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+++ {"tags": ["jupyterlite_sphinx_strip"]}

```{eval-rst}
.. notebooklite:: hypothesis_normaltest.md
   :new_tab: True
```

(hypothesis_normaltest)=

+++

# Normal test

The {func}`scipy.stats.normaltest` function tests the null hypothesis that a
sample comes from a normal distribution. It is based on D’Agostino and
Pearson’s [^1] [^2] test that combines skew and kurtosis to produce an omnibus
test of normality.

Suppose we wish to infer from measurements whether the weights of adult human
males in a medical study are not normally distributed [^3]. The weights (lbs)
are recorded in the array `x` below.

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

The normality test {func}`scipy.stats.normaltest` of [^1] and [^2] begins by
computing a statistic based on the sample skewness and kurtosis.

```{code-cell}
from scipy import stats
res = stats.normaltest(x)
res.statistic
```

(The test warns that our sample has too few observations to perform the test.
We'll return to this at the end of the example.) 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 this
normality test, the null distribution for very large samples is the chi-squared
distribution with two degrees of freedom.

```{code-cell}
import matplotlib.pyplot as plt
dist = stats.chi2(df=2)
stat_vals = np.linspace(0, 16, 100)
pdf = dist.pdf(stat_vals)
fig, ax = plt.subplots(figsize=(8, 5))

def plot(ax):  # we'll reuse this
    ax.plot(stat_vals, pdf)
    ax.set_title("Normality Test Null Distribution")
    ax.set_xlabel("statistic")
    ax.set_ylabel("probability density")

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.

```{code-cell}
fig, ax = plt.subplots(figsize=(8, 5))
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, (13.5, 5e-4), (14, 5e-3), arrowprops=props)
i = stat_vals >= res.statistic  # index more extreme statistic values
ax.fill_between(stat_vals[i], y1=0, y2=pdf[i])
ax.set_xlim(8, 16)
ax.set_ylim(0, 0.01)
plt.show()
```

```{code-cell}
res.pvalue
```

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 [^4] 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.
This is the reason we received a warning at the beginning of the example; our
sample is quite small. In this case,
{class}`scipy.stats.monte_carlo_test` may provide a more accurate, albeit
stochastic, approximation of the exact p-value.

```{code-cell}
def statistic(x, axis):
    # Get only the `normaltest` statistic; ignore approximate p-value
    return stats.normaltest(x, axis=axis).statistic
res = stats.monte_carlo_test(x, stats.norm.rvs, statistic,
                             alternative='greater')
fig, ax = plt.subplots(figsize=(8, 5))
plot(ax)
ax.hist(res.null_distribution, np.linspace(0, 25, 50),
        density=True)
ax.legend(['asymptotic approximation (many observations)',
           'Monte Carlo approximation (11 observations)'])
ax.set_xlim(0, 14)
plt.show()
```

```{code-cell}
res.pvalue
```

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 [^5].

## References

[^1]: D'Agostino, R. B. (1971). An omnibus test of normality for moderate and
large sample size. Biometrika, 58, 341-348
[^2]: D'Agostino, R. and Pearson, E. S. (1973). Tests for departure from 
normality. Biometrika, 60, 613-622.
[^3]: Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for
normality (complete samples). Biometrika, 52(3/4), 591-611.
[^4]: 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.
[^5]: Panagiotakos, D. B. (2008). The value of p-value in biomedical research.
The open cardiovascular medicine journal, 2, 97.
