# adopy.tasks.psi¶

Psychometric function is to figure out whether a subject can perceive a signal with varying levels of magnitude. The function has one design variable for the intensity of a stimulus, $$x$$; the model has four model parameters: guess rate $$\gamma$$, lapse rate $$\delta$$, threshold $$\alpha$$, and slope $$\beta$$.

class adopy.tasks.psi.Task2AFC

Bases: adopy.base._task.Task

The Task class for a simple 2-Alternative Forced Choice (2AFC) task with a single design variable, the magnitude of a stimulus.

Design variables
• stimulus ($$x$$) - magnitude of a stimulus

Responses
• choice ($$y$$) - index of the chosen option

Examples

>>> from adopy.tasks.psi import Task2AFC
['stimulus']
[0, 1]


## Model¶

class adopy.tasks.psi.ModelLogistic

Bases: adopy.tasks.psi._ModelPsi

The psychometric function using logistic function.

\begin{split}\begin{align} F(x; \alpha, \beta) &= \frac{1}{1 + \exp\left(-\beta (x - \mu) \right)} \\ \Psi \left( x \mid \alpha, \beta, \gamma, \delta \right) &= \gamma + (1 - \gamma - \delta) \cdot F(x; \alpha, \beta) \end{align}\end{split}
Model parameters
• threshold ($$\alpha$$) - indifference point where the probability of a positive response equals to 0.5.

• slope ($$\beta$$) - slope of a tangent line on the threshold point ($$\beta > 0$$)

• guess_rate ($$\gamma$$) - leftward asymptote of the psychometric function ($$0 < \gamma < 1$$)

• lapse_rate ($$\delta$$) - rightward asymptote of the psychometric function($$0 < \delta < 1$$)

Examples

>>> from adopy.tasks.psi import ModelLogistic
>>> model = ModelLogistic()
>>> model.params
['threshold', 'slope', 'guess_rate', 'lapse_rate']

compute(choice, stimulus, guess_rate, lapse_rate, threshold, slope)

Compute log likelihood of obtaining responses with given designs and model parameters. The function provide the same result as the argument func given in the initialization. If the likelihood function is not given for the model, it returns the log probability of a random noise.

Warning

Since the version 0.4.0, compute() function should compute the log likelihood, instead of the probability of a binary response variable. Also, it should include the response variables as arguments. These changes might break existing codes using the previous versions of ADOpy.

Changed in version 0.4.0: Provide the log likelihood instead of the probability of a binary response.

class adopy.tasks.psi.ModelWeibull

Bases: adopy.tasks.psi._ModelPsi

The psychometric function using log Weibull (Gumbel) cumulative distribution function.

\begin{split}\begin{align} F(x; \mu, \beta) &= CDF_\text{Gumbel_l} \left( \beta (x - \mu) \right) \\ \Psi \left( x \mid \alpha, \beta, \gamma, \delta \right) &= \gamma + (1 - \gamma - \delta) \cdot F(x; \alpha, \beta) \end{align}\end{split}
Model parameters
• threshold ($$\alpha$$) - indifference point where the probability of a positive response equals to 0.5.

• slope ($$\beta$$) - slope of a tangent line on the threshold point ($$\beta > 0$$)

• guess_rate ($$\gamma$$) - leftward asymptote of the psychometric function ($$0 < \gamma < 1$$)

• lapse_rate ($$\delta$$) - rightward asymptote of the psychometric function($$0 < \delta < 1$$)

Examples

>>> from adopy.tasks.psi import ModelLogistic
>>> model = ModelLogistic()
>>> model.params
['threshold', 'slope', 'guess_rate', 'lapse_rate']

compute(choice, stimulus, guess_rate, lapse_rate, threshold, slope)

Compute log likelihood of obtaining responses with given designs and model parameters. The function provide the same result as the argument func given in the initialization. If the likelihood function is not given for the model, it returns the log probability of a random noise.

Warning

Since the version 0.4.0, compute() function should compute the log likelihood, instead of the probability of a binary response variable. Also, it should include the response variables as arguments. These changes might break existing codes using the previous versions of ADOpy.

Changed in version 0.4.0: Provide the log likelihood instead of the probability of a binary response.

class adopy.tasks.psi.ModelProbit

Bases: adopy.tasks.psi._ModelPsi

The psychometric function using Probit function (Normal cumulative distribution function).

\begin{split}\begin{align} F(x; \mu, \beta) &= CDF_\text{Normal} \left( \beta (x - \mu) \right) \\ \Psi \left( x \mid \alpha, \beta, \gamma, \delta \right) &= \gamma + (1 - \gamma - \delta) \cdot F(x; \alpha, \beta) \end{align}\end{split}
Model parameters
• threshold ($$\alpha$$) - indifference point where the probability of a positive response equals to 0.5.

• slope ($$\beta$$) - slope of a tangent line on the threshold point ($$\beta > 0$$)

• guess_rate ($$\gamma$$) - leftward asymptote of the psychometric function ($$0 < \gamma < 1$$)

• lapse_rate ($$\delta$$) - rightward asymptote of the psychometric function($$0 < \delta < 1$$)

Examples

>>> from adopy.tasks.psi import ModelProbit
>>> model = ModelProbit()
>>> model.params
['threshold', 'slope', 'guess_rate', 'lapse_rate']

compute(choice, stimulus, guess_rate, lapse_rate, threshold, slope)

Compute log likelihood of obtaining responses with given designs and model parameters. The function provide the same result as the argument func given in the initialization. If the likelihood function is not given for the model, it returns the log probability of a random noise.

Warning

Since the version 0.4.0, compute() function should compute the log likelihood, instead of the probability of a binary response variable. Also, it should include the response variables as arguments. These changes might break existing codes using the previous versions of ADOpy.

Changed in version 0.4.0: Provide the log likelihood instead of the probability of a binary response.

## Engine¶

class adopy.tasks.psi.EnginePsi(model, grid_design, grid_param, d_step=1, **kwargs)

Bases: adopy.base._engine.Engine

The Engine class for the psychometric function estimation. It can be only used for Task2AFC.

property d_step

Step size on index to compute $$\Delta$$ for the staircase method.

get_design(kind='optimal')

Choose a design with a given type.

• optimal: an optimal design $$d^*$$ that maximizes the information for fitting model parameters.

• staircase: Choose the stimulus $$s$$ as below:

$\begin{split}x_t = \begin{cases} x_{t-1} - \Delta & \text{if } y_{t-1} = 1 \\ x_{t-1} + 2 \Delta & \text{if } y_{t-1} = 0 \end{cases}\end{split}$

where $$\Delta$$ is determined by d_step which is the step-size change on the grid index.

• random: a design randomly chosen.

Parameters

kind ({‘optimal’, ‘staircase’, ‘random’}, optional) – Type of a design to choose

Returns

design – A chosen design vector