Bayesian Correction for Attenuation of Correlation in Multi-Trial Spike Count Data

Abstract

When correlation is measured in the presence of noise, its value is decreased. In single-neuron recording experiments, for example, the correlation of selectivity indices in a pair of tasks may be assessed across neurons, but, because the number of trials is limited, the measured index values for each neuron will be noisy. This attenuates the correlation. A correction for such attenuation was proposed by Spearman more than 100 yr ago, and more recent work has shown how confidence intervals may be constructed to supplement the correction. In this paper, we propose an alternative Bayesian correction. A simulation study shows that this approach can be far superior to Spearman's, both in accuracy of the correction and in coverage of the resulting confidence intervals. We demonstrate the usefulness of this technology by applying it to a set of data obtained from the frontal cortex of a macaque monkey while performing serial order and variable reward saccade tasks. There the correction results in a substantial increase in the correlation across neurons in the two tasks.

INTRODUCTION

A central theme in the statistical analysis of neuronal data is the appropriate accounting for uncertainty. This often involves the inclusion of sources of variability that might otherwise be omitted. Although in some cases taking into account additional sources of variability may decrease the magnitude of an effect (Behseta et al. 2005), in other cases, the effect of interest may actually increase. An important example of this second situation involves estimation of correlation in the presence of noise. Suppose θ and ξ are random variables having a positive correlation ρ_θξ and ɛ and δ are independent “noise” random variables that corrupt the measurement of θ and ξ producing X = θ + ɛ and Y = ξ + δ. A simple mathematical argument (see appendix) shows that

where ρ_XY is the correlation between X and Y. In words, the presence of noise decreases the correlation. Thus in many circumstances, a measured correlation will underestimate the strength of the actual correlation between two variables. However, if the likely magnitude of the noise is known, it becomes possible to correct the estimate. The purpose of this note is to provide a Bayesian correction and to show that it can produce good results when examining correlations derived from multi-trial spike counts.

We apply the method in the context of single-neuronal recording experiments. Broadly speaking, it is sometimes necessary to compare the selectivity of a neuron for a particular variable across two task contexts. For example, one might wish to compare shape selectivity across blocks of trials in which the shape has different colors (Edwards et al. 2003) or compare selectivity for saccade direction across blocks of trials in which the saccade is selected according to different rules (Olson et al. 2000). It is also sometimes necessary to compare selectivity for two different variables as measured in separate task contexts. For example, we might wish to compare selectivity for the direction of motion of a visual stimulus viewed during passive fixation with selectivity for saccade direction in a task requiring eye movements (Horwitz and Newsome 2001). The standard approach to making such comparisons is to compute, for multiple neurons, index 1 in context 1 and index 2 in context 2 and then to compute the correlation between the two indices across neurons. The correlation may be statistically significant but smaller than one might expect, which raises the question: is the small correlation due to a genuine discordance between the two forms of selectivity, or is it due to noise arising from random trial-to-trial variability in the neuronal firing rates? This is the kind of question the methods of this article are designed to answer.

The idea of introducing a “correction for attenuation” of the correlation goes back at least to Spearman (1904). He did not at that time, however, have the technology to provide confidence intervals associated with his proposed technique. Frost and Thompson (2000) reviewed some solutions to the problem of constructing confidence intervals for the slope of a noise-corrupted regression line, and Charles (2005) gave procedures for obtaining confidence intervals for the correlation based on Spearman's formula. We performed a computer simulation study to compare the Bayesian correction with Spearman's correction and the Bayesian confidence intervals with those based on Spearman's correction. We found the Bayesian method to be far superior. We then applied the method to data from the frontal cortex of a macaque monkey recorded while the monkey was performing serial order and variable reward saccade tasks.

METHODS

Notation

Let X_i = θ_i + ɛ_i, and Y_i = ξ_i + δ_i, where θ_i and ξ_i represent the underlying true values, and ɛ_i and δ_i are the associated error values (representing noise) for the ith observation, for i = 1, 2,…, n. Let represent the variance for X_i and Y_i respectively. To construct the Bayesian model (see following text), we assume that ɛ_i ∼ N(0,σ_ɛi²), and δ_i ∼ N(0,σ_δi²). In neurophysiological applications, may be estimated from repeated trials and then treated as known, separately for each neuron, as we discuss in data analysis and also in the appendix. This is referred to as inhomogeneous variances. Otherwise, in the case of homogeneous variances, since we will use the notation Finally, we let and σ_ξ² be the means and the variances of θ_i and ξ_i, respectively.

Spearman's method

Spearman (1904) tackled the problem of correcting the attenuation of the correlation coefficient through a series of intuitive steps (see Charles 2005). Spearman's formula for attenuation correction is given by

(1)

where ρ_θξ is the corrected correlation, ρ_XY is the correlation between X and Y, and ρ_XX, and ρ_YY are known as reliabilities, defined as The derivation of Eq. 1 is given in the appendix.

In practice, we estimate (1) with the sample corrected correlation as follows

(2)

where r_XY is the usual (Pearson) correlation given by in which x̄ and ȳ are the sample means, and s_x and s_y are the sample SDs of X and Y. Also According to Spearman (1904), to calculate in Eq. 2, one would need repeated measurements of X_i and Y_i for each i = 1,…, n. This is important because in some applications, such as in data analysis in the following text, the quantities X_i and Y_i are indices derived from complete sets of trials; repeated measurements are not available.

It is possible for to be >1. Spearman was aware of this, but the method was received quite unfavorably by Karl Pearson (1904), who firmly believed that correlation formulas that yield values larger than one are essentially faulty. The Bayesian method we introduce in the following text does not suffer from this defect.

Confidence intervals for Spearman's method

The most commonly applied confidence interval for ρ_θξ is as follows (Charles 2005; Hunter and Schmidt 1990; Winne and Belfrey 1982)

(3)

in which by L and U, we refer to the lower and upper bounds of the confidence interval for ρ_xy. To calculate L and U in Eq. 3, one can use the asymptotic confidence interval of the so-called Fisher's z transformation (Fisher 1924). In general, if ρ and r are the theoretical and the sample correlations, respectively, then it can be shown that converges in distribution to a normally distributed random variable with mean 0 and variance (1 − ρ²)², and that the asymptotic variance of which does not depend on r (see page 52 in DasGupta 2008). Consequently, the lower and the upper bounds of a confidence interval for are

where z_(1−α/2) is the 100(1 − α/2)-th percentile of the standard normal distribution. Using the inverse transformation to restate the confidence bounds in terms of r we obtain

Specifically, for data (x₁, y₁),…, (x_n, x_n), putting L and U defined in the preceding text in Eq. 3 provides the usual confidence interval based on Spearman's method. The values of L_z and U_z based on Fisher's z-transformation can be obtained using standard software. For example, to calculate L and U in Matlab, one can use the command corrcoef. The command cor.test in the statistical software R will also produce L and U.

Measurement error methods

The problem of correlation attenuation due to noise can be viewed from a simple linear regression perspective. For a thorough review, see the books by Carroll et al. (2006), and Fuller (1987), and the article by Frost and Thompson (2000). Consider the linear regression model

for i = 1,…, n, with the familiar assumption Suppose that instead of observing values for the predictor θ, we record noisy values where ɛ_i is the measurement error for the ith observation. Consequently, we fit

Note that unlike previous sections, measurement error is only assumed for the predictor X.

The problem may be formulated further with two assumptions: and where σ_θ² and σ_ɛ² represent the so-called between objects and within objects uncertainty, respectively. As shown in the appendix, one can estimate the true slope of the regression line β*, via

where β̂ is the least-squares estimator of β, and is the reliability or as introduced in the previous text. Note that similar to Spearman's technique, the regression-based model assumes homogeneous variances. As shown in the appendix, when the measurement error is also considered for the response variable, the regression-based method and Spearman's approach will produce the same correction for attenuation. As with Spearman's technology, regression-based techniques are constructed under the assumption that there are repeated measurements in each (Y_i,X_i), for i = 1,…, n.

Bayesian method

To address the problem of correlation attenuation, we form a random-effects model, also called a hierarchical model. Following the notation in the previous text, let X_i = θ_i + ɛ_i and Y_i = ξ_i + δ_i, where θ_i and ξ_i represent the underlying true values, and ɛ_i and δ_i are the associated error values for the ith observation, i = 1, 2,…, n. In the first stage of the model, we have: where σ_ɛi, and σ_δi are estimated for repeated trials and then treated as known for each i = 1,…, n.

In the second stage of the model, we join to the distributional assumption for the pairs (X_i, Y_i), the additional specification

in which

where be the means and the variances of θ_i and ξ_i respectively.

The quantity we seek is Because we have formulated the model in this way, and because the uncertainty of measurements of interest is explicitly incorporated into the estimate of ρ_θξ will be adjusted for those sources of variation. We employ Bayesian methods (Casella and George 1992; Gelman et al. 1995; Wilks et al. 1995) to estimate ρ_θξ via its posterior distribution. We use conjugate priors for μ and Σ. This will facilitate the application of conventional software (such as WinBugs).

We put independent univariate normal priors on In both cases, we use diffuse Gaussian priors with mean 0 and a large variance, σ². An Inverted-Wishart prior with the scale matrix R, and degrees of freedom d with the density is used for the covariance matrix Σ. To avoid injecting substantial information, we use a very diffuse prior on Σ. Following Kass and Natarajan (2006), we set d = 2 df, and we take the scale matrix R to be the harmonic mean of the observed covariance matrices where S_i is a 2 × 2 observed covariance matrix associated with neuron i, obtained from trial-to-trial variation in the spike counts.

By sampling from the posterior distribution of Σ, one can form a confidence interval (often called a Bayesian “credible interval”) for the correlation coefficient ρ. Thus a confidence interval for ρ having coverage probability approximately 1 − α is

where ρ, and ρ¹⁻ are the th and the1−th quantiles of the sample from the posterior distribution of ρ.

The hierarchical model used here has the property that the resulting posterior means for (θ_i,ξ_i) are weighted combinations of the observed values (X_i,Y_i) and their weighted means (with weights defined in terms of the relative values of the variances, so that more precisely measured observations have greater weight). This is usually referred to as “shrinkage” because the spread of the values of posterior means for (θ_i,ξ_i) will be smaller than the spread of the data (X_i,Y_i). This phenomenon is discussed in Bayesian texts, such as Gelman et al. (1995) and will be illustrated in Fig. 1.

FIG. 1.

Plots of reward- vs. rank-selective indices, before and after Bayesian correction. Top: uncorrected indices. The x axis represents the index value for the the serial order saccade task. This is obtained through I_rank = (f₃ − f₁)/(f₃ + f₁), where f₁ and f₃ were the mean firing rates measured at the times of the 1st and 3rd saccades, respectively. The y axis indicates the index of selectivity for the size of the anticipated reward: I_reward = (f_b − f_s)/(f_b + f_z) where f_b and f_s were the firing rates during the postcue delay period on big-reward and small-reward trials, respectively. Bottom: plot of posterior means representing values after correction for noise.

RESULTS

Data analysis

We demonstrate the effect of the Bayesian correction of the attenuation using data from an experiment we carried out recently. The aim of the experiment was to characterize the neural mechanisms that underlie serial order performance. We trained monkeys to perform eye movements in a given order signaled by a cue. For example, one cue carried the instruction: look up, then right, then left. Monitoring neuronal activity in frontal cortex during task performance, we found that many neurons fire at different rates during different stages of the task, with some firing at the highest rate during the first, some during the second, and some during the third stage. We refer to this property as rank selectivity. Rank-selective neurons might genuinely be sensitive to the monkey's stage in the sequence. Alternatively, they might be sensitive to some correlated factor. One such factor is expectation of reward. Reward (a drop of juice) was delivered only after all three movements had been completed. Thus as the stage of the trial progressed from one to three, the expectation of reward might have increased.

To test whether rank-selective neurons were sensitive to the size of the anticipated reward, we trained the same monkeys to perform a task in which a visual cue presented at the beginning of the trial signaled to the monkey whether he would receive one drop or three drops of juice after a fixed interval. We reasoned that neuronal activity related to expectation of reward would be greater after the promise of three drops than after the promise of one. We monitored the activity of 54 neurons during the performance of both the serial order task and the variable reward task. We then computed an index of rank selectivity in the serial-order task and of selectivity for the size of the anticipated reward in the variable reward task for each neuron. The rank index was where f₁ and f₃ were the mean firing rates measured at the times of the first and third saccades, respectively. To obtain the rank index for a given neuron, we collected data from ∼25 trials for each of the nine possible rank-direction combinations (3 directions of eye movements: rightward, upward, or leftward; and 3 ranks: 1st, 2nd, or 3rd). The reward index was where f_b and f_s were the firing rates during the post cue delay period on big- and small-reward trials, respectively. To obtain reward index for a given neuron, we collected data from ∼20 trials for each of the six possible reward-direction combinations (3 directions of eye movements: rightward, upward or leftward; and 2 sizes of anticipated reward: 1 or 3 drops of juice).

On carrying out a conventional correlation analysis, we found that the two measures were positively correlated but that the effect was not strong (r = 0.49). We speculated that the low correlation was in part due to trial-to-trial uncertainty of neuronal firing rate that would affect the correlation even when all controllable factors (such as rank and the size of the anticipated reward) are held constant. We were surprised at the low degree of the measured correlation between the rank and reward indices because we knew that the expectation of reward increases over the course of a serial-order trial and that neuronal activity in the SEF is affected by the expectation of reward (Roesch and Olson 2003). We wondered if our estimate of the correlation between the two indices had been attenuated by noise arising from random trial-to-trial variations in neuronal activity.

To implement the proposed Bayesian technology, we let I_reward and I_rank play the role of neurons as in the preceding text. Consequently, we would need to estimate and the variances of respectively. The following relationship may be used to calculate per neuron

(4)

where μ_f1, τ_f1², μ_f2, and τ_f2² are the means and the variances of f₁ and f₂ in I = (f₁ − f₂)/(f₁ + f₂), respectively. A similar formulation can be used to estimate σ_δ². The derivation of Eq. 4 is given in the appendix. Also considering the magnitude of X, and Y values, we let σ² = 80 for the variance of normal priors on μ_θ and μ_ξ.

In Fig. 1, we demonstrate the dramatic effect of the Bayesian attenuation correction for 54 recorded neurons. On the top, we give the scatterplot for the pairs of indices prior to the application of the Bayesian method. The x and y axes are represented by I_rank and I_reward, respectively. The bottom part contains the scatterplot for the posterior estimation of the same set of indices. The correlation for the attenuated data are 0.49. The corrected correlation using the Bayesian method increased dramatically to 0.83 with the confidence interval (0.77,0.88). For this data, Spearman's method gave an estimate of 0.85 but the 95% confidence interval was (0.65,0.99). Not only is the Bayesian interval much smaller, but from our results of the simulation study, we expect it to have a probability coverage much closer to 0.95.

Simulation study

METHODS.

To investigate the accuracy of the proposed Bayesian method, we performed simulation studies repeating the data structure of the previous section under three scenarios. In scenario 1 (Fig. 2), we considered the truth to be a realization of a bivariate normal distribution with

As a result of bivariate normality, the Bayesian method assumes a linear relationship between θ and ξ. However, one might not expect much distortion from this assumption. Correlation is itself a measure of linear association, so a method based on linearity might be expected to behave reasonably well even when the underlying relationship is nonlinear. We used scenario 2 as an initial check on this intuition. Scenario 2 is shown in Fig. 3. In scenario 2, we assumed that the true data points were built around the quadratic equation y = 1.1x₂ + 0.6x − 0.004, evaluated at x ∈ (−0.6, 0.6), after adding random noise N(0,1) to each point.

FIG. 2.

Simulation scenario 1. True indices are obtained by simulating from a bivariate normal distribution.

FIG. 3.

Simulation scenario 2. Data points are constructed around the quadratic equation y = 1.1x² + 0.gx − 0.004, evaluated at x ∈ (−0.6,0.6), after adding random noise N(0,1) to each point.

As may be seen in Fig. 3, scenario 2 provides a strikingly nonlinear relationship between θ and ξ, clearly violating the linearity assumption imbedded in the Bayesian method. Finally, in scenario 3, we took the true values to be the data shown in Fig. 1, bottom, that is, the data obtained from implementing the Bayesian procedure.

The general algorithm for simulating neuronal data in all scenarios can be explained through the following steps: 1) let the bivariate data in Figs. 2 and 3 and Fig. 1, bottom, represent true values for I_rank and I_reward in each scenario. 2) Considering that manufacture true values for λ₁, λ₂, λ₃, λ₄, associated with each pair of true (I_rank, I_reward). 3) Knowing λs, randomly simulate 1,000 sets of 200 Poisson counts: 200 = 50 (neurons) × 4 (λs). 4) For each randomly generated dataset, calculate the associated simulation values of I_rank and I_reward . 5) Implement both the Bayesian technology and Spearman's approach on the simulated data. Assess the goodness of each technology.

To implement step 2, we generated a pair of Poisson intensities, one for each condition. Because we began the simulation process by setting true values for I, we would need to calculate true λ values by applying a reverse procedure. Due to the fact that we manufactured a pair of such that they match the equation for I. For example, when I = 0.08, we may take and when −0.036, we let (see the supplementary material for an extended example.)

In each scenario, to imitate the multi-trial nature of neuronal recordings, we simulated Poisson data with n = 15,30,60,120, where n is the number of trials. For step 3, we generated Poisson counts using the following general idea. Let λ be the true firing rate for a given experimental condition and suppose that interest lies in creating multiple trials of spike occurrences over the span of 1 s. Let n be the number of trials. Then nλ is the expected value of the Poisson counts for that condition.

After simulating data for each scenario, we kept track of the mean square error as well as the SE of the point estimator using the usual correlation (r_XY), the corrected r_XY or r_θξ as shown in formula 1, and the proposed Bayesian method. Additionally, we calculated the coverage of the true correlation in the simulated confidence intervals, using: the z-transformed uncorrected correlation coefficient, the corrected z method as presented in the previous text, and Our proposed Bayesian method.

To calculate the coverage we use

where M is the number of simulated data sets. To calculate the mean squared error, we set MSE = E(ρ̂ − ρ)², where ρ̂ is the point estimator for ρ. For example, in the Bayesian case, since the posterior draws of ρ were fairly symmetric, the posterior mode and the posterior mean were extremely close. Consequently we used the posterior mode as the point estimator. Note that MSE is generally approximately proportional to 1/n. Therefore the practical implication is that if one method reduces MSE by a factor of say 2, this is roughly equivalent to doubling the sample size.

Simulation results are summarized in Tables 1–6. Each table contains the results for the z-transformed method, the corrected z method, and the Bayesian method. The method that produces smaller mean square error is more desirable. Confidence intervals were constructed to have ∼0.95 probability of coverage. Therefore good performance of each method would be indicated by coverage probability close to 0.95.

TABLE 1.

MSE for simulations under scenario 1 (normal data)

	rXY	Spearman	Bayesian
n = 15	0.036	0.025	0.0037
n = 30	0.015	0.01	0.0016
n = 60	0.0061	0.0051	0.00091
n = 120	0.0026	0.0025	0.00049

For example, the MSE of r_XY when n = 15 was 0.036. In every case, the Bayesian estimator has much smaller MSE than Spearman's. MSE, mean squared error.

TABLE 6.

Simulations under scenario 3 (based on the SEF data)

	z	Corrected z	Bayesian
n = 15	0.012	0.91	0.93
n = 30	0.07	0.92	0.94
n = 60	0.39	0.96	0.95
n = 120	0.68	0.96	0.95

Entries as in Table 2. In all cases, the Bayesian technique provides accurate coverage probability with much less varibility across datasets.

In each scenario, 25,000 sets of Poisson data were simulated. The results of the mean squared error calculations are given in Table 1. They indicate that the hierarchical Bayesian method yields a huge reduction in mean squared error compared with both of the r_XY and r_θξ, the corrected r_XY method. An explanation for the improvement of the hierarchical method over the other two techniques is that the Bayesian approach, because of its multi-level structure, allows for proper accounting for uncertainty due to error. To give an example, the estimated mean squared error achieved by the Bayesian method at n = 30 is about half of the mean squared error of the other two methods at n = 120. In other words, use of the Bayesian method is roughly equivalent to eightfold increase in sample size with Spearman's method. Further calculations (not included in Tables 1–6) revealed results: mean square error of the Bayesian estimator at n = 15 is nearly equal to the mean squared error at n = 90 when the corrected r_XY method is performed; both in terms of the mean square error and the SE, the Bayesian method at n = 45, outperformed Spearman's technique even at a considerably large sample size of n = 120; and, for smaller n (n = 15,30), the mean squared error of the Bayesian technique was 10 times smaller than the mean squared error of the Spearman's method.

As shown in Table 2, both transformed-z, and the corrected technique tend to either underestimate (for small n), or overestimate (for large n), the true coverage of 0.95. The overestimation phenomenon reveals itself sharply especially in the case of larger trial numbers (n ≥ 30). In contrast, the coverage of the Bayesian method is consistently close to 0.95 regardless of the sample size. Additionally, simulation standard errors of the Bayesian method are universally smaller than the ones obtained from the other two methods. For instance, the SE of the coverage at n = 120 is 0.008 for the z-transformed method, 0.009 for the corrected z method, and 0.0008 for the Bayesian method, a reduction of >10 times. This is a suggestion the SE of the coverage in Spearman's method is highly affected when a few of the simulated data sets behave erratically.

TABLE 2.

Simulations under scenario 1 (normal data)

	z	Corrected z	Bayesian
n = 15	0.82	0.93	0.94
n = 30	0.97	0.95	0.95
n = 60	0.99	0.99	0.95
n = 120	1.00	1.00	0.95

Coverage probability of the Fisher's z, corrected z, and Bayesian, 95% confidence intervals. For different sample size values, the Bayesian estimator has a coverage probability closer to 95% than the corrected z. The coverage standard errors are all small. For example, the SE of the Bayesian method is 0.007.

Tables 3 and 4 summarize the results for scenario 2. Again, the proposed Bayesian method outperforms the other methods in terms of the mean squared error, as well as the coverage values. As shown in Table 3, the mean squared error of the Bayesian method at n = 30, which is 0.001, cannot be achieved by either of the other two methods even at a significantly larger trial size of n = 120. According to Tables 3 and 4, the Bayesian coverage is far more accurate than the others. Similarly to scenario 1, both the z-transformed and its corrected version have the undesirable property of underestimating the coverage for data sets with smaller to moderate trial numbers (n ≤ 30), and overestimating the coverage for the ones with even moderate trial sizes (n ≥ 60). The results of the scenario 3 (Tables 5 and 6) are fully consistent with the findings of the other two scenarios.

TABLE 3.

MSE for simulations under scenario 2 (banana-shaped data)

	rxy	Spearman	Bayesian
n = 15	0.091	0.034	0.0072
n = 30	0.035	0.012	0.0013
n = 60	0.014	0.0052	0.00086
n = 120	0.0036	0.0026	0.00035

In all cases, the Bayesian method has much smaller MSE than Spearman's entries as in Table 1.

TABLE 4.

Simulations under Scenario 2 (banana-shaped data)

	z	Corrected z	Bayesian
n = 15	0.092	0.85	0.94
n = 30	0.15	0.92	0.95
n = 60	0.61	0.96	0.95
n = 120	0.96	0.97	0.95

Entries as in Table 2. In all cases, the Bayesian technique provides accurate coverage probability with much less varibility across datasets.

TABLE 5.

MSE for simulations under scenario 3 (based on the SEF data)

	rxy	Spearman	Bayesian
n = 15	0.011	0.042	0.0098
n = 30	0.055	0.019	0.0049
n = 60	0.017	0.0061	0.00099
n = 120	0.009	0.0011	0.00047

The results of simulations 1 and 2 are repeated. For different sample size values, the Bayesian method produces a much smaller MSE.

In addition to the above mentioned simulation studies, Poisson data were generated from a true scenario with extremely low correlation, r ≈ 0. We performed simulations for n = 15 and n = 60. We did this additional simulation to study the performance of the Bayesian correction method when there is no correlation. As with the previous simulations, for both sample sizes, the posterior coverage of 0 was extremely close to 0.95 for the Bayesian test at the 0.05 level. Meanwhile, the simulation's mean squared error was close to 0.01 for n = 15, and 0.001 for n = 60. This provides an example when the Bayesian methodology does not cause an artificial increase in the correlation.

DISCUSSION

Spearman's idea of correcting for attenuation of the correlation coefficient has been around for >100 yr. It deserves wider recognition in neurophysiological practice, but Spearman's method is flawed. In the first place, it can produce correlations larger than 1. Second, Spearman's reliability r_xx is only well-defined when the variability within items (here, within neurons) is the same for every item; but the variance of such things as a firing-rate index is likely to change across neurons. Third, as our simulation results showed, it is much less accurate than the Bayesian method we introduced: for normally distributed variables Spearman's method may require 10 times as many trials for comparable accuracy, while in the non-normal case Spearman's method is even worse. Finally, while confidence intervals based on Spearman's method have been developed, they are also much less accurate than the Bayesian intervals we recommend.

The Bayesian method we applied is easily implemented with existing software (see the appendix). In some situations, practitioners worry that the prior information injected via Bayes' Theorem may introduce substantial biases. The priors we used were chosen to be diffuse (not highly informative) and have been shown to have good properties in previously-published work (Kass and Natarajan 2006). The simulation study reported here indicates that the method performs well. From our analysis of SEF data we conclude that the method can have a substantial impact on neurophysiological findings.

GRANTS

This work was supported by National Institutes of Health Grants RO1 MH-064537-04 to R. E. Kass, 1P50MH-084053-01 and 1RO1EY-018620-01 to C. R. Olson.

APPENDIX

Correlation decreases when noise is added

Let θ and ξ be two independent random variables with ɛ and δ as their associated noise. Let X = θ + ɛ, and Y = ξ + δ Because of independence, one can write

Derivation of Spearman's formula

Suppose that we observe pairs of X and Y s, with X = θ + ɛ, and Y = ξ + δ. Note from the definition of correlation that

Also due to independence between θ, and ξ (previous appendix), we have

therefore

let

Spearman referred to ρ_xx, the proportion of the observed variance of X's that is due to variance among true values θs, as the reliability of X's. Similarly the reliability of Y's is defined as

Then Spearman's formula for correcting attenuation in correlation is

(A1)

Relationship between regression-based approach and Spearman's correction

First, assume that and consider the regression model The covariance matrix of (X, Y) is

Let Clearly, the reliability in Spearman's notation.

Suppose now that we are interested in finding the relationship between ρ_Yθ, the true correlation, and ρ_XY, the attenuated one through the regression model. We have

(A2)

on the other hand

(A3)

Substituting Eq. A2 in A3 gives

One can easily expand this result to the case in which Y_i = ξ_i + δ_i, where ξ_i = α + βX_i. Note that

therefore

which is Spearman's formula.

Derivation of Formula 4

The derivation of σ_ξ² and to σ_δ² is a simple application of the delta method (for details, see section 9.9 in Wasserman 2004.) Let X and Y be two random variables. Consider the function . We have

Similarly, Let and E(Y) = μ₂, Var(X) = σ₁², and Var(Y) = σ₂². Then assuming independence between X and Y, one can write

Normal hierarchical modeling in WinBugs

WinBugs is a freeware that can be downloaded from the BUGS Project website at http://www.mrc-bsu.cam.ac.uk/bugs/. The following provides a general way of coding hierarchical normal models as explained in the previous text (Bayesian Method).

model

{ for (i in 1 : N) (

theta {i, 1 : 2} dmnorm (mu []. signal [.])

for (j in 1 :T) (

Y [i. j]dnorm (theta [i, j]. se[i, j])

}

}

mu [1 : 2] dnorm (mean [], prec[,])

sigma [1 : 2, 1 : 2]dwish (r[,], 2)

R. inv [1:2,1:2]< -inverse (sigma [,])

}

In the preceding, N represents the number of objects (neurons), T = 2 is the dimensionality, Y[i, j] represents the experimental data. In our data analysis data analysis), Y[i, j] would be the ith index for the variable reward task (j = 1), and serial-order task (j = 2), theta[i,j] refers to the vector (θ_i,ξ_i), se[i,j] represents σ_ɛ² (j = 1), and σ_δ² (j = 2) for the ith observation, mu[], and Sigma[,] are the mean vector μ, and the covariance matrix Σ, as shown in Bayesian model, respectively. Finally, mu[1:2], and R.inv[1:2,1:2] are the priors for the mean vector μ, and the covariance matrix Σ. To run this model, the data need to be formatted appropriately. As an example, to generate the posterior values in the lower panel of Fig. 1, the data need to be formatted as a 54 × 2 array. Additionally, an array of size 54 × 2 of se[i,j] values is needed to run the model.