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Showing posts with the label confidence interval

Metastability and Eye Movements: A Dynamical‑Systems Interpretation

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  1. Why metastability is a natural language for gaze behavior The idea that perception, attention, and neural activity evolve through transiently stable states has deep roots in cognitive science and neuroscience. Concepts such as attractors , basins , noise‑induced transitions , and escape times appear in work on perceptual switching, decision‑making, and neural population dynamics (Kelso, 1995; Rabinovich et al., 2008; Deco & Jirsa, 2012). Eye movements, especially the alternation between fixations and saccades ,   fit remarkably well into this metastable picture. The Freidlin–Wentzell theory of rare events (Freidlin & Wentzell, 2012) provides a rigorous mathematical language for these intuitions. 2. Fixations as metastable states  Consider the gaze position X t as the state of a stochastic dynamical system. During a fixation the  gaze remains confined to a small region, microsaccades and noise generate small fluctuations, and the gaze tends to return...
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R function conf_int_Accuracy

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  R function conf_int_Accuracy <- function(GMTobs, s, nDays = 2, nAnalysts = 2, nPlates = 2, nReplicates = 3, FoldDilution = 2, Threshold = 1, alpha = 0.05) {   # INPUT:   # GMTobs: vector of observed geometric means at each fold dilution   # s: vector of the standard deviations of the log-transformed replicates   # nDays: number of experimental days (default = 2)   # nAnalysts: number of analysts performing the tests in validation (default = 2)   # nPlates: number of plates used per analyst per day (default = 2)   # nReplicates: number of measurements that each analyst performs per plate per day   # FoldDilution: step of the serial dilution (default = 2)   # Threshold: critical threshold of the log-difference between the observed and the true mean (default=1)   # alpha: significance level (default=0.05)   #   # OUTPUT:   # vector of Relative Accuracy calculated at each fold dilution and its confidence interval. In...
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Quantifying a biossay: confidence interval of the accuracy

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A bioassay validation is a set of procedures that ensure the precision and accuracy of the test results. Precision refers to the proximity of results to each other, and accuracy is the proximity of measurements results to the true value. Here we want to focus on the accuracy and the building of a confidence interval around it.   Let's design the validation experiments with three possible sources of variability, day, analyst, and plate having nd levels, na levels, and np levels, respectively. Each analyst performs nm measurements per plate per day on each sample. The samples will be serially diluted by a factor of two, resulting in dl dilution levels from 1/1 to 1/(2^(dl-1)). So, the total number of measurements, N, for each dilution level will N=nd*na*np*nm.   To calculate the accuracy , A, we can follow the FDA recommendation [1] of comparing the mean of the measured values with the "true" value, and define accuracy as that is the ratio between the observed geometric m...
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The confidence interval of variance and standard deviation

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  In statistics, a confidence interval (CI) is a range of values that is likely to contain the true value of a parameter with a certain level of confidence. For example, a 95% CI means that 95% of the time, the true value of the parameter will fall within the interval.   One of the parameters that we often want to estimate is the variance or the standard deviation of a population. The variance measures how spread out the data are around the mean, and the standard deviation is the square root of the variance. Both are useful indicators of variability and dispersion in a data set.   However, in most cases, we do not have access to the entire population, but only to a sample. Therefore, we need to use sample statistics to estimate the population parameters. The sample variance and the sample standard deviation are calculated as follows:   s$^{2}$ =  (1/(n-1)) * sum((x$_i$ - m)$^{2}$)  for i = 1 to n   s = sqrt(s$^{2}$)   where n i...
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R function ci_sigma

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The function  ci_sigma  evaluates the  confidence interval of variance and standard deviation   for populations that are approximately normal.   ci_sigm a   takes two arguments: x : a vector of data. alpha : significance level (default = 0.05). The function checks if the input data is approximately normal using the Shapiro-Francia test from the  nortest  package. If the hypothesis of normality is not satisfied at the given significance level, it suggests trying a log-transformation and performs the Shapiro-Francia test again. If the hypothesis of normality is satisfied, it calculates critical values from the chi-squared distribution and uses them to calculate confidence intervals for both variance and standard deviation. The function returns a list containing five elements: VAR : variance of input data. SD : standard deviation of input data. CI.VAR : confidence interval for variance. CI.SD : ...
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Diagnosing Singular Fits in Linear Mixed Models: Practical Examples, Code, and Alternatives

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  Introduction This post is the practical follow‑up to When Linear Mixed Models Don’t Converge , where I discussed the theoretical reasons why convergence issues often arise in linear mixed models. Here, we move from theory to practice: simulated examples, diagnostic tools, interpretation of singular fits, and practical modeling alternatives. If you haven’t read the first part yet, you may want to start there for the conceptual foundations. Figure. Conceptual map showing how model complexity and available information interact Legend of Figure 1. Green = identifiable models; yellow = risk of singularity; red = non‑identifiable models. A Concrete Example Using R: When a Random-Slope Model Becomes Singular Let’s simulate a simple dataset with only 8 groups and a very small random‑slope variance. This is a classic scenario where the model converges numerically but is statistically non‑identifiable. ```r set.seed(123) n_groups <- 8 n_per_group <- 20 group <- factor(rep(1:n_gro...
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Understanding Anaerobic Threshold (VT2) and VO2 Max in Endurance Training

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  Introduction: The Science Behind Ventilatory Thresholds VT1 vs. VT2: Defining Energy Transitions p (Aerobic Threshold) : The transition from lipid metawwwpp (fat-burning) to a mixed metabolism, where both carbohydrates and fats fuel muscle activity. V0T2 (Anaerobic Threshold) : The point where the  shifts to a predominantly carbohydrate-based wmetabolism , leading to rapid lactate waccumpulation and increased reliance on anaerobic energy pathways. www w In p VT1 and VT2 Wwww ewAgep & Sex Younger athletes generally exhibit higher VT2 values , while aging naturally reduces aerobicw capacity. Men tend to have a higher VO2 Max due to greater lung capacity and muscle mass, but women can achieve similar endurance levels through optimized training. Body Composition & Health Status High body fat percentage may reduce VT2 efficiency, as excess weight increases ...
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Unit Root Testing in Practice: A Tutorial on ADF–KPSS, Specification, and Diagnostics

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  Stationary Signals   A stationary signal is defined by the stability of its statistical properties over time. Specifically, this implies that the signal’s mean, variance, and autocorrelation function remain consistent, regardless of the time in which the signal is evaluated. There are two main types of stationarity: ·        Strict-sense stationarity refers to the property of a signal in which all statistical characteristics remain unchanged over time. This encompasses all moments of the distribution, indicating that the entire probability distribution remains constant throughout the observed period. ·        Wide-sense stationarity requires that the first two moments, specifically the mean and variance, remain constant over time. The autocorrelation function is dependent only on the time difference between two points, not on the exact time points themselves.   Importance in Signal...
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