Non-given variables and unobservable relationships between variables are categorized as hidden variables.  Identifying hidden variables is an important step in forming and testing a hypothesis.

I. Definition

In a scientific inquiry process with many variables, some variables are obvious, but others are either not given or cannot be determined directly.  Some relationships between variables are easily observed, but other relationships don’t show up during experimentation.  Both non-given variables and unobservable relationships between variables are categorized as hidden variables.  These hidden variables are related to other given variables and sometimes affect the dependent variable being tested.  Thus, identifying hidden variables is an important step in forming and testing a hypothesis.

II. Simplified examples of hidden variables

1. Question ID: 20200100100

A zoologist travels to Africa to study the natural breeding environment of giraffes.  While there, he notices a type of tall tree that produces a special fruit that only grows at the top of the tree.  He also notices that giraffes that frequently eat this fruit appear to be stronger and taller than those who cannot reach the fruit.  He concludes that the fruit contains rich nutrients which make the giraffes that eat the fruit grow stronger and taller. Which one of the following statements do you agree with?

A. When a giraffe frequently eats this special fruit, it grows stronger and taller.
B. The nutrients in the fruit can help the giraffe grow stronger and taller.
C. Both A and B are correct.
D. The result is not sufficient to demonstrate that eating the fruit causes a giraffe to grow stronger and taller.
E. None of the above statements is reasonable.

Note:  This question asks the student to determine the relationship between eating a special fruit and the strength and height of the giraffes.  The fact that giraffes that frequently eat this fruit appear to be stronger and taller is presented obviously.  However, the condition that only tall giraffes are able to reach the fruit is hidden.  In other words, the short and tall giraffes have different heights by nature, regardless of whether they eat the fruit or not.  If students cannot discover this hidden variable, they may choose a wrong answer.

2. Yellowstone Park

Park patrols in Yellow Stone National Park noticed that sightings of wild wolves by visitors have gradually increased.  A possible link is that the forest is recovering from a fire two decades ago.  In order to more accurately determine the wild wolf population, tourists were encouraged to report their sightings of the wild wolves starting from the beginning of last year.  During the last year, the total reported spotting incidents quadrupled compared to the previous year’s data.  Based on this evidence, which one of the following statements do you agree with?

A. The population of wild wolves has increased significantly.
B. The recovery of the forest has caused the wild wolf population to increase.
C. With more people watching for wild wolves, the data collected is more accurate.
D. All of the above statements A, B and C are right.
E. The result is not sufficient to conclude that the wild wolf population has increased.
F. None of the above is a good conclusion.

Note: In this question, students are asked whether or not (and why) the wild wolf population has increased.  From the total reported sightings, it seems that the wolf population has increased significantly.  However, there is a hidden variable:  the number of tourists watching for the wolves.  This number is not the same each year, so the number of wolf sightings cannot be compared.  Another hidden factor is that tourists were not encouraged to report sightings in past years; it could be that the wolf population and number of sightings remains the same but more of those sightings are being reported.

III. Review of hidden variables in research

Hidden variables are those that can affect the outcome of an experiment but that are not obvious in the context of the problem. Experimenters need to think carefully to identify hidden variables.

Lawson, Alkhoury, Benford, Clark, and Falconer (2000) refer to hidden variables as “unseen entities.”  He notes that no matter how long an experiment is observed, certain aspects of the experiment can never been seen even though they have very observable effects.  Such variables include osmosis, combustion, air pressure, genes, molecules, photosynthesis, and biogeochemical cycles.  These can be considered “theoretical concepts” rather than “descriptive concepts” as their influence cannot be derived from direct observation.  This requires a higher level of reasoning for students during the formation of a hypothesis.  The student cannot invent an explanation for an unseen theoretical entity until an experiment yields some puzzling observations that need an explanation.

Research on hidden variables is also popular in physics and statistics.

In physics, hidden variable theory tries to explain away the statistical nature of quantum mechanics.  Hidden variables correspond to aspects of physical reality, which could, in principle, be measured but may not be for practical reasons.  In this context, the term hidden variable is commonly used to reflect the fact that the variables are really there but are not directly observed.  (http://en.wikipedia.org/wiki/Hidden_variables)

In statistics, hidden variables (or latent variables), are variables that are not directly observed but rather are inferred (through a mathematical model) from other variables that are observed (directly measured).  Hidden variables sometimes correspond to abstract concepts such as categories, behavioral or mental states, or data structures.  The terms hypothetical variables or hypothetical constructs may be used in these contexts to reflect the fact that some variables are not observed despite their interactions with several of the observed variables.
(http://en.wikipedia.org/wiki/Latent_variable) (Elidan, Lotner, Friedman, & Koller, 2001)

An example of hidden variables in Lawson’s research

A student put a drop of blood on a microscope slide and then looked at the blood under a microscope.  As you can see in the diagram below, the magnified red blood cells look like little round balls.  After adding a few drops of salt water to the drop of blood, the student noticed that the cells appeared to become smaller.

This observation raises an interesting question:  Why do the red blood cells appear smaller?

Here are two possible explanations:

1. Salt ions (Na+ and Cl-) push on the cell membranes and make the cells appear smaller.
2. Water molecules are attracted to the salt ions so the water molecules move out of the cells and leave the cells smaller.

To test these explanations, the student used some salt water, a very accurate weighing device, and some water-filled plastic bags, and assumed the plastic behaves just like red-blood-cell membranes. The experiment involved carefully weighing a water-filled bag in a salt solution for ten minutes and then reweighing the bag.

What result of the experiment would best show that explanation I is probably wrong?

1. The bag loses weight
2. The bag weighs the same
3. The bag appears smaller

What result of the experiment would best show that explanation II is probably wrong?

1. The bag loses weight
2. The bag weighs the same
3. The bag appears smaller

Note: The water molecules and salt ions cannot be seen by the student, so they are considered unseen entities.  The student needs to test the hypothesis that the ions leave cells because of their attraction to water molecules, which can be difficult for the student to visualize because these variables cannot be directly observed.

IV. Importance of hidden variables

1. The importance of hidden variables in science

Historically, hidden variable theories were promoted by a minority of physicists who argued that quantum mechanics was “incomplete.”  Albert Einstein, the most famous proponent of hidden variables, famously insisted that, "I am convinced God does not play dice," but whether he objected to the statistical nature of quantum mechanics is disputed.  Einstein, Podolsky, and Rosen argued that "elements of reality" (hidden variables) must be included in the description of quantum mechanics in order to explain entanglement without the concept of action at a distance.

For students, identifying hidden variables is a scientific reasoning skill that requires creativity.  If a student can only see the obvious variables but has no sense of the hidden (but important) variables, the student is not displaying adequate creativity.

2. The importance of hidden variables in society

Hidden variables exist all around us, and they can’t be ignored.  Problems can arise where those investigating the issue feel they have checked every factor that may affect the outcome, but they can’t solve the problem because they are neglecting a hidden variable.  For example, although two twins grow up in the same family, go to the same school, play the same games, and have the same set of friends, their personalities are different.  What make them different?  There may be some hidden variables that research have yet to discover.

V. References

Gal Elidan, Noam Lotner, Nir Friedman and Daphne Koller. Discovering Hidden Variables: A Structure-Based Approach. In Leen et al. (2001), pages 479-485.

Anton E. Lawson, Brian Clark, Erin Cramer-Meldrum, Kathleen A. Falconer, Jeffrey M. Sequist, Yong-Ju Kwon. Development of Scientiﬁc Reasoning in College Biology: Do Two Levels of General Hypothesis-Testing Skills Exist? Journal of Research in Science Teaching, vol. 37, no. 1, pp. 81-101 (2000)

Lawson, A.E., Drake, N., Johnson, J., Kwon, Y.J., & Scarpone, C. (2000). How good are students at testing alternative explanations involving unseen entities? The American Biology Teacher, 62(4), 246-252.

Anton E. Lawson, Souheir Alkhoury, Russell Benford, Brian R. Clark, Kathleen A. Falconer. What Kinds of Scientific Concepts Exist? Concept Construction and Intellectual Development in College Biology. Journal of Research in Science Teaching, vol. 37, no. 9, pp. 996-1018 (2000)