Measurement error attenuates only in the one-regressor case

With a true regressor of variance s11 measured with classical error of variance su2, the one-regressor population slope is \(\beta_1 s_{11}/(s_{11}+s_{u}^2)\): attenuation by the reliability ratio (Research.P5.Attenuation.single_attenuation). With a second, error-free regressor of variance s22 and covariance s12, the population coefficients are \(b_1 = \beta_1 (s_{11}s_{22}-s_{12}^2)/D^*\) and \(b_2 = \beta_2 + \beta_1 s_u^2 s_{12}/D^*\) with \(D^* = (s_{11}+s_u^2)s_{22}-s_{12}^2\) (two_regressor_coeffs): the error-free coefficient is biased unless \(\beta_1\), \(s_u^2\) or \(s_{12}\) is zero (error_free_biased_iff), and the bias has the sign of \(\beta_1 s_{12}\) (bias_sign), away from zero as often as toward it.

Usage

morie_attenuation_two_regressor(beta1, beta2, s11, s22, s12 = 0, su2 = 0)

Arguments

beta1, beta2

True coefficients of the error-prone and the error-free regressor.

s11, s22

Variances of the two true regressors.

s12

Their covariance; must satisfy s12^2 < s11 * s22.

su2

Variance of the classical measurement error on the first regressor.

Value

A list: slope_single (the one-regressor slope on the observed regressor), reliability (s11 / (s11 + su2)), b1, b2 (the two-regressor population coefficients), bias_b2 (b2 - beta2), bias_b2_sign and dstar.

References

Weisburd D, Britt C (2014). Statistics in Criminal Justice, 4th ed., ch. 2. Springer.

Examples

morie_attenuation_two_regressor(beta1 = 1, beta2 = 0.5, s11 = 1, s22 = 1, s12 = 0.6, su2 = 0.5)
#> $slope_single
#> [1] 0.6666667
#> 
#> $reliability
#> [1] 0.6666667
#> 
#> $b1
#> [1] 0.5614035
#> 
#> $b2
#> [1] 0.7631579
#> 
#> $bias_b2
#> [1] 0.2631579
#> 
#> $bias_b2_sign
#> [1] "positive"
#> 
#> $dstar
#> [1] 1.14
#> 
#> attr(,"theorem")
#> [1] "Research.P5.Attenuation.two_regressor_coeffs"