pystatpower.mean.independent.inequality ¶

FUNCTION	DESCRIPTION
`solve_power`	Calculate the statistical power.
`solve_size`	Estimate the required sample size.
`solve_diff`	Estimate the required mean difference.
`solve_treatment_mean`	Estimate the required mean in the treatment group.
`solve_reference_mean`	Estimate the required mean in the reference group.
`solve_treatment_std`	Estimate the required standard deviation in the treatment group.
`solve_reference_std`	Estimate the required standard deviation in the reference group.

solve_power ¶

solve_power(
    *,
    treatment_mean: float | None = None,
    reference_mean: float | None = None,
    diff: float | None = None,
    treatment_std: float | None = None,
    reference_std: float | None = None,
    std: float | None = None,
    treatment_size: int,
    reference_size: int,
    alternative: Literal["two-sided", "greater", "less"] = "two-sided",
    alpha: float = 0.05,
    dist: Literal["z", "t"] = "t",
    equal_var: bool = False,
    approx_t_method: Literal["welch", "satterthwaite"] = "welch",
) -> float

Calculate the statistical power.

PARAMETER	DESCRIPTION
`treatment_mean`	Mean in the treatment group. If `diff` is omitted, this parameter is required along with `reference_mean`. TYPE: `float \| None` DEFAULT: `None`
`reference_mean`	Mean in the reference group. If `diff` is omitted, this parameter is required along with `treatment_mean`. TYPE: `float \| None` DEFAULT: `None`
`diff`	Mean difference between treatment and reference group. If both `treatment_mean` and `reference_mean` are not specified, this parameter is required. TYPE: `float \| None` DEFAULT: `None`
`treatment_std`	Standard deviation in the treatment group. TYPE: `float \| None` DEFAULT: `None`
`reference_std`	Standard deviation in the reference group. TYPE: `float \| None` DEFAULT: `None`
`std`	Standard deviation in both groups. This is a convenience parameter that will override `treatment_std` and `reference_std` when `dist` is `z` and `equal_var` is `True`. If you specify `dist` as `z` and `equal_var` as `True`, you can just specify `std` instead of `treatment_std` and `reference_std`. Internally, the value of `std` will be treated as the standard deviation of both the treatment and reference groups. TYPE: `float \| None` DEFAULT: `None`
`treatment_size`	Sample size in the treatment group. TYPE: `int`
`reference_size`	Sample size in the reference group. TYPE: `int`
`alternative`	Type of the alternative hypothesis. If `alternative` is `'two-sided'`, the alternative hypothesis \(\mu_1 \neq \mu_2\) If `alternative` is `'greater'`, the alternative hypothesis \(\mu_1 > \mu_2\) If `alternative` is `'less'`, the alternative hypothesis \(\mu_1 < \mu_2\) TYPE: `Literal['two-sided', 'greater', 'less']` DEFAULT: `'two-sided'`
`alpha`	Significance level. If `alternative` is `'two-sided'`, `alpha` represents the two-sided significance level. If `alternative` is `'greater'` or `'less'`, `alpha` represents the one-sided significance level. TYPE: `float` DEFAULT: `0.05`
`dist`	The distribution used for the test. `'z'`: Standard normal distribution. `'t'`: Student's or non-central t distribution. TYPE: `Literal['z', 't']` DEFAULT: `'t'`
`equal_var`	Whether to assume equal variances between treatment and reference groups. `True`: Variances are assumed equal. `False`: Variances are assumed unequal. TYPE: `bool` DEFAULT: `False`
`approx_t_method`	Approximate t-test method. It is used when `dist` is `'t'` and `equal_var` = `False`. `'welch'`: Welch's approximate t-test (1947). `'satterthwaite'`: Satterthwaite's approximate t-test (1946). TYPE: `Literal['welch', 'satterthwaite']` DEFAULT: `'welch'`

RETURNS	DESCRIPTION
`float`	The calculated statistical power of the test.

RAISES	DESCRIPTION
`ValueError`	If all of `diff`, `treatment_mean` and `reference_mean` are omitted.
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and all `treatment_std`, `reference_std` and `std` are omitted.
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and both `treatment_std` and `reference_std` are provided, but they are not equal.

solve_size ¶

solve_size(
    *,
    treatment_mean: float | None = None,
    reference_mean: float | None = None,
    diff: float | None = None,
    treatment_std: float | None = None,
    reference_std: float | None = None,
    std: float | None = None,
    ratio: float = 1,
    alternative: Literal["two-sided", "greater", "less"] = "two-sided",
    alpha: float = 0.05,
    power: float = 0.8,
    dist: Literal["z", "t"] = "t",
    equal_var: bool = False,
    approx_t_method: Literal["welch", "satterthwaite"] = "welch",
) -> tuple[int, int]

Estimate the required sample size.

PARAMETER	DESCRIPTION
`treatment_mean`	Mean in the treatment group. If `diff` is omitted, this parameter is required along with `reference_mean`. TYPE: `float \| None` DEFAULT: `None`
`reference_mean`	Mean in the reference group. If `diff` is omitted, this parameter is required along with `treatment_mean`. TYPE: `float \| None` DEFAULT: `None`
`diff`	Mean difference between treatment and reference group. If both `treatment_mean` and `reference_mean` are not specified, this parameter is required. TYPE: `float \| None` DEFAULT: `None`
`treatment_std`	Standard deviation in the treatment group. TYPE: `float \| None` DEFAULT: `None`
`reference_std`	Standard deviation in the reference group. TYPE: `float \| None` DEFAULT: `None`
`std`	Standard deviation in both groups. This is a convenience parameter that will override `treatment_std` and `reference_std` when `dist` is `z` and `equal_var` is `True`. If you specify `dist` as `z` and `equal_var` as `True`, you can just specify `std` instead of `treatment_std` and `reference_std`. Internally, the value of `std` will be treated as the standard deviation of both the treatment and reference groups. TYPE: `float \| None` DEFAULT: `None`
`ratio`	Ratio of sample sizes in the treatment and reference groups. TYPE: `float` DEFAULT: `1`
`alternative`	Type of the alternative hypothesis. If `alternative` is `'two-sided'`, the alternative hypothesis \(\mu_1 \neq \mu_2\) If `alternative` is `'greater'`, the alternative hypothesis \(\mu_1 > \mu_2\) If `alternative` is `'less'`, the alternative hypothesis \(\mu_1 < \mu_2\) TYPE: `Literal['two-sided', 'greater', 'less']` DEFAULT: `'two-sided'`
`alpha`	Significance level. If `alternative` is `'two-sided'`, `alpha` represents the two-sided significance level. If `alternative` is `'greater'` or `'less'`, `alpha` represents the one-sided significance level. TYPE: `float` DEFAULT: `0.05`
`power`	Desired statistical power. TYPE: `float` DEFAULT: `0.8`
`dist`	The distribution used for the test. `'z'`: Standard normal distribution. `'t'`: Student's or non-central t distribution. TYPE: `Literal['z', 't']` DEFAULT: `'t'`
`equal_var`	Whether to assume equal variances between treatment and reference groups. `True`: Variances are assumed equal. `False`: Variances are assumed unequal. TYPE: `bool` DEFAULT: `False`
`approx_t_method`	Approximate t-test method. It is used when `dist` is `'t'` and `equal_var` = `False`. `'welch'`: Welch's approximate t-test (1947). `'satterthwaite'`: Satterthwaite's approximate t-test (1946). TYPE: `Literal['welch', 'satterthwaite']` DEFAULT: `'welch'`

RETURNS	DESCRIPTION
`tuple[int, int]`	The required sample sizes in the treatment and reference groups, respectively.

RAISES	DESCRIPTION
`ValueError`	If all of `diff`, `treatment_mean` and `reference_mean` are omitted.
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and all `treatment_std`, `reference_std` and `std` are omitted.
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and both `treatment_std` and `reference_std` are provided, but they are not equal.

solve_diff ¶

solve_diff(
    *,
    treatment_std: float | None = None,
    reference_std: float | None = None,
    std: float | None = None,
    treatment_size: int,
    reference_size: int,
    alternative: Literal["two-sided", "greater", "less"] = "two-sided",
    alpha: float = 0.05,
    power: float = 0.8,
    dist: Literal["z", "t"] = "t",
    equal_var: bool = False,
    approx_t_method: Literal["welch", "satterthwaite"] = "welch",
    direction: Literal["greater", "less"] | None = None,
) -> float

Estimate the required mean difference.

PARAMETER	DESCRIPTION
`treatment_std`	Standard deviation in the treatment group. TYPE: `float \| None` DEFAULT: `None`
`reference_std`	Standard deviation in the reference group. TYPE: `float \| None` DEFAULT: `None`
`std`	Standard deviation in both groups. This is a convenience parameter that will override `treatment_std` and `reference_std` when `dist` is `z` and `equal_var` is `True`. If you specify `dist` as `z` and `equal_var` as `True`, you can just specify `std` instead of `treatment_std` and `reference_std`. Internally, the value of `std` will be treated as the standard deviation of both the treatment and reference groups. TYPE: `float \| None` DEFAULT: `None`
`treatment_size`	Sample size in the treatment group. TYPE: `int`
`reference_size`	Sample size in the reference group. TYPE: `int`
`alternative`	Type of the alternative hypothesis. If `alternative` is `'two-sided'`, the alternative hypothesis \(\mu_1 \neq \mu_2\) If `alternative` is `'greater'`, the alternative hypothesis \(\mu_1 > \mu_2\) If `alternative` is `'less'`, the alternative hypothesis \(\mu_1 < \mu_2\) TYPE: `Literal['two-sided', 'greater', 'less']` DEFAULT: `'two-sided'`
`alpha`	Significance level. If `alternative` is `'two-sided'`, `alpha` represents the two-sided significance level. If `alternative` is `'greater'` or `'less'`, `alpha` represents the one-sided significance level. TYPE: `float` DEFAULT: `0.05`
`power`	Expected statistical power. 0.8 is a commonly used value for statistical power. TYPE: `float` DEFAULT: `0.8`
`dist`	The distribution used for the test. `'z'`: Standard normal distribution. `'t'`: Student's or non-central t distribution. TYPE: `Literal['z', 't']` DEFAULT: `'t'`
`equal_var`	Whether to assume equal variances between treatment and reference groups. `True`: Variances are assumed equal. `False`: Variances are assumed unequal. TYPE: `bool` DEFAULT: `False`
`approx_t_method`	Approximate t-test method. It is used when `dist` is `'t'` and `equal_var` = `False`. `'welch'`: Welch's approximate t-test (1947). `'satterthwaite'`: Satterthwaite's approximate t-test (1946). TYPE: `Literal['welch', 'satterthwaite']` DEFAULT: `'welch'`
`direction`	The search direction for the mean difference relative to zero. `'greater'`: Search for the mean difference that is greater than zero. `'less'`: Search for the mean difference that is less than zero. Note If `alternative` is `'two-sided'`, the parameter `direction` is required. If `alternative` is `'greater'`, the search direction is automatically inferred to be `'greater'`, and the parameter `direction` is ignored. If `alternative` is `'less'`, the search direction is automatically inferred to be `'less'`, and the parameter `direction` is ignored. TYPE: `Literal['greater', 'less'] \| None` DEFAULT: `None`

RETURNS	DESCRIPTION
`float`	The required difference between the treatment and reference means.

RAISES	DESCRIPTION
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and all `treatment_std`, `reference_std` and `std` are omitted.
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and both `treatment_std` and `reference_std` are provided, but they are not equal.
`ValueError`	If `alternative` is `'two-sided'` and `direction` is omitted.

solve_treatment_mean ¶

solve_treatment_mean(
    *,
    reference_mean: float,
    treatment_std: float | None = None,
    reference_std: float | None = None,
    std: float | None = None,
    treatment_size: int,
    reference_size: int,
    alternative: Literal["two-sided", "greater", "less"] = "two-sided",
    alpha: float = 0.05,
    power: float = 0.8,
    dist: Literal["z", "t"] = "t",
    equal_var: bool = False,
    approx_t_method: Literal["welch", "satterthwaite"] = "welch",
    direction: Literal["greater", "less"] | None = None,
) -> float

Estimate the required mean in the treatment group.

PARAMETER	DESCRIPTION
`reference_mean`	Mean in the reference group. TYPE: `float`
`treatment_std`	Standard deviation in the treatment group. TYPE: `float \| None` DEFAULT: `None`
`reference_std`	Standard deviation in the reference group. TYPE: `float \| None` DEFAULT: `None`
`std`	Standard deviation in both groups. This is a convenience parameter that will override `treatment_std` and `reference_std` when `dist` is `z` and `equal_var` is `True`. If you specify `dist` as `z` and `equal_var` as `True`, you can just specify `std` instead of `treatment_std` and `reference_std`. Internally, the value of `std` will be treated as the standard deviation of both the treatment and reference groups. TYPE: `float \| None` DEFAULT: `None`
`treatment_size`	Sample size in the treatment group. TYPE: `int`
`reference_size`	Sample size in the reference group. TYPE: `int`
`alternative`	Type of the alternative hypothesis. If `alternative` is `'two-sided'`, the alternative hypothesis \(\mu_1 \neq \mu_2\) If `alternative` is `'greater'`, the alternative hypothesis \(\mu_1 > \mu_2\) If `alternative` is `'less'`, the alternative hypothesis \(\mu_1 < \mu_2\) TYPE: `Literal['two-sided', 'greater', 'less']` DEFAULT: `'two-sided'`
`alpha`	Significance level. If `alternative` is `'two-sided'`, `alpha` represents the two-sided significance level. If `alternative` is `'greater'` or `'less'`, `alpha` represents the one-sided significance level. TYPE: `float` DEFAULT: `0.05`
`power`	Expected statistical power. 0.8 is a commonly used value for statistical power. TYPE: `float` DEFAULT: `0.8`
`dist`	The distribution used for the test. `'z'`: Standard normal distribution. `'t'`: Student's or non-central t distribution. TYPE: `Literal['z', 't']` DEFAULT: `'t'`
`equal_var`	Whether to assume equal variances between treatment and reference groups. `True`: Variances are assumed equal. `False`: Variances are assumed unequal. TYPE: `bool` DEFAULT: `False`
`approx_t_method`	Approximate t-test method. It is used when `dist` is `'t'` and `equal_var` = `False`. `'welch'`: Welch's approximate t-test (1947). `'satterthwaite'`: Satterthwaite's approximate t-test (1946). TYPE: `Literal['welch', 'satterthwaite']` DEFAULT: `'welch'`
`direction`	The search direction for the treatment mean relative to the reference mean. `'greater'`: Search for the treatment mean that is greater than the reference mean. `'less'`: Search for the treatment mean that is less than the reference mean. Note If `alternative` is `'two-sided'`, the parameter `direction` is required. If `alternative` is `'greater'`, the search direction is automatically inferred to be `'greater'`, and the parameter `direction` is ignored. If `alternative` is `'less'`, the search direction is automatically inferred to be `'less'`, and the parameter `direction` is ignored. TYPE: `Literal['greater', 'less'] \| None` DEFAULT: `None`

RETURNS	DESCRIPTION
`float`	The required mean in the treatment group.

RAISES	DESCRIPTION
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and all `treatment_std`, `reference_std` and `std` are omitted.
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and both `treatment_std` and `reference_std` are provided, but they are not equal.
`ValueError`	If `alternative` is `'two-sided'` and `direction` is omitted.

solve_reference_mean ¶

solve_reference_mean(
    *,
    treatment_mean: float,
    treatment_std: float | None = None,
    reference_std: float | None = None,
    std: float | None = None,
    treatment_size: int,
    reference_size: int,
    alternative: Literal["two-sided", "greater", "less"] = "two-sided",
    alpha: float = 0.05,
    power: float = 0.8,
    dist: Literal["z", "t"] = "t",
    equal_var: bool = False,
    approx_t_method: Literal["welch", "satterthwaite"] = "welch",
    direction: Literal["greater", "less"] | None = None,
) -> float

Estimate the required mean in the reference group.

PARAMETER	DESCRIPTION
`treatment_mean`	Mean in the treatment group. TYPE: `float`
`treatment_std`	Standard deviation in the treatment group. TYPE: `float \| None` DEFAULT: `None`
`reference_std`	Standard deviation in the reference group. TYPE: `float \| None` DEFAULT: `None`
`std`	Standard deviation in both groups. This is a convenience parameter that will override `treatment_std` and `reference_std` when `dist` is `z` and `equal_var` is `True`. If you specify `dist` as `z` and `equal_var` as `True`, you can just specify `std` instead of `treatment_std` and `reference_std`. Internally, the value of `std` will be treated as the standard deviation of both the treatment and reference groups. TYPE: `float \| None` DEFAULT: `None`
`treatment_size`	Sample size in the treatment group. TYPE: `int`
`reference_size`	Sample size in the reference group. TYPE: `int`
`alternative`	Type of the alternative hypothesis. If `alternative` is `'two-sided'`, the alternative hypothesis \(\mu_1 \neq \mu_2\) If `alternative` is `'greater'`, the alternative hypothesis \(\mu_1 > \mu_2\) If `alternative` is `'less'`, the alternative hypothesis \(\mu_1 < \mu_2\) TYPE: `Literal['two-sided', 'greater', 'less']` DEFAULT: `'two-sided'`
`alpha`	Significance level. If `alternative` is `'two-sided'`, `alpha` represents the two-sided significance level. If `alternative` is `'greater'` or `'less'`, `alpha` represents the one-sided significance level. TYPE: `float` DEFAULT: `0.05`
`power`	Expected statistical power. 0.8 is a commonly used value for statistical power. TYPE: `float` DEFAULT: `0.8`
`dist`	The distribution used for the test. `'z'`: Standard normal distribution. `'t'`: Student's or non-central t distribution. TYPE: `Literal['z', 't']` DEFAULT: `'t'`
`equal_var`	Whether to assume equal variances between treatment and reference groups. `True`: Variances are assumed equal. `False`: Variances are assumed unequal. TYPE: `bool` DEFAULT: `False`
`approx_t_method`	Approximate t-test method. It is used when `dist` is `'t'` and `equal_var` = `False`. `'welch'`: Welch's approximate t-test (1947). `'satterthwaite'`: Satterthwaite's approximate t-test (1946). TYPE: `Literal['welch', 'satterthwaite']` DEFAULT: `'welch'`
`direction`	The search direction for the reference mean relative to the treatment mean. `'greater'`: Search for the reference mean that is greater than the treatment mean. `'less'`: Search for the reference mean that is less than the treatment mean. Note If `alternative` is `'two-sided'`, the parameter `direction` is required. If `alternative` is `'greater'`, the search direction is automatically inferred to be `'less'`, and the parameter `direction` is ignored. If `alternative` is `'less'`, the search direction is automatically inferred to be `'greater'`, and the parameter `direction` is ignored. TYPE: `Literal['greater', 'less'] \| None` DEFAULT: `None`

RETURNS	DESCRIPTION
`float`	The required mean in the treatment group.

RAISES	DESCRIPTION
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and all `treatment_std`, `reference_std` and `std` are omitted.
`ValueError`	If `dist` is `z` and `equal_var` is `True`, and both `treatment_std` and `reference_std` are provided, but they are not equal.
`ValueError`	If `alternative` is `'two-sided'` and `direction` is omitted.

solve_treatment_std ¶

solve_treatment_std(
    *,
    treatment_mean: float | None = None,
    reference_mean: float | None = None,
    diff: float | None = None,
    reference_std: float | None = None,
    treatment_size: int,
    reference_size: int,
    alternative: Literal["two-sided", "greater", "less"] = "two-sided",
    alpha: float = 0.05,
    power: float = 0.8,
    dist: Literal["z", "t"] = "t",
    equal_var: bool = True,
    approx_t_method: Literal["welch", "satterthwaite"] = "welch",
) -> float

Estimate the required standard deviation in the treatment group.

PARAMETER	DESCRIPTION
`treatment_mean`	Mean in the treatment group. If `diff` is omitted, this parameter is required along with `reference_mean`. TYPE: `float \| None` DEFAULT: `None`
`reference_mean`	Mean in the reference group. If `diff` is omitted, this parameter is required along with `treatment_mean`. TYPE: `float \| None` DEFAULT: `None`
`diff`	Mean difference between treatment and reference group. If both `treatment_mean` and `reference_mean` are not specified, this parameter is required. TYPE: `float \| None` DEFAULT: `None`
`reference_std`	Standard deviation in the reference group. If `dist` is `z` and `equal_var` is `True`, this parameter is ignored. If `dist` is `z` and `equal_var` is `False`, this parameter is required. If `dist` is `t` and `equal_var` is `True`, this parameter is optional. If specified, this value is used to calculate the standard error of mean difference. If `dist` is `t` and `equal_var` is `False`, this parameter is required. TYPE: `float \| None` DEFAULT: `None`
`treatment_size`	Sample size in the treatment group. TYPE: `int`
`reference_size`	Sample size in the reference group. TYPE: `int`
`alternative`	Type of the alternative hypothesis. If `alternative` is `'two-sided'`, the alternative hypothesis \(\mu_1 \neq \mu_2\) If `alternative` is `'greater'`, the alternative hypothesis \(\mu_1 > \mu_2\) If `alternative` is `'less'`, the alternative hypothesis \(\mu_1 < \mu_2\) TYPE: `Literal['two-sided', 'greater', 'less']` DEFAULT: `'two-sided'`
`alpha`	Significance level. If `alternative` is `'two-sided'`, `alpha` represents the two-sided significance level. If `alternative` is `'greater'` or `'less'`, `alpha` represents the one-sided significance level. TYPE: `float` DEFAULT: `0.05`
`power`	Expected statistical power. 0.8 is a commonly used value for statistical power. TYPE: `float` DEFAULT: `0.8`
`dist`	The distribution used for the test. `'z'`: Standard normal distribution. `'t'`: Student's or non-central t distribution. TYPE: `Literal['z', 't']` DEFAULT: `'t'`
`equal_var`	Whether to assume equal variances between treatment and reference groups. `True`: Variances are assumed equal. `False`: Variances are assumed unequal. TYPE: `bool` DEFAULT: `True`
`approx_t_method`	Approximate t-test method. It is used when `dist` is `'t'` and `equal_var` = `False`. `'welch'`: Welch's approximate t-test (1947). `'satterthwaite'`: Satterthwaite's approximate t-test (1946). TYPE: `Literal['welch', 'satterthwaite']` DEFAULT: `'welch'`

RETURNS	DESCRIPTION
`float`	The required standard deviation in the treatment group.

RAISES	DESCRIPTION
`ValueError`	If all of `diff`, `treatment_mean` and `reference_mean` are omitted.
`ValueError`	If `equal_var` is `False`, and `reference_std` is omitted.

solve_reference_std ¶

solve_reference_std(
    *,
    treatment_mean: float | None = None,
    reference_mean: float | None = None,
    diff: float | None = None,
    treatment_std: float | None = None,
    treatment_size: int,
    reference_size: int,
    alternative: Literal["two-sided", "greater", "less"] = "two-sided",
    alpha: float = 0.05,
    power: float = 0.8,
    dist: Literal["z", "t"] = "t",
    equal_var: bool = True,
    approx_t_method: Literal["welch", "satterthwaite"] = "welch",
) -> float

Estimate the required standard deviation in the reference group.

PARAMETER	DESCRIPTION
`treatment_mean`	Mean in the treatment group. If `diff` is omitted, this parameter is required along with `reference_mean`. TYPE: `float \| None` DEFAULT: `None`
`reference_mean`	Mean in the reference group. If `diff` is omitted, this parameter is required along with `treatment_mean`. TYPE: `float \| None` DEFAULT: `None`
`diff`	Mean difference between treatment and reference group. If both `treatment_mean` and `reference_mean` are not specified, this parameter is required. TYPE: `float \| None` DEFAULT: `None`
`treatment_std`	Standard deviation in the treatment group. If `dist` is `z` and `equal_var` is `True`, this parameter is ignored. If `dist` is `z` and `equal_var` is `False`, this parameter is required. If `dist` is `t` and `equal_var` is `True`, this parameter is optional. If specified, this value is used to calculate the standard error of mean difference. If `dist` is `t` and `equal_var` is `False`, this parameter is required. TYPE: `float \| None` DEFAULT: `None`
`treatment_size`	Sample size in the treatment group. TYPE: `int`
`reference_size`	Sample size in the reference group. TYPE: `int`
`alternative`	Type of the alternative hypothesis. If `alternative` is `'two-sided'`, the alternative hypothesis \(\mu_1 \neq \mu_2\) If `alternative` is `'greater'`, the alternative hypothesis \(\mu_1 > \mu_2\) If `alternative` is `'less'`, the alternative hypothesis \(\mu_1 < \mu_2\) TYPE: `Literal['two-sided', 'greater', 'less']` DEFAULT: `'two-sided'`
`alpha`	Significance level. If `alternative` is `'two-sided'`, `alpha` represents the two-sided significance level. If `alternative` is `'greater'` or `'less'`, `alpha` represents the one-sided significance level. TYPE: `float` DEFAULT: `0.05`
`power`	Expected statistical power. 0.8 is a commonly used value for statistical power. TYPE: `float` DEFAULT: `0.8`
`dist`	The distribution used for the test. `'z'`: Standard normal distribution. `'t'`: Student's or non-central t distribution. TYPE: `Literal['z', 't']` DEFAULT: `'t'`
`equal_var`	Whether to assume equal variances between treatment and reference groups. `True`: Variances are assumed equal. `False`: Variances are assumed unequal. TYPE: `bool` DEFAULT: `True`
`approx_t_method`	Approximate t-test method. It is used when `dist` is `'t'` and `equal_var` = `False`. `'welch'`: Welch's approximate t-test (1947). `'satterthwaite'`: Satterthwaite's approximate t-test (1946). TYPE: `Literal['welch', 'satterthwaite']` DEFAULT: `'welch'`

RETURNS	DESCRIPTION
`float`	The required standard deviation in the reference group.

RAISES	DESCRIPTION
`ValueError`	If all of `diff`, `treatment_mean` and `reference_mean` are omitted.
`ValueError`	If `equal_var` is `False`, and `treatment_std` is omitted.