In some clinical studies, the spread of the data
may be more relevant than the average of the data. E.g., when we
assess how a drug reaches various organs, variability of drug
concentrations is important, as in some cases too little and in
other cases dangerously high levels get through. Also,
variabilities in drug response may be important. For example, the
spread of glucose levels of a slow-release-insulin is
important.
One Sample Variability Analysis
For testing whether the standard deviation (or
variance) of a sample is significantly different from the standard
deviation (or variance) to be expected the chi-square test with
multiple degrees of freedom is adequate. The test statistic, the
chi-square-value (= χ2–value) is calculated according
to
(n = sample size, s = standard deviation,
s2 = variance sample, σ = expected standard
deviation, σ2 = expected variance).
For example, the aminoglycoside compound
gentamicin has a small therapeutic index. The standard deviation of
50 measurements is used as a criterion for variability. Adequate
variability is accepted if the standard deviation is less than 7
μg/l. In our sample a standard deviation of 9 μg/l is
observed.
The test procedure is given.
The chi-square table (page 32) shows that, for 50
− 1 = 49 degrees of freedom, we will find a p-value <
0.01. This sample’s standard deviation is significantly larger
than that required. This means that the variability in plasma
gentamicin concentrations is larger than acceptable.
Two Sample Variability Test
F-tests can be applied to test if the
variabilities of two samples are significantly different from one
another. The division sum of the samples’ variances (larger
variance/smaller variance) is used for the analysis. For example,
two formulas of gentamicin produce the following standard
deviations of plasma concentrations.
with degrees of freedom (dfs) for
|
Patients (n)
|
Standard deviation (SD) (μg/l)
|
|
|---|---|---|
|
Formula-A
|
10
|
3.0
|
|
Formula-B
|
15
|
2.0
|
The F-table on the next page shows that an
F-value of at least 3.01 is required not to reject the null -
hypothesis. Our F-value is 2.25 and, so, the p-value is > 0.05.
No significant difference between the two formulas can be
demonstrated. This F-test is given on the next page.
F-Table
|
df
of denominator
|
2-tailed P-value
|
1-tailed P-value
|
Degrees of freedom (df) of the numerator
|
||||||||||||
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
15
|
25
|
500
|
|||
|
1
|
0.05
|
0.025
|
647.8
|
799.5
|
864.2
|
899.6
|
921.8
|
937.1
|
948.2
|
956.6
|
963.3
|
968.6
|
984.9
|
998.1
|
1017.0
|
|
1
|
0.10
|
0.205
|
161.4
|
199.5
|
215.7
|
224.6
|
230.2
|
234.0
|
236.8
|
238.9
|
240.5
|
241.9
|
245.9
|
249.3
|
254.1
|
|
2
|
0.05
|
0.025
|
38.51
|
39.00
|
39.17
|
39.25
|
39.30
|
39.33
|
39.36
|
39.37
|
39.39
|
39.40
|
39.43
|
39.46
|
39.50
|
|
2
|
0.10
|
0.05
|
18.51
|
19.00
|
19.16
|
19.25
|
19.13
|
19.33
|
19.35
|
19.37
|
19.38
|
19.40
|
19.43
|
19.46
|
19.49
|
|
3
|
0.05
|
0.025
|
17.44
|
16.04
|
15.44
|
15.10
|
14.88
|
14.73
|
14.62
|
14.54
|
14.47
|
14.42
|
14.25
|
14.12
|
13.99
|
|
3
|
0.10
|
0.05
|
10.13
|
9.55
|
9.28
|
9.12
|
9.01
|
8.94
|
8.89
|
8.85
|
8.81
|
8.79
|
8.70
|
8.63
|
8.53
|
|
4
|
0.05
|
0.025
|
12.22
|
10.65
|
9.98
|
9.60
|
9.36
|
9.20
|
9.07
|
8.98
|
8.90
|
8.84
|
8.66
|
8.50
|
8.27
|
|
4
|
0.10
|
0.05
|
7.71
|
6.94
|
6.59
|
6.39
|
6.26
|
6.16
|
6.09
|
6.04
|
6.00
|
5.96
|
5.86
|
5.77
|
5.64
|
|
5
|
0.05
|
0.025
|
10.01
|
8.43
|
7.76
|
7.39
|
7.15
|
6.98
|
6.85
|
6.76
|
6.68
|
6.62
|
6.43
|
6.27
|
6.03
|
|
5
|
0.10
|
0.05
|
6.61
|
5.79
|
5.41
|
5.19
|
5.05
|
4.95
|
6.88
|
4.82
|
4.77
|
4.74
|
4.62
|
4.52
|
4.37
|
|
6
|
0.05
|
0.025
|
8.81
|
7.26
|
6.60
|
6.23
|
5.99
|
5.82
|
5.70
|
5.60
|
5.52
|
5.46
|
5.27
|
5.11
|
4.86
|
|
6
|
0.10
|
0.05
|
5.99
|
5.14
|
4.76
|
4.53
|
4.39
|
4.28
|
4.21
|
4.15
|
4.10
|
4.06
|
3.94
|
3.83
|
3.68
|
|
7
|
0.05
|
0.025
|
8.07
|
6.54
|
5.89
|
5.52
|
5.29
|
5.12
|
4.99
|
4.90
|
4.82
|
4.76
|
4.57
|
4.40
|
4.16
|
|
7
|
0.10
|
0.05
|
5.59
|
4.74
|
4.35
|
4.12
|
3.97
|
3.87
|
3.79
|
3.73
|
3.68
|
3.64
|
3.51
|
3.40
|
3.24
|
|
8
|
0.05
|
0.025
|
7.57
|
6.06
|
5.42
|
5.05
|
4.82
|
4.65
|
4.53
|
4.43
|
4.36
|
4.30
|
4.10
|
3.94
|
3.68
|
|
8
|
0.10
|
0.05
|
5.32
|
4.46
|
4.07
|
3.84
|
3.69
|
3.58
|
3.50
|
3.44
|
3.39
|
3.35
|
3.22
|
3.11
|
2.94
|
|
9
|
0.05
|
0.025
|
7.21
|
5.71
|
5.08
|
4.72
|
4.48
|
4.32
|
4.20
|
4.10
|
4.03
|
3.96
|
3.77
|
3.60
|
3.35
|
|
9
|
0.10
|
0.05
|
5.12
|
4.26
|
3.86
|
3.63
|
3.48
|
3.37
|
3.29
|
3.23
|
3.18
|
3.14
|
3.01
|
2.89
|
2.72
|
|
10
|
0.05
|
0.025
|
6.94
|
5.46
|
4.83
|
4.47
|
4.24
|
4.07
|
3.95
|
3.85
|
3.78
|
3.72
|
3.52
|
3.35
|
3.09
|
|
10
|
0.10
|
0.05
|
4.96
|
4.10
|
3.71
|
3.48
|
3.33
|
3.22
|
3.14
|
3.07
|
3.02
|
2.98
|
2.85
|
2.73
|
2.55
|
|
15
|
0.05
|
0.025
|
6.20
|
4.77
|
4.15
|
3.80
|
3.58
|
3.41
|
3.29
|
3.20
|
3.12
|
3.06
|
2.86
|
2.69
|
2.41
|
|
15
|
0.10
|
0.05
|
4.54
|
3.68
|
3.29
|
3.06
|
2.90
|
2.79
|
2.71
|
2.64
|
2.59
|
2.54
|
2.40
|
2.28
|
2.08
|
|
20
|
0.05
|
0.025
|
5.87
|
4.46
|
3.86
|
3.51
|
3.29
|
3.13
|
3.01
|
2.91
|
2.84
|
2.77
|
2.57
|
2.40
|
2.10
|
|
20
|
0.10
|
0.05
|
4.35
|
3.49
|
3.10
|
2.87
|
2.71
|
2.60
|
2.51
|
2.45
|
2.39
|
2.35
|
2.20
|
2.07
|
1.86
|
|
30
|
0.05
|
0.025
|
5.57
|
4.18
|
3.59
|
3.25
|
3.03
|
2.87
|
2.75
|
2.65
|
2.57
|
2.51
|
2.31
|
2.12
|
1.81
|
|
30
|
0.10
|
0.05
|
4.17
|
3.32
|
2.92
|
2.69
|
2.53
|
2.42
|
2.33
|
2.27
|
2.21
|
2.16
|
2.01
|
1.88
|
1.64
|
|
50
|
0.05
|
0.025
|
5.34
|
3.97
|
3.39
|
3.05
|
2.83
|
2.67
|
2.55
|
2.46
|
2.38
|
2.32
|
2.11
|
1.92
|
1.57
|
|
50
|
0.10
|
0.05
|
4.03
|
3.18
|
2.79
|
2.56
|
2.40
|
2.29
|
2.20
|
2.13
|
2.07
|
2.03
|
1.87
|
1.73
|
1.46
|
|
100
|
0.05
|
0.025
|
5.18
|
3.83
|
3.25
|
2.92
|
2.70
|
2.54
|
2.42
|
2.32
|
2.24
|
2.18
|
1.97
|
1.77
|
1.38
|
|
100
|
0.10
|
0.05
|
3.94
|
3.09
|
2.70
|
2.46
|
2.31
|
2.19
|
2.10
|
2.03
|
1.97
|
1.93
|
1.77
|
1.62
|
1.31
|
|
1000
|
0.05
|
0.025
|
5.04
|
3.70
|
3.13
|
2.80
|
2.58
|
2.42
|
2.30
|
2.20
|
2.13
|
2.06
|
1.85
|
1.64
|
1.16
|
|
1000
|
0.10
|
0.05
|
3.85
|
3.00
|
2.61
|
2.38
|
2.22
|
2.11
|
2.02
|
1.95
|
1.89
|
1.84
|
1.68
|
1.52
|
1.13
|