Power can be defined as statistical conclusive
force. A study result is often expressed in the form of the mean
result and its standard deviation (SD) or standard error (SE). With
the mean result getting larger and the standard error getting
smaller, the study obtains increasing power.
What is the power of the underneath study?
A blood pressure study shows a mean decrease in
blood pressure of 10.8 mm Hg with a standard error of 3.0 mm Hg.
Results from study samples are often given in grams, liters, Euros,
mm Hg etc. For the calculation of power we have to standardize our
study result, which means that the mean result has to be divided by
its own standard error:
The t-values are found in the t-table, can be
looked upon as standardized results of all kinds of studies.
In our blood pressure study the t-value =
10.8/3.0 = 3.6. The unit of the t-value is not mm Hg, but rather
SE-units. The question is: what power does the study have, if we
assume a type I error (alpha) = 5% and a sample size of n =
20.
The question is: what is the power of this study
if we assume a type I error (alpha) of 5%, and will have a sample
size of n = 20.
A.
90% < power < 95%,
B.
power > 80%,
C.
power < 75%,
D.
power > 75%.
n = 20 indicates 20−2 = 18 degrees of freedom
in the case of two groups of ten patients each.
We will use the following power equation (prob =
probability, z = value on the z-line (the x-axis of the
t-distribution)
So, there is a very good power here. See below
for explanation of the calculation.
Explanation of the above calculation.
The t-table on the next page is a more detailed
version of the t-table of page 21, and is adequate for power
calculations. The degrees of freedom are in the left column and
correlate with the sample size of a study. With large samples the
frequency distribution of the data will be a little bit narrower,
and that is corrected in the table. The t-values are to be looked
upon as mean results of studies, but not expressed in mmol/l,
kilograms, but in so-called SE-units (Standard error units), that
are obtained by dividing your mean result by its own standard
error. With a t-value of 3.6 and 18 degrees of freedom
t−t1 equals 1.5. This value is between 1.330 and 1.734.
Look right up at the upper row for finding beta (type II error =
the chance of finding no difference where there is one). We are
between 0.1 and 0.05 (10% and 5%). This is an adequate estimate of
the type II error. The power equals 100% − beta = between 90% and
95% in our example.
t-Table
|
Q =
0.4
|
0.25
|
0.1
|
0.05
|
0.025
|
0.01
|
0.005
|
0.001
|
|
|
v
|
2Q = 0.8
|
0.5
|
0.2
|
0.1
|
0.05
|
0.02
|
0.01
|
0.002
|
|
1
|
0.325
|
1.000
|
3.078
|
6.314
|
12.706
|
31.821
|
63.657
|
318.31
|
|
2
|
0.289
|
0.816
|
1.886
|
2.920
|
4.303
|
6.965
|
9.925
|
22.326
|
|
3
|
0.277
|
0.765
|
1.638
|
2.353
|
3.182
|
4.547
|
5.841
|
10.213
|
|
4
|
0.171
|
0.741
|
1.533
|
2.132
|
2.776
|
3.747
|
4.604
|
7.173
|
|
5
|
0.267
|
0.727
|
1.476
|
2.015
|
2.571
|
3.365
|
4.032
|
5.893
|
|
6
|
0.265
|
0.718
|
1.440
|
1.943
|
2.447
|
3.143
|
3.707
|
5.208
|
|
7
|
0.263
|
0.711
|
1.415
|
1.895
|
2.365
|
2.998
|
3.499
|
4.785
|
|
8
|
0.262
|
0.706
|
1.397
|
1.860
|
2.306
|
2.896
|
3.355
|
4.501
|
|
9
|
0.261
|
0.703
|
1.383
|
1.833
|
2.262
|
2.821
|
3.250
|
4.297
|
|
10
|
0.261
|
0.700
|
1.372
|
1.812
|
2.228
|
2.764
|
3.169
|
4.144
|
|
11
|
0.269
|
0.697
|
1.363
|
1.796
|
2.201
|
2.718
|
3.106
|
4.025
|
|
12
|
0.269
|
0.695
|
1.356
|
1.782
|
2.179
|
2.681
|
3.055
|
3.930
|
|
13
|
0.259
|
0.694
|
1.350
|
1.771
|
2.160
|
2.650
|
3.012
|
3.852
|
|
14
|
0.258
|
0.692
|
1.345
|
1.761
|
2.145
|
2.624
|
2.977
|
3.787
|
|
15
|
0.258
|
0.691
|
1.341
|
1.753
|
2.131
|
2.602
|
2.947
|
3.733
|
|
16
|
0.258
|
0.690
|
1.337
|
1.746
|
2.120
|
2.583
|
2.921
|
3.686
|
|
17
|
0.257
|
0.689
|
1.333
|
1.740
|
2.110
|
2.567
|
2.898
|
3.646
|
|
18
|
0.257
|
0.688
|
1.330
|
1.734
|
2.101
|
2.552
|
2.878
|
3.610
|
|
19
|
0.257
|
0.688
|
1.328
|
1.729
|
2.093
|
2.539
|
2.861
|
3.579
|
|
20
|
0.257
|
0.687
|
1.325
|
1.725
|
2.086
|
2.528
|
2.845
|
3.552
|
|
21
|
0.257
|
0.686
|
1.323
|
1.721
|
2.080
|
2.518
|
2.831
|
3.527
|
|
22
|
0.256
|
0.686
|
1.321
|
1.717
|
2.074
|
2.508
|
2.819
|
3.505
|
|
23
|
0.256
|
0.685
|
1.319
|
1.714
|
2.069
|
2.600
|
2.807
|
3.485
|
|
24
|
0.256
|
0.685
|
1.318
|
1.711
|
2.064
|
2.492
|
2.797
|
3.467
|
|
25
|
0.256
|
0.684
|
1,316
|
1.708
|
2.060
|
2.485
|
2.787
|
3.450
|
|
26
|
0.256
|
0.654
|
1,315
|
1.706
|
2.056
|
2.479
|
2.779
|
3.435
|
|
27
|
0.256
|
0.684
|
1,314
|
1.701
|
2.052
|
2.473
|
2.771
|
3.421
|
|
28
|
0.256
|
0.683
|
1,313
|
1.701
|
2.048
|
2.467
|
2.763
|
3.408
|
|
29
|
0.256
|
0.683
|
1.311
|
1.699
|
2.045
|
2.462
|
2.756
|
3.396
|
|
30
|
0.256
|
0.683
|
1.310
|
1.697
|
2.042
|
2.457
|
2.750
|
3.385
|
|
40
|
0.255
|
0.681
|
1.303
|
1.684
|
2.021
|
2.423
|
2.704
|
3.307
|
|
60
|
0.254
|
0.679
|
1.296
|
1.671
|
2.000
|
2.390
|
2.660
|
3.232
|
|
120
|
0.254
|
0.677
|
1.289
|
1.658
|
1.950
|
2.358
|
2.617
|
3.160
|
|
∞
|
0.253
|
0.674
|
1.282
|
1.645
|
1.960
|
2.326
|
2.576
|
3.090
|