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tea_tasting.multiplicity #

Multiple hypothesis testing.

MultipleComparisonsResults #

Bases: DictsReprMixin, UserDict[object, ExperimentResult]

Multiple comparisons result.

to_arrow() #

Convert the object to a PyArrow Table.

Source code in src/tea_tasting/utils.py 
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@_cache_method
def to_arrow(self) -> pa.Table:
    """Convert the object to a PyArrow Table."""
    return pa.Table.from_pylist(self.to_dicts())

to_dicts() #

Convert the result to a sequence of dictionaries.

Source code in src/tea_tasting/multiplicity.py 
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@tea_tasting.utils._cache_method
def to_dicts(self) -> tuple[dict[str, object], ...]:
    """Convert the result to a sequence of dictionaries."""
    return tuple(
        {"comparison": str(comparison)} | metric_result
        for comparison, experiment_result in self.items()
        for metric_result in experiment_result.to_dicts()
    )

to_html(keys=None, formatter=get_and_format_num, *, max_rows=None, align=None, indent=None) #

Convert the object to HTML.

Default formatting rules:

  • If a name starts with "rel_" or equals to "power" consider it a percentage value. Round percentage values to 2 significant digits, multiply by 100 and add "%".
  • Round other values to 3 significant values.
  • If value is less than 0.001 or is greater than or equal to 10_000_000, format it in exponential presentation.
  • If a name ends with "_ci", consider it a confidence interval. Look up for attributes "{name}_lower" and "{name}_upper", and format the interval as "[{lower_bound}, {upper_bound}]".

Parameters:

Name Type Description Default
keys Sequence[str] | None

Keys to convert. If a key is not defined in the dictionary it's assumed to be None.

 None
formatter Callable[[dict[str, object], str], str]

Custom formatter function. It should accept a dictionary of metric result attributes and an attribute name, and return a formatted attribute value.

 get_and_format_num
max_rows int | None

Maximum number of rows to convert. If None, the default value will be used. If 0 or less, all rows will be converted.

 None
align Literal['auto', 'left', 'right'] | None

Column alignment mode:

  • "auto": left-align keys in default_text_keys, right-align all other keys.
  • "left": left-align all columns.
  • "right": right-align all columns.

If None, the default value will be used.

 None
indent str | None

Whitespace to insert for each indentation level. If None, do not indent.

 None

Returns:

Type Description
str

A table with results rendered as HTML.
Source code in src/tea_tasting/utils.py 
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def to_html(
    self,
    keys: Sequence[str] | None = None,
    formatter: Callable[[dict[str, object], str], str] = get_and_format_num,
    *,
    max_rows: int | None = None,
    align: Literal["auto", "left", "right"] | None = None,
    indent: str | None = None,
) -> str:
    """Convert the object to HTML.

    Default formatting rules:

    - If a name starts with `"rel_"` or equals to `"power"` consider it
        a percentage value. Round percentage values to 2 significant digits,
        multiply by `100` and add `"%"`.
    - Round other values to 3 significant values.
    - If value is less than `0.001` or is greater than or equal to `10_000_000`,
        format it in exponential presentation.
    - If a name ends with `"_ci"`, consider it a confidence interval.
        Look up for attributes `"{name}_lower"` and `"{name}_upper"`,
        and format the interval as `"[{lower_bound}, {upper_bound}]"`.

    Args:
        keys: Keys to convert. If a key is not defined in the dictionary
            it's assumed to be `None`.
        formatter: Custom formatter function. It should accept a dictionary
            of metric result attributes and an attribute name, and return
            a formatted attribute value.
        max_rows: Maximum number of rows to convert.
            If `None`, the default value will be used.
            If `0` or less, all rows will be converted.
        align: Column alignment mode:

            - `"auto"`: left-align keys in `default_text_keys`,
              right-align all other keys.
            - `"left"`: left-align all columns.
            - `"right"`: right-align all columns.

            If `None`, the default value will be used.
        indent: Whitespace to insert for each indentation level. If `None`,
            do not indent.

    Returns:
        A table with results rendered as HTML.
    """
    if keys is None:
        keys = self.default_keys
    if max_rows is None:
        max_rows = self.default_max_rows
    align = (
        self.default_align
        if align is None
        else check_scalar(align, "align", typ=str, in_={"auto", "left", "right"})
    )

    def get_cell_attrs(key: str) -> dict[str, str]:
        if align == "auto" and key in self.default_text_keys:
            return {"style": "text-align: left;"}
        return {}

    table = ET.Element(
        "table",
        {
            "class": "dataframe",
            "style": f"text-align: {'left' if align == 'left' else 'right'};",
        },
    )
    thead = ET.SubElement(table, "thead")
    thead_tr = ET.SubElement(thead, "tr")
    for key in keys:
        th = ET.SubElement(thead_tr, "th", get_cell_attrs(key))
        th.text = key
    tbody = ET.SubElement(table, "tbody")
    for pretty_dict in self.to_pretty_dicts(keys, formatter, max_rows=max_rows):
        tr = ET.SubElement(tbody, "tr")
        for key in keys:
            td = ET.SubElement(tr, "td", get_cell_attrs(key))
            td.text = pretty_dict[key]
    if indent is not None:
        ET.indent(table, space=indent)
    return ET.tostring(table, encoding="unicode", method="html")

to_markdown(keys=None, formatter=get_and_format_num, *, max_rows=None, align=None) #

Convert the object to a Markdown table.

This is a convenience wrapper around to_string(table_format="markdown").

Parameters:

Name Type Description Default
keys Sequence[str] | None

Keys to convert. If a key is not defined in the dictionary it's assumed to be None.

 None
formatter Callable[[dict[str, object], str], str]

Custom formatter function. It should accept a dictionary of metric result attributes and an attribute name, and return a formatted attribute value.

 get_and_format_num
max_rows int | None

Maximum number of rows to convert. If None, the default value will be used. If 0 or less, all rows will be converted.

 None
align Literal['auto', 'left', 'right'] | None

Column alignment mode:

  • "auto": left-align keys in default_text_keys, right-align all other keys.
  • "left": left-align all columns.
  • "right": right-align all columns.

If None, the default value will be used.

 None

Returns:

Type Description
str

A table with results rendered as Markdown.
Source code in src/tea_tasting/utils.py 
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def to_markdown(
    self,
    keys: Sequence[str] | None = None,
    formatter: Callable[[dict[str, object], str], str] = get_and_format_num,
    *,
    max_rows: int | None = None,
    align: Literal["auto", "left", "right"] | None = None,
) -> str:
    """Convert the object to a Markdown table.

    This is a convenience wrapper around `to_string(table_format="markdown")`.

    Args:
        keys: Keys to convert. If a key is not defined in the dictionary
            it's assumed to be `None`.
        formatter: Custom formatter function. It should accept a dictionary
            of metric result attributes and an attribute name, and return
            a formatted attribute value.
        max_rows: Maximum number of rows to convert.
            If `None`, the default value will be used.
            If `0` or less, all rows will be converted.
        align: Column alignment mode:

            - `"auto"`: left-align keys in `default_text_keys`,
              right-align all other keys.
            - `"left"`: left-align all columns.
            - `"right"`: right-align all columns.

            If `None`, the default value will be used.

    Returns:
        A table with results rendered as Markdown.
    """
    return self.to_string(
        keys,
        formatter,
        max_rows=max_rows,
        align=align,
        table_format="markdown",
    )

to_pandas() #

Convert the object to a Pandas DataFrame.

Source code in src/tea_tasting/utils.py 
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@_cache_method
def to_pandas(self) -> pd.DataFrame:
    """Convert the object to a Pandas DataFrame."""
    import pandas as pd  # noqa: PLC0415
    return pd.DataFrame.from_records(self.to_dicts())

to_polars() #

Convert the object to a Polars DataFrame.

Source code in src/tea_tasting/utils.py 
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@_cache_method
def to_polars(self) -> pl.DataFrame:
    """Convert the object to a Polars DataFrame."""
    import polars as pl  # noqa: PLC0415
    return pl.from_dicts(self.to_dicts())

to_pretty_dicts(keys=None, formatter=get_and_format_num, *, max_rows=None) #

Convert the object to a list of dictionaries with formatted values.

Default formatting rules:

  • If a name starts with "rel_" or equals to "power" consider it a percentage value. Round percentage values to 2 significant digits, multiply by 100 and add "%".
  • Round other values to 3 significant values.
  • If value is less than 0.001 or is greater than or equal to 10_000_000, format it in exponential presentation.
  • If a name ends with "_ci", consider it a confidence interval. Look up for attributes "{name}_lower" and "{name}_upper", and format the interval as "[{lower_bound}, {upper_bound}]".

Parameters:

Name Type Description Default
keys Sequence[str] | None

Keys to convert. If a key is not defined in the dictionary it's assumed to be None.

 None
formatter Callable[[dict[str, object], str], str]

Custom formatter function. It should accept a dictionary of metric result attributes and an attribute name, and return a formatted attribute value.

 get_and_format_num
max_rows int | None

Maximum number of rows to convert. If None, the default value will be used. If 0 or less, all rows will be converted.

 None

Returns:

Type Description
list[dict[str, str]]

List of dictionaries with formatted values.
Source code in src/tea_tasting/utils.py 
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def to_pretty_dicts(
    self,
    keys: Sequence[str] | None = None,
    formatter: Callable[[dict[str, object], str], str] = get_and_format_num,
    *,
    max_rows: int | None = None,
) -> list[dict[str, str]]:
    """Convert the object to a list of dictionaries with formatted values.

    Default formatting rules:

    - If a name starts with `"rel_"` or equals to `"power"` consider it
        a percentage value. Round percentage values to 2 significant digits,
        multiply by `100` and add `"%"`.
    - Round other values to 3 significant values.
    - If value is less than `0.001` or is greater than or equal to `10_000_000`,
        format it in exponential presentation.
    - If a name ends with `"_ci"`, consider it a confidence interval.
        Look up for attributes `"{name}_lower"` and `"{name}_upper"`,
        and format the interval as `"[{lower_bound}, {upper_bound}]"`.

    Args:
        keys: Keys to convert. If a key is not defined in the dictionary
            it's assumed to be `None`.
        formatter: Custom formatter function. It should accept a dictionary
            of metric result attributes and an attribute name, and return
            a formatted attribute value.
        max_rows: Maximum number of rows to convert.
            If `None`, the default value will be used.
            If `0` or less, all rows will be converted.

    Returns:
        List of dictionaries with formatted values.
    """
    pretty_dicts, _, _ = self._to_pretty_dicts(
        keys,
        formatter,
        max_rows=max_rows,
        escape_markdown=False,
    )
    return pretty_dicts

to_string(keys=None, formatter=get_and_format_num, *, max_rows=None, align=None, table_format='plain') #

Convert the object to a string.

Default formatting rules:

  • If a name starts with "rel_" or equals to "power" consider it a percentage value. Round percentage values to 2 significant digits, multiply by 100 and add "%".
  • Round other values to 3 significant values.
  • If value is less than 0.001 or is greater than or equal to 10_000_000, format it in exponential presentation.
  • If a name ends with "_ci", consider it a confidence interval. Look up for attributes "{name}_lower" and "{name}_upper", and format the interval as "[{lower_bound}, {upper_bound}]".

Parameters:

Name Type Description Default
keys Sequence[str] | None

Keys to convert. If a key is not defined in the dictionary it's assumed to be None.

 None
formatter Callable[[dict[str, object], str], str]

Custom formatter function. It should accept a dictionary of metric result attributes and an attribute name, and return a formatted attribute value.

 get_and_format_num
max_rows int | None

Maximum number of rows to convert. If None, the default value will be used. If 0 or less, all rows will be converted.

 None
align Literal['auto', 'left', 'right'] | None

Column alignment mode:

  • "auto": left-align keys in default_text_keys, right-align all other keys.
  • "left": left-align all columns.
  • "right": right-align all columns.

If None, the default value will be used.

 None
table_format Literal['plain', 'markdown']

Output table format:

  • "plain": plain text table.
  • "markdown": Markdown table.
 'plain'

Returns:

Type Description
str

A table with results rendered as string.
Source code in src/tea_tasting/utils.py 
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def to_string(
    self,
    keys: Sequence[str] | None = None,
    formatter: Callable[[dict[str, object], str], str] = get_and_format_num,
    *,
    max_rows: int | None = None,
    align: Literal["auto", "left", "right"] | None = None,
    table_format: Literal["plain", "markdown"] = "plain",
) -> str:
    """Convert the object to a string.

    Default formatting rules:

    - If a name starts with `"rel_"` or equals to `"power"` consider it
        a percentage value. Round percentage values to 2 significant digits,
        multiply by `100` and add `"%"`.
    - Round other values to 3 significant values.
    - If value is less than `0.001` or is greater than or equal to `10_000_000`,
        format it in exponential presentation.
    - If a name ends with `"_ci"`, consider it a confidence interval.
        Look up for attributes `"{name}_lower"` and `"{name}_upper"`,
        and format the interval as `"[{lower_bound}, {upper_bound}]"`.

    Args:
        keys: Keys to convert. If a key is not defined in the dictionary
            it's assumed to be `None`.
        formatter: Custom formatter function. It should accept a dictionary
            of metric result attributes and an attribute name, and return
            a formatted attribute value.
        max_rows: Maximum number of rows to convert.
            If `None`, the default value will be used.
            If `0` or less, all rows will be converted.
        align: Column alignment mode:

            - `"auto"`: left-align keys in `default_text_keys`,
              right-align all other keys.
            - `"left"`: left-align all columns.
            - `"right"`: right-align all columns.

            If `None`, the default value will be used.
        table_format: Output table format:

            - `"plain"`: plain text table.
            - `"markdown"`: Markdown table.

    Returns:
        A table with results rendered as string.
    """
    if keys is None:
        keys = self.default_keys
    if max_rows is None:
        max_rows = self.default_max_rows
    align = (
        self.default_align
        if align is None
        else check_scalar(align, "align", typ=str, in_={"auto", "left", "right"})
    )
    table_format = check_scalar(
        table_format,
        "table_format",
        typ=str,
        in_={"plain", "markdown"},
    )

    left_aligned_keys: set[str]
    if align == "auto":
        left_aligned_keys = set(self.default_text_keys)
    elif align == "left":
        left_aligned_keys = set(keys)
    else:
        left_aligned_keys = set()

    def justify(key: str, val: str, fillchar: str = " ") -> str:
        if key in left_aligned_keys:
            return val.ljust(widths[key], fillchar)
        return val.rjust(widths[key], fillchar)

    pretty_dicts, widths, key_labels = self._to_pretty_dicts(
        keys,
        formatter,
        max_rows=max_rows,
        escape_markdown=table_format == "markdown",
    )
    if table_format == "plain":
        rows = [" ".join(justify(key, key) for key in keys).rstrip()]
        rows.extend(
            " ".join(justify(key, pretty_dict[key]) for key in keys).rstrip()
            for pretty_dict in pretty_dicts
        )
        return "\n".join(rows)

    rows = [
        f"| {' | '.join(justify(key, key_labels[key]) for key in keys)} |",
        f"| {' | '.join(justify(key, ':', '-') for key in keys)} |",
    ]
    rows.extend(
        f"| {' | '.join(justify(key, pretty_dict[key]) for key in keys)} |"
        for pretty_dict in pretty_dicts
    )
    return "\n".join(rows)

with_defaults(*, keys=None, max_rows=None, align=None) #

Copies the object and sets the new default parameters.

Parameters:

Name Type Description Default
keys Sequence[str] | None

New default keys for the methods to_pretty_dicts, to_string, to_markdown, and to_html.

 None
max_rows int | None

New default max_rows for the methods to_pretty_dicts, to_string, to_markdown, and to_html.

 None
align Literal['auto', 'left', 'right'] | None

New default align for the methods to_string, to_markdown, and to_html.

 None

Returns:

Type Description
DictsReprMixinT

A copy of the object with the new default parameters.
Source code in src/tea_tasting/utils.py 
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def with_defaults(
    self: DictsReprMixinT,
    *,
    keys: Sequence[str] | None = None,
    max_rows: int | None = None,
    align: Literal["auto", "left", "right"] | None = None,
) -> DictsReprMixinT:
    """Copies the object and sets the new default parameters.

    Args:
        keys: New default `keys` for the methods `to_pretty_dicts`, `to_string`,
            `to_markdown`, and `to_html`.
        max_rows: New default `max_rows` for the methods `to_pretty_dicts`,
            `to_string`, `to_markdown`, and `to_html`.
        align: New default `align` for the methods `to_string`, `to_markdown`,
            and `to_html`.

    Returns:
        A copy of the object with the new default parameters.
    """
    new_instance = self.__class__.__new__(self.__class__)
    new_instance.__dict__.update(self.__dict__)  # type: ignore
    new_instance._cache = None
    if keys is not None:
        new_instance.default_keys = keys
    if max_rows is not None:
        new_instance.default_max_rows = max_rows
    if align is not None:
        new_instance.default_align = check_scalar(
            align,
            "align",
            typ=str,
            in_={"auto", "left", "right"},
        )
    return new_instance

with_keys(keys) #

Copies the object and sets the new default keys.

Parameters:

Name Type Description Default
keys Sequence[str]

New default keys for the methods to_pretty_dicts, to_string, to_markdown, and to_html.

 required

Returns:

Type Description
DictsReprMixinT

A copy of the object with the new default keys.
Source code in src/tea_tasting/utils.py 
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def with_keys(self: DictsReprMixinT, keys: Sequence[str]) -> DictsReprMixinT:
    """Copies the object and sets the new default `keys`.

    Args:
        keys: New default `keys` for the methods `to_pretty_dicts`, `to_string`,
            `to_markdown`, and `to_html`.

    Returns:
        A copy of the object with the new default `keys`.
    """
    return self.with_defaults(keys=keys)

with_max_rows(max_rows) #

Copies the object and sets the new default max_rows.

Parameters:

Name Type Description Default
max_rows int

New default max_rows for the methods to_pretty_dicts, to_string, to_markdown, and to_html.

 required

Returns:

Type Description
DictsReprMixinT

A copy of the object with the new default max_rows.
Source code in src/tea_tasting/utils.py 
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def with_max_rows(self: DictsReprMixinT, max_rows: int) -> DictsReprMixinT:
    """Copies the object and sets the new default `max_rows`.

    Args:
        max_rows: New default `max_rows` for the methods `to_pretty_dicts`,
            `to_string`, `to_markdown`, and `to_html`.

    Returns:
        A copy of the object with the new default `max_rows`.
    """
    return self.with_defaults(max_rows=max_rows)

adjust_fdr(experiment_results, metrics=None, *, alpha=None, arbitrary_dependence=False) #

Adjust p-value and alpha to control the false discovery rate (FDR).

The number of hypotheses tested is the total number of metrics included in the comparison in all experiment results. For example, if there are 3 experiments with 2 metrics in each, the number of hypotheses is 6.

The function performs one of the following corrections, depending on parameters:

  • Benjamini-Hochberg procedure, assuming non-negative correlation between hypotheses (arbitrary_dependence=False).
  • Benjamini-Yekutieli procedure, assuming arbitrary dependence between hypotheses (arbitrary_dependence=True).

The function adds the following attributes to the results:

  • pvalue_adj: The adjusted p-value, which should be compared with the unadjusted FDR (alpha).
  • alpha_adj: The adjusted FDR, which should be compared with the unadjusted p-value (pvalue).
  • null_rejected: A binary indicator (0 or 1) that shows whether the null hypothesis is rejected.

Parameters:

Name Type Description Default
experiment_results ExperimentResult | Mapping[object, ExperimentResult]

Experiment results.

 required
metrics str | set[str] | Sequence[str] | None

Metrics included in the comparison. If None, all metrics are included.

 None
alpha float | None

Significance level. If None, the value from global settings is used.

 None
arbitrary_dependence bool

If True, arbitrary dependence between hypotheses is assumed and Benjamini-Yekutieli procedure is performed. If False, non-negative correlation between hypotheses is assumed and Benjamini-Hochberg procedure is performed.

 False

Returns:

Type Description
MultipleComparisonsResults

The experiment results with adjusted p-values and alphas.
Parameter defaults

Default for parameter alpha can be changed using the config_context and set_config functions. See the Global configuration reference for details.

References

Examples:

>>> import polars as pl
>>> import tea_tasting as tt

>>> data = pl.concat((
...     tt.make_users_data(
...         rng=42,
...         orders_uplift=0.10,
...         revenue_uplift=0.15,
...         return_type="polars",
...     ),
...     tt.make_users_data(
...         rng=21,
...         orders_uplift=0.15,
...         revenue_uplift=0.20,
...         return_type="polars",
...     )
...         .filter(pl.col("variant").eq(1))
...         .with_columns(variant=pl.lit(2, pl.Int64)),
... ))
>>> data
shape: (6_046, 5)
┌──────┬─────────┬──────────┬────────┬─────────┐
│ user ┆ variant ┆ sessions ┆ orders ┆ revenue │
│ ---  ┆ ---     ┆ ---      ┆ ---    ┆ ---     │
│ i64  ┆ i64     ┆ i64      ┆ i64    ┆ f64     │
╞══════╪═════════╪══════════╪════════╪═════════╡
│ 0    ┆ 1       ┆ 2        ┆ 1      ┆ 9.58    │
│ 1    ┆ 0       ┆ 2        ┆ 1      ┆ 6.43    │
│ 2    ┆ 1       ┆ 2        ┆ 1      ┆ 8.3     │
│ 3    ┆ 1       ┆ 2        ┆ 1      ┆ 16.65   │
│ 4    ┆ 0       ┆ 1        ┆ 1      ┆ 7.14    │
│ …    ┆ …       ┆ …        ┆ …      ┆ …       │
│ 3989 ┆ 2       ┆ 4        ┆ 4      ┆ 34.93   │
│ 3991 ┆ 2       ┆ 1        ┆ 0      ┆ 0.0     │
│ 3992 ┆ 2       ┆ 3        ┆ 3      ┆ 27.96   │
│ 3994 ┆ 2       ┆ 2        ┆ 1      ┆ 17.22   │
│ 3998 ┆ 2       ┆ 3        ┆ 0      ┆ 0.0     │
└──────┴─────────┴──────────┴────────┴─────────┘

>>> experiment = tt.Experiment(
...     sessions_per_user=tt.Mean("sessions"),
...     orders_per_session=tt.RatioOfMeans("orders", "sessions"),
...     orders_per_user=tt.Mean("orders"),
...     revenue_per_user=tt.Mean("revenue"),
... )

>>> # Results without correction.
>>> results = experiment.analyze(data, control=0, all_variants=True)
>>> results
variants metric             control treatment rel_effect_size rel_effect_size_ci  pvalue
(0, 1)   sessions_per_user     2.00      1.98          -0.66%      [-3.7%, 2.5%]   0.674
(0, 1)   orders_per_session   0.266     0.289            8.8%      [-0.89%, 19%]  0.0762
(0, 1)   orders_per_user      0.530     0.573            8.0%       [-2.0%, 19%]   0.118
(0, 1)   revenue_per_user      5.24      5.99             14%        [2.1%, 28%]  0.0211
(0, 2)   sessions_per_user     2.00      2.02           0.98%      [-2.1%, 4.1%]   0.532
(0, 2)   orders_per_session   0.266     0.295             11%        [1.2%, 22%]  0.0273
(0, 2)   orders_per_user      0.530     0.594             12%        [1.7%, 23%]  0.0213
(0, 2)   revenue_per_user      5.24      6.25             19%        [6.6%, 33%] 0.00218

>>> # Success metrics.
>>> metrics = {"orders_per_user", "revenue_per_user"}

>>> # Benjamini-Hochberg procedure,
>>> # assuming non-negative correlation between hypotheses.
>>> adjusted_results_fdr = tt.adjust_fdr(results, metrics)
>>> adjusted_results_fdr
comparison metric           control treatment rel_effect_size  pvalue pvalue_adj
(0, 1)     orders_per_user    0.530     0.573            8.0%   0.118      0.118
(0, 1)     revenue_per_user    5.24      5.99             14%  0.0211     0.0284
(0, 2)     orders_per_user    0.530     0.594             12%  0.0213     0.0284
(0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.00872

>>> # The adjusted confidence level alpha.
>>> adjusted_results_fdr.with_keys((
...     "comparison",
...     "metric",
...     "control",
...     "treatment",
...     "rel_effect_size",
...     "pvalue",
...     "alpha_adj",
... ))
comparison metric           control treatment rel_effect_size  pvalue alpha_adj
(0, 1)     orders_per_user    0.530     0.573            8.0%   0.118    0.0500
(0, 1)     revenue_per_user    5.24      5.99             14%  0.0211    0.0375
(0, 2)     orders_per_user    0.530     0.594             12%  0.0213    0.0375
(0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.0375

>>> # Benjamini-Yekutieli procedure,
>>> # assuming arbitrary dependence between hypotheses.
>>> tt.adjust_fdr(results, metrics, arbitrary_dependence=True)
comparison metric           control treatment rel_effect_size  pvalue pvalue_adj
(0, 1)     orders_per_user    0.530     0.573            8.0%   0.118      0.245
(0, 1)     revenue_per_user    5.24      5.99             14%  0.0211     0.0592
(0, 2)     orders_per_user    0.530     0.594             12%  0.0213     0.0592
(0, 2)     revenue_per_user    5.24      6.25             19% 0.00218     0.0182
Source code in src/tea_tasting/multiplicity.py 
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def adjust_fdr(
    experiment_results: tea_tasting.experiment.ExperimentResult | Mapping[
        object, tea_tasting.experiment.ExperimentResult],
    metrics: str | set[str] | Sequence[str] | None = None,
    *,
    alpha: float | None = None,
    arbitrary_dependence: bool = False,
) -> MultipleComparisonsResults:
    """Adjust p-value and alpha to control the false discovery rate (FDR).

    The number of hypotheses tested is the total number of metrics included in
    the comparison in all experiment results. For example, if there are
    3 experiments with 2 metrics in each, the number of hypotheses is 6.

    The function performs one of the following corrections, depending on parameters:

    - Benjamini-Hochberg procedure, assuming non-negative correlation between
        hypotheses (`arbitrary_dependence=False`).
    - Benjamini-Yekutieli procedure, assuming arbitrary dependence between
        hypotheses (`arbitrary_dependence=True`).

    The function adds the following attributes to the results:

    - `pvalue_adj`: The adjusted p-value, which should be compared with
        the unadjusted FDR (`alpha`).
    - `alpha_adj`: The adjusted FDR, which should be compared with the unadjusted
        p-value (`pvalue`).
    - `null_rejected`: A binary indicator (`0` or `1`) that shows whether
        the null hypothesis is rejected.

    Args:
        experiment_results: Experiment results.
        metrics: Metrics included in the comparison.
            If `None`, all metrics are included.
        alpha: Significance level. If `None`, the value from global settings is used.
        arbitrary_dependence: If `True`, arbitrary dependence between hypotheses
            is assumed and Benjamini-Yekutieli procedure is performed.
            If `False`, non-negative correlation between hypotheses is assumed
            and Benjamini-Hochberg procedure is performed.

    Returns:
        The experiment results with adjusted p-values and alphas.

    Parameter defaults:
        Default for parameter `alpha` can be changed using the `config_context`
        and `set_config` functions.
        See the [Global configuration](https://tea-tasting.e10v.me/api/config/)
        reference for details.

    References:
        - [Multiple comparisons problem](https://en.wikipedia.org/wiki/Multiple_comparisons_problem).
        - [False discovery rate](https://en.wikipedia.org/wiki/False_discovery_rate).

    Examples:
        ```pycon
        >>> import polars as pl
        >>> import tea_tasting as tt

        >>> data = pl.concat((
        ...     tt.make_users_data(
        ...         rng=42,
        ...         orders_uplift=0.10,
        ...         revenue_uplift=0.15,
        ...         return_type="polars",
        ...     ),
        ...     tt.make_users_data(
        ...         rng=21,
        ...         orders_uplift=0.15,
        ...         revenue_uplift=0.20,
        ...         return_type="polars",
        ...     )
        ...         .filter(pl.col("variant").eq(1))
        ...         .with_columns(variant=pl.lit(2, pl.Int64)),
        ... ))
        >>> data
        shape: (6_046, 5)
        ┌──────┬─────────┬──────────┬────────┬─────────┐
        │ user ┆ variant ┆ sessions ┆ orders ┆ revenue │
        │ ---  ┆ ---     ┆ ---      ┆ ---    ┆ ---     │
        │ i64  ┆ i64     ┆ i64      ┆ i64    ┆ f64     │
        ╞══════╪═════════╪══════════╪════════╪═════════╡
        │ 0    ┆ 1       ┆ 2        ┆ 1      ┆ 9.58    │
        │ 1    ┆ 0       ┆ 2        ┆ 1      ┆ 6.43    │
        │ 2    ┆ 1       ┆ 2        ┆ 1      ┆ 8.3     │
        │ 3    ┆ 1       ┆ 2        ┆ 1      ┆ 16.65   │
        │ 4    ┆ 0       ┆ 1        ┆ 1      ┆ 7.14    │
        │ …    ┆ …       ┆ …        ┆ …      ┆ …       │
        │ 3989 ┆ 2       ┆ 4        ┆ 4      ┆ 34.93   │
        │ 3991 ┆ 2       ┆ 1        ┆ 0      ┆ 0.0     │
        │ 3992 ┆ 2       ┆ 3        ┆ 3      ┆ 27.96   │
        │ 3994 ┆ 2       ┆ 2        ┆ 1      ┆ 17.22   │
        │ 3998 ┆ 2       ┆ 3        ┆ 0      ┆ 0.0     │
        └──────┴─────────┴──────────┴────────┴─────────┘

        >>> experiment = tt.Experiment(
        ...     sessions_per_user=tt.Mean("sessions"),
        ...     orders_per_session=tt.RatioOfMeans("orders", "sessions"),
        ...     orders_per_user=tt.Mean("orders"),
        ...     revenue_per_user=tt.Mean("revenue"),
        ... )

        >>> # Results without correction.
        >>> results = experiment.analyze(data, control=0, all_variants=True)
        >>> results
        variants metric             control treatment rel_effect_size rel_effect_size_ci  pvalue
        (0, 1)   sessions_per_user     2.00      1.98          -0.66%      [-3.7%, 2.5%]   0.674
        (0, 1)   orders_per_session   0.266     0.289            8.8%      [-0.89%, 19%]  0.0762
        (0, 1)   orders_per_user      0.530     0.573            8.0%       [-2.0%, 19%]   0.118
        (0, 1)   revenue_per_user      5.24      5.99             14%        [2.1%, 28%]  0.0211
        (0, 2)   sessions_per_user     2.00      2.02           0.98%      [-2.1%, 4.1%]   0.532
        (0, 2)   orders_per_session   0.266     0.295             11%        [1.2%, 22%]  0.0273
        (0, 2)   orders_per_user      0.530     0.594             12%        [1.7%, 23%]  0.0213
        (0, 2)   revenue_per_user      5.24      6.25             19%        [6.6%, 33%] 0.00218

        >>> # Success metrics.
        >>> metrics = {"orders_per_user", "revenue_per_user"}

        >>> # Benjamini-Hochberg procedure,
        >>> # assuming non-negative correlation between hypotheses.
        >>> adjusted_results_fdr = tt.adjust_fdr(results, metrics)
        >>> adjusted_results_fdr
        comparison metric           control treatment rel_effect_size  pvalue pvalue_adj
        (0, 1)     orders_per_user    0.530     0.573            8.0%   0.118      0.118
        (0, 1)     revenue_per_user    5.24      5.99             14%  0.0211     0.0284
        (0, 2)     orders_per_user    0.530     0.594             12%  0.0213     0.0284
        (0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.00872

        >>> # The adjusted confidence level alpha.
        >>> adjusted_results_fdr.with_keys((
        ...     "comparison",
        ...     "metric",
        ...     "control",
        ...     "treatment",
        ...     "rel_effect_size",
        ...     "pvalue",
        ...     "alpha_adj",
        ... ))
        comparison metric           control treatment rel_effect_size  pvalue alpha_adj
        (0, 1)     orders_per_user    0.530     0.573            8.0%   0.118    0.0500
        (0, 1)     revenue_per_user    5.24      5.99             14%  0.0211    0.0375
        (0, 2)     orders_per_user    0.530     0.594             12%  0.0213    0.0375
        (0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.0375

        >>> # Benjamini-Yekutieli procedure,
        >>> # assuming arbitrary dependence between hypotheses.
        >>> tt.adjust_fdr(results, metrics, arbitrary_dependence=True)
        comparison metric           control treatment rel_effect_size  pvalue pvalue_adj
        (0, 1)     orders_per_user    0.530     0.573            8.0%   0.118      0.245
        (0, 1)     revenue_per_user    5.24      5.99             14%  0.0211     0.0592
        (0, 2)     orders_per_user    0.530     0.594             12%  0.0213     0.0592
        (0, 2)     revenue_per_user    5.24      6.25             19% 0.00218     0.0182

        ```
    """  # noqa: E501
    alpha = (
        tea_tasting.utils.auto_check(alpha, "alpha")
        if alpha is not None
        else tea_tasting.config.get_config("alpha")
    )
    arbitrary_dependence = tea_tasting.utils.check_scalar(
        arbitrary_dependence, "arbitrary_dependence", typ=bool)

    # results and metric_results refer to the same dicts.
    results, metric_results = _copy_results(experiment_results, metrics)
    method = _Benjamini(
        alpha=alpha,  # type: ignore
        m=len(metric_results),
        arbitrary_dependence=arbitrary_dependence,
    )
    # In-place update.
    _hochberg_stepup(metric_results, method.adjust)

    return MultipleComparisonsResults(results)

adjust_fwer(experiment_results, metrics=None, *, alpha=None, arbitrary_dependence=False, method='sidak') #

Adjust p-value and alpha to control the family-wise error rate (FWER).

The number of hypotheses tested is the total number of metrics included in the comparison in all experiment results. For example, if there are 3 experiments with 2 metrics in each, the number of hypotheses is 6.

The function performs one of the following procedures, depending on parameters:

  • Hochberg's step-up procedure, assuming non-negative correlation between hypotheses (arbitrary_dependence=False).
  • Holm's step-down procedure, assuming arbitrary dependence between hypotheses (arbitrary_dependence=True).

The function adds the following attributes to the results:

  • pvalue_adj: The adjusted p-value, which should be compared with the unadjusted FDR (alpha).
  • alpha_adj: The adjusted FWER, which should be compared with the unadjusted p-value (pvalue).
  • null_rejected: A binary indicator (0 or 1) that shows whether the null hypothesis is rejected.

Parameters:

Name Type Description Default
experiment_results ExperimentResult | Mapping[object, ExperimentResult]

Experiment results.

 required
metrics str | set[str] | Sequence[str] | None

Metrics included in the comparison. If None, all metrics are included.

 None
alpha float | None

Significance level. If None, the value from global settings is used.

 None
arbitrary_dependence bool

If True, arbitrary dependence between hypotheses is assumed and Holm's step-down procedure is performed. If False, non-negative correlation between hypotheses is assumed and Hochberg's step-up procedure is performed.

 False
method Literal['bonferroni', 'sidak']

Correction method, Bonferroni ("bonferroni") or Šidák ("sidak").

 'sidak'

Returns:

Type Description
MultipleComparisonsResults

The experiment results with adjusted p-values and alphas.
Parameter defaults

Default for parameter alpha can be changed using the config_context and set_config functions. See the Global configuration reference for details.

References

Examples:

>>> import polars as pl
>>> import tea_tasting as tt

>>> data = pl.concat((
...     tt.make_users_data(
...         rng=42,
...         orders_uplift=0.10,
...         revenue_uplift=0.15,
...         return_type="polars",
...     ),
...     tt.make_users_data(
...         rng=21,
...         orders_uplift=0.15,
...         revenue_uplift=0.20,
...         return_type="polars",
...     )
...         .filter(pl.col("variant").eq(1))
...         .with_columns(variant=pl.lit(2, pl.Int64)),
... ))
>>> data
shape: (6_046, 5)
┌──────┬─────────┬──────────┬────────┬─────────┐
│ user ┆ variant ┆ sessions ┆ orders ┆ revenue │
│ ---  ┆ ---     ┆ ---      ┆ ---    ┆ ---     │
│ i64  ┆ i64     ┆ i64      ┆ i64    ┆ f64     │
╞══════╪═════════╪══════════╪════════╪═════════╡
│ 0    ┆ 1       ┆ 2        ┆ 1      ┆ 9.58    │
│ 1    ┆ 0       ┆ 2        ┆ 1      ┆ 6.43    │
│ 2    ┆ 1       ┆ 2        ┆ 1      ┆ 8.3     │
│ 3    ┆ 1       ┆ 2        ┆ 1      ┆ 16.65   │
│ 4    ┆ 0       ┆ 1        ┆ 1      ┆ 7.14    │
│ …    ┆ …       ┆ …        ┆ …      ┆ …       │
│ 3989 ┆ 2       ┆ 4        ┆ 4      ┆ 34.93   │
│ 3991 ┆ 2       ┆ 1        ┆ 0      ┆ 0.0     │
│ 3992 ┆ 2       ┆ 3        ┆ 3      ┆ 27.96   │
│ 3994 ┆ 2       ┆ 2        ┆ 1      ┆ 17.22   │
│ 3998 ┆ 2       ┆ 3        ┆ 0      ┆ 0.0     │
└──────┴─────────┴──────────┴────────┴─────────┘

>>> experiment = tt.Experiment(
...     sessions_per_user=tt.Mean("sessions"),
...     orders_per_session=tt.RatioOfMeans("orders", "sessions"),
...     orders_per_user=tt.Mean("orders"),
...     revenue_per_user=tt.Mean("revenue"),
... )

>>> # Results without correction.
>>> results = experiment.analyze(data, control=0, all_variants=True)
>>> results
variants metric             control treatment rel_effect_size rel_effect_size_ci  pvalue
(0, 1)   sessions_per_user     2.00      1.98          -0.66%      [-3.7%, 2.5%]   0.674
(0, 1)   orders_per_session   0.266     0.289            8.8%      [-0.89%, 19%]  0.0762
(0, 1)   orders_per_user      0.530     0.573            8.0%       [-2.0%, 19%]   0.118
(0, 1)   revenue_per_user      5.24      5.99             14%        [2.1%, 28%]  0.0211
(0, 2)   sessions_per_user     2.00      2.02           0.98%      [-2.1%, 4.1%]   0.532
(0, 2)   orders_per_session   0.266     0.295             11%        [1.2%, 22%]  0.0273
(0, 2)   orders_per_user      0.530     0.594             12%        [1.7%, 23%]  0.0213
(0, 2)   revenue_per_user      5.24      6.25             19%        [6.6%, 33%] 0.00218

>>> # Success metrics.
>>> metrics = {"orders_per_user", "revenue_per_user"}

>>> # Hochberg's step-up procedure with Šidák correction,
>>> # assuming non-negative correlation between hypotheses.
>>> adjusted_results_fwer = tt.adjust_fwer(results, metrics)
>>> adjusted_results_fwer
comparison metric           control treatment rel_effect_size  pvalue pvalue_adj
(0, 1)     orders_per_user    0.530     0.573            8.0%   0.118      0.118
(0, 1)     revenue_per_user    5.24      5.99             14%  0.0211     0.0422
(0, 2)     orders_per_user    0.530     0.594             12%  0.0213     0.0422
(0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.00869

>>> # The adjusted confidence level alpha.
>>> adjusted_results_fwer.with_keys((
...     "comparison",
...     "metric",
...     "control",
...     "treatment",
...     "rel_effect_size",
...     "pvalue",
...     "alpha_adj",
... ))
comparison metric           control treatment rel_effect_size  pvalue alpha_adj
(0, 1)     orders_per_user    0.530     0.573            8.0%   0.118    0.0500
(0, 1)     revenue_per_user    5.24      5.99             14%  0.0211    0.0253
(0, 2)     orders_per_user    0.530     0.594             12%  0.0213    0.0253
(0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.0253

>>> # Holm's step-down procedure with Bonferroni correction,
>>> # assuming arbitrary dependence between hypotheses.
>>> tt.adjust_fwer(
...     results,
...     metrics,
...     arbitrary_dependence=True,
...     method="bonferroni",
... )
comparison metric           control treatment rel_effect_size  pvalue pvalue_adj
(0, 1)     orders_per_user    0.530     0.573            8.0%   0.118      0.118
(0, 1)     revenue_per_user    5.24      5.99             14%  0.0211     0.0634
(0, 2)     orders_per_user    0.530     0.594             12%  0.0213     0.0634
(0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.00872
Source code in src/tea_tasting/multiplicity.py 
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def adjust_fwer(
    experiment_results: tea_tasting.experiment.ExperimentResult | Mapping[
        object, tea_tasting.experiment.ExperimentResult],
    metrics: str | set[str] | Sequence[str] | None = None,
    *,
    alpha: float | None = None,
    arbitrary_dependence: bool = False,
    method: Literal["bonferroni", "sidak"] = "sidak",
) -> MultipleComparisonsResults:
    """Adjust p-value and alpha to control the family-wise error rate (FWER).

    The number of hypotheses tested is the total number of metrics included in
    the comparison in all experiment results. For example, if there are
    3 experiments with 2 metrics in each, the number of hypotheses is 6.

    The function performs one of the following procedures, depending on parameters:

    - Hochberg's step-up procedure, assuming non-negative correlation between
        hypotheses (`arbitrary_dependence=False`).
    - Holm's step-down procedure, assuming arbitrary dependence between
        hypotheses (`arbitrary_dependence=True`).

    The function adds the following attributes to the results:

    - `pvalue_adj`: The adjusted p-value, which should be compared with
        the unadjusted FDR (`alpha`).
    - `alpha_adj`: The adjusted FWER, which should be compared with the unadjusted
        p-value (`pvalue`).
    - `null_rejected`: A binary indicator (`0` or `1`) that shows whether
        the null hypothesis is rejected.

    Args:
        experiment_results: Experiment results.
        metrics: Metrics included in the comparison.
            If `None`, all metrics are included.
        alpha: Significance level. If `None`, the value from global settings is used.
        arbitrary_dependence: If `True`, arbitrary dependence between hypotheses
            is assumed and Holm's step-down procedure is performed.
            If `False`, non-negative correlation between hypotheses is assumed
            and Hochberg's step-up procedure is performed.
        method: Correction method, Bonferroni (`"bonferroni"`) or Šidák (`"sidak"`).

    Returns:
        The experiment results with adjusted p-values and alphas.

    Parameter defaults:
        Default for parameter `alpha` can be changed using the `config_context`
        and `set_config` functions.
        See the [Global configuration](https://tea-tasting.e10v.me/api/config/)
        reference for details.

    References:
        - [Multiple comparisons problem](https://en.wikipedia.org/wiki/Multiple_comparisons_problem).
        - [Family-wise error rate](https://en.wikipedia.org/wiki/Family-wise_error_rate).
        - [Holm–Bonferroni method](https://en.wikipedia.org/wiki/Holm%E2%80%93Bonferroni_method).

    Examples:
        ```pycon
        >>> import polars as pl
        >>> import tea_tasting as tt

        >>> data = pl.concat((
        ...     tt.make_users_data(
        ...         rng=42,
        ...         orders_uplift=0.10,
        ...         revenue_uplift=0.15,
        ...         return_type="polars",
        ...     ),
        ...     tt.make_users_data(
        ...         rng=21,
        ...         orders_uplift=0.15,
        ...         revenue_uplift=0.20,
        ...         return_type="polars",
        ...     )
        ...         .filter(pl.col("variant").eq(1))
        ...         .with_columns(variant=pl.lit(2, pl.Int64)),
        ... ))
        >>> data
        shape: (6_046, 5)
        ┌──────┬─────────┬──────────┬────────┬─────────┐
        │ user ┆ variant ┆ sessions ┆ orders ┆ revenue │
        │ ---  ┆ ---     ┆ ---      ┆ ---    ┆ ---     │
        │ i64  ┆ i64     ┆ i64      ┆ i64    ┆ f64     │
        ╞══════╪═════════╪══════════╪════════╪═════════╡
        │ 0    ┆ 1       ┆ 2        ┆ 1      ┆ 9.58    │
        │ 1    ┆ 0       ┆ 2        ┆ 1      ┆ 6.43    │
        │ 2    ┆ 1       ┆ 2        ┆ 1      ┆ 8.3     │
        │ 3    ┆ 1       ┆ 2        ┆ 1      ┆ 16.65   │
        │ 4    ┆ 0       ┆ 1        ┆ 1      ┆ 7.14    │
        │ …    ┆ …       ┆ …        ┆ …      ┆ …       │
        │ 3989 ┆ 2       ┆ 4        ┆ 4      ┆ 34.93   │
        │ 3991 ┆ 2       ┆ 1        ┆ 0      ┆ 0.0     │
        │ 3992 ┆ 2       ┆ 3        ┆ 3      ┆ 27.96   │
        │ 3994 ┆ 2       ┆ 2        ┆ 1      ┆ 17.22   │
        │ 3998 ┆ 2       ┆ 3        ┆ 0      ┆ 0.0     │
        └──────┴─────────┴──────────┴────────┴─────────┘

        >>> experiment = tt.Experiment(
        ...     sessions_per_user=tt.Mean("sessions"),
        ...     orders_per_session=tt.RatioOfMeans("orders", "sessions"),
        ...     orders_per_user=tt.Mean("orders"),
        ...     revenue_per_user=tt.Mean("revenue"),
        ... )

        >>> # Results without correction.
        >>> results = experiment.analyze(data, control=0, all_variants=True)
        >>> results
        variants metric             control treatment rel_effect_size rel_effect_size_ci  pvalue
        (0, 1)   sessions_per_user     2.00      1.98          -0.66%      [-3.7%, 2.5%]   0.674
        (0, 1)   orders_per_session   0.266     0.289            8.8%      [-0.89%, 19%]  0.0762
        (0, 1)   orders_per_user      0.530     0.573            8.0%       [-2.0%, 19%]   0.118
        (0, 1)   revenue_per_user      5.24      5.99             14%        [2.1%, 28%]  0.0211
        (0, 2)   sessions_per_user     2.00      2.02           0.98%      [-2.1%, 4.1%]   0.532
        (0, 2)   orders_per_session   0.266     0.295             11%        [1.2%, 22%]  0.0273
        (0, 2)   orders_per_user      0.530     0.594             12%        [1.7%, 23%]  0.0213
        (0, 2)   revenue_per_user      5.24      6.25             19%        [6.6%, 33%] 0.00218

        >>> # Success metrics.
        >>> metrics = {"orders_per_user", "revenue_per_user"}

        >>> # Hochberg's step-up procedure with Šidák correction,
        >>> # assuming non-negative correlation between hypotheses.
        >>> adjusted_results_fwer = tt.adjust_fwer(results, metrics)
        >>> adjusted_results_fwer
        comparison metric           control treatment rel_effect_size  pvalue pvalue_adj
        (0, 1)     orders_per_user    0.530     0.573            8.0%   0.118      0.118
        (0, 1)     revenue_per_user    5.24      5.99             14%  0.0211     0.0422
        (0, 2)     orders_per_user    0.530     0.594             12%  0.0213     0.0422
        (0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.00869

        >>> # The adjusted confidence level alpha.
        >>> adjusted_results_fwer.with_keys((
        ...     "comparison",
        ...     "metric",
        ...     "control",
        ...     "treatment",
        ...     "rel_effect_size",
        ...     "pvalue",
        ...     "alpha_adj",
        ... ))
        comparison metric           control treatment rel_effect_size  pvalue alpha_adj
        (0, 1)     orders_per_user    0.530     0.573            8.0%   0.118    0.0500
        (0, 1)     revenue_per_user    5.24      5.99             14%  0.0211    0.0253
        (0, 2)     orders_per_user    0.530     0.594             12%  0.0213    0.0253
        (0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.0253

        >>> # Holm's step-down procedure with Bonferroni correction,
        >>> # assuming arbitrary dependence between hypotheses.
        >>> tt.adjust_fwer(
        ...     results,
        ...     metrics,
        ...     arbitrary_dependence=True,
        ...     method="bonferroni",
        ... )
        comparison metric           control treatment rel_effect_size  pvalue pvalue_adj
        (0, 1)     orders_per_user    0.530     0.573            8.0%   0.118      0.118
        (0, 1)     revenue_per_user    5.24      5.99             14%  0.0211     0.0634
        (0, 2)     orders_per_user    0.530     0.594             12%  0.0213     0.0634
        (0, 2)     revenue_per_user    5.24      6.25             19% 0.00218    0.00872

        ```
    """  # noqa: E501, RUF002
    alpha = (
        tea_tasting.utils.auto_check(alpha, "alpha")
        if alpha is not None
        else tea_tasting.config.get_config("alpha")
    )
    method = tea_tasting.utils.check_scalar(
        method, "method", typ=str, in_={"sidak", "bonferroni"})
    arbitrary_dependence = tea_tasting.utils.check_scalar(
        arbitrary_dependence, "arbitrary_dependence", typ=bool)

    # results and metric_results refer to the same dicts.
    results, metric_results = _copy_results(experiment_results, metrics)
    method_cls = _Sidak if method == "sidak" else _Bonferroni
    method_ = method_cls(alpha=alpha, m=len(metric_results))  # type: ignore
    procedure = _holm_stepdown if arbitrary_dependence else _hochberg_stepup
    # In-place update.
    procedure(metric_results, method_.adjust)

    return MultipleComparisonsResults(results)