Download Conversion of Common Test Statistics to r and d Values Statistic r

Conversion of Common Test Statistics to r and d Values
Statistic
r-value
d-value
_______________________________________________________________________
t
1.
2.
3.
4.
(t
t
2
2
2t
df
+ df)
z
z2
z2 + N
2
F dfn = 1
F
F + dfd
F
dfd
2
F dfn > 1
dfn F
d fnF + d fd
d fnF
dfd
2z
N
χ2
5.
χ2 df = 1
χ2
2
N - χ2
N
χ2
6.
7.
2
χ2 + N
χ2 df > 1
r
N
4r2
1 - r2
r
d
χ2
2
2
d
4 +d
8.
______________________________________________________________________
d
Note. dfn = degrees of freeedom for the numerator, dfd = degrees of freedom for the denominator. Tables taken
after Friedman (1982) and Wolf (1986).
Cohen suggested computing:
d=
(Y1 − Y2 )
( s12 + s22 ) / 2
Hedges and Olkin (1985) suggested an adjusted d,
Hedges’ g =
(Y1 − Y2 )
(( n1 − 1) s12
+ ( n2 − 1) s22 ) /((n1
.
d =
⎡ ⎡
⎤⎤
3
1− ⎢
⎢
⎥ ⎥ and
( s12 + s22 ) / 2 ⎣ ⎣ 4( n1 + n2 ) − 9 ⎦ ⎦
(Y1 − Y2 )
⎡ ⎡
⎤⎤
3
⎢1 − ⎢
⎥⎥
+ n2 − 2)) ⎣ ⎣ 4( n1 + n2 ) − 9 ⎦ ⎦
Common Critical Values from the Normal Distribution
for Quick Approximate Power Analysis
Power
0.70
0.80
0.90
Distance
-0.525
-0.842
-1.282
α = 0.05
1-tailed
2-tailed
(1.645)
(1.960)
2.170
2.485
2.487
2.802
2.927
3.242
α = 0.01
1-tailed
2-tailed
(2.326)
(2.576)
2.846
3.101
3.168
3.418
3.608
3.858
For a 2-group design, approximate per group sample size (nj) for a given α and level of
Statistical Power (1-β) for the can be solved as:
2
2 zcv
, where zcv2 is the critical value from the Table above, d is a standardized mean
2
d
(Y − Y )
difference, d = 1 2 , and s is an assumed standard deviation. As pointed out above, various
s
nj ≥
metrics have been proposed. In general, the use of Cohen’s d the adjusted d, or Hedges’ g will
lead to approximately the same result.
For example, suppose a study reports the control group had a mean of YC =10, the treatment
group had a mean of YT =12 and the pooled standard deviation was s = 1.5.
Then the standardized mean difference would be: d = (12.7-11.8)/1.5 = 0.6.
For a future study to have 70% Power (1-β = 0.70) for a 2-tailed test at α = 0.05
The approximate necessary per group sample size would be:
nj ≥
2(2.4852 )
2(6.175225)
≥
≥ 34.3 ≈ 35.
2
0.36
0.6
For a future study to have 80% Power (1-β = 0.80) for a 2-tailed test at α = 0.01
The approximate necessary per group sample size would be:
nj ≥
2(3.4182 )
2(11.682724)
≥
≥ 64.9 ≈ 65.
2
0.36
0.6
Reversing this process, if a researcher knew that he could only obtain 100 total subjects
(nj = 50 per group), then we could solve for an approximate minimum effect size (d):
d ≥
zcv
nj
2
Thus, if the research desired 80% Power (1-β = 0.80) for a 2-tailed test at α = 0.05
d ≥
2.802 2.802
≥
≥ 0.5604 would be the approximate necessary effect size.
5
50
2
To double check this enter the effect size of d = 0.5604 the critical value for 80% Power
(1-β = 0.80) for a 2-tailed test at α = 0.05 into
nj ≥
2
2 zcv
2(2.8022 )
15.702408
≥
≥ 50
≥
2
2
0.31404816
d
0.5604