<p>Those responding make some very good points about being specific about operational definitions and such. However, I've argued -- in an evolutionary psych textbook -- that there is a probably a reason why discomfort from cold and electric shock have power laws with exponents greater than perceived brightness (at low stimulus intensities), for example. It's adaptive to be highly sensitive to intense levels of things that can kill or harm you -- to avoid frostbite or death from electrocution, for example. This would also suggest that the psychophysical exponents for normally harmless dimensions such as brightness and loudness would go up for very bright lights or very loud noises (that could damage your eyes or ears, respectively).</p>

<p>Psychologist Martie Haselton has made very similar arguments in her research on "error management theory." Roy Baumeister has made similar points in his Psychological Bulletin paper entitled something like 'Bad is stronger than good."   </p>

<p>Brett Pelham    </p>

CC BY-SA 3.0
2015-09-16T17:10:37.610Z
2015-09-16T17:10:37.610Z
Brett Pelham
<p>Mainly: the choice of variables.</p>

<p>Beware that the exponent depends on the choice of parameters; or even worse - relations can change from power-law to linear, logarithmic or exponential, with redefining variables.</p>

<p>E.g. you can measure sound volume in either amplitude, energy density (square of the amplitude) or dB (logarithm of the first one). None of it is privileged. Moreover, "psychological magnitude" needs exact definition, for the same reason (so not only to specify which type of reaction is that but what is the exact physical quantity being measured).</p>

<p>In general, if you consider only qualitative parameters, then you can look at only at qualitative behaviour of the relation (e.g. if it is monotonous). To calculate exponents (or even - to claim that a certain relation is a power function) it is not enough.</p>

<p>Of course, there are many adaptations to make sensitivity higher (e.g. edge detection) or lower (e.g. to see well in lighting conditions differing by order of magnitude, in terms of light intensity). But to compare their exponents, for the comparison to make sense, you need to have a very good justification for the right choice of variables.</p>

CC BY-SA 3.0
2012-10-25T09:13:55.227Z
Thanks for the information Piotr. Do you have any citations/references to papers that have considered the issues that you discussed?
Jake Westfall
CC BY-SA 3.0
2012-10-25T09:54:48.403Z
Well, it's a piece of any standard physics introduction that: you cannot compare different variables (e.g. what is more: one meter or one kilogram) unless you have relations between them. And both depending on the choice of variables, and of the constrains, you obtain vary different relations between "physical quantities". For example, take "heat" as a function of "electricity" - i takes no sense. Even if restrict ourselves to "heat" = energy generated per second (W), (I - current, U - voltage, R - resistance) then it can be W(U) = I U, W(U) = U^2/R, W(I) = I U, W(I) = R I^2, ...
Piotr Migdal
CC BY-SA 3.0
2012-10-25T09:58:15.967Z
Now I'm away from may books, but it should be is something like "Feynman lectures on physics" or, if you want something more specific, look at [dimensional analysis](http://en.wikipedia.org/wiki/Dimensional_analysis) (there you see very well that also the _constrains_ change relations drastically).
Piotr Migdal
CC BY-SA 3.0
2012-10-25T15:54:46.340Z
The point about units is simple enough and I'm not questioning it. My request is whether you are aware of papers specifically from psychophysics, or psychology more generally, that bring up these dimensional issues in response to the psychophysics work from the OP. It is clear that table and figures like those I posted above invite comparisons in the exponents across stimuli. So if it is the case that such comparisons truly are of little or no warrant, then I would expect people to have made this basic point in writing somewhere in the psychophysics literature. Do you see what I mean?
Jake Westfall
CC BY-SA 3.0
2012-10-25T22:17:09.980Z
I don't know psychological papers on it. But well, it's fundamental stuff and the Nature does not care if psychologist know it or not. (Compare as if some guys used 2+2=5 claiming "hey, but there is no paper _in our field_ disproving that".) I know some well more subtle papers (e.g. [arXiv:1102.4101](http://arxiv.org/abs/1102.4101) when it is argued that a certain statistical relation in data-drive sociology is not necessary a power law). Also, I bet that most of mathematical psychologists know are very well aware of it. Sadly, I have doubts whether standard psychologists are enough educated.
Piotr Migdal
CC BY-SA 3.0
2012-10-25T22:31:09.003Z
Okay, so then maybe the reason I can't find any discussion in the psychophysics literature about the questions I raised in the OP is because it is just implicitly and universally understood among psychophysicists that you Don't Ask That, for the reasons that you point out concerning units. If this is the case then I guess my question is answered.
Jake Westfall
CC BY-SA 3.0
2012-10-27T15:04:40.120Z
@Jake It's not that unlikely that for some it's "so obvious, there is no point of writing a paper about it" and the others are not aware of that issue (anyway, it requires a certain proficiency in natural sciences), with not that many in between. But hey - maybe there is place for you to clarify.  Are you writing a paper when you want to refer to material of the question that you asked?
Piotr Migdal
CC BY-SA 3.0
2012-10-27T20:52:31.680Z
I am moving this discussion over to the Cognitive Sciences chat room ("The Axon Terminal").
Jake Westfall
CC BY-SA 3.0
2012-10-22T20:48:19.093Z
2012-10-22T20:48:19.093Z
Piotr Migdal
<p>Following the work of Stanley Stevens, psychophysical functions of stimulus intensity are commonly assumed to follow power laws, as illustrated below:
<img src="https://i.sstatic.net/oXCq3.png" alt="power law exponents graph"></p>

<p>This appears to be true for a wide variety of different types of stimuli and sensations. A table that is often presented to illustrate this generality is from <a href="http://books.google.com/books/about/Psychophysics_introduction_to_its_percep.html?id=e_VqAAAAMAAJ" rel="noreferrer">Stevens, 1975</a>:</p>

<p><img src="https://i.sstatic.net/XJfjB.png" alt="power law exponents table"></p>

<p>However, it is not obvious from perusing this table what kinds of stimuli tend to be associated with what kinds of power exponents -- in particular, what kinds of stimuli tend to be associated with <em>positively accelerated</em> psychophysical functions of intensity vs. <em>negatively accelerated</em> psychophysical functions of intensity. Indeed, the exponents in the table above appear to be distributed rather haphazardly to me, and I have not seen any mention of work suggesting it is otherwise in my admittedly limited literature search.</p>

<p><strong>So my question is:</strong> Is anyone aware of research that attempts to characterize what kinds of stimuli tend to be associated with what kinds of psychophysical functions of stimulus intensity? If so, what is the basic summary from this research, and what are some suggested readings? As hinted above, I am particularly interested in stimulus features that predict positively vs. negatively accelerated psychophysical functions of stimulus intensity, but references to and descriptions of more specific research questions are also quite welcome.</p>

<p><strong>Update:</strong> I thought it might help to say just a little bit more about exactly what I'm interested in finding out here, in as clear and concise a way as I know how. What I am looking for are any references to review papers and/or research articles that are about (or even that just mention) predicting the form of the psychophysical power law across a range of different stimuli, using stimulus-level predictors. It is hard to extract any general conclusions about this from Stevens's table above, given that the power law exponent appears to vary widely, sometimes being >1 and sometimes &lt;1, <em>even for stimuli on the same psychological continuum</em>, as in the cases of "Taste" and "Warmth." The point of the table (and indeed the body of research that it is based on) is just to show that these functions follow power laws of some kind or other -- while this is mildly interesting, what I really want to know is how well we can predict what <em>kinds</em> of power laws different stimuli will follow. Please let me know if I can offer any additional clarification.</p>

CC BY-SA 3.0
2012-10-21T03:48:06.093Z
**Speculation:** (I couldn't find any research to confirm my intuition about this) **Negative exponent** is for sensory information that's integrated into contextual processing, it's adaptive, so it has less of an effect as more is present and the mind adapts to the stimulus.

**Positive exponent** is more related to an alert system.  There's a threshold at which it takes off (similar to a transistor).  **Note:** Wikipedia documents some strong criticisms of Steven's Power Law.
Keegan Keplinger
CC BY-SA 3.0
2012-10-21T06:23:22.483Z
Thanks for the comment; I welcome speculation. I am aware of some of the criticisms of the evidence for universal power functions, but just to be clear to other readers: I would prefer *not* to have this thread veer off into that discussion if it can be helped. Whether the most accurate psychophysical function is actually power, log, exponential, or whatever, the important point for the purposes of this thread is just that some types of stimuli are characterized by more quickly increasing slopes than others, and I am interested in the stimulus factors that influence this variability.
Jake Westfall
<perception><sensation><psychophysics>
What stimulus features determine the psychophysical power law exponent?
Jake Westfall
# What stimulus features determine the psychophysical power law exponent?

(Asked by: username_0 on 2012-10-14)

Following the work of Stanley Stevens, psychophysical functions of stimulus intensity are commonly assumed to follow power laws, as illustrated below:
![power law exponents graph](https://i.sstatic.net/oXCq3.png)

This appears to be true for a wide variety of different types of stimuli and sensations. A table that is often presented to illustrate this generality is from [Stevens, 1975](http://books.google.com/books/about/Psychophysics_introduction_to_its_percep.html?id=e_VqAAAAMAAJ):

![power law exponents table](https://i.sstatic.net/XJfjB.png)

However, it is not obvious from perusing this table what kinds of stimuli tend to be associated with what kinds of power exponents -- in particular, what kinds of stimuli tend to be associated with *positively accelerated* psychophysical functions of intensity vs. *negatively accelerated* psychophysical functions of intensity. Indeed, the exponents in the table above appear to be distributed rather haphazardly to me, and I have not seen any mention of work suggesting it is otherwise in my admittedly limited literature search.

**So my question is:** Is anyone aware of research that attempts to characterize what kinds of stimuli tend to be associated with what kinds of psychophysical functions of stimulus intensity? If so, what is the basic summary from this research, and what are some suggested readings? As hinted above, I am particularly interested in stimulus features that predict positively vs. negatively accelerated psychophysical functions of stimulus intensity, but references to and descriptions of more specific research questions are also quite welcome.

**Update:** I thought it might help to say just a little bit more about exactly what I'm interested in finding out here, in as clear and concise a way as I know how. What I am looking for are any references to review papers and/or research articles that are about (or even that just mention) predicting the form of the psychophysical power law across a range of different stimuli, using stimulus-level predictors. It is hard to extract any general conclusions about this from Stevens's table above, given that the power law exponent appears to vary widely, sometimes being >1 and sometimes <1, *even for stimuli on the same psychological continuum*, as in the cases of "Taste" and "Warmth." The point of the table (and indeed the body of research that it is based on) is just to show that these functions follow power laws of some kind or other -- while this is mildly interesting, what I really want to know is how well we can predict what *kinds* of power laws different stimuli will follow. Please let me know if I can offer any additional clarification.

**Tags:** perception, sensation, psychophysics

> **Comments:**
> **username_1**: **Speculation:** (I couldn't find any research to confirm my intuition about this) **Negative exponent** is for sensory information that's integrated into contextual processing, it's adaptive, so it has less of an effect as more is present and the mind adapts to the stimulus.  **Positive exponent** is more related to an alert system.  There's a threshold at which it takes off (similar to a transistor).  **Note:** Wikipedia documents some strong criticisms of Steven's Power Law.  
> **username_0**: Thanks for the comment; I welcome speculation. I am aware of some of the criticisms of the evidence for universal power functions, but just to be clear to other readers: I would prefer *not* to have this thread veer off into that discussion if it can be helped. Whether the most accurate psychophysical function is actually power, log, exponential, or whatever, the important point for the purposes of this thread is just that some types of stimuli are characterized by more quickly increasing slopes than others, and I am interested in the stimulus factors that influence this variability.  

---

## 2 answers

### Answer by username_3 on 2012-10-22. Score: 5 | Accepted answer

Mainly: the choice of variables.

Beware that the exponent depends on the choice of parameters; or even worse - relations can change from power-law to linear, logarithmic or exponential, with redefining variables.

E.g. you can measure sound volume in either amplitude, energy density (square of the amplitude) or dB (logarithm of the first one). None of it is privileged. Moreover, "psychological magnitude" needs exact definition, for the same reason (so not only to specify which type of reaction is that but what is the exact physical quantity being measured).

In general, if you consider only qualitative parameters, then you can look at only at qualitative behaviour of the relation (e.g. if it is monotonous). To calculate exponents (or even - to claim that a certain relation is a power function) it is not enough.

Of course, there are many adaptations to make sensitivity higher (e.g. edge detection) or lower (e.g. to see well in lighting conditions differing by order of magnitude, in terms of light intensity). But to compare their exponents, for the comparison to make sense, you need to have a very good justification for the right choice of variables.

> **Comments:**
> **username_0**: Thanks for the information Piotr. Do you have any citations/references to papers that have considered the issues that you discussed?  
> **username_3**: Well, it's a piece of any standard physics introduction that: you cannot compare different variables (e.g. what is more: one meter or one kilogram) unless you have relations between them. And both depending on the choice of variables, and of the constrains, you obtain vary different relations between "physical quantities". For example, take "heat" as a function of "electricity" - i takes no sense. Even if restrict ourselves to "heat" = energy generated per second (W), (I - current, U - voltage, R - resistance) then it can be W(U) = I U, W(U) = U^2/R, W(I) = I U, W(I) = R I^2, ...  
> **username_3**: Now I'm away from may books, but it should be is something like "Feynman lectures on physics" or, if you want something more specific, look at [dimensional analysis](http://en.wikipedia.org/wiki/Dimensional_analysis) (there you see very well that also the _constrains_ change relations drastically).  
> **username_0**: The point about units is simple enough and I'm not questioning it. My request is whether you are aware of papers specifically from psychophysics, or psychology more generally, that bring up these dimensional issues in response to the psychophysics work from the OP. It is clear that table and figures like those I posted above invite comparisons in the exponents across stimuli. So if it is the case that such comparisons truly are of little or no warrant, then I would expect people to have made this basic point in writing somewhere in the psychophysics literature. Do you see what I mean?  
> **username_3**: I don't know psychological papers on it. But well, it's fundamental stuff and the Nature does not care if psychologist know it or not. (Compare as if some guys used 2+2=5 claiming "hey, but there is no paper _in our field_ disproving that".) I know some well more subtle papers (e.g. [arXiv:1102.4101](http://arxiv.org/abs/1102.4101) when it is argued that a certain statistical relation in data-drive sociology is not necessary a power law). Also, I bet that most of mathematical psychologists know are very well aware of it. Sadly, I have doubts whether standard psychologists are enough educated.  
> **username_0**: Okay, so then maybe the reason I can't find any discussion in the psychophysics literature about the questions I raised in the OP is because it is just implicitly and universally understood among psychophysicists that you Don't Ask That, for the reasons that you point out concerning units. If this is the case then I guess my question is answered.  
> **username_3**: @Jake It's not that unlikely that for some it's "so obvious, there is no point of writing a paper about it" and the others are not aware of that issue (anyway, it requires a certain proficiency in natural sciences), with not that many in between. But hey - maybe there is place for you to clarify.  Are you writing a paper when you want to refer to material of the question that you asked?  
> **username_0**: I am moving this discussion over to the Cognitive Sciences chat room ("The Axon Terminal").  

---

### Answer by username_2 on 2015-09-16. Score: 1

Those responding make some very good points about being specific about operational definitions and such. However, I've argued -- in an evolutionary psych textbook -- that there is a probably a reason why discomfort from cold and electric shock have power laws with exponents greater than perceived brightness (at low stimulus intensities), for example. It's adaptive to be highly sensitive to intense levels of things that can kill or harm you -- to avoid frostbite or death from electrocution, for example. This would also suggest that the psychophysical exponents for normally harmless dimensions such as brightness and loudness would go up for very bright lights or very loud noises (that could damage your eyes or ears, respectively).

Psychologist Martie Haselton has made very similar arguments in her research on "error management theory." Roy Baumeister has made similar points in his Psychological Bulletin paper entitled something like 'Bad is stronger than good."

username_2


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