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Some of the images in this writeup are part of Ana Swanson's Wonk Blot post How a dog sees a rainbow, and 12 other images that explain how we see color at the Washington Post.

Download CC library swatches

by Jessica Brassard, 3 Feb 2019

In an audience of 8 men and 8 women, chances are 50% that at least one has some degree of color blindness^{1,2}. When encoding information or designing content, use colors that is color-blind safe.

^{1}About 8% of males and 0.5% of females are affected with some kind of color blindness in populations of European descent (wikipedia, Worldwide prevalence of red-green color deficiency, JOSAA). The rate for other races is lower Asians and Africans is lower (Caucasian Boys Show Highest Prevalence of Color Blindness Among Preschoolers, AAO).

^{2}The probability that among `N=8` men and `N=8` women at least one person is affected by color blindness is `P(men,women) = P(8,8) = 1 - (1-0.08)^8 * (1-0.005)^8 = 0.51`. For `N=34` (i.e., 68 people in total), this probability is `P(34,34)=0.95`. Because the rate of color blindness in women is so low, for most groups of mixed gender we can approximate the probability by only counting the men. For example, in a group of 17 women the probability that at least one of them is color blind is `P(0,17) = 0.082`, which is the same probability as for 1 man, `P(1,0)`.

Color Oracle is a good and free color blindness simulator for Windows, Mac and Linux.

You can download the RGB transformation table for deuteranopia, protanopia and tritanopia. It is available for all (r,g,b) colors in steps of 5 in each of the channels. The mapping for all other RGB colors can be interpolated.

Transformation for *all* 16.8 million RGB colors (interpolated from the table above) are also available independently for each type of color blindness:
deuteranopia,
protanopia,
and
tritanopia.

The normal human eye is a 3-channel color detector^{3}. There are three types of photoreceptors, each sensitive to a different part of the spectrum. Their combined response to a given wavelength produces a unique response that is the basis of the perception of color.

^{3}Compared to hearing, the color vision is a primitive detector. While we can hear thousands of distinct frequencies and process them simultaneously, we have only three independent color inputs. While the ear can distinguish pure tones from complex sounds that have multiple frequencies the eye is relatively unsophisticated in separating a color sensation into its three constituent primary stimuli.

People with color blindness have one of the photo receptor groups either reduced in number or entirely missing. With only two groups of photoreceptors, the perception of hue is drastically altered.

For example, in *deuteranopia*, the most common type of color blindness, the medium (M) wavelength photoreceptors are reduced in number or missing. This results in the loss of perceived difference between reds and greens because only one group of photoreceptors (L) are sensitive to the wavelengths of these colors. The spectrum appears to be split into two hues along the blue-green boundary (see figure below).

Visible light is in the range of 390-700 nm. The exact definition of the upper limit varies, with some sources giving as high as 760 nm. Shorter wavelengths are absorbed by the cornea (<295nm) and lens (315-390nm). Some near infrared light also reaches the retina (760-1400nm).

The opposite condition to color blindness exists too—tetrachromacy. In this case, an individual has an extra type of color receptor which improves discrimination in the red part of the spectrum. While the anatomy of their retina can be described, how true tetrachromats subjectively perceive color is unknown. And, perhaps, even unknowable.

Tetrachromacy is common in other animals, such as fish (e.g. goldfish, zebrafish) and birds (e.g. finch, starling). The dimensionality of the perceived color space isn't necessarily proportional to the number of different receptors. If the signal from 3 color receptors are combined by the brain and each processor has a weighted response to a broad range of wavelengths, then a color can be modeled by a point in 3-dimensional space, in which the receptors are the axes. This system can perceive a large number of colors.

In the extreme case where the receptors respond to a very narrow range, of which none overlap with the other, a color is one of three points in a 1-dimensional space. This sytem can perceive only 3 colors.

For example, although the mantis shrimp has 12 different color receptors, the receptors work independently, their color discrimination is poorer than ours.

If you use Color Oracle to transform your screen colors to simulate color blindness, you can see that none of the equivalent swatches in one kind of color blindness are equivalent in another. This is particularly interesting when applied to a duotone image which is drawn using equivalent colors. In the figure below^{4}, a row of Mr. Spocks disappears (or is difficult to see) to people with color blindness.

^{4}In tribute to Leonard Nimoy, 1931–2015

To people with color blindness, some colors appear the same. This equivalence can be used to identify distinct colors which are unique to those with normal and color blind vision.

The seven colors (and black) in the figure below are perceived as reasonably distinct by both normal and color blind individuals. The table on the left is reproduced from Nature Method's Points of View: Color blindness by Bang Wong.

For more tips about designing with color blindness in mind, see Color Universal Design (CUD) — How to make figures and presentations that are friendly to Colorblind people.

The figure below shows the mapping of different colors to six different grades of each of the two hues seen by deuteranopes. It offers more distinct options than the 7-color palette above.

Even more color choices for color blindess, including colors that map onto greys.

If you're looking to encode quantitative information, I suggest using the subset of Brewer palettes that are safe for color blindess (e.g. pink-yellow-green, brown-blue-green).

*You can look back there to explain things,
but the explanation disappears.
You'll never find it there.
Things are not explained by the past.
They're explained by what happens now.
—Alan Watts*

A Markov chain is a probabilistic model that is used to model how a system changes over time as a series of transitions between states. Each transition is assigned a probability that defines the chance of the system changing from one state to another.

Together with the states, these transitions probabilities define a stochastic model with the Markov property: transition probabilities only depend on the current state—the future is independent of the past if the present is known.

Once the transition probabilities are defined in matrix form, it is easy to predict the distribution of future states of the system. We cover concepts of aperiodicity, irreducibility, limiting and stationary distributions and absorption.

This column is the first part of a series and pairs particularly well with Alan Watts and Blond:ish.

Grewal, J., Krzywinski, M. & Altman, N. (2019) Points of significance: Markov Chains. *Nature Methods* **16**:663–664.

*Places to go and nobody to see.*

Exquisitely detailed maps of places on the Moon, comets and asteroids in the Solar System and stars, deep-sky objects and exoplanets in the northern and southern sky. All maps are zoomable.

Quantile regression explores the effect of one or more predictors on quantiles of the response. It can answer questions such as "What is the weight of 90% of individuals of a given height?"

Unlike in traditional mean regression methods, no assumptions about the distribution of the response are required, which makes it practical, robust and amenable to skewed distributions.

Quantile regression is also very useful when extremes are interesting or when the response variance varies with the predictors.

Das, K., Krzywinski, M. & Altman, N. (2019) Points of significance: Quantile regression. *Nature Methods* **16**:451–452.

Altman, N. & Krzywinski, M. (2015) Points of significance: Simple linear regression. *Nature Methods* **12**:999–1000.

Outliers can degrade the fit of linear regression models when the estimation is performed using the ordinary least squares. The impact of outliers can be mitigated with methods that provide robust inference and greater reliability in the presence of anomalous values.

We discuss MM-estimation and show how it can be used to keep your fitting sane and reliable.

Greco, L., Luta, G., Krzywinski, M. & Altman, N. (2019) Points of significance: Analyzing outliers: Robust methods to the rescue. *Nature Methods* **16**:275–276.

Altman, N. & Krzywinski, M. (2016) Points of significance: Analyzing outliers: Influential or nuisance. Nature Methods 13:281–282.