Related Links:
Sequential Analysis Introduction and Explained Page
Sequential Unpaired Difference Between Two Counts Program Page
Sequential Unpaired Difference Between Two Means Program Page
Sequential Unpaired Difference Between Two Ordinal Arrays Program Page
Sequential Unpaired Difference Between Two Proportions Program Page
Sequential Unpaired Difference Between Two Survival Rates Program Page
Introduction
Counts
Means
Proportions
Ordinal Arrays
Survivals
Reference
Methodological Considerations
Terminology
Settings
General discussions on sequential analysis are presented in
Sequential Analysis Explained and will not be repeated here.
This page briefly explains the Triangular Test that was developed in the late
1980s and 1990s by Whitehead. For those interested in obtaining a full
understanding of the theories and methodologies of the Triangular Test, Whitehead's
text book (see reference) is highly recommended.
The Triangular test is a sequential statistical method for comparing two groups, based on the relationship
between Fisher's information V (expression of the quantity of data) and the
efficiency score Z (expression of the effect size). The calculation of V and Z
depends on the nature of the measurements concerned, but the interpretations
of their relationship are the same. V and Z can be calculated at any time
during the study.
Statistical borders are drawn that allows the researcher to make one of 3 decision
whenever the data is reviewed. These are to continue with the experiment, to reject the null
hypothesis and stop the experiment, or to accept the null hypothesis and stop the experiment.
The baseline and two borders forms a triangle. While the V / Z plot remains within
the triangle, no decision should be made other than to collect more data. If the
plot crosses the outer border, then the null hypothesis can be rejected
(significant difference exists). If the plot crosses the inner border than
the null hypothesis can be accepted (no significant difference).
The primary straight line borders are calculated on the assumption that the
data will be reviewed after the data is obtained from every case as they occur.
These borders are narrowed to become more powerful if the
data are reviewed less frequently, the extent of the narrowing depending
on the sample size between the reviews. The final borders, with periodic
narrowing, looks like a Christmas tree.
The methods of calculating the effects size for different types of data, and how
the borders are defined, will not be explained in details in these pages. These
are well described in Whitehead's book, and the algorithms can be easily obtained in published
papers (see reference)
Please note that the coordinates defined in these pages are based on a 2 tailed
test (detecting a difference in either direction). If a one
tail test is to be used (one group more than the other but not interested if
it is the other way around) then the type I Error (α) used should be doubled (eg. 0.1 instead of 0.05).
Please also note that the stopping borders are not affected by the ratio of sample size between
the two groups, as these are calculated from α, β(1power), and the effect size θ.
Discrepancies between the two sample sizes however affects the calculation of V and Z from the data,
so will alter the predicted sample size requirement at the planning stage.
At the end of the analysis, a termination test can be done by calculating the
final effect size Theta T=Z/sqrt(V). For the null hypothesis, T0=0 and
SD(T0)=sqrt(V). The normalized z test can therefore be used to test whether T
deviates from null.
In his book, Whitehead presented 5 models, for normally distributed means,
Poisson distributed counts, binomially distributed proportions, survival rates,
and nonparametric ordinal arrays. These are discussed individually as in
the following sections.
Parameters during planning:
Results
 V : = is the Fisher Information, representing the amount of information available
 Z : = is the Efficiency Score, representing the Effect Size of the comparison
 Z(sig for significant) : = The value of Z at V the outside of which (away from the x axis) a
decision to reject the null hypothesis can be made
 Z(nsig for not significant) : = The value of Z at V inside of which (closer to the x axis) a
decision to accept the null hypothesis to be made
 The significant and not significant values are not the same as the borders drawn at the time of planning,
but are also governed by the number of cases between each review.
 In the two tailed model, two sets of borders and significant values are calculated, one for group 1 > group 2
(borders above the x axis), and the other for group 1 < group 2 (borders below the x axis). In the one
tailed model, one set of these borders is irrelevant and should be removed.
Default Settings
The Two Tailed Model : Calculations and plotting in the Unpaired Sequential series of programs from StatsToDo, by default, assumes
the two tailed model, to test the difference between two groups, regardless of which group has greater outcome value.
Many studies are however one tailed, to detect whether a particular group has a greater outcome value than the other,
but not interested in the reverse. In these cases, the following parameters and results should be modified, as follows :
 The Type I Error parameter needs to be doubled (e.g. use 0.1 instead of 0.05). This will reduce the sample size needed.
 The two results tables should be modified, the significant borders representing the side that is not in the model should
be deleted.
 The bitmap needs to be modified, so that the half (top or bottom) of the plots not in the model
is deleted. This can be done after the initial calculation, by tweaking the
MacroPlot codes (removing those rows that plot the unnecessary lines and rescale the bitmap).
Using the same data, but assume two or one tail, will produce the following two plots. Please note the following
 Doubling α changes the stopping border, and the data line now crosses the decision line much earlier
 The lower borders are now irrelevant and can be removed, the plot can then be rescaled.
Ratio of the two groups sizes: The default value for the ratio is 1, that there is an equal allocation to the
two groups, but this may not be suitable in situation where one group needs to be very much bigger than the other
(for example, when one of the experimental treatments is dangerous or expensive, and a 2 to 1 or 3 to 1 ratio of case
allocation to the two groups is planned). When the sample sizes are substantially different, the maximum sample size
required and the decision borders are affected, particularly where the model is two counts, two proportions, or two survivals.
In these cases the one sided model should be used, as the decision borders for the two sides of the model (group 1>group 2
or group 1 < group 2) are not the same.
Data Input Output
Example
Input : In addition to α, power, and ratio that are common inputs to all models, the following inputs are used
 Lamda (λ) : is the averaged event rate (λ=k/n) where k is the observed number of events
(pregnancy, asthma attacks, car crashes, falls, number of cells), and n is number of units of observation (number of
women years, number of children months, per thousand cars per year, per hundred beds per month in a hospital,
number of microlitres of body fluid). λ_{1} and λ_{2} are the event rates of group 1 and 2
the study is design to detect.
 Theta (θ) is the effect size, and calculated from λ_{1} and λ_{2}
according to equation 3.12, p. 37
 The data : are in 4 columns, separated by spaces or tabs. They are the number of units of observation and
number of events observed in group 1, followed by that in group 2 (n_{1} k_{1} n_{2} k_{2}).
Each row represents the accumulated numbers found at each review, in temporal order.
Output : V, Z, and the Z values to reject or accept the null hypothesis are common to all data types, and in
this model are calculated from the following observations
 n1 and n2 are the number of observation units in the two groups
 k1 and k2 are the number of observed events in the two groups
 Z and V are calculated according to equations 3.10 (p.35) and 3.11 (p.36)
 Z(sig) and Z(nsig) are adjusted according to equation 4.8 p. 82
θ = 0.6931
Maximum number of subjects = 679
Border (significant difference) Z = 6.1778 + 0.2425V
Border (no significant difference) Z = 6.1778 + 0.7274V

We wish to compare a special breathing exercise against no exercise as they affect
the frequency of asthma attacks in children. We will conclude that the exercise
reduces asthma attack if it can half the frequency from 20 per 100 children months to 10.
We therefore set the two λ values to 20 asthma attacks per 100 children months (λ1=0.2) for the control group,
and 10 (λ2=0.1) for the exercise group. We designate α=0.05 and power=0.8,
and will allocate the case into two equal size groups (r=1). The borders as calculated
are shown above and to the right.
We recruited the research subjects, and randomized them into two groups to
receive or not receive the exercise. After completion of the treatment (or equivalent
time if not treated), the child is followed up for 1 month, and any asthmatic
attack is recorded. The plan is to review the data available every 2 months
until the study is completed.
 Control  Tmt 
Review  cases  count  cases  count 
2 month  50  10  53  8 
4 month  100  18  99  15 
6 month  150  28  160  25 
The results at the end of 6 months (third reviews) are presented in the table
to the left, and the Triangular plot shown to the right.
As the test is 2 sided, two triangles, one for group 1>group 2, and the other
for group 2>group 1.
The straight line borders are for use if the data is reviewed after every case,
and the inward Christmas tree like narrowing are border adjustments when
there are more than 1 case added between reviews, the extent of the narrowing
depends on the sample size between reviews.
The relationship between V and Z are shown in the line joining the dots.
 At the first review at 2 months, there were 50 children months in the control group
who had between them 10 asthma attacks (λ=10/50=0.2), and 53 children months
who received the exercise who had 8 asthma attacks between them (λ=8/53=0.15) (V=4.5 , Z=1.3).
The resulting plot was within the borders, and the decision was to continue to collect data.
 At the second review at 4 months, the number of completed cases have
increased. There were now 100 children months in the control group who had between
them 18 asthma attacks (λ=18/100=0.18), and 99 children months who received the exercise
who had 15 asthma attacks between them (λ=15/99=0.15). (V=8.2 , Z=1.4).As the
resulting plot remained within the borders, the decision was to continue to collect data.
 At the third review at 6 months, there were now 150 children months in the control group who had between
them 28 asthma attacks (λ=28/150=0.19), and 160 children months who received the exercise
who had 25 asthma attacks between them (λ=25/160=0.16). (V=13.2 , Z=2.4).The plot now
traversed both inner borders, and the study could be terminated, with the conclusion
drawn that the incidents of asthma in the two groups were not significantly different.
The terminal analysis showed effect size(Z/sqrt(V))=0.6473, p=0.2587. The sample size
for the two groups combined was 310, less than half of the maximum number of
cases required for the Triangular Test, and well short of the 436 cases sample size
if the study was to be a fixed sample one. This is because the difference between the
two groups in reality was very much less than anticipated, and the null
hypothesis could be accepted earlier.
Data Input / Output
Example
Input : In addition to α, power, and ratio that are common inputs to all models, the following inputs are used
 Difference Between means = mean(group 1)  mean(group 2). The study is then designed to detect this difference.
 Within Group Standard Deviation is the expected background or population Standard Deviation. This is usually
not available at the planning stage, and some sort of guesstimate is commonly used.
 Theta (θ) = difference between means / SD
 The data are in 6 columns, separated by spaces or tabs. They are sample size, mean, and Standard Deviation found in
group 1, followed by the same for group 2 (n_{1} mean_{1} SD_{1} n_{2} mean_{2}
SD_{2}).
Each row represents the accumulated numbers found at each review, in temporal order.
Output : V, Z, and the Z values to reject or accept the null hypothesis are common to all data types, and in
this model are calculated from the following observations
 n1 and n2 are the number of cases in the two groups
 mean1 and mean2 are the mean values found in the two groups
 sd1 and sd2 are the Standard Deviations found in the two groups
 Z and V are calculated according to equations 3.36 and 3.37 (p.50)
 Z(sig) and Z(nsig) are adjusted according to equation 4.8 p. 82
Effect size (diff / sd) = 0.6931
Maximum number of subjects = 110
Border (significant difference) Z = 6.4232 + 0.23325V
Border (no significant difference) Z = 6.4232 + 0.6996V

We wish to compare the at term birth weight of babies, between boys and girls.
We assume that birth weights are normally distributed, with a standard deviation of
0.3Kgs. We would be happy to decide that boys weighed more than girls if the
difference is 0.2kg or more. We designate α=0.05 and power=0.8,
and will anticipate that the number of boys and girls delivered are roughly the same
so we set the ratio to 1. The borders as calculated are shown above and to the right.
 Boys  Girls 
Review  n  mean  sd  n  mean  sd 
week 1  10  3.8  0.2  10  3.6  0.3 
week 2  20  3.7  0.3  22  3.5  0.2 
week 3  28  3.8  0.3  30  3.6  0.3 
We decided to weigh all the babies (in Kgs) delivered each week, and review the data weekly.
The results at the end of 3 weeks (third reviews) are presented in the table
to the left, and the Triangular plot shown to the right.
As the test is 2 sided, two triangles, one for boys>girls, and the other
for girls>boys.
The straight line borders are for use if the data is reviewed after every case,
and the inward Christmas tree like narrowing are border adjustments when
there are more than 1 case added between reviews, the extent of the narrowing
depends on the sample size between reviews.
The relationship between V and Z are shown in the line joining the dots.
 At the first review after 1 week, there were 10 boys with mean=3.8 and sd=0.2,
and 10 girls mean 3.6 and sd=0.3, V=4.6 and Z=3.8. The resulting plot has reached the inner border
of the lower triangle, and at this point the conclusion that girls are not
significantly heavier than boys could be drawn. However the plot was still
within the borders of the upper triangle, so no decision could be made as to
whether boys are heavier than girls, so the correct decision was to continue to collect data.
 At the second review after 2 weeks, the number of babies delivered has
increased. There were now 20 boys with mean=3.7 and sd=0.3, and 22 girls with
mean=3.5 and sd=0.2, V=9,7 and Z=7.9. The plot has now reached the adjusted border, and at this point
the study could and should have stopped, and the conclusion that boys are heavier
than girls could be made.
 The extension of the study to the third week was unnecessary, and it merely
confirmed the finding of the second review.
The terminal analysis at the end of the second week showed effect size(Z/sqrt(V))=2.5255
p= 0.0058. The sample size at that time was 20, less than the 110 maximum
required and well short of the 71 cases required for a fixed sample size study.
Data Input / Output
Example
Input : In addition to α, power, and ratio that are common inputs to all models, the following inputs are used
 Proportion + is the number of positives in the index of interest divided by the total number p=n(pos) / (n(pos)+n(neg)).
In the example, the study is designed to detect an effect size where group 1 has 50% positives and group 2 20%
 Theta (θ) is the effect size, and calculated according to equation 3.3 p 32
 The data : are in 4 columns, separated by spaces or tabs. They are numbers positive in group 1 and group 2,
then numbers negative in group 1 and group 2 (n_{1}(pos) n_{2}(pos) n_{1}(neg) n_{2}(neg)). Each row represents the numbers
found at each review, in temporal order.
Output : V, Z, and the Z values to reject or accept the null hypothesis are common to all data types, and in
this model are calculated from the following observations
 Pos1, Pos2, Neg1, and Neg2 are the number of cases where the attribute of interest is positive or negative in the two groups
 Prop1 and Prop2 are the proportions positive in the two groups, where prop = Pos/Neg
 Z and V are calculated according to equations 3.8 and 3.7 (p.33)
 Z(sig) and Z(nsig) are adjusted according to equation 4.8 p. 82
Effect size log(odds ratio) = 1.3863
Maximum number of subjects = 112
Border (significant difference) Z = 3.0889 + 0.484V
Border (no significant difference) Z = 3.0889 + 1.4548V

The transport department noticed that only 20% of teenagers pass their driving test
in the first attempt. It was decided to introduce a training course, but this
is expensive, so the department wishes to go ahead only if this can increase the
pass rate to at least 50%.
We designate α=0.05 and power=0.8, the two proportions 0.5 and 0.2, and
set the ratio of the two groups to 1. The borders as calculated are shown above and to the right.
 Passed  Failed 
Review  Training  Control  Training  Control 
Month 1  4  2  4  7 
Month 2  7  4  8  11 
Month 4  15  6  15  24 
Teenagers were recruited to the trial as they applied for testing, and randomly
allocated to training before the test. A decision to review the result monthly
was made. The results at the end of 3 months (third reviews) are presented in the table
to the left, and the Triangular plot shown to the right.
As the test is 2 sided, two triangles, one for trained>controls, and the other
for controls>trained.
The straight line borders are for use if the data is reviewed after every case,
and the inward Christmas tree like narrowing are border adjustments when
there are more than 1 case added between reviews, the extent of the narrowing
depends on the sample size between reviews.
The relationship between V and Z are shown in the line joining the dots.
 At the first review after 1 month, there were 8 teenagers trained, and 4(50%)
passed the driving test. There were 9 controls and 2 (22%) passed. (V=0.97, Z=1.18). The
Triangular Test plot at this time impinged on the inner border of the lower triangle,
so a conclusion that those trained did not do worse than the control could be made.
However the plot was within the upper triangle, so no conclusion could be made
whether those trained did better, so data collection had to continue.
 At the second review after 2 months, the number trained was 15 and 7 (47%)
passed. There were 15 controls 4(27%) of whom passed. (V=1.74, Z=1.50). The plot remained within
the upper triangle so data collection continues. As the plot was nowhere
near the border, it was decided to improve the power by not reviewing the data
at the 3^{rd} month, rather to do so in 2 month's time at month 4.
 At the third review 4 months after the study began, the number trained
was 30 and 15 (50%) of whom passed the test. There were also 30 controls,
6 (20%) of whom passed. (V=3.41 , Z=4.50). The plot now crosses the adjusted outer border of the
upper triangle, so the study can be terminated, and the conclusion made that
the training course significantly improved the pass rate of the test.
The terminal analysis at the end of the second week showed effect size(Z/sqrt(V))=2.436
p= 0.007. The sample size at that time was 60, less than the 112 maximum
required and short of the 72 cases required for a fixed sample size study.
Data Input / Output
Example
Input : In addition to α, power, and ratio that are common inputs to all models, the following inputs are used
 Number of divisions is the number of cells in the arrays to be compared. Examples are 3 division in some
pain scale (0=no pain, 1=some pain, 2=severe pain) or the 5 division Likert Scale (0=Strongly Disagree, 1=Disagree,
2=Neutral, 3=Agree, and 4=Strongly Agree).
 Expected Proportions is the table of expected proportions with two columns for the two groups, and the number of rows
the number of divisions. Each cell contains the proportion or probability for that division in that group, so that
each column sums to the totality of 1. This defines what the study is designed to detect.
 Theta (θ) is the effect size, and calculated from the table of proportions,
according to equation 3.25 p. 46
 The data : are in 2 columns, separated by spaces or tabs. The number of rows is the multiple of
the number of reviews and number of divisions (rows=reviews x divisions) Each cell contains the number of cases
in that group for that division in that review.
Output : V, Z, and the Z values to reject or accept the null hypothesis are common to all data types, and in
this model are calculated from the following observations
 n1 and n2 are the number of observation units in the two groups
 mean1 and mean2 are the mean values observed in the two groups
 Z and V are calculated according to equations 3.27 and 3.28 (p.47)
 Z(sig) and Z(nsig) are adjusted according to equation 4.8 p. 82
 Tmt  Control 
none  0.3  0.05 
Moderate  0.65  0.8 
Severe  0.05  0.15 
We wish to compare the effect of an analgesics with no analgesics on pain
experienced during a minor surgical procedure. We will measure pain experienced
in a 3 point scale (0=none, 1=mild pain, 2=severe pain).
Effect size θ = 2.1
Maximum number of subjects = 55
Border (significant difference) Z = 2.0419 + 0.7336V
Border (no significant difference) Z = 2.0419 + 2.2007V

We will consider the analgesic effective if the response is as good as or better
than that set out to test is as shown above and to the left. Without analgesics
(column 2, control), we anticipate 5% to have no pain, 80% moderate pain, and 15% with severe
pain. With analgesics (column 1, intervention), we hope 30% will experience no pain, 65% some pain, and
5% severe pain.
The borders as calculated are shown above and to the right.
Review  Pain  Tmt  Control 
week 1  none  2  0 
 moderate  3  5 
 severe  0  2 
week 2  none  3  0 
 moderate  9  10 
 severe  1  3 
week 3  none  8  1 
 moderate  10  12 
 severe  1  5 
We decided to recruit suitable subjects from the surgical list, randomize
participants to receiving or not receiving the analgesics, and obtain their
response at the end of the operation. We planned to review the data on a weekly basis.
The results at the end of 3 weeks (third reviews) are presented in the table
to the left, and the Triangular plot shown to the right.
As the test is 2 sided, two triangles, one for treated>controls, and the other
for control>treated.
The straight line borders are for use if the data is reviewed after every case,
and the inward Christmas tree like narrowing are border adjustments when
there are more than 1 case added between reviews, the extent of the narrowing
depends on the sample size between reviews.
The relationship between V and Z are shown in the line joining the dots.
 At the first review after 1 week, there were 5 treated subjects, 2 (40%)
had no pain and 3 (60%) had moderate pain. There were also 7 control subjects,
none (0%) were pain free, 5 (71%) had moderate pain and 2 (29%) had severe pain. (V=0.6 , Z=1.5).
The resulting plot has crossed the inner border of the upper triangle, and
at this point the conclusion those treated had no greater pain than the control
cases could be drawn. However the plot was still within the borders of the
lower triangle, so no decision could be made as to those treated had less pain.
 At the second review after 2 weeks, there were now 13 treated cases,
3 (23%) were pain free, 9 (69%) moderate pain, and 1 (8%) severe pain. There
were also 13 control cases, none (0%) were pain free, 10 (77%) had moderate pain,
and 3 (23%) severe pain. (V=1.3 , Z=2.1). The Triangular Test plot however remained within the lower
triangle, so more data needed to be collected.
 At the third review after 3 weeks, there were now 19 treated cases,
8 (42%) were pain free, 10 (53%) with moderate pain, and 1 (5%) with severe
pain. There were 18 control cases, 1 (6%) was pain free, 12 (67%) had moderate pain,
and 5 (17%) severe pain. (V=2.3 , Z=4.3). The Triangular Test plot now crosses the outer border
of the lower triangle, so the study can terminate, and the conclusion drawn
that those receiving the analgesics had significantly less pain.
The terminal analysis at the end of the second week showed effect size(Z/sqrt(V))=2.8407
p= 0.002. The sample size at that time was 37, less than the 55 maximum
required but 2 more than the 35 required for a fixed sample size study. This shows
that, although the averaged sample size required for the Triangular Test is less than
that in a fixed sample size study, there are circumstances when the same or greater sample size
will be used.
Also, when the study is reviewed, the Triangular Test plot at the end of the
second week was so close to statistical significance that an argument can be made to
break protocol, and make the third review sooner. Had this happened, a significant
result using a smaller sample size might have been achieved.
Data Input / Output
Example
Input Survival rates were initially used to define survival from cancer
at a specified time after diagnosis or commencement of treatment. The related statistics however are applicable to any
time related events.
Output :
V, Z, and the Z values to reject or accept the null hypothesis are common to all data types, and in
this model are calculated from the following observations
 TObs1 and TObs2 are the number of cases in the two groups that are in the study at that review
 TEvs1 and TEvs2 are the number of cases that survived in the two groups at that review
 Surv1 and Surv2 are the adjusted survival rates in the two groups at that review (taking into consideration of numbers lost to
follow up, numbers not in the study long enough to be included, and the different times events occurred)
 Z and V are calculated according to equations 3.20 and 3.21 (p.42)
 Z(sig) and Z(nsig) are adjusted according to equation 4.8 p. 82
Effect size θ = 0.66
Maximum number of subjects = 277
Border (significant difference) Z = 6.4449 + 0.2324V
Border (no significant difference) Z = 6.4449 + 0.6972V

The current 3 year survival rate for a particular cancer is 30% (0.3). We wish to
compare the survival rate between the current treatment (control) and the use of
a new drug (Tmt). We will accept the new drug to be better if it can improve
the 3 year survival rate to 50% (0.5). We designate α=0.05, power=0.8,
the two survival rates to compare as 0.3 and 0.5, and the number of intervals = 3
for 3 year survival rate. We will allocate equal numbers to the two groups so that the ratio = 1.
The borders as calculated are shown above and to the right.
We decided to recruit suitable subjects for the study. We will perform the first review
3 years after the start of the study, then repeat the review on a yearly basis.
The data entered is as shown in the table to the right. Detailed explanation of the data input and results produced are as follows
  Control  Treated 
  died  survived  died  survived 
Review 1  year 1  3  30  3  20 
 year 2  6  20  4  10 
 year 3  5  11  2  6 
Review 2  year 1  8  40  5  45 
 year 2  12  25  10  32 
 year 3  8  15  9  20 
Review 3  year 1  20  60  8  62 
 year 2  16  40  11  41 
 year 3  15  20  11  25 
 Rows 1, 2, and 3 are 3 years of death and survival at the first review.
 Row 1 : first review, year 1
 In control group, there were a total of 33 cases, 3 died and 30 survived year 1.
 In treatment group, there were a total of 23 cases, 2 died and 20 survived year 1.
 Row 2 : first review year 2
 In control group, of the 30 that survived year 1, 26 died or survived in year 2. Four cases that were either
lost to follow up, or had only been in the study for less than 2 year but had not died, and therefore not counted.
Of these 26, 6 died during year 2, and 20 survived year 2
 In treatment group, of the 20 that survived, 14 died or survived year 2. Six cases were either lost to follow up
or had been in the study less than 2 years but had not died, and therefore not counted. Of these 14, 4 died and 10 survived.
 Row 3 : first review year 3
 In control group, of the 20 that survived years 2, 16 died or survived in year 3. Four cases that were either
loss to follow up or had been in the study less than 3 years but had not died, so were not counted.
Of these 16, 5 died and 11 survived.
 In treatment group, of the 10 that survived year 2, 8 died or survived year 3. Two cases that were either lost
to follow up or had not been in the study for 3 years but had not died, and so were not counted. Of these 8,
2 died and 6 survived.
 At the first review, the calculation is that survival rate in the control group is 0.46, and in the treatment group
0.48, v=5.3 and z=0.7. As the decision line had not been crossed, the study continues
 Rows 4, 5, and 6 are the accumulative 3 years survival figures at the second review. The numbers consists of all the
records of the first review, and what happened to those that survived since, plus all the new cases that
have entered the study.
 Row 4 : second review, year 1
 In control group, the accumulated data now showed 48 cases, 15 new cases added to the 33 that were present
at the first review. Of these 48, 8 died 40 survived. The 8 deaths were 3 from the first review, plus another
5 from the new cases.
 In treatment group, the accumulated data is now 50 cases, 27 new cases added to the 23 that were present in the first review.
Of these 50, 5 died and 45 survived. The 5 deaths included the 3 from the first review, plus another 2 from the new cases.
 Row 5 : second review year 2
 In control group, of the 40 that survived there were 37 that died or survived year 2. These included new cases
that joined the study since, plus some of the old cases that have now died or survived year 2, minus those loss to follow up
or not have been in the study for 2 completed years but had not died, so were not counted. Of these 37, 12
died and 25 survived year 2. The 12 deaths included the 6 who died in year 2 of the first review, plus another
6 new deaths.
 In treatment group, of the 45 that survived year 1, 42 died or survived in year 2.These included new cases
that joined the study since, plus some of the old cases that have now died or survived year 2, minus those
loss to follow up or not have been in the study for 2 completed years but had not died, so were not counted.
Of these 42, 10 died and 32 survived year 2. The 10 deaths included the 4 that died in year 2 during the first
review, plus another 6 new deaths.
 Row 6 : second review year 3
 In control group, of the 25 cases that survived year 2, there were 23 cases that died or survived year 3.
These included new cases that joined the study since, plus some of the old cases that have now died or survived
year 3, minus those loss to follow up or not have been in the study for 3 completed years but had not died,
so were not counted. Of these 23, 8 died and 15 survived. The 8 deaths included the 5 that died during the first
review, plus another 3 new deaths.
 In treatment group, of the 32 that survived year 2, 29 died or survived in year 3. These included new cases
that joined the study since, plus some of the old cases that have now died or survived year 3, minus those
loss to follow up or not have been in the study for 3 completed years but had not died, so were not counted.
Of these 29, 9 died and 20 survived year 3. The 9 deaths included the 2 that died in the first review, plus
another 7 new deaths.
 At the second review, the calculation is that survival rate in the control group is 0.37, and in the treatment
group 0.47, v=12.8 and z=4.3.
 Rows 7, 8, and 9 are the 3 years survival figures at the third review. The numbers consists of all the records of
the second review, and what happened to those that survived since, plus all the new cases that have entered the study.
 Row 7 : third review, year 1
 In control group, the accumulated data now showed 80 cases, 32 new cases added to the 48 that were present at
the second review. Of these 80, 20 died 60 survived. The 20 deaths were 8 from the first and second reviews,
plus another 12 from new cases.
 In treatment group, the accumulated data is now 70 cases, 20 new cases added to the 50 present in the second review.
Of these 8 died and and 62 survived. The 8 deaths included the 5 from the first and second reviews, plus another 3
from the new cases.
 Row 8 : third review year 2
 In control group, of the 60 that survived there were there were 56 that died or survived year 2. These included new cases
that joined the study since, plus some of the old cases that have now died or survived year 2, minus those loss to follow up
or not have been in the study for 2 completed years but had not died, so were not counted. Of these 56, 16 died
and 40 survived year 2. The 16 deaths included the 12 who died in year 2 of the first and second reviews, plus another
4 new deaths.
 In treatment group, of the 62 that survived year 1, 52 died or survived in year 2. These included new cases
that joined the study since, plus some of the old cases that have now died or survived year 2, minus those loss to follow up
or not have been in the study for 2 completed years but had not died, so were not counted. Of these 52,
11 died and 41 survived year 2. The 11 deaths included the 10 that died in year 2 during the first and second reviews,
plus another 1 new death.
 Row 9 : third review year 3
 In control group, of the 40 cases that survived year 2, there were 35 cases that died or survived year 3.
These included new cases that joined the study since, plus some of the old cases that have now died or survived
year 3, minus those loss to follow up
or not have been in the study for 3 completed years but had not died, so were not counted. Of these 35,
15 died and 20 survived. The 15 deaths included the 8 that died during the first and second reviews, plus another 7
new deaths.
 In treatment group, of the 41 that survived year 2, 36 died or survived in year 3. These included new cases
that joined the study since, plus some of the old cases that have now died or survived year 3, minus those loss
to follow up or not have been in the study for 3 completed years but had not died, so were not counted. Of these 36,
11 died and 25 survived year 3. The 11 deaths included the 9 that died in the first and second reviews, plus another
2 new deaths.
 At the third review, the calculation is that survival rate in the control group is 0.31, and in the treatment group 0.48, v=20.0
and z=10.6. At this point, the sequential line crosses the significance line, and the study stops with the rejection of the
Null Hypothesis.
 Survival rate year 3 
 Control  Tmt 
Review1  0.4808  0.4658 
Review2  0.3672  0.4729 
Review3  0.3061  0.4849 
The survival rates in the 3 review periods are shown in the table to the right, and the sequential plots are as
shown in the diagram to the left.
As the test is 2 sided, two triangles, one for treated>controls, and the other
for control>treated.
The straight line borders are for use if the data is reviewed after every case,
and the inward Christmas tree like narrowing are border adjustments when
there are more than 1 case added between reviews, the extent of the narrowing
depends on the sample size between reviews.
The relationship between V and Z are shown in the line joining the dots.
 At the first review after 3 years, the 3 year survival rates were 48%
and 47% for the control and treatment groups. (V=5.3, Z=0.7). The Triangular plot has not crossed
any of the borders, and the only decision possible is to continue with the study.
 At the second review 1 year later, the 3 year survival rates were 37%
and 47% for the control and treatment groups. (V=12.8, Z=4.3). The Triangular plot has
now cross the inner border of the upper triangle, so that the decision that group 1 survival rate
is not significantly greater than group 2 can be made. However, no decision can as yet be made
whether survival rate in group 2 is greater than group 1. The decision is to continue with the study.
 At the third review 1 year later (5 years after the start of the study),
the 3 year survival rates were 31% and 49% for the control and treatment groups. (V=20.0, Z=10.6).
The Triangular plot has now reached the outer border of the lower triangle, so the study
can be terminated, and the conclusion that the new treatment (group 2) offers a higher
3 year survival rate than control (group 1) can be made.
The terminal analysis at the end of the second week showed effect size(Z/sqrt(V))=2.375
p= 0.0088. The sample size is difficult to state, as subjects entered and remained in the
study for different periods of time. However there were 80 control and 70 treated
in the study, a sample size of 150, in excess of the 71 calculated for a fixed
sample size study.
Whitehead John (1992). The Design and Analysis of Sequential Clinical Trials
(Revised 2nd. Edition) . John Wiley & Sons Ltd., Chichester, ISBN 0 47197550 8.
