Applied Statistics Bo Markussen

Statistical methods for the Biosciences November, 2021

Exercises for Day 1

Used datasets and R scripts can be downloaded in a ZIP archive from the
Absalon page (Applied Statistics) or from
http://www.math.ku.dk/~pdq668/SmB/material/day1.zip
Exercise 1.1: Significance vs. importance.
In the lecture I propose the following four main questions1 to be answered
by the statistical analysis of a dataset:
1. Is there an effect?
2. Where is the effect?
3. What is the effect?
4. Can the conclusions be trusted?
The founder of modern statistics R.A. Fisher once wrote:
“It is the magnitude of treatment differences that is of primary
importance, not their statistical significance”
Which of the four questions listed above are concerned with significance and
with magnitude respectively? Do you agree with Fisher?

Exercise 1.2: Datasets, variables and observations.

Often data is organized in tables in a laboratory diary or in an Excel
sheet. Below you see four examples from the biosciences. For each example
discuss the following questions, and summarize your conclusions in a Table-
of-Variables:
(a) How many observations have been made?
(b) What are the variables in the experiment?
(c) What are the variable types (nominal, ordinal, interval, ratio)?
(d) What do you think is the relevant question to be answered by the
statistical analysis?
(e) Which variable would you use as the response?
1
In some situations the word “effect” should be replaced by “association” in these
questions.

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Statistical methods for the Biosciences November, 2021

Data example 1: In an experiment concerning the effect of antibiotic and

vitamin additives on growth 12 rats were given two different levels of an-
tibiotic and two different levels of vitamin in their diet, and the growth was
measured over some time period. The following table shows the measure-
ments for all 12 rats.
Level of vitamin
Level of antibiotic 0 5
0 1.30 1.19 1.08 1.26 1.21 1.19
40 1.05 1.00 1.05 1.52 1.56 1.55
Hint: There are three variables in this example.
Data example 2: An experiment made by Anders Juel Møller (KVL) com-
pared two chilling methods (tunnel-chilling and fast-chilling) of pork meat.
24 porks were sampled from two pH groups (high and low pH). After slaugh-
tering the 24 porks were divided into two sides. One side was tunnel-chilled,
the other fast-chilled. After some time the tenderness of the 48 meat pieces
was measured. The measurements are displayed in the following table.
Pork pH Tunnel Fast
1 low 7.22 5.56
2 low 3.11 3.33
3 low 7.44 7.00
4 low 4.33 4.89
5 low 6.78 6.56
6 low 5.56 5.67
7 low 7.33 6.33
8 low 4.22 5.67
9 low 3.89 4.00
10 low 5.78 5.56
11 low 6.44 5.67
12 low 8.00 5.33
13 high 8.44 8.44
14 high 7.11 6.00
15 high 6.00 5.78
16 high 7.56 7.67
17 high 5.11 4.56
18 high 8.67 8.00
19 high 5.78 7.67
20 high 6.11 5.67
21 high 7.44 7.56
22 high 7.67 6.11
23 high 8.00 8.22
24 high 8.78 8.44

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Statistical methods for the Biosciences November, 2021

Data example 3: In an experiment comparing the difference between two

different diets 20 persons participated. By randomization 10 persons were
assigned to each diet and every week a weight gain or weight loss was ob-
served. The observations are the number of weeks where the diet resulted in
a weight loss for each of the 20 persons in the experiment. The table below
displays the results for a period of eight weeks showing the number of persons
for each combination of diet and weeks with weight loss.
Weeks with weight loss
0 1 2 3 4 5 6 7 8
Diet 1 1 0 2 0 1 1 2 0 3
Diet 2 2 1 0 1 2 1 2 1 0

Hint: The observations are perhaps not what they seem at first sight. How
many observations are there here?

Data example 4: In an experiment concerning the influence of stress on

metabolism in rats the regulation of 96 genes were measured using the qPCR
method. A total of 47 rats were allocated to 8 groups as shown in the
following table.

Group 1 2 3 4 5 6 7 8
Number of rats 6 5 6 6 6 6 6 6
Sex male male male male female female female female
Stabeling single single group group single single group group
Food additive no yes no yes no yes no yes

In each group the average gene regulation was measured on a logarithmic

scale. The following table shows the measurements for 8 genes.

Group
Gene 1 2 3 4 5 6 7 8
Abcb1b 5.554 4.49 4.85 5.076 7.416 6.684 7.524 6.894
Abcb1 5.334 5.55 5.53 4.656 3.456 3.134 3.894 3.004
Abcb4 1.134 1.19 1.51 1.406 1.916 1.454 2.054 1.684
Abcc1 8.114 8.01 8.86 8.466 8.316 7.104 7.884 6.644
Abp1 8.224 8.68 9.24 8.676 11.406 8.504 10.604 8.214
Adh1 −2.996 −3.38 −2.92 −3.214 −3.964 −4.216 −3.766 −4.416
Adh4 2.944 3.10 3.24 3.786 2.456 2.474 2.154 2.494
Ahr 3.624 3.62 4.56 4.976 3.136 3.334 3.014 3.294

The experiment was made by Tina Vicky Alstrup Hansen (UCPH-LIFE).

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Statistical methods for the Biosciences November, 2021

Exercise 1.3: Simple but excellent features in RStudio.

The purpose of this exercise is to show some small details in RStudio
as well as my favourite method of starting RStudio. I have a Windows 10
laptop, and the RStudio icon is attached to my “process bar” in the lower-left
corner of my desktop. Alternatively I might have had the RStudio icon on
the desktop itself, or in the “programs folder” in the start-menu. I’ll assume
that you have similar access to RStudio on your Windows, Mac, or Linux
laptop. Now please start RStudio, and do the following step-by-step (I hope
it works, let’s see. . . ).

1. Look at the Environment 2 in the upper-right window. Are the any

variables?

2. In any case, let’s make two new variables and a data frame by executing
the following 3 lines in the Console in the lower-left window:

x <- rnorm(100)
y <- x+rnorm(100)
z <- data.frame(x1=x,x2=y)

Now you should have (at least) 3 objects in your Environment: Two
Values called “x” and “y”, and one Data called “z”.

3. Click on the object called “z” in order to get a display in the Editor
in the upper-left window. You may always do this later on to see the
data you have inside R. Pretty neat3 , right?

4. Clear the Environment by executing the following line in the Console:

rm(list=ls())

Alternatively, you can also clear the Environment by clicking the “broom”
icon in the upper-right window.

5. Now let save an empty(!) R-dataset to the folder you are using for the
computer exercises today: Click on the Files-menu in the lower-right
window, and browse the file system to locate the folder (sometimes
2
In early versions of RStudio this was called the Workspace. Essentially its the same
thing, but now you have the option of choosing different environments.
3
In my view as a teacher this feature matches the major pedagogical point of Excel
and JMP (easy to use statistical software, available from www.kunet.dk), namely the
possibility to see the datasets.

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Statistical methods for the Biosciences November, 2021

also called a “directory”) you want to use. E.g. it might be a subfolder

called “Day 1” in a folder called “Statistics course” (you, of course,
may have used different names).

6. Click on “Set As Working Directory” inside the More-submenu in the

lower-right window.

7. Now close RStudio. Since the workspace has been changed (in the
steps you did above), RStudio will ask you whether you want to save
the workspace. Click “Save” (or possibly “Yes”) to do so.
Of course you can also save your data without closing RStudio. Either
by clicking the “floppy-disk” icon in the Environment window, or by
executing the save.image() command in the Console.

8. Now, use your operating system (Windows, Mac, or Linux) to browse to

the folder you chose above. Then there should be a file called something
like “.RData”. This file contains the empty workspace you just saved.

9. If you (double) click on this file, then it hopefully activates RStudio4

at the present folder on your laptop.

Of course this way of saving the working environment makes even more sense
when you save a non-empty environment with the variables you are working
on. In my daily work I have relevant R datasets for each of the projects I’m
working on. And to continue my work I simple double click on the R datasets
in order to resume my work inside the correct folder.

Exercise 1.4: Getting data into R.

In this exercise we continue practising the usage of RStudio. The first
step in a statistical analysis is to get the data into R. So let’s try this using
the second data example from Exercise 1.2.
If you have data in a plain text file (ASCII file), then it is straightforward
to import the data. Let’s try this using the file dataExample2.txt available
from the ZIP archive day1.zip5

1. Click on “From Text (base)...” in the Import Dataset-submenu in the

Environment window.
4
Alternatively it might start the classical R console (RGui). If so, then you should use
the operating system to associate .RData files with RStudio instead.
5
You need to “unzip” the ZIP archive, i.e. extract the files inside the archive, before
you can import the files into R. Ok?

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Statistical methods for the Biosciences November, 2021

2. Browse your file system to locate the file dataExample2.txt and click
“Open” (or something similar) to open the file.
3. A window with an Import Wizard should appear. The file dataExample2.txt
contains a heading with the variable names, data entries are separated
by Tab(-ulator) signs, and decimals are given by commas.
4. Try out the different possibilities in the left part of the Wizard, and
see how the data frame in the lower part changes.
5. Click “Import” when things look right.
Please notice that the import Wizard actually generates some R code in the
Console (which is also available in the History in the upper-right window).
Thus, you may insert this R code in your R program instead of using the
import Wizard. In the long run this is much easier6 .
In practice it is more common to have data recorded in an Excel sheet. So
let’s try to read the Excel file named dataExample2.xls, which is assumed
to be extracted from the zip archive day1.zip.
1. Make sure that the possibility “From Excel...” is available in the Import
Dataset-submenu in the Environment window. If this is not the case,
then you probably should update RStudio.
2. Import dataExample2.xls by using the “From Excel...” interface.
And again, for future usage you might want to use the generated R
code instead of using the import Wizard.
3. Now that we have a dataset inside R let’s also try to make some graph-
ics. Execute the following R commands7 in the Console:

library(ggplot2)
ggplot(dataExample2,aes(x=Fast,y=Tunnel,col=pH)) +
geom_point() +
geom_abline(intercept=0,slope=1) +
ggtitle("Tenderness of pork meat")

Can you decipher the purpose of the R code? And what can you infer
about the data from the resulting plot?

6
And also necessary when you knit R Markdown documents!
7
If the ggplot2-package is not available, then you need install it via “Install” inside
Packages in the lower-right window.

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Applied Statistics Bo Markussen
Statistical methods for the Biosciences November, 2021

Exercise 1.5: Hypertension in diabetic patients.

Before commencing on the statistical methods we introduce yet another
R technicality. So far we have seen data encoded in text-files, Excel sheets,
and R scripts. But of course R also has a format for saving data, namely in
RData-files8 . If RStudio is already open, then you may read RData files us-
ing the “open file” icon in the Environment window, or by using the load()
function from the Console. If RStudio is not open, then you may open RStu-
dio together with the RData file by double clicking on the file (in Windows).
The data for this exercise is available in the file hypertension.RData,
and also in an Excel sheet (just in case you need it, which you shouldn’t).

An experiment on 19 diabetic patients was conducted in order to compare

the effects of two drugs called Drug E and Drug N on the treatment of high
blood pressure. The experiment is a cross-over study. This means that all
patients try both drugs in two different study periods. Both study periods
lasted for 14 days. In between the two study periods was a wash-out period,
which also lasted for 14 days. The patients were randomly assigned to two
groups called E/N and N/E. The patients in the E/N-group received drug E
in the first study period and drug N in the second study period. The patients
in the N/E-group received drug N in the first study period and drug E in the
second study period.
The systolic and the diastolic blood pressure was measured for all the
patients at the beginning and the end of both study periods. In this exercise
we will only analyse the observations of the systolic blood pressure. These
observations are shown in the table on the next page.
8
We already have worked with RData-files in Exercise 1.3.

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Statistical methods for the Biosciences November, 2021

Systolic blood pressure

Patient id Treatment order Baseline 1 End 1 Baseline 2 End 2
9 Drug E, Drug N 124 136 120 145
21 Drug E, Drug N 120 132 138 126
8 Drug E, Drug N 115 96 111 91
12 Drug E, Drug N 134 118 123 123
16 Drug E, Drug N 131 106 111 123
19 Drug E, Drug N 119 108 113 112
20 Drug E, Drug N 124 112 108 112
24 Drug E, Drug N 127 113 121 143
13 Drug N, Drug E 113 113 107 97
17 Drug N, Drug E 132 109 122 119
18 Drug N, Drug E 129 133 139 130
23 Drug N, Drug E 124 120 127 118
25 Drug N, Drug E 112 103 112 121
10 Drug N, Drug E 124 112 128 122
11 Drug N, Drug E 144 154 156 137
14 Drug N, Drug E 134 118 122 109
15 Drug N, Drug E 119 118 115 114
22 Drug N, Drug E 123 123 114 108
26 Drug N, Drug E 122 123 124 120

The R dataset hypertension.RData contains the dataset. Beside the

raw observations encoded in the variables patient, order, baseline1, end1,
baseline2 and end2 five new variables called change1, change2, average, diff
and E diff N have been defined.

ˆ The variable change1 contains the change of blood pressure over study
period 1.

ˆ The variable change2 contains the change of blood pressure over study
period 2.

ˆ The variable average contains the average change of the blood pressure
over both study periods.

ˆ The variable diff contains the difference of the changes of blood pres-
sure between study period 1 and study period 2.

ˆ The variable E diff N contains the difference of the changes of the blood
pressure between the study periods given drug E and drug N.

To analyse the dataset for the cross-over study the following four T-tests may
be performed:

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Statistical methods for the Biosciences November, 2021

ˆ Two sample T-test comparing E diff N in the E/N-group against the

N/E-group.
ˆ Two sample T-test comparing average in the E/N-group against the
N/E-group.
ˆ Two sample T-test comparing diff in the E/N-group against the N/E-
group.
ˆ One sample T-test comparing E diff N against the mean value 0.
Two of these T-tests do the actual comparison between the effects of drug
E and drug N. These tests, however, are only valid when the following two
problems do not occur:
Problem 1: A spill-over (also called a carry-over) from study period 1 to study
period 2. A possible explanation for such an effect is that the drug
given in study period 1 still has an effect in study period 2.
Problem 2: An interaction between the effects of the drugs and the study periods.
For instance that the effect of drug E for some strange reason is larger
in study period 1 than in study period 2.
The two remaining T-tests are done to validate that these two problems have
not occurred.
a) Which of the four T-tests listed above do the drug comparison, and
which T-tests validates against problem 1 and 2?
Help to get started: If the drugs have different effects and if there is
a spill-over from period 1 to period 2, then the difference between the
changes in the E- and the N-period will depend on the order the drugs
were given.
b) Perform the relevant T-tests. Remember to validate the underlying
normality assumption before you make the T-tests. What is the con-
clusion from these tests?

Remark: I would probably not do the statistical analysis using all these T-
tests. Instead I would do the analysis using a random effect model. We will
return to this on course day 5.
Reference: Bradstreet, T.E. (1994) “Favorite Data Sets from Early Phases of
Drug Research - Part 3.” Proceedings of the Section on Statistical Education
of the American Statistical Association.

End of Exercises.