Spaghetti Plot

The first time when I used the term "Spaghetti" was for one of the pharmacokinetic studies where I would like to see the time-concentration curves for all individuals plotted on the same panel. The figure on the right side is an example of a Spaghetti plot from simulated data.

I don't think there is any formal definition for Spaghetti Plot, but this term refers to the plot for visualizing the trajectories for all individual subjects. The name “spaghetti plot” is called because it looks a bit like spaghetti noodles thrown on a wall.



The funny thing is that one time when I used the term 'spaghetti plot', I was asked not to use this term since it sounded like 'not formal'. Instead of using 'spaghetti plot', I had to change it to 'indivudual plots' or something like that. As a matter of fact, this term is actually used pretty often in pharmacokinetic studies and also in longitudinal studies.

In longitudinal studies, the spaghetti plot is used to visualize the trajectories or patterns or time trends. The spaghetti plot is typically used in the situation that the # of subjects is not too large and is generated for each group (if there is two treatment groups, there will be one spaghetti plot for each treatment group).

Spaghetti plot can be easily generated by software such as R and SAS. In SAS, the following statement can be used:

symbol1 value = circle color = black interpol = join repeat = 5;
proc gplot;
plot y*time = id / nolegend;
run;

Where y is the desired variable we would like to visualize; time is the time or visit; id is the subject #.

Some further readings:
1. UCLA: How can I visualize longitudinal data in SAS?
2. A oral contraceptive drug interaction study
3. A lecture notes by derived variable summaries
4. Quantitative Methods for Tracking Cognitive Change 3 Years After CABG

Williams Design

Williams Design is a special case of orthogonal latin squares design. It is a high-crossover design and typically used in Phase I studies. Due to the limitation of the # of subjects, we would like to achieve the balance and maximize the comparisons with the smallest # of subjects.

A Williams design possesses balance property and requires fewer sequences and periods. If the number of treatments (n) is an odd number, there will be 2 x n number of sequences. If the number of treatments (n) is an even number, there will be n number of sequences. The example below is a Williams Design with a 4 by 4 crossover (four treatments, four sequences, and also four periods).

Let A, B, C, and D stand for four different treatments, a Williams Design will be arranged as:

A D B C
B A C D
C B D A
D C A B

Notice that each treatment only occurs one time in one sequence, in one period. Furthermore, each treatment only follow another treatment one time. For example, treatment D following treatment B only one time in all sequences.

Several years ago, I wrote a paper on generating the randomization schedule using SAS. I illustrated an example for Williams Design.

There is a new paper by Wang et al specifically discussing about "The Construction of a Williams Design and Randomization in Cross-Over Clinical Trials using SAS"

Williams Design is deliberated in detail in the books "Design and Analysis of Clinical Trials" and Design and Analysis of Bioavailability and Bioequivalence Studies" by Chow and Liu

Williams Design is not purely used in Phase I or bioavailabity studies. I participated in a study with drug abuse area where a Williams design was used. It looks like that other people also uses Williams Design in drug abuse research.

Protocol Amendment after IND

In clinical development, filing of IND (Investigational New Drug) is an important milestone. FDA is required by the Modernization Act to respond in writing to an IND sponsor within 30 calendar days of receipt of the sponsor’s IND filing including the clinical study protocol(s). If the clinical study is not put on hold, the sponsor can start all clinical work including the patient enrollment.

After the initial IND is approved, how to oversee the IND if the sponsor makes significant changes to the study protocol?

First of all, any changes in the research protocol (protocol amendment or administrative letter) or patient informed consent form must be approved by the IRB (institutional Review Board) before the investigator or any sub-investigators put those changes into effect

Secondly, the protocol amendment needs to be submitted to FDA (immediately or through IND annual report). According to 21CFR312.30, the following requirements are stated:

"(b) Changes in a protocol. (1) A sponsor shall submit a protocol

amendment describing any change in a Phase 1 protocol that significantly
affects the safety of subjects or any change in a Phase 2 or 3 protocol
that significantly affects the safety of subjects, the scope of the
investigation, or the scientific quality of the study. Examples of
changes requiring an amendment under this paragraph include:
 (i) Any increase in drug dosage or duration of exposure of
individual subjects to the drug beyond that in the current protocol, or
any significant increase in the number of subjects under study.
 (ii) Any significant change in the design of a protocol (such as the
addition or dropping of a control group).
 (iii) The addition of a new test or procedure that is intended to
improve monitoring for, or reduce the risk of, a side effect or adverse
event; or the dropping of a test intended to monitor safety.
 (2)(i) A protocol change under paragraph (b)(1) of this section may
be made provided two conditions are met:
 (a) The sponsor has submitted the change to FDA for its review; and
 (b) The change has been approved by the IRB with responsibility for
review and approval of the study. The sponsor may comply with these two
conditions in either order.
 (ii) Notwithstanding paragraph (b)(2)(i) of this section, a protocol
change intended to eliminate an apparent immediate hazard to subjects
may be implemented immediately provided FDA is subsequently notified by
protocol amendment and the reviewing IRB is notified in accordance with
Sec. 56.104(c)."

IN FDA's compliance program guidance manual on 'clinical investigators and sponsor investigators', 

there are the following statements:

"Protocol changes/amendments. During the course of a study, a protocol may be formally changed 

by the sponsor. Such a change is usually prospectively planned and implemented in a systematic 

fashion through a protocol amendment. Protocol amendments must be reviewed and approved by 

the IRB, prior to implementation, and submitted to FDA. "

Not all protocol changes require the submission of a formal protocol amendment,
however, the sponsor's reporting responsibility depends on the nature of the
change.  In practice, many companies adopt a conservative approach by reporting
virtually all protocol changes.

Double Dummy Technique

Double dummy is a technique for retaining the blind when administering supplies in a clinical trial, when the two treatments cannot be made identical. Supplies are prepared for Treatment A (active and indistinguishable placebo) and for Treatment B (active and indistinguishable placebo). Subjects then take two sets of treatment; either A (active) and B (placebo), or A (placebo) and B (active).

Double dummy is a method of blinding where both treatment groups may receive placebo. For example, one group may receive Treatment A and the placebo of Treatment B; the other group would receive Treatment B and the placebo of Treatment A.

The figure on the left side is a double-dummy example for a two treatmetn arm scenario. The figure on the right side is a double-dummy example for a three-arm scenario. To maintain the blinding, subjects in each arm will take one tablet and one capsule. In the example on the right side table, subject in placebo arm will take one placebo tablet and one placebo capsule.

Group t-test or Chi-square test based on the summary data

Sometimes, the only data we have is the summary data (mean, standard deviation, # of subjects). Can we use the summary data (instead of the raw data) to calculate the statistical and p-values?

Yes, we can.

Below is an example for group t-test. I illustrate two methods for calculating the p-values based on the summary data.

In the method 1, we will use the SAS procedure PROC TTEST. The only trick thing is to enter the summary data in a data set with an SAS internal variable _STAT_ for the indicator of the summary statistics. The program below is self-explanatory.

data summary;
length _stat_ $4;
input week $ _STAT_ $ value@@;
datalines;
w1 n 7
w1 mean -2.6
w1 std 1.13
w2 n 5
w2 mean -1.2
w2 std 0.45
;
proc print;run;
proc ttest data=summary;
class week;
var value;
run;


Another way is to use the formula.


The correct formula for calculating the t value for group t-test is shown on the right side Where m=0 with degree freedom of n1+n2-2. To compare means from two independent samples with n1 and n2 observations to a value m, this formula can also be used.

where s**2 is the pooled variance

s**2 = [((n1-1)s1**2+(n2-1)s2**2)/(n1+n2-2)]

and s1**2 and s2**2 are the sample variances of the two groups. The use of this t statistic depends on the assumption that sigma1**2=sigma2**2, where sigma1**2 and sigma2**2 are the population variances of the two groups.

*Method #2;
data ttest;
input n1 mean1 sd1 n2 mean2 sd2;
s2 = (((n1-1)*sd1**2+(n2-1)*sd2**2)/(n1+n2-2));
s =sqrt(s2);
denominator = s * sqrt((1/n1) + (1/n2));
df = n1+n2-2;
t = (mean1 - mean2)/denominator;
p = (1-probt(abs(t),df))*2;
datalines;
7 -2.6 1.13
5 -1.2 0.45
;
run;
proc print;
run;

It will be even easier if the summary data is # of counts or frequency data. we can use SAS PROC FREQ option WEIGHT to indicate that data is for # of counts instead of the original individual data. The SAS codes will be something like:

data disease;
do exposure=1 to 2;
do disease=1 to 2;
input index@;
output;
end;
end;
cards;
23 32
17 15
;
proc freq data=disease;
tables exposure*disease/chisq;
weight index;
run;