> # Inflation of Type I error rate when IVs are measured with error
> 
> ##### Set parameters #####
> 
> n <- 250
> gamma0 <- 1 ; gamma1 <- 1 ; gamma2 <- 0
> explvar <- 0.75 # Xi1 explains this much of Y
> corr <- 0.80    # Between Xi1 and Xi2 
> reli1 <- 0.90   # Reliability of x1 (measuring xi1) 
> reli2 <- .90    # Reliability of x2 (measuring xi2) 
> ####   Compute secondary parameters ####
> stdev <- gamma1*sqrt((1-explvar)/explvar) # SD of regression error 
> sdverr1 <- sqrt((1-reli1)/reli1)   # SD of error in Xi1 meas 
> sdverr2 <- sqrt((1-reli2)/reli2)   # SD of error in Xi2 meas 
> a <- sqrt((1+corr)/2) # Coefficient for making Xi1 and Xi2 correlated 
> set.seed(9999)
> 
> ##### Generate random data #####
> 
> Z1 <- rnorm(n); Z2 <- rnorm(n)
> Xi1 <- a*Z1 + sqrt(1-a*a)*Z2
> Xi2 <- a*Z1 - sqrt(1-a*a)*Z2
> zeta <- stdev*rnorm(n)
> delta1 <- sdverr1*rnorm(n)
> delta2 <- sdverr2*rnorm(n)
> Y <- gamma0 + gamma1*Xi1 + gamma2*Xi2 + zeta
> x1 <- Xi1 + delta1
> x2 <- Xi2 + delta2
> 
> summary(lm(Y ~ x1+x2))

Call:
lm(formula = Y ~ x1 + x2)

Residuals:
      Min        1Q    Median        3Q       Max 
-1.643991 -0.382166  0.006572  0.450898  1.582773 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.02052    0.03848  26.523  < 2e-16 ***
x1           0.77194    0.05359  14.403  < 2e-16 ***
x2           0.17195    0.05311   3.238  0.00137 ** 
---
Signif. codes:  0 Ô***Õ 0.001 Ô**Õ 0.01 Ô*Õ 0.05 Ô.Õ 0.1 Ô Õ 1 

Residual standard error: 0.607 on 247 degrees of freedom
Multiple R-Squared: 0.7256,	Adjusted R-squared: 0.7234 
F-statistic: 326.6 on 2 and 247 DF,  p-value: < 2.2e-16 

> 
> summary(lm(Y ~ Xi1+Xi2))

Call:
lm(formula = Y ~ Xi1 + Xi2)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.39838 -0.33283  0.01543  0.34493  1.22230 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.03403    0.03149  32.835   <2e-16 ***
Xi1          1.04490    0.05282  19.781   <2e-16 ***
Xi2         -0.02403    0.05308  -0.453    0.651    
---
Signif. codes:  0 Ô***Õ 0.001 Ô**Õ 0.01 Ô*Õ 0.05 Ô.Õ 0.1 Ô Õ 1 

Residual standard error: 0.4962 on 247 degrees of freedom
Multiple R-Squared: 0.8166,	Adjusted R-squared: 0.8151 
F-statistic: 549.9 on 2 and 247 DF,  p-value: < 2.2e-16 

> 
> anova(lm(Y ~ x1+x2))
Analysis of Variance Table

Response: Y
           Df  Sum Sq Mean Sq F value    Pr(>F)    
x1          1 236.737 236.737 642.625 < 2.2e-16 ***
x2          1   3.862   3.862  10.483  0.001370 ** 
Residuals 247  90.992   0.368                      
---
Signif. codes:  0 Ô***Õ 0.001 Ô**Õ 0.01 Ô*Õ 0.05 Ô.Õ 0.1 Ô Õ 1 
> anova(lm(Y ~ x1+x2))[2,5]
[1] 0.001369597
> anova(lm(Y ~ x1+x2))[2,5]<0.05
[1] TRUE
> 
> nsim <- 20
> for(i in 1:nsim)
+ {
+     Z1 <- rnorm(n); Z2 <- rnorm(n)
+     Xi1 <- a*Z1 + sqrt(1-a*a)*Z2
+     Xi2 <- a*Z1 - sqrt(1-a*a)*Z2
+     zeta <- stdev*rnorm(n)
+     delta1 <- sdverr1*rnorm(n)
+     delta2 <- sdverr2*rnorm(n)
+     Y <- gamma0 + gamma1*Xi1 + gamma2*Xi2 + zeta
+     x1 <- Xi1 + delta1
+     x2 <- Xi2 + delta2
+     print(anova(lm(Y ~ x1+x2))[2,5]<0.05)
+ }
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] FALSE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] FALSE
[1] TRUE
[1] FALSE
[1] TRUE
>