# plotChiSquaredFit

Plot goodnesss-of-fit for variance regression

## Description

`plotChiSquaredFit(`

plots
the empirical CDF of the chi-squared probabilities of the ratio
between the observed and the estimated variance stratified by count
levels into five equal-sized bins. Use this plot to assess the goodness-of-fit.`test`

)

`test`

, an output of the `nbintest`

function, is a `NegativeBinomialTest`

object.
It contains results from an unpaired hypothesis test for two independent
samples.

**Note**

If the `'VarianceLink'`

name-value pair argument
was set to `'Identity'`

when you ran `nbintest`

,
then the chi-squared probability is computed using the ratio between
the observed variance to the mean.

`plotChiSquaredFit(test,`

uses
a name-value pair argument.`Name,Value`

)

returns
handles to axes.`H`

= plotChiSquaredFit(___)

## Examples

### Perform unpaired hypothesis test for short-read count data

This example shows how to perform an unpaired hypothesis test for synthetic short-read count data from two different biological conditions.

The data in this example contains synthetic gene count data for 5000 genes, representing two different biological conditions, such as diseased and normal cells. For each condition, there are five samples. Only 10% of the genes (500 genes) are differentially expressed. Specifically, half of them (250 genes) are exactly 3-fold overexpressed. The other 250 genes are 3-fold underexpressed. The rest of the gene expression data is generated from the same negative binomial distribution for both conditions. Each sample also has a different size factor (that is, the coverage or sampling depth).

Load the data.

load('nbintest_data.mat','K','H0');

The variable `K`

contains gene count data. The rows represent genes, and the columns represent samples. In this case, the first five columns represent samples from the first condition. The other five columns represent samples from the second condition. Display the first few rows of `K`

.

K(1:5,:)

`ans = `*5×10*
13683 14140 8281 14309 12208 8045 9446 11317 14597 14592
16028 16805 9813 16486 14076 9901 10927 13348 16999 17036
814 862 492 910 758 521 573 753 870 936
15870 16453 9857 16454 14267 9671 10997 13624 17151 17205
9422 9393 5734 9598 8174 5381 6315 7752 9869 9795

In this example, the null hypothesis is true when the gene is not differentially expressed. The variable `H0`

contains boolean indicators that indicate for which genes the null hypothesis is true (marked as 1). In other words, H0 contains known labels that you will use later to compare with predicted results.

sum(H0)

ans = 4500

Out of 5000 genes, 4500 are not differentially expressed in this synthetic data.

Run an unpaired hypothesis test for samples from two conditions using `nbintest`

. The assumption is that the data came from a negative binomial distribution, where the variance is linked to the mean via a locally-regressed smooth function of the mean as described in [1] by setting `'VarianceLink'`

to `'LocalRegression'`

.

tLocal = nbintest(K(:,1:5),K(:,6:10),'VarianceLink','LocalRegression');

Use `plotVarianceLink`

to plot a scatter plot for each experimental condition (for X and Y conditions), with the sample variance on the common scale versus the estimate of the condition-dependent mean. Use a linear scale for both axes. Include curves for all other linkage options by setting `'Compare'`

to `true`

.

plotVarianceLink(tLocal,'Scale','linear','Compare',true)

The `Identity`

line represents the Poisson model, where the variance is identical to the mean as described in [3]. Observe that the data seems to be overdispersed (that is, most points are above the `Identity`

line). The `Constant`

line represents the negative binomial model, where the variance is the sum of the shot noise term (mean) and a constant multiplied by the squared mean as described in [2]. The `Local Regression`

and `Constant`

linkage options appear to fit better with the overdispersed data.

Use `plotChiSquaredFit`

to assess the goodness-of-fit for variance regression. It plots the empirical CDF (ecdf) of the chi-squared probabilities. The probabilities are the ratio between the observed and the estimated variance stratified by short-read count levels into five equal-sized bins.

plotChiSquaredFit(tLocal)

Each figure shows five ecdf curves. Each curve represents one of the five short-read count levels. For instance, the blue line represents the ecdf curve for a low short-read counts between 0 and 1264. The red line represents high counts (more than 11438).

One way to interpret the curves is to check if the ecdf curves are above the diagonal line. If they are above the line, then the variance is overestimated. If they are below the line, then the variance is underestimated. In both figures, the variance seems to be correctly estimated for higher counts (that is, the red line follows the diagonal line), but slightly overestimated for lower count levels.

To assess the performance of the hypothesis test, construct a confusion matrix using the known labels and the predicted p-values.

confusionmat(H0,(tLocal.pValue > .001))

`ans = `*2×2*
493 7
5 4495

Out of 500 differentially expressed genes, 493 are correctly predicted (true positives) and 7 of them are incorrectly predicted as not-differentially expressed genes (false negatives). Out of 4500 genes that are not differentially expressed, 4495 are correctly predicted (true negatives) and 5 of them are incorrectly predicted as differentially expressed genes (false positives).

For a comparison, run the hypothesis test again assuming that counts are modeled by the Poisson distribution, where the variance is identical to the mean.

tPoisson = nbintest(K(:,1:5),K(:,6:10),'VarianceLink','Identity');

Plot the ecdf curves. Observe that all the curves are below the diagonal line, implying that the variance is underestimated. Therefore, the negative binomial model fits the data better.

plotChiSquaredFit(tPoisson)

## Input Arguments

`test`

— Unpaired hypothesis test result

`NegativeBinomialTest`

object (default)

Unpaired hypothesis test results, specified as a `NegativeBinomialTest`

object.
test is returned by the `nbintest`

function.

### Name-Value Arguments

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

**Example:**

`'NumBins',4`

`NumBins`

— Number of equal-sized bins

`5`

(default) | positive integer

Number of equal-sized bins, specified as a comma-separated pair
consisting of `'NumBins'`

and a positive integer.

**Example: **`'NumBins',3`

## Output Arguments

`H`

— Handles to axes

vector of handles

Handles to axes, specified as a vector of handles.

## See Also

`nbintest`

| `NegativeBinomialTest`

| `plotVarianceLink`

| `mattest`

**Introduced in R2014b**

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