUE, message=FALSE}
## The leukemia dataset is provided as a list with 3 elements. 
## An explaination of these can be found using the command:
## help(leukemia)

library(plsgenomics, quietly = TRUE)
data(leukemia)

## Extract the expression matrix.
expression <- leukemia$X 

## Run PCA and extract the principal components.
pca <- prcomp(expression, scale = TRUE)
mat <- pca$x

## Extract the grouping variable (coded as 1 (ALL) and 2 (AML)).
classes <- ifelse(leukemia$Y == 1, "ALL", "AML")
```

```{r, echo = FALSE}
gTable <- as.data.frame(table(classes))
knitr::kable(gTable, 
    digits = 2, 
    align = c(rep('l', 2)), 
    caption = "Classes argument table."
)
```

```{r ALLprojection, fig.align='center', fig.width=10, fig.height=10}
ob <- pcp(mat, classes, normalize = TRUE)
plot(ob)
```

```{r ALLclassification, fig.align='center', fig.width=8, fig.height=6}
cl <- classify(ob)
plot(cl)
```

```{r ALLpermutation, fig.align='center', message = FALSE}
iterations <- 100
pe <- permute(mat = mat, iter = iterations, classes = classes, projmethod = "pcp")
pvalue(pe)
conf.int(pe)
```

As we can see from the results, at alpha 0.05, we would reject the null 
hypothesis concluding that there is a seperation beteen the AML and ALL class
clusters. Despite this, by examining the confidence interval, we see that 
the upper limit is greater than 0.05 which should give us pause before 
rejecting the null at alpha = 0.05. Although not run here, 10000 iterations 
results in p = < 1e-04 with the confidence interval of CI99%-low = 0.00000 and
CI99%-high = 0.00052 giving us greater confidence to reject the null.

# Common questions
**_Do I have to use dimensionality reduced 
data as input to ClusterSignificance._**

Well, technically no, although ClusterSignificance was designed to 
function downstream of dimensionality reduction algorithms and we 
see this being the most frequent use-case for ClusterSignificance. 
One of the resons for this is, it is often the case that datasets 
containing a large number of features, such as those originating from
high throughput experiments, will often contain features that cannot 
discriminate between the class seperations of interest. This is often 
accompanied by "background noise" arising during the sample preperation. 
Dimensionality reduction thus serves here as a feature selection method, 
identifying sources of large variation within the data. Despite this, there 
are some potential pitfalls with dimensionality reduction which are 
highlighted in the next question. 

**_What does the assumption "The obtained principal curve accurately 
depicts the data with respect to the class separations under consideration" 
mean?_**

We believe the principal curve method to be a robust method that will perform 
well in the majority of situations we predict ClusterSignificance to be used 
in. Despite this, the principal curve method is an unsupervised method and 
therefore, it is not necessarily the case that it will be drawn in an optimal
way to access the class seperations of interest. We currently have some 
functionality allowing the user to regulate the smoothness of the principal 
curve. This can be accomplished using the `df` argument to the `pcp` function
**BUT** you must supply the same argument to the `permute` function or your 
results will not be valid. We recommend altering the `df` argument as a last
resort and to only be used under situations where no other alternative is 
available.

**_What does the assumption "The dimensionality reduction method was 
sufficiently capable of detecting the dissimilarities, characterized
by features in high-dimensional space, that correspond to the class 
separations under investigation" mean?_**

Dimensionality reduction can be said to detect dissimilarities, characterized
by features in high-dimensional space although, there is no garantee that the 
dissimilarities it detects will be those discriminating the class seperations 
you are interested in. In addition, linear dimensionality reduction methods, 
such as PCA, may not provide a useful representation of the data when the 
discriminatory features between classes under investigation vary in a 
non-linear fashion.

**_What does the assumption "The joint distribution remains invariant 
under all rearrangements of the subscripts." mean?_**

This is a standard assumption for permutation tests which means that 
the data must exhibit "exchangeability". A basic description of 
exchangeability can be found 
[here](https://en.wikipedia.org/wiki/Exchangeable_random_variables). 
_Extensions Of The Concept Of Exchangeability And Their Applications_
Good (2012) provides a detailed discussion on this topic.

**_Does ClusterSignificance actually perform a permutaiton test?_**

Despite the name of one of ClusterSignificance's stages being _permutation_,
strictly speaking, ClusterSignificance is not designed to perform
a permutation test. Due to the fact that a real permutation test tests all
possible combinations without replacment, which is often infeasible, most 
users will actually be performing a monte carlo test. Instead of permuting 
all possible combinations to form the null distribution (or permutation 
distribution), a monte carlo test samples the permutation distribution 
and uses this sample as the null distribution. For this reason, a p-value 
from a monte carlo test is actually a esitmate of the true p-value. For 
this reason we provide the `conf.int` function that will calculate the 
confidence interval for the p-value estimate. It is theoretically, 
possible to perform a permutation test with ClusterSignificance by 
providing your own permutation matrix instead of letting 
ClusterSignificance calculate one for you which we discuss in the 
<a href="#advanced-usage">Advanced usage</a> section.

**_How many permutations should I run?_**

The complete answer to this question is too in-depth to answer here. 
We would recommend 10^4 permutations as a starting point. Ideally, the 
permutation test returns a exact p-value (i.e. not p-value < X), which 
means that at least one permuted max score was greater than the max score
for the real data, although this is not always possible. Furthermore, it 
is unncessary to run more permutations than possible class combinations,
which provides one concrete way to calculate the number of ideal 
permutations. Despite this, it will often challange your computers
memory to calculate this number if many combinations are possible.
Under such circumstances, it is common that the permutation 
distribution is randomly sampled and the more times this 
sampling occurs, the nearer your estimate of the p-value will
come to the true p-value (see the question above). A confidence interval
can be calculated for the reported p-value using the `conf.int` function.

**_Why does the prncipal curve change shape between steps 3 and 4
of the projection phase?_**

The line is actually the same shape, it is the scale of the X-axis 
that changes. This is necessary due to the fact that as the points 
move to one dimension more space is needed on the x-axis to show 
all the points.

**_Why are the points projected diagonally onto the line in step 4
of the Mlp projection?_**

Step 4 shows the orthogonal projection onto the regression line
and appears visually to not be perpendicular to the line. This is
due to the automatic scaling of the y-axis which is necessary 
when large differences in proportions between the x and y dimensions
would otherwise cause data to disappear outside of the margins.


# Advanced usage
## Concatenate permutation results
When running the permutation step of the Pcp method, it is possible that
all iterations are not successfully completed, as can be seen with the message
upon permutation completion. Due to the fact that, the p-value is calculated
not by the number of iterations input by the user but, instead, with the
number of iterations that successfully complete, this will not be an issue
for most users. However, if it is the case that a specific number of iterations
is required by the user it is possible to concatenate multiple permutation
runs.  

```{r concatPermuts, message=FALSE, eval=TRUE}
mat <- mlpMatrix
classes <- rownames(mlpMatrix)
iterations <- 10

pe1 <- permute(mat=mat, iter=iterations, classes=classes, projmethod="pcp")
pe2 <- permute(mat=mat, iter=iterations, classes=classes, projmethod="pcp")
pe3 <- permute(mat=mat, iter=iterations, classes=classes, projmethod="pcp")
pvalue(pe1)

pe <- c(pe1, pe2, pe3)
pvalue(pe)
```

## User defined permutation matrix
In some cases, e.g. PCA, it may be desirable to define a permutation matrix
rather than allowing ClusterSignificance to do this automatically. For
example, when determining the separation of clusters post-PCA, a more
valid approach may be to perform PCA on a sampled matrix of the original
values, after which this is provided to ClusterSignificance, rather than
allowing ClusterSignificance to randomly sample the principal components.
This can be accomplished by providing the *user.permutations* argument
to the permute method. 

```{r userDefinedPermats, eval=TRUE}
##define permMatrix function
.pca <- function(y, classes, uniq.groups, mat) {
    pca <- prcomp(
        sapply(1:ncol(mat[classes %in% uniq.groups[, y], ]), 
            function(i)
                mat[classes %in% uniq.groups[, y], i] <- 
                    sample(
                        mat[classes %in% uniq.groups[, y], i], 
                        replace=FALSE
                    )
        ), scale=TRUE
    )
    return(pca$x)
}

permMatrix <- function(iterations, classes, mat) {
    uniq.groups <- combn(unique(classes), 2)
    permats <- lapply(1:ncol(uniq.groups), 
        function(y)
            lapply(1:iterations, 
                function(x)
                    .pca(y, classes, uniq.groups, mat)
            )
    )
    return(permats)
}

set.seed(3)
mat <- pcpMatrix
classes <- rownames(pcpMatrix)
iterations = 100
permats <- permMatrix(iterations, classes, mat)
```

The permutation matrix is a list (we can call this list 1) with one 
element per comparison. Each of the elements in list 1 holds another 
list (we can call this list 2). The number of elements in list 2 is 
equal to the number of iterations and, thus, each element in list 2 
is a permuted matrix. We can then provide this permutation matrix to 
ClusterSignificance in the following manner:

```{r runUserDefinedMatrix, eval=FALSE}
pe <- permute(
    mat = mat, 
    iter=100, 
    classes=classes, 
    projmethod="pcp",
    userPermats=permats
)
```

# Conclusion

The ClusterSignificance package can estimate a p-value for a separation 
projected on a line. Typically, the package might be used post-PCA-analysis, 
but clusters coming from other sources can also be tested.

# Session info

```{r echo=TRUE, eval=TRUE}
sessionInfo()
```