Linear bivariate regression: One group, size of slope

A linear regression is used to estimate the parameters a, b of a linear relationship Y = a + bX between the dependent variable Y and the independent variable X. X is assumed to be a set of fixed values, whereas Y_i is modeled as a random variable: Y_i = a + bX_i + ε_i, where ε_i denotes normally distributed random errors with mean 0 and standard deviation σ_i. A common assumption also adopted here is that all σ_i's are identical, that is σ_i = σ. The standard deviation of the error is also called the standard deviation of the residuals.

A common task in linear regression analysis is to test whether the slope b is identical to a fixed value b₀ or not. The null and the two-sided alternative hypotheses are:

H₀ : b − b₀ = 0
H₁ : b − b₀ ≠ 0.

Effect size index

Slope H1, the slope b of the linear relationship assumed under H₁ is used as effect size measure. To fully specify the effect size, the following additional inputs must be given:

Slope H0.

This is the slope b₀ assumed under H₀.

Std dev σ x.

This is the standard deviation σ_x of the values in X and must be > 0.

Std dev σ y.

This is the standard deviation σ_y > 0 of the Y values. Important relationships of σ_y to other relevant measures are:

σ_y = (bσ_x)/ρ
σ_y = σ/√(1 − ρ²)

where σ denotes the standard deviation of the residuals Y_i − (aX + b) and ρ the correlation coefficient between X and Y.

Pressing the Determine button on the left side of the effect size label in the main window opens the effect size drawer. The effect size drawer may be used to determine the standard deviation σ_y > 0 of the Y values (Std dev σ y) and/or the slope b of the linear relationship assumed under H₁ (Slope H1) from other values based on the two equations given above.


Below you see the effect size drawer with different combinations of input and output values in different input modes. The input variables are placed on the left side of the arrow (=>), the output variables on the right side. The input values must conform to the usual restrictions, that is, σ > 0, σ_x > 0, σ_y > 0, −1 < ρ < 1. In addition, σ_y = (bσ_x)/ρ (see above) together with the restriction on ρ implies the additional restriction −1 < b · σ_x /σ_y < 1.


Clicking on the Calculate and transfer to main window button copies the values given in Std dev σ_x, Std dev σ_y, and Slope H1 to the corresponding input fields in the main window.

Options

This test has no options.

Examples

We replicate an example given on page 593 in Dupont and Plummer (1998). The question investigated in this example is whether the actual average time spent per day exercising is related to the body mass index (BMI) after 6 months on a training program.

The estimated standard deviation of exercise time of participants is σ_x = 7.5 minutes. From a previous study the standard deviation of the BMI in the group of participants is estimated to be σ_y = 4. The sample size is N = 100.  A standard level of α = 0.05 is assumed. We want to determine the power with which a slope b₀ = −0.0667 of the regression line (corresponding to a drop of BMI by 2 per 30 min/day exercise) can be detected.

Select

Type of power analysis: Post hoc

Input

Tail(s): Two
Slope H1: -0.0667
α err prob: 0.05
Total sample size: 100
Slope H0: 0
Std dev σ x: 7.5
Std dev σ y: 4

Output

Noncentrality parameter δ: 1.260522
Critical t : -1.984467
Denominator df: 98
Power (1- β ): 0.238969

The output shows that the power of this test is about 0.24. This confirms the value estimated by Dupont and Plummer (1998, p. 596) for this example.

Relation to Multiple Regression: Omnibus

The present procedure is a special case of the Multiple Regression procedure, or better: a different interface to the same procedure using more convenient variables. To show this, we demonstrate how the Multiple Regression: Omnibus (R² deviation from zero) procedure can be used to compute results of the example above.

First, we determine R² = ρ² from the relation b = ρ · σ_y /σ_x, which implies ρ² = (b · σ_x /σ_y)² . Entering (-0.0667*7.5/4)² into the G*Power calculator gives ρ² = 0.01564062. Then we perform these selections and inputs:

Select

Multiple Regression: Omnibus (R² deviation from zero)
Type of power analysis: Post hoc

Input

Effect size f²: 0.01588914 (calculated from ρ² = 0.01564062 using the effect size drawer)
α err prob: 0.05
Total sample size: 100
Number of  predictors: 1

Output

Noncentrality parameter λ: 1.588915
Critical F : 3.938111
Numerator df: 1
Denominator df: 98
Power (1- β ): 0.238969

Thus, we get exactly the same power that we got using the Linear Regression: Size of slope, one group procedure. Also note that λ = δ² = 1.260522² = 1.588915, and that F = t² = -1.984467 = 3.93811.

Related tests

Correlation: Point biserial model
Linear multiple regression: R² deviation from zero

Implementation notes

The H₀ distribution is the central F distribution with numerator degrees of freedom df₁ = 1 and denominator degrees of freedom df₂ = N − 2, where N is the sample size. The H₁ distribution is the noncentral F distribution with the same degrees of freedom and noncentrality parameter λ = N([σ_x(b − b₀)]/σ)².

Validation

The results were checked against the values produced by by PASS (Hintze, 2006). Perfect correspondance was found.

References

Dupont, W. D., & Plummer, W. D., Jr. (1998). Power and sample size calculations for studies involving linear regression. Controlled Clinical Trials, 19, 589-601.

Hintze, J. (2006). NCSS, PASS, and GESS. Kaysville, Utah: NCSS.