Strictly Parallel Tests Classical Test Theory Results

* Where x1 is the observed score on test 1 and x2 is the observed score on test 2.

* t1 = t2 which is the true score for students (which is equal because they are strictly positive)

* e1, e2, are uncorrelated with each other or t1 and t2 and have the same standard deviation.

* Observed scores x1 and x2 are calculated as x1 = t1 + e1, x2 = t2 + e2

* Classical test theory tells us in strictly parralell test corr(x1,t1)^2 = corr(x2,t2)^2 = corr(x1,x2)

* corr(x1,t1)^2 = {E[(x1-E(x1))(t1-E(t1))]/sd(x1)sd(t1)}^2

* corr(x1,t1)^2 = {E[(t1+e1-E(t1+e1))(t1-E(t1))]/sd(x1)sd(t1)}^2

* corr(x1,t1)^2 = {E[(t1+e1-E(t1))(t1-E(t1))]/sd(x1)sd(t1)}^2

* corr(x1,t1)^2 = {E[(t1+E(t1))(t1-E(t1)) + e1(t1-E(t1))]/sd(x1)sd(t1)}^2

* corr(x1,t1)^2 = {(sd(t1) + 0)/sd(x1)sd(t1)}^2

* corr(x1,t1)^2 = {sd(t1)/sd(x1)sd(t1)}^2

* Since the tests are parrellel sd(e1)=sd(e2) & sd(x1)=sd(x2) along with corr(e1,t1)=0 and corr(e2,t2)=0 implies.

* corr(x1,t1)^2 = {sd(t1)/sd(x1)sd(t1)}^2 = {sd(t2)/sd(x2)sd(t2)}^2 = corr(x2,t2)^2

* Let's see what this looks like:

clear
set obs 10000
* Imagine we have test results back for 1000 students

gen t1 = rnormal()
gen e1 = rnormal()
  gen x1 = t1+e1

gen t2 = t1
gen e2 = rnormal()
  gen x2 = t2+e2


qui corr x1 t1
di r(rho)^2

qui corr x2 t2
di r(rho)^2

* Good, our simulation confirms our algebra.

* Other results are:

qui corr x1 x2
di r(rho) " = corr x2 t2"

local corr_x1x2 = r(rho)

sum t1
local var_t1 = r(sd)^2

sum x1
local var_x1 = r(sd)^2

di "var(t) = `var_t1' = var(x)*corr(x1x2) = `=`var_x1'*`corr_x1x2''"

* And

sum e1
local var_e1 = r(sd)^2

di "var(e) = `var_e1' = var(x)*(1-corr(x1x2)) = `=`var_x1'*(1-`corr_x1x2')'"

* Note, all of these equalities are approximate because we are use a random simulated sample estimates of our parameters.