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# Jarque-Bera goodness of fit test

Suppose we wish to infer from measurements whether the weights of adult human males in a medical study are not normally distributed [1]. The weights (lbs) are recorded in the array `x` below.

    import numpy as np
    x = np.array([148, 154, 158, 160, 161, 162, 166, 170, 182, 195, 236])

The Jarque-Bera test `scipy.stats.jarque_bera` begins by computing a statistic based on the sample skewness and kurtosis.

    from scipy import stats
    res = stats.jarque_bera(x)
    res.statistic

    np.float64(6.982848237344646)

Because the normal distribution has zero skewness and zero (“excess” or “Fisher”) kurtosis, the value of this statistic tends to be low for samples drawn from a normal distribution.

The test is performed by comparing the observed value of the statistic against the null distribution: the distribution of statistic values derived under the null hypothesis that the weights were drawn from a normal distribution.

For the Jarque-Bera test, the null distribution for very large samples is the `chi-squared distribution` with two degrees of freedom.

    import matplotlib.pyplot as plt

    dist = stats.chi2(df=2)
    jb_val = np.linspace(0, 11, 100)
    pdf = dist.pdf(jb_val)
    fig, ax = plt.subplots(figsize=(8, 5))

    def jb_plot(ax):  # we'll reuse this
        ax.plot(jb_val, pdf)
        ax.set_title("Jarque-Bera Null Distribution")
        ax.set_xlabel("statistic")
        ax.set_ylabel("probability density")

    jb_plot(ax)
    plt.show();

The comparison is quantified by the p-value: the proportion of values in the null distribution greater than or equal to the observed value of the statistic.

    fig, ax = plt.subplots(figsize=(8, 5))
    jb_plot(ax)
    pvalue = dist.sf(res.statistic)
    annotation = (f'p-value={pvalue:.6f}\n(shaded area)')
    props = dict(facecolor='black', width=1, headwidth=5, headlength=8)
    _ = ax.annotate(annotation, (7.5, 0.01), (8, 0.05), arrowprops=props)
    i = jb_val >= res.statistic  # indices of more extreme statistic values
    ax.fill_between(jb_val[i], y1=0, y2=pdf[i])
    ax.set_xlim(0, 11)
    ax.set_ylim(0, 0.3)
    plt.show()

    res.pvalue

    np.float64(0.03045746622458189)

If the p-value is “small” - that is, if there is a low probability of sampling data from a normally distributed population that produces such an extreme value of the statistic - this may be taken as evidence against the null hypothesis in favor of the alternative: the weights were not drawn from a normal distribution. Note that:

- The inverse is not true; that is, the test is not used to provide evidence for the null hypothesis.

- The threshold for values that will be considered “small” is a choice that should be made before the data is analyzed [2] with consideration of the risks of both false positives (incorrectly rejecting the null hypothesis) and false negatives (failure to reject a false null hypothesis).

Note that the chi-squared distribution provides an asymptotic approximation of the null distribution; it is only accurate for samples with many observations. For small samples like ours, `scipy.stats.monte_carlo_test` may provide a more accurate, albeit stochastic, approximation of the exact p-value.

    def statistic(x, axis):
        # underlying calculation of the Jarque Bera statistic
        s = stats.skew(x, axis=axis)
        k = stats.kurtosis(x, axis=axis)
        return x.shape[axis]/6 * (s**2 + k**2/4)

    res = stats.monte_carlo_test(x, stats.norm.rvs, statistic,
                                 alternative='greater')
    fig, ax = plt.subplots(figsize=(8, 5))
    jb_plot(ax)
    ax.hist(res.null_distribution, np.linspace(0, 10, 50),
            density=True)
    ax.legend(['asymptotic approximation (many observations)',
               'Monte Carlo approximation (11 observations)'])
    plt.show()

    res.pvalue

    np.float64(0.0068)

Furthermore, despite their stochastic nature, p-values computed in this way can be used to exactly control the rate of false rejections of the null hypothesis [3].

## References

[1]

Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611

[2]

Phipson, B. and Smyth, G. K. (2010) Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn. Statistical Applications in Genetics and Molecular Biology 9.1.

[3]

Panagiotakos, D. B. (2008). The value of p-value in biomedical research. The open cardiovascular medicine journal, 2, 97.