**Appendix for DataColada[39].**

**Derivation that if
test-retest correlation
for a dependent variable is r<.5, **

**subtracting baseline lowers
power.**

By Uri Simonsohn

June 17, 2015

Let’s consider a two-cell design,
treatment vs
control, with dependent variable: *y*

Let y_{2}^{t} and y_{2}^{c}
be the means for treatment and control respectively in the *after* period.

Let y_{1}^{t} and y_{1}^{c}
be the means for treatment and control respectively in the *before* period.

The between subject difference

(1) B= y_{2}^{t} - y_{2}^{c}

^{ }

The mixed-design test subtracts the
baseline

(2) M=
y_{2}^{t} - y_{2}^{c} - (y_{1}^{t}
– y_{1}^{c})

* baseline*

The expected difference is the same,
E(B)=E(M), because
with random assignment we have E(y_{1}^{t}
– y_{1}^{c})=0

This makes sense, we don’t expect
differences at
baseline, so we expect the same with B or M

How about the standard error of B and M?

Let’s make things easy. Assume all
variances are the
same:

(3) VAR(y_{2}^{t})=VAR(y_{1}^{t})=VAR(y_{2}^{c})=VAR(y_{1}^{c})=V

(4) COV(y_{2}^{t},
y_{1}^{t})=COV(y_{2}^{c}, y_{1}^{c})=C

(note: because of random assignment COV(y_{2}^{c},
y_{2}^{t})= COV(y_{1}^{c}, y_{1}^{t})=0)

Recall the high-school formula for
variance of sum of
random variables:

(5) VAR(a-b)=VAR(a)+VAR(b)-2COV(a,b)

We want to compute the variance of the B
(between) and
M (mixed design) estimates:

VAR(B)=VAR[(y_{2}^{t}
- y_{2}^{c})]]

=2V

VAR(M)=VAR[(Y2-Y1)
– (X2-X1)]

4V-4C

Mixed and Between subject design have the
same sample
size and the same effect size, hence Mixed has more power iff its
variance is
smaller than Between’s.

For VAR(B)>VAR(M) we need

2V>4V-4C

Which occurs if

4C>2V

Which occurs if

C/V>1/2

C/V, the covariance over the variance, is
the
correlation, so:

The Mixed design has a smaller variance
and hence
greater power iff r>.5