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

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

(hypothesis_bartlett)=

+++

# Bartlett's test for equal variances

In [^1], the influence of vitamin C on the tooth growth of guinea pigs was
investigated. In a control study, 60 subjects were divided into small dose,
medium dose, and large dose groups that received daily doses of 0.5, 1.0 and
2.0 mg of vitamin C, respectively. After 42 days, the tooth growth was measured.

The `small_dose`, `medium_dose`, and `large_dose` arrays below record
tooth growth measurements of the three groups in microns.

```{code-cell}
import numpy as np
small_dose = np.array([
    4.2, 11.5, 7.3, 5.8, 6.4, 10, 11.2, 11.2, 5.2, 7,
    15.2, 21.5, 17.6, 9.7, 14.5, 10, 8.2, 9.4, 16.5, 9.7
])
medium_dose = np.array([
    16.5, 16.5, 15.2, 17.3, 22.5, 17.3, 13.6, 14.5, 18.8, 15.5,
    19.7, 23.3, 23.6, 26.4, 20, 25.2, 25.8, 21.2, 14.5, 27.3
])
large_dose = np.array([
    23.6, 18.5, 33.9, 25.5, 26.4, 32.5, 26.7, 21.5, 23.3, 29.5,
    25.5, 26.4, 22.4, 24.5, 24.8, 30.9, 26.4, 27.3, 29.4, 23
])
```

The {func}`scipy.stats.bartlett` statistic is sensitive to differences in
variances between the samples.

```{code-cell}
from scipy import stats
res = stats.bartlett(small_dose, medium_dose, large_dose)
res.statistic
```

The value of the statistic tends to be high when there is a large difference in
variances.

We can test for inequality of variance among the groups by comparing the
observed value of the statistic against the null distribution: the distribution
of statistic values derived under the null hypothesis that the population
variances of the three groups are equal.

For this test, the null distribution follows the
{ref}`chi-square distribution <continuous-chi2>` as shown below.

```{code-cell}
import matplotlib.pyplot as plt
k = 3  # number of samples
dist = dist = stats.chi2(df=k-1)
val = np.linspace(0, 5, 100)
pdf = dist.pdf(val)
fig, ax = plt.subplots(figsize=(8, 5))

def plot(ax):  # we'll reuse this
    ax.plot(val, pdf, color='C0')
    ax.set_title("Bartlett Test Null Distribution")
    ax.set_xlabel("statistic")
    ax.set_ylabel("probability density")
    ax.set_xlim(0, 5)
    ax.set_ylim(0, 1)

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:.3f}\n(shaded area)')
props = dict(facecolor='black', width=1, headwidth=5, headlength=8)
_ = ax.annotate(annotation, (1.5, 0.22), (2.25, 0.3), arrowprops=props)
i = val >= res.statistic
ax.fill_between(val[i], y1=0, y2=pdf[i], color='C0')
plt.show()
```

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

If the p-value is "small" - that is, if there is a low probability of sampling
data from distributions with identical variances 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 variances of the groups are not
equal. 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).
- Small p-values are not evidence for a *large* effect; rather, they can
  only provide evidence for a "significant" effect, meaning that they are
  unlikely to have occurred under the null hypothesis.

Note that the chi-square distribution provides the null distribution when the
observations are normally distributed. For small samples drawn from non-normal
populations, it may be more appropriate to perform a permutation test: Under the
null hypothesis that all three samples were drawn from the same population, each
of the measurements is equally likely to have been observed in any of the three
samples. Therefore, we can form a randomized null distribution by calculating
the statistic under many randomly-generated partitionings of the observations
into the three samples.

```{code-cell}
def statistic(*samples):
    return stats.bartlett(*samples).statistic
ref = stats.permutation_test(
    (small_dose, medium_dose, large_dose), statistic,
    permutation_type='independent', alternative='greater'
)
fig, ax = plt.subplots(figsize=(8, 5))
plot(ax)
bins = np.linspace(0, 5, 25)
ax.hist(
    ref.null_distribution, bins=bins, density=True, facecolor="C1"
)
ax.legend(['asymptotic approximation\n(many observations)',
           'randomized null distribution'])
plot(ax)
plt.show()
```

```{code-cell}
ref.pvalue  # randomized test p-value
```

Note that there is significant disagreement between the p-value calculated here
and the asymptotic approximation returned by {func}`scipy.stats.bartlett` above.
The statistical inferences that can be drawn rigorously from a permutation test
are limited; nonetheless, they may be the preferred approach in many
circumstances [^3].

## References

[^1]: Bliss, C.I. (1952), The Statistics of Bioassay: With Special Reference to
the Vitamins, pp 499-503. {doi}`10.1016/C2013-0-12584-6`
[^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]: Ludbrook, J., & Dudley, H. (1998). Why permutation tests are superior to
t and F tests in biomedical research. The American Statistician, 52(2), 127-132.
