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Introduction 1STT
Method validation Accuracy = Trueness & Precision This file addresses the basic statistical concepts for the validation of method accuracy; accuracy is understood as joint indtrueness and precision (see [1], Terms Worksheet). Precision Precision (and imprecision) are qualitative concepts (see Terminology Sheet). Experimental estimates of imprecision are the Variance or Standard Deviation of a distribution of measurement results.EXCEL functions: VAR; STDEV; ANOVA. Calculations: Relative Standard Deviation in % or Coefficient of Variation (R.S.D., or CV), 100 * (STDEV/AVERAGE) Trueness Trueness is a qualitative concept, too. Trueness is related to the concept of systematic error. The experimental estimate of trueness is the [instrumental] Bias, which is the "average of replicate indications minus a reference quantity value". EXCEL function: AVERAGE. Calculations: Difference, or %-difference. Accuracy As the above, accuracy is a qualitative concept. It's experimental estimate is the Measurement Error (Error), which is the "measured quantity value minus a reference quantity value". Note: often called Total Error. Calculations: Difference, or %-difference.
Disclaimer
EXCEL SettingsActivate the Add-Ins (Tools>Add-Ins):
Analysis ToolPak & Analysis ToolPak -VBA.
Macro security (Tools>Macro>Security):
Medium or low.
This file addresses the basic statistical concepts for the validation of method accuracy; accuracy is understood as joint index of
of a distribution of measurement results.
Calculations: Relative Standard Deviation in % or Coefficient of Variation (R.S.D., or CV),
Trueness is a qualitative concept, too. Trueness is related to the concept of systematic error. , which is the "average of replicate indications minus a
), which is the "measured quantity value minus a reference quantity
Introduction 2STT
Method validation - The problem of finite number of measurements Method validation can be done only with a finite number of measurements. Consequently, all experimental data are only experimental ESTIMATES have statistical CONFIDENCE INTERVALS. Confidence intervals STDEV: involvement of the CHI-square statistics; for imprecision. AVERAGE: involvement of the Student's t-statistics; for trueness. CENTILE: involvement of Normal statistics and t-statistics; for total error. Note 1: for the total error, tolerance or prediction intervals may be of additional interest.Note 2: method comparison studies are an important tool for estimating total error. The involved statistics (for example, reghere. EXCEL functions: CHIDIST; CHIINV; TDIST; TINV; NORMDIST; NORMSDIST; NORMSINV. Significance testing Due to the nature of validation, we have to compare the confidence intervals of our estimates with requirements. In statisticTests involved are the "One-sample" F-test and t-test (note: the "reference quantity value" is assumed to be errorEXCEL functions: EXCEL has no in-built one-sample tests, however, they can be constructed with EXCEL using the above functions. Power analysis A method should not get validated by chance. Therefore, we need an idea i) how many measurements we need for validating a requnder assumed method performance (for example, stable CV = 7%); ii) which CV can we tolerate when the limit is 15% and we appestimating the CV (duplicates on 10 different days). Statistics involved are the Power of the F-test and the t-test. EXCEL functions: EXCEL has no tools for power analysis, however, the power of some tests can be estimated with simulation expwith the free "R"-software and G*Power software. Tolerance interval One worksheet shortly presents the calculation of "rest bias" from tolerance interval considerations. Validation ―Confirmation, through the provision of objective evidence, that requirements for a specific intended use or application haveSystems— Requirements, International Organization for Standardization (ISO), Geneva, 2000).
Method validation can be done only with a finite number of measurements. Consequently, all experimental data are only estimates of true population parameters: →ALL
may be of additional interest. Note 2: method comparison studies are an important tool for estimating total error. The involved statistics (for example, regression analysis) are complex and not presented
EXCEL functions: CHIDIST; CHIINV; TDIST; TINV; NORMDIST; NORMSDIST; NORMSINV.
Due to the nature of validation, we have to compare the confidence intervals of our estimates with requirements. In statistical terms, this is done with significance tests. (note: the "reference quantity value" is assumed to be error-free).
sample tests, however, they can be constructed with EXCEL using the above functions.
A method should not get validated by chance. Therefore, we need an idea i) how many measurements we need for validating a requirement (for example, CV limit = 15%) under assumed method performance (for example, stable CV = 7%); ii) which CV can we tolerate when the limit is 15% and we apply a a certain experimental design for
EXCEL functions: EXCEL has no tools for power analysis, however, the power of some tests can be estimated with simulation experiments. Power analysis will be done
One worksheet shortly presents the calculation of "rest bias" from tolerance interval considerations.
―Confirmation, through the provision of objective evidence, that requirements for a specific intended use or application have been fulfilled‖ (ISO 9001, Quality Management Requirements, International Organization for Standardization (ISO), Geneva, 2000).
TerminologySTT
Terminology [1] Accuracy Closeness of agreement between a measured quantity value and a true quantity value of the measurand. Trueness Closeness of agreement between the average of an infinite number of replicate measured quantity values and a reference quanti Precision Closeness of agreement between indications or measured quantity values obtained by replicate measurements on the same or simi Measurement error (error) Measured quantity value minus a reference quantity value. Note: often called Total Error. Systematic [measurement] error Component of measurement error that in replicate measurements remains constant or varies in a predictable manner.Bias [of measurement] Estimate of a systematic measurement error. Instrumental bias Average of replicate indications minus a reference quantity value. Random [measurement] error Component of measurement error that in replicate measurements varies in an unpredictable manner.NOTE 2 Random measurement errors of a set of replicate measurements form a distribution that can be summarized by its expectazero, and its variance. [1] ISO/IEC Guide 99:2007. International vocabulary of metrology – Basic and general concepts and associated terms (VIM). InternStandardization: Geneva, 2007 (VIM = Vocabulaire International de Métrologie). [1] JCGM 200:2008. International vocabulary of metrology – Basic and general concepts and associated terms (VIM). International (BIPM); Joint Committee for Guides in Metrology (JCGM): Paris, 2008. Freely available at: http://www.bipm.org/en/publications/guides/vim.html
Closeness of agreement between a measured quantity value and a true quantity value of the measurand.
Closeness of agreement between the average of an infinite number of replicate measured quantity values and a reference quantity value.
Closeness of agreement between indications or measured quantity values obtained by replicate measurements on the same or similar objects under specified conditions.
Component of measurement error that in replicate measurements remains constant or varies in a predictable manner.
Component of measurement error that in replicate measurements varies in an unpredictable manner. NOTE 2 Random measurement errors of a set of replicate measurements form a distribution that can be summarized by its expectation, which is generally assumed to be
Basic and general concepts and associated terms (VIM). International Organization for
Basic and general concepts and associated terms (VIM). International Bureau of Weights and Measures
RequirementsSTT
Requirements We will work with the requirements for bioanalytical method validation [1]: CVmax = 15% (20% at the limit of quantitation, LoQ). Maximum Bias (Bmax) = 15% (20% at the limit of quantitation, LoQ). $Maximum Total error (TEmax): 1.96 x CVmax = 29.4% Note: In document [1], TEmax is indirectly defined by the "in-study" quality control rule: 66.7% of the results should be withinOne may slightly change that to 68.3%, because this corresponds to +/-1s. $We use the Westgard concept for calculating CV/B ratios that satisfy a TEmax of 29.4%: TE = B + 1.96 multiplier gradually from 1.96 to 1.65 when introducing bias [8]. By doing so, we keep the results outside TE at 5% (moving fTE = B (0→15) + (1.96→1.65) s (15→8.76); for B = 15, s (8.76→0): TE = 29.4→15. Note: If we would allow Bmax and CVmax happen at the same time, TEmax would be 39.75% (=15 + 1.65 x 15). 1 Guidance for Industry: Bioanalytical Method Validation, US Department of Health andHumanServices, Food and Drug Administratand Research (CBER), Rockville, MD, 2001. 2 Hoffman D, Kringle R. A total error approach for the validation of quantitative analytical methods. Pharm Res 2007;24(6):11 3 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Carey RN, Garber CC. Concepts and practices in the evaluation of clinical chemistTechnol. 1978;44(8):803-13. 4 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Garber CC, Carey RN. Concepts and practices in the evaluation of clinical chemistacceptability. Am J Med Technol. 1978;44(7):727-42. 5 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Garber CC, Carey RN. Concepts and practices in the evaluation of clinical chemistTechnol. 1978;44(6):552-71. 6 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Carey RN, Garber CC. Concepts and practices in the evaluation of clinical chemistprocedures. Am J Med Technol. 1978;44(5):420-30. 7 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Carey RN, Garber CC. Concepts and practices in the evaluation of clinical chemistapproach. Am J Med Technol. 1978;44(4):290-300. 8 Stöckl D, Thienpont LM. About the z-multiplier in total error calculations. Clin Chem Lab Med 2008;46(11):1648
study" quality control rule: 66.7% of the results should be within +/-15% (4 results out of 6, 4-6-15 rule) [2].
$We use the Westgard concept for calculating CV/B ratios that satisfy a TEmax of 29.4%: TE = B + 1.96 s [3 - 7]. However, we modify that concept by changing the z-multiplier gradually from 1.96 to 1.65 when introducing bias [8]. By doing so, we keep the results outside TE at 5% (moving from 2-sided 5% to 1-sided 5%).
Note: If we would allow Bmax and CVmax happen at the same time, TEmax would be 39.75% (=15 + 1.65 x 15).
1 Guidance for Industry: Bioanalytical Method Validation, US Department of Health andHumanServices, Food and Drug Administration, Center for Biologics Evaluation
2 Hoffman D, Kringle R. A total error approach for the validation of quantitative analytical methods. Pharm Res 2007;24(6):1157-64.
3 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Carey RN, Garber CC. Concepts and practices in the evaluation of clinical chemistry methods. V. Applications. Am J Med
4 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Garber CC, Carey RN. Concepts and practices in the evaluation of clinical chemistry methods: IV. Decisions of
5 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Garber CC, Carey RN. Concepts and practices in the evaluation of clinical chemistry methods. III: Statistics. Am J Med
6 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Carey RN, Garber CC. Concepts and practices in the evaluation of clinical chemistry methods. II. Experimental
7 Westgard JO, de Vos DJ, Hunt MR, Quam EF, Carey RN, Garber CC. Concepts and practices in the evaluation of clinical chemistry methods. I. Background and
multiplier in total error calculations. Clin Chem Lab Med 2008;46(11):1648-9.
Total error
Combinations of CV & B that fulfill TE = 29.4% (TE = Bias + z * CV)
according to TE = B (0→15) & z (1.96→1.65), s (15→8.8).
Note: Other fields work with TE = Bias + 1 * SD (z 1→0.475)
The corresponding TE curves are distinctly diffferent.
STT
0
10
20
30
0 5 10 15
IBia
sI
(%)
..
CV (%)
[8.76;15]
[15;0]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
CV (fractions of TE)
Bia
s (fr
actio
ns o
f TE
) ..
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
CV (fractions of TE)
Bia
s (fr
actio
ns o
f TE
) ..
Generic 15% 15
SE/RE SE RE SE RE
0.00 0 1.000 0.0 15
0.10 0.1 0.992 1.5 14.89
0.20 0.2 0.978 3.0 14.67
0.31 0.3 0.955 4.5 14.32
0.43 0.4 0.921 6.0 13.82
0.57 0.5 0.879 7.5 13.18
0.73 0.6 0.827 9.0 12.40
0.91 0.7 0.765 10.5 11.48
1.13 0.8 0.705 12.0 10.58
1.40 0.9 0.644 13.5 9.67
1.7132 1 0.584 15.0 8.76 8.756
2.10 1.1 0.523 16.5 7.844
2.60 1.2 0.462 18.0 6.932
3.24 1.3 0.401 19.5 6.021
4.11 1.4 0.341 21.0 5.109
5.36 1.5 0.280 22.5 4.198
7.30 1.6 0.219 24.0 3.286
10.74 1.7 0.158 25.5 2.375
18.45 1.8 0.098 27.0 1.463
51.67 1.9 0.037 28.5 0.552
#DIV/0! 1.96 0.000 29.4 0
See also
A total error approach for the validation of quantitative analytical methods
Hoffman D, Kringle R. PHARMACEUTICAL RESEARCH 2007;24:1157-64.
Power Analysis
Finding n for a given CV, or a CV for given n
CVmax = 15%
a-error consideration
One-sample F -test (1-tailed!) One-sample F -test (1-tailed!)
Limit 15.0 Limit 15.0
CVexp 3.40 n = 3 CVexp 9.12 n = 10
CHI2
0.10 CHI2
3.33
p Value 0.0501 p Value 0.0501
Limit 15.0 Limit 15.0
CVexp 6.32 n = 5 CVexp 10.95 n = 20
CHI2
0.71 CHI2
10.12
p Value 0.0499 p Value 0.0500
With 3 measurements, the maximum CV would be 3.4%.
With 20 measurements, the maximum CV would be 11%.
However, if the maximum CV is present during validation,
the chance to validate the method is only 50%!
Tolerance interval (0.95, 0.95); TE limit = 29.4% (NO bias!)
2-sided (k-limit = 1.96)
n TE k s Compare s(a)
10 29.4 3.38 8.70 roughly to 9.11
20 29.4 2.75 10.69 a-error 10.94
k-values from ISO 3207-1975
b-error consideration = Power
We want the method validated in 95% of the cases.
We investigate the Power of the 1-sided 1-sample F -test (EXCEL has no power analysis).
G*Power (http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/)
Chi-square (1-sided, a = 0.05) Power = 0.95
n Limit "Power"-CV
3 15 1.96
5 15 4.11
10 15 6.65
20 15 8.69
40 15 10.29
With an actual CV of 8.69% and a limit CV of 15%, we would validate the method in 95% of the cases.
(see also Worksheet Power Chi2-test)
STT
n df
3 2
5 4
10 9
20 19
We investigate the Power of the 1-sided 1-sample F -test (EXCEL has no power analysis).
With an actual CV of 8.69% and a limit CV of 15%, we would validate the method in 95% of the cases.
1-sample F
SD limit 15 225 (Var0)
n 5 10 20 100
Effect-size Chi2-test
Var0/Var1 Power Power Power Power Var 1
1.00 0.050 0.050 0.050 0.050 225
1.10 0.071 0.081 0.096 0.175 205
1.20 0.095 0.119 0.157 0.380 188
1.30 0.121 0.162 0.229 0.601 173
1.40 0.148 0.209 0.308 0.777 161
1.50 0.176 0.257 0.389 0.890 150
1.60 0.204 0.306 0.467 0.950 141
1.70 0.233 0.354 0.540 0.979 132
1.80 0.261 0.401 0.607 0.992 125
1.90 0.288 0.446 0.666 0.997 118
2.00 0.315 0.489 0.718 0.999 113
2.10 0.340 0.528 0.763 1.000 107
2.20 0.365 0.566 0.801 1.000 102
2.30 0.389 0.600 0.833 1.000 98
2.40 0.412 0.632 0.860 1.000 94
2.50 0.434 0.661 0.883 1.000 90
2.60 0.456 0.688 0.902 1.000 87
2.70 0.476 0.713 0.918 1.000 83
2.80 0.495 0.736 0.931 1.000 80
2.90 0.513 0.756 0.943 1.000 78
3.00 0.531 0.775 0.952 1.000 75
3.10 0.548 0.793 0.959 1.000 73
3.20 0.564 0.809 0.966 1.000 70
3.30 0.579 0.823 0.971 1.000 68
3.40 0.593 0.836 0.976 1.000 66
3.50 0.607 0.849 0.979 1.000 64
3.60 0.621 0.860 0.983 1.000 63
3.70 0.633 0.870 0.985 1.000 61
3.80 0.645 0.879 0.987 1.000 59
3.90 0.657 0.888 0.989 1.000 58
4.00 0.668 0.896 0.991 1.000 56
4.10 0.678 0.903 0.992 1.000 55
4.20 0.688 0.910 0.993 1.000 54
4.30 0.698 0.916 0.994 1.000 52
4.40 0.707 0.921 0.995 1.000 51
4.50 0.716 0.927 0.996 1.000 50
4.60 0.724 0.931 0.996 1.000 49
4.70 0.732 0.936 0.997 1.000 48
4.80 0.740 0.940 0.997 1.000 47
4.90 0.747 0.944 0.998 1.000 46
5.00 0.755 0.947 0.998 1.000 45
5.10 0.761 0.950 0.998 1.000 44
5.20 0.768 0.953 0.998 1.000 43
5.30 0.774 0.956 0.999 1.000 42
5.40 0.780 0.959 0.999 1.000 42
5.50 0.786 0.961 0.999 1.000 41
5.60 0.792 0.963 0.999 1.000 40
5.70 0.797 0.966 0.999 1.000 39
5.80 0.802 0.967 0.999 1.000 39
5.90 0.807 0.969 0.999 1.000 38
STT
6.00 0.812 0.971 0.999 1.000 38
6.10 0.817 0.973 0.999 1.000 37
6.20 0.821 0.974 1.000 1.000 36
6.30 0.826 0.975 1.000 1.000 36
6.40 0.830 0.977 1.000 1.000 35
6.50 0.834 0.978 1.000 1.000 35
6.60 0.838 0.979 1.000 1.000 34
6.70 0.841 0.980 1.000 1.000 34
6.80 0.845 0.981 1.000 1.000 33
6.90 0.849 0.982 1.000 1.000 33
7.00 0.852 0.983 1.000 1.000 32
7.1 0.855 0.984 1.000 1.000 32
7.2 0.858 0.985 1.000 1.000 31
7.3 0.861 0.985 1.000 1.000 31
7.4 0.864 0.986 1.000 1.000 30
7.5 0.867 0.987 1.000 1.000 30
7.6 0.870 0.987 1.000 1.000 30
7.7 0.873 0.988 1.000 1.000 29
7.8 0.875 0.989 1.000 1.000 29
7.90 0.878 0.989 1.000 1.000 28
8.00 0.880 0.990 1.000 1.000 28
8.10 0.883 0.990 1.000 1.000 28
8.20 0.885 0.990 1.000 1.000 27
8.30 0.887 0.991 1.000 1.000 27
8.40 0.890 0.991 1.000 1.000 27
8.50 0.892 0.992 1.000 1.000 26
8.60 0.894 0.992 1.000 1.000 26
8.70 0.896 0.992 1.000 1.000 26
8.80 0.898 0.993 1.000 1.000 26
8.90 0.900 0.993 1.000 1.000 25
9.00 0.901 0.993 1.000 1.000 25
9.10 0.903 0.994 1.000 1.000 25
9.20 0.905 0.994 1.000 1.000 24
9.30 0.907 0.994 1.000 1.000 24
9.40 0.908 0.994 1.000 1.000 24
9.50 0.910 0.994 1.000 1.000 24
9.60 0.912 0.995 1.000 1.000 23
9.70 0.913 0.995 1.000 1.000 23
9.80 0.915 0.995 1.000 1.000 23
9.90 0.916 0.995 1.000 1.000 23
10.00 0.917 0.995 1.000 1.000 23
10.10 0.919 0.996 1.000 1.000 22
10.20 0.920 0.996 1.000 1.000 22
10.30 0.922 0.996 1.000 1.000 22
10.40 0.923 0.996 1.000 1.000 22
10.50 0.924 0.996 1.000 1.000 21
10.60 0.925 0.996 1.000 1.000 21
10.70 0.926 0.996 1.000 1.000 21
10.80 0.928 0.997 1.000 1.000 21
10.90 0.929 0.997 1.000 1.000 21
11.00 0.930 0.997 1.000 1.000 20
11.10 0.931 0.997 1.000 1.000 20
11.20 0.932 0.997 1.000 1.000 20
11.30 0.933 0.997 1.000 1.000 20
11.40 0.934 0.997 1.000 1.000 20
11.50 0.935 0.997 1.000 1.000 20
11.60 0.936 0.997 1.000 1.000 19
11.70 0.937 0.998 1.000 1.000 19
11.80 0.938 0.998 1.000 1.000 19
11.90 0.939 0.998 1.000 1.000 19
12.00 0.940 0.998 1.000 1.000 19
12.10 0.941 0.998 1.000 1.000 19
12.20 0.941 0.998 1.000 1.000 18
12.30 0.942 0.998 1.000 1.000 18
12.40 0.943 0.998 1.000 1.000 18
12.50 0.944 0.998 1.000 1.000 18
12.60 0.945 0.998 1.000 1.000 18
12.70 0.945 0.998 1.000 1.000 18
12.80 0.946 0.998 1.000 1.000 18
12.90 0.947 0.998 1.000 1.000 17
13.00 0.948 0.998 1.000 1.000 17
13.10 0.948 0.998 1.000 1.000 17
13.20 0.949 0.998 1.000 1.000 17
13.30 0.950 0.999 1.000 1.000 17
13.40 0.950 0.999 1.000 1.000 17
13.50 0.951 0.999 1.000 1.000 17
13.60 0.952 0.999 1.000 1.000 17
13.70 0.952 0.999 1.000 1.000 16
13.80 0.953 0.999 1.000 1.000 16
13.90 0.953 0.999 1.000 1.000 16
14.00 0.954 0.999 1.000 1.000 16
14.10 0.955 0.999 1.000 1.000 16
14.20 0.955 0.999 1.000 1.000 16
14.30 0.956 0.999 1.000 1.000 16
14.40 0.956 0.999 1.000 1.000 16
14.50 0.957 0.999 1.000 1.000 16
14.60 0.957 0.999 1.000 1.000 15
14.70 0.958 0.999 1.000 1.000 15
14.80 0.958 0.999 1.000 1.000 15
14.90 0.959 0.999 1.000 1.000 15
15.00 0.959 0.999 1.000 1.000 15
1-sided
G*Power software (http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/).
SD1
15.0
14.3
13.7
13.2
12.7
12.2
11.9
11.5
11.2
10.9
10.6
10.4
10.1
9.9
9.7
9.5
9.3
9.1
9.0
8.8
8.7
8.5
8.4
8.3
8.1
8.0
7.9
7.8
7.7
7.6
7.5
7.4
7.3
7.2
7.2
7.1
7.0
6.9
6.8
6.8
6.7
6.6
6.6
6.5
6.5
6.4
6.3
6.3
6.2
6.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.02.04.06.08.010.012.014.016.0
SD
6.1
6.1
6.0
6.0
5.9
5.9
5.8
5.8
5.8
5.7
5.7
5.6
5.6
5.6
5.5
5.5
5.4
5.4
5.4
5.3
5.3
5.3
5.2
5.2
5.2
5.1
5.1
5.1
5.1
5.0
5.0
5.0
4.9
4.9
4.9
4.9
4.8
4.8
4.8
4.8
4.7
4.7
4.7
4.7
4.7
4.6
4.6
4.6
4.6
4.5
4.5
4.5
4.5
4.5
4.4
4.4
4.4
4.4
4.4
4.3
4.3
4.3
4.3
4.3
4.3
4.2
4.2
4.2
4.2
4.2
4.2
4.1
4.1
4.1
4.1
4.1
4.1
4.1
4.0
4.0
4.0
4.0
4.0
4.0
4.0
3.9
3.9
3.9
3.9
3.9
3.9
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
Po
we
r
Power Analysis
Finding Bias for a given n , with CV: see Worksheet Power CV
a and b considerations (maximum bias = 15%)
n = 10 n = 20
CV 6.65 8.69 Exact values by power of the non-central one-sample t- test (95% power)
df CI-a 15 - CI-a CI-b 15 - CI-b "R" software: http://www.r-project.org/
9 3.85 11.1 7.52 7.5 Piface: http://www.cs.uiowa.edu/~rlenth/Power/
19 3.36 11.6 6.64 8.4 http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/
With 20 measurements, we can tolerate a bias of 11.6% (a-error) and 8.4% (b-error).
STT
Exact values by power of the non-central one-sample t- test (95% power)
"R" software: http://www.r-project.org/
Piface: http://www.cs.uiowa.edu/~rlenth/Power/
http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/
1-sample t
SD from power calculations (limit 15)
SD 4.11 6.65 8.69 11.9
n 5 10 20 100 5
Effect-size t-test
Diff/SD Power Power Power Power Difference
0.00 0.050 0.050 0.050 0.050 0.000
0.01 0.052 0.053 0.055 0.061 0.041
0.02 0.054 0.056 0.060 0.074 0.082
0.03 0.056 0.060 0.065 0.089 0.123
0.04 0.058 0.063 0.070 0.106 0.164
0.05 0.061 0.067 0.076 0.125 0.206
0.06 0.063 0.071 0.083 0.147 0.247
0.07 0.065 0.075 0.090 0.171 0.288
0.08 0.068 0.079 0.097 0.198 0.329
0.09 0.070 0.084 0.104 0.226 0.370
0.10 0.073 0.088 0.112 0.257 0.411
0.11 0.075 0.093 0.121 0.290 0.452
0.12 0.078 0.098 0.130 0.325 0.493
0.13 0.081 0.103 0.139 0.362 0.534
0.14 0.084 0.109 0.149 0.400 0.575
0.15 0.087 0.114 0.159 0.438 0.617
0.16 0.090 0.120 0.170 0.478 0.658
0.17 0.093 0.126 0.181 0.517 0.699
0.18 0.096 0.132 0.193 0.557 0.740
0.19 0.099 0.138 0.205 0.596 0.781
0.20 0.102 0.145 0.217 0.634 0.822
0.21 0.106 0.152 0.230 0.670 0.863
0.22 0.109 0.159 0.243 0.705 0.904
0.23 0.113 0.166 0.257 0.739 0.945
0.24 0.116 0.173 0.271 0.770 0.986
0.25 0.120 0.181 0.286 0.799 1.028
0.26 0.124 0.188 0.300 0.826 1.069
0.27 0.128 0.196 0.315 0.850 1.110
0.28 0.132 0.205 0.331 0.872 1.151
0.29 0.136 0.213 0.347 0.892 1.192
0.30 0.140 0.222 0.363 0.909 1.233
0.31 0.144 0.230 0.379 0.924 1.274
0.32 0.148 0.239 0.396 0.937 1.315
0.33 0.153 0.249 0.412 0.949 1.356
0.34 0.157 0.258 0.429 0.958 1.397
0.35 0.162 0.267 0.446 0.966 1.439
0.36 0.166 0.277 0.463 0.973 1.480
0.37 0.171 0.287 0.480 0.979 1.521
0.38 0.176 0.297 0.497 0.983 1.562
0.39 0.181 0.307 0.515 0.987 1.603
0.40 0.185 0.317 0.532 0.990 1.644
0.41 0.190 0.328 0.549 0.992 1.685
0.42 0.196 0.339 0.566 0.994 1.726
0.43 0.201 0.349 0.583 0.996 1.767
0.44 0.206 0.360 0.600 0.997 1.808
STT
0.45 0.211 0.371 0.616 0.998 1.850
0.46 0.217 0.382 0.632 0.998 1.891
0.47 0.222 0.393 0.649 0.999 1.932
0.48 0.228 0.405 0.664 0.999 1.973
0.49 0.233 0.416 0.680 0.999 2.014
0.50 0.239 0.427 0.695 1.000 2.055
0.51 0.245 0.439 0.710 1.000 2.096
0.52 0.251 0.450 0.725 1.000 2.137
0.53 0.256 0.462 0.739 1.000 2.178
0.54 0.262 0.473 0.753 1.000 2.219
0.55 0.268 0.485 0.766 1.000 2.261
0.56 0.275 0.497 0.779 1.000 2.302
0.57 0.281 0.508 0.792 1.000 2.343
0.58 0.287 0.520 0.804 1.000 2.384
0.59 0.293 0.531 0.815 1.000 2.425
0.60 0.300 0.543 0.827 1.000 2.466
0.61 0.306 0.554 0.837 1.000 2.507
0.62 0.313 0.566 0.848 1.000 2.548
0.63 0.319 0.577 0.858 1.000 2.589
0.64 0.326 0.589 0.867 1.000 2.630
0.65 0.332 0.600 0.876 1.000 2.672
0.66 0.339 0.611 0.885 1.000 2.713
0.67 0.346 0.622 0.893 1.000 2.754
0.68 0.352 0.633 0.901 1.000 2.795
0.69 0.359 0.644 0.908 1.000 2.836
0.70 0.366 0.655 0.915 1.000 2.877
0.71 0.373 0.665 0.921 1.000 2.918
0.72 0.380 0.676 0.927 1.000 2.959
0.73 0.387 0.686 0.933 1.000 3.000
0.74 0.394 0.697 0.939 1.000 3.041
0.75 0.401 0.707 0.944 1.000 3.083
0.76 0.408 0.716 0.948 1.000 3.124
0.77 0.415 0.726 0.953 1.000 3.165
0.78 0.422 0.736 0.957 1.000 3.206
0.79 0.429 0.745 0.961 1.000 3.247
0.80 0.436 0.754 0.964 1.000 3.288
0.81 0.444 0.763 0.967 1.000 3.329
0.82 0.451 0.772 0.970 1.000 3.370
0.83 0.458 0.781 0.973 1.000 3.411
0.84 0.465 0.789 0.976 1.000 3.452
0.85 0.472 0.798 0.978 1.000 3.494
0.86 0.480 0.806 0.980 1.000 3.535
0.87 0.487 0.814 0.982 1.000 3.576
0.88 0.494 0.821 0.984 1.000 3.617
0.89 0.501 0.829 0.986 1.000 3.658
0.90 0.508 0.836 0.987 1.000 3.699
0.91 0.516 0.843 0.989 1.000 3.740
0.92 0.523 0.850 0.990 1.000 3.781
0.93 0.530 0.857 0.991 1.000 3.822
0.94 0.537 0.863 0.992 1.000 3.863
0.95 0.544 0.869 0.993 1.000 3.905
0.96 0.551 0.875 0.994 1.000 3.946
0.97 0.559 0.881 0.994 1.000 3.987
0.98 0.566 0.887 0.995 1.000 4.028
0.99 0.573 0.892 0.996 1.000 4.069
1.00 0.580 0.898 0.996 1.000 4.110
1.01 0.587 0.903 0.997 1.000 4.151
1.02 0.594 0.907 0.997 1.000 4.192
1.03 0.601 0.912 0.997 1.000 4.233
1.04 0.608 0.917 0.998 1.000 4.274
1.05 0.614 0.921 0.998 1.000 4.316
1.06 0.621 0.925 0.998 1.000 4.357
1.07 0.628 0.929 0.998 1.000 4.398
1.08 0.635 0.933 0.999 1.000 4.439
1.09 0.642 0.937 0.999 1.000 4.480
1.10 0.648 0.940 0.999 1.000 4.521
1.11 0.655 0.944 0.999 1.000 4.562
1.12 0.661 0.947 0.999 1.000 4.603
1.13 0.668 0.950 0.999 1.000 4.644
1.14 0.674 0.953 0.999 1.000 4.685
1.15 0.681 0.956 1.000 1.000 4.727
1.16 0.687 0.958 1.000 1.000 4.768
1.17 0.693 0.961 1.000 1.000 4.809
1.18 0.700 0.963 1.000 1.000 4.850
1.19 0.706 0.965 1.000 1.000 4.891
1.20 0.712 0.967 1.000 1.000 4.932
1.21 0.718 0.970 1.000 1.000 4.973
1.22 0.724 0.971 1.000 1.000 5.014
1.23 0.730 0.973 1.000 1.000 5.055
1.24 0.736 0.975 1.000 1.000 5.096
1.25 0.741 0.977 1.000 1.000 5.138
1.26 0.747 0.978 1.000 1.000 5.179
1.27 0.753 0.980 1.000 1.000 5.220
1.28 0.758 0.981 1.000 1.000 5.261
1.29 0.764 0.982 1.000 1.000 5.302
1.30 0.769 0.984 1.000 1.000 5.343
1.31 0.775 0.985 1.000 1.000 5.384
1.32 0.780 0.986 1.000 1.000 5.425
1.33 0.785 0.987 1.000 1.000 5.466
1.34 0.790 0.988 1.000 1.000 5.507
1.35 0.795 0.989 1.000 1.000 5.549
1.36 0.800 0.989 1.000 1.000 5.590
1.37 0.805 0.990 1.000 1.000 5.631
1.38 0.810 0.991 1.000 1.000 5.672
1.39 0.815 0.992 1.000 1.000 5.713
1.40 0.819 0.992 1.000 1.000 5.754
1.41 0.824 0.993 1.000 1.000 5.795
1.42 0.829 0.993 1.000 1.000 5.836
1.43 0.833 0.994 1.000 1.000 5.877
1.44 0.837 0.994 1.000 1.000 5.918
1.45 0.842 0.995 1.000 1.000 5.960
1.46 0.846 0.995 1.000 1.000 6.001
1.47 0.850 0.996 1.000 1.000 6.042
1.48 0.854 0.996 1.000 1.000 6.083
1.49 0.858 0.996 1.000 1.000 6.124
1.50 0.862 0.997 1.000 1.000 6.165
1.51 0.866 0.997 1.000 1.000 6.206
1.52 0.870 0.997 1.000 1.000 6.247
1.53 0.873 0.997 1.000 1.000 6.288
1.54 0.877 0.998 1.000 1.000 6.329
1.55 0.880 0.998 1.000 1.000 6.371
1.56 0.884 0.998 1.000 1.000 6.412
1.57 0.887 0.998 1.000 1.000 6.453
1.58 0.891 0.998 1.000 1.000 6.494
1.59 0.894 0.998 1.000 1.000 6.535
1.60 0.897 0.999 1.000 1.000 6.576
1.61 0.900 0.999 1.000 1.000 6.617
1.62 0.903 0.999 1.000 1.000 6.658
1.63 0.906 0.999 1.000 1.000 6.699
1.64 0.909 0.999 1.000 1.000 6.740
1.65 0.912 0.999 1.000 1.000 6.782
1.66 0.914 0.999 1.000 1.000 6.823
1.67 0.917 0.999 1.000 1.000 6.864
1.68 0.920 0.999 1.000 1.000 6.905
1.69 0.922 0.999 1.000 1.000 6.946
1.70 0.925 0.999 1.000 1.000 6.987
1.71 0.927 1.000 1.000 1.000 7.028
1.72 0.930 1.000 1.000 1.000 7.069
1.73 0.932 1.000 1.000 1.000 7.110
1.74 0.934 1.000 1.000 1.000 7.151
1.75 0.936 1.000 1.000 1.000 7.193
1.76 0.939 1.000 1.000 1.000 7.234
1.77 0.941 1.000 1.000 1.000 7.275
1.78 0.943 1.000 1.000 1.000 7.316
1.79 0.945 1.000 1.000 1.000 7.357
1.80 0.947 1.000 1.000 1.000 7.398
1.81 0.948 1.000 1.000 1.000 7.439
1.82 0.950 1.000 1.000 1.000 7.480
1.83 0.952 1.000 1.000 1.000 7.521
1.84 0.954 1.000 1.000 1.000 7.562
1.85 0.955 1.000 1.000 1.000 7.604
1.86 0.957 1.000 1.000 1.000 7.645
1.87 0.958 1.000 1.000 1.000 7.686
1.88 0.960 1.000 1.000 1.000 7.727
1.89 0.961 1.000 1.000 1.000 7.768
1.90 0.963 1.000 1.000 1.000 7.809
1.91 0.964 1.000 1.000 1.000 7.850
1.92 0.966 1.000 1.000 1.000 7.891
1.93 0.967 1.000 1.000 1.000 7.932
1.94 0.968 1.000 1.000 1.000 7.973
1.95 0.969 1.000 1.000 1.000 8.015
1.96 0.971 1.000 1.000 1.000 8.056
1.97 0.972 1.000 1.000 1.000 8.097
1.98 0.973 1.000 1.000 1.000 8.138
1.99 0.974 1.000 1.000 1.000 8.179
2.00 0.975 1.000 1.000 1.000 8.220
1-sided
G*Power software (http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/).
10 20 100
Difference Difference Difference
0.000 0.000 0.000
0.067 0.087 0.119
0.133 0.174 0.238
0.200 0.261 0.357
0.266 0.348 0.476
0.333 0.435 0.595
0.399 0.521 0.714
0.466 0.608 0.833
0.532 0.695 0.952
0.599 0.782 1.071
0.665 0.869 1.190
0.732 0.956 1.309
0.798 1.043 1.428
0.865 1.130 1.547
0.931 1.217 1.666
0.998 1.304 1.785
1.064 1.390 1.904
1.131 1.477 2.023
1.197 1.564 2.142
1.264 1.651 2.261
1.330 1.738 2.380
1.397 1.825 2.499
1.463 1.912 2.618
1.530 1.999 2.737
1.596 2.086 2.856
1.663 2.173 2.975
1.729 2.259 3.094
1.796 2.346 3.213
1.862 2.433 3.332
1.929 2.520 3.451
1.995 2.607 3.570
2.062 2.694 3.689
2.128 2.781 3.808
2.195 2.868 3.927
2.261 2.955 4.046
2.328 3.042 4.165
2.394 3.128 4.284
2.461 3.215 4.403
2.527 3.302 4.522
2.594 3.389 4.641
2.660 3.476 4.760
2.727 3.563 4.879
2.793 3.650 4.998
2.860 3.737 5.117
2.926 3.824 5.236
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 2.0 4.0 6.0
Po
wer
Difference
2.993 3.911 5.355
3.059 3.997 5.474
3.126 4.084 5.593
3.192 4.171 5.712
3.259 4.258 5.831
3.325 4.345 5.950
3.392 4.432 6.069
3.458 4.519 6.188
3.525 4.606 6.307
3.591 4.693 6.426
3.658 4.780 6.545
3.724 4.866 6.664
3.791 4.953 6.783
3.857 5.040 6.902
3.924 5.127 7.021
3.990 5.214 7.140
4.057 5.301 7.259
4.123 5.388 7.378
4.190 5.475 7.497
4.256 5.562 7.616
4.323 5.649 7.735
4.389 5.735 7.854
4.456 5.822 7.973
4.522 5.909 8.092
4.589 5.996 8.211
4.655 6.083 8.330
4.722 6.170 8.449
4.788 6.257 8.568
4.855 6.344 8.687
4.921 6.431 8.806
4.988 6.518 8.925
5.054 6.604 9.044
5.121 6.691 9.163
5.187 6.778 9.282
5.254 6.865 9.401
5.320 6.952 9.520
5.387 7.039 9.639
5.453 7.126 9.758
5.520 7.213 9.877
5.586 7.300 9.996
5.653 7.387 10.115
5.719 7.473 10.234
5.786 7.560 10.353
5.852 7.647 10.472
5.919 7.734 10.591
5.985 7.821 10.710
6.052 7.908 10.829
6.118 7.995 10.948
6.185 8.082 11.067
6.251 8.169 11.186
6.318 8.256 11.305
6.384 8.342 11.424
6.451 8.429 11.543
6.517 8.516 11.662
6.584 8.603 11.781
6.650 8.690 11.900
6.717 8.777 12.019
6.783 8.864 12.138
6.850 8.951 12.257
6.916 9.038 12.376
6.983 9.125 12.495
7.049 9.211 12.614
7.116 9.298 12.733
7.182 9.385 12.852
7.249 9.472 12.971
7.315 9.559 13.090
7.382 9.646 13.209
7.448 9.733 13.328
7.515 9.820 13.447
7.581 9.907 13.566
7.648 9.994 13.685
7.714 10.080 13.804
7.781 10.167 13.923
7.847 10.254 14.042
7.914 10.341 14.161
7.980 10.428 14.280
8.047 10.515 14.399
8.113 10.602 14.518
8.180 10.689 14.637
8.246 10.776 14.756
8.313 10.863 14.875
8.379 10.949 14.994
8.446 11.036 15.113
8.512 11.123 15.232
8.579 11.210 15.351
8.645 11.297 15.470
8.712 11.384 15.589
8.778 11.471 15.708
8.845 11.558 15.827
8.911 11.645 15.946
8.978 11.732 16.065
9.044 11.818 16.184
9.111 11.905 16.303
9.177 11.992 16.422
9.244 12.079 16.541
9.310 12.166 16.660
9.377 12.253 16.779
9.443 12.340 16.898
9.510 12.427 17.017
9.576 12.514 17.136
9.643 12.601 17.255
9.709 12.687 17.374
9.776 12.774 17.493
9.842 12.861 17.612
9.909 12.948 17.731
9.975 13.035 17.850
10.042 13.122 17.969
10.108 13.209 18.088
10.175 13.296 18.207
10.241 13.383 18.326
10.308 13.470 18.445
10.374 13.556 18.564
10.441 13.643 18.683
10.507 13.730 18.802
10.574 13.817 18.921
10.640 13.904 19.040
10.707 13.991 19.159
10.773 14.078 19.278
10.840 14.165 19.397
10.906 14.252 19.516
10.973 14.339 19.635
11.039 14.425 19.754
11.106 14.512 19.873
11.172 14.599 19.992
11.239 14.686 20.111
11.305 14.773 20.230
11.372 14.860 20.349
11.438 14.947 20.468
11.505 15.034 20.587
11.571 15.121 20.706
11.638 15.208 20.825
11.704 15.294 20.944
11.771 15.381 21.063
11.837 15.468 21.182
11.904 15.555 21.301
11.970 15.642 21.420
12.037 15.729 21.539
12.103 15.816 21.658
12.170 15.903 21.777
12.236 15.990 21.896
12.303 16.077 22.015
12.369 16.163 22.134
12.436 16.250 22.253
12.502 16.337 22.372
12.569 16.424 22.491
12.635 16.511 22.610
12.702 16.598 22.729
12.768 16.685 22.848
12.835 16.772 22.967
12.901 16.859 23.086
12.968 16.946 23.205
13.034 17.032 23.324
13.101 17.119 23.443
13.167 17.206 23.562
13.234 17.293 23.681
13.300 17.380 23.800
G*Power software (http://www.psycho.uni-duesseldorf.de/aap/projects/gpower/).
Different n
Fixed SD 8.69 (n = 20)
8.0 10.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 5.0 10.0 15.0
Po
wer
Difference
20.0
Tolerance interval
Finding Bias for a given n , Tolerance interval considerations
Tolerance interval (0.95, 0.95); TE limit = 29.4% ($)
1-sided 1-sided
n Beta CV k k x s Bias
10 6.65 2.91 19.4 10.0
20 8.69 2.40 20.9 8.5
k-values from ISO 3207-1975
$29.4 = 1.96*15% CV
With 20 measurements, we can tolerate a bias of 8.5%.
1-sided b-content interval (0.95, 0.95) (for ANOVA); TE limit = 29.4%, values from Data-Worksheet
This example is based on the Worksheets Precision and Data (ANOVA calculations)
Hoffman D, Kringle R. Pharm Res 2007;24:1157-64.s t
2 73.8 H1 1.707 Help numbers
dfB 9 H2 1.538 12.606
dfE 10 MSB 76.30 1.026
Ne 18.86 MSE 71.29
Z0.95 1.645 J (rep/run) 2
CHI2(0.95,9) 3.325 J-1 1 Not relevant for this application
CHI2(0.95,10) 3.940 I (runs) 10 Actual mean 102
Target Bias
1-sided b-content (Mean +/-) 21.3 100 8.1
With 20 measurements, we can tolerate a bias of 8.1%.
Note 1
The "rest bias" is calculated by use of the TE-limit of 29.4%.
Note 2
With n = 20, all 3 approaches deliver a bias limit of approx. 8%
when we use the "b"-CV for calculation.
STT
1-sided b-content interval (0.95, 0.95) (for ANOVA); TE limit = 29.4%, values from Data-Worksheet
This example is based on the Worksheets Precision and Data (ANOVA calculations)
Note, simplified calculations, better values in:β-Expectation and β-Content Tolerance Limits for Balanced One-Way ANOVA Random Model
Robert W. Mee, Technometrics, Vol. 26, No. 3 (1984), pp. 251-254
Not relevant for this application
PrecisionSTT
Requirement We will work with the CV requirement for bioanalytical method validation [1]: CVmax = 15% (20% at the limit of quantitation, understood as total CV, consisting of a within-day (within-run) and a between-day (betweenwork with the standard deviation (s) at a level of 100 (= m). This corresponds then directly to the CV. Experimental design The experimental design for estimating imprecision should strive for balanced degrees of freedom (df) for withinFurther, the number of measurement results should be sufficiently high for obtaining reasonable confidence intervals. We choodesign: Days = 10 (= k); Replicates = 2 (per day = n); Total = 20 (the guidance documents require 5 [1] or 3 [2] measurements, only).The data are simulated with the EXCEL-function "Random Number Generation" (Tools>Data Analysis) in such a way to obtain a total of approximately 8.69. Three data sets are simulated: sb = 0, sw = 8.69, m = 100 (directly 2 x 10) sb = 6.14, sw = 6.14, m = 100 (first the 10 between, then the replicates: 10 times, "between values", sb = 7.8, sw = 3.8, m = 100 (see case 2) Depending on the ratio sb/sw, the df's for st approach the total df's (maximum 19) or the between df's (minimum 9). The imprecision components are calculated by ANOVA (dynamically programmed), the df's with the Satterthwaite approximation =[(n 1 Guidance for Industry: Bioanalytical Method Validation, US Department of Health andHumanServices, Food and Drug AdministratEvaluation and Research (CBER), Rockville, MD, 2001. 2 ICH Topic Q2 (R1), Validation of Analytical Procedures: Text and Methodology, International Conference on Harmonization (ICRequirements for Registration of Pharmaceuticals for Human Use, Geneva, 2005. 3 Stöckl D, D'Hondt H, Thienpont LM. Method validation across the disciplines-Critical investigation of major validation criteriprotocols. J Chromatogr B Analyt Technol Biomed Life Sci 2008;Dec 31[Epub ahead of print].
We will work with the CV requirement for bioanalytical method validation [1]: CVmax = 15% (20% at the limit of quantitation, LoQ). This CVmax is day (between-run) component. For the ease of the EXCEL-simulations, we will
). This corresponds then directly to the CV.
The experimental design for estimating imprecision should strive for balanced degrees of freedom (df) for within- (sw) and between-day (sb) estimates [3]. Further, the number of measurement results should be sufficiently high for obtaining reasonable confidence intervals. We choose for following experimental
); Total = 20 (the guidance documents require 5 [1] or 3 [2] measurements, only). function "Random Number Generation" (Tools>Data Analysis) in such a way to obtain a total standard deviation (st)
= 100 (first the 10 between, then the replicates: 10 times, "between values", sw and n = 2)
approach the total df's (maximum 19) or the between df's (minimum 9). The imprecision components are calculated by ANOVA (dynamically programmed), the df's with the Satterthwaite approximation =[(n -1)xMSw+MSb]^2/{[(n - 1)xMSw^2]/k+MSb^2/(k-1)}.
1 Guidance for Industry: Bioanalytical Method Validation, US Department of Health andHumanServices, Food and Drug Administration, Center for Biologics
2 ICH Topic Q2 (R1), Validation of Analytical Procedures: Text and Methodology, International Conference on Harmonization (ICH) of Technical
Critical investigation of major validation criteria and associated experimental protocols. J Chromatogr B Analyt Technol Biomed Life Sci 2008;Dec 31[Epub ahead of print].
Data
ANOVA (equal sample sizes!)
Daily
Day A B VAR Mean
1 109.8 99.1 57.245 104.45
2 103.0 105.2 2.42 104.1
3 91.7 104.5 81.92 98.1
4 114.6 92.6 242 103.6
5 99.4 95.1 9.245 97.25
6 108.3 113.0 11.045 110.65
7 114.0 106.4 28.88 110.2
8 98.1 107.0 39.605 102.55
9 92.0 107.0 112.5 99.5
10 98.0 82.0 128 90AVERAGETotal 102.040
COUNTRep 2
Source of Variation SS df MSBetween Groups 686.6880 9 76.299
Within Groups 712.8600 10 71.286
Satterthwaite 18.86Total 1399.55 19
ANOVA (equal sample sizes!)
Daily
Day A B VAR Mean
1 106.8 100.5 19.845 103.65
2 95.4 103.6 33.62 99.5
3 102.0 84.3 156.645 93.15
4 107.2 122.3 114.005 114.75
5 104.3 103.0 0.845 103.65
6 95.5 97.2 1.445 96.35
7 115.9 108.6 26.645 112.25
8 103.7 99.7 8 101.7
9 93.6 91.0 3.38 92.3
10 101.4 102.6 0.72 102AVERAGETotal 101.930
COUNTRep 2
Source of Variation SS df MS
Between Groups 967.3920 9 107.488
Within Groups 365.1500 10 36.515
Satterthwaite 15Total 1332.542 19
ANOVA (equal sample sizes!)
Daily
Replicate
Replicate
Replicate
STT
Day A B VAR Mean
1 106.2 101.7 10.125 103.95
2 85.8 91.0 13.52 88.4
3 101.2 108.0 23.12 104.6
4 101.0 108.2 25.92 104.6
5 103.1 109.0 17.405 106.05
6 109.6 118.0 35.28 113.8
7 83.6 88.0 9.68 85.8
8 100.6 99.4 0.72 100
9 101.7 104.9 5.12 103.3
10 99.6 103.6 8 101.6AVERAGETotal 101.210
COUNTRep 2
Source of Variation SS df MS
Between Groups 1239.948 9 137.772
Within Groups 148.890 10 14.889
Satterthwaite 11Total 1388.838 19
Help 1 Help 2
5.81 57.245
4.24 2.42
15.52 81.92
2.43 242
22.94 9.245
74.13 11.045
66.59 28.88
0.26 39.605
6.45 112.5
144.96 128
8.44 s w
1.58 s b Set s b to 0 when a negative SQRT results!
8.59 s t
Standard error: s/sqrt(n)1.928 s t, Satterthwaite+1
1.953 s b and s t
Help 1 Help 2
2.96 19.845
5.90 33.62
77.09 156.645
164.35 114.005
2.96 0.845
31.14 1.445
106.50 26.645
0.05 8
92.74 3.38
0.00 0.72
6.04 s w
5.96 s b
8.49 s t
Standard error: s/sqrt(n)2.146 s t, Satterthwaite+1
2.318 s b and s t
Help 1 Help 2
7.51 10.125
164.10 13.52
11.49 23.12
11.49 25.92
23.43 17.405
158.51 35.28
237.47 9.68
1.46 0.72
4.37 5.12
0.15 8
3.86 s w
7.84 s b
8.74 s t
Standard error: s/sqrt(n)2.529 s t, Satterthwaite+1
2.625 s b and s t
