Metrics

Background on the colocalization metrics the plugin computes (Pearson, Spearman, Li ICQ, the Manders overlap coefficient and tM1/tM2), the Costes auto-threshold used for Manders, and the single-pair diagnostics.

Documentation index: Home · Usage · Metrics · Python API

For a deeper treatment, see the ImageJ colocalization analysis page that this plugin took its design cues from.

Co-occurrence versus correlation

"Colocalization" is an umbrella term that covers two distinct phenomena (Aaron, Taylor & Chew, 2018):

• Correlation — do the intensities of the two channels vary together? Answered by Pearson (PCC), Spearman (SRCC) and Li ICQ.
• Co-occurrence — what fraction of one channel's signal spatially overlaps the other's, regardless of how the intensities relate? Answered by the Manders overlap coefficient (r, k1, k2) and the thresholded Manders coefficients (tM1/tM2).

The two are independent: an image can show high co-occurrence with low correlation, or vice versa. Manders' coefficients are sometimes loosely called "correlation coefficients" — they are not; they measure co-occurrence. Which metric is appropriate depends on the biological question, so the plugin reports both families side by side.

Pearson (PCC)

The standard linear correlation coefficient between paired pixel intensities, restricted to the analysed region.

\[ \mathrm{PCC} = \frac{\sum_i (a_i - \bar a)(b_i - \bar b)} {\sqrt{\sum_i (a_i - \bar a)^2 \sum_i (b_i - \bar b)^2}} \]

Range: −1 (perfect anti-correlation) to +1 (perfect correlation), with 0 meaning no linear relationship.

When to use it. As a quick first look at whether two channels co-vary linearly. PCC is unaffected by uniform scaling or offsets, so it tolerates differences in absolute brightness between channels.

Watch out for. - Saturated pixels skew PCC towards 0; clip your acquisition appropriately. - Strong background offsets reduce PCC; consider region-restricting to the foreground or use SRCC. - A single bright outlier pair can dominate the result.

We delegate the computation to skimage.measure.pearson_corr_coeff, which returns the coefficient and a two-tailed p-value (both surfaced in the results table).

Spearman (SRCC)

The Pearson correlation of the ranks of the intensities, rather than the intensities themselves.

When to use it. Whenever the relationship between channels is monotonic but not necessarily linear (e.g. saturating sensor responses, gamma-corrected images). SRCC is also robust to outliers - a single bright pixel can't pull the rank away.

Watch out for. - Like PCC, SRCC compares only paired pixels. If one channel is noisy background everywhere except where the other is bright, SRCC can be misleading. A region mask helps.

We use scipy.stats.spearmanr on the masked, flattened intensities. Both the rank correlation coefficient and the p-value are returned.

The plugin defaults to Spearman only because it gives a sensible result on a wider range of microscopy data than PCC alone.

Li ICQ

The Intensity Correlation Quotient introduced by Li et al. (2004). For each pixel i form the covariance contribution P_i = (a_i - mean(a)) * (b_i - mean(b)): this product is positive when both channels co-vary at that pixel (both above or both below their means) and negative when they anti-vary. ICQ is the fraction of non-zero P_i that are positive, re-centred to lie on [-0.5, 0.5]:

\[ \mathrm{ICQ} = \frac{|\{i : P_i > 0\}|}{N} - 0.5 \]

Range: −0.5 (perfectly segregated staining) → 0 (random) → +0.5 (perfectly dependent staining). Unlike PCC and Spearman, ICQ measures sign agreement with respect to the channel means rather than magnitude correlation, which makes it robust to many of the intensity artefacts that affect PCC (offsets, soft saturation) while still being sensitive to dependent staining.

When to use it. Reported alongside PCC/SRCC as a quick sanity check on the shape of the dependence: an ICQ near 0 with a high PCC is a warning that the PCC is being driven by a few extreme pixel pairs rather than a population-wide co-variation.

Watch out for. ICQ does not have a standard p-value, only the scalar; we report the value alone. Like the other correlation metrics it is restricted to the analysed region (whole image, shape, or label).

We compute it locally in _metrics.li_icq (no scikit-image equivalent).

Overlap coefficient (r, k1, k2)

The overlap coefficient of Manders et al. (1992) and its two split components. Unlike tM1/tM2 below, these need no threshold - they are computed directly from the raw intensity products over the region:

\[ r = \frac{\sum_i a_i b_i}{\sqrt{\sum_i a_i^2 \; \sum_i b_i^2}} \qquad k_1 = \frac{\sum_i a_i b_i}{\sum_i a_i^2} \qquad k_2 = \frac{\sum_i a_i b_i}{\sum_i b_i^2} \]

Range of r: 0 to 1 for non-negative intensities. It is essentially the Pearson coefficient computed without mean-subtraction, which makes it insensitive to a difference in average brightness between the two channels - but also means it cannot report anti-correlation.

k1 and k2 split that co-occurrence per channel: k1 is dominated by pixels where channel A is bright, k2 by where B is bright. Their asymmetry diagnoses which channel drives the overlap.

When to use it. As a brightness-insensitive companion to PCC, particularly when the two channels were acquired at very different gain. The split k1/k2 help when one channel is much sparser than the other.

Watch out for. - r hides the sign of any relationship; always read it next to PCC or ICQ, never alone. - Like tM1/tM2 it assumes non-negative intensities. If background subtraction has pushed either channel below zero, r/k1/k2 are undefined and reported as blank/NaN — use PCC, SRCC or ICQ, which are all well defined on signed data. - Any coefficient whose denominator is zero (an all-zero channel within the region) is reported as a blank/NaN cell. - All three are accumulated in float64 regardless of the image dtype, so 8- and 16-bit images give the same answer as their float equivalents.

Computed locally in _metrics.overlap; the three values populate the overlap, k1 and k2 columns.

Manders (tM1/tM2)

A co-occurrence measure (not a correlation — see above). Two coefficients, tM1 and tM2, that ask: "what fraction of the intensity in one channel is co-located with above-threshold signal in the other channel?"

\[ \mathrm{tM_1} = \frac{\sum_i a_i \cdot \mathbb{1}[b_i > T_b]}{\sum_i a_i} \qquad \mathrm{tM_2} = \frac{\sum_i b_i \cdot \mathbb{1}[a_i > T_a]}{\sum_i b_i} \]

They are written tM1/tM2 (rather than plain M1/M2) to signal that a threshold T is always applied — un-thresholded Manders coefficients over a whole image are trivially close to 1.

Range: 0 to 1 each. They are not percentages - interpret them as "the fraction of channel A's signal that overlaps with channel B's above-threshold pixels", and similarly for tM2. Asymmetry between tM1 and tM2 is meaningful and expected when one channel is sparser than the other.

When to use it. When you care about co-occurrence of signal rather than the correlation of intensities. Two channels can have low PCC but high tM1 and tM2 if their bright pixels reliably overlap but with widely varying intensities.

Watch out for. - Manders is sensitive to the choice of threshold. Use Costes (below) if you don't have a principled way to set it manually. - High background offsets confuse Costes - pre-process or set thresholds manually. - Both assume non-negative intensities. If background subtraction has pushed a channel below zero, the coefficient that sums that channel is reported as blank/NaN (its "fraction of the total" reading breaks down once values can cancel), while PCC, SRCC and ICQ remain valid.

Note that tM1 and tM2 are decided independently, so one can be blank while the other carries a number. Each looks at a different denominator: tM1 divides by the total of channel A and only gates on T_b, and tM2 mirrors that. So a region where B is empty gives tM1 = 0 (a real measurement: none of A co-occurs with B) alongside a blank tM2 (nothing to divide by) - and a channel with no auto-threshold blanks only the coefficient that needed that threshold. A blank cell always means "not measurable", never "measured as zero".

Computed in _metrics.manders directly from the intensity sums.

Costes' auto-threshold

Manders' coefficients depend on a per-channel threshold separating "signal" from "background". Picking the threshold by eye is subjective; the iterative method introduced by Costes et al. (2004) gives a reproducible answer that is widely cited in the colocalization literature.

Algorithm (as implemented in _metrics.costes_threshold, matched to Fiji's Coloc 2 AutoThresholdRegression):

1. Fit an orthogonal (total-least-squares) regression line b = m·a + c over the masked pixels - not an ordinary least-squares fit. Orthogonal regression treats the two channels symmetrically (neither is the independent variable) and avoids the slope bias OLS shows when the predictor channel is noisy.
2. Move a candidate threshold along that line by bisection, keeping the pair on the line (T_b = m·T_a + c). The channel that is stepped is the one giving finer resolution - channel A when |m| < 1, else channel B.
3. At each candidate, compute the Pearson correlation of the below-threshold pixels (a < T_a or b < T_b). Bisect downward while that correlation is zero or positive, and upward when it is negative or undefined, converging on the threshold where the background pixels stop correlating.

When no threshold can be fitted — fewer than two pixels, a constant channel, or a regression slope that is non-positive or undefined (no linear co-occurrence to threshold for) — the thresholds and the resulting tM1/tM2 are reported as blank/NaN, and the note below the table says so. Fiji's Coloc 2 instead falls back to (max(a), max(b)) here; that yields tM1 = tM2 = exactly 0.00, which is easy to misread as a measured "nothing co-occurs" rather than "nothing could be measured", so this plugin deliberately diverges.

The selected thresholds appear in the threshold_a / threshold_b columns of the results table and as red reference lines on the scatter plot. The orthogonal regression line is overlaid as a cyan dashed line (via _metrics.costes_regression) so you can see the axis the auto-threshold was found along.

The Costes randomisation significance test is available on the Diagnostics tab (see below).

Other auto-thresholds (Otsu, Li, …)

Besides Costes, the threshold method can be any of the standard histogram thresholds - Otsu, Li, Triangle, Yen, Mean, IsoData (the skimage.filters.threshold_* family, as ImageJ's Auto Threshold offers in JACoP B). Each channel is thresholded independently from its own intensity histogram, giving the thresholded Manders coefficients (tM1/tM2). These are a good choice when the two channels have clearly bimodal (signal vs background) histograms; Costes is preferable when the relationship between the channels is the thing you want the threshold to respect. A channel with no contrast (constant within the region) has no defined auto-threshold, so its tM1/tM2 are reported as blank/NaN.

Diagnostics

The metrics above each collapse colocalization to a number per region. The Diagnostics tab instead produces a plot for a single channel pair (optionally restricted to a region), to inspect the shape of the relationship. Each is computed in the napari-free _diagnostics module.

Costes randomization (significance test)

Answers "could this PCC have arisen by chance?". Channel B is scrambled in blocks n_iter times - destroying spatial co-occurrence while preserving each channel's intensity histogram and local texture - and the PCC is recomputed each time to build a null distribution. The observed PCC is plotted against that null, with a right-tailed p-value (#{null ≥ observed} + 1) / (n_iter + 1) and a z-score. A genuine colocalization sits far to the right of the null. The block size should approximate the point-spread-function width. Works on 2D and 3D images.

Van Steensel cross-correlation function (CCF)

Shifts channel B relative to A by ±max_shift pixels and plots the PCC at each shift. A peak at shift 0 indicates colocalization; a central trough with side peaks indicates mutual exclusion (the channels are anti-located by roughly the shift of the side peaks). The summary reports the peak Pearson r and the shift it occurs at.

Li ICA

The scatter behind the Li ICQ scalar: for each channel, the per-pixel covariance product (A−Ā)(B−B̄) is plotted against that channel's intensity. Dependent staining produces a characteristic rightward (positive-product) skew that grows with intensity; random or segregated staining stays centred on zero. The ICQ value is shown alongside.

Choosing a metric

Question you're asking	Use
"Are the two channels linearly correlated?"	Pearson
"Is the relationship monotonic but not necessarily linear?"	Spearman
"Do the channels co-vary in the same direction relative to their means, ignoring magnitude?"	Li ICQ
"How much do the signals overlap, ignoring brightness differences and without a threshold?"	Overlap r (k1/k2)
"What fraction of A's signal sits where B is bright?"	Manders tM1
"...and vice versa?"	Manders tM2
"Just give me a robust default"	Spearman

PCC, SRCC and ICQ measure correlation; the overlap coefficient (r, k1, k2) and Manders (tM1/tM2) measure co-occurrence. They answer different questions and disagree often - which is precisely why reporting more than one is good practice.

References

• Aaron, J.S., Taylor, A.B. & Chew, T.-L. (2018). Image co-localization – co-occurrence versus correlation. J. Cell Sci. 131(3), jcs211847.
• Bolte, S. & Cordelières, F.P. (2006). A guided tour into subcellular colocalization analysis in light microscopy. J. Microsc. 224(3), 213-232.
• Costes, S.V. et al. (2004). Automatic and quantitative measurement of protein-protein colocalization in live cells. Biophys. J. 86(6), 3993-4003.
• Dunn, K.W., Kamocka, M.M., McDonald, J.H. (2011). A practical guide to evaluating colocalization in biological microscopy. Am. J. Physiol. Cell Physiol. 300(4), C723-C742.
• Li, Q. et al. (2004). A Syntaxin 1, Galpha(o), and N-type Calcium Channel Complex at a Presynaptic Nerve Terminal: Analysis by Quantitative Immunocolocalization. J. Neurosci. 24(16), 4070-4081.
• ImageJ colocalization analysis
• ImageJ Coloc 2 plugin