Example #1
0
def diagnostics(data, params, nafc=2, sigmoid="logistic", core="ab", cuts=None, gammaislambda=False):
    # here we need to hack stuff, since data can be either 'real' data, or just
    # a list of intensities, or just an empty sequence

    # in order to remain compatible with psipy we must check for an empty
    # sequence here, and return a specially crafted return value in that case.
    # sorry..
    if op.isSequenceType(data) and len(data) == 0:
        pmf, nparams = sfu.make_pmf(
            sfr.PsiData([0], [0], [0], 1), nafc, sigmoid, core, None, gammaislambda=gammaislambda
        )
        thres = np.array([pmf.getThres(params, cut) for cut in sfu.get_cuts(cuts)])
        slope = np.array([pmf.getSlope(params, th) for th in thres])
        return np.array([]), np.array([]), 0.0, thres, np.nan, np.nan

    shape = np.shape(np.array(data))
    intensities_only = False
    if len(shape) == 1:
        # just intensities, make a dataset with k and n all zero
        k = n = [0] * shape[0]
        data = [[xx, kk, nn] for xx, kk, nn in zip(data, k, n)]
        intensities_only = True
    else:
        # data is 'real', just do nothing
        pass

    dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, None, gammaislambda=gammaislambda)
    cuts = sfu.get_cuts(cuts)
    params = sfu.get_params(params, nparams)
    predicted = np.array([pmf.evaluate(intensity, params) for intensity in dataset.getIntensities()])

    if intensities_only:
        return predicted
    else:
        deviance_residuals = pmf.getDevianceResiduals(params, dataset)
        deviance = pmf.deviance(params, dataset)
        thres = np.array([pmf.getThres(params, cut) for cut in cuts])
        slope = np.array([pmf.getSlope(params, th) for th in thres])
        rpd = pmf.getRpd(deviance_residuals, params, dataset)
        rkd = pmf.getRkd(deviance_residuals, dataset)
        return predicted, deviance_residuals, deviance, thres, slope, rpd, rkd
Example #2
0
def mapestimate(data, nafc=2, sigmoid="logistic", core="ab", priors=None, cuts=None, start=None, gammaislambda=False):

    dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)

    cuts = sfu.get_cuts(cuts)

    opt = sfr.PsiOptimizer(pmf, dataset)
    estimate = opt.optimize(pmf, dataset, sfu.get_start(start, nparams) if start is not None else None)
    H = pmf.ddnegllikeli(estimate, dataset)
    thres = [pmf.getThres(estimate, c) for c in cuts]
    slope = [pmf.getSlope(estimate, th) for th in thres]
    deviance = pmf.deviance(estimate, dataset)

    # convert to numpy stuff
    estimate = np.array(estimate)
    fisher = np.zeros((nparams, nparams))
    for (i, j) in ((i, j) for i in xrange(nparams) for j in xrange(nparams)):
        fisher[i, j] = sfr.doublep_value(H(i, j))
    thres = np.array(thres)
    slope = np.array(slope)
    deviance = np.array(deviance)

    return estimate, fisher, thres, slope, deviance
Example #3
0
 def test_get_cuts(self):
     # this used to cause an error since
     # operator.isNumberType() on an ndarry is always true
     cuts = np.array([1.0, 2.0, 3.0])
     sfu.get_cuts(cuts)
Example #4
0
def bootstrap(
    data,
    start=None,
    nsamples=2000,
    nafc=2,
    sigmoid="logistic",
    core="ab",
    priors=None,
    cuts=None,
    parametric=True,
    gammaislambda=False,
):

    dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)

    cuts = sfu.get_cuts(cuts)
    ncuts = len(cuts)
    if start is not None:
        start = sfu.get_start(start, nparams)

    bs_list = sfr.bootstrap(nsamples, dataset, pmf, cuts, start, True, parametric)
    jk_list = sfr.jackknifedata(dataset, pmf)

    nblocks = dataset.getNblocks()

    # construct the massive tuple of return values
    samples = np.zeros((nsamples, nblocks), dtype=np.int32)
    estimates = np.zeros((nsamples, nparams))
    deviance = np.zeros((nsamples))
    thres = np.zeros((nsamples, ncuts))
    slope = np.zeros((nsamples, ncuts))
    Rpd = np.zeros((nsamples))
    Rkd = np.zeros((nsamples))
    for row_index in xrange(nsamples):
        samples[row_index] = bs_list.getData(row_index)
        estimates[row_index] = bs_list.getEst(row_index)
        deviance[row_index] = bs_list.getdeviance(row_index)
        thres[row_index] = [bs_list.getThres_byPos(row_index, j) for j in xrange(ncuts)]
        slope[row_index] = [bs_list.getSlope_byPos(row_index, j) for j in xrange(ncuts)]
        Rpd[row_index] = bs_list.getRpd(row_index)
        Rkd[row_index] = bs_list.getRkd(row_index)

    thacc = np.zeros((ncuts))
    thbias = np.zeros((ncuts))
    slacc = np.zeros((ncuts))
    slbias = np.zeros((ncuts))
    for cut in xrange(ncuts):
        thacc[cut] = bs_list.getAcc_t(cut)
        thbias[cut] = bs_list.getBias_t(cut)
        slacc[cut] = bs_list.getAcc_t(cut)
        slbias[cut] = bs_list.getBias_t(cut)

    ci_lower = sfr.vector_double(nparams)
    ci_upper = sfr.vector_double(nparams)

    for param in xrange(nparams):
        ci_lower[param] = bs_list.getPercentile(0.025, param)
        ci_upper[param] = bs_list.getPercentile(0.975, param)

    outliers = np.zeros((nblocks), dtype=np.bool)
    influential = np.zeros((nblocks))

    for block in xrange(nblocks):
        outliers[block] = jk_list.outlier(block)
        influential[block] = jk_list.influential(block, ci_lower, ci_upper)

    return samples, estimates, deviance, thres, thbias, thacc, slope, slbias, slacc, Rpd, Rkd, outliers, influential
def diagnostics(data, params, nafc=2, sigmoid='logistic', core='ab', cuts=None, gammaislambda=False):
    """ Some diagnostic statistics for a psychometric function fit.

    This function is a bit messy since it has three functions depending on the
    type of the `data` argument.

    Parameters
    ----------

    data : variable
        real data : A list of lists or an array of data.
            The first column should be stimulus intensity, the second column should
            be number of correct responses (in 2AFC) or number of yes- responses (in
            Yes/No), the third column should be number of trials. See also: the examples
            section below.
        intensities : sequence of floats
            The x-values of the psychometric function, then we obtain only the
            predicted values.
        no data : empty sequence
            In this case we evaluate the psychometric function at the cuts. All
            other return values are then irrelevant.

    params : sequence of len nparams
        parameter vector at which the diagnostic information should be evaluated

    nafc : int
        Number of responses alternatives for nAFC tasks. If nafc==1 a Yes/No task is
        assumed.

    sigmoid : string
        Name of the sigmoid to be fitted. Valid sigmoids include:
                logistic    (1+exp(-x))**-1 [Default]
                gauss       Phi(x)
                gumbel_l    1 - exp(-exp(x))
                gumbel_r    exp(-exp(-x))
                exponential x>0: 1 - exp(-x); else: 0
                cauchy      atan(x)/pi + 0.5
                id          x; only useful in conjunction with NakaRushton core
        See `swignifit.utility.available_sigmoids()` for all available sigmoids.

    core : string
        \"core\"-type of the psychometric function. Valid choices include:
                ab          (x-a)/b [Default]
                mw%g        midpoint and width, with "%g" a number larger than 0 and less than 0.5. 
                            mw%g corresponds to a parameterization in terms of midpoint and width of
                            the rising part of the sigmoid. This width is defined as the length of the
                            interval on which the sigmoidal part reaches from "%g" to 1-"%g".
                linear      a+b*x
                log         a+b*log(x)
                weibull     2*s*m*(log(x)-log(m))/log(2) + log(log(2)) 
                            This will give you a weibull if combined with the gumbel_l sigmoid and a
                            reverse weibull if combined with the gumbel_r sigmoid.
                poly        (x/a)**b   Will give you a weibull if combined with an exp sigmoid
                NakaRushton The Naka-Rushton nonlinearity; should only be used with an id core
        See `swignifit.utility.available_cores()` for all available cores.

    cuts : sequence of floats
        Cuts at which thresholds should be determined.  That is if cuts =
        (.25,.5,.75), thresholds (F^{-1} ( 0.25 ), F^{-1} ( 0.5 ), F^{-1} ( 0.75
        )) are returned.  Here F^{-1} denotes the inverse of the function
        specified by sigmoid. If cuts==None, this is modified to cuts=[0.5].

    Output
    ------

    (predicted, deviance_residuals, deviance, thres, Rpd, Rkd)

    predicted : numpy array of length nblocks
        predicted values associated with the respective stimulus intensities

    deviance_residuals : numpy array of length nblocks
        deviance residuals of the data

    deviance float
        deviance of the data

    thres : numpy array length ncuts
        the model prediction at the cuts

    Rpd : float
        correlation between predicted performance and deviance residuals

    Rkd : float
        correlation between block index and deviance residuals

    Example
    -------
    >>> x = [float(2*k) for k in xrange(6)]
    >>> k = [34,32,40,48,50,48]
    >>> n = [50]*6
    >>> d = [[xx,kk,nn] for xx,kk,nn in zip(x,k,n)]
    >>> prm = [2.75, 1.45, 0.015]
    >>> pred,di,D,thres,slope,Rpd,Rkd = diagnostics(d,prm)
    >>> D
    8.0748485860836254
    >>> di[0]
    1.6893279652591433
    >>> Rpd
    -0.19344675783032755

    """

    # here we need to hack stuff, since data can be either 'real' data, or just
    # a list of intensities, or just an empty sequence

    # in order to remain compatible with psipy we must check for an empty
    # sequence here, and return a specially crafted return value in that case.
    # sorry..
    # TODO after removal of psipy we can probably change this.
    if op.isSequenceType(data) and len(data) == 0:
        pmf, nparams =  sfu.make_pmf(sfr.PsiData([0],[0],[0],1), nafc, sigmoid, core, None, gammaislambda=gammaislambda )
        thres = np.array([pmf.getThres(params, cut) for cut in sfu.get_cuts(cuts)])
        slope = np.array([pmf.getSlope(params, th ) for th in thres])
        return np.array([]), np.array([]), 0.0, thres, np.nan, np.nan

    shape = np.shape(np.array(data))
    intensities_only = False
    if len(shape) == 1:
        # just intensities, make a dataset with k and n all zero
        k = n = [0] * shape[0]
        data  = [[xx,kk,nn] for xx,kk,nn in zip(data,k,n)]
        intensities_only = True
    else:
        # data is 'real', just do nothing
        pass

    dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, None, gammaislambda=gammaislambda)
    cuts = sfu.get_cuts(cuts)
    params = sfu.get_params(params, nparams)
    predicted = np.array([pmf.evaluate(intensity, params) for intensity in
            dataset.getIntensities()])

    if intensities_only:
        return predicted
    else:
        deviance_residuals = pmf.getDevianceResiduals(params, dataset)
        deviance = pmf.deviance(params, dataset)
        thres = np.array([pmf.getThres(params, cut) for cut in cuts])
        slope = np.array([pmf.getSlope(params, th ) for th in thres])
        rpd = pmf.getRpd(deviance_residuals, params, dataset)
        rkd = pmf.getRkd(deviance_residuals, dataset)
        return predicted, deviance_residuals, deviance, thres, slope, rpd, rkd
def mapestimate ( data, nafc=2, sigmoid='logistic', core='ab', priors=None,
        cuts = None, start=None, gammaislambda=False):
    """ MAP or constrained maximum likelihood estimation for a psychometric function.

    Parameters
    ----------

    data : A list of lists or an array of data.
        The first column should be stimulus intensity, the second column should
        be number of correct responses (in 2AFC) or number of yes- responses (in
        Yes/No), the third column should be number of trials. See also: the examples
        section below.

    nafc : int
        Number of responses alternatives for nAFC tasks. If nafc==1 a Yes/No task is
        assumed.

    sigmoid : string
        Name of the sigmoid to be fitted. Valid sigmoids include:
                logistic    (1+exp(-x))**-1 [Default]
                gauss       Phi(x)
                gumbel_l    1 - exp(-exp(x))
                gumbel_r    exp(-exp(-x))
                exponential x>0: 1 - exp(-x); else: 0
                cauchy      atan(x)/pi + 0.5
                id          x; only useful in conjunction with NakaRushton core
        See `swignifit.utility.available_sigmoids()` for all available sigmoids.

    core : string
        \"core\"-type of the psychometric function. Valid choices include:
                ab          (x-a)/b [Default]
                mw%g        midpoint and width, with "%g" a number larger than 0 and less than 0.5. 
                            mw%g corresponds to a parameterization in terms of midpoint and width of
                            the rising part of the sigmoid. This width is defined as the length of the
                            interval on which the sigmoidal part reaches from "%g" to 1-"%g".
                linear      a+b*x
                log         a+b*log(x)
                weibull     2*s*m*(log(x)-log(m))/log(2) + log(log(2)) 
                            This will give you a weibull if combined with the gumbel_l sigmoid and a
                            reverse weibull if combined with the gumbel_r sigmoid.
                poly        (x/a)**b   Will give you a weibull if combined with an exp sigmoid
                NakaRushton The Naka-Rushton nonlinearity; should only be used with an id core
        See `swignifit.utility.available_cores()` for all available cores.

    priors : sequence of strings length number of parameters
        Prior distributions on the parameters of the psychometric function.
        These are expressed in the form of a list of prior names.
        Valid prior choices include:
                Uniform(%g,%g)   Uniform distribution on an interval
                Gauss(%g,%g)     Gaussian distribution with mean and standard deviation
                Beta(%g,%g)      Beta distribution
                Gamma(%g,%g)     Gamma distribution
                nGamma(%g,%g)    Gamma distribution on the negative axis
                invGamma(%g,%g)  inverse Gamma distribution
                ninvGamma(%g,%g) inverse Gamma distribution on the negative axis
                if an invalid prior or `None` is selected, no constraints are imposed at all.
        See `swignifit.utility.available_priors()` for all available sigmoids.

        if an invalid prior is selected, no constraints are imposed on that
        parameter resulting in an improper prior distribution.

    cuts : sequence of floats
        Cuts at which thresholds should be determined.  That is if cuts =
        (.25,.5,.75), thresholds (F^{-1} ( 0.25 ), F^{-1} ( 0.5 ), F^{-1} ( 0.75
        )) are returned.  Here F^{-1} denotes the inverse of the function
        specified by sigmoid. If cuts==None, this is modified to cuts=[0.5].

    start : sequence of floats of length number of model parameters
        Values at which to start the optimization, if None the starting value is
        determined using a coarse grid search.

    Output
    ------

    estimate, fisher, thres, slope, deviance

    estimate : numpy array length nparams
        the map/cml estimate

    fisher : numpy array shape (nparams, nparams)
        the fisher matrix

    thres : numpy array length ncuts
        the model prediction at the cuts

    slope : numpy array length ncuts
        the gradient of the psychometric function at the cuts

    deviance : numpy array length 1
        the deviance for the estimate

    Example
    -------
    >>> x = [float(2*k) for k in xrange(6)]
    >>> k = [34,32,40,48,50,48]
    >>> n = [50]*6
    >>> d = [[xx,kk,nn] for xx,kk,nn in zip(x,k,n)]
    >>> priors = ('flat','flat','Uniform(0,0.1)')
    >>> estimate, fisher, thres, slope, deviance = mapestimate ( d, priors=priors )
    >>> estimate
    array([ 2.75180624,  1.45717745,  0.01555658])
    >>> deviance
    array(8.0713313642328242)

    """

    dataset, pmf, nparams = sfu.make_dataset_and_pmf(data,
            nafc, sigmoid, core, priors, gammaislambda=gammaislambda)

    cuts = sfu.get_cuts(cuts)

    opt = sfr.PsiOptimizer(pmf, dataset)
    estimate = opt.optimize(pmf, dataset, sfu.get_start(start, nparams) if start is not
            None else None)
    H = pmf.ddnegllikeli(estimate, dataset)
    thres = [pmf.getThres(estimate, c) for c in cuts]
    slope = [pmf.getSlope(estimate, th) for th in thres]
    deviance = pmf.deviance(estimate, dataset)

    # convert to numpy stuff
    estimate = np.array(estimate)
    fisher = np.zeros((nparams,nparams))
    for (i,j) in ((i,j) for i in xrange(nparams) for j in xrange(nparams)):
        fisher[i,j] = sfr.doublep_value(H(i,j))
    thres = np.array(thres)
    slope = np.array(slope)
    deviance = np.array(deviance)

    return estimate, fisher, thres, slope, deviance
def bootstrap(data, start=None, nsamples=2000, nafc=2, sigmoid="logistic",
        core="ab", priors=None, cuts=None, parametric=True, gammaislambda=False ):
    """ Parametric bootstrap of a psychometric function.

    Parameters
    ----------

    data : A list of lists or an array of data.
        The first column should be stimulus intensity, the second column should
        be number of correct responses (in 2AFC) or number of yes- responses (in
        Yes/No), the third column should be number of trials. See also: the examples
        section below.

    start : sequence of floats of length number of model parameters
        Generating values for the bootstrap samples. If this is None, the
        generating value will be the MAP estimate. Length should be 4 for Yes/No
        and 3 for nAFC.

    nsamples : number
        Number of bootstrap samples to be drawn.

    nafc : int
        Number of alternatives for nAFC tasks. If nafc==1 a Yes/No task is
        assumed.

    sigmoid : string
        Name of the sigmoid to be fitted. Valid sigmoids include:
                logistic    (1+exp(-x))**-1 [Default]
                gauss       Phi(x)
                gumbel_l    1 - exp(-exp(x))
                gumbel_r    exp(-exp(-x))
                exponential x>0: 1 - exp(-x); else: 0
                cauchy      atan(x)/pi + 0.5
                id          x; only useful in conjunction with NakaRushton core
        See `swignifit.utility.available_sigmoids()` for all available sigmoids.

    core : string
        \"core\"-type of the psychometric function. Valid choices include:
                ab          (x-a)/b [Default]
                mw%g        midpoint and width, with "%g" a number larger than 0 and less than 0.5. 
                            mw%g corresponds to a parameterization in terms of midpoint and width of
                            the rising part of the sigmoid. This width is defined as the length of the
                            interval on which the sigmoidal part reaches from "%g" to 1-"%g".
                linear      a+b*x
                log         a+b*log(x)
                weibull     2*s*m*(log(x)-log(m))/log(2) + log(log(2)) 
                            This will give you a weibull if combined with the gumbel_l sigmoid and a
                            reverse weibull if combined with the gumbel_r sigmoid.
                poly        (x/a)**b   Will give you a weibull if combined with an exp sigmoid
                NakaRushton The Naka-Rushton nonlinearity; should only be used with an id core
        See `swignifit.utility.available_cores()` for all available cores.

    priors : sequence of strings length number of parameters
        Constraints on the likelihood estimation. These are expressed in the form of a list of
        prior names. Valid prior choices include:
                Uniform(%g,%g)   Uniform distribution on an interval
                Gauss(%g,%g)     Gaussian distribution with mean and standard deviation
                Beta(%g,%g)      Beta distribution
                Gamma(%g,%g)     Gamma distribution
                nGamma(%g,%g)    Gamma distribution on the negative axis
                invGamma(%g,%g)  inverse Gamma distribution
                ninvGamma(%g,%g) inverse Gamma distribution on the negative axis
                if an invalid prior or `None` is selected, no constraints are imposed at all.
        See `swignifit.utility.available_priors()` for all available sigmoids.

    cuts : a single number or a sequence of numbers.
        Cuts indicating the performances that should be considered 'threshold'
        performances. This means that in a 2AFC task, cuts==0.5 the 'threshold'
        is somewhere around 75%% correct performance, depending on the lapse
        rate parametric boolean to indicate whether or not the bootstrap
        procedure should be parametric or not.

    parametric : boolean
        If `True` do parametric, otherwise do a non-parametric bootstrap.

    gammaislambda : boolean
        Set the gamma == lambda prior.

    Returns
    -------

    (samples,estimates,deviance,
    threshold, th_bias, th_acceleration,
    slope, slope_bias, slope_accelerateion
    Rkd,Rpd,outliers,influential)

    samples : numpy array, shape: (nsamples, nblocks)
        the bootstrap sampled data

    estimates : numpy array, shape: (nsamples, nblocks)
        estimated parameters associated with the data sets

    deviance : numpy array, length: nsamples
        deviances for the bootstraped datasets

    threshold : numpy array, shape: (nsamples, ncuts)
        thresholds/cuts for each bootstraped datasets

    th_bias : numpy array, shape: (ncuts)
        the bias term associated with the threshold

    th_acc : numpy array, shape: (ncuts)
        the acceleration constant associated with the threshold

    slope : numpy array, shape: (nsamples, ncuts)
        slope at each cuts for each bootstraped datasets

    sl_bias : numpy array, shape: (ncuts)
        bias term associated with the slope

    sl_acc : numpy array, shape: (ncuts)
        acceleration term associated with the slope

    Rkd : numpy array, length: nsamples
        correlations between block index and deviance residuals

    Rpd : numpy array, length: nsamples
        correlations between model prediction and deviance residuals

    outliers : numpy array of booleans, length nblocks
        points that are outliers

    influential : numpy array of booleans, length nblocks
        points that are influential observations

    Example
    -------
    >>> x = [float(2*k) for k in xrange(6)]
    >>> k = [34,32,40,48,50,48]
    >>> n = [50]*6
    >>> d = [[xx,kk,nn] for xx,kk,nn in zip(x,k,n)]
    >>> priors = ('flat','flat','Uniform(0,0.1)')
    >>> samples,est,D,thres,thbias,thacc,slope,slbias,slacc,Rkd,Rpd,out,influ \
            = bootstrap(d,nsamples=2000,priors=priors)
    >>> np.mean(est[:,0])
    2.7547034408466811
    >>> mean(est[:,1])
    1.4057297989923003

    """
    dataset, pmf, nparams = sfu.make_dataset_and_pmf(data, nafc, sigmoid, core, priors, gammaislambda=gammaislambda)

    cuts = sfu.get_cuts(cuts)
    ncuts = len(cuts)
    if start is not None:
        start = sfu.get_start(start, nparams)

    bs_list = sfr.bootstrap(nsamples, dataset, pmf, cuts, start, True, parametric)
    jk_list = sfr.jackknifedata(dataset, pmf)

    nblocks = dataset.getNblocks()

    # construct the massive tuple of return values
    samples = np.zeros((nsamples, nblocks), dtype=np.int32)
    estimates = np.zeros((nsamples, nparams))
    deviance = np.zeros((nsamples))
    thres = np.zeros((nsamples, ncuts))
    slope = np.zeros((nsamples, ncuts))
    Rpd = np.zeros((nsamples))
    Rkd = np.zeros((nsamples))
    for row_index in xrange(nsamples):
        samples[row_index] = bs_list.getData(row_index)
        estimates[row_index] = bs_list.getEst(row_index)
        deviance[row_index] = bs_list.getdeviance(row_index)
        thres[row_index] = [bs_list.getThres_byPos(row_index, j) for j in xrange(ncuts)]
        slope[row_index] = [bs_list.getSlope_byPos(row_index, j) for j in xrange(ncuts)]
        Rpd[row_index] = bs_list.getRpd(row_index)
        Rkd[row_index] = bs_list.getRkd(row_index)

    thacc = np.zeros((ncuts))
    thbias = np.zeros((ncuts))
    slacc = np.zeros((ncuts))
    slbias = np.zeros((ncuts))
    for cut in xrange(ncuts):
        thacc[cut] = bs_list.getAcc_t(cut)
        thbias[cut] = bs_list.getBias_t(cut)
        slacc[cut] = bs_list.getAcc_s(cut)
        slbias[cut] = bs_list.getBias_s(cut)

    ci_lower = sfr.vector_double(nparams)
    ci_upper = sfr.vector_double(nparams)

    for param in xrange(nparams):
        ci_lower[param] = bs_list.getPercentile(0.025, param)
        ci_upper[param] = bs_list.getPercentile(0.975, param)

    outliers = np.zeros((nblocks), dtype=np.bool)
    influential = np.zeros((nblocks))

    for block in xrange(nblocks):
        outliers[block] = jk_list.outlier(block)
        influential[block] = jk_list.influential(block, ci_lower, ci_upper)

    return samples, estimates, deviance, thres, thbias, thacc, slope, slbias, slacc, Rpd, Rkd, outliers, influential