The Malmquist Bias
Published on May 22, 2006 at 6:29 pm.
2 Comments.
Filed under astronomy, cosmology.
What is the Malmquist Bias? It sounds like the title of some sort of action novel, or a spy thriller, or some such. Well, no, it is simply a term from astronomy. It is a special case of what in science we call a selection effect in data. The Malmquist bias is named for the Swedish astronomer Gunner Malmquist (February 21, 1893, to June 27, 1982) who first described this effect in data.
Let me first explain a selection effect for the non-scientists reading this. Science is supposed to be objective, and experiments are supposed to control for as many variables as possible. However, sometimes the way that you make measurements can affect the data. Lets give some examples of this effect outside of the natural sciences. For example, if you do a written survey to assess literacy, you find that just about everyone can read. Well, duh. Only the people who can read actually could fill out the survey! If you want to determine how Americans feel about gun control, you’d likely find a very different result if you surveyed attendees of a national Republican convention than you would if you surveyed attendees of a national Democratic convention. If you wanted to find whether or not college graduates believe that they got a good value for their education, you would want to survey people who graduated from college. How would you go about finding a list of such people? You could find a list of people who defaulted on student loans and survey them. You could also find a list of contributers to an alumni association and survey them. Both sets of people would be list of college graduates. But, you would likely get different results from your surveys. These are examples of where the experimental sample was in some way flawed. All three cases involved samples selected for ease of getting the sample. However, some self selection has already occurred prior to the experimental sampling. The results, therefore, did not answer the original question, but rather answered the question as applied to the subset studied.
The Malmquist bias is a sort of selection effect based upon observations. In astronomy, we generally just have to look at the data that we can collect. We are not able to actively experiment and influence or interact with our system. We just look. This has its own limitations and imposes a particular type of selection effects. We see things best that are more easily seen. For example, it we were biolgists studying a forest and wanted to know what was the most abundant organism in the forest. Well, the first response is TREES! Well, the trees are obvious, of course. The trees sort of define the forest. They are the first thing that you see. You can see the trees from far away. An aerial survey shows the trees. But when you get closer, you see that there are more birds than trees. So, would birds be the dominant lifeform? No, because if you get even closer, you see that insects vastly outnumber the birds. Then, insects are the most abundant thing in the forest, right? No, again. Bacteria outnumber the insects even more than the insects outnumber the birds. However, the bacteria are so much smaller than anything else that it is very difficult to observe them. You certainly can’t see them from satellite or aerial photos!Â
As with the forest example, the Malmquist bias is a selection effect that comes from how easy or how hard it is to see different things. In particular, the Malmquist bias says that low luminosity objects are under-represented in data from more distant samples. That means that if you are a professional astronomer, you need to address this effect before your conclusions can be believable. I usually see the Malmquist bias discussed in papers about galaxies, but Malmquist himself proposed the effect for stars. Let me explain.
Suppose that you want to know what is the most common type of star. That is simple, you might say. Just look at all the stars that you see in the sky, find out what type of star they are, plot the data in a histogram of some sort, and see what category comes up with the biggest number. Simple! Also, wrong. For you see, the more distant an object is, the dimmer that it appears. Distant stars look much dimmer than nearby ones of the same type. So stars that are so dim as to be hard to see even if they are nearby, you won’t be able to see at all if they are far away. Stars that are very bright nearby will still be bright enough to be seen so far away that you’ll be able to see them at distances so great that all other stars are too dim to see. What this means is that you undercount the dim objects. We know that red dwarf stars are the most abundant type of star, but if you are walking outdoors at night with a friend who is an astronomer, don’t ask him/her to show you one! Even the nearby ones are too dim to see with the naked eye. And most are even too dim to see through the eyepiece with telescopes that most amateur astronomers use. Yet, we can study the nearest stars and see that they are the most common type. Though we don’t see them at great distances, we believe that they are there in the same numbers as they are in our part of the galaxy. If that is the case, then red dwarf stars are by far the most numerous star in the galaxy, even if they are not the most common type of star observed.
Likewise, the Malmquist bias works with galaxies. When we look at distant galaxy clusters, we tend to see only the large bright members of the galaxy cluster. The smaller ones are too dim to easily detect. We find lots of the smaller galaxies in our Local Group of galaxies. However, we don’t see them in very distant galaxy clusters because they are small and dim, and just to hard to spot at those great distances. They should still be there, though. We might assume that they are there in the same numbers as in our Local Group. Hmm. Well, this might be a problem. For you see, the farther away that you look, the farther back in time that you look. Are the numbers of dwarf galaxies constant, or does the number get smaller over time as they are consumed through collisions with the larger galaxies? Perhaps small clouds of gas collapse to form new dwarf galaxies at some point. There are some proponents of both ideas. The problem is that we have a tough time finding the dwarf galaxies at great distances. So here, the Malmquist bias rears its head, and it is harder to put to rest when you are talking about cosmological distances, and the corresponding differences in the age of the samples studied. The more distant galaxy samples exist at a younger time in the life of the universe than do the nearby samples. So, does the lack of observation mean that there were fewer dwarf galaxies, or is this a Malmquist bias effect? If it is a Malmquist bias, what does that mean, if anything? You see the problem. It sure makes things complicated.
So, to refresh your memory. The Malquist Bias is an observational selection effect in which the dimmer objects are under-represented in samples of data from great distances. This is because the farther away, the dimmer that something appears. If it is dim to start with, then it just too dim to see at great distances. You have to then infer the presence of the dim objects at great distances. If you don’t feel comfortable doing that, then you can only compare data samples made as similar distances. For example, if you are studying galaxy groups, you can only compare galaxy groups at distances of 1500 Mpc (Mpc, or megaparsec, is 3,260,000 lightyears) with other galaxy groups located at about 1500 Mpc. A comparison of a 1500 Mpc galaxy group with one located at only 10 Mpc is not really valid unless you address the Malmquist bias and whether or not undercounting the dwarf galaxies will affect your conclusions.
-Astroprof
Robert Clarke on May 15, 2007 at 10:43 am: 1
Thank you. Although it has been applied to astronomy/cosmology here, it is an important aspect of almost any kind of research and of course, is often never mentioned in marketing, historical, consumer etc. research. Illustrates how the relating of research to prove almost anything can be accomplished. Just ignore the Malmquist Bias.
7 November 09 (pm) « blueollie on November 7, 2009 at 4:53 pm: 2
[…] the Malmquist Bias was discussed: this is the bias which states that, in situations where one is limited by signal […]