Statistics forToxicology-Testing-WiSe2425.pdf

Statistics for Toxicology

1. Statistics in Toxicology II
Testing
Kirsten Schorning
Department of Statistics,
TU Dortmund University
Winter semester 2024/25

2. 1 Introduction
Introduction
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 1

3. 1 Introduction 1.1 Overview
1.1 Lecturer
JProf. Dr. Kirsten Schorning (lecture)
Mathematics Building, Room 725
Email: [email protected]
Study of Mathematics with minor in Computer linguistics in
Bochum;
JProfessor for Mathematical Statistics, Department of Statistics, TU
Dortmund University; with research areas: Statistical methods for
toxicology, optimal design
Leonie Schuermeyer (exercises)
Mathematics Building, Room 916
Email: [email protected]
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 2

4. 1 Introduction 1.1 Overview
1.1 Organization
Course: Statistics in toxicology (testing)
Lecture dates:
Monday, 10:15 - 11:45 in C/HS 1
Exercise dates:
Wednesday, 8.30-10.00 in M/E 27
Moodle: Statistics in Toxicology II, WiSe 24/25
Website:
https://moodle.tu-dortmund.de/course/view.php?id=48552
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 3

5. 1 Introduction 1.1 Overview
1.1 Organization
Admission to exam
For admission to the written exam, 50% of the points of the graded
exercise sheet 4 must be achieved. The submission of sheet 4 is
mandatory, and it will be corrected in detail.
The exercises will consist of theoretical and software exercises.
Further information can be found on the Moodle page.
Admission to exam via BOSS system, deadline TBA via Moodle.
Written exam date
One hour exam
Room and data: TBA in Moodle
Written retry exam date
One hour exam
Room and data: TBA in Moodle
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 4

6. 1 Introduction 1.1 Overview
1.1 Organization
Modules
BS 14, BS 15 "Spezialgebiete", "Quantitative Methoden"
MS 6, MS 7 "Spezialgebiete"
MD E2-12 "Applications - Elective Courses"
Prerequisites
For BS: Statistik I - III
For MD: MD Req4 Probability, MD Req5 Inference, MD Req6
Linear models
No deep knowledge in genetics and biology, but at least a strong
interest
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 5

7. 1 Introduction 1.1 Overview
1.1 Content
The main topic of the lecture is the analysis of data from toxicology,
especially dose-response measurements. Specic topics are:
Proof of hazard using simultaneous comparisons with a negative
control
Multiple testing
Tests for normally distributed endpoints and for proportions
Trend tests
Analysis of long-term eects in cancer studies
Survival analysis, tests for survival endpoint
Analysis of eects in mutagenicity assays
Mixture distributions
EM algorithm
Multiple testing for genomic data
Family-wise error rate
False discovery rate
Clinical trials phase I studies
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 6

8. 1 Introduction 1.1 Overview
1.1 Literature
Main book for this course:
Hothorn, L.A.: Statistics in Toxicology Using R, Chapman and
Hall/CRC, 2016 (SiTUR)
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 7

9. 1 Introduction 1.1 Overview
1.1 Motivation
There are three kinds of lies - lies, damned lies, and
statistics (Leonard Henry Courtney, 1832-1918)
To guess is cheap, to guess wrong is expensive (Chinese
proverb)
Statistics is (also) fun
Statistics is (also) intuition
Statistics is (also) surprise
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 8

10. 1 Introduction 1.1 Overview
1.1 Toxicology
Denitions
a science that deals with poisons and their eect and with the
problems involved (such as clinical, industrial, or legal problems)
(Merriam-Webster, 22.11.2023)
Toxicology is a scientic discipline, overlapping with biology,
chemistry, pharmacology, and medicine, that involves the study of
the adverse eects of chemical substances on living organisms and
the practice of diagnosing and treating exposures to toxins and
toxicants. The relationship between dose and its eects on the
exposed organism is of high signicance in toxicology. Factors that
inuence chemical toxicity include the dosage, duration of exposure
(whether it is acute or chronic), route of exposure, species, age, sex,
and environment. Toxicologists are experts on poisons and
poisoning. There is a movement for evidence-based toxicology as
part of the larger movement towards evidence-based practices.
(Wikipedia, 05.10.2024)
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 9

11. 1 Introduction 1.1 Overview
1.1 Toxicology
Denitions
Toxicology is a eld of science that helps us understand the harmful
eects that chemicals, substances, or situations can have on people,
animals, and the environment. Some refer to toxicology as the
"Science of Safety" because, as a eld, it has evolved from a science
focused on studying poisons and adverse eects of chemical
exposures to a science devoted to studying safety.
(NIH, https://www.niehs.nih.gov/health/topics/science/toxicology,
05.10.2024)
The contribution (or general role) of statistics respectively data
analysis is typically neglected, even in extended denitions.
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 10

12. 1 Introduction 1.1 Overview
1.1 Statistics for toxicology
Statistics for toxicology: For statisticians in its basic form, maybe
standard, but for toxicologists, often a great challenge!
communication is key!
Central question still open:signicant test of a criterion for hazard
or even more importantis non-signicance a criterion for safety?
Dierence between statistical signicance and biological relevance
Hypothesis testing or modeling?
Arguments pro testing (SiTUR book)
Guidelines often require hypothesis tests
Modelling not possible for only 2-3 dierent doses
Modelling is only correct when certain assumptions are fullled
Hypotheses tests often preferred in scientic publications
Arguments pro modeling
If enough doses and replicates are available and functional form can
be justied, then estimates are more precise (also at non-tested
doses)
Flexible frameworks exist (such as MCPMod)
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 11

13. 1 Introduction 1.1 Overview
1.1 Statistics for toxicology
Research project at TU Dortmund (rst phase 04/2021-09/2025):
DFG Research Training Group (RTG) 2624 "Biostatistical methods for
high-dimensional data in toxicology"
https://grk2624.statistik.tu-dortmund.de/en/
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 12

14. 1 Introduction 1.2 Principles
1.2 Evaluation of short-term repeated toxicity
studies
General principles
Compound is often administered between 4 weeks and 3 months,
but other exposition times are possible
Type of endpoints (with dierent scales of measurements):
continuous (e.g., hemoglobin, a protein in blood that carries oxygen)
rates (proportions of histopathological ndings)
ordered categorical data (e.g., graded histopathological ndings)
Typical design: Negative controlCand dosesD1; : : : ;Dk, often for
k= 3 doses (concentrations)
Small sample size per dose is a common feature in toxicology
For the simple comparison dose-control (for dierent doses) the
following tests are "standard" and will be discussed in this course:
Unadjusted two-sample test
Dunnett-type tests without order restriction
Williams-type tests with order restriction
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 13

15. 1 Introduction 1.2 Principles
1.2 Data visualization: Barplots and boxplots
Common presentations and visualizations
Tables with measures for groups (doses): Means, standard
deviations, standard error, sample sizes
Barplots (barcharts) (with standard error, sometimes standard
deviation)
Disadvantages of barplots
Assumption of normal distribution not necessarily fullled
Individual data points not shown, also no extreme values
Alternatives:
Plots including all individual data points
Boxplots (median, quartile, extreme values): not yet used much in
toxicology, but provide additional information regarding variability,
symmetry, and potential outliers
Letters of signicance (compact letter display, generated e.g. with
insert-and-absorb algorithm, see second next slide)
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 14

16. 1 Introduction 1.2 Principles
1.2 Data visualization: Barplots and boxplots
Measures of variability
Given are data pointsx1; : : : ;xnwith mean x
Standard deviation:sd=
r
1
n1
nP
i=1
(xix)
2
, measures variability of
individual data points
Standard error:
s
d
p
n
, measures variability of x
MAD: median absolute deviation from median, more robust
alternative forsd
Boxplots
Many dierent ways to construct boxplots
Box indicates quartiles, with median (second quartile) as line within
the box
Whiskers extended to smallest value within 1.5 IQR of the lower
quartile and to largest value within 1.5 IQR of the upper quartile
More extreme points typically plotted as single points
Attention: For very small sample size (in toxicology, e.g., often 3
values) overinterpretation of ve measures, for very large sample size
overinterpretation of extreme values (not necessarily outliers!)
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17. 1 Introduction 1.2 Principles
1.2 Data visualization: Letters of signicance
Idea of letters of signicance
Show statistically signicant dierences between variables
For all variables with the same letter, the dierence between the
means isnotstatistically signicant
More precisely: If two treatments (doses) do not share any letter,
then (and only then) they are statistically dierent
Algorithm: Insert-and-absorb algorithm (Piepho, 2004)
Given is a set ofk(k1)=2 signicance statements (often in the
form ofp-values) corresponding to all pairwise comparisons, wherek
is the number of treatments, adjusted for multiplicity where necessary
For every signicant pair, perform a series of steps as described on
the next slide
Advantages and disadvantages
Often understandable presentation of groups of similar treatments
Worst-case runtime is exponential!for very large numbers of
treatments (which is rare) not always suitable
Exactp-value is unknown, unless included (e.g. in a table)
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 16

18. 1 Introduction 1.2 Principles
1.2 Data visualization: Compact letter display
Insert-and-absorb algorithm (Piepho, 2004,
https://doi.org/10.1198/1061860043515), see also exercises1. Generate a column connecting all treatments.
2. For each signicant comparison do:
For each column currently in the display do:
If the column connects the two signicantly different treatments then do:
- Duplicate the column.
- In the rst of the two columns delete the letter corresponding to the one
treatment. If possible, absorb the column into another column.
- In the secondof the two columnsdeletethe lettercorrespondingto the other
treatment. If possible, absorb the column into another column.
End.
End.
End.
Fig. 1:
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19. 1 Introduction 1.2 Principles
1.2 Presentation
Presentation of data using a table
Fig. 2:
corresponding standard errors, DMSO (dimethyl sulfoxide) is a solvent
that is often used as control, tBOOH (tert-Butyl hydroperoxide) is an
organic compound whose eects are of interest
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20. 1 Introduction 1.2 Principles
1.2 Presentation
Presentation of data using a barplot
Fig. 3:
corresponding standard errors (only upwards), and letters of signicance
(equal letters indicate non-signicant dierence between groups)
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21. 1 Introduction 1.2 Principles
1.2 Presentation
Presentation of data with individual points Fig. 4:
means (logarithmic scale!)
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22. 1 Introduction 1.2 Principles
1.2 Presentations
Presentation of data with boxplots
Fig. 5:
dierences indicated by lines with stars
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23. 1 Introduction 1.2 Principles
1.2 Presentation
Comparison of two graphical presentations (next slide)
Data represent litter-specic pup weights (of newborn rats)
Litter: a number of young animals born to an animal at one time
Barplot visualizes only means and standard deviations
Boxplot shows individual data (dots, jittered), parametric and
non-parametric summary measures (mean, standard deviation (both
next to boxplot), median, interquartile range), and per-litter
structure (including group-specic number of litters and number of
animals)
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24. 1 Introduction 1.2 Principles
1.2 Presentation
Fig. 6:
(left) and barplots plus standard deviation (right), for the same dataset
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25. 1 Introduction 1.2 Principles
1.2 Presentation of test results
Presentation styles
Rejection/non-rejection ofH0, letters indicate non-distinguishable
treatment groups
Rejection ofH0for three-levels (0.05, 0.01, 0.001), visualized by
stars (,,)
p-value (p)
condence interval
Stars are still common in toxicology, also p-values are frequently used
p-value has philosophical disadvantage
A small value does not proveH1, but only demonstrates thatH0
does not t well to the data
In other words: ap-value is the probability of observing a sample
statistic that is at least as extreme as the sample statistic when one
assumes that the null hypothesis is true
Ap-value refers toP(datajH0), not toP(H0jdata) !
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26. 1 Introduction 1.2 Principles
1.2 Presentation of test results
Condence intervals have the advantage that they contain
information about
rejection/non-rejection ofH0, by inclusion/non-inclusion of the value
ofH0(e.g., 0 for dierence and 1 for ratio)
interpretation of biological relevance, by the distance of the
condence limits to this value ofH0, in terms of the measured unit
(dierence) or percentage change (ratio)
the directional decision (whether increasing or decreasing eects
occur)
Condence limits are also recommended in guidelines for clinical
trials (ICH E9), for toxicological studies no such recommendations
exist yet (SiTUR)
For condence intervals an adequate eect size is required
(dierence, ratio, odds ratio)
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 25

27. 1 Introduction 1.2 Principles
1.2 Eect sizes
Letibe the proportion of events in groupi, typically estimated by
^i=
Yi
ni
, whereniis the size of groupiandYithe number of events in
groupi.
Therisk dierencebetween groupiand control group 0 is dened as
i0
Therelative riskbetween groupiand control group 0 is dened as
i
0
Theodds ratiobetween groupiand control group 0 is dened as
i=(1i)
0=(10)
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28. 1 Introduction 1.2 Principles
1.2 Presentation of test results
Example data set
Data set with outcome (serum) triglyceride (Blutfette)
High value is risk factor for cardiovascular diseases
Sodium dichromate dihydrate given to F344 rats (inbreeding strain
of rats, dates back to the year 1920), often used in cancer research
and toxicology research
One control group, ve (experimental) doses (62:5, 125, 250, 500,
1000) in a 13-week-study
Figure on next slide:
Left: Boxplots and letters
Right: Boxplots, individual values (with jitter), Dunnett-type
condence intervals (see chapter 2) and stars (for dierence between
corresponding dose and control)
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29. 1 Introduction 1.2 Principles
1.2 Presentation of test results
Fig. 7:
boxplots with letter-type representation of signicances, right: boxplots with
Dunnett-type stars (for simultaneous comparison against control)
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30. 1 Introduction 1.2 Principles
1.2 Proof of hazard or proof of safety?
Main question: Is a compound (new drug, chemical) in general
harmless, harmless up to a specied dose, or harmful?
Typical problem when testing withp-values:
For large sample sizes, potentially even small (biologically
non-relevant) eects are statistically signicant
For small sample sizes, potentially even large (biologically) relevant
eects are not statistically signicant
Important insight:Absence of evidence is not evidence of absence!
This means: If the result is non-signicance, one cannot draw the
conclusion that there is no eect!
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 29

31. 1 Introduction 1.2 Principles
1.2 Sample size
Minimal required sample size for a toxicological assay is specied in
many guidelines (e.g., 3 or 5 per dose)
A minimal required sample size helps to increase the power of a test
(e.g., for the proof of hazard), but depends on the variance of the
outcome values, the test level, and assumptions like normality, for
the proof of safety also on a tolerance threshold#
Note: Properties of tests often do not hold for very small sample
size (e.g., asymptotic distribution is normal distribution)
Problematic interpretation ofp-value: Direct function of sample
size, thus be careful with the interpretation in unplanned studies
!Simulation study: Analysis of data sets with the same values for mean
and variance, but dierent sample size 5, 10, 15 (in R: the function
ermvnormsimulates normal distributed data with exact values for
parameters)
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32. 1 Introduction 1.2 Principles
1.2 Two-step test decisions
In the case of potential variance heterogeneity (dierent variances
for dierent doses) often rst a pretest on variance heterogeneity is
performed and then (depending on the result) a corresponding test
for the desired eect(s) is used
Problems with this procedure
Levelmight not be kept
Nonparametric tests might be unsuitable in case of variance
heterogeneity
Conditional tests are often problematic
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 31

33. 1 Introduction 1.2 Principles
1.2 Importance of control groups
Negative control group is typically required in toxicology!
Historical controls might be useful, but often unclear how similar
they are to simultaneous control experiments (e.g., due to batch
eects)
Positive controls are not common, but can also be useful
to demonstrate that an assay is sensitive (changes in outcome
variable can actually be achieved)
to demonstrate the relevance of a change versus negative control
(using a non-inferiority test w.r.t. positive control)
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 32

34. 1 Introduction 1.2 Principles
1.2 Importance of control groups
Example: Micronucleus data
6 treatments: negative control (Vehicle, C{), Hydro30, Hydro50,
Hydro75, Hydro100, positive control (Cycle25)
Dunnett-type comparison: all 4 doses of Hydro once tested versus
C{ and once versus C+
Description of the gure on next slide:
upperp-value for comparison against C{ (vehicle)
lowerp-value for comparison against C+ (non-inferiority against C+)
result, for example for Hydro 100: signicant increase compared to
C{ (p= 0:001), but no signicant inferiority against C+ (p= 0:639)
p-value for Cycle25 corresponds to t-test for C{ versus C+ (here
p<0:01)
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 33

35. 1 Introduction 1.2 Principles
1.2 Importance of control groups
upperp-value
comparison
against vehicle C{
lowerp-value
comparison
against C+
p-value for
Cycle25
corresponds to
t-test for C{
versus C+
Fig. 8:
positive control
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 34

36. 1 Introduction 1.2 Principles
1.2 Statistical signicance and biological relevance
Statistical signicance versus biological relevance:
Dierence between the two terms is important
Example (simulated data, ve doses D1-D5, negative control NC)
Biological relevance, when estimated eecteis above ana priori
dened relevance thresholdc(herec= 2:1):
D1-NC: Statistically not signicant, biological relevance unclear
D2-NC: Statistically signicant (0<e<c), biologically not relevant
D3-NC: Statistically signicant (0<e), not signicantly below
thresholdc, biological relevance unclear
D4-NC: Statistically signicant, probably biologically relevant
D5-NC: Statistically signicant (c<e), biologically relevant
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 35

37. 1 Introduction 1.2 Principles
1.2 Statistical signicance and biological relevance
Fig. 9:
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 36

38. 1 Introduction 1.2 Principles
1.2 Proof of hazard using two-sample comparisons
Proof of hazard is the typical (most frequent) test situation:
Null hypothesisH0corresponds to equality of eects and is rejected
or not rejected
For normal distributed continuous endpoint: often t-test
(respectively Welch test) is used, sometimes also Wilcoxon test
Example:
F344 rats, treated 13 weeks with sodium dichromate dihydrate,
6 dierent doses
18 endpoints, including BUN (blood urea nitrigen), Creat (serum
creatinin), ALB (albumin), SerumGlucose (serum glucose),
CreatKinase (creatine kinase), ALT (anilin aminotransferase)
Condence intervals of Welch test: scale-specic
Alternatives
Condence intervals for ratio-to-control comparisons
Condence intervals for log-normal data
Nonparametric condence intervals for ratio-to-control comparisons
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 37

39. 1 Introduction 1.2 Principles
1.2 Proof of hazard using two-sample comparisons
Datasetclin: 13-week study with sodium dichromate dihydrate
administered to rats, 6 doses, 10 animals per dose, and 18 biological
endpoints for each animal
In R:mclin: Data set joined with commandmelt
R code:str
(mclin)
## data.frame: 1080 obs. of 3 variables:
## $ dose : Factor w/ 6 levels "0","62.5","125",..: 1 1 1 1 1 1 1 11 1 ...
## $ variable: Factor w/ 18 levels "BUN","Creat",..: 1 1 1 1 1 11 1 1 1 ...
## $ value : num 17.3 15 15.7 16.5 16.7 15.2 16.2 16.6 17.9 16.7 ...
levels
(mclin
$
dose);
levels
(mclin
$
variable)
## [1] "0" "62.5" "125" "250" "500" "1000"
## [1] "BUN" "Creat" "TP " "ALB "
## [5] "SerumGlucose" "CreatKinase" "ALT" "SDH"
## [9] "ALP" "Nuc" "TBA" "Cholesterol"
## [13] "Triglyceride" "Chloride" "Sodium" "Calc"
## [17] "PHOS" "Potassium"
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40. 1 Introduction 1.2 Principles
1.2 Proof of hazard using two-sample comparisons
In this example many comparisons are of interest, and simultaneous
condence intervals are suitable (required)
Rat example: Four types of condence intervals are calculated
Welch t-test condence intervals based on original data (classical,
symmetric in additive sense)
Ratio-to-control condence intervals, suitable e.g. for skewed
distributions, related to logarithmic transformation (symmetric in
multiplicative sense)
Ratio-to-control condence intervals (Fieller) for log-normal
endpoints (alternative to transformation)
Non-parametric ratio-to-control condence intervals
(Hodges-Lehmann, based on ranks and test of median dierences, as
in Wilcoxon test)
Result: In general similar results in all 4 cases, but dierent types of
symmetry and also dierent lengths of condence intervals,
especially close to dierence 0
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 39

41. 1 Introduction 1.2 Principles
1.2 Condence intervals Welch t−test confidence intervals
 
Potassium : 1000 − 0
Potassium : 500 − 0
Potassium : 250 − 0
Potassium : 125 − 0
Potassium : 62.5 − 0
PHOS : 1000 − 0
PHOS : 500 − 0
PHOS : 250 − 0
PHOS : 125 − 0
PHOS : 62.5 − 0
Calc : 1000 − 0
Calc : 500 − 0
Calc : 250 − 0
Calc : 125 − 0
Calc : 62.5 − 0
Sodium : 1000 − 0
Sodium : 500 − 0
Sodium : 250 − 0
Sodium : 125 − 0
Sodium : 62.5 − 0
Chloride : 1000 − 0
Chloride : 500 − 0
Chloride : 250 − 0
Chloride : 125 − 0
Chloride : 62.5 − 0
Triglyceride : 1000 − 0
Triglyceride : 500 − 0
Triglyceride : 250 − 0
Triglyceride : 125 − 0
Triglyceride : 62.5 − 0
Cholesterol : 1000 − 0
Cholesterol : 500 − 0
Cholesterol : 250 − 0
Cholesterol : 125 − 0
Cholesterol : 62.5 − 0
TBA : 1000 − 0
TBA : 500 − 0
TBA : 250 − 0
TBA : 125 − 0
TBA : 62.5 − 0
Nuc : 1000 − 0
Nuc : 500 − 0
Nuc : 250 − 0
Nuc : 125 − 0
Nuc : 62.5 − 0
ALP : 1000 − 0
ALP : 500 − 0
ALP : 250 − 0
ALP : 125 − 0
ALP : 62.5 − 0
SDH : 1000 − 0
SDH : 500 − 0
SDH : 250 − 0
SDH : 125 − 0
SDH : 62.5 − 0
ALT : 1000 − 0
ALT : 500 − 0
ALT : 250 − 0
ALT : 125 − 0
ALT : 62.5 − 0
CreatKinase : 1000 − 0
CreatKinase : 500 − 0
CreatKinase : 250 − 0
CreatKinase : 125 − 0
CreatKinase : 62.5 − 0
SerumGlucose : 1000 − 0
SerumGlucose : 500 − 0
SerumGlucose : 250 − 0
SerumGlucose : 125 − 0
SerumGlucose : 62.5 − 0
ALB  : 1000 − 0
ALB  : 500 − 0
ALB  : 250 − 0
ALB  : 125 − 0
ALB  : 62.5 − 0
TP  : 1000 − 0
TP  : 500 − 0
TP  : 250 − 0
TP  : 125 − 0
TP  : 62.5 − 0
Creat : 1000 − 0
Creat : 500 − 0
Creat : 250 − 0
Creat : 125 − 0
Creat : 62.5 − 0
BUN : 1000 − 0
BUN : 500 − 0
BUN : 250 − 0
BUN : 125 − 0
BUN : 62.5 − 0
−100 0 100 200 300 400 500
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42. 1 Introduction 1.2 Principles
1.2 Condence intervals Ratio−to−control confidence intervals
 
Potassium : 1000 / 0
Potassium : 500 / 0
Potassium : 250 / 0
Potassium : 125 / 0
Potassium : 62.5 / 0
PHOS : 1000 / 0
PHOS : 500 / 0
PHOS : 250 / 0
PHOS : 125 / 0
PHOS : 62.5 / 0
Calc : 1000 / 0
Calc : 500 / 0
Calc : 250 / 0
Calc : 125 / 0
Calc : 62.5 / 0
Sodium : 1000 / 0
Sodium : 500 / 0
Sodium : 250 / 0
Sodium : 125 / 0
Sodium : 62.5 / 0
Chloride : 1000 / 0
Chloride : 500 / 0
Chloride : 250 / 0
Chloride : 125 / 0
Chloride : 62.5 / 0
Triglyceride : 1000 / 0
Triglyceride : 500 / 0
Triglyceride : 250 / 0
Triglyceride : 125 / 0
Triglyceride : 62.5 / 0
Cholesterol : 1000 / 0
Cholesterol : 500 / 0
Cholesterol : 250 / 0
Cholesterol : 125 / 0
Cholesterol : 62.5 / 0
TBA : 1000 / 0
TBA : 500 / 0
TBA : 250 / 0
TBA : 125 / 0
TBA : 62.5 / 0
Nuc : 1000 / 0
Nuc : 500 / 0
Nuc : 250 / 0
Nuc : 125 / 0
Nuc : 62.5 / 0
ALP : 1000 / 0
ALP : 500 / 0
ALP : 250 / 0
ALP : 125 / 0
ALP : 62.5 / 0
SDH : 1000 / 0
SDH : 500 / 0
SDH : 250 / 0
SDH : 125 / 0
SDH : 62.5 / 0
ALT : 1000 / 0
ALT : 500 / 0
ALT : 250 / 0
ALT : 125 / 0
ALT : 62.5 / 0
CreatKinase : 1000 / 0
CreatKinase : 500 / 0
CreatKinase : 250 / 0
CreatKinase : 125 / 0
CreatKinase : 62.5 / 0
SerumGlucose : 1000 / 0
SerumGlucose : 500 / 0
SerumGlucose : 250 / 0
SerumGlucose : 125 / 0
SerumGlucose : 62.5 / 0
ALB  : 1000 / 0
ALB  : 500 / 0
ALB  : 250 / 0
ALB  : 125 / 0
ALB  : 62.5 / 0
TP  : 1000 / 0
TP  : 500 / 0
TP  : 250 / 0
TP  : 125 / 0
TP  : 62.5 / 0
Creat : 1000 / 0
Creat : 500 / 0
Creat : 250 / 0
Creat : 125 / 0
Creat : 62.5 / 0
BUN : 1000 / 0
BUN : 500 / 0
BUN : 250 / 0
BUN : 125 / 0
BUN : 62.5 / 0
0 2 4 6 8
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 41

43. 1 Introduction 1.2 Principles
1.2 Condence intervals Ratios−to−control for log−normal endpoints
 
Potassium : 1000 / 0
Potassium : 500 / 0
Potassium : 250 / 0
Potassium : 125 / 0
Potassium : 62.5 / 0
PHOS : 1000 / 0
PHOS : 500 / 0
PHOS : 250 / 0
PHOS : 125 / 0
PHOS : 62.5 / 0
Calc : 1000 / 0
Calc : 500 / 0
Calc : 250 / 0
Calc : 125 / 0
Calc : 62.5 / 0
Sodium : 1000 / 0
Sodium : 500 / 0
Sodium : 250 / 0
Sodium : 125 / 0
Sodium : 62.5 / 0
Chloride : 1000 / 0
Chloride : 500 / 0
Chloride : 250 / 0
Chloride : 125 / 0
Chloride : 62.5 / 0
Triglyceride : 1000 / 0
Triglyceride : 500 / 0
Triglyceride : 250 / 0
Triglyceride : 125 / 0
Triglyceride : 62.5 / 0
Cholesterol : 1000 / 0
Cholesterol : 500 / 0
Cholesterol : 250 / 0
Cholesterol : 125 / 0
Cholesterol : 62.5 / 0
TBA : 1000 / 0
TBA : 500 / 0
TBA : 250 / 0
TBA : 125 / 0
TBA : 62.5 / 0
Nuc : 1000 / 0
Nuc : 500 / 0
Nuc : 250 / 0
Nuc : 125 / 0
Nuc : 62.5 / 0
ALP : 1000 / 0
ALP : 500 / 0
ALP : 250 / 0
ALP : 125 / 0
ALP : 62.5 / 0
SDH : 1000 / 0
SDH : 500 / 0
SDH : 250 / 0
SDH : 125 / 0
SDH : 62.5 / 0
ALT : 1000 / 0
ALT : 500 / 0
ALT : 250 / 0
ALT : 125 / 0
ALT : 62.5 / 0
CreatKinase : 1000 / 0
CreatKinase : 500 / 0
CreatKinase : 250 / 0
CreatKinase : 125 / 0
CreatKinase : 62.5 / 0
SerumGlucose : 1000 / 0
SerumGlucose : 500 / 0
SerumGlucose : 250 / 0
SerumGlucose : 125 / 0
SerumGlucose : 62.5 / 0
ALB  : 1000 / 0
ALB  : 500 / 0
ALB  : 250 / 0
ALB  : 125 / 0
ALB  : 62.5 / 0
TP  : 1000 / 0
TP  : 500 / 0
TP  : 250 / 0
TP  : 125 / 0
TP  : 62.5 / 0
Creat : 1000 / 0
Creat : 500 / 0
Creat : 250 / 0
Creat : 125 / 0
Creat : 62.5 / 0
BUN : 1000 / 0
BUN : 500 / 0
BUN : 250 / 0
BUN : 125 / 0
BUN : 62.5 / 0
0 2 4 6 8 10 12
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 42

44. 1 Introduction 1.2 Principles
1.2 Condence intervals Nonparametric ratio−to−control comparisons
 
Potassium : 1000 / 0
Potassium : 500 / 0
Potassium : 250 / 0
Potassium : 125 / 0
Potassium : 62.5 / 0
PHOS : 1000 / 0
PHOS : 500 / 0
PHOS : 250 / 0
PHOS : 125 / 0
PHOS : 62.5 / 0
Calc : 1000 / 0
Calc : 500 / 0
Calc : 250 / 0
Calc : 125 / 0
Calc : 62.5 / 0
Sodium : 1000 / 0
Sodium : 500 / 0
Sodium : 250 / 0
Sodium : 125 / 0
Sodium : 62.5 / 0
Chloride : 1000 / 0
Chloride : 500 / 0
Chloride : 250 / 0
Chloride : 125 / 0
Chloride : 62.5 / 0
Triglyceride : 1000 / 0
Triglyceride : 500 / 0
Triglyceride : 250 / 0
Triglyceride : 125 / 0
Triglyceride : 62.5 / 0
Cholesterol : 1000 / 0
Cholesterol : 500 / 0
Cholesterol : 250 / 0
Cholesterol : 125 / 0
Cholesterol : 62.5 / 0
TBA : 1000 / 0
TBA : 500 / 0
TBA : 250 / 0
TBA : 125 / 0
TBA : 62.5 / 0
Nuc : 1000 / 0
Nuc : 500 / 0
Nuc : 250 / 0
Nuc : 125 / 0
Nuc : 62.5 / 0
ALP : 1000 / 0
ALP : 500 / 0
ALP : 250 / 0
ALP : 125 / 0
ALP : 62.5 / 0
SDH : 1000 / 0
SDH : 500 / 0
SDH : 250 / 0
SDH : 125 / 0
SDH : 62.5 / 0
ALT : 1000 / 0
ALT : 500 / 0
ALT : 250 / 0
ALT : 125 / 0
ALT : 62.5 / 0
CreatKinase : 1000 / 0
CreatKinase : 500 / 0
CreatKinase : 250 / 0
CreatKinase : 125 / 0
CreatKinase : 62.5 / 0
SerumGlucose : 1000 / 0
SerumGlucose : 500 / 0
SerumGlucose : 250 / 0
SerumGlucose : 125 / 0
SerumGlucose : 62.5 / 0
ALB  : 1000 / 0
ALB  : 500 / 0
ALB  : 250 / 0
ALB  : 125 / 0
ALB  : 62.5 / 0
TP  : 1000 / 0
TP  : 500 / 0
TP  : 250 / 0
TP  : 125 / 0
TP  : 62.5 / 0
BUN : 1000 / 0
BUN : 500 / 0
BUN : 250 / 0
BUN : 125 / 0
BUN : 62.5 / 0
0 2 4 6 8 10
Kirsten Schorning:Statistics in Toxicology II Winter semester 2024/25 43