pyomarker.models.test_retest.real.bland_altman

Tools for classical repeatability calculations

BlandAltman

between_subject_mean_squares(x1, x2) staticmethod

Between subject, mean squares of paired measurements.

Definition

\[ bsms = \frac{2}{N} \sum_{i=1}^{N} (\bar{x}_{i} - \mu)^{2} \]

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required

Returns:

Name	Type	Description
float	float	bsms.

between_subject_standard_deviation(x1, x2) staticmethod

Between subject standard deviation of paired measurements.

Definition

\[ \sigma_{b} = \sqrt{(bsms - wsms) / 2} \]

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required

Returns:

Name	Type	Description
float	float	The standard deviation.

coefficient_of_repeatability(x1, x2, ci=0.9) staticmethod

The coefficient of repeatability of paired baseline measurements.

Definition

\[ r = 1.96 \times \sqrt{2} \times s_{w} \]

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required
ci	float	The confidence interval. Must be between 0 and 1. Defaults to 0.9.	0.9

Returns:

Name	Type	Description
float	float	The coefficient of repeatability.
NDArray	NDArray	The coefficient of repeatability confidence interval. Structure: (lower, upper).

coefficient_of_variation(x1, x2) staticmethod

The coefficient of variation on raw paired baseline measurements.

Definition

\[ CoV = \sigma_{w} / \mu \times 100\% \]

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required

Returns:

Name	Type	Description
float	float	The coefficient of variation.

fit(x1, x2)

Fit the Bland-Altman model to data

Note: This class assumes that the data are from two timepoints only.

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required

Returns:

Name	Type	Description
BlandAltman	BlandAltman	A reference to self.

intraclass_correlation_coefficient(x1, x2, ci=0.9) staticmethod

The intra-class correlation coefficient of paired baseline measurements.

Definition

\[ ICC = \frac{\sigma_{b}^{2}}{\sigma_{b}^{2} + \sigma_{w}^{2}} \]

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required
ci	float	The confidence interval. Must be between 0 and 1. Defaults to 0.9.	0.9

Returns:

Name	Type	Description
float	float	The coefficient of repeatability.
NDArray	NDArray	The coefficient of repeatability confidence interval. Structure: (lower, upper).

metrics()

Calculate summary of repeatability metrics.

Returns:

Type	Description
Dict[str, Any]	Dict[str, Any]: The metrics dictionary.

population_mean(x1, x2) staticmethod

Population mean of paired measurements.

Definition

\[ \mu = \frac{1}{N} \sum_{i=1}^{N} \frac{1}{2}\,(x_{1i} + x_{2i}) \]

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required

Returns:

Name	Type	Description
float	float	The population mean.

ratio_coefficient_of_variation(x1, x2) staticmethod

The coefficient of variation on log-transformed paired baseline measurements.

Definition

$$ CoVr = \sqrt{e^{\sigma_{w}^{2}} - 1} \times 100\% $$ where \(\sigma_{w}\) is the within subject standard deviation of the logarithm of paired baseline measurements.

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required

Raises:

Type	Description
ValueError	If any elements of x1 or x2 are less than or equal to 0.

Returns:

Name	Type	Description
float	float	The (ratio) coefficient of variation.

ratio_limits_of_agreement(x1, x2, ci=0.9) staticmethod

The limits of agreement derive from log-transformed paired baseline measurements.

Definition

$$ LoA = [\exp(±1.96\times \sqrt{2} \times s_{w}) - 1] \times 100\% $$ where \(\sigma_{w}\) is the within subject standard deviation of the logarithm of paired baseline measurements.

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required
ci	float	The confidence interval. Must be between 0 and 1. Defaults to 0.9.	0.9

Raises:

Type	Description
ValueError	If any elements of x1 or x2 are less than or equal to 0.

Returns:

Name	Type	Description
NDArray	NDArray	The (ratio) limits of agreement. Structure: (negative, positive).
NDArray	NDArray	The (ratio) limits of agreement confidence interval. Structure: (negative/lower, negative/upper, positive/lower, positive/upper).

within_subject_mean(x1, x2) staticmethod

Per subject mean.

Definition

\[ \bar{x}_{i} = \frac{1}{2}\,(x_{1i} + x_{2i}) \]

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required

Returns:

Name	Type	Description
NDArray	NDArray	1D array within subject means.

within_subject_mean_squares(x1, x2) staticmethod

Within subject, mean squares of paired measurements.

Definition

\[ wsms = \frac{1}{N} \sum_{i=1}^{N} (x_{1i} - \bar{x}_{i})^{2} + (x_{2i} - \bar{x}_{i})^{2} \]

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required

Returns:

Name	Type	Description
float	float	wsms

within_subject_standard_deviation(x1, x2, ci=0.9) staticmethod

Within subject standard deviation of paired measurements.

Definition

\[ \sigma_{w} = \sqrt{\frac{1}{2N}\sum_{i=1}^{N}(x_{2i} - x_{1i})^{2}} \]

Parameters:

Name	Type	Description	Default
x1	NDArray	1D array of baseline measurements.	required
x2	NDArray	1D array of repeat baseline measurements.	required
ci	float	The confidence interval. Must be between 0 and 1. Defaults to 0.9.	0.9

Returns:

Name	Type	Description
float	float	The within subject standard deviation.
NDArray	NDArray	The within subject standard deviation confidence interval. Structure: (lower, upper).