# Interpretation of the Pearson's colocalization parameter

## Definition

### Pearson's coefficient

We follow the usual naming convention in CoLocalization for the two compared data channels: R for the first channel, G for the second channel. They are also know as red and green channels, independently of the WaveLength they have actually registered. The pixel values in the channels are Ri and Gi respectively, with i the pixel index. This holds for datasets of any dimension, the index i running from zero to N - 1 , N the total number of image elements.

The Pearson's coefficient rp is a single number, defined as

with Ravg and Gavg the averages of all the Ri and Gi values respectively. The summations are over all the VoXels in the image. The coefficient describes how well the red and green channels are related by a linear equation (G = a R + b), but without saying anything about which is that equation. It will be equal to 1 if, for instance, all the red voxels are exactly double in intensity than the green ones, but also if the ratio is exactly 1.33, 0.7, or any other factor (plus an optional additive constant b). The closer rp is to +/- 1.0, the more linearly related all the voxels intensities in the two channels are. The real linear relationship doesn't affect the coefficient. Therefore, if the green channel detector measures with less intensity output than the red channel's, you can still calculate correlations.

The coefficient is the ratio between the CoVariance of the channels and the product of their Standard Deviations.

(The covariance is defined as that sum in the numerator divided by the total number of pixels N, but another 1/N appears in the denominator when
multiplying the standard deviations, and they cancel each other, that's why there's no N appearing in the above equation).

### Object Pearson's coefficient

This new Pearson's coefficient is available from Huygens version 3.6 and higher. It is the same as the original Pearson's coefficient, with the difference that the averages are not calculated over all the image voxels. Only the voxels which are not background (i.e. part of an object, hence the name Object Pearson's coefficient) are taken into account for the average. In this way the Object Pearson coefficient is not biased anymore by large background areas.

### Spearman coefficient

Since version 3.7 the Spearman coefficient is added to the colocalization coefficients. The Spearman coefficient is the Pearson coefficient, but based on intensity ranks, instead of intensity values. Intensity rank is the position of an intensity value, if all intensity values of the image were ordered. If there are ties in this ordering the intensity values that are tied get their averaged intensity rank. Then, the Pearson coefficient is calculated over the intensity ranks.

The advantage of calculating the coefficients based on ranks, is that it measures any monotonic dependencies between two color channels, instead of only linear dependencies which the Pearson coefficient can measure.

The Spearman has the same properties as the Pearson.

## Interpretation

The higher this figure, the more co-dependent both channels are. When the signal in one channel varies from place to place, the other one varies accordingly: they go together.

You can interpret the statistical figure like this: if e.g. rp = 0.80, then its squared value (0.80² = 0.64 = 64 %) is the percentage of the variance of the green channel that can be explained by the changes of the red channel, and vice versa. The rest (100 - 64 = 36 %) of the variances is "independent" from the other channel's.

As many other statistical figures, it makes more sense when comparing different results. An image (or a region in it) with r = 0.89 is more
correlated than another with r = 0.78, for example.

### Negative values

The coefficient ranges from -1 to 1. A value of 1 shows that a linear equation describes the relationship perfectly and positively, with all data points lying on the same line and with Y increasing with X. A score of -1 shows that all data points lie on a single line but that Y increases as X decreases. A value of 0 shows that a linear model is inappropriate, that there is no linear relationship between the variables. (Wikipedia).

A negative value closer to -1 will appear to have therefore difficult interpretation in the context of CoLocalization. Nevertheless, r < 0 basically means that the two labeled targets repel one another. The more R is present, the less G appears there. That is interesting, to study the repulsion of two labeled targets, it could be ContraLocalization...

## Local values

The coefficient rp parametrizes the full image, while a Pearson's Colocalization Map parametrizes the colocalization per VoXel, so there is no summation all over the image. A single value is calculated per voxel creating a 3D map, that can be represented in a 3D image for example in the form of an iso-colocalization surface. In the Colocalization Analyzer all the (global image) colocalization coefficients are always returned, but only one map is calculated and displayed, depending on which was the user's selection.

## Background dependence

As the deviations are measured respect to the average value of each channel, the existence of a data BackGround doesn't affect this parameter. (The background would be reflected in the additive constant b of the linear equation relating both channels and, as we mentioned, the actual equation does not affect the coefficient).

However, to handle user specified background values in a consistent manner across all colocalization coefficients computed by the Colocalization Analyzer, these background values are taken into account:

If you specify background values (one per channel) in the ColocalizationAnalyzer, those values will be subtracted from all the pixel intensities when calculating colocalization. Pixel values below the background values will be set to zero instead of becoming negative. Because of this substitution with a larger number (zero) setting a user specified background value does change the resulting Pearson's coefficient.

If this is unwanted, both background values should be set to zero. Background removal should be done prior to CoLocalization analysis. It is part of the Image Restoration. See BackGround.

## Intensity dependance

As explained above, this coefficient is not affected by channels sensitivities (relative channel strength).

Spurious intensity variations inside channels will of course affect the calculations...

See Colocalization Theory.

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