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PSYCH_GLOSS glossary of terms, struct fieldnames and common variable names
2AFC an 2-alternative forced-choice experimental paradigm, in which
the observer selects one of 2 stimuli per trial. Similarly 4AFC,
8AFC, nAFC.
alpha parameter of the underlying psychometric function F. Together,
alpha and beta determine the horizontal displacement of the
curve, and its slope. alpha is the first element of the parameter
vector theta.
beta parameter of the underlying psychometric function F. Together,
alpha and beta determine the horizontal displacement of the
curve, and its slope. beta is the second element of the parameter
vector theta.
BCa the bias-corrected accelerated method of obtaining bootstrap
confidence intervals. For most problems, the coverage of BCa
intervals can be shown to exhibit better convergence than that of
unadjusted bootstrap percentile intervals. See Davison, AC & Hinkley,
DV (1997): Bootstrap methods and their application; Cambridge: CUP,
and Efron, B & Tibshirani, RJ (1993): An Introduction to the Bootstrap;
New York: Chapman & Hall.
bootstrap a Monte Carlo method for estimating variability. A large number
of simulated data sets are generated from a distribution that is
assumed to approximate the true distribution underlying the data
(in our implementation, we use the maximum-likelihood fitted
function of form psi in order to generate data). Whatever process
was carried out on the data to obtain an estimate (e.g. fitting a
function and obtaining a threshold) is carried out on each of the
simulated data sets, to obtain an expected distribution of
estimates.
bootstrap the inaccuracy of a bootstrap variability estimate that arises
error because of a discrepancy between the estimated or assumed
bootstrap generating function and the true distribution.
conf short for "confidence levels" which is our imprecise shorthand
for "the cumulative probability value corresponding to a
confidence interval boundary". Our default values for conf are
[0.023, 0.159, 0.841, 0.977] because they provide confidence
intervals whose coverage is familiar: if the variable in question
were Gaussian, they would give us [-2, -1, +1, +2] standard
deviations from the mean.
confLimMethod should read 'BCa', indicating that confidence limits in the 'lims'
fields were obtained by the BCa method
corr linear correlation coefficient
cpe cumulative probability estimate: for any measure z, this is an
estimate of the integral from -infinity to z of the probability
density function for Z. For a right-tailed test, significance is
equal to cpe. For a left-tailed test, significance = 1-cpe.
cuts the probability levels at which thresholds or slopes are
calculated, given in the (0, 1) range of F.
d a vector of length K giving deviance residuals for each block.
D deviance summary statistic ( = sum(d.^2)). This is the first
statistical measure returned by the PSIGNIFIT engine.
dat data set: each row is an observation. May be expressed as
[x y n], [x r n] or [x r w].
deriv derivative of the attributes of interest (parameters, thresholds
or slopes) with respect to each of the parameters (our convention is
for columns to denote different attributes, for example thresholds at
different cut levels, and for rows to denote different parameters).
Derivatives are evaluated at the maximum-likelihood estimated
or initial parameter values. Used to calculate "lff" (see below) in the
BCa method.
deviance each residual is equal to the square root of the deviance
residuals calculated for one of the data points in isolation, signed
according to the direction of the difference between observed
performance and model prediction. The sum of squared deviance
residuals equals overall deviance, D.
est initial estimate of something (parameters, thresholds, slopes).
Usually this is the maximum-likelihood estimate from a fit, but
sometimes the user supplies a hypothesis explicity - in which
case est refers to the values derived from the hypothesized
distribution.
F underlying psychometric function. Relates stimulus intensity x to
the probability that the psychological mechanism of interest can
detect the stimulus, in the absence of stimulus-independent
errors or lucky guesses. See the MATLAB function PSYCHF.
gamma parameter of the psychometric performance function psi,
determining the lower bound of predicted performance: psi(x) >=
gamma for all x. Its value corresponds to predicted performance
in the absence of a stimulus. In nAFC paradigms, gamma is usually
fixed at the reciprocal of the number of intervals per trial. In
yes/no paradigms, it is usually small (< 0.5). gamma is the third
element of the parameter vector theta.
k a vector of length K denoting the chronological index for each
block in the data set
K number of blocks in the data set (= length(n))
lambda parameter of the psychometric performance function psi,
determining the upper bound of predicted performance: psi(x) <=
1-lambda for all x. 1-lambda is the predicted performance level
for an arbitrarily large stimulus. lambda is typically small
(<0.05) because it is generally assumed that observers do not
make stimulus-independent errors at high rates. lambda is the
fourth element in the parameter vector theta.
ldot: the derivative of log-likelihood, with respect to each of the
parameters, evaluated at the MLE, for each of the bootstrap data
sets. Thus ldot has R rows and four columns (one for each
parameter). It is used to obtain BCa confidence interval limits,
and is output by the PSIGNIFIT engine.
lims a matrix whose columns refer to different estimates and whose
rows correspond to different elements of conf. Each element is
the estimate whose cpe in the bootstrap distribution is equal to
the appropriate element of conf. The method used to obtain
the confidence limits is indicated by the field
'confLimMethod' - usually it will be the BCa method.
lff: the least-favourable direction(s) in parameter space for
inference about a variable or variables. It is used to obtain BCa
confidence interval limits. In our format, it is a matrix with
one column for each variable, and four rows indicating the
components of the least-favourable direction in the dimensions of
the four parameters. A least-favourable direction vector should
be calculated for each parameter, threshold or slope estimate -
see Davison, AC & Hinkley, DV (1997): Bootstrap methods and their
application; Cambridge: CUP, pp206-7 and p249.
"log slope" gradient of the psychometric function with respect to log10(x).
This can be calculated as threshold * slope * log(10), or by
passing the option 'log' into FINDSLOPE. See the entries for
"threshold" and "slope".
m number of points in parameter space at which simulations are
repeated during sensitivity analysis
n a vector of length K denoting the number of trials in each block
of the data set
N total number of observations in data set (= sum(n))
nAFC see 2AFC
p a vector of length K denoting a model's prediction for the
expected values of y
PA denotes parameters
parameters alpha, beta, gamma and lambda.
psi psychometric performance function, relating stimulus intensity x
to the probability of a correct or positive response. A common
form for predicting performance in a single psychophysical
experiment is
p = psi(x; {alpha, beta, gamma, lambda}) =
gamma + (1 - gamma -lambda) F(x; {alpha, beta})
See the MATLAB function PSI.
r a vector of length K denoting the number of correct (or positive)
responses in each block of the data set (= y ./ n).
R number of simulations performed
r_pd correlation coefficient between p and d (model predictions and
signed deviance residuals). Used as a statistical check on the
functional form of one's model, (usually psi). This is the second
statistical measure returned by the PSIGNIFIT engine.
r_kd correlation coefficient between k and d (chronological indices
and signed deviance residuals) excluding those points for which
y == 0 or y == 1. Used as a statistical check on any change in
the observer's performance over time (between blocks). This is
the third statistical measure returned by the PSIGNIFIT engine.
sensitivity a way of examining the severity of bootstrap error. Our technique
analysis is to re-run the bootstrap m times, with different parameter sets
for the generating function. The m new parameter set lie on the
(sens) boundary of a region in alpha-beta space. The default is to take 8
points that lie on the boundary of a joint confidence region of
a given coverage in parameter space. The shape of the region
is likelihood-based (all points on the skin have the same deviance value
with respect to the original data set). The points' precise locations are
chosen by an algorithm that uses the original bootstrap distribution of
parameters, and aims to spread out the points' directions in the alpha/beta
plane while exploring the extremes of variation in alpha and beta within the
region (gamma and lambda, if they are free parameters, may be varied in
order to accomplish this aim). At the end of sensitivity analysis we report
the "worst-case" variability estimate (see "worst" below).
shape the functional form of F: in the current implementation, this can
be Weibull, logistic, cumulative Gaussian, Gumbel or linear.
sim matrix of simulated values: each row is a different simulation,
and each column is a different variable.
SL denotes slopes
slope gradient of the psychometric function with respect to x,
evaluated at a particular threshold value for x. The "slope at
0.5" would therefore usually refer to the value of dF/dx
evaluated at the point at which F(x) = 0.5. Slopes can also be
calculated in the context of psi (so the "75% slope" would be
d(psi)/dx evaluated where psi(x) = 0.75). See the entry for
"threshold" below.
TH denotes thresholds
theta [alpha beta gamma lambda].
threshold inverse of the psychometric function with respect to x.
The "threshold at 0.5" would usually refer to F^-1(0.5). This is
a threshold in the context of the underlying psychometric
function F, which is the default measurement in FINDTHRESHOLD and
FINDSLOPE. By passing the option 'performance' into these two
functions, thresholds can instead be calculated in the context of
the psychometric performance function psi. So the "75%
performance threshold" would be psi^-1(0.75) and the "75%
performance slope" would be the derivative of psi at that point.
Note, however, that the PSIGNIFIT engine can only calculate BCa
confidence limits for "underlying" thresholds and slopes (inverse of F).
w a vector of length K denoting the number of incorrect (or
negative) responses in each block of the data set (= n - r).
worst-case a matrix with the same format as "lims": for each column (i.e.
bootstrap each variable) confidence limits are listed. For a certain
limit variable t (a threshold, for example), let us use t_0 to denote
the value of t in the bootstrap generating function, and u_0 to
(worst) denote, say, the upper limit of a confidence interval obtained by
the bootstrap method. In sensitivity analysis, we perform m
additional bootstraps: each one has a different generating
function, so each one has a different initial value for t:
t_1.....t_m. The m bootstraps yield m estimates for the upper
confidence interval limit, u_1....u_m. Now, finally, we can
define the "worst case" bootstrap limit u_worst:
u_worst = t_0 + max([u_0-t_0, u_1-t_1, ......u_m-t_m])
So, the difference between u_worst and t_0 is the same as the
largest difference between u and t encountered during sensitivity
analysis.
x a vector of length K denoting the stimulus value for each block
in the data set.
y a vector of length K denoting the proportion of correct responses
for each block in the data set (= r ./ n).
yes/no any single-interval experimental paradigm, in which the
observer sees one stimulus per trial.
Part of the psignifit standalone distribution version 2.5.6.
Copyright (c) J.Hill 1999-2005.
Please read the LICENSE and NO WARRANTY statement in Legal.txt
mailto:psignifit@bootstrap-software.org
http://bootstrap-software.org/psignifit/
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