Bivariate data
Bivariate data – Graphical & statistical techniques
Graphical techniques
• Scatter-plot
• Difference-plot
• Residual-plot
• Krouwer-plot
• Influences on the plots (data-range; subgroups; outliers; scaling)
• Influences of random- and systematic errors on the plots
• Linearity
• Specifications in plots
Combined graphical/statistical techniques
• The Bland & Altman approach
Correlation
• The statistical model
• Correlation in method comparison
• Non-parametric correlation
Regression
• Ordinary linear regression (OLR)
• Deming regression
• Passing-Bablok regression (non-parametric)
• Weighted regression
• Regression & method comparison
• Regression & calibration
Datasets; GraphBivariate-EXCEL;
Correlation&Regression; CorrRegr-EXCEL; Bland&Altman
Statistics & graphics for the laboratory
87
Graphical techniques
The scatter-plot
Construction of the axes
• x-axis: comparative method (A)
• y-axis: test method (B)
• line of equality ( y = x ): - - Usually, both axes extend
from 0 to the highest result
Glucose (mmol/l)
25
20
15
10
5
0
0
5
10
15
Glucose (mmol/l)
20
25
The absolute difference plot
Construction of the x-axis
• Hierarchically higher (A) and lower method (B)
– x-axis: hierarchically higher method (A)
• Hierarchically equivalent methods
– x-axis: (A + B)/2
Construction of the y-axis
–y = B - A
Absolute bias (mmol/l)
6
– y-axis is freely scalable
– x-axis bisects the y-axis at 0
4
2
0
-2
0
5
10
15
20
25
-4
-6
Glucose (mmol/l)
Statistics & graphics for the laboratory
88
Graphical techniques
The percent (%) difference plot
Construction of the x-axis
• Hierarchically higher (A) and lower method (B)
– x-axis: hierarchically higher method (A)
• Hierarchically equivalent methods
– x-axis: (A + B)/2
Construction of the y-axis
– y = [(B - A)/A]x100, or [(B - A)/0.5x(A + B)]x100
30
– y-axis is freely scalable
– x-axis bisects the y-axis at 0
Percent bias (%)
20
10
0
-10
0
5
10
15
20
25
-20
-30
Glucose (mmol/l)
The residuals plot
Construction of the axes
• x-axis: comparison method (A)
• y-axis: regression (OLR) residuals: yi - ŷ
– y-axis is freely scalable
– x-axis bisects the y-axis at 0
Residuals (mmol/l)
6
4
2
0
-2
0
5
10
15
20
25
-4
-6
Glucose (mmol/l)
Statistics & graphics for the laboratory
89
Graphical techniques
The Krouwer plot (for the % bias)
Construction of the axes
• x-axis: %-bias
• y-axis: “folded” cumulative percentage
"Folded" cumulative %
50%
40%
30%
20%
10%
0%
-15
-10
-5
0
5
Percent bias (%)
10
15
Construction of the Krouwer plot
"Folded cumulative percentage"
Cumulative frequency
1
Fold this area
over an angle
of 180°
0,8
0,6
0,4
0,2
Folded cum. frequency
0
-5
-4 -3
-2 -1 0 1
Multiple of s
2
3
4
5
-5
-4 -3
-2 -1 0 1 2
Multiple of s
3
4
5
0,5
0,4
0,3
0,2
0,1
0
Statistics & graphics for the laboratory
90
Graphical techniques
Characteristics of the plots
Scatter plot (with the line y = x)
• Simple construction: the same for methods with the same/different hierarchy
• Good overview about the data through the comparison with the y = x line
“Difference” plots (absolute, %, residuals)
• y-axis is freely scalable
• Construction depends on method hierarchy
• The residuals plot can only be constructed with knowledge of regression data
 Not a pure graphical technique, but: useful for the judgement of linearity
(shown later)
Krouwer plot
• Gives an overview about the general distribution of errors
• Information is lost about the concentration-dependency of errors
Graphical presentation of a method comparison
First conclusion
• There are several different types of graphics for the interpretation of method
comparison studies
• The residuals- and the Krouwer plot are useful for the interpretation of special
aspects of a method comparison (linearity, respectively, error distribution)
• The scatter plot (y = x included) and the absolute and %-difference plot give the
best overview about method comparison data
 More detailed investigation with those 3
Influences on the plots
• Range of the results
• “Subgroups”
• Outliers
• Scaling of the axes
Statistics & graphics for the laboratory
91
Graphical techniques
Influence of the range
Influence of the range on the different plots
140
130
120
120
130
140
150
y=x
25
50
75 100 125 150
9
10
5
0
-5120
4
Absolute bias
Glucose (mmol/l)
6
y=x
5
0
5
10
15
6
3
0
-3120
20
25
130
140
150
-6
-9
Sodium (mmol/l)
20
0
150
-15
25
10
140
-10
Sodium (mmol/l)
15
130
Percent bias (%)
15
150
0
Percent bias plot
Sodium (mmol/l)
30
2
0
-2 0
5
10
15
20
25
-4
Percent bias (%)
150
125
100
75
50
25
0
Absolute bias plot
Absolute bias
Sodium (mmol/l)
Scatter plot
-6
20
10
0
-10 0
5
10
15
20
25
-20
-30
Glucose (mmol/l)
Glucose (mmol/l)
Glucose (mmol/l)
- Graphical resolution
resolution ofof
thethe
scatter
plot: plot:
worseworse
than the
bias the
plots.
-Graphical
scatter
than
bias plots.
- The resolution of the scatter plot can be improved by an insert.
-The resolution of the scatter plot can be improved by an insert.
Influence of the range on the different plots
Scatter plot
0
4000
400
y=x
800
2000
1000
0
0
5000
10000
-1000
0
0
5000
10000
-2000
log Absolute bias
log Estradiol (pmol/l)
3
2
log y = log x
0
1
2
3
4
0
5
5000
10000
-25
Estradiol (pmol/l)
0,30
4
log Estradiol (pmol/l)
0
Estradiol (pmol/l)
5
0
25
-50
Estradiol (pmol/l)
1
Percent bias (%)
0
50
0,15
0,00
0
1
2
3
4
5
-0,15
log Percent bias (%)
400
6000
Percent bias plot
2000
800
8000
Absolute bias
Estradiol (pmol/l)
10000
Absolute bias plot
-0,30
15
10
5
0
-5 0
1
2
3
4
5
-10
-15
log Estradiol (pmol/l)
log Estradiol (pmol/l)
- Graphical resolution of the scatter plot: worse than the bias plots.
- Scatter plot:
improve resolution
by insert
or logarithmic
scale
-Graphical
resolution
of the scatter
plot:
worse than
the bias plots.
-Scatter plot: improve resolution by an insert or logarithmic scale.
Don't expect that "one size fits all"
Statistics & graphics for the laboratory
DatasetsMethComp
92
Graphical techniques
Subgroups
Note: y-axis of the difference plot is freely scalable! Therefore, its graphical
resolution, usually, is better than the one of the scatter plot
 A “subgroup” is easier to see in the %-difference plot
than in the scatter plot
40
30
Percent bias (%)
Glucose (mmol/l)
25
20
15
10
5
20
10
0
-10 0
5
10
15
20
25
-20
-30
0
0
5
10
15
Glucose (mmol/l)
20
-40
25
Glucose (mmol/l)
Outliers
Glucose “normal”
PERCENT BIAS PLOT
20
15
10
5
0
2
0
0
15
20
25
-4
25
10
0
0
10
Absolute bias
15
15
10
5
0
25
10
15
20
25
-20
Glucose (mmol/l)
80
5
0
-5 0
5
-10
Glucose (mmol/l)
20
5 10 15 20
Glucose (mmol/l)
10
-2
25
0
5
5
10
15
20
-10
-15
25
Percent bias (%)
5 10 15 20
Glucose (mmol/l)
20
Percent bias (%)
4
0
Glucose (mmol/l)
ABSOLUTE BIAS PLOT
Absolute bias
Glucose (mmol/l)
SCATTER PLOT
25
40
0
0
5
10
15
20
25
-40
-80
Glucose (mmol/l)
Glucose (mmol/l)
Glucose “with outliers”
Outliers have no influence on the resolution of the scatter plot, but reduce
the resolution of the difference plots.
 Scatter plot more robust than difference plots
Statistics & graphics for the laboratory
93
Graphical techniques
Scaling
30
A
1
0
-1 0
5
10 15 20 25
-2
-3
20
15
B
10
0
-10 0
5
10 15 20 25
-20
-30
Glucose (mmol/l)
Absolute bias
2
Absolute bias
Absolute bias
3
10
C
5
0
-5 0
5
10 15 20 25
-10
-15
Glucose (mmol/l)
Glucose (mmol/l)
y-scaling …
Effect…
A: "as the data are"
Good resolution, but x- & y-axis
cannot be compared directly
B: free
Good/poor agreement
can be manipulated graphically
C: identical (graphical distance)
x and y scaling
Loss of resolution,
but better direct comparison possible
Graphs and errors
Random errors
• SD constant (small range; e.g., sodium)
• CV constant (medium range; e.g., glucose)
• SD/CV variable (wide range; e.g., estradiol)
CV constant/SD decreasing down to a
certain concentration, then
SD/CV
Common situation
CV
SD
SD constant and CV increasing
Concentration
Statistics & graphics for the laboratory
94
Graphical techniques
Graphs and errors
Systematic errors
-Constant
-Proportional
-Combination
(constant/proportional)
-Non-linearity
Graphs and errors
Examples
Systematic errors
•y=x
• y = 1.1 • x
•y=x+1
Random errors
• General examples with CV = 2% and SD = 0.1
Case 1: y = x
Scatter plot
CV =
2%
1
30
20
10
0
0
0
0
SD =
0.1
Abs. Diff. plot
10 20 30
10
20
30
-1
1
30
20
10
0
0
0
0
10 20 30
-1
10
20
30
% Diff. plot
10
5
0
-5 0
-10
10
5
0
-5 0
-10
10 20 30
10 20 30
From: D. Stöckl. Ann Clin Biochem 1996;33:575-7
Statistics & graphics for the laboratory
95
Graphical techniques
Graphs and errors
What could be noted?
For case 1: y = x
Better resolution of the difference plots
Scatter plot
• At constant CV, typical V-form of the random error limits
• At constant SD, parallel limits for random error
Absolute difference plot
• At constant CV, typical V-form of the random error limits
• At constant SD, parallel limits for random error
%-difference plot
• At constant CV, parallel limits for random error
• At constant SD, typical hyperbolic limits for random error
Case 2: y = 1.1 • x
Scatter plot
CV =
2%
30
20
10
0
0
SD =
0.1
Abs. Diff. plot
10 20 30
3
2
1
0
-1
-2 0
-3
10 20 30
3
2
1
0
-1
-2 0
-3
30
20
10
0
0
10
10
20 30
20 30
% Diff. plot
20
10
0
-10 0
-20
30
15
0
-15 0
-30
10 20 30
10 20 30
Case 3: y = x + 1
Scatter plot
CV =
2%
30
20
10
0
0
SD =
0.1
10 20 30
2
1
0
-1 0
-2
10 20 30
2
1
0
-1 0
-2
30
20
10
0
0
Abs. Diff. plot
10
10
20 30
20 30
% Diff. plot
120
60
0
-60 0
-120
120
60
0
-60 0
-120
10 20 30
10 20 30
Statistics & graphics for the laboratory
96
Graphical techniques
Graphs and errors
What could be noted?
(additionally to y = x)
A large proportional error
• Deteriorates the resolution of the absolute difference plot
• Has no influence on the %-difference plot
A large constant error
(as compared to the random error)
• Has no influence on the absolute difference plot
• The hyperbolic error limits in the %-difference plot become “one-sided”
Summary
The difference plots, generally, have a better resolution than the scatter plot
The scatter plot is robust against all sorts of errors
The limits for random error are
• V-shaped (constant CV)
• parallel (constant SD)
The absolute difference plot is robust against constant errors, but sensitive
to proportional errors (loss of resolution)
The limits for random error are
• V-shaped (constant CV)
• parallel (constant SD)
The %-difference plot is robust against proportional errors,
but sensitive towards constant errors
The limits for random error are
• parallel (CV constant and no constant error),
• 2-sided hyperbolic (SD constant and no const. error), or
• 1-sided hyperbolic (existence of a relatively big constant error)
Statistics & graphics for the laboratory
97
Graphical techniques
Linearity
Judgement of linearity
Consider the following ways
Regression analysis
20
15
10
5
0
25
4
y = 1,0697x 0,6433
20
15
10
5
0
0
Residuals plot
Residual (mmol/l)
25
Glucose (mmol/l)
Glucose (mmol/l)
Scatter plot
5 10 15 20 25
Glucose (mmol/l)
0
5 10 15 20 25
Glucose (mmol/l)
2
0
0
5
10 15 20 25
-2
-4
Glucose (mmol/l)
Best with regression (residuals plot)
For a broad range
Logarithmic
5000
10000
4000
Routine
Routine
1000
100
10
3000
2000
1000
0
1
1
10
100
1000
ID-GC/MS
10000
0
1000
2000 3000
ID-GC/MS
4000
5000
Easier with a logarithmic plot
 Conclusion: "no size fits all"
Statistics & graphics for the laboratory
98
Graphical techniques
Specifications
Specifications are needed for the interpretation of a method comparison.
We look for specifications in
• The scatter plot
• The absolute difference plot
• The %-difference plot
The scatter plot
Routine (mmol/l)
25
20
15
10
5
0
0
5
10
15
20
25
Reference (mmol/l)
Sort of specification:
Constant: 1 mmol/l
Proportional: 10%
The %-difference plot
20
3
2
1
0
-1
0
5
10
15
20
-2
25
Routine (% difference)
Routine (abs. difference)
The absolute difference plot
15
10
5
0
-5 0
5
10
15
20
25
-10
-15
-20
-3
Reference (mmol/l)
Sort of specification:
Constant: 1 mmol/l
Proportional: 10%
Reference (mmol/l)
Sort of specification:
Constant: 1 mmol/l
Proportional: 10%
Statistics & graphics for the laboratory
99
Graphical techniques
Specifications
Glucometer XYZ (mmol/l)
"Error grid analysis" (glucose)
25
Sort of specification: complex
Base assumptions
A
A
C
20
E
15
B
B
10
D
5
1. Values <3.9 mmol/l to increase
2. Values >10 mmol/l to lower
3. Deviations from the reference
up to 20% are acceptable, or,
both values (glucometer and reference)
are <3.9 mmol/l
D
Interpretation of the areas:
C
0
0
E
5
10
15
20
Reference (mmol/l)
25
A: clinically accurate
B: clinically irrelevant deviation of >20%
C: possible unnecessary overcorrection
D: glucometer produces dangerous errors
E: wrong treatment
Adapted from: Clarke et al. Diabetes Care 1987;10:622-8
Summary
The scatter plot is useful for all sorts of specifications
The limits for specifications (around y = x) are
• parallel (absolute specification)
• or V-shaped (% specification)
The absolute difference plot is most appropriate for absolute specifications
The limits for specifications (around 0) are
• parallel (absolute specification)
• or V-shaped (% specification)
The %-difference plot is most appropriate for % specifications
The limits for specifications (around 0) are
• parallel (% specification)
• 2-sided hyperbolic (absolute specification)
Annex
• More examples
• Examples sorted according to plot-type
Statistics & graphics for the laboratory
100
Exercises
GraphBivariate-EXCEL
This file is a template for a
Scatter plot (with line of equality)
Absolute bias plot (x-axis with hierarchichally higher method, only)
% bias plot (x-axis with hierarchichally higher method, only)
Absolute bias plot (x-axis with average x&y)
% bias plot (x-axis with average x&y)
Residuals plot
It may be adapted to the needs of the user.
This file can also be used to reproduce most of the plots in this tutorial by using
the datasets in:
Datasets(Method comparison: Sodium, Glucose, Estradiol)
Statistics & graphics for the laboratory
101
Graphical techniques
Annex – More examples
Case 4: y = x + 0.05
Scatter plot
CV =
2%
30
20
10
0
1
0
0
0
SD =
0.1
Abs. Diff. plot
10 20 30
10
20 30
-1
1
30
20
10
0
0
0
0
10 20 30
10
20 30
-1
% Diff. plot
10
5
0
-5 0
-10
10
5
0
-5 0
-10
10 20 30
10 20 30
Case 5: y = 1.1 • x + 1
Scatter plot
CV =
2%
30
20
10
0
0
SD =
0.1
10 20 30
6
3
0
-3 0
-6
10 20 30
6
3
0
-3 0
-6
30
20
10
0
0
Abs. Diff. plot
10
10
20 30
20 30
% Diff. plot
120
60
0
-60 0
-120
120
60
0
-60 0
-120
10 20 30
10 20 30
Statistics & graphics for the laboratory
102
Graphical techniques
Scatter-plot
y=x
y = 1,1x
y = x + 0,05
y=x+1
y = 1,1x + 1
CV = 2%
30
20
10
0
30
20
10
0
0
30
20
10
0
0
10 20 30
30
20
10
0
10 20 30
0
10 20 30
30
20
10
0
0
10 20 30
0
10 20 30
0
10 20 30
SD = 0.1
30
20
10
0
30
20
10
0
0
10 20 30
30
20
10
0
0
30
20
10
0
10 20 30
0
10 20 30
30
20
10
0
0
10 20 30
Absolute bias-plot
y=x
y = 1,1x
y = x + 0,05
y=x+1
y = 1,1x + 1
CV = 2%
1
0
0
10
-1
3
2
1
0
-1
20 30 -2 0
-3
1
0
10
20
30
0
10
-1
2
1
0
20 30 -1 0
-2
10
6
3
0
20 30 -3 0
-6
10
20
30
10
20 30
SD = 0.1
1
0
0
-1
10
3
2
1
0
-1
20 30 -2 0
-3
1
0
10
20 30
0
10
20
-1
2
1
0
30 -1 0
-2
10
6
3
0
20 30 -3 0
-6
% bias-plot
y=x
y = 1,1x
y = x + 0,05
y=x+1
y = 1,1x + 1
CV = 2%
10
5
0
-5 0
-10
20
10
0
-10
10 20 30
0
-20
10
5
0
-5
10 20 30
0
-10
120
60
0
-60
10 20 30
0
-120
120
60
0
-60
10 20 30
0
-120
10
5
0
10 20 30 -5 0
-10
120
60
0
10 20 30 -60 0
-120
120
60
0
10 20 30 -60 0
-120
10 20 30
SD = 0.1
10
5
0
-5 0
-10
30
15
0
10 20 30 -15 0
-30
Statistics & graphics for the laboratory
10 20 30
103
Notes
Notes
Statistics & graphics for the laboratory
104
Combined graphical/statistical techniques
Combined graphical/statistical techniques
The Bland&Altman approach for the interpretation of method comparison
studies
References
• Altman DG, Bland JM. Measurement in medicine: the analysis of method
comparison studies. Statistician 1983;32:307-17.
• Bland JM, Altman DG. Statistical methods for assessing agreement between two
methods of clinical measurement. Lancet 1986;307-10.
• Bland JM, Altman DG. Measuring agreement in method comparison studies. Stat
Methods Med Res 1999;8:135-60.
Approach
The goal of the Bland & Altman approach is to compare the outcome of method
comparison studies in terms of systematic (SE) and total error (TE) with quality
specifications for systematic (SEspec) and total error (TEspec).
Calculations
This requires the following calculations (note: the B&A symbols are used here)
-Mean difference (đ) and its 95% confidence limits (CL) (equivalent to SE)
-1.96 SDdiff and its CL (equivalent to TE)
SDdiff = standard deviation of the differences between the methods
Those are to be compared with the specifications in the following way:
• đ ± CL  SEspec and
• 1.96 SDdiff ± CL  TEspec,
Graphics
At the same time, Bland&Altman recommended to present the data in an absolute
bias plot including the lines for đ and 1.96 SDdiff.
Original plot
Adapted from:
Bland JM, Altman DG.
Lancet 1986;i:307-10.
Statistics & graphics for the laboratory
105
Combined graphical/statistical techniques
Combined graphical/statistical techniques
Limitations of the original plot
• Does not recognize different method hierarchies
Same hierarchy: x = (A+B)/2
Different hierarchy: x = A
• In many cases a %-bias plot is more appropriate. In that connection, it is better to
calculate the 1.96 CV values, because the SD is often increasing with the level, so
that no mean SD exists.
• Does not include confidence limits
• Does not include TE/SE-specifications
Statistics & graphics for the laboratory
106
Combined graphical/statistical techniques
Combined graphical/statistical techniques
3
2
1
0
-1
%-bias plot
20
0
5
10
15
20
25
-2
-3
Routine (% difference)
Routine (abs. difference)
Remember: Quality specifications in the
Absolute bias plot
15
10
5
0
-5 0
5
10
15
20
25
-10
-15
-20
Reference (mmol/l)
Sort of specification:
Constant: 1 mmol/l
Proportional: 10%
Reference (mmol/l)
Sort of specification:
Constant: 1 mmol/l
Proportional: 10%
Limitations of the original plot
• Does not recognize different method hierarchies
• In many cases a %-bias plot is more appropriate
• Does not include confidence limits
• Does not include TE/SE-specifications
Because of these limitations, it is recommended to use an "extended"
Bland&Altman plot (see next page)
See also following references
• Stöckl D. Beyond the myths of difference plots [letter]. Ann Clin Biochem
1996;33:575-7.
• Dewitte K, Fierens C, Stöckl D, Thienpont LM. Application of the Bland-Altman
plot for the interpretation of method-comparison studies: a critical investigation of
its practice. Clin Chem 2002;48:799-801.
• Stöckl D, Rodríguez Cabaleiro D, Van Uytfanghe K, Thienpont LM. Interpreting
method comparison studies by use of the bland-altman plot: reflecting the
importance of sample size by incorporating confidence limits and predefined error
limits in the graphic. Clin Chem 2004;50:2216-8.
Statistics & graphics for the laboratory
107
Combined graphical/statistical techniques
Bland & Altman plot – Expanded
Recommendations
• Construct the x-axis according to the hierarchy of the methods
• Choose a bias-plot (absolute, %) that fits your data
• Use the "extended" version of the plot (+specifications and CL's)
(the 1-sided limits are chosen because the comparison is versus a specification).
• Be aware of the meaning of the calculated estimates "mean" bias or "mean"
SD/CV
20
Routine - Reference (%)
15
Mean Diff.
10
5
±1,96
CVdiff.
0
±95%
CL's
-5
3
4
5
6
7
8
9
SE limit
-10
TE limit
-15
-20
Reference (mmol/l)
Bland&Altman
This file contains a template for the Bland&Altman plot with pre-programmed
confidence limits and entries for the SE and TE specifications. It may be adapted
to the needs of the user.
Statistics & graphics for the laboratory
108
Notes
Notes
Statistics & graphics for the laboratory
109
Notes
Notes
Statistics & graphics for the laboratory
110
Correlation and Regression
Correlation and Regression
Correlation
• The statistical model
• Correlation in method comparison
• Non-parametric correlation
Regression
• Ordinary linear rgression (OLR)
• Deming regression
• Passing-Bablok regression (non-parametric)
• Weighted regression
• Regression & method comparison
• Regression & calibration
Statistics & graphics for the laboratory
111
Correlation
Correlation and regression
Correlation
• Correlation concerns association between variables, e. g. serum cholesterol and
indicators of heart disease.
• Correlation is a descriptive measure that does not allow conclusions concerning causal
relationships.
• Correlation is also used together with regression (method comparison studies)
Comparison Correlation <> Regression
Regression model: one variable (the dependent variable, y) is a function of
another variable (the independent variable, x)
• Example: Blood pressure may be considered a function of age
Correlation model: both variables are random effects factors
• Example: Human arm and leg lengths are correlated
Univariate and multivariate correlation
• Univariate (simple): between two variables
• Multivariate: between several variables and an outcome measure, e.g. between serum
cholesterol and triglyceride and an indicator of heart disease
Univariate correlation and relationships of data
• Linear relationship (often implicitly assumed)
• A curvilinear relationship, e.g. a polynomial model
• Cyclical relationship
Linear correlation – Computations
Pearson´s product – moment correlation coefficient r
Computation from the cross product and sums of squared deviations from the mean
values:
r is dimensionless and can
take values from -1 to +1
• The correlation coefficient can be computed regardless of variable distribution types.
• Associated significance tests depend on the type of distributions and are valid for the
bivariate normal distribution.
Coefficient of determination (r2)
Squaring r gives the Coefficient of determination which tells us the proportion of
variance that the two variables have in common. For a height-weight example, r = 0.807
and squaring r gives 0.6512, which means that the height of a person explains 65% of
the person’s weight; the other 35% could probably be explained by other factors,
perhaps nature and nurture.
Statistics & graphics for the laboratory
112
Correlation
Hypothesis testing and r
Testing against zero:
Standard error of r :
SEr = [(1-r2)/(N-2)]0.5
t-test for significance against zero:
t = (r – 0)/SEr with (N-2) degrees of freedom
Non-zero correlation
r is transformed to z = 0.5 ln[(1+r)/(1-r)] (Fisher´s z-transformation, which yields a
symmetric normal-like distribution)
SE of z: [1/(N-3)]0.5
Hypothesis testing and confidence intervals are based on the z-transformation
n
1
2
3
4
5
6
7
8
9
10
15
20
30
50
100
500
Probability
95%
99%
0.997
1.000
0.950
0.990
0.878
0.959
0.811
0.917
0.754
0.874
0.707
0.834
0.666
0.798
0.632
0.765
0.602
0.735
0.576
0.708
0.482
0.606
0.423
0.537
0.349
0.449
0.273
0.354
0.195
0.254
0.088
0.115
Note: n = n - 2
r-critical
Critical values for r
1
0.8
0.6
0.4
0.2
0
-0.2 0
-0.4
-0.6
-0.8
-1
99%
95%
20
40
60
80
100
n-2
Correlation and P
• A weak correlation may be highly statistically significant given a large N, e.g. as
observed in large epidemiological studies.
• The clinical importance of a given degree of correlation depends on the situation.
Statistics & graphics for the laboratory
113
Correlation
Correlation
Meaning of the Pearson correlation coefficient, r
• Measure for the strength of linear correlation
r=0
r=0
r = +1
r = -1
• Becomes smaller (e.g., <1) when dispersion in y occurs
 r is a measure for random analytical error
r = 0,9507
r = +1
Correlation in method comparison studies
30
A
20
10
r = 0,9969
0
0
10
20
Test method
Test method
• Systematic errors have no influence on r
30
B
20
10
30
0
C
20
10
r = 0,9969
0
0
10
20
30
Reference method
10
20
30
Reference method
Test method
Test method
Reference method
30
r = 0,9969
0
A: No SE
B: Constant SE
C: Proportional SE
D: Constant &
proportional SE
30
D
20
10
r = 0,9969
0
0
10
20
30
Reference method
Statistics & graphics for the laboratory
114
Correlation
Correlation in method comparison studies
175
150
150
125
125
r = 0,786
100
100
150
Test
175
Test
Test
The Pearson correlation coefficient, r
• Influence of the data range:
– r: increases with the range
– Inclusion of extreme values: artificial improvement of r
100
r = 0,963
100
125 150
Reference
175
100
r = 0,962
50
125 150
Reference
175
50
100
150
Reference
Conclusion: correlation in method comparison studies
The Pearson correlation coefficient, r is, as measure for method
comparability, difficult to interpret
• r depends on the range of x-values. The greater the range is, the higher are
the values for r
• r is not influenced by systematic errors
• Often, much too small values of r (e.g., r = 0.8) are judged as a good
correlation in method comparison
Some advocate to use r as indicator for proper data distribution before
applying linear regression and recommend for this purpose r-values >0.975
(small range) or >0.99 (wide range).
However, when several methods are compared with the same data-set, r is a
useful index for ranking the methods.
Nonparametric correlation
The parametric correlation coefficient is sensitive towards outliers.
Nonparametric correlation coefficients (Spearman or Kendall) are more robust and
are calculated on the basis of the ordered (ranked) observations.
The computation principle is an assessment of how well the rank order of the
second variable corresponds to the rank order of the first variable.
Statistics & graphics for the laboratory
115
Notes
Notes
Statistics & graphics for the laboratory
116
Regression
Regression
Linear regression procedures
– Linear regression procedures assume a linear relationship between 2
variables (e.g., 2 methods): yi = a0 + b • xi (a0 = intercept; b = slope)
– Slope and intercept of the regression line are determined by minimizing the
sum of the squared distances between the data points and the regression line
(parametric procedures)
Linear regression in method comparison gives information on:
– Constant systematic error (intercept)
– Proportional systematic error (slope)
– Random error (SDy/x)
Non-linear or curvilinear regression procedures
– Minimize the sum of squares of the residuals on the basis of any clear
mathematical relationship (polynomial, logarithmic, etc.) between two
methods
– In the easiest case, the curve can be approximated by several linear
regression calculations performed over different ranges of x (e.g., the low,
middle, and high range)
– Curvilinear regression is most adequate for calibration purposes, either for the
dose/response case, or for calibration of a routine method through method
comparison with a reference method
Statistics & graphics for the laboratory
117
Regression
Regression
Linear regression procedures: Overview
Ordinary least-squares regression (OLR)
• Weighted variant
Deming regression
• Weighted variant
Passing Bablok regression (non-parametric)
Ordinary least-squares regression (OLR)
Assumptions (see also figure):
• x: error-free, which implicates that SDax = 0
• y : measurement uncertainty is present, with assumption that SDay is
constant throughout the measurement range and normally distributed.
Statistics & graphics for the laboratory
118
Regression
Ordinary least-squares regression (OLR)
Computations
xm = Σ xi/N ym = Σ yi/N
u = Σ(xi - xm)2
q = Σ(yi - ym)2
p = Σ (xi - xm)(yi - ym)
Statistical estimates of OLR
• Slope (b), SE(b) & 0.95-confidence limits (CLs) for b
• Intercept (a0), SE(a0) & 0.95-CLs for a0
• Standard error of the y-estimate (SDy/x)
• Regression residuals
• 95% prediction interval for single points
b = p/u
a 0 = ym - b x m
OLR: REMARKS
• OLR minimizes the sum of the squares of the y-residuals (= deviations of yi from
the regression line in y-direction)
• The regression line will pass through a centroid point that is the mean of x and
the mean of y
• Disadvantage of OLR is its sensitivity towards outliers (i.e. extreme values of x or
big residuals in the y-direction)
• OLR gives biased slope estimate in case of narrow range and measurement
error in x
Linear regression estimates: graphical presentation
OLR: Limitation SDay constant
SDay is normally not constant but increases with increasing values of x (when
measurement values are distributed over a decade or more). This is reflected in
the residual plot by a trend towards increasing scatter at high levels.
>Because of the latter, weighted forms of linear regression have been
introduced.
Statistics & graphics for the laboratory
119
Regression
Weighted OLR
CV
SD
SD/CV
Weighted linear regression
In a weighted regression procedure, SDa
is regarded proportional to a function
of the level x (=c h(x)) and weights are:
• wi = 1/h(xi)2
Weights are inversely proportional to SDay
Concentration
Concentration
Centroid of a weighted regression line
25
20
Routine
The centroid point of the
weighted regression line does not pass
through the mean of x and y
but lies more at the side of the origin
CV
SD
SD/CV
Example:
Proportional relationship
between SDa and xi (Fig., upper part):
wi = 1/(xi)2
Ppossibly truncated at a low limit
(Fig., lower part)
Weighted
mean
15
10
5
Mean
0
0
10
15
Reference
20
25
1.2
Expected slope
Slope
Further limitations of OLR
Biased slope estimate in case
of narrow range (x)
and analytical error (a) in x
ß´= ß/(1+λ)
λ = ơa2/ ơX2
This leads to incorrect
testing of significance
5
0.8
Slope from OLR
0.4
0.0
0.4
0.8
1.2
SDa/SDx
Limitation: error (a) in x
In method comparison studies, the assumption of an error-free x is often not valid.
For that reason, regression techniques have been developed that allow error in
both variables (x & y): >e.g., Deming regression
Statistics & graphics for the laboratory
120
Regression
Deming regression
Assumptions:
• x & y: measurement errors may be present in both, with SDax = SDay or SDax
and SDay related (SDax/SDay)
• SDax and SDay: constant throughout the measurement range and normally
distributed
Deming regression estimates a straight line by minimization of the sum of squared
distances at an angle to the regression line dependent on the relation between the
x and y precision, resulting in an estimate without bias (contrarily to OLR that
gives a biased slope estimate in case of SDax0)
Graphical representation of the assumptions:
measurement uncertainty in both x and y
The model assumed in
Deming regression analysis
Computation of the Deming regression line
Minimization of the weighted sum of squared
distances to the line:
S = [(x - X)2 + (y-Y)2 ];  = SDax2/ SDay2
which provides the solution:
b =[(λq - u)+[(u- λq)2+4 λp2]0.5]/ 2λp
a0 = ym - b xm
Computation of standard errors SE(a0) and SE(b) requires a specialized
procedure, e.g. the jackknife method.
Jackknife principle
Computerized resampling principle. Sampling variation is simulated by
consecutively withdrawing one (x, y) of the set with recalculation of estimates
From the dispersion of estimates, SEs are derived.
Statistics & graphics for the laboratory
121
Regression
Weighted Deming regression
Model assumed in weighted
Deming regression analysis
Higher efficiency in case of proportional measurement uncertainty (constant CV),
reflected by more homogenous scatter of standardized residuals and smaller SEs
of slope and intercept estimates.
Method comparison example (Datasets-MethComp)
(n = 50; Statistics: CBstat, K. Linnet)
Deming regression:
Slope: b = 1.053
SE(b) = 0.023; 0.02< P < 0.05
Intercept: a0 = -0.22
SE(a0) = 0.19; n.s. from 0
No significant deviation from linearity
No outliers
Residuals show increased scatter
at high levels: poor model fit.
Weighted Deming regression:
Slope: b = 1.032
SE(b) = 0.012; 0.01< P < 0.02
Intercept: a0 = -0.01
SE(a0) = 0.07; n.s. from 0
Homogeneous scatter of residuals:
Better model fit.
weighted
Weighted versus unweighted Deming
Smaller SEs of slope and intercept:
SE(b) = 0.012 versus 0.023 unweighted
SE(a0) = 0.07 versus 0.19 unweighted
Statistics & graphics for the laboratory
122
Regression
Deming regression
Weighted versus unweighted Deming regression
Given measurement values distributed over a decade or more, the analytical SD
seldom is constant but varies often proportionally, so that the CV% is about
constant.
In this case, it is advantageous to apply a weighted analysis, which provides
lower SEs of estimates.
Weighted Deming regression analysis covers the probably most
commonly occurring data situation in method comparison studies.
Passing-Bablok regression
Assumptions:
• Passing-Bablok regression is a non-parametric method, making no assumptions
about distribution of errors. May be used in case of constant or proportional error
• It assumes that the ratio SDax/SDay is equal to the slope
Passing-Bablok regression uses the slopes between any two data points xi/yi to
calculate the slope of the regression line. The intercept is estimated so that at
least half of the data points are located above or on the regression line and at
least half the data points below or on the regression line.
An advantage of Passing-Bablok regression is its robustness to outliers.
A disadvantage are the broader confidence intervals (due to the nature of nonparametric procedures).
Geometrical interpretation of regression techniques
(minimizing residuals)
20
a
Method Y
Method Y
20
90°
10
d
0
90°+d
10
d
0
0
20
10
Method X
20
0
20
c
Method Y
Method Y
b
10
d
d
0
10
Method X
d
20
2
S1,2
10
1
S2,3
S1,3
3
0
0
10
Method X
20
0
10
Method X
20
Statistics & graphics for the laboratory
123
Regression
Linear regression with CBstat – Summary of output data
• Slope (b), SE(b) & 0.95-confidence limits (CLs) for b
• Intercept (a0), SE(a0) & 0.95-CLs for a0
• Standard error of the y-estimate (SDy/x)
• Correlation coefficient (+ P-value)
• Outlier identification (4s)
• Scatter plot with regression line, 0.95-confidence region and x = y line
• Residuals plot (normalized)
• Additional: runs test for linearity
Residuals for linearity testing
Runs test:
Sequences of residuals with the
same sign are counted and
related to critical limits (= testing
of randomness of residuals)
Relationship correlation & regression
r is related to the regression slope(s):
• r = [byx bxy ]0.5: r is the geometric mean of the two regression slopes
• byx = r SDy/SDx: i.e. r is a rescaled version of the regression slope (identity given
SDy = SDx)
r is related to SDy/x: r2 ~ 1 – SD2y/x/SD2ay
Linear regression – In method comparison
Calculation of a bias (DC)
10
Bias may consist of:
• Constant part (0)
(e.g. fixed matrix effect)
• Proportional part (ß-1)
(e.g. calibration difference)
DC = YC – XC = 0 + (b – 1) • XC
YC
b
DC
5
0
0
0
5
Statistics & graphics for the laboratory
XC
10
124
Regression
Linear regression – In method comparison
Confidence interval of a bias or
systematic difference (SE)
Prediction interval
Statistical significance of estimates:
• Does the slope deviate from 1? t = (b-1)/SE(b)
 Indication for proportional error
• Does the intercept deviate from 0? t = (a0 - 0)/SE(a0)
 Indication for constant error
• SDy/x (from OLR)
 Measure for random error
• Are the data pairs linearly related?
Additional in CBstat
Runs test or visual inspection of residuals plot
 Indication for (non)linear relationship
– Further application
 CI for SE at a critical concentration
25
25
20
20
y (mmol/l)
y (mmol/l)
Statistical test for slope equal to 1
(95% confidence limits to consider)
15
10
5
0
15
10
5
0
0
5
10 15 20 25
x (mmol/l)
Simulation: CV 5%
Regression
y = 0.9443x – 0.1521
95% CLs for slope
0.9106 – 0.9780
Significantly different
from 1
0
5
10 15 20 25
x (mmol/l)
Simulation: CV = 15%
Regression
y = 0.9414x – 0.1233
95% CLs for slope
0.8656 – 1.0172
NOT significantly different
from 1: high RE!
Statistics & graphics for the laboratory
125
Regression
Interpretation of SDy/x in method comparison
• SDy/x is a measure for the random error component in method comparison, i.e. in
both x and y. Thus: SDy/x is related to the expected total imprecision: SDy/x2 =
SDay2 + b2 SDax2
• Given proportional analytical errors (and intercept around 0), approximately:
CVy/x2 = CVay2 + CVax2 or CVy/x = 2 CVay for CVay = CVax
• If only imprecision effects play a role in the method comparison, SDy/x2  SDay2 +
b2 SDax2 (to convert SDy/x into CVy/x, take value of y)
If SDy/x2 >> SDay2 + b2 SDax2
 Proof of sample-related effects (see exercises)
Routine (mmol/L)
Comparison of regression procedures in practice
6,0
All regression procedures …
5,5
– Ordinary least-squares regression (OLR)
5,0
– Deming regression (DR)
4,5
– Passing-Bablok regression (PBR)
4,0
… give nearly the same results
Note: r = 0.993
3,5
3,0
3,0
3,5
4,0
4,5
5,0
5,5
6,0
Reference (mmol/L)
Routine (mmol/L)
6,0
The regression procedures …
5,5
– OLR, DR, PBR
5,0
… give different results
Note: r = 0.871
4,5
4,0
3,5
3,0
3,0
3,5
4,0
4,5
5,0
5,5
6,0
Reference (mmol/L)
Conclusion
Choose the "statistically best" regression method?
Answer: No, look for analytical reasons of the poor comparability!
Notice also that the 95% CLs of the slope are: 1.08 – 1.44
Statistics & graphics for the laboratory
126
Regression
Regression: Examples from the practice
When different regression procedures give different results …
– OLR (red): y = 0.750 x – 0.006 (r = 0.996)
– Passing Bablok (blue): y = 0.686 x + 0.022
… look whether the data are linear!
 The residuals plot demonstrates non-linearity
0,2
3
Residuals
Routine
4
2
1
0,1
0
-0,1
0
0
1
2
3
Reference
4
0
1
2
3
4
-0,2
Reference
CI for SE at a critical concentration
Therapeutic interval for drug assay: 300 – 2000 nmol/L
Delta = Ŷ-X = a0 + (b-1)X = 20.3 +(1.014 – 1)X
• X = 300 : Delta = 24.5 ; SE(Delta) = 9.5
Significance test:
texp = (Delta – 0)/SE = 2.6 ; tcrit[0.05;n –2) = 1.998 significant
• X = 2000: Delta = 48.9 ; SE(Delta) = 34.2
t = 1.4 not significant
Conclusion:
• At the lower decision point, a statistical significant difference exists, but it is
judged to be clinically unimportant
• At the upper decision point, no difference of statistical significance
• The assays can be interchanged without clinical consequences
Statistics & graphics for the laboratory
127
Regression
Regression & correlation in method comparison
Summary
• Perform correlation analysis before [r-values >0.975 (small range) or >0.99 (wide
range)]
• In case of a method comparison of methods of the same hierarchy, regression
techniques, that take the error in x and y into account, should be used. We
recommend Deming regression.
• Classical OLR is only applicable in case of method comparison with a reference
method or in the calibration case (weighed-in concentrations).
• Regression data are the more unreliable the greater the random error and the
smaller the data range are.
• Often forgotten data from regression analysis are the 95% confidence limits of
slope and intercept.
• Linear regression always results in a line, even when the data are not linear.
Therefore, linear regression data always should be accompanied by a graphical
presentation of results (scatter plot with x = y line or residuals plot) and the
indication of the number of observations. The graph should be visually inspected
for adequate range, distribution of data, and linearity
Regression analysis provides information about:
• constant systematic difference (intercept)
• proportional systematic difference (slope)
• random error (SDy/x from OLR)
• sample-related effects (SDy/x >>>SDay2 + b2SDax2)
Regression software
CBstat (K. Linnet): A Windows program
• (weighted) OLR
• (weighted) Deming regression
• Passing Bablok regression
(www.cbstat.com)
MedCalc
(www.medcalc.com)
EP-Evaluator (D. G. Rhoads Ass., USA)
(www.dgrhoads.com)
Analyse-It (Excel-plug-in)
(www.analyse-it.com)
Statistics & graphics for the laboratory
128
Regression
Regression and calibration
Calculations
• Concentration of unknown and its random error
• Limit of Detection (LoD)
Graphical model
• S = Signal
• Yb = Signal of blank
via regression = intercept a
• Sb = Sy/x
• LoD = a + 3 Sy/x
Concentration (x0) and its random error (Sx0)
NOTE: do not confuse with x at zero (0) concentration!
Calculate x0 from signal (y0) via regression equation y = bx + a  x0 = (y0 – a)/b
Sx0: approximation:
m = number of measurements of unknown
n = number of calibration points
The confidence interval of x0 is:
CI = ± t(n-2, ) • Sx0
Calculation of LoD
• Yb = "Signal of blank" via regression = intercept a
• Sb = "Standard deviation of blank" = Sy/x
"Signal" LoD = a + 3 Sy/x
Calculate CLoD via regression equation.
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129
Data transformation
CAVE log transformation
Introduction of non-linearity by data transformation in method comparison
and commutability studies.
Stöckl D, Thienpont LM. Clin Chem Lab Med 2008;46:1784-5.
6
y = 1.0994x - 0.3849
300
250
200
150
100
50
Routine method (lnAU)..
Routine method (AU)..
350
0
y = 1.0113x + 0.0339
5
4
3
2
1
0
50 100 150 200 250 300 350
1
Reference method (AU)
3
4
5
6
6
y = 0.9995x + 14.65
300
250
200
150
100
50
0
Routine method (lnAU)..
350
Routine method (AU)..
2
Reference method (lnAU)
5
4
3
y = -0.0108x 3 + 0.21x 2
- 0.376x + 3.075
2
1
0
50 100 150 200 250 300 350
Reference method (AU)
1
2
3
4
5
6
Reference method (lnAU)
Statistics & graphics for the laboratory
130
Exercises
CorrRegr-EXCEL
This EXCEL-file describes the advantages and disadvantages of the different
EXCEL options for performing correlation and regression analysis.
These options are:
1. With the fx icon
2. With Tools>Data Analysis
3. With a figure
It also contains a worksheet with the additional regression features
-95% confidence interval of the slope
-95% prediction interval
Correlation&Regression
This tutorial contains interactive exercises for self-education in:
-Correlation, and
-Regression
Worksheet correlation shows
the influence of dispersion, slope, intercept, and range on r.
Worksheet regression1 shows
the influence of dispersion, slope, and intercept on the standard errors of slope,
intercept, and Sy/x.
Worksheet regression2 shows
the influence of the range on the standard errors of slope, intercept, and Sy/x (this
example is constructed with a constant SD over the range).
Note
r and r-square are given for information, only.
Datasets (Method comparison: Weighted Deming, PractRegr1, PractRegr2)
Statistics & graphics for the laboratory
131
Annex
Introduction
EXCEL® requirements
The "Data Analysis" Add-in
• In the "Worksheet Menu Bar", under
• Tools
 "Data Analysis" should appear
If it is not present,
• Click "Tools" and Add-Ins
 Activate Analysis ToolPak & Analysis ToolPak - VBA
… if not present in "Add-Ins"
Install them from the EXCEL or Office package
• "Add-ins"
Statistics & graphics for the laboratory
132
Annex
Tips to create EXCEL®-figures
Create a figure: "Chart-wizard"
Data&DataPresentation
("Figure")
Modify a figure with:
• "Chart-wizard"
• "Chart-menu"
• Double-click (left) on an element
Move or size
• Left mouse click depressed:
Notice the full squares!
• Shift & left click:
Notice the empty squares!
: move with 
: size with right click > Format object,
: or, direct with the "Format" menu
Statistics & graphics for the laboratory
133
Annex
Tips to create EXCEL®-figures (ctd.)
Make your own "templates": Activate figure
• Chart>Chart type>Custom types>User defined>Add
IMPORTANT: Scale names and sizes are kept too!
Layout tips for EXCEL-figures
• Not more than 8 columns (standard width 8,43)
• Not more than 22 rows (standard height 12,75)
• Font: minimum 16 (14), bold preferred
• Use thick lines
• Symbol size 6 or 7
• Click off autoscaling
• Click "Don't move or size with cells"
Statistics & graphics for the laboratory
134
Annex
Copy EXCEL®-figures into PowerPoint
Windows 98 with Office 2000 experience
• Copy & paste direct if animation is intended
• Copy & paste direct, then >Copy>Delete>Edit>Paste special: Picture
(Enhanced metafile: "EMF") = Easy magnification without loss of quality
• Copy & paste direct, then >Copy>Delete>Edit>Paste special: GIF, preferably
keep 100% size: often looks more attractive
• Note: Often preferred to copy the cell-range where the figure is placed ("What
you see is what you get": colours, layout)
• Adding text: often preferable in PowerPoint!
Print EXCEL®-figures from PowerPoint
Windows 98 with Office 2000 experience
Note: Printing of ppt-Figures may pose problems.
Check the print early if you want to make handouts!
EMF figures ("Cells direct", then EMF) print well
• In the absence of Gaussian-type lines
• Incorporate text preferably in the .ppt slide and copy both as EMF
[Bigger] GIF figures ("Cells direct", then GIF)
• Advantage: better print of Gaussian-type lines
• Problems: Incorporated text and scales have poor resolution, can be
improved by
• Paste direct, then GIF, then add text (& axes, eventually) in ppt, then copy &
paste special both as EMF
• Note: Overpaste of figure scales with .ppt text fields works only with GIF, but
not with EMF.
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Annex
Copy EXCEL®-figures into PowerPoint – Examples
Example: 5 columns, 16 rows, font 16 & 14 bold
Cells direct
12
8
4
117,2
106,3
0
84,5
Bin
Frequency-polygon
Frequency
117,2
106,3
95,4
12
8
4
0
84,5
Frequency
Frequency-polygon
95,4
Direct
Bin
Cells direct & EMF
Cells direct & GIF
12
8
4
117,2
106,3
95,4
0
84,5
Frequency
Frequency-polygon
Bin
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Annex
Statistical resources
Glossary of statistical terms
http://linkage.rockefeller.edu/wli/glossary/stat.html
http://www.statsoft.com/textbook/glosfra.html
http://www.stats.gla.ac.uk/steps/glossary/index.html (most practical)
http://davidmlane.com/hyperstat/glossary.html
http://stat-www.berkeley.edu/~stark/SticiGui/Text/gloss.htm
Interesting educational resources
http://www.ruf.rice.edu/%7Elane/rvls.html
http://www.math.uah.edu/stat/index.xml
http://cast.massey.ac.nz/
http://www.anu.edu.au/nceph/surfstat/surfstat-home/surfstat.html (with progress
tests!)
http://www.stat.vt.edu/~sundar/java/applets/
http://www.kuleuven.ac.be/ucs/java/version2.0/Content.htm
http://www.seeingstatistics.com/seeing1999/resources/opening.html (many
possibilities, own data!)
http://www.margaret.net/statistics/p02.htm
http://bmj.bmjjournals.com/collections/statsbk/index.shtml
http://science.widener.edu/svb/stats/stats.html
http://www.vam.org.uk/vamstatdemo/demolist.asp
http://www.stat.sc.edu/~west/applets/tdemo1.html (t-distribution)
http://www.visualstatistics.net/ (t-distribution for EXCEL!)
http://www.stat.uiowa.edu/~rlenth/Power/ (power)
Statistical software
General
http://www.spss.com/sigmastat/
http://www.sas.com/technologies/analytics/statistics/index.html
http://www.stata.com/
http://www.minitab.com/
http://www.graphpad.com/ (also educational!)
"Laboratory statistics"
http://www.medcalc.be
http://www.cbstat.com
http://www.analyse-it.com
http://www.dgrhoads.com/
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Annex
Statistical resources
Books
•Biometry: The Principles and Practice of Statistics in Biological Research. Robert
R. Sokal, F. James Rohlf
•Statistics and Chemometrics for Analytical Chemistry. James N. Miller, Jane C.
Miller
•Clinical Investigation and Statistics in Laboratory Medicine. Richard Jones, Brian
Payne
•Statistics at Square One. Ninth Edition. T D V Swinscow
(see also: http://bmj.bmjjournals.com/collections/statsbk/index.shtml)
•http://www.statsoft.com/textbook/stathome.html
•http://davidmlane.com/hyperstat/
•http://faculty.vassar.edu/lowry/webtext.html
•http://www.tufts.edu/~gdallal/LHSP.HTM
Books (PDF) from the net
•Analyzing Data with GraphPad Prism. A companion to GraphPad Prism version 3
(graphpad.com).
•The InStat Guide to Choosing and Interpreting Statistical Tests. A manual for
GraphPad InStat Version 3 (graphpad.com).
•NIST/SEMATECH e-Handbook of Statistical Methods
(http://www.itl.nist.gov/div898/handbook/)
Statistics & graphics for the laboratory
138
Statistical tables
Outlier testing: Dixon Q-test (one-tailed)
x1  x2  x3 ...  xn
n
P = 0.01
P = 0.05
3
4
5
6
7
8
9
10
11
12
13
14
16
18
20
25
0.988
0.889
0.780
0.698
0.637
0.683
0.635
0.597
0.679
0.642
0.615
0.641
0.595
0.561
0.535
0.489
0.941
0.765
0.642
0.560
0.507
0.554
0.512
0.477
0.576
0.546
0.521
0.546
0.507
0.475
0.450
0.406
Test quotient
(x n - x n-1)/(x n-x 1)
(x n - x n-1)/(x n-x 2)
(x n - x n-2)/(x n-x 2)
(x n - x n-2)/(x n-x 3)
From: Rohlf FJ, Sokal RR. Statistical tables. 3rd ed.
New York: WH Freedman & Co.: 1995.
Factors for control limits
of range rules
n
R0.01
R0.05
2
3
4
6
8
10
12
16
20
3.64
4.12
4.40
4.76
4.99
5.16
5.29
5.50
5.65
2.77
3.31
3.63
4.03
4.29
4.47
4.62
4.85
5.01
From: Pearson ES. The probability integral of the range of samples
of n observations from a normal population. I. Forew ord and tables.
Biometrika 1942;32:301.
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139
Cochran C – Critical values
Statistics & graphics for the laboratory
140
Annex
Presenter's publications & courses related to the topic
Publications
•Stöckl D. Beyond the myths of difference plots [letter]. Ann Clin Biochem
1996;33:575-7.
•Stöckl D. Difference versus mean plots [reply]. Ann Clin Biochem 1997;34:571.
•Hyltoft Petersen P, Stöckl D, Blaabjerg O, Pedersen B, Birkemose E, Thienpont
L, Flensted Lassen J, Kjeldsen J. Graphical interpretation of analytical data from
comparison of a field method with a reference method by use of difference plots
[opinion]. Clin Chem 1997;43:2039-46.
•Stöckl D, Dewitte K, Thienpont LM. Validity of linear regression in method
comparison studies: is it limited by the statistical model or the quality of the
analytical input data? Clin Chem 1998;44:2340-6.
•Stöckl D, Dewitte K, Fierens C, Thienpont LM. Evaluating clinical accuracy of
systems for self-monitoring of blood glucose by error grid analysis. Comment on
constructing the “upper A-line”. Diabetes Care 2000;11:1711-2.
•Dewitte K, Fierens C, Stöckl D, Thienpont LM. Application of the Bland-Altman
plot for interpretation of method-comparison studies: a critical investigation of its
practice. Clin Chem 2002;48:799-801;discussion 801-2.
•Cabaleiro DR, Stöckl D, Thienpont LM. Error messages when calculating chisquare statistics with microsoft EXCEL. Clin Chem Lab Med 2004;42:243.
•Stöckl D, Rodríguez Cabaleiro D, Van Uytfanghe K, Thienpont LM. Interpreting
method comparison studies by use of the bland-altman plot: reflecting the
importance of sample size by incorporating confidence limits and predefined error
limits in the graphic. Clin Chem 2004;50:2216-8.
•Stöckl D, Rodríguez Cabaleiro D, Thienpont LM. Peculiarities and problems with
the EXCEL F-test. Clin Chem Lab Med 2004:42:273.
Courses
•Analytical quality in the medical laboratory: Concepts for method selection,
evaluation, and control. In cooperation with Belgian Association of Laboratory
Technologists and Hogeschool Gent (Gent, Belgium, 1998).
•Practice-oriented strategies for the development and evaluation of analytical
methods. FOCUS: Graphical and statistical techniques for the interpretation of
method comparison studies (academical year 2000/1). In cooperation with Prof.
LM Thienpont (University of Ghent).
•Graphical and statistical techniques for the interpretation of method comparison
studies. 14th IFCC European Congress of Clinical Chemistry and Laboratory
Medicine - Euromedlab 2001 (Prague, Czech Republic).
•Graphical techniques for the intepretation of method comparison studies.
Education days for Clinical Biochemists: Method validation. Odense, Denmark: 1720 December 2001.
•Educational course on biostatistics. 18th International Congress of Clinical
Chemistry and Laboratory Medicine IFCC Worldlab 2002 (Kyoto, Japan).
•Statistical and graphical techniques for the intepretation of method comparison
studies. 15th IFCC European Congress of Clinical Chemistry and Laboratory
Medicine - Euromedlab 2003 (Barcelona, Spain).
•Statistical and graphical tools for the medical laboratory – A problem oriented
journey from test utility to internal quality control. 2003 (Bratislava, Slovakia).
•Statistical and graphical tools for the laboratory – from test utility to IQC. Full-day
Workshop, AACC 2004.
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