Analyses and statistical tests

Equations for both the Kalman filter for both a randomly changing and correlated stimulus which we use in the current chapter and a fixed weighted average model are fully outlined in chapter 2, pages 44-46.

Confidence intervals.

We report standard (parametric) 95 % confidence intervals of the mean.

Correction for potential ambiguity in responses using circular stimuli.

Recording responses and associated error from circular stimuli using a straight response bar can potentially be problematic. The reason for this problem is because the response line participants use to signal their estimate of orientation partially spans the diameter of the circle and points to two different angles. For example, a line at 60° also lies at 240° at the other side of the circle. This means that if an orientation on a trial is presented at 60° and the subject responds at say 65°, it is possible to record a response of both 65 giving 5° of error or 245° giving 180° of error. The way we and others (Fischer & Whitney, 2014) deal with this issue is to assume the response angle closest to the true trial orientation angle, as the angle the participant meant to signal and calculate the minimum angle difference

difference between response and trial orientation and converting this to vector form using sin and cos and then converting this value back to an angle using the Matlab four-quadrant inverse tangent function (Mathworks Inc, 2012).

Proximal variance calibration calculations and relationship to Kalman filter equations. To calculate proximal variance from our proximal calibration procedure we fit cumulative normal psychometric functions to the participants’ responses data for both the 5% and 20% contrast conditions using the Palamedes toolbox’s fitted variance parameter (Prins & Kingdom (2009). We take the variance (degs2_{) of the width of the slope of fitted cumulative normal psychometric function} as our measure of proximal variance i.e. a narrower slope indicates a less variable response (for example, a slope with a variance of 20 (deg2_{) indicates a less variable perception of the stimulus in} comparison to a slope with variance of 120 (degs2_{) for the 5% contrast condition). To provide an} estimate of proximal variance for a single stimulus presentation used in the main experimental conditions we divide the square root of the variance by two. Importantly, the variance we record represents the vital quantity of measurement variance (σ2

m) or the reliability of the observer’s measurement in the Kalman filter equations and can be used to form an estimate of participants Kalman gain for a response to a single stimulus presentation in conjunction with a figure for distal variance (see equation 7, page 48). To test for differences between proximal variances across participants for both contrast conditions we perform repeated measure t tests.

Estimated Kalman gains, Model fitted Kalman gains and fitted weights to participant data. To calculate an estimated level of Kalman gain at the group level, we took our mean proximal variance measurements from our proximal variance calibration experiment and pre-set mean distal variance figures and followed the Kalman gain equation (see methods equations page 48). It should be noted, that due to the nature of presenting random orientations which produces a uniform distribution (high variance) and a Gaussian random walk (lower variance) the level of distal variance changes between each block and for each participant in each experimental session. This meant that the distal variance data was not the same for every participant. This meant that any statistical tests we ran on estimated Kalman gains for individual participants might not have provided unbiased comparisons so this aspect of our analysis is only intended to provide an approximate estimate of Kalman gains at a group level. In addition to estimating approximate Kalman gains, we computationally modelled participant Kalman gains.

Modelling of participant Kalman gains was carried out by running a nonlinear squares fit to ascertain which level of Kalman gain explained participant’s responses (corrected for circular stimuli) most appropriately. A least squares model fit is a mathematical procedure for finding the best fitting regression slope or curve to a given set of data points. Here, this works by plotting a regression model based on the equations for estimation in the Kalman filter (chapter 2, page 43, and equation 6) at a

number of Kalman gain levels to participants’ responses until the regression slope with the least residual squared error is found. The gain at which point is reached is taken as the participants’ model fitted Kalman gain

Our modelling analysis of Kalman gain produces a specific level of Kalman gain in each subject and therefore is suitable for statistical comparison. Statistical analysis of model fitted Kalman gains between contrast conditions is carried out using paired sample t tests (Bonferroni-corrected). We report 95% confidence intervals. We also modelled participant responses with a fixed weighted average simulation model by running another nonlinear least squares fit to find the best fitting weights from the preceding 6 trial orientations to the current participant response.

Serial dependence calculations

To assess the magnitude of serial dependence across contrast conditions and experiments we

calculated serial dependence based on Fischer & Whitney (2014). We first calculated and plotted the error in degrees on each individual trial on the Y axis (corrected for circular responses). Error is calculated as the participant response minus the actual stimulus orientation. Positive errors indicate responses clockwise of the true current trial orientation and negative errors indicating responses anti clockwise of the true trial orientation. Next, we plotted the relative orientation of the current stimulus in comparison to the previous stimulus orientation. Relative orientation is calculated as the previous (n-1) stimulus orientation minus the current stimulus orientation. To measure the magnitude of serial dependence over all trials we fit standard regression slopes corrected for symmetrical and circular stimuli to error and relative orientation. By this measure serial dependence is directly related to the steepness and intercept of the regression slope. In both experiments, statistical analysis of regression slopes used paired t tests (Bonferroni corrected).

N-back serial dependence analysis.

Previous studies have shown that participant’s responses are not only serially dependent on the immediately preceding trial but that responses also depend on trials presented over the last 10-15 seconds (Fischer & Whitney, 2014). To assess how serial dependence changed as a function of time in our experiments we measured serial dependence between not just the current and immediately

previous trial but between the current and previous six trials. This was calculated in the same way as serial dependence but instead of only comparing error on the current trial against the relative

orientation of the current trial compared to immediately previous trial but also error on the current trial compared to the relative orientation of the previous six trials. Again, we fit regression slopes corrected for circular and symmetrical stimuli to participant’s data and report standard deviations and 95% confidence intervals.