Category Archives: statistics

Experimenter effects in replication efforts

I mentioned the issue of data quality before but reading Richard Morey’s interesting post about standardised effect sizes the other day made me think about this again. Yesterday I gave a lecture discussing Bem’s infamous precognition study and the meta-analysis he recently published of the replication attempts. I hadn’t looked very closely at the meta-analysis data before but for my lecture I produced the following figure:Bem-Meta

This shows the standardised effect size for each of the 90 results in that meta-analysis split into four categories. On the left in red we have the ten results by Bem himself (nine of which are his original study and one is a replication of one of them by himself). Next, in orange we have what they call ‘exact replications’ in the meta-analysis, that is, replications that used his program/materials. In blue we have ‘non-exact replications’ – those that sought to replicate the paradigms but didn’t use his materials. Finally, on the right in black we have what I called ‘different’ experiments. These are at best conceptual replications because they also test whether precognition exists but use different experiment protocols. The hexagrams denote the means across all the experiments in each category (these are non-weighted means but it’s not that important for this post).

While the means for all categories are evidently greater than zero, the most notable thing should be that Bem’s findings are dramatically different from the rest. While the mean effect size in the other categories are below or barely at 0.1 and there is considerable spread beyond zero in all of them, all ten of Bem’s results are above zero and, with one exception, above 0.1. This is certainly very unusual and there are all sorts of reasons we could discuss for why this might be…

But let’s not. Instead let’s assume for the sake of this post that there is indeed such a thing as precognition and that Daryl Bem simply knows how to get people to experience it. I doubt that this is a plausible explanation in this particular case – but I would argue that for many kinds of experiments such “experimenter effects” are probably notable. In an fMRI experiment different labs may differ considerably in how well they control participants’ head motion or even simply in terms of the image quality of the MRI scans. In psychophysical experiments different experimenters may differ in how well they explain the task to participants or how meticulous they are in ensuring that they really understood the instructions, etc. In fact, the quality of the methods surely must matter in all experiments, whether they are in astronomy, microbiology, or social priming. Now this argument has been made in many forms, most infamously perhaps in Jason Mitchell’s essay “On the emptiness of failed replications” that drew much ire from many corners. You may disagree with Mitchell on many things but not on the fact that good methods are crucial. What he gets wrong is laying the blame for failed replications solely at the feet of “replicators”. Who is to say that the original authors didn’t bungle something up?

However, it is true that all good science should seek to reduce noise from irrelevant factors to obtain as clean observations as possible of the effect of interest. Using again Bem’s precognition experiments as an example, we could hypothesise that he indeed had a way to relax participants to unlock their true precognitive potential that others seeking to replicate his findings did not. If that were true (I’m willing to bet a fair amount of money that it isn’t but that’s not the point), if true, this would indeed mean that most of the replications – failed or successful – in his meta-analysis are only of low scientific value. All of these experiments are more contaminated by noise confounds than his experiments; thus only he provides clean measurements. Standardised effect sizes like Cohen’s d divide the absolute raw effect by a measure of uncertainty or dispersion in the data. The dispersion is a direct consequence of the noise factors involved. So it should be unsurprising that the effect size is greater for experimenters that are better at eliminating unnecessary noise.

Statistical inference seeks to estimate the population effect size from a limited sample. Thus, a meta-analytic effect size is an estimate of the “true” effect size from a set of replications. But since this population effect includes the noise from all the different experimenters, it does not actually reflect the true effect? The true effect is people’s inherent precognitive ability. The meta-analytic effect size estimate is spoiling that with all the rubbish others pile on with their sloppy Psi experimentation skills. Surely we want to know the former not the latter? Again, for precognition most of us will probably agree that this is unlikely – it seems more trivially explained by some Bem-related artifact – but in many situations this is a very valid point: Imagine one researcher manages to produce a cure for some debilitating disease but others fail to replicate it. I’d bet that most people wouldn’t run around shouting “Failed replication!”, “Publication bias!”, “P-hacking!” but would want to know what makes the original experiment – the one with the working drug – different from the rest.

The way I see that, meta-analysis of large scale replications is not the right way to deal with this problem. Meta-analysis of one lab’s replications are worthwhile, especially as a way to summarise a set of conceptually related experiments – but then you need to take them with a grain of salt because they aren’t independent replications. But large-scale meta-analysis across different labs don’t really tell us all that much. They simply don’t estimate the effect size that really matters. The same applies to replication efforts (and I know I’ve said this before). This is the point on which I have always sympathised with Jason Mitchell: you cannot conclude a lot from a failed replication. A successful replication that nonetheless demonstrates that the original claim is false is another story but simply failing to replicate some effect only tells you that something is (probably) different between the original and the replication. It does not tell you what the difference is.

Sure, it’s hard to make that point when you have a large-scale project like Brian Nosek’s “Estimating the reproducibility of psychological science” (I believe this is a misnomer because they mean replicability not reproducibility – but that’s another debate). Our methods sections are supposed to allow independent replication. The fact that so few of their attempts produced significant replications is a great cause for concern. It seems doubtful that all of the original authors knew what they were doing and so few of the “replicators” did. But in my view, there are many situations where this is not the case.

I’m not necessarily saying that large-scale meta-analysis is entirely worthless but I am skeptical that we can draw many firm conclusions from it. In cases where there is reasonable doubt about differences in data quality or experimenter effects, you need to test these differences. I’ve repeatedly said that I have little patience for claims about “hidden moderators”. You can posit moderating effects all you want but they are not helpful unless you test them. The same principle applies here. Rather than publishing one big meta-analysis after another showing that some effect is probably untrue or, as Psi researchers are wont to do, in an effort to prove that precognition, presentiment, clairvoyance or whatever are real, I’d like to see more attempts to rule out these confounds.

In my opinion the only way to do this is through adversarial collaboration. If an honest skeptic can observe Bem conduct his experiments, inspect his materials, and analyse the data for themselves and yet he still manages to produce these findings, that would go a much longer way convincing me that these effects are real than any meta-analysis ever could.

Humans are dirty test tubes

 

 

What is selectivity?

TL;DR: Claiming that something is “selective” implies knowledge of the stimulus dimension to which it is tuned. It also does not apply to simple intensity codes because selectivity requires a tuning preference.

TuningVsSensitivity
Unless you can tell me where on the x-axis “pain” is relative to “rejection” or whatever else dACC may respond to, you can’t really know that this brain area behaves according to either relationship.

This post is a just little rant about an age-old pet peeve of mine: neuroscience studies claiming they found that some brain area is selective for some stimulus/task/mental state etc. This issue recently resurfaced in my mind because of an interesting exchange between Tal Yarkoni on the one hand and Matt Lieberman and Naomi Eisenberger on the other. The latter recently published a study in PNAS suggesting that dACC is “selective for pain”. Yarkoni wrote a detailed rebuttal to their claims criticising the way they inferred selectivity. I recommend following this on-going discourse (response by Lieberman & Eisenberger and a  response to their response by Yarkoni). I won’t go into any depth on it here. Rather I want to make a more general comment on why I feel the term selectivity is frequently misused.

Neuroimaging methods like fMRI allow researchers to localise brain regions that respond preferentially to particular stimuli or tasks. This “blobology” is becoming less common now that our field has matured and many experiments are more sophisticated. Nevertheless, localising brain regions that respond more to one experimental condition than another will probably remain a common sight in the neuroimaging literature for a long time to come, even if the most typical such use is probably functional localisers to limit the regions of interest in more complex experiments.

Anyway, in blobological studies, results are frequently reported as showing that your blob is “selective” for something or other: not only are we supposed to believe that the dACC is selective for pain, but also that the FFA is selective for images of faces, the LOC is selective for intact objects, and the MT+ complex is selective for motion. While some of these claims may be correct, they are not demonstrated by blobology using any kind of method. In fact, Lieberman and Eisenberger write in their response to Yarkoni:

“We’ve never seen a response to one of these papers that says they were wrong to make these claims because they didn’t test for the thousands of other things the region of interest might respond to.  Thus the weak form of selectivity, the version we were using, can be stated this way:

Selectivityweak: The dACC is selective for pain, if pain is a more reliable source of dACC activation than the other terms of interest (executive, conflict, salience).”

Perhaps there has never been a response saying that these were wrong but I think there should have. What is selectivity? While the dictionary defines “selective” as a synonym of

“discriminating, particular, discerning”

the term has long been established in neuroscience. Take the Nobel Prize winning research by Hubel and Wiesel in the 1960s for example. They discovered that neurons in the visual cortex are selective to the orientation of simple bar stimuli. So for instance a horizontal bar may drive a neuron to fire strongly while a vertical one would not. In the neurophysiology literature one would say that this particular neuron has an “preference” for horizontal bars.

This, however, does not mean that the neuron is “selective” for horizontal bars. The firing rate in response to oblique bars is likely somewhere between the rate for horizontal and vertical ones. The distinction between preference and selectivity may seem like semantic quibbling but it’s not. They are referring to very different concepts and, as I will try to argue, this distinction is important. But let’s not jump ahead.

“Strong” selectivity (to use Lieberman and Eisenberger’s terminology) implies that the neuron only responds to horizontal bars but not much else. In the following images, the contrast of each oriented grating denotes how much of a neuronal response it would produce.

High
“Strong” orientation selectivity

A “weakly” selective neuron might respond similarly across a wide range of orientations:

Low
“Weak” orientation selectivity

An even less selective neuron would show very similar responses to all orientations and a completely non-selective neuron would respond at the same rate to any visual stimulus regardless of its orientation.

None
Almost no orientation selectivity

This means we can measure responses of the neuron across the whole range of orientations (or, more generally speaking, across a range of different stimulus values). Thus we can determine not only the preferred orientation but also how strongly the exact orientation modulates neuronal responses. In other words, we can quantify how discerning, how selective the neuron is for orientation. The key point here is that in all these examples the stimulus dimension for which the neuron is selective is the orientation of bar stimuli. The neuron prefers horizontal. It is selective for orientation.

So at least according to how neurophysiologists have defined selectivity over the past half century, the term refers to how much varying the stimulus along this dimension affects the neuronal response. I hope you will agree now that this does not apply to many of the claims of selectivity in the neuroimaging literature: an experiment that shows stronger responses in FFA to faces than houses shows that FFA prefers faces. It does not show that it doesn’t also prefer other stimuli, a point already discussed by Tarkoni and Lieberman and Eisenberger. More importantly, it does not demonstrate that FFA is selective for faces.

Rather the stimulus dimension for which there may be selectivity here could be loosely categorised as “visual objects” or even just “images”. Face-selectivity implies that the region is sensitive to changing the face identity (or possibly also some other attribute specific to faces). Now, I believe there is fairly good evidence that FFA does just that – however, simply comparing the response to faces to non-face images does not and cannot possibly demonstrate this. Only comparing responses (or response patterns) to different faces can achieve that.

Why does this matter? Is this really not mere semantics? No, because from all of this follows that to demonstrate selectivity requires systematic manipulation of the stimulus space. You must map out how changing the stimulus modulates responses by a neuron or brain region or whatever. You cannot make any claim about selectivity without any concept of how different stimuli relate to each other. For orientation that is simple but for more complex objects it is not. Comparing faces to houses or body parts or animals or tools or whatnot cannot achieve that, not unless you can tell me exactly why one category (say, houses?) should be more similar to faces than another (say, cars?). There are many possible models that could relate different categories. It could be based on low-level visual similarity, semantic similarity, conceptual similarity, etc. There are studies investigating exactly that, for example by using representational similarity analysis – but a discussion of this is outside the scope of this post. The point is that simply randomly comparing different object categories, no matter how many thousands, does not by itself tell you about selectivity.

Hopefully, by now I have convinced you why the claim that the dACC is selective for pain cannot possibly be correct, at least not if it is based only on a blobological method comparing responses to an arbitrary set of stimuli. I am not even sure that selectivity for pain is even conceptually possible. It would imply that there are mental states or tasks that are just not quite pain but not really something else yet either, and that there is a systematic relationship between that and dACC responses. Perhaps this is possible, I don’t know. Either way, no fMRI or NeuroSynth or other analysis comparing pain and rejection and conflict resolution or whatever can demonstrate this.

While we’re at it, showing that responses in dACC covary with the intensity of pain would not confirm selectivity for pain either. All that this shows is that dACC is responsive to pain. Because stimulus selectivity and preference go hand in hand. Selectivity for pain would imply that this region preferentially responds to a particular pain level but less so to pain that is stronger or weaker.

I added another paragraph because of a discussion I had about this last point on social media: selectivity implies that a neuron or brain area is tuned to a stimulus space and it can only exist if there is also a stimulus preference. An intensity code merely implies that responses increase as the stimulus quantity is increased. While the steepness of this increase can vary and tells you about sensitivity, such a neuron has no stimulus preference because the response either saturates (thus losing sensitivity beyond a certain level) or increases linearly (which is probably biologically implausible). This is mechanistically different from selectivity with different consequences on how the stimulus dimension is represented and how it affects behaviour. A brain area may very well be sensitive to contrast, to pain intensity, or confidence – but unless the code allows you to infer the exact stimulus level from the response it isn’t selective.

Does correlation imply prediction?

TL;DR: Leave-one-out cross-validation is a bad way for testing the predictive power of linear correlation/regression.

Correlation or regression analysis are popular tools in neuroscience and psychology research for analysing individual differences. It fits a model (most typically a linear relationship between two measures) to infer whether the variability in some measure is related to the variability in another measure. Revealing such relationships can help understand the underlying mechanisms. We and others used it in previous studies to test specific mechanistic hypotheses linking brain structure/function and behaviour. It also forms the backbone of twin studies of heritability that in turn can implicate genetic and experiential factors in some trait. Most importantly, in my personal view individual differences are interesting because they acknowledge the fact that every human being is unique rather than simply treating variability as noise and averaging across large groups people.

But typically every report of a correlational finding will be followed by someone zealously pointing out that “Correlation does not imply causation”. And doubtless it is very important to keep that in mind. A statistical association between two variables may simply reflect the fact that they are both related to a third, unknown factor or a correlation may just be a fluke.

Another problem is that the titles of studies using correlation analysis sometimes use what I like to call “smooth narrative” style. Saying that some behaviour is “predicted by” or  “depends on” some brain measure makes for far more accessible and interesting reading that dryly talking about statistical correlations. However, it doesn’t sit well with a lot of people, in part because such language may imply a causal link that the results don’t actually support. Jack Gallant seems to regularly point out on Twitter that the term “prediction” should only ever be used when a predictive model is built on some data set but the validity is tested on an independent data set.

Recently I came across an interesting PubPeer thread debating this question. In this one commenter pointed out that the title of the study under discussion, “V1 surface size predicts GABA concentration“, was unjustified because this relationship explains only about 7% of the variance when using a leave-one-out cross-validation procedure. In this procedure all data points except one are used to fit the regression and the final point is then used to evaluate the fit of the model. This procedure is then repeated n-fold using every data point as evaluation data once.

Taken at face value this approach sounds very appealing because it uses independent data for making predictions and for testing them. Replication is a cornerstone of science and in some respects cross-validation is an internal replication. So surely this is a great idea? Naive as I am I have long had a strong affinity for this idea.

Cross-validation underestimates predictive power

But not so fast. These notions fail to address two important issues (both of which some commenters on that thread already pointed out): first, it is unclear what amount of variance a model should explain to be important. 7% is not very much but it can nevertheless be of substantial theoretical value. The amount of variance that can realistically be explained by any model is limited by the noise in the data that arises from measurement error or other distortions. So in fact many studies using cross-validation to estimate the variance explained by some models (often in the context of model comparison) instead report the amount of explainable variance accounted for by the model. To derive this one must first estimate the noise ceiling, that is, the realistic maximum of variance that can possibly be explained. This depends on the univariate variability of the measures themselves.

Second, the cross-validation approach is based on the assumption that the observed sample, which is then subdivided into model-fitting and evaluation sets, is a good representation of the population parameters the analysis is attempting to infer. As such, the cross-validation estimate also comes with an error (this issue is also discussed by another blog post mentioned in that discussion thread). What we are usually interested in when we conduct scientific studies is to make an inference about the whole population, say a conclusion that can be broadly generalised to any human brain, not just the handful of undergraduate students included in our experiments. This does not really fit the logic of cross-validation because the evaluation is by definition only based on the same sample we collected.

Because I am a filthy, theory-challenged experimentalist, I decided to simulate this (and I apologise to all my Bayesian friends for yet again conditioning on the truth here…). For a range of sample sizes between n=3 and n=300 I drew a sample with from a population with a fixed correlation of rho=0.7 and performed leave-one-out cross-validation to quantify the variance explained by it (using the squared correlation between predicted and observed values). I also performed a standard regression analysis and quantified the variance explained by that. At each sample size I did this 1000 times and then calculated the mean variance explained for each approach. Here are the results:

rho=7

What is immediately clear is that the results strongly depend on the sample size. Let’s begin with the blue line. This represents the variance explained by the standard regression analysis on the whole observed sample. The dotted, black, horizontal line denotes the true effect size, that is, the variance explained by the population correlation (so R^2=49%). The blue line starts off well above the true effect but then converges on it. This means that at small sample sizes, especially below n=10, the observed sample inflates how much variance is explained by the fitted model.

Next look at the red line. This denotes the variance explained by the leave-one-out cross-validation procedure. This also starts off above the true population effect and follows the decline of the observed correlation. But then it actually undershoots and goes well below the true effect size. Only then it gradually increases again and converges on the true effect. So at sample sizes that are most realistic in individual differences research, n=20-100ish, this cross-validation approach underestimates how much variance a regression model can explain and thus in fact undervalues the predictive power of the model.

The error bars in this plot denote +/- 1 standard deviation across the simulations at each sample size. So as one would expect, the variability across simulations is considerable when sample size is small, especially when n <= 10. These sample sizes are maybe unusually small but certainly not unrealistically small. I have seen publications calculating correlations on such small samples. The good news here is that even with such small samples on average the effect may not be inflated massively (let’s assume for the moment that publication bias or p-hacking etc are not an issue). However, cross-validation is not reliable under these conditions.

A correlation of rho=0.7 is unusually strong for most research. So I repeated this simulation analysis using a perhaps more realistic effect size of rho=0.3. Here is the plot:rho=3

Now we see a hint of something fascinating: the variance explained by the cross-validation approach actually subtly exceeds that of the observed sample correlation. They again converge on the true population level of 9% when the sample size reaches n=50. Actually there is again an undershoot but it is negligible. But at least for small samples with n <= 10 the cross-validation certainly doesn’t perform any better than the observed correlation. Both massively overestimate the effect size.

When the null hypothesis is true…

So if this is what is happening at a reasonably realistic rho=0.3, what about when the null hypothesis is true? This is what is shown in here (I apologise for the error bars extending into the impossible negative range but I’m too lazy to add that contingency to the code…):

rho=0

The problem we saw hinted at above for rho=0.3 is exacerbated here. As before, the variance explained for the observed sample correlation is considerably inflated when sample size is small. However, for the cross-validated result this situation is much worse. Even at a sample size of n=300 the variance explained by the cross-validation is greater than 10%. If you read the PubPeer discussion I mentioned, you’ll see that I discussed this issue. This result occurs because when the null hypothesis is true – or the true effect is very weak – the cross-validation will produce significant correlations between the inadequately fitted model predictions and the actual observed values. These correlations can be positive or  negative (that is, the predictions systematically go in the wrong direction) but because the variance explained is calculated by squaring the correlation coefficient they turn into numbers substantially greater than 0%.

As I discussed in that thread, there is another way to calculate the variance explained by the cross-validation. I won’t go into detail on this but unlike the simpler approach I employed here this does not limit the variance explained to fall between 0-100%. While the estimates are numerically different, the pattern of results is qualitatively the same. At smaller sample sizes the variance explained by cross-validation systematically underestimates the true variance explained.

When the interocular traumatic test is significant…

My last example is the opposite scenario. While we already looked at an unusually strong correlation, I decided to also simulate a case where the effect should be blatantly obvious. Here rho=0.9:

rho=9

Unsurprisingly, the results are similar as those seen for rho=0.7 but now the observed correlation is already doing a pretty decent job at reaching the nominal level of 81% variance explained. Still, the cross-validation underperforms at small sample sizes. In this situation, this actually seems to be a problem. It is rare that one would observe a correlation of this magnitude in psychological or biological sciences but if so chances are good that the sample size is small in that case. Often the reason for this may be that correlation estimates are inflated at small sample sizes but that’s not the point here. The point is that leave-one-out cross-validation won’t tell you. It underestimates the association even if it is real.

Where does all this leave us?

It is not my intention to rule out cross-validation. It can be a good approach for testing models and is often used successfully in the context of model comparison or classification analysis. In fact, as the debate about circular inference in neuroscience a few years ago illustrated, there are situations where it is essential that independent data are used. Cross-validation is a great way to deal with overfitting. Just don’t let yourself be misled into believing it can tell you something it doesn’t. I know it is superficially appealing and I had played with it previously for just that reason – but this exercise has convinced me that it’s not as bullet-proof is one might think.

Obviously, validation of a model with independent data is a great idea. A good approach is to collect a whole independent replication sample but this is expensive and may not always be feasible. Also, if a direct replication is performed it seems better that this is acquired independently by different researchers. A collaborative project could do this in which each group uses the data acquired by the other group to test their predictive model. But that again is not something that is likely to become regular practice anytime soon.

In the meantime we can also remember that performing typical statistical inference is a good approach after all. Its whole point is to infer the properties of the whole population from a sample. When used properly it tends to do a good job at that. Obviously, we should take measures to improve its validity, such as increasing power by using larger samples and/or better measurements. I know I am baysed but Bayesian hypothesis tests seem superior at ensuring validity than traditional significance testing. Registered Reports can probably also help and certainly should reduce the skew by publication bias and flexible analyses.

Wrapping up

So, does correlation imply prediction? I think so. Statistically this is precisely what it does. It uses one measure (or multiple measures) to make predictions of another measure. The key point is not whether calling it a prediction is valid but whether the prediction is sufficiently accurate to be important. The answer to this question actually depends considerably on what we are trying to do. A correlation explaining 10-20% of the variance in a small sample is not going to be a clear biomarker for anything. I sure as hell wouldn’t want any medical or judicial decisions to be based solely on such an association. But it may very well be very informative about mechanisms. It is a clearly detectable effect even with the naked eye.

In the context of these analysis, a better way than quantifying the variance explained is to calculate the root mean squared deviation (essentially the error bar) of the prediction. This provides an actually much more direct index of how accurately one variable predicts another. The next step – and I know I sound like a broken record – should be to confirm that these effects are actually scientifically plausible. This mantra is true for individual differences research as much as it is for Bem’s precognition and social priming experiments where I mentioned it before. Are the differences in neural transmission speed or neurotransmitter concentration implied by these correlation results realistic based on what we know about the brain?  These are the kinds of predictions we should actually care about in these discussions.

 

Visualising group data

Recently I have been thinking a bit about what the best way is to represent group data. The most typical way this is done is by showing summary statistics (usually the mean) and error bars (usually standard errors) either in bar plots or in plots with lines and symbols. A lot of people seem to think this is not an appropriate way to visualise results because it obscures the data distribution and also whether outliers may influence the results. One reason prompting me to think about this is that in at least one of our MSc courses students are explicitly told by course tutors that they should be plotting individual subject data. It is certainly true that close inspection of your data is always important – but I am not convinced that it is the only and best way to represent all sorts of data. In particular, looking at the results from an experiment of a recent student of mine you wouldn’t make heads or tails from just plotting individual data. Part of the reason is that most of the studies we do use within-subject designs and standard ways of plotting individual data points can actually be misleading. There are probably better ones, and perhaps my next post will deal with that.

For now though I want to only consider group data which were actually derived from between-subject or at least mixed designs. A recently published study in Psychological Science reported that sad people are worse at discriminating colours along the blue-yellow colour axis but not along the red-green colour axis. This sparked a lot of discussion on Twitter and in the blogosphere, for example this post by Andrew Gelman and also this one by Daniel Lakeland. Publications like this tend to attract a lot of coverage by mainstream media and this was no exception. This then further fuels the rage of skeptical researchers :P. There are a lot of things to debate here, from the fact that the study authors interpret a difference between differences as significant without testing the interaction, the potential inadequacy of the general procedure for measuring perceptual differences (raw accuracy rather than a visual threshold measure), and also that outliers may contribute to the main result. I won’t go into this discussion but I thought this data set (which to the authors’ credit is publicly available) would be a good example for my musings.

So here I am representing the data from their first study by plotting it in four different ways. The first plot, in the upper left, is a bar plot showing the means and standard errors for the different experimental conditions. The main result in the article is that the difference between control and sadness is significant for discriminating colours along the blue-yellow axis (the two bars on the left).

Slide1

And judging by the bar graph you could certainly be forgiven for thinking so (I am using the same truncated scale used in the original article). The error bars seem reasonably well separated and this comparison is in fact statistically significant at p=0.0147 on a parametric independent sample t-test or p=0.0096 on a Mann-Whitney U-test (let’s ignore the issue of the interaction for this example).

Now consider the plot in the upper right though. Here we have the individual data points for the different groups and conditions. To give an impression of how the data are distributed, I added a little Gaussian noise to the x-position of each point. The data are evidently quite discrete due to the relatively small number of trials used to calculate the accuracy for every subject. Looking at the data in this way does not seem to give a very clear impression that there is a substantial difference between the control and sadness groups in either colour condition. The most noticeable difference is that there is one subject in the sadness group whose accuracy is not matched with any counterpart in the control group, at 0.58 accuracy. Is this an outlier pulling the result?

Next I generated a box-and-whisker plot in the lower left panel. The boxes in these plots denote the inter-quartile range (IQR, i.e. between 25th and 75th percentile of the data), the red lines indicate the medians, the error bars denote a range of 1.5 times the IQR beyond the percentiles (although it is curtailed when there are no data points beyond that range as by the ceiling at 1), and the red crosses are outliers that fall outside this range. The triangular notches surrounding the medians are a way to represent uncertainty and if they do not overlap (as is the case for the blue-yellow data) this suggests a difference between medians at the 5% significance level. Clearly the data point at 0.58 accuracy in the sadness group is considered an outlier in this plot although it is not the only one.

Finally, I also wrote a Matlab function to create cat-eye plots (Wikipedia calls those violin plots – personally they look mostly like bottles, amphoras or vases to me – or, in this case, like balloons). This is shown in the lower right panel. These plots show the distribution of the data in each condition smoothed by a kernel density function. The filled circles indicate the median, the vertical lines the inter-quartile range, and the asterisk the mean. Plots like this seem to be becoming more popular lately. They do have the nice feature that they give a fairly direct impression of how the data are distributed. It seems fairly clear that these are not normal distributions, which probably has largely to do with the ceiling effect: as accuracy cannot be higher than 1 the distributions are truncated there. The critical data set, the blue-yellow discrimination for the sadness group, has a fairly thick tail towards the bottom which is at least partially due to that outlier. This all suggests that the traditional t-test was inappropriate here but then again we did see a significant difference on the U-test. And certainly, visual inspection still suggests that there may be a difference here.

Next I decided to see what happens if I remove this outlier at 0.58. For consistency, I also removed their data from the red-green data set. This change does not alter the statistical inference in a qualitative way even though the p-values increase slightly. The t-test is still significant at p=0.0259 and the U-test at p=0.014.

Slide2

Again, the bar graph shows a fairly noticeable difference. The scatter plot of the individual data points on the other hand now hardly seems to show any difference. Both the whisker and the cat-eye plot seem to still show qualitatively similar results as when the outlier is included. There seems to be a difference in medians for the blue-yellow data set. The cat-eye plot makes is more apparent that the tail of the distribution for the sadness group is quite heavy something that isn’t that clear in the whisker plot.

Finally, I decided to simulate a new data set with a similar pattern of results but in which I knew the ground truth. All four data sets contained 50 data points that were chosen from a Gaussian distribution with mean of 70 and standard deviation of 10 (I am a moron and therefore generated these on a scale of percent rather than proportion correct – and now I’m too lazy to replot all this just to correct it. It doesn’t matter really). For the control group in the blue-yellow condition I added an offset of 5 while in the sadness group I subtracted 5. This means that there is a significant difference (t-test: p=0.0017; U-test: 0.0042).

Slide3

Now all four types of plot fairly clearly reflect this difference between control and sadness groups. The bar graph in particular clearly reflects the true population means in each group. But even in the scatter plot the difference is clearly apparent even though the distributions overlap considerably. The difference seems a lot less obvious in the whisker and cat-eye plots however. The notches in the whisker plot do not overlap although they seem to be very close. The difference seems to be more visually striking for the cat-eye plot but it isn’t immediately apparent from the plot how much confidence this should instill in this result.

Conclusions & Confusions

My preliminary conclusion is that all of this is actually more confusing than I thought. I am inclined to agree that the bar graph (or a similar symbol and error bar plot) seems to overstate the strength of the evidence somewhat (although one should note that this is partly because of the truncated y-scales that such plots usually employ). On the other hand, showing the individual subject data does seem to understate the results considerably except when the effect is pretty strong. So perhaps things like whisker or cat-eye (violin/bottle/balloon) plots are the most suitable but in my view they also aren’t as intuitive as some people seem to suggest. Obviously, I am not the first person who has thought about these things nor have I spent an extraordinarily long time thinking about it. It might be useful to conduct a experiment/survey in which people have to judge the strength of effects based on different kinds of plot. Anyway, in general I would be very curious to hear other people’s thoughts.

The Matlab code and data file for these examples can be found here.