I’ve been getting a lot of views this month from Belarus-  I’m just asking out of curiosity, what is it Belarusians have found interesting?

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Doing science.

Sometime, somewhere, somehow, my work will make a meaningful difference in the course of things. It will matter. I will receive no accolades, fame, fortune, nor security from it. I may even be dead when it happens. But some day my work will be used to connect two unconnected dots in a pattern and permit great discovery. He or she who connects all of the pieces will, likely, not recognize the importance of the individual parts in their contribution to what he or she has built, but they will be there nevertheless and my life’s labor will not have been in vain. That is the best that a good scientist can hope for.

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Explaining a pre-exposure effect that is enhanced with a change in context.

I think I know why I obtained the effects in my previous set of experiments that I obtained, where the effect of pre-exposure to a CS was even more pronounced when the context was changed between phases. 

I have, at least, made the effect go away where there is no detectable effect of a context change on the pre-exposed groups. 

I may even have a small loss of latent inhibition with a context change, but if so it is a small effect and cannot be detected statistically with Ns of 20.

I am optimistic about my potential explanation- I could have uncovered a new effect that, up until this point in time, could have remained unknown due to methodological limitations. 

Another experiment is ready to go and hopefully will be underway soon.  With luck, it will be finished in the next few weeks. 

Right or wrong on my idea, I’ll be posting more detail and data when have more than cross-experiment comparisons with which to work.


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The Learning Laboratory

The video below shows the human learning laboratory here at the University of the Basque country.  It is equipped with four experiment stations, and an eye-tracking station.  Each station has a Dell Optiplex computer running an ATI 2400hd video card and a 22in monitor.  

The eye tracker is a RED 250 system from SMI Vision.

There is an additional Optiplex system for general lab use (e.g., Office programs, internet).

Be sure to click the settings icon (shaped like a gear) to bump the playback quality up a bit.

The UPV Learning Laboratory

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A pre-exposure effect that is enhanced with a change in context-

This post is going to describe an effect that looks like latent inhibition, but rather than being attenuated by a context switch, it is enhanced.

The post is going to be a long one.  A lot to write, and I’m not spending much time on revisions so it may read like crap.  If you see something completely daft (other than my results!), please point it out.

I’m no expert in latent inhibition, but I got involved with it tangentially through an interest in perceptual learning.  In a previous method (spacebird) I had demonstrated a latent inhibition effect, and that it was, as expected, context specific (Nelson & Sanjuan, 2006).

Using my present methodology I began using an SMI-Vision eye tracker to monitor visual attention to events in my game.  Research with animals has shown that attention to novel stimuli decreases with exposure, and then returns, at least briefly, once that stimulus is conditioned where it predicts some other event.

I thought it would be nice to validate the data produced by the eye tracker in my method against these animal data.  The idea was to conduct two experiments and then have a modest publication from it.  The first experiment would be a latent inhibition experiment where a sensor is pre-exposed or not, between subjects, and then paired with an attacking spaceship. 

The second experiment would be to assess the extent to which my effect was context dependent and how visual attention to the CS correlated with that dependency. 

I have described Experiment 1 already, but I am going to do it again in this post briefly.  

It was in “Experiment 2” where things began to go weird as several short posts lamented. 

Today, I am going to show the experimental designs and results that I have been alluding to in those posts.

Experiment 1 demonstrated latent inhibition.  

Experiment 2 was to demonstrate its context-specificity, which it did in a very odd way.  Due to the odd results I conducted Experiment 3 to determine whether or not my procedure might be inadvertently producing Conditioned Inhibition.  In short, it was not. 

Then, I conducted Experiments 4 & 5 using the same design as Experiment 2, with some potentially important procedural changes that had no effect.  

Here is the plan for today. First, I am going to show Experiment 1 again here, briefly. Then, I will show Experiments 2, 4, & 5 as their data can be combined into one set of figures.  Next, I will present Experiment 3 followed by two more brief experiments to show that my method doesn’t just produce strange results in general.

Finally, I’ll speculate on what I think might be going on and mention the experiment currently underway.

I’ve described the methodology in detail in other posts, and will briefly summarize it here with some slides I’ve prepared for a talk.


The sensor is shown below as the illumination of the Red oval in the top middle of the panel at the bottom of the screen.  In this screenshot the US is present and attacking the station.



Experiment 1

Participants all receive response acquisition training (RA) and transport to a galaxy for patrol.  Here they receive six pre-exposures to R (Latent Inhibition) or exposure to the contexts (Control).  Both groups then received 10 acquisition trials with R.


Experiment 1 results

Responding: Participants performed practically no keyboard responses to the CS during pre-exposure, as would be expected.  The graph shows responses per second (y axis) for each of the 20 seconds of pre exposure on each trial (x axis)


Gazing: Participants in the latent inhibition group did look at the sensor when it was illuminated.  That orienting declined both within the stimulus and between presentations.  The figure below shows seconds spent gazing at the CS (y-axis) for each second of each trial (x axis) during pre-exposure.   The glowing effect corresponds roughly to a standard-error of the mean.


When not looking at the CS, the groups looked at the screen which was divided into four zones. The top left was designated “US Zone” as it was the area of the screen where the US would be appearing.  The other zones were labeled TR, BL, and BR (TR = top right…).  BL and BR were mutually exclusive of the area designated as “Sensor”


When not looking at the sensor, participants had an unconditioned tendency to look up at the US zone, probably due to the animated flames on the weapon.  The figure collapses across seconds.  There were no differences between the groups that could not be explained by the observations above that the latent inhibition group spent more time looking at the sensor than the control group (more time looking at the sensor = less time looking at the rest of the screen).



During conditioning, when the CS was paired with an attacking spaceship, responding to the CS prior to the arrival of the spaceship was acquired more slowly in the latent inhibition group.  The graph shows responses per second (Y axis) on each of the first 5 seconds of the CS prior to the arrival of the US on each trial (x axis).

The Latent Inhibition group was significantly slower on all trials, except trial 9.  The effect was robust.


The gaze data showed that the control group looked at the sensor more on the first trial, but any differences thereafter were weak and of suspect reliability.  Looking at the first second of gazing, there was some small increase in the latent inhibition group from the first trial.  The figure shows seconds gazing at the CS during its first 5-s of presentation (before the US) on each trial).


Unless the gaze data are grossly insensitive, it seems that visual attention to the CS can hardly account for the full behavioral differences observed.

When not gazing at the CS, the participants directed their attention to the US zone.  Although there was an unconditioned tendency to look there, during conditioning that tendency appeared to increase, and increase equally between the Latent inhibition and Control groups.


Next, I analyzed responding during the US when both the CS and US were present.

Responding increased from the level observed during the CS alone, then decreased slightly over the 15 seconds that the US was present.  The drop off on the final second was due to the US exiting the screen during that second.

What was revealed was that the latent inhibition group responded significantly more than that control group.  That result can perhaps be interpreted as a “surprise” effect.  If the appearance of the attacking spaceship is more surprising, it might be expected to elicit greater responding.


I am not reporting any gaze data as when the US was present, time spent looking at anything else was negligible.

In summary, Experiment 1 showed:

  • Visual attention to the CS declined both within the CS and between trials during pre exposure.
  • Visual attention to the pre-exposed CS recovered slightly during conditioning.
  • Pre exposure produced a profound retardation of response acquisition.
  • Pre exposure enhanced the response to the US, suggesting that the US was more surprising (e.g., less-well predicted by the CS)
  • Participants learn where the US will appear, and that did not appear affected by pre exposure.


Experiment 2 was to demonstrate that the latent-inhibition effect I saw in Experiment 1 was context specific, and assess the extent to which visual attention to the CS correlated in some way with that contextual control.

There was one procedural change.  Because participants had an unconditioned tendency to look at the top left of the screen, I did not want to use the spaceship that appeared in that corner.  Instead of using the “Stellarian” pictured as the winged ship in the left panel of the figure below, that entered from the top left corner, I used the “Juk Destroyer” which entered from the bottom right.  The US Zone was now the area where participants in Experiment 1 least preferred to look.


The design was straightforward, and was the same design for Experiments 4 and 5.  I’ll get to the full set of differences between the experiments later.  For now, I will say that there was no eye-tracking in Experiment 5.

Thus, behavioral data will be collapsed across the three experiments ( Experiment by XXXX analyses showed no interactions of the manipulations with Experiment).  Gaze data will be presented for Experiments 2 and 4 combined.


A: & B: are contexts, R a red sensor CS, and + represents an attack by the Juk Destroyer spaceship. 

Two Latent Inhibition groups received pre-exposure to the Red sensor while the Control group received exposure to the context alone.

LI Same then received conditioning in the same context as where pre-exposure took place.

LI-Different received conditioning in a different context than where pre-exposure took place.

The two contexts used are shown below.  Each group received the same exposure to the contexts, and within each group the contexts were equally exposed and counterbalanced.


Experiments 2,4, & 5: Results

Responding: Unlike Experiment 1, there was some slight unconditional tendency to respond when the CS was presented in pre-exposure. There were no differences between the LI Same and LI Different groups.


Gaze: Like Experiment 1, participants showed both within CS and between-trial decrements in orienting to the CS during pre exposure, with no differences between the two LI groups.


When not gazing at the CS, participants tended to gaze at the upper portions of the screen.  The figure below is collapsed across seconds, thus the effect, as figured, appears more prevalent in the control group.  It was also present in the LI groups, but more so in the later seconds of the CS presentation when participants were not so strongly orienting to the CS.  

Nevertheless, the important part of the figure is that there was no preference for the US Zone as shown in the first experiment.


The next figure shows the response data from the conditioning phase.  The figure shows responses per second for each second of the CS prior to the arrival of the US on each trial.

As in Experiment 1, responding in the control group was acquired quickly and robustly.  Responding in the LI groups was profoundly hindered by the pre-exposure.

The prediction was that responding would be acquired more rapidly in the LI Different group than in the LI Same group, demonstrating that the pre-exposure effect was at least somewhat context dependent.

The data showed that the effect was indeed somewhat context dependent, but not as expected!   A change of context between pre-exposure and conditioning enhanced the effect of pre exposure on interfering with response acquisition.


The effect was present in all experiments, and presented here collapsed across experiments where individual groups have an N greater than 60.  Acquisition of responding was retarded after pre exposure, and that effect was more pronounced when pre exposure and conditioning occurred in different contexts.

The next figure shows Gazing at the CS during the first five seconds of the CS before the arrival of the US on each trial.

There were no differences whatsoever in gazing at the CS between the LI Same and LI Different groups.  Both pre-exposure groups gazed less than the Control group. Again, it appears as though differences observed in the behavioral data cannot be explained by differences in visual attention to the CS.

The right side of the figure shows the Group x Seconds interaction that was revealed by the overall analysis.  The control group, having never seen the stimulus, attended to it longer than the LI Groups, with a significant difference on the second Second.


When not looking at the CS, the participants looked at the screen, in particular at the area where the US was to appear and this effect emerged over trials.  The next figure shows those gaze data, collapsed across seconds on each trial.


The analyses showed some differences between the control and LI Same group in looking at the US zone, but the differences were not consistent.  On one trial the control group may look more, and on another the LI Same group may look more.  In comparison to the LI Different group, both the Control and LI Same groups showed more looks at the US Zone on two different trials than the LI Different group.  

I’m not confident at this point in making much about these observed few differences.  If anything, they would suggest that, like the behavioral data, the LI Different group had more difficulty learning to expect the US. 

But, the data shown next contradict that possibility.

The figure below shows the response to the US and CS on each second of the US’s presence.  In Experiment 1 the LI group had a greater response to the US than the Control group, suggesting that the US might have been more surprising in that condition.

In these experiments, that difference occurred again, but only for the LI Same group.  The control and LI Different groups predominately responded identically, and the LI Same group showed greater responding than either of the others.  


That result would suggest that participants in the LI Different group may be expecting the US just as well as in the Control group (i.e., a loss of latent inhibition) and that the response decrement observed in the behavior was a difficulty not in learning to expect the US, but a difficulty in learning the response.

Another interesting aspect of these data, and those from Experiment 1, is that the stronger response in the LI-Same group over that of the Control group is apparent on early trials, even trial one here.

Thus, the differences, at least initially, may not be based on how well the CS predicts the US, but how well the CS predicts “something” or “nothing.”    In Hall’s analysis of latent inhibition, pre-exposure teaches the organism that “nothing” is going to occur.  Thus, in the pre-exposed groups the fact that something occurs, regardless of whether it is a spaceship or not, is more surprising and, in this case where it is a spaceship, generates more responding.

Such an effect is not present in the LI Different group.  That suggests again that Latent inhibition, in the sense of the pre-exposure interfering with being able to learn that the US is forthcoming, is being lost and that the behavioral deficit is a problem with learning the response.

Experiments 2, 4, & 5- differences.

Experiment 2 used the method as described throughout this blog. The sensor was illuminated, embedded in a sensor panel, and the participant’s gaze was monitored with an Eye tracker. In short, the sensor appeared as in the figure below.


Many theories that account for latent inhibition (e.g., Wagner, 1981) assume that the effect is due to the context becoming associated with the CS, thus the CS is not surprising in that context which reduces its ability to be associated with other stimuli, such as the US. Thus, if the CS is presented in another context it is not expected there and is surprising again and can be associated with the US.

Although it would not account for the greater retardation observed in the different context, I thought that perhaps my odd results might have to do with having the sensor embedded in a panel, which was present in every context. Perhaps the panel was, in some way, blocking the ability of the CS and Context to be associated.

So, Experiment 4 simply did away with the panel. There was nothing present until the sensor appeared, floating dis-embodied in the screen as in the figure below.


But- that had no effect on the result.

At that point I wondered whether the eye-tracker might be placing some peculiar demand characteristic on the participant that was affecting the result. So, in experiment 5 I used the same stimuli as in Experiment 4, the dis-embodied sensor, but conducted the experiment on a different set of computers, identical, but with no eye tracker. Again, that had no effect on the results.

If anything occurred across the three experiments it was that the effect where acquisition was even more delayed in the different context got bigger. However, the analysis showed that to not be the case.

Experiment 2, 4 & 5 summary.

  • Replicated experiment 1
    • Visual attention to the CS declined during pre exposure.
    • Visual attention to the CS was slightly greater to the control group than the latent inhibition groups during conditioning.
    • Responding to the CS was acquired more slowly following pre exposure.
    • Participants learned to look at the area of the screen where the US would arrive, and that did not vary consistently between groups.
    • Responding to the US was more vigorous in the LI Same condition than in the control condition, suggesting the US might be more surprising in the LI Same condition.
  • New findings
    • Some weak response to the CS was observed during pre-exposure.
    • A change of context between pre-exposure and conditioning served to enhance the acquisition deficit.
    • There were no differences in visual attention to the CS between the LI Same and LI different groups, although there were other behavioral differences.
    • The response to the US was more vigorous in the LI Same than in the LI different group, suggesting that perhaps some aspect of latent inhibition was lost resulting in the US being less surprising in the LI Different group.
    • Whether the Sensor was embedded in a panel, or free floating against the context, had no impact.
    • Use of the eye tracker appeared to have no impact.

Experiment 3

After getting the odd behavioral result in Experiment 2 where acquisition was more profoundly affected by pre-exposure when conducted in a different context, I decided to investigate whether my procedure was producing conditioned inhibition rather than latent inhibition.

In all the experiments there was a response acquisition phase where participants were trained to respond to all four spaceships.  Next, there was a substantial change of context between response acquisition and the experimental phases, but it is possible that participants were expecting spaceships during pre-exposure which did not occur, making the CS inhibitory.   My previous work has (Bouton & Nelson, 1994; Nelson & Bouton, 1997) has shown conditioned inhibition to transfer robustly across contexts. 

Even if the procedure produced conditioned inhibition, it would not explain why it was more present in the different context, but it would at least provide some new information on the problem.

The design is presented below.

E3 design

Group Latent Inhibition received pre exposure to the Red sensor, exactly as Group LI Same in the previous experiment (2). That is, two contexts were used, even though all training took place in the same context, to match the experiment in all details related to pre-exposure.  Group External Inhibition was treated exactly as the Control in Experiment 2.

Next, both groups received conditioning with a green sensor to use as a target in a summation test.  There were only six conditioning trials so as to not make responding too strong to easily observe an effect.

Finally, both groups received four trials where either the Green sensor alone, or the Red and Green sensors together were illuminated (G- or RG- was between subjects).

Any difference between G and RG in the External Inhibition group is just that- external inhibition. It is the effect of a novel stimulus on the test.

If R is a conditioned inhibitor, the difference between G and RG on the test should be greater in the Latent Inhibition group than in the External Inhibition group.

The eye-tracker was not used as I was interested in the keyboard response data.

Experiment 3 Results

As in the previous experiments, there was some slight response to the sensor during pre-exposure.


During conditioning, it is possible that there could be generalization from Red to Green, where we would see acquisition be slower in the Latent Inhibition group.  However, that was not the case.  There were no differences between the groups in their acquisition of responding to Green, suggestive of little-to-no generalization.


On the test, both groups showed a relatively timed set of responses to green.  Responding increased up until around second 5 when the US would ordinarily appear, and then began to decrease even though the CS was still present.

Combining Red with Green did attenuate responding but it did so equally in the two groups.  If anything is to be made of any differences that can be found between the groups it is that the difference between G and RG is greater in the External inhibition group than in the Latent Inhibition Group. That is, if anything, pre-exposure seemed to reduce external inhibition.   The results are not compatible with the pre-exposure procedure having produced conditioned inhibition.

The figure shows responses per second on all 20 seconds of the CS presentation for each trial of the test.  Responding to Green is shown in Green and responding to the Red and Green sensors is shown in Yellow.  Left panel shows the Latent Inhibition group and the right panel shows the External Inhibition group.



So, do my contexts not work as expected?  Perhaps there is some idiosyncrasy to my method where context changes do not work as expected.

That does not seem to be the case.  Consider the data below.  Here, participants received conditioning with Red followed by extinction in either the same or different context as where conditioning took place.  The figure shows the response to the CS during all 20 seconds of its presence on each trial.

Both groups show the timed responding where responding increases up until the time where the US was expected (at the gap shown after second 5) and then begins to decrease.

The group receiving extinction in a different context (ABA) showed less responding than the group receiving extinction in the same context (AAA).  Thus, conditioning is somewhat context specific with this method which is consistent with observations with some other methods.


After extinction, all groups were then tested back in Context A.  Thus, renewal is expected in the ABA condition as conditioning and extinction took place in different contexts. That is exactly the result that was obtained.


Thus, conditioning is somewhat context specific in the method, and extinction is highly context specific in the method.  There does not seem to be any particular systematic way that the contexts misbehave in the method.

Those data above were collected without the eye tracker.  The data below were collected using the eye tracker, and are fundamentally unchanged aside from being slightly more noisy due to having fewer participants.   Thus, consistent with my interpretation from Experiments 2,4,& 5, the eye tracker itself does not tend to lead to different behavior in participants.



So- what is going on?

I’m not sure.  But, I have one idea I am pursuing.

In the first experiment there was no response observed to the CS during pre-exposure.  In the later experiments, using a different US, there was a response observed to the CS.

The CS was Red, and the Spaceship US being used was also Red.  It could be that there is generalization of a sort between the Red CS and the Red spaceship to which they have been taught to respond.  Thus, when they see the Red CS they respond on the key to which they have been trained to respond to the Red Spaceship.

It could be that the participants did this behavior in the first experiment as well, but I did not record responding on the red-spaceship key because that ship was not being used so I could not have seen it.

That effect of generalization may facilitate learning the response to the CS as it comes to predict the red spaceship.

Thus, responding in the control group reflects normal conditioning (that which is not latently inhibited) plus whatever facilitation arises from both the CS and Spaceship being Red.I’ll call this CS-US similarity facilitation.

I’ll assume that pre-exposure habituated the response to the Red CS, and produced latent inhibition (however you want to conceptualize the process responsible for it).  

Perhaps that facilitation effect of both the CS and US being red recovers during conditioning.  Thus, at least in LI Same, we see latent inhibition (reduced acquisition) that is somewhat offset by this facilitation effect.

Perhaps this recovery of facilitation is not observed in the LI Different condition.  In LI Same, participants are in a context where they have responded to the CS before, but in LI different that responding has habituated (habituation is largely context independent), and they are in a context where they have never  responded to the CS.  Perhaps the responding based on the CS’s similarity to the US is somehow less able to be recovered in the different context.

So, I could be losing latent inhibition ( as might be suggested by the response-to-the-US data), but also losing the recovery of the CS-US similarity facilitation.   Depending on the relative strengths of these two processes, responding in LI Different could come out to be less than in LI Same.

Its not much, but its all I can come up with at the moment.  I am currently conducting Experiment 5 (no sensor panel, no eye tracking) again, but using a Green CS which shares no strong similarity to any of the spaceship USs.   That should, hopefully, eliminate the responding that occurs to the CS as the result of the CS and US being the same color, and perhaps impact the context-switch effect if the recovery of the CS-US similarity facilitation does depend on the stimulus being in the context where they first responded based on that similarity.

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Two dimensional ANOVA- can it be done, and would you want to?

In this post I am going to try to answer the questions above.  Regarding the first question, I’ll explain where the question began for me.  Then, I will review the meaning of the terms Variance and Standard Deviation, and review the basic steps in Analysis of Variance (ANOVA).  Finally, I’ll show how I think ANOVA can be extended to two dimensions, and, to address the second question, show why the whole process, was a big bust.

Can Analysis of Variance be generalized to multi-dimensions?  That is the question that I’ve been mulling lately, and making random facebook posts in my angst the last week or so.

The question comes from my eye-data heat maps.  The two figures below show where people look on the screen during my video-game task. Each point is a record of looking at that spot on the screen.  The brighter / hotter colored points indicate more looks accumulating over time in that area.

The top figure is one group, the bottom is a different group, treated differently.  The difference in how they were treated is not important. US Left_p1_SigTrials_PRE

US Right_P1_SigTrials_CS

Ignore the hot spot in the circle.  When not looking in the circle, it is pretty clear to me that the groups are looking at the screen differently.  The bottom group tends to look to the right, particularly up at the weapon while the top group tends to look around on the left side of the screen.

To analyze something like these patterns the screen is divided up into regions (a priori preferably) and the number of looks (or time spent looking) by each subject in each region is counted and analyzed. 

I thought it would be nice to take the raw data themselves, the XY coordinate pairs, and determine if the groups differ in where they are looking when not looking in the circle.

Individually, it would be a (relatively) simple matter to determine if the groups differ in their average X coordinates, or average Y coordinates.  I could simply do two separate analyses of variance, one on X, and one on Y.

When you do an analysis of variance, you compare the variance within groups of scores to the variance between groups of scores.  The idea is that if everything is random, the variance within a group of scores should be the same as the variance between a group of scores. 

That is… Between Variance / Within Variance = 1.

Variance is a statistic used to estimate variability, how much numbers tend to change.  It is calculated simply.  A score, X, is subtracted from the Mean of the scores, that difference is squared, that is done for every number, and all of those are added up and divided by N-1.  The variance is the average squared difference from the mean

Sum( X- MeanX)^2 / N – 1

The first part of the equation

Sum (x-MeanX)^2

is called the Sum of Squares, or SS.  It is the sum of squared deviations around the mean.

The denominator, N-1, is called the “degrees of freedom” or DF.  I’ll talk more about degrees of freedom further down.  (there are good reasons to use N-1 here instead of N that have to do with estimating populations- stuff  that we don’t need to get into here).

The square root of the variance is called the standard deviation.  It is something very similar to the average deviation of a score from the mean- not exactly though, (the sqrt of the average of squares is not the same as the average of the sqrt of squares) but close enough.

In the end, the standard deviation

Sqrt( Sum (X-MeanX)^2 / N-1 )

and the Absolute deviation

sum( abs(X-MeanX))/N)

are almost always linearly related where Standard deviation = 1.253314137 * Absolute deviation.

Why all the squaring?  Why not just work with absolute differences?

The answer some might give is “It gets rid of negative numbers” which is a simple-minded answer people give either because they are ignorant, or don’t feel like the student is ready for the real reason.

Truth be told, I’m somewhat on the ignorant side. I really do not fully follow the sum total of the real reasons- in part it has to do with some esoteric, but important, arguments between Fisher and Eddington.  You can get into those meaty reasons here if you are so inclined. 

Now me, I like to make up my own reasons and state them with reason and authority so they appear to make sense and are not challenged.

I have always thought of the squaring as having to do with Pythagoras.

C = sqrt( A^2 + B^2)

The length of the hypotenuse of a triangle is equal to the square root of the sum of the squares of each leg.

If you have a point at X1,Y1, and another point at X2,Y2, the length of A is  X1-X2 and the length of B is Y1-Y2. 

C = sqrt((X1-X2)^2+(Y1-Y2)^2)

So, assume you only have one variable, drop Y, and you have

C = sqrt(X1-X2)^2

do that for a bunch of X points, substituting MeanX for X2, and you are just calculating the sum of squares, and begin to see the standard deviation emerging.   The standard deviation relates differences to distances, at least in my mind.

Now, back to ANOVA.  Suppose you have two groups of numbers and want to know if they differ more than you’d expect by chance.  You could do an ANOVA (given many assumptions..).

Consider the imaginary X data below for two groups.
.99 1.73
.78 2.11
.4 1.92
3.07 3.87
-.24 .37

The first thing we would do is begin calculating all our sums of squares.  We start considering all the numbers together as one big group.

Sum of Squares total = SS Total = Sum(X-GM)^2  where GM =Grand Mean.  GM = 1.5

SStotal = 14.98

Next, we’d calculate the Sum of squares for each group-

Sum(X-Lm)^2 where Lm = Local group mean.  The mean for the first group is 1, and that for the second group is 2. 

SSgroup1 = 6.23 and for SSgroup2 = 6.25.  Add them together and we get 12.48

Next, we would calculate the Sum of Squares between the groups.  To do that, we use the means and take the size of the groups into account.

SSbetween = sum( n * (lm-Gm)^2)  where n is the size of the group which is being compared to the grand mean.

SSbetween = 5*(1-1.5)^2+5*(2-1.5)^2 = 2.5 

Now we collect N-1 for all the sums of squares to use in the divisions to create variances.

Total we have 10 numbers, 9 degrees of freedom.

Within the first group we have 5 numbers, so 4 degrees of freedom, same for the second group, so we have 8 degrees of freedom within our groups.

We have two groups, so we have 1 degree of freedom between our groups.  Lets arrange all that into the standard ANOVA summary table.

Sum of Squares DF Variance F P
Between 2.5 1 2.5 1.60 .24
Within 12.48 8 1.59
Total 14.98 9

Notice that between + within = total, and that is as it should be.  Now, we’d divide each SS by its relevant DF and get the variances for Within and Between, then divide between by within to get F. 

F says that the variance between the groups is 1.6 times bigger than the variance within a group. With a little Excel magic (Fdist()) we can say that would happen with a probability of .24 just by chance.   We really don’t have any evidence that the groups are different.

Lets now imagine that we had measured the groups on a separate variable, Y.  The data for Y look like…

2.18 3.55
.71 1.36
2.41 2.35
.11 2.36
-.42 4.59

The mean for the first group is 1 and the mean for the second is 2.84.   If we then do an analysis of variance we would obtain the following.

SS DF Variance F P
Between 8.5 1 8.5 5.44 .048
Within 12.5 8 1.56
Total 21 9

With Y, we have 5.44 times as much variance between the groups as we have within the groups- and that would happen with a probability of .048 by chance.  We are reasonably safe in saying that the groups are different.  They are significantly different, so to speak.

Now imagine both variables, X and Y together.

Group 1 Group 2
Scores X Y X Y
.99 2.18 1.73 3.55
.78 .71 2.11 1.36
.4 2.41 1.92 2.35
3.07 .11 3.87 2.36
-.24 -.42 .37 4.59
Means 1 1 2 2.84
Standard Deviations 1.25 1.25 1.25 1.25

Group 1 forms a cluster of points at x=1, y=1 with a standard deviation (~average distance from the mean) of 1.25.

Group 2 forms a cluster of points at x=2, y=2.84, with a standard deviation of 1.25.

You might imagine, as I did, those two point clusters as the following diagram.  Each circle is centered at a group mean and has a radius of 1.25. When I began to do that, I got excited.  Would it not be nice to be able to analyze the two clusters as a whole using ANOVA. It seemed to make sense to me.

XY Together Overlap

Why would it be nice?  Well I thought- look at the overlap when you just consider X


There is a lot of X overlap, thus it is no wonder we were unable to conclude that the groups were different when looking only at X.

Now look at the overlap in Y-


There is much less overlap in Y, thus, we were able to conclude that yes, we do have two different groups of numbers here.

But look at the amount of overlap when X and Y are considered together-


Not much overlap at all there-  if we could test the differences based on both dimensions at once, we should have a more powerful test, as there is clearly less overlap.

The hypotenuse is longer than either leg, always.

That glorious thought is what made me believe the analysis would be more powerful.  But it turns out to be the problem that I did not consider, that ultimately burst my balloon.  I had to go through the steps to get there, so I’m not letting you off the hook with the punchline- let us proceed.

How might this ANOVA be done.  Its all Pythagoras I thought. Don’t work with differences, work with distances.

Rather than compute sums of squares beginning with the square of the difference from the mean of the variables, one variable at a time, compute the sums of squares as the square of the distance from the mean of the points.

Remember Pythagoras from above…

C = sqrt( A^2 + B^2)

…where A and B are the lengths of the legs of the triangle, computed as the difference between two points.

C = sqrt( (X1-X2)^2+(Y1-Y2)^2)

One of our points will simply always be a mean, the grand mean of points, or the mean of points in each group.

The Grand mean of points would be the mean of the X coordinates and the mean of the Y coordinates, or (1.5, 1.92).

Sum of squares total would be

sum( ( x – GMx)^2 + (y – GMy)^2 )

If we do that, we get SS Total of 35.98.

Then, we’d get SS Within- where we do

sum( (x-Lmx)^2 +(y-Lmy)^2)

for each group and add those together and we’d get 24.98.

Finally, we’d do the sum of squares for Between.  Expanding our equation we’d have

sum (n1*(LmX-GMx)^2+n2*(LmY – GMy)^2)

And we’d get 11.

Now, we need to get our degrees of freedom.  Degrees of freedom tell you how many numbers are free to vary.  I have 4 numbers in my head.  That’s all I’ve told you, and given that, I can change any or all of the 4 numbers- it makes no difference to you.  But, if I tell you the numbers average 6, then I am no longer free to change the numbers willy nilly.  I can change 3 of them, but the fourth must be something specific to obtain a mean of 6.  When you describe a group of numbers, you lose a degree of freedom.

In our one-variable ANOVA we had 9 degrees of freedom total.  But now we are analyzing two variables together, so would we have 18?  9 for the X, and 9 for the Y?  The answer is no- we would still have 9.  Each XY combination describes a point, and if either X or Y change, the point is different.  So, there are not double the degrees of freedom.

Our summary table would look like the following:

SS DF Variance F P
Between 11 1 11 3.52 .097
Within 24.98 8 3.12
Total 35.98 9

There are two things that got my attention here.  The sums of squares are just the sums of squares from the X analysis added those from the Y analysis.  After looking at the formula, I had a “duh” reaction, as the formula simply add the squared deviations for X to those for Y.

But, my next reaction was the F.  The variance between the groups of points is 3.52 times as much as the variance within a group of points, and that would happen with a probability of .097 by chance-  we could not conclude that the groups are significantly different.

What gives?  The analysis of Y said the distributions were different, and the figures show that there is less overlap between the distributions when both variables are considered at once.  The F here should be bigger than in the analysis of Y alone, and the P should have been smaller.

Take the square root of the within variance in the analysis above.  It is 1.77.   What that is saying is that the standard deviation of the distances is 1.77.   The standard deviation of the differences among either X or Y alone was 1.25.

I drew the circles with a radius of 1.25, but that wasn’t right.  The average difference in the X direction was 1.25, and the average difference in the Y direction was 1.25, thus, the standard deviations describe the length of the legs of a triangle.  The radius of the circle would be the hypotenuse.

R = sqrt (1.25^2+1.25^2) = 1.77

So, the circles should have really been drawn as below.TrueMultiVariateOverlap

Now, the overlap is considerable.

How can that be I wondered- how can the standard deviation of X or Y be changing depending on the other variable?  The X and Y extents overlap considerably more than in the earlier figures.

When we only consider X, we are collapsing across Y.  When we only consider Y we are collapsing across X.  Given the circular distributions extreme values of X get less frequent with extreme values of Y, and vice versa.  So, when we collapse across either variable we are increasing the frequency of probable values more so than the frequency of improbable values.  Thus, the distributions get more peaked, and the standard deviation gets smaller.

Consider the two distributions below.

    3                    3

  2 3 4              2 3 4

1 2 3 4 5       1 2 3 4 5

Each has a standard deviation of 1.22.

Now, group them together (collapse over whatever distinguishes he two distributions) and you’d have


  2 3 4

  2  3 4

1 2 3 4 5

1 2 3 4 5

and you’d have a standard deviation of 1.18.

In summary, I think ANOVA can be extended to 2 variables by working with distances rather than differences, but you wouldn’t want to because the hypotenuse is always longer than the legs, thus the overlap in the distributions will be larger when both variables are considered together in a Euclidian combination than considered alone.

Of course, this might all be completely innacurate BS, but thats the best I can do with it.  I’m a psychologist Jim, not a statistician..

What I need for my heatmaps is some more sophisticated cluster analyses, such as the Ordering Points to Identify Clustering Structure algorithm.  I’ve neither the time nor knowledge resources at the moment to begin that, but I see it coming in the future.

Posted in Eye Tracking, Statistics | Tagged , , , , , | 14 Comments

Latent inhibition is not context specific & eye tracker demands?

With my new method, I have robust evidence that it produces latent inhibition.  The original goal of that experiment was to show that the eye-tracker can be shown to monitor visual attention in conditioning, validating it with a phenomenon where attention is widely assumed to be operating.   In addition, I sought to assess whether visual attention was restored with a context-change.  That is, latent inhibition is supposed to be context specific, and if it is an attention phenomenon, attention should be restored.

The problem I encountered was that the latent inhibition was not context specific.  This lack of effect was a problem because I have good evidence that the contexts are discriminated.

So, I ran another experiment to ensure that my latent inhibition procedure did not make the stimulus a conditioned inhibitor- and it did not.

At that point, I assumed that some of the visual aspects of the game (e.g., the sensor panel) were becoming associated with the sensors, which according to Wagner (e.g., 1981) as well as McLaren & Mackintosh (2000) would make my latent inhibition context independent.  So, I removed the sensor panel, allowing the sensors to appear only on the background, and ran the experiment again.  The result replicated- latent inhibition was not context specific.  If anything, across the two experiments latent inhibition appears especially present in the different context.

The evidence I have that the contexts are discriminated comes from an experiment on “renewal.”   There, I show that the conditioning is slightly affected by a change in context.  Extinction is faster in a different context.  And I show that extinction is highly context specific. When the extinguished CS is tested outside the context where extinction took place, robust responding is observed.

So- why is my latent inhibition context independent (in other studies I have shown it to be context specific in humans)?   

I noticed the obvious at this point, because I could see nothing else. 

The experiments on renewal were conducted without the eye tracker. 

The experiments on latent inhibition were conducted with the eye tracker.   

Could the eye-tracker be the culprit?  Could it place some demand characteristic on the participant such that context-switch effects disappear in my methods?

I am now concurrently running the latent inhibition experiment, again, but without the eye tracker and the renewal experiments, again, with the eye tracker.  

When I get those data, I’ll be getting all these experiments ready for a formal paper submission and putting them up here.

Posted in Learning Theory, Miscellaneous foreshadowing, The Learning Game | Tagged , , , | 1 Comment