LinearRegression

linear_regression.LinearRegression(self, alpha, p, phi=1)

Covariate adjusted t-tests from Lindon et al. (2024).

Coefficients and covariance matrix are calculated using the Recursive Least Squares algorithm.

• Parent class: Inference
• Example

Parameters

Name	Type	Description	Default
alpha	float	Probability of Type I error \(\alpha\).	required
p	int	Number of covariates \(p\).	required
phi	float	Prior scale \(\phi\).	1

Attributes

Name	Description
Xty	Sum of products of covariates and response \(X^T y\).
beta	Estimate of regression coefficients \(\hat{\beta}\).
covariance	Estimate of covariance matrix \(\hat{\Sigma}\).
phi	Prior scale \(\phi\).
yty	Sum of squared response values \(y^T y\).

Methods

Name	Description
nu	Degrees of freedom.
predict	Predict values for given covariates.
sigma	Estimate the standard deviation of the error term.
sse	Compute the Sum of Squared Errors (SSE).
standard_errors	Estimate the standard errors of the coefficients.
t_stats	Calculate the t statistics of the coefficients.
update	Update the model with new data.
z2	Calculate the squared z-scores.

nu

linear_regression.LinearRegression.nu()

Degrees of freedom.

Returns

Name	Type	Description
	int

predict

linear_regression.LinearRegression.predict(X)

Predict values for given covariates.

Parameters

Name	Type	Description	Default
X	np.ndarray	Matrix of covariates.	required

Returns

Name	Type	Description
	np.ndarray

sigma

linear_regression.LinearRegression.sigma()

Estimate the standard deviation of the error term.

Returns

Name	Type	Description
	float

sse

linear_regression.LinearRegression.sse()

Compute the Sum of Squared Errors (SSE).

Returns

Name	Type	Description
	float

standard_errors

linear_regression.LinearRegression.standard_errors()

Estimate the standard errors of the coefficients.

Returns

Name	Type	Description
	np.ndarray

t_stats

linear_regression.LinearRegression.t_stats()

Calculate the t statistics of the coefficients.

Returns

Name	Type	Description
	np.ndarray

update

linear_regression.LinearRegression.update(yx)

Update the model with new data.

Parameters

Name	Type	Description	Default
yx	np.ndarray	Array of response and covariate values \([y, x_1, \dots, x_p]\).	required

z2

linear_regression.LinearRegression.z2()

Calculate the squared z-scores.

Returns

Name	Type	Description
	np.ndarray

References

Lindon, Michael, Dae Woong Ham, Martin Tingley, and Iavor Bojinov. 2024. “Anytime-Valid Linear Models and Regression Adjusted Causal Inference in Randomized Experiments.” https://arxiv.org/abs/2210.08589.