Exemplo n.º 1
0
def ansari(x,y):
    """Determine if the scale parameter for two distributions with equal
    medians is the same using the Ansari-Bradley statistic.

    Specifically, compute the AB statistic and the probability of error
    that the null hypothesis is true but rejected with the computed
    statistic as the critical value.

    One can reject the null hypothesis that the ratio of variances is 1 if
    returned probability of error is small (say < 0.05)
    """
    x,y = asarray(x),asarray(y)
    n = len(x)
    m = len(y)
    if (m < 1):
        raise ValueError, "Not enough other observations."
    if (n < 1):
        raise ValueError, "Not enough test observations."
    N = m+n
    xy = r_[x,y]  # combine
    rank = stats.rankdata(xy)
    symrank = amin(array((rank,N-rank+1)),0)
    AB = sum(symrank[:n],axis=0)
    uxy = unique(xy)
    repeats = (len(uxy) != len(xy))
    exact = ((m<55) and (n<55) and not repeats)
    if repeats and ((m < 55)  or (n < 55)):
        print "Ties preclude use of exact statistic."
    if exact:
        astart, a1, ifault = statlib.gscale(n,m)
        ind = AB-astart
        total = sum(a1,axis=0)
        if ind < len(a1)/2.0:
            cind = int(ceil(ind))
            if (ind == cind):
                pval = 2.0*sum(a1[:cind+1],axis=0)/total
            else:
                pval = 2.0*sum(a1[:cind],axis=0)/total
        else:
            find = int(floor(ind))
            if (ind == floor(ind)):
                pval = 2.0*sum(a1[find:],axis=0)/total
            else:
                pval = 2.0*sum(a1[find+1:],axis=0)/total
        return AB, min(1.0,pval)

    # otherwise compute normal approximation
    if N % 2:  # N odd
        mnAB = n*(N+1.0)**2 / 4.0 / N
        varAB = n*m*(N+1.0)*(3+N**2)/(48.0*N**2)
    else:
        mnAB = n*(N+2.0)/4.0
        varAB = m*n*(N+2)*(N-2.0)/48/(N-1.0)
    if repeats:   # adjust variance estimates
        # compute sum(tj * rj**2,axis=0)
        fac = sum(symrank**2,axis=0)
        if N % 2: # N odd
            varAB = m*n*(16*N*fac-(N+1)**4)/(16.0 * N**2 * (N-1))
        else:  # N even
            varAB = m*n*(16*fac-N*(N+2)**2)/(16.0 * N * (N-1))
    z = (AB - mnAB)/sqrt(varAB)
    pval = (1-distributions.norm.cdf(abs(z)))*2.0
    return AB, pval
Exemplo n.º 2
0
def ansari(x,y):
    """
    Perform the Ansari-Bradley test for equal scale parameters

    The Ansari-Bradley test is a non-parametric test for the equality
    of the scale parameter of the distributions from which two
    samples were drawn.

    Parameters
    ----------
    x, y : array_like
        arrays of sample data

    Returns
    -------
    AB : float
        The Ansari-Bradley test statistic
    p-value : float
        The p-value of the hypothesis test

    See Also
    --------
    fligner : A non-parametric test for the equality of k variances
    mood : A non-parametric test for the equality of two scale parameters

    Notes
    -----
    The p-value given is exact when the sample sizes are both less than
    55 and there are no ties, otherwise a normal approximation for the
    p-value is used.

    References
    ----------
    .. [1] Sprent, Peter and N.C. Smeeton.  Applied nonparametric statistical
           methods.  3rd ed. Chapman and Hall/CRC. 2001.  Section 5.8.2.

    """
    x,y = asarray(x),asarray(y)
    n = len(x)
    m = len(y)
    if m < 1:
        raise ValueError("Not enough other observations.")
    if n < 1:
        raise ValueError("Not enough test observations.")
    N = m+n
    xy = r_[x,y]  # combine
    rank = stats.rankdata(xy)
    symrank = amin(array((rank,N-rank+1)),0)
    AB = sum(symrank[:n],axis=0)
    uxy = unique(xy)
    repeats = (len(uxy) != len(xy))
    exact = ((m<55) and (n<55) and not repeats)
    if repeats and ((m < 55)  or (n < 55)):
        warnings.warn("Ties preclude use of exact statistic.")
    if exact:
        astart, a1, ifault = statlib.gscale(n,m)
        ind = AB-astart
        total = sum(a1,axis=0)
        if ind < len(a1)/2.0:
            cind = int(ceil(ind))
            if (ind == cind):
                pval = 2.0*sum(a1[:cind+1],axis=0)/total
            else:
                pval = 2.0*sum(a1[:cind],axis=0)/total
        else:
            find = int(floor(ind))
            if (ind == floor(ind)):
                pval = 2.0*sum(a1[find:],axis=0)/total
            else:
                pval = 2.0*sum(a1[find+1:],axis=0)/total
        return AB, min(1.0,pval)

    # otherwise compute normal approximation
    if N % 2:  # N odd
        mnAB = n*(N+1.0)**2 / 4.0 / N
        varAB = n*m*(N+1.0)*(3+N**2)/(48.0*N**2)
    else:
        mnAB = n*(N+2.0)/4.0
        varAB = m*n*(N+2)*(N-2.0)/48/(N-1.0)
    if repeats:   # adjust variance estimates
        # compute sum(tj * rj**2,axis=0)
        fac = sum(symrank**2,axis=0)
        if N % 2: # N odd
            varAB = m*n*(16*N*fac-(N+1)**4)/(16.0 * N**2 * (N-1))
        else:  # N even
            varAB = m*n*(16*fac-N*(N+2)**2)/(16.0 * N * (N-1))
    z = (AB - mnAB)/sqrt(varAB)
    pval = distributions.norm.sf(abs(z)) * 2.0
    return AB, pval
Exemplo n.º 3
0
def ansari(x,y):
    """Determine if the scale parameter for two distributions with equal
    medians is the same using the Ansari-Bradley statistic.

    Specifically, compute the AB statistic and the probability of error
    that the null hypothesis is true but rejected with the computed
    statistic as the critical value.

    One can reject the null hypothesis that the ratio of variances is 1 if
    returned probability of error is small (say < 0.05)
    """
    x,y = asarray(x),asarray(y)
    n = len(x)
    m = len(y)
    if (m < 1):
        raise ValueError, "Not enough other observations."
    if (n < 1):
        raise ValueError, "Not enough test observations."
    N = m+n
    xy = r_[x,y]  # combine
    rank = stats.rankdata(xy)
    symrank = amin(array((rank,N-rank+1)),0)
    AB = sum(symrank[:n],axis=0)
    uxy = unique(xy)
    repeats = (len(uxy) != len(xy))
    exact = ((m<55) and (n<55) and not repeats)
    if repeats and ((m < 55)  or (n < 55)):
        warnings.warn("Ties preclude use of exact statistic.")
    if exact:
        astart, a1, ifault = statlib.gscale(n,m)
        ind = AB-astart
        total = sum(a1,axis=0)
        if ind < len(a1)/2.0:
            cind = int(ceil(ind))
            if (ind == cind):
                pval = 2.0*sum(a1[:cind+1],axis=0)/total
            else:
                pval = 2.0*sum(a1[:cind],axis=0)/total
        else:
            find = int(floor(ind))
            if (ind == floor(ind)):
                pval = 2.0*sum(a1[find:],axis=0)/total
            else:
                pval = 2.0*sum(a1[find+1:],axis=0)/total
        return AB, min(1.0,pval)

    # otherwise compute normal approximation
    if N % 2:  # N odd
        mnAB = n*(N+1.0)**2 / 4.0 / N
        varAB = n*m*(N+1.0)*(3+N**2)/(48.0*N**2)
    else:
        mnAB = n*(N+2.0)/4.0
        varAB = m*n*(N+2)*(N-2.0)/48/(N-1.0)
    if repeats:   # adjust variance estimates
        # compute sum(tj * rj**2,axis=0)
        fac = sum(symrank**2,axis=0)
        if N % 2: # N odd
            varAB = m*n*(16*N*fac-(N+1)**4)/(16.0 * N**2 * (N-1))
        else:  # N even
            varAB = m*n*(16*fac-N*(N+2)**2)/(16.0 * N * (N-1))
    z = (AB - mnAB)/sqrt(varAB)
    pval = (1-distributions.norm.cdf(abs(z)))*2.0
    return AB, pval