Linear bivariate regression: Two groups, difference between slopes

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.

Having determined the linear relationships between X and Y in two groups: Y = a₁ + b₁X, Y = a₂ + b₂X, we may ask whether the slopes b₁ and b₂ are identical. The null and the two-sided alternative hypotheses are

H₀ : b₁ − b₂ = 0
H₁ : b₁ − b₂ ≠ 0.

Effect size index

The absolute value of the difference between the slopes |∆slope| = |b₁ − b₂| (Slope H1) is used as effect size. To fully specify the effect size, the following additional inputs must be given:

Std dev residual σ

The standard deviation σ > 0 of the residuals in the combined data set (i.e. the square root of the weighted sum of the residual variances in the two data sets): If σ²_r1 and σ²_r2 denote the variance of the residuals r₁ = (a₁ + b₁ X₁) − Y₁ and r₂ = (a² + b² X²) − Y₂ in Groups 1 nd 2, and n₁, n₂ denote the sample sizes in Groups 1 and 2, then

Std dev σ_x1

The standard deviation σ_x1 > 0 of the X values in Group 1.

Std dev σ_x2

The standard deviation σ_x2 > 0 of the X values in Group 2.

Important relationships between the standard deviations σ_xi of X_i, σ_yi of Y_i, the slopes b_i of the regression lines, and the correlation coefficient ρ_i between X_i and Y_i are:

σ_yi = (b_i σ_xi)/ρ_i
σ_yi = σ_ri / √(1 − ρ²_i)

where σ_i denotes the standard deviation of the residuals Y_i − (b_i X + a_i).

The effect size dialog may be used to determine Std dev residual σ (the standard deviation of the residuals) and |∆ slope| (the absolute value of the difference between the slopes, |b₁ − b₂|) from other values based on the equations above. Pressing the Determine button on the left side of the effect size label in the main window opens the effect size drawer.




The input variables are located on the left side of the arrow '=>', the output variables are located on the right side. The input values must conform to the usual restrictions, that is, σ_xi > 0, σ_x2 > 0, −1 < ρ_i < 1. In addition, the fact that σ_yi = (b_i σ_xi)/ρ_i, together with the restriction on ρ_i implies the additional restriction −1 < b·σ_xi/σ_yi < 1.

Clicking on the Calculate and transfer to main window button copies the values given in Std dev σ_x1, Std dev σ_x2, Std dev residual σ, Allocation ratio N2/N1, and |∆slope| to the corresponding input fields in the main window.

Options

This test has no options.

Examples

We replicate an example given on page 594 in Dupont and Plummer (1998) that refers to an example in Armitage, Berry, and Matthews, 2002 (p. 325). The data and relevant statistics are shown in below. Note: Contrary to Dupont and Plummer (1998), we here consider the data as hypothesized true values and normalize the variance by N, not by (N − 1).




The relation of age and vital capacity for two groups of men working in the cadmium industry is investigated. Group 1 includes n₁ = 28 workers with less than 10 years of cadmium exposure. Group 2 includes n₂ = 44 workers never exposed to cadmium.

The standard deviation of the ages in both groups are σ_x1 = 9.029 and σ_x2 = 11.87. Regressing vital capacity on age gives the following slopes of the regression lines: β₁ = −0.04653 and β₂ = −0.03061. To calculate the pooled standard deviation of the residuals we use the effect size drawer. We use sx, sy, slope => residual s, r input mode and insert the values given above, the standard deviations σ_y1, σ_y2 of y (capacity) as given the table above, and the allocation ratio n₂/n₁ = 44/28 = 1.571428. This results in a pooled standard deviation of the residuals of σ = 0.5578413 (see the effect size drawer above).

We want to recruit enough workers to detect a true difference in slope of |(−0.03061) − (−0.04653)| = 0.01592 with power of .80, α = 0.05 and the same allocation ratio to the two groups as in the sample data.

Select

Type of power analysis: A priori

Input

Tail(s): Two
|∆ slope|: 0.01592
α err prob: 0.05
Power (1- β ): 0.80
Allocation ratio N2/N1: 1.571428
Std dev residual σ: 0.5578413
Std dev σ x1: 9.02914
Std dev σ x2: 11.86779

Output

Noncentrality parameter δ: 2.811598
Critical t: 1.965697
Df: 415
Sample size group 1: 163
Sample size group 2: 256
Total sample size: 419
Actual power: 0.800980

The output shows that we need 419 workers in total, with 163 in Group 1 and 256 in Group 2. These values are close to those reported in Dupont and Plummer (1998, p. 596) for this example (166 + 261 = 427). The slight difference is due to the fact that Dupont and Plummer normalized the variances by N − 1, and their using shifted central t distributions instead of non-central t distributions.

Relation to Multiple Regression: Special

The present procedure is essentially a special case of the multiple regression procedure, but provides a more convenient interface. To show this, we demonstrate how the multiple regression procedure can be used to compute the example above (see also Dupont and Plummer, 1998, p.597). First, the data are combined and extended into a data set of size n₁ + n₂ = 28 + 44 = 72. With respect to this combined data set we define the following variables (vectors of length 72):

• y contains the measured vital capacity
• x₁ contains the age data
• x₂ codes group membership (0 = not exposed, 1 = exposed)
• x₃ contains the element-wise product of x₁ and x₂

The multiple regression model

y = β0 + β1 x1 + β2 x2 + β3 x3 + εi

reduces to y = β0 + β1 x1 and y = (β0 + β2 x2) + (β1 + β3) x1 for unexposed and exposed workers, respectively. In this model, β3 represents the difference in slope between both groups, which is assumed to be zero under the null hypothesis. Thus, the above model reduces to

y = β0 + β1 x1 + β2 x2 + εi

if the null hypothesis is true. 

f ² = (R²₁ − R²₀)/(1 − R²₁) = 0.3243 − 0.3115 1 − 0.3243 = 0.018870 75= 0.3115.