Lab 2 – Functions and Antibiotics
Date: August 30, 2011
Assignment and Report Due Date: September 6, 2011
Goal: In this lab you will learn about a model for antibiotic efficiency and see how it relates to data. We
will consider the relationship between dosage and efficiency during antibiotic treatment of an infection.
During the course of the lab you will learn two new commands: function() and rep(). First, you will learn
how to define and use functions in R. You will then use this knowledge to plot a given model and see
how it relates to experimental data.
New commands
function() – defines a function and it’s inputs
rep() – creates a vector with repeated entries, i.e. (1,1,1,1,1)
Introduction to Functions in R
During last week’s lab we learned how to create input and output vectors in order to plot functions.
There is actually an easier way to do this! We can explicitly define functions in R using the function
command. Recall that last week we plotted the line f(x) = 2*x + 1 using the following code.
We first defined the slope and y-intercept.
>m=2
>b=1
Next we defined the input vector.
> x = 1:5
Then we defined the output vector using the function f(x) = m*x + b.
> f = m*x + b
Finally we plotted the result.
> plot(x,f,type = “o”)
Now let’s see how we can do this using function definitions. Instead of just defining an output vector f,
we will define the actual function which can be used to give us multiple outputs. Let’s call this function
F.
> F = function(input){m*input + b}
Let’s talk through the pieces of this definition. In the parentheses we put our input which for now I have
called “input”. Inside the curly brackets we put the formula for what we are going to do to the input to
get output. In this case we are plotting a line and using the slope-intercept form of a line as our formula
to get output.
NOTE: remember variables can be called anything you would like as long as you are consistent within
the function definition. We could have also defined the function in this way.
> F = function(x){m*x + b}
So now let’s see why this is useful. Imagine we want to know the output when our input = 1.
> F(1)
This gives the output when our input is one. Before when we wanted to plug in the value of 1 for our
input we typed the following.
> m*1 + b
Function definitions make evaluating functions for given values much easier.
We can also easily define a set of input and output vectors very simply in order to plot them.
> x = 1:5
> f = F(x)
> plot(x,f,type=”o”)
Now let’s change the input vector to see a different part of the line.
> x = -10:0
> plot(x,F(x),type=”o”)
We can even call the function F inside the plot command to simplify our life more since we know
F(x) will give us the output values we want.
What if we want to plot a different line by changing the values of m and b. Since m and b are included in
our definition of F, this is very simple. First redefine m and b. Then just replot the function.
> m = -1
>b=0
> plot(x,F(x),type=”o”)
As you can see, R knows just how to handle this since we have used functions. We only had to define
our function once and then it becomes simple to evaluate and plot the function as we change
parameters or change the vector of input values.
In the lecture you discussed point-slope forms of lines by using a function of the following form.
Here we must define the parameters
and to plot this line. Try using the following parameters.
> m = -65
> x0 = 8
> y0 = 305
Define your input vector, x.
> x = 8:13
Define the point-slope function.
> F = function(x){m*(x-x0)+y0}
Plot the function
> plot(x,F(x),type=”o”)
What could this line represent? Give an interpretation that could relate to everyday life.
Let’s try one more function to make sure we get it. This time we will plot a parabola which has three
parameters: a,h, and k. For fun we will let our input be called w.
>a=1
>h=0
>k= 0
> parabola = function(w){a*(w-h)^2+k}
First let’s evaluate our function “parabola” at a couple points and see what we get.
> parabola(-3)
> parabola(2)
Let’s define a new input vector over which to plot this parabola. We’ll call this input vector “y”.
> y = seq(from=-3, to = 3, by = .1)
> plot(y,parabola(y),type = “o”)
There is our parabola! Change the values of the parameters and see what happens to the parabola.
If you have any questions with function definitions or how to use them please ask now.
Biology Background
Physicians treat many patients for otitis media, commonly known as an ear infection. When the middle
ear becomes infected with bacteria it can cause pressure and pain. Bacteria such as Streptococcus
pneumonia or Pseudomona aeruginosa may be the culprits behind painful earaches. As you know,
antibiotics are used to treat bacterial infections. Amoxicillin is often prescribed to children and adults
for such infections. Amoxicillin inhibits the synthesis of the cell wall in bacteria which ultimately kills the
bacteria. It has been proven a very effective medication for treating otitis media.
A natural question to ask is how does antibiotic dosage affect the amount of bacteria killed? This can be
thought of as antibiotic efficacy. How much antibiotic is needed to efficiently kill the bacteria causing an
infection? If you give a patient 250mg of amoxicillin, how many bacteria will that kill? How about if you
give them a dose of 500mg? Will that kill twice as much as a dose of 250mg? Take a minute to think
about what a plot might look like if you were to draw it on the following figure.
Model describing Antibiotic Efficacy
We will use the following function to describe antibiotic efficacy.
In this equations our input, d, represents the dose of antibiotic and the output, E(d), is how we will
measure efficacy, the number of bacteria killed in response to the antibiotic. There are two parameters
in this equation: M and k.
Let’s define this function in R and plot it to see what it looks like. Before the function definition, define
the parameters as shown.
> M = 50
>k=5
> E = function(d){M*d^2/(k^2*M^2+d^2)}
Let’s first set up a blank plot of the appropriate size and label it.
> plot(c(0,1100),c(0,90),type="n",ann=FALSE)
> title(main = "Antibiotic Efficacy",xlab = "Amoxicillin Dosage (mg)",ylab = "# of Dead Bacteria
(*10^8)")
Since our input values are dosages, we will define them to be in a realistic dosage range. Patients are
generally described less than 1000 mg/day so we will set our dosage vector to range from 0 to 1100.
> DoseVector = seq(from=0,to=1100,by=25)
>lines(DoseVector,E(DoseVector),col=”blue”)
Think about this curve. Why is it a good model for antibiotic efficacy?
Let’s now see how our model relates to some sample data.
Hypothetical Experiment
Suppose a research lab conducted a study on the efficacy of amoxicillin. For six months, the lab
monitored patients who present with otitis media. Each patient was given the following dose of
amoxicillin for their first day of treatment. The amount of dead bacteria was recorded before treatment
continued on the second day. The table below shows this data for 60 patients.
PATIENT DATA
# Dead
Dose Bacteria
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
5.3
6.21
9.22
8.99
9.35
9.97
6.63
6.95
7.32
4.77
4.83
5.2
3.81
4.03
4.1
# Dead
Dose Bacteria
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
9.75
11.91
15.08
18.75
19.23
22.52
24.82
24.89
25
22.39
22.84
23.74
19.27
20.07
20.81
# Dead
Dose Bacteria
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
7.36
9.01
15.12
23.74
26.93
28.08
40.13
41.4
42.57
47.74
48.27
49.02
49.92
49.99
50
# Dead
Dose Bacteria
750
750
750
750
750
750
750
750
750
750
750
750
750
750
750
8.07
11.44
14.94
20.05
21.03
28.34
40.24
42.81
46.9
59.21
59.76
60.25
67.55
68.26
69.77
Let’s add these data points to our plot in R. We will first plot the points for patients who were given
100mg. In order to do this we can use the rep() command to make a vector with repeated dosages.
First let’s see what rep() does.
> rep(1,times=5)
As you can see, this creates a vector with five ones.
We will now use rep when plotting our data points since we have repeated dosages.
> points(rep(100,times=15),c(5.13,6.21,9.22,8.99,9.35,9.97,6.63,6.95,7.32,4.77,4.83,5.2,3.81,
4.03,4.1),pch = 3)
> points(rep(250,times=15),c(9.75,11.91,15.08,18.75,19.23,22.52,24.82,24.89,25,22.39,22.84,
23.74,19.27,20.07,20.81),pch = 3)
> points(rep(500,times=15),c(7.36,9.01,15.12,23.74,26.93,28.08,40.13,41.4,42.57,47.74,
48.27,49.02,49.92,49.99,50),pch = 3)
> points(rep(750,times=15),c(8.07,11.44,14.94,20.05,21.03,28.34,40.24,42.81,46.9,59.21,
59.76,60.25,67.55,68.26,69.77),pch = 3)
You should now have your data points plotted as black + signs on your plot. Does our model(the blue
curve) look like a good fit for the experimental data? Why or why not?
What could be missing in our model? Why doesn’t the data fit into the curve we plotted before?
STOP here and make sure you answer the questions before
continuing!!! They will be needed for your report.
Our model has not taken into account body mass! The data we have been given in the table also does
not tell us how big these patients were that were receiving the specified doses of amoxicillin. They
could be infants or large sumo wrestlers.
Our model actually does include body mass in the parameter, M, which we set to be 50 when we
originally plotted our function. This body mass is measured in kilograms so what we have plotted so far
represents a curve describing antibiotic efficacy for a person weighing 50kg or about 110 lbs. We now
need to see how our model changes for different parameter values. We will add lines to our existing
plot to show what the curves look like for different body masses. We will use the following values of
mass to represent different groups of people and will plot each line in a different color.
M = 10
Babies
M = 25
Children
M = 50
Small Adults
M = 75
Medium Adults
M = 100 Large Adults
> M = 10
> lines(DoseVector,E(DoseVector), col=2)
> M = 25
> lines(DoseVector,E(DoseVector), col = 3)
> M = 50
> lines(DoseVector,E(DoseVector),col=4)
> M = 75
> lines(DoseVector,E(DoseVector),col=5)
> M = 100
> lines(DoseVector,E(DoseVector),col=6)
We have now plotted curves for five different body masses. You can see that the curves for different
body masses are very different which could account for the varying data points we plotted from the
table given.
Let’s now look at a data table where we include the mass of patients.
Dose
mass
#Dead
Dose
mass
#Dead
Dose
mass
#Dead
Dose
mass
#Dead
100
5.74
5.3
250
10.15
9.75
500
7.4
7.36
750
8.08
8.07
100
6.96
6.21
250
12.68
11.91
500
9.09
9.01
750
11.44
11.44
100
13.31
9.22
250
16.78
15.08
500
15.48
15.12
750
15.09
14.94
100
31.94
8.99
250
22.58
18.75
500
25.25
23.74
750
20.42
20.05
100
28.93
9.35
250
23.47
19.23
500
29.24
26.93
750
21.47
21.03
100
21.71
9.97
250
31.41
22.52
500
30.74
28.08
750
29.43
28.34
100
52.75
6.63
250
44.38
24.82
500
50.28
40.13
750
43.65
40.24
100
49.48
6.95
250
45.59
24.89
500
53.06
41.4
750
47.01
42.81
100
45.9
7.32
250
49.08
25
500
55.86
42.57
750
52.68
46.9
100
78.86
4.77
250
80.66
22.39
500
73.62
47.74
750
73.38
59.21
100
77.75
4.83
250
76.91
22.84
500
76.56
48.27
750
74.5
59.76
100
71.35
5.2
250
69.15
23.74
500
81.86
49.02
750
75.52
60.25
100
101.01
3.81
250
106.19
19.27
500
94.39
49.92
750
94.16
67.55
100
95.14
4.03
250
99.39
20.07
500
98.04
49.99
750
96.53
68.26
100
93.15
4.1
250
93.32
20.81
500
99.96
50
750
102.08
69.77
We will now color code these points when plotting them according to their masses. Use the following
color codes as shown in the table below.
mass <= 20
Babies
col = 2
20 < mass <= 40 Children
col = 3
40 < mass <= 60 Small Adults
col = 4
60 < mass <= 80 Medium Adults col = 5
80 < mass
Large Adults
col = 6
We will plot all the values for the baby masses first as an example of how to plot the other points.
> points(c(100,100,100,250,250,250,500,500,500,750,750,750),c(5.3,6.21,9.22,9.75,11.91,15.08,
7.36,9.01,15.02,8.07,11.44,14.94),pch=3,col=2)
You should now have your data points for baby masses plotted in red.
>points(c(100,100,100,250,250,250,500,500,500,750,750,750),c(8.99,9.35,9.97,18.75,19.23,
22.52,23.74,26.93,28.08,20.05,21.03,28.34),pch=3,col=3)
You should now have your data points for baby masses plotted in green.
>points(c(100,100,100,250,250,250,500,500,500,750,750,750),c(6.63,6.95,7.32,24.82,24.89,25,
40.13,41.4,42.75,40.24,42.81,46.9),pch=3,col=4)
You should now have your data points for small adult masses plotted in blue.
Follow these examples to plot the data points for medium adults (light blue) and large adults (magenta).
How does the data seem to match with our model now? Does it appear our model is a good fit for the
data collected?
What have you learned about models and how they relate to data from this lab?