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Presentation prepared by Drug Regulations – a not for
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the latest in Pharmaceuticals.
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 Case Study formulation details which will be
used in the presentation
 Definitions
 Terminology
 Full factorial designs
◦ m-factor ANOVA
 Fractional factorial designs
 Multi-factorial designs
Component Function Unit
( mg/tablet)
Unit
( % W/W)
Acetriptan, USP Active 20 10
Lactose Monohydrate, NF Filler 64-86 32-43
Microcrystalline Cellulose
(MCC), NF
Filler 72-92 36-46
Croscarmellose Sodium
(CCS), NF
Disintegrant 2-10 1-5
Magnesium Stearate, NF* Lubricant 2-6 1-3
Talc, NF Glidant/Lubricant 1-10 0.5-5
Total tablet weight 200 100
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Compression of the lubricated granules into thablets.
Lubrication of the blend with lubricants
Milling of the ribbons from roller compaction stage
Roller Compaction
Roller Compaction of blend from earlier step
Manufacturing Process
Blending prior to roller compaction Mixing of Active Ingredient & Excipients
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 In this case study we will evaluate the blending process for blend
uniformity by analyzing the content of active ingredient : Acetriptan.
 Based on the knowledge of similar formulations and processes it was
decided to study following factors
 Response Variable : Concentration of Acetriptan in blend. ( Blend
Uniformity)
Factor Input Variables High Level Low Level
A Mixing Time 5 min (+) 3 Min (-)
B Mag. Stearate
concentration
2.0 % (+) 1.0 % (-)
C Talc
concentration
2 % (+) 1 % (-)
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 Experiment:
 A test or series of tests where the experimenter
makes purposeful changes to input variables of
a process or system so that we can observe or
identify the reasons for changes in the output
responses.
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 Design of Experiment:
 Design of Experiments: is concerned with the
planning and conduct of experiments to analyze
the resulting data so that we obtain valid and
objective conclusions.
 Determine which variables are the most influential in
a process or system
 Determine where to set the inputs so the output is
always near the desired state
 Determine where to set the inputs so the output
variability is minimized
 Determine where to set the inputs so the influence of
uncontrollable factors is minimized (robust design)
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 Treatments are the different procedures we want to
compare.
◦ Treatment : Lubrication time of 3 and 5 minutes , Magnesium
Concentration : 1 % and 2 % , Talc Concentration : 1 % and 2 %.
 Experimental units are the things to which we apply the
treatments.
◦ Blend to be lubricated
 Responses are outcomes that we observe after applying a
treatment to an experimental unit.
◦ Assay / Content uniformity of the blend
 Interaction Effect
◦ Effect of one input factor depends on level of another input factor
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 Measurement units (or response units) are
the actual objects on which the response is
measured.
 These may differ from the experimental units.
 In the lubrication example
◦ Treatment is : Lubrication time of 3 min and 5 min
◦ Experimental Unit : The lubrication blend to which
the treatment is applied.
◦ Response : Blend uniformity
◦ Response Unit : Per cent of Acetriptan content
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 Distinction between experimental units and
measurement units.
 Consider an educational study, where six classrooms of
25 first graders each are assigned at random to two
different reading programs, with all the first graders
evaluated via a common reading exam at the end of the
school year.
 Are there
 six experimental units (the classrooms) or
 150 (the students)?
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 One way to determine the experimental unit is via the consideration
that an experimental unit should be able to receive any treatment.
 Thus if students were the experimental units, we could see more
than one reading program in any treatment.
 However, the nature of the experiment makes it clear that all the
students in the classroom receive the same program, so the
classroom as a whole is the experimental unit.
 We don’t measure how a classroom reads, though; we measure how
students read. Thus students are the measurement units for this
experiment.
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 Blinding occurs when the evaluators of a
response do not know which treatment was given
to which unit.
 Blinding helps prevent bias in the evaluation,
even unconscious bias from well-intentioned
evaluators.
 Double blinding occurs when both the evaluators
of the response and the (human subject)
experimental units do not know the assignment
of treatments to units.
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 Control has several different uses in design.
First, an experiment is controlled because we
as experimenters assign treatments to
experimental units. Otherwise, we would have
an observational study.
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 Second, a control treatment is a “standard”
treatment that is used as a baseline or basis of
comparison for the other treatments.
 This control treatment might be the treatment in
common use, or
 It might be a null treatment (no treatment at all).
 For example, a study of new pain killing drugs
could use a standard pain killer as a control
treatment.
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 Factors combine to form treatments.
 For example, in the experiment to study lubrication
process following 3 factors can be evaluated.
 Mixing Time : 3 min and 5 min
 Magnesium Stearate Concentration : 1.0 % and 2.0 %
 Talc Concentration : 1 % and 2 %
 One treatment is a combination of mixing time ,
Magnesium Stearate Concentration and Talc
Concentration.
 We can vary each factor separately.
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 We can vary each factor separately.
 Individual settings for each factor are called
levels of the factor.
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Factor High Level Low Level
A Mixing Time 5 min (+) 3 Min (-)
B Mag. Stearate
concentration
2.0 % (+) 1.0 % (-)
C Talc
concentration
2 % (+) 1 % (-)
 Confounding occurs when the effect of one factor or treatment cannot
be distinguished from that of another factor or treatment.
 The two factors or treatments are said to be confounded.
 Except in very special circumstances, confounding should be avoided.
 Consider using a concentration of Mag. Stearate of 1.0 % with 3 min of
mixing and concentration of 2.0 % with 5 min of mixing.
 In this experiment, we cannot distinguish the effect of mixing from the
effect of Mag. Stearate concentration on blend uniformity.
 Mixing time effect and Mag. Stearate concentration effect are
confounded.
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 One Factor at a Time
◦ Pros : Straight-forward, easily understood
◦ Cons: Impossible to address interactions
◦ Tends to “over collect” data, not efficient sample
sizes
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◦ Case Study Example :
◦ Effect of Mixing Time (A) on Blend Uniformity :
 Keep level of other factors B & C : Concentration of Magnesium
Stearate and Talc constant and vary the Factor A : Mixing time
◦ Effect of Talc Concentration on Blend Uniformity
 Keep Magnesium Stearate concentration and the mixing time
constant and change the concentration of Talc
◦ Effect of Magnesium Concentration on Blend Uniformity
 Keep Talc concentration constant mixing time constant and vary
the concentration of Magnesium Stearate.
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 Comparative designs to:
 Choose between alternatives, with narrow scope, suitable for
an initial comparison
 Completely Randomized Designs
 Choose between alternatives, with broad scope, suitable for a
confirmatory comparison
 Randomized Block Design
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 Here we consider completely randomized designs that have one primary factor.
 Mixing time in the Lubrication process
 The experiment compares the values of a response variable based on the different
levels of that primary factor.
 Response Variable : Assay in CU analysis
 Different Levels of Primary Factor : Mixing time of 3 , 5 and 7 minutes
 For completely randomized designs, the levels of the primary factor are randomly
assigned to the experimental units.
 By randomization, we mean that the run sequence of the experimental units is
determined randomly.
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 For this example, there are 3 levels of the primary factor with each level to be run 2 times, then
there are 6 factorial possible run sequences (or 6! ways to order the experimental trials).
 Because of the replication, the number of unique orderings is 90 (since 90 = 6!/(2!*2!*2!)).
 An example of an unrandomized design would be to always run 2 replications for the first level,
then 2 for the second level, and finally 2 for the third level.
 To randomize the runs, one way would be to put 6 slips of paper in a box with 2 having level 1, 2
having level 2, and 2 having level 3.
 Before each run, one of the slips would be drawn blindly from the box and the level selected would
be used for the next run of the experiment.
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 In practice, the randomization is typically performed by a computer program.
 However, the randomization can also be generated from random number tables or by
some physical mechanism (e.g., drawing the slips of paper).
 All completely randomized designs with one primary factor are defined by 3 numbers:
 k = number of factors (= 1 for these designs)
L = number of levels
n = number of replications
 and the total sample size (number of runs) is N = k x L x n.
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 A typical example of a completely randomized design is the
following:
 k = 1 factor (X1)
L = 4 levels of that single factor (called "1", "2", "3", and "4")
n = 3 replications per level
N = 4 levels * 3 replications per level = 12 runs
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X1
3
1
4
2
2
1
3
4
1
2
4
3
 A completely randomized design may look like
this
 Note that in this example there are 12!/(3!*3!*3!*3!) = 369,600 ways to run the experiment, all
equally likely to be picked by a randomization procedure.
 The model for the response is
 Yi,j =μ+ Ti +random error
 with
 Yi,j being any observation for which X1 = i
 (i and j denote the level of the factor and the replication within the level of the factor, respectively)
 μ (or mu) is the general location parameter
Ti is the effect of having treatment level i
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Estimate
for μ:
Y¯ = the average of
all the data
Estimate for Ti: Y¯i−Y¯with Y¯i = average of
all Y for which X1 = i.
Statistical tests for levels of X1 is One way ANOVA
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 Separates total variation observed in a set of
measurements into:
1. Variation within one system
 Due to random measurement errors
2. Variation between systems
 ( Different Mixing Time , Or Different concentration of
Magnesium Stearate or Different concentration of Talc)
 Due to real differences + random error
 Is variation(2) statistically > variation(1)?
 One-factor experimental design
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Variation Alternatives Error Total
Sum of squares SSA SSE SST
Deg freedom k -1 k(n -1) kn -1
Mean square sa
2
= SSA (k -1) se
2
= SSE [k(n -1)]
Computed F sa
2
se
2
Tabulated F F[1-a;(k-1),k(n-1)]
 Blocking
◦ Designed to improve precision of comparisons
◦ Used to reduce or eliminate nuisance factors
 Nuisance Factor
◦ A nuisance factor is a “design factor that probably
has an effect on the response but we are not
interested in that effect”
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 Nuisance factors are those that may affect the measured result, but are not
of primary interest.
 For example, in applying a treatment, nuisance factors might be the specific
operator who prepared the treatment, the time of day the experiment was
run, and the room temperature.
 All experiments have nuisance factors.
 The experimenter will typically need to spend some time deciding which
nuisance factors are important enough to keep track of or control, if
possible, during the experiment.
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 Nuisance Factors, Types ⇒ Cures
◦ Known and controllable ⇒ Use blocking to
systematically eliminate the effect
◦ Known but uncontrollable ⇒ If it can be measured,
use Analysis of Covariance (ANCOVA)
◦ Unknown and uncontrollable ⇒ Randomization is the
insurance
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 When we can control nuisance factors, an important technique known as
blocking can be used to reduce or eliminate the contribution to
experimental error contributed by nuisance factors.
 The basic concept is to create homogeneous blocks in which the nuisance
factors are held constant and the factor of interest is allowed to vary.
 Within blocks, it is possible to assess the effect of different levels of the
factor of interest without having to worry about variations due to changes
of the block factors, which are accounted for in the analysis.
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 A nuisance factor is used as a blocking factor if
every level of the primary factor occurs the
same number of times with each level of the
nuisance factor.
 The analysis of the experiment will focus on
the effect of varying levels of the primary
factor within each block of the experiment.
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 The general rule is:
 "Block what you can, randomize what you cannot."
 Blocking is used to remove the effects of a few of the
most important nuisance variables. Randomization is
then used to reduce the contaminating effects of the
remaining nuisance variables.
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 One useful way to look at a randomized block
experiment is to consider it as a collection of
completely randomized experiments, each run
within one of the blocks of the total
experiment.
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 Case Study
 Case Study Formulation : Lubrication Experiments
 Assume that two batches of Active Ingredients are used for the experiments.
 Though each batch of active ingredient will be tested and meet specifications
before being used in the experiments , there could be some difference due to
batch to batch variability.
 This variability may be small and we may not be interested in determining its
effect on the response factor.
 Therefore different batches of Active ingredients can be considered as a nuisance
factor.
 Completely randomizing the experiments may result in potential serious
problems.
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 Case Study
 It is quite possible that Active ingredients from different
batches may have slightly different surface & flow properties ,
 This may affect the distribution of the active in pre-roller
compaction blending process which in turn may affect the
uniformity of the active in roller compresses granules and
therefore in the lubricated blend.
 Therefore if we do not obtain a uniform blend in the
lubrication stage we may not know whether it is due the
lubrication process or whether due to the variation from the
Active ingredient.
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 Cased Study
 The experimental error will reflect both random error and variability between
different batches of the active ingredient.
 We would like to make the experimental error as small as possible i.e. remove the
variability between different batches of active ingredients from the experimental
error.
 A design that would accomplish this requires conducting all experiments with
each of the Active Ingredient batches.
 In this example , each batch of Active ingredient would form a block.
 This statistical design is called “ Randomized complete Block Design”
 The word complete indicates that each block –Active Ingredient Batch contains all
the treatments.
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 Case Study Formulation : Lubrication Experiments
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Block 1 :
Supplier A
Block 2:
Supplier B
Experiment No. from DOE
2 6
4 3
3 5
7 7
8 2
1 4
5 1
 Case Study
 By using this design, the blocks or one active Ingredient batch is
used to compare other factors ( Mixing speed , Concentration of
Magnesium Stearate and Talc.)
 This design strategy improves the accuracy of the comparison
among the three factors by eliminating the variability among the
active ingredient batches.
 Within a block the order in which each experiment is conducted is
based on randomization.
 The statics of this design is similar to the paired “t-test”
 Randomized complete block design is a generalization of that
concept. ( RCBD)
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 RCBD is one of the most widely used experimental
designs.
 Similar but Machinery or Unit of test equipment are
often different in their operating characteristics
and would be a typical blocking factors.
 Batches of Raw Materials , People and time are
common nuisance source of variability.
 These can be systematically controlled through
blocking.
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 Blocking
 Blocking can also be used in situations which do not involve
nuisance factors. e.g.
 In a chemical reaction catalyst feed rate is to be studied to
determine effect on viscosity of polymer.
 There are several factors such as raw material source ,temperature ,
operator raw material purity are very difficult to control in full scale
process.
 Catalyst feed rate can be tested in blocks where each block consist
of combination of some uncontrollable factors.
 Blocks can be used to test the robustness of the process variable the
feed rate to conditions which can not be easily controlled.
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Source of
Variation
Sum of
Squares
Degrees of
Freedom
Mean Square F 0
Treatments SS Treatments a-1 SS Treatments
---------------
( a-1)
MS Treatments
---------------
MS E
Block SS Blocks b-1 SS Blocks
-----------
( b-1)
Error SS E (a-1) (b-1) SS E
--------------
( a-1) (b-1)
Total SS T N-1
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 Latin square designs, and the related Graeco-Latin square and
Hyper-Graeco-Latin square designs, are a special type of
comparative design.
 There is a single factor of primary interest, typically called the
treatment factor, and several nuisance factors.
 For Latin square designs there are 2 nuisance factors, for Graeco-
Latin square designs there are 3 nuisance factors, and for Hyper-
Graeco-Latin square designs there are 4 nuisance factors.
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 The nuisance factors are used as blocking variables.
1. For Latin square designs, the 2 nuisance factors are divided into a tabular grid with
the property that each row and each column receive each treatment exactly once.
2. As with the Latin square design, a Graeco-Latin square design is a kxk tabular grid
in which k is the number of levels of the treatment factor. However, it uses 3
blocking variables instead of the 2 used by the standard Latin square design.
3. A Hyper-Graeco-Latin square design is also a kxk tabular grid with k denoting the
number of levels of the treatment factor. However, it uses 4 blocking variables
instead of the 2 used by the standard Latin square design.
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 The advantages of Latin square designs are:
1. They handle the case when we have several nuisance
factors and we either cannot combine them into a
single factor or we wish to keep them separate.
2. They allow experiments with a relatively small
number of runs.
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 The disadvantages are:
1. The number of levels of each blocking variable must
equal the number of levels of the treatment factor.
2. The Latin square model assumes that there are no
interactions between the blocking variables or between
the treatment variable and the blocking variable.
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 Note that Latin square designs are equivalent to
specific fractional factorial designs
 (e.g., the 4x4 Latin square design is equivalent to a
43-1 Fractional Design design).
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 Designs for Latin squares with 3-, 4-, and 5-level
factors are given next. These designs show what the
treatment combinations should be for each run.
 When using any of these designs, be sure to
randomize the treatment units and trial order, as
much as the design allows.
 For example, one recommendation is that a Latin
square design be randomly selected from those
available, then randomize the run order.
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 Screening designs to identify which factors/effects
are important
o when you have 2 - 4 factors and can perform a full factorial
o when you have more than 3 factors and want to begin with
as small a design as possible
o when you have some qualitative factors, or you have some
quantitative factors that are known to have a non-
monotonic effect
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 You can also restrict the term screening design to
the case where you are trying to extract the most
important factors from a large (say > 5) list of initial
factors (usually a fractional factorial design).
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 The following is an example of a full factorial design with 3 factors that also
illustrates replication , randomization and Center points
 Suppose that we wish to improve the yield of a polishing operation. The three
inputs (factors) that are considered important to the operation are Speed (X1),
Feed (X2), and Depth (X3). We want to ascertain the relative importance of each of
these factors on Yield (Y).
 Speed, Feed and Depth can all be varied continuously along their respective
scales, from a low to a high setting. Yield is observed to vary smoothly when
progressive changes are made to the inputs. This leads us to believe that the
ultimate response surface for Y will be smooth.
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 High ( +) , Low (-) and Standard ( 0) Setting for a Polishing Operation
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 We want to try various combinations of these
settings so as to establish the best way to run the
polisher.
 There are eight different ways of combining high and
low settings of Speed, Feed, and Depth.
 These eight are shown at the corners of the following
diagram.
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 A 23 Full Factorial Design with runs in Standard
order
 Running the entire design more than once makes for easier data analysis
because, for each run (i.e., `corner of the design box') we obtain an
average value of the response as well as some idea about the dispersion
(variability, consistency) of the response at that setting.
 One of the usual analysis assumptions is that the response dispersion is
uniform across the experimental space.
 The technical term is `homogeneity of variance'. Replication allows us to
check this assumption and possibly find the setting combinations that give
inconsistent yields, allowing us to avoid that area of the factor space.
 We now have constructed a design table for a two-level full factorial in
three factors, replicated twice.
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 A 23 Full Factorial Design replicated twice & with runs in
Standard order
 If we now ran the design as is, in the order shown,
we would have two deficiencies, namely:
1. No randomization, and
2. No Center points.
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 The more freely one can randomize experimental runs, the more insurance one has against
extraneous factors possibly affecting the results, and hence perhaps wasting our experimental time
and effort.
 For example, consider the `Depth' column: the settings of Depth, in standard order, follow a `four
low, four high, four low, four high' pattern.
 Suppose now that four settings are run in the day and four at night, and that (unknown to the
experimenter) ambient temperature in the polishing shop affects Yield.
 We would run the experiment over two days and two nights and conclude that Depth influenced
Yield, when in fact ambient temperature was the significant influence.
 So the moral is: Randomize experimental runs as much as possible.
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 A 23 Full Factorial Design replicated twice & with runs in Randomized order
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 This design would be
improved by adding
at least 3 Center
point runs placed at
the beginning,
middle and end of
the experiment. The
final design matrix is
shown alongside.
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 We often need to eliminate the influence of extraneous factors when running an experiment. We
do this by "blocking".
 Previously, blocking was introduced when Randomized Block Design were discussed.
 There we were concerned with one factor in the presence of one of more nuisance factors.
 In this section we look at a general approach that enables us to divide 2-level factorial experiments
into blocks.
 For example, assume we anticipate predictable shifts will occur while an experiment is being run.
 This might happen when one has to change to a new batch of raw materials halfway through the
experiment.
 The effect of the change in raw materials is well known, and we want to eliminate its influence on
the subsequent data analysis.
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 In this case, we need to divide our experiment into two halves (2 blocks),
one with the first raw material batch and the other with the new batch.
 The division has to balance out the effect of the materials change in such
a way as to eliminate its influence on the analysis, and we do this by
blocking.
 An eight-run 23 full factorial has to be blocked into two groups of four
runs each.
 Consider the design `box' for the 23 full factorial. Blocking can be achieved
by assigning the first block to the dark-shaded corners and the second
block to the open circle corners.
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Blocking Scheme for a 23 Using Alternate Corners
Three-factor
 This works because we are in fact assigning the `estimation' of the (unwanted) blocking
effect to the three-factor interaction, and because of the special property of two-level
designs called orthogonality.
 That is, the three-factor interaction is "confounded" with the block effect as will be seen
shortly.
 Orthogonality guarantees that we can always estimate the effect of one factor or interaction
clear of any influence due to any other factor or interaction.
 Orthogonality is a very desirable property in DOE and this is a major reason why two-level
factorials are so popular and successful.
 Formally, consider the 23 design table with the three-factor interaction column added.
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 Two Blocks for a 23 Design
 Rows that have a `-1' in the three-factor interaction column are assigned to `Block I' (rows 1,
4, 6, 7), while the other rows are assigned to `Block II' (rows 2, 3, 5, 8).
 Note that the Block I rows are the open circle corners of the design `box' above; Block II are
dark-shaded corners.
 The general rule for blocking is: use one or a combination of high-order interaction columns
to construct blocks.
 This gives us a formal way of blocking complex designs.
 Apart from simple cases in which you can design your own blocks, your statistical/DOE
software will do the blocking if asked, but you do need to understand the principle behind it.
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 The price you pay for blocking by using high-order interaction columns is that you
can no longer distinguish the high-order interaction(s) from the blocking effect -
they have been “Confounded”`,' or “ aliased”.
 In fact, the blocking effect is now the sum of the blocking effect and the high-
order interaction effect.
 This is fine as long as our assumption about negligible high-order interactions
holds true, which it usually does.
 Within a block, center point runs are assigned as if the block were a separate
experiment - which in a sense it is Randamization takes place within a block as it
would for any non-blocked DOE.
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 The price you pay for blocking by using high-order interaction columns is that you
can no longer distinguish the high-order interaction(s) from the blocking effect -
they have been “Confounded”`,' or “ aliased”.
 In fact, the blocking effect is now the sum of the blocking effect and the high-
order interaction effect.
 This is fine as long as our assumption about negligible high-order interactions
holds true, which it usually does.
 Within a block, center point runs are assigned as if the block were a separate
experiment - which in a sense it is Randamization takes place within a block as it
would for any non-blocked DOE.
www.drugragulations.org 83
 The ASQC (1983) Glossary & Tables for Statistical Quality Control defines fractional
factorial design in the following way:
 "A factorial experiment in which only an adequately chosen fraction of the
treatment combinations required for the complete factorial experiment is selected
to be run."
 Even if the number of factors, k, in a design is small, the 2k runs specified for a full
factorial can quickly become very large.
 For example, 26 = 64 runs is for a two-level, full factorial design with six factors. To
this design we need to add a good number of centerpoint runs and we can thus
quickly run up a very large resource requirement for runs with only a modest
number of factors.
www.drugragulations.org 84
 The solution to this problem is to use only a fraction of the runs specified by the
full factorial design.
 Which runs to make and which to leave out is the subject of interest here.
 In general, we pick a fraction such as ½, ¼, etc. of the runs called for by the full
factorial.
 We use various strategies that ensure an appropriate choice of runs.
 The following slides will show you how to choose an appropriate fraction of a full
factorial design to suit your purpose at hand. Properly chosen fractional factorial
designs for 2-level experiments have the desirable properties of being
both balanced and orthogonal.
www.drugragulations.org 85
 Consider the two-level, full factorial design for three factors, namely the 23 design.
 This implies eight runs (not counting replications or center points). Graphically, as
shown earlier, we can represent the 23 design by the following cube:
 A 23 Full Factorial Design;
Factors X1, X2, X3. (The arrows show the direction of increase of the factors. Numbers
`1' through '8' at the corners of the design cube reference the 'Standard Order' of runs)
www.drugragulations.org 86
 A 23 Two-level, Full Factorial Design Table Showing Runs in
'Standard Order,' Plus Observations
www.drugragulations.org 87
X1 X2 X3 Y
1 -1 -1 -1 y1 = 33
2 +1 -1 -1 y2 = 63
3 -1 +1 -1 y3 = 41
4 +1 +1 -1 Y4 = 57
5 -1 -1 +1 y5 = 57
6 +1 -1 +1 y6 = 51
7 -1 +1 +1 y7 = 59
8 +1 +1 +1 y8 = 53
 The right-most column of the table lists 'y1' through 'y8' to indicate the responses
measured for the experimental runs when listed in standard order.
 For example, `y1' is the response (i.e., output) observed when the three factors
were all run at their 'low' setting. The numbers entered in the 'y' column will be
used to illustrate calculations of effects.
 From the entries in the table we are able to compute all 'effects' such as main
effects, first-order 'interaction' effects, etc.
 For example, to compute the main effect estimate 'c1' of factor X1, we compute the
average response at all runs withX1 at the 'high' setting, namely
(1/4)(y2 + y4 + y6 + y8), minus the average response of all runs with X1 set at 'low,'
namely (1/4)(y1 + y3 + y5 + y7). That is,
www.drugragulations.org 88
 For example, to compute the main effect estimate 'c1' of factor X1, we compute the
average response at all runs withX1 at the 'high' setting, namely
(1/4)(y2 + y4 + y6 + y8), minus the average response of all runs with X1 set at 'low,'
namely (1/4)(y1 + y3 + y5 + y7). That is,
www.drugragulations.org 89
c1 = (1/4) (y2 + y4 + y6 + y8) -
(1/4) (y1 + y3 + y5 + y7)
= (1/4) (63+57+51+53 ) -
(1/4) (33+41+57+59)
= 8.5
 Suppose, however, that we only have enough resources to do four runs. Is it still
possible to estimate the main effect for X1? Or any other main effect? The answer
is yes, and there are even different choices of the four runs that will accomplish
this.
 For example, suppose we select only the four light (unshaded) corners of the
design cube. Using these four runs (1, 4, 6 and 7), we can still compute c1 as
follows:
www.drugragulations.org 90
c1 = (1/2) (y4 + y6) - (1/2) (y1 + y7)
= (1/2) (57+51) - (1/2) (33+59)
= 8
www.drugragulations.org 91
Simarly, we would compute c2, the effect due to X2, as
c2 = (1/2) (y4 + y7) - (1/2) (y1 + y6)
= (1/2) (57+59) - (1/2) (33+51)
= 16
Finally, the computation of c3 for the effect due to X3 would be
c3 = (1/2) (y6 + y7) - (1/2) (y1 + y4)
= (1/2) (51+59) - (1/2) (33+57)
= 10
www.drugragulations.org 92
 We could also have used the four dark (shaded) corners of the
design cube for our runs and obtained similiar, but slightly
different, estimates for the main effects.
 In either case, we would have used half the number of runs
that the full factorial requires.
 The half fraction we used is a new design written as 23-1. Note
that 23-1 = 23/2 = 22 = 4, which is the number of runs in this
half-fraction design
www.drugragulations.org 93
94
DoE : number of experiments
D O E : Number of experiments
95
 With n experiments, you can calculate the
coefficients for n-1 factors and interactions
◦ For 2 factors, Factorial design requires 22 = 4
experiments, you can calculate the coefficients for
3 factors and interactions : A, B and interaction AB
(green color), it’s not interesting to erase 1
experiment and loose informations on possible
interaction between A and B.
96
 With n experiments, you can calculate the
coefficients for n-1 factors and interactions
◦ For 3 factors, Factorial design requires 23 = 8
experiments, you can calculate the coefficients for
7 factors and interactions : A, B, C and interactions
AB, AC, BC and ABC. Using semi Factorial design (22
= 4 experiments) informations on possible
interaction are also lost (red color).
97
 With n experiments, you can calculate the
coefficients for n-1 factors and interactions
◦ For more than 3 factors, the number of experiments
should be limited to the number of factors tested + the
number of single interactions. I never found (up to now)
significant triple interactions (ABC), Why loose time,
money for a large number of such interactions ABC…F,
ABD…F, ACD…F, AED…F…(yellow color)
98
Trial A B C
1 Lo Lo Lo
2 Lo Lo Hi
3 Lo Hi Lo
4 Lo Hi Hi
5 Hi Lo Lo
6 Hi Lo Hi
7 Hi Hi Lo
8 Hi Hi Hi
www.drugragulations.org
 Replication
◦ Independent Repeat run of each factor combination
◦ Permits estimation of experimental error
◦ Estimation of Error : Determines if observed
differences in data are statistically different.
◦ Permits more precise estimates of the sample
statistics
Replication is not repeated measurements.
www.drugragulations.org 99
 Replication
◦ Replication is not repeated measurements.
 Case Study Formulation : Acetriptan Tablets
◦ In this example we are studying the effect of lubrication time on
blend uniformity by assaying the blend.
◦ Blend analysis at lubrication stage after 3 and 5 minutes of mixing
◦ Samples are taken at each time interval from 6 different locations
◦ At each location 2 samples are taken
◦ For replicating this experiment you need have another experiment
in which another bend is mixed for 5 minutes , samples are
withdrawn at 3 and 5 minutes time interval for analysis.
www.drugragulations.org
10
0
 Randomization
◦ Corner stone of statistical experiments by design.
◦ Insures that observations or errors are more likely
to be independent
◦ Helps “average out” effects of extraneous factors
◦ Special designs when complete randomization not
feasible
www.drugragulations.org
10
1
 Randomization means
◦ Allocation of experimental material and
◦ The order in which the individual experimental runs are performed
◦ ARE RNADOMLY DETERMIED.
 Case Study Formulation
◦ For the formulation development if a 23 experiment is being done in
replicate , this will involve 8 experiments to be done in replicate i.e 16
experiments.
◦ Randomization means randomizing the experiments as well as
randomizing the material used for experiments i.e. different batches of
actives and excipient materials. ( If different batches are used)
www.drugragulations.org
10
2
 Randomization
◦ Computer programs are used in selecting and constructing experimental designs.
◦ These programs present the experimental run in random order.
◦ This is created by using a random number generator
◦ Even in such cases it is essential to assign
 Experimental material
 Operators
 Gauges
 Measurement devices
◦ This allocation also needs to be randomized if each can not be kept constant.
◦ If complete randomization is not possible there are statistical methods to deal with
restriction on randomization.
www.drugragulations.org
10
3
 A good experimental design must
◦ Avoid systematic error
◦ Be precise
◦ Allow estimation of error
◦ Have broad validity.
www.drugragulations.org
10
4
 Comparative experiments estimate
differences in response between treatments.
 If experiment has systematic error, then
comparisons will be biased, no matter how
precise our measurements are or how many
experimental units we use.
www.drugragulations.org
10
5
 Case Study Example : Consider following situations
 The samples for blend uniformity are analyzed as follows:
www.drugragulations.org
10
6
Case Study
Number
All Experiments
with 3 min
mixing time
All Experiments
with 5 min
mixing time
1 Spectrophotome
ter
Perkin Elmer
Model X : Unit 1
Perkin Elmer
Model X : Unit 2
2 Analyst Analyst A Analyst B
3 Brand of filter
used during
filtration of final
solution
Type X
Brand A
Type X
Brand B
 We will not know if there is any difference in blend
uniformity whether it is due to different mixing times
( 3 & 5 min) , or the difference is it is due to
◦ Case study 1 : Use of different Spectrophotometers
 Differences in calibration
◦ Case study 2 : Use of different analyst
 Expertise level of analyst A and Analyst B could be different
◦ Case study 3: Use of different brand of filters
 Extraction of “leachables” may be different for brand A and
brand B
www.drugragulations.org
10
7
 Experimental Error is the random variation present in all
experimental results.
 Different experimental units will give different responses to
the same treatment.
 It is often true that applying the same treatment over and over
again to the same unit will result in different responses in
different trials.
 Experimental error does not refer to conducting the wrong
experiment or dropping test tubes.
www.drugragulations.org
10
8
 Even without systematic error, there will be random error in the
responses.
 This will lead to random error in the treatment comparisons.
 Experiments designed to increase precision are precise when this
random error in “treatment comparisons” is small.
 Precision depends on the
◦ Size of the random errors in the “responses” ( ie assay value
◦ The number of experimental units used, and
◦ The experimental design used.
 There are several designs to improve precision.
www.drugragulations.org
10
9
 Experiments must be designed so that we
have an estimate of the size of the random
error.
 This permits statistical inference:
◦ Confidence intervals or
◦ Tests of significance.
 We cannot do inference without an estimate
error.
www.drugragulations.org
11
0
 An experiment may have several randomized
features in addition to the assignment of
treatments to units.
 Randomization is one of the most important
elements of a well-designed experiment.
www.drugragulations.org
11
1
 We defined confounding as occurring when
the effect of one factor or treatment cannot
be distinguished from that of another factor
or treatment.
 Randomization helps prevent confounding.
 Let’s start by looking at the trouble that can
happen when we don’t randomize.
www.drugragulations.org
11
2
 Consider a new drug treatment for coronary artery
disease.
 We wish to compare this drug treatment with bypass
surgery, which is costly and invasive.
 We have 100 patients in our pool of volunteers that
have agreed via informed consent to participate in our
study;
 They need to be assigned to the two treatments.
 We then measure five-year survival as a response
www.drugragulations.org
11
3
 What sort of trouble can happen if we fail to randomize?
 Bypass surgery is a major operation, and patients with severe disease may
not be strong enough to survive the operation.
 It might thus be tempting to assign the stronger patients to surgery and
the weaker patients to the drug therapy.
 This confounds strength of the patient with treatment differences ( Surgery
or New Drug)
 The drug therapy would likely have a lower survival rate because it is
getting the weakest patients, even if the drug therapy is every bit as good
as the surgery.
www.drugragulations.org
11
4
 Alternatively, perhaps only small quantities of the drug are available early
in the experiment, so that we assign more of the early patients to surgery,
and more of the later patients to drug therapy.
 There will be a problem if the early patients are somehow different from
the later patients. For example, the earlier patients might be from your
own practice, and the later patients might be recruited from other doctors
and hospitals.
 The patients could differ by age, socioeconomic status, and other factors
that are known to be associated with survival.
www.drugragulations.org
11
5
 Here is how randomization has helps us.
 No matter which features of the population of experimental units are
associated with our response, our randomizations put approximately half
the patients with these features in each treatment group.
◦ Approximately half the men get the drug;
◦ Approximately half the older patients get the drug;
◦ Approximately half the stronger patients get the drug; and so on.
 These are not exactly 50/50 splits, but the deviation from an even split
follows rules of probability that we can use when making inference about
the treatments.
www.drugragulations.org
11
6
 This example is, of course, an oversimplification.
 A real experimental design would include
considerations for age, gender, health status, and
so on.
 The beauty of randomization is that it helps
prevent confounding, even for factors that we do
not know are important.
www.drugragulations.org
11
7
 We have taken a very simplistic view of
experiments;
 “assign treatments to units and then measure
responses” hides a multitude of potential steps
and choices that will need to be made.
 Many of these additional steps can be randomized,
as they could also lead to confounding.
www.drugragulations.org
11
8
 If the experimental units are not used simultaneously,
you can randomize the order in which they are used.
 If the experimental units are not used at the same
location, you can randomize the locations at which they
are used.
 If you use more than one measuring instrument for
determining response, you can randomize which units
are measured on which instruments.
www.drugragulations.org
11
9
12
0
 Factor A – a input levels
 Factor B – b input levels
 n measurements for each input combination
 abn total measurements
Copyright 2004 David J. Lilja 121
Factor A
1 2 … j … a
FactorB
1 … … … … … …
2 … … … … … …
… … … … … … …
i … … … yijk … …
… … … … … … …
b … … … … … …
n replications
122
 Each individual
measurement is
composition of
◦ Overall mean
◦ Effect of
alternatives
◦ Measurement
errors
errortmeasuremen
Atodueeffect
meanoverall..
..




ij
i
ijiij
e
y
eyy


123
 Each individual
measurement is
composition of
◦ Overall mean
◦ Effects
◦ Interactions
◦ Measurement
errors
errortmeasuremen
BandAofninteractiotodueeffect
Btodueeffect
Atodueeffect
meanoverall...
...






ijk
ij
j
i
ijkijjiijk
e
y
eyy




12
4
 As before, use sum-of-squares identity
SST = SSA + SSB + SSAB + SSE
 Degrees of freedom
◦ df(SSA) = a – 1
◦ df(SSB) = b – 1
◦ df(SSAB) = (a – 1)(b – 1)
◦ df(SSE) = ab(n – 1)
◦ df(SST) = abn - 1
12
5
)]1(),1)(1(;1[)]1(),1(;1[)]1(),1(;1[
222222
2222
Tabulated
Computed
)]1([)]1)(1[()1()1(squareMean
)1()1)(1(11freedomDeg
squaresofSum
ErrorABBA




nabbanabbnaba
eababebbeaa
eabba
FFFF
ssFssFssFF
nabSSEsbaSSABsbSSBsaSSAs
nabbaba
SSESSABSSBSSA

12
6
 If n=1
◦ Only one measurement of each configuration
 Can then be shown that
◦ SSAB = SST – SSA – SSB
 Since
◦ SSE = SST – SSA – SSB – SSAB
 We have
◦ SSE = 0
12
7
 Thus, when n=1
◦ SSE = 0
◦ → No information about measurement errors
 Cannot separate effect due to interactions
from measurement noise
 Must replicate each experiment at least twice
12
8
 Replication
◦ Completely re-run experiment with same input
levels
◦ Used to determine impact of measurement error
 Recognition and statement of the problem in non
statistical language
◦ Selection of factors, levels, ranges
◦ Selection of response variables
◦ Choice of experimental design
◦ Performance of the experiment
◦ Statistical analysis of the data
◦ Conclusions and recommendations
www.drugragulations.org
12
9
 Use team’s non-statistical knowledge of the problem to:
◦ Choose factors
◦ Determine proper levels
◦ Decide number of replications
◦ Interpret results
 Keep the design and analysis as simple as
 possible
 Recognize the difference between practical and statistical significance
 Be prepared to iterate – commit no more than 25% of available resources
to first series
www.drugragulations.org
13
0

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Pharmaceutical Design of Experiments for Beginners

  • 1. www.drugragulations.org 1 Presentation prepared by Drug Regulations – a not for profit organization. Visit www.drugregulations.org for the latest in Pharmaceuticals.
  • 2. 2  Case Study formulation details which will be used in the presentation  Definitions  Terminology  Full factorial designs ◦ m-factor ANOVA  Fractional factorial designs  Multi-factorial designs
  • 3. Component Function Unit ( mg/tablet) Unit ( % W/W) Acetriptan, USP Active 20 10 Lactose Monohydrate, NF Filler 64-86 32-43 Microcrystalline Cellulose (MCC), NF Filler 72-92 36-46 Croscarmellose Sodium (CCS), NF Disintegrant 2-10 1-5 Magnesium Stearate, NF* Lubricant 2-6 1-3 Talc, NF Glidant/Lubricant 1-10 0.5-5 Total tablet weight 200 100 www.drugregulations.org 3
  • 4. www.drugregulations.org 4 Compression of the lubricated granules into thablets. Lubrication of the blend with lubricants Milling of the ribbons from roller compaction stage Roller Compaction Roller Compaction of blend from earlier step Manufacturing Process Blending prior to roller compaction Mixing of Active Ingredient & Excipients
  • 5. 5  In this case study we will evaluate the blending process for blend uniformity by analyzing the content of active ingredient : Acetriptan.  Based on the knowledge of similar formulations and processes it was decided to study following factors  Response Variable : Concentration of Acetriptan in blend. ( Blend Uniformity) Factor Input Variables High Level Low Level A Mixing Time 5 min (+) 3 Min (-) B Mag. Stearate concentration 2.0 % (+) 1.0 % (-) C Talc concentration 2 % (+) 1 % (-)
  • 6. 6  Experiment:  A test or series of tests where the experimenter makes purposeful changes to input variables of a process or system so that we can observe or identify the reasons for changes in the output responses.
  • 7. 7  Design of Experiment:  Design of Experiments: is concerned with the planning and conduct of experiments to analyze the resulting data so that we obtain valid and objective conclusions.
  • 8.  Determine which variables are the most influential in a process or system  Determine where to set the inputs so the output is always near the desired state  Determine where to set the inputs so the output variability is minimized  Determine where to set the inputs so the influence of uncontrollable factors is minimized (robust design) www.drugragulations.org 8
  • 9.  Treatments are the different procedures we want to compare. ◦ Treatment : Lubrication time of 3 and 5 minutes , Magnesium Concentration : 1 % and 2 % , Talc Concentration : 1 % and 2 %.  Experimental units are the things to which we apply the treatments. ◦ Blend to be lubricated  Responses are outcomes that we observe after applying a treatment to an experimental unit. ◦ Assay / Content uniformity of the blend  Interaction Effect ◦ Effect of one input factor depends on level of another input factor www.drugragulations.org 9
  • 10.  Measurement units (or response units) are the actual objects on which the response is measured.  These may differ from the experimental units.  In the lubrication example ◦ Treatment is : Lubrication time of 3 min and 5 min ◦ Experimental Unit : The lubrication blend to which the treatment is applied. ◦ Response : Blend uniformity ◦ Response Unit : Per cent of Acetriptan content www.drugragulations.org 10
  • 11.  Distinction between experimental units and measurement units.  Consider an educational study, where six classrooms of 25 first graders each are assigned at random to two different reading programs, with all the first graders evaluated via a common reading exam at the end of the school year.  Are there  six experimental units (the classrooms) or  150 (the students)? www.drugragulations.org 11
  • 12.  One way to determine the experimental unit is via the consideration that an experimental unit should be able to receive any treatment.  Thus if students were the experimental units, we could see more than one reading program in any treatment.  However, the nature of the experiment makes it clear that all the students in the classroom receive the same program, so the classroom as a whole is the experimental unit.  We don’t measure how a classroom reads, though; we measure how students read. Thus students are the measurement units for this experiment. www.drugragulations.org 12
  • 13.  Blinding occurs when the evaluators of a response do not know which treatment was given to which unit.  Blinding helps prevent bias in the evaluation, even unconscious bias from well-intentioned evaluators.  Double blinding occurs when both the evaluators of the response and the (human subject) experimental units do not know the assignment of treatments to units. www.drugragulations.org 13
  • 14.  Control has several different uses in design. First, an experiment is controlled because we as experimenters assign treatments to experimental units. Otherwise, we would have an observational study. www.drugragulations.org 14
  • 15.  Second, a control treatment is a “standard” treatment that is used as a baseline or basis of comparison for the other treatments.  This control treatment might be the treatment in common use, or  It might be a null treatment (no treatment at all).  For example, a study of new pain killing drugs could use a standard pain killer as a control treatment. www.drugragulations.org 15
  • 16.  Factors combine to form treatments.  For example, in the experiment to study lubrication process following 3 factors can be evaluated.  Mixing Time : 3 min and 5 min  Magnesium Stearate Concentration : 1.0 % and 2.0 %  Talc Concentration : 1 % and 2 %  One treatment is a combination of mixing time , Magnesium Stearate Concentration and Talc Concentration.  We can vary each factor separately. www.drugragulations.org 16
  • 17.  We can vary each factor separately.  Individual settings for each factor are called levels of the factor. www.drugragulations.org 17 Factor High Level Low Level A Mixing Time 5 min (+) 3 Min (-) B Mag. Stearate concentration 2.0 % (+) 1.0 % (-) C Talc concentration 2 % (+) 1 % (-)
  • 18.  Confounding occurs when the effect of one factor or treatment cannot be distinguished from that of another factor or treatment.  The two factors or treatments are said to be confounded.  Except in very special circumstances, confounding should be avoided.  Consider using a concentration of Mag. Stearate of 1.0 % with 3 min of mixing and concentration of 2.0 % with 5 min of mixing.  In this experiment, we cannot distinguish the effect of mixing from the effect of Mag. Stearate concentration on blend uniformity.  Mixing time effect and Mag. Stearate concentration effect are confounded. www.drugragulations.org 18
  • 19.  One Factor at a Time ◦ Pros : Straight-forward, easily understood ◦ Cons: Impossible to address interactions ◦ Tends to “over collect” data, not efficient sample sizes www.drugragulations.org 19
  • 20. ◦ Case Study Example : ◦ Effect of Mixing Time (A) on Blend Uniformity :  Keep level of other factors B & C : Concentration of Magnesium Stearate and Talc constant and vary the Factor A : Mixing time ◦ Effect of Talc Concentration on Blend Uniformity  Keep Magnesium Stearate concentration and the mixing time constant and change the concentration of Talc ◦ Effect of Magnesium Concentration on Blend Uniformity  Keep Talc concentration constant mixing time constant and vary the concentration of Magnesium Stearate. www.drugragulations.org 20
  • 21.  Comparative designs to:  Choose between alternatives, with narrow scope, suitable for an initial comparison  Completely Randomized Designs  Choose between alternatives, with broad scope, suitable for a confirmatory comparison  Randomized Block Design www.drugragulations.org 21
  • 22.  Here we consider completely randomized designs that have one primary factor.  Mixing time in the Lubrication process  The experiment compares the values of a response variable based on the different levels of that primary factor.  Response Variable : Assay in CU analysis  Different Levels of Primary Factor : Mixing time of 3 , 5 and 7 minutes  For completely randomized designs, the levels of the primary factor are randomly assigned to the experimental units.  By randomization, we mean that the run sequence of the experimental units is determined randomly. www.drugragulations.org 22
  • 23.  For this example, there are 3 levels of the primary factor with each level to be run 2 times, then there are 6 factorial possible run sequences (or 6! ways to order the experimental trials).  Because of the replication, the number of unique orderings is 90 (since 90 = 6!/(2!*2!*2!)).  An example of an unrandomized design would be to always run 2 replications for the first level, then 2 for the second level, and finally 2 for the third level.  To randomize the runs, one way would be to put 6 slips of paper in a box with 2 having level 1, 2 having level 2, and 2 having level 3.  Before each run, one of the slips would be drawn blindly from the box and the level selected would be used for the next run of the experiment. www.drugragulations.org 23
  • 24.  In practice, the randomization is typically performed by a computer program.  However, the randomization can also be generated from random number tables or by some physical mechanism (e.g., drawing the slips of paper).  All completely randomized designs with one primary factor are defined by 3 numbers:  k = number of factors (= 1 for these designs) L = number of levels n = number of replications  and the total sample size (number of runs) is N = k x L x n. www.drugragulations.org 24
  • 25.  A typical example of a completely randomized design is the following:  k = 1 factor (X1) L = 4 levels of that single factor (called "1", "2", "3", and "4") n = 3 replications per level N = 4 levels * 3 replications per level = 12 runs www.drugragulations.org 25
  • 26. www.drugragulations.org 26 X1 3 1 4 2 2 1 3 4 1 2 4 3  A completely randomized design may look like this
  • 27.  Note that in this example there are 12!/(3!*3!*3!*3!) = 369,600 ways to run the experiment, all equally likely to be picked by a randomization procedure.  The model for the response is  Yi,j =μ+ Ti +random error  with  Yi,j being any observation for which X1 = i  (i and j denote the level of the factor and the replication within the level of the factor, respectively)  μ (or mu) is the general location parameter Ti is the effect of having treatment level i www.drugragulations.org 27
  • 28. www.drugragulations.org 28 Estimate for μ: Y¯ = the average of all the data Estimate for Ti: Y¯i−Y¯with Y¯i = average of all Y for which X1 = i. Statistical tests for levels of X1 is One way ANOVA
  • 29. 29  Separates total variation observed in a set of measurements into: 1. Variation within one system  Due to random measurement errors 2. Variation between systems  ( Different Mixing Time , Or Different concentration of Magnesium Stearate or Different concentration of Talc)  Due to real differences + random error  Is variation(2) statistically > variation(1)?  One-factor experimental design
  • 30. 30 Variation Alternatives Error Total Sum of squares SSA SSE SST Deg freedom k -1 k(n -1) kn -1 Mean square sa 2 = SSA (k -1) se 2 = SSE [k(n -1)] Computed F sa 2 se 2 Tabulated F F[1-a;(k-1),k(n-1)]
  • 31.  Blocking ◦ Designed to improve precision of comparisons ◦ Used to reduce or eliminate nuisance factors  Nuisance Factor ◦ A nuisance factor is a “design factor that probably has an effect on the response but we are not interested in that effect” www.drugragulations.org 31
  • 32.  Nuisance factors are those that may affect the measured result, but are not of primary interest.  For example, in applying a treatment, nuisance factors might be the specific operator who prepared the treatment, the time of day the experiment was run, and the room temperature.  All experiments have nuisance factors.  The experimenter will typically need to spend some time deciding which nuisance factors are important enough to keep track of or control, if possible, during the experiment. www.drugragulations.org 32
  • 33.  Nuisance Factors, Types ⇒ Cures ◦ Known and controllable ⇒ Use blocking to systematically eliminate the effect ◦ Known but uncontrollable ⇒ If it can be measured, use Analysis of Covariance (ANCOVA) ◦ Unknown and uncontrollable ⇒ Randomization is the insurance www.drugragulations.org 33
  • 34.  When we can control nuisance factors, an important technique known as blocking can be used to reduce or eliminate the contribution to experimental error contributed by nuisance factors.  The basic concept is to create homogeneous blocks in which the nuisance factors are held constant and the factor of interest is allowed to vary.  Within blocks, it is possible to assess the effect of different levels of the factor of interest without having to worry about variations due to changes of the block factors, which are accounted for in the analysis. www.drugragulations.org 34
  • 35.  A nuisance factor is used as a blocking factor if every level of the primary factor occurs the same number of times with each level of the nuisance factor.  The analysis of the experiment will focus on the effect of varying levels of the primary factor within each block of the experiment. www.drugragulations.org 35
  • 36.  The general rule is:  "Block what you can, randomize what you cannot."  Blocking is used to remove the effects of a few of the most important nuisance variables. Randomization is then used to reduce the contaminating effects of the remaining nuisance variables. www.drugragulations.org 36
  • 37.  One useful way to look at a randomized block experiment is to consider it as a collection of completely randomized experiments, each run within one of the blocks of the total experiment. www.drugragulations.org 37
  • 39.  Case Study  Case Study Formulation : Lubrication Experiments  Assume that two batches of Active Ingredients are used for the experiments.  Though each batch of active ingredient will be tested and meet specifications before being used in the experiments , there could be some difference due to batch to batch variability.  This variability may be small and we may not be interested in determining its effect on the response factor.  Therefore different batches of Active ingredients can be considered as a nuisance factor.  Completely randomizing the experiments may result in potential serious problems. www.drugragulations.org 39
  • 40.  Case Study  It is quite possible that Active ingredients from different batches may have slightly different surface & flow properties ,  This may affect the distribution of the active in pre-roller compaction blending process which in turn may affect the uniformity of the active in roller compresses granules and therefore in the lubricated blend.  Therefore if we do not obtain a uniform blend in the lubrication stage we may not know whether it is due the lubrication process or whether due to the variation from the Active ingredient. www.drugragulations.org 40
  • 41.  Cased Study  The experimental error will reflect both random error and variability between different batches of the active ingredient.  We would like to make the experimental error as small as possible i.e. remove the variability between different batches of active ingredients from the experimental error.  A design that would accomplish this requires conducting all experiments with each of the Active Ingredient batches.  In this example , each batch of Active ingredient would form a block.  This statistical design is called “ Randomized complete Block Design”  The word complete indicates that each block –Active Ingredient Batch contains all the treatments. www.drugragulations.org 41
  • 42.  Case Study Formulation : Lubrication Experiments www.drugragulations.org 42 Block 1 : Supplier A Block 2: Supplier B Experiment No. from DOE 2 6 4 3 3 5 7 7 8 2 1 4 5 1
  • 43.  Case Study  By using this design, the blocks or one active Ingredient batch is used to compare other factors ( Mixing speed , Concentration of Magnesium Stearate and Talc.)  This design strategy improves the accuracy of the comparison among the three factors by eliminating the variability among the active ingredient batches.  Within a block the order in which each experiment is conducted is based on randomization.  The statics of this design is similar to the paired “t-test”  Randomized complete block design is a generalization of that concept. ( RCBD) www.drugragulations.org 43
  • 44.  RCBD is one of the most widely used experimental designs.  Similar but Machinery or Unit of test equipment are often different in their operating characteristics and would be a typical blocking factors.  Batches of Raw Materials , People and time are common nuisance source of variability.  These can be systematically controlled through blocking. www.drugragulations.org 44
  • 45.  Blocking  Blocking can also be used in situations which do not involve nuisance factors. e.g.  In a chemical reaction catalyst feed rate is to be studied to determine effect on viscosity of polymer.  There are several factors such as raw material source ,temperature , operator raw material purity are very difficult to control in full scale process.  Catalyst feed rate can be tested in blocks where each block consist of combination of some uncontrollable factors.  Blocks can be used to test the robustness of the process variable the feed rate to conditions which can not be easily controlled. www.drugragulations.org 45
  • 48. www.drugragulations.org 48 Source of Variation Sum of Squares Degrees of Freedom Mean Square F 0 Treatments SS Treatments a-1 SS Treatments --------------- ( a-1) MS Treatments --------------- MS E Block SS Blocks b-1 SS Blocks ----------- ( b-1) Error SS E (a-1) (b-1) SS E -------------- ( a-1) (b-1) Total SS T N-1
  • 50.  Latin square designs, and the related Graeco-Latin square and Hyper-Graeco-Latin square designs, are a special type of comparative design.  There is a single factor of primary interest, typically called the treatment factor, and several nuisance factors.  For Latin square designs there are 2 nuisance factors, for Graeco- Latin square designs there are 3 nuisance factors, and for Hyper- Graeco-Latin square designs there are 4 nuisance factors. www.drugragulations.org 50
  • 51.  The nuisance factors are used as blocking variables. 1. For Latin square designs, the 2 nuisance factors are divided into a tabular grid with the property that each row and each column receive each treatment exactly once. 2. As with the Latin square design, a Graeco-Latin square design is a kxk tabular grid in which k is the number of levels of the treatment factor. However, it uses 3 blocking variables instead of the 2 used by the standard Latin square design. 3. A Hyper-Graeco-Latin square design is also a kxk tabular grid with k denoting the number of levels of the treatment factor. However, it uses 4 blocking variables instead of the 2 used by the standard Latin square design. www.drugragulations.org 51
  • 52.  The advantages of Latin square designs are: 1. They handle the case when we have several nuisance factors and we either cannot combine them into a single factor or we wish to keep them separate. 2. They allow experiments with a relatively small number of runs. www.drugragulations.org 52
  • 53.  The disadvantages are: 1. The number of levels of each blocking variable must equal the number of levels of the treatment factor. 2. The Latin square model assumes that there are no interactions between the blocking variables or between the treatment variable and the blocking variable. www.drugragulations.org 53
  • 54. www.drugragulations.org 54  Note that Latin square designs are equivalent to specific fractional factorial designs  (e.g., the 4x4 Latin square design is equivalent to a 43-1 Fractional Design design).
  • 58.  Designs for Latin squares with 3-, 4-, and 5-level factors are given next. These designs show what the treatment combinations should be for each run.  When using any of these designs, be sure to randomize the treatment units and trial order, as much as the design allows.  For example, one recommendation is that a Latin square design be randomly selected from those available, then randomize the run order. www.drugragulations.org 58
  • 59. 59
  • 60. 60
  • 61.  Screening designs to identify which factors/effects are important o when you have 2 - 4 factors and can perform a full factorial o when you have more than 3 factors and want to begin with as small a design as possible o when you have some qualitative factors, or you have some quantitative factors that are known to have a non- monotonic effect www.drugragulations.org 61
  • 62.  You can also restrict the term screening design to the case where you are trying to extract the most important factors from a large (say > 5) list of initial factors (usually a fractional factorial design). www.drugragulations.org 62
  • 64.  The following is an example of a full factorial design with 3 factors that also illustrates replication , randomization and Center points  Suppose that we wish to improve the yield of a polishing operation. The three inputs (factors) that are considered important to the operation are Speed (X1), Feed (X2), and Depth (X3). We want to ascertain the relative importance of each of these factors on Yield (Y).  Speed, Feed and Depth can all be varied continuously along their respective scales, from a low to a high setting. Yield is observed to vary smoothly when progressive changes are made to the inputs. This leads us to believe that the ultimate response surface for Y will be smooth. www.drugragulations.org 64
  • 65.  High ( +) , Low (-) and Standard ( 0) Setting for a Polishing Operation www.drugragulations.org 65
  • 66.  We want to try various combinations of these settings so as to establish the best way to run the polisher.  There are eight different ways of combining high and low settings of Speed, Feed, and Depth.  These eight are shown at the corners of the following diagram. www.drugragulations.org 66
  • 69. www.drugragulations.org 69  A 23 Full Factorial Design with runs in Standard order
  • 70.  Running the entire design more than once makes for easier data analysis because, for each run (i.e., `corner of the design box') we obtain an average value of the response as well as some idea about the dispersion (variability, consistency) of the response at that setting.  One of the usual analysis assumptions is that the response dispersion is uniform across the experimental space.  The technical term is `homogeneity of variance'. Replication allows us to check this assumption and possibly find the setting combinations that give inconsistent yields, allowing us to avoid that area of the factor space.  We now have constructed a design table for a two-level full factorial in three factors, replicated twice. www.drugragulations.org 70
  • 71. www.drugragulations.org 71  A 23 Full Factorial Design replicated twice & with runs in Standard order
  • 72.  If we now ran the design as is, in the order shown, we would have two deficiencies, namely: 1. No randomization, and 2. No Center points. www.drugragulations.org 72
  • 73.  The more freely one can randomize experimental runs, the more insurance one has against extraneous factors possibly affecting the results, and hence perhaps wasting our experimental time and effort.  For example, consider the `Depth' column: the settings of Depth, in standard order, follow a `four low, four high, four low, four high' pattern.  Suppose now that four settings are run in the day and four at night, and that (unknown to the experimenter) ambient temperature in the polishing shop affects Yield.  We would run the experiment over two days and two nights and conclude that Depth influenced Yield, when in fact ambient temperature was the significant influence.  So the moral is: Randomize experimental runs as much as possible. www.drugragulations.org 73
  • 74.  A 23 Full Factorial Design replicated twice & with runs in Randomized order www.drugragulations.org 74
  • 75.  This design would be improved by adding at least 3 Center point runs placed at the beginning, middle and end of the experiment. The final design matrix is shown alongside. www.drugragulations.org 75
  • 76.  We often need to eliminate the influence of extraneous factors when running an experiment. We do this by "blocking".  Previously, blocking was introduced when Randomized Block Design were discussed.  There we were concerned with one factor in the presence of one of more nuisance factors.  In this section we look at a general approach that enables us to divide 2-level factorial experiments into blocks.  For example, assume we anticipate predictable shifts will occur while an experiment is being run.  This might happen when one has to change to a new batch of raw materials halfway through the experiment.  The effect of the change in raw materials is well known, and we want to eliminate its influence on the subsequent data analysis. www.drugragulations.org 76
  • 77.  In this case, we need to divide our experiment into two halves (2 blocks), one with the first raw material batch and the other with the new batch.  The division has to balance out the effect of the materials change in such a way as to eliminate its influence on the analysis, and we do this by blocking.  An eight-run 23 full factorial has to be blocked into two groups of four runs each.  Consider the design `box' for the 23 full factorial. Blocking can be achieved by assigning the first block to the dark-shaded corners and the second block to the open circle corners. www.drugragulations.org 77
  • 78. www.drugragulations.org 78 Blocking Scheme for a 23 Using Alternate Corners Three-factor
  • 79.  This works because we are in fact assigning the `estimation' of the (unwanted) blocking effect to the three-factor interaction, and because of the special property of two-level designs called orthogonality.  That is, the three-factor interaction is "confounded" with the block effect as will be seen shortly.  Orthogonality guarantees that we can always estimate the effect of one factor or interaction clear of any influence due to any other factor or interaction.  Orthogonality is a very desirable property in DOE and this is a major reason why two-level factorials are so popular and successful.  Formally, consider the 23 design table with the three-factor interaction column added. www.drugragulations.org 79
  • 80. www.drugragulations.org 80  Two Blocks for a 23 Design
  • 81.  Rows that have a `-1' in the three-factor interaction column are assigned to `Block I' (rows 1, 4, 6, 7), while the other rows are assigned to `Block II' (rows 2, 3, 5, 8).  Note that the Block I rows are the open circle corners of the design `box' above; Block II are dark-shaded corners.  The general rule for blocking is: use one or a combination of high-order interaction columns to construct blocks.  This gives us a formal way of blocking complex designs.  Apart from simple cases in which you can design your own blocks, your statistical/DOE software will do the blocking if asked, but you do need to understand the principle behind it. www.drugragulations.org 81
  • 82.  The price you pay for blocking by using high-order interaction columns is that you can no longer distinguish the high-order interaction(s) from the blocking effect - they have been “Confounded”`,' or “ aliased”.  In fact, the blocking effect is now the sum of the blocking effect and the high- order interaction effect.  This is fine as long as our assumption about negligible high-order interactions holds true, which it usually does.  Within a block, center point runs are assigned as if the block were a separate experiment - which in a sense it is Randamization takes place within a block as it would for any non-blocked DOE. www.drugragulations.org 82
  • 83.  The price you pay for blocking by using high-order interaction columns is that you can no longer distinguish the high-order interaction(s) from the blocking effect - they have been “Confounded”`,' or “ aliased”.  In fact, the blocking effect is now the sum of the blocking effect and the high- order interaction effect.  This is fine as long as our assumption about negligible high-order interactions holds true, which it usually does.  Within a block, center point runs are assigned as if the block were a separate experiment - which in a sense it is Randamization takes place within a block as it would for any non-blocked DOE. www.drugragulations.org 83
  • 84.  The ASQC (1983) Glossary & Tables for Statistical Quality Control defines fractional factorial design in the following way:  "A factorial experiment in which only an adequately chosen fraction of the treatment combinations required for the complete factorial experiment is selected to be run."  Even if the number of factors, k, in a design is small, the 2k runs specified for a full factorial can quickly become very large.  For example, 26 = 64 runs is for a two-level, full factorial design with six factors. To this design we need to add a good number of centerpoint runs and we can thus quickly run up a very large resource requirement for runs with only a modest number of factors. www.drugragulations.org 84
  • 85.  The solution to this problem is to use only a fraction of the runs specified by the full factorial design.  Which runs to make and which to leave out is the subject of interest here.  In general, we pick a fraction such as ½, ¼, etc. of the runs called for by the full factorial.  We use various strategies that ensure an appropriate choice of runs.  The following slides will show you how to choose an appropriate fraction of a full factorial design to suit your purpose at hand. Properly chosen fractional factorial designs for 2-level experiments have the desirable properties of being both balanced and orthogonal. www.drugragulations.org 85
  • 86.  Consider the two-level, full factorial design for three factors, namely the 23 design.  This implies eight runs (not counting replications or center points). Graphically, as shown earlier, we can represent the 23 design by the following cube:  A 23 Full Factorial Design; Factors X1, X2, X3. (The arrows show the direction of increase of the factors. Numbers `1' through '8' at the corners of the design cube reference the 'Standard Order' of runs) www.drugragulations.org 86
  • 87.  A 23 Two-level, Full Factorial Design Table Showing Runs in 'Standard Order,' Plus Observations www.drugragulations.org 87 X1 X2 X3 Y 1 -1 -1 -1 y1 = 33 2 +1 -1 -1 y2 = 63 3 -1 +1 -1 y3 = 41 4 +1 +1 -1 Y4 = 57 5 -1 -1 +1 y5 = 57 6 +1 -1 +1 y6 = 51 7 -1 +1 +1 y7 = 59 8 +1 +1 +1 y8 = 53
  • 88.  The right-most column of the table lists 'y1' through 'y8' to indicate the responses measured for the experimental runs when listed in standard order.  For example, `y1' is the response (i.e., output) observed when the three factors were all run at their 'low' setting. The numbers entered in the 'y' column will be used to illustrate calculations of effects.  From the entries in the table we are able to compute all 'effects' such as main effects, first-order 'interaction' effects, etc.  For example, to compute the main effect estimate 'c1' of factor X1, we compute the average response at all runs withX1 at the 'high' setting, namely (1/4)(y2 + y4 + y6 + y8), minus the average response of all runs with X1 set at 'low,' namely (1/4)(y1 + y3 + y5 + y7). That is, www.drugragulations.org 88
  • 89.  For example, to compute the main effect estimate 'c1' of factor X1, we compute the average response at all runs withX1 at the 'high' setting, namely (1/4)(y2 + y4 + y6 + y8), minus the average response of all runs with X1 set at 'low,' namely (1/4)(y1 + y3 + y5 + y7). That is, www.drugragulations.org 89 c1 = (1/4) (y2 + y4 + y6 + y8) - (1/4) (y1 + y3 + y5 + y7) = (1/4) (63+57+51+53 ) - (1/4) (33+41+57+59) = 8.5
  • 90.  Suppose, however, that we only have enough resources to do four runs. Is it still possible to estimate the main effect for X1? Or any other main effect? The answer is yes, and there are even different choices of the four runs that will accomplish this.  For example, suppose we select only the four light (unshaded) corners of the design cube. Using these four runs (1, 4, 6 and 7), we can still compute c1 as follows: www.drugragulations.org 90 c1 = (1/2) (y4 + y6) - (1/2) (y1 + y7) = (1/2) (57+51) - (1/2) (33+59) = 8
  • 91. www.drugragulations.org 91 Simarly, we would compute c2, the effect due to X2, as c2 = (1/2) (y4 + y7) - (1/2) (y1 + y6) = (1/2) (57+59) - (1/2) (33+51) = 16 Finally, the computation of c3 for the effect due to X3 would be c3 = (1/2) (y6 + y7) - (1/2) (y1 + y4) = (1/2) (51+59) - (1/2) (33+57) = 10
  • 92. www.drugragulations.org 92  We could also have used the four dark (shaded) corners of the design cube for our runs and obtained similiar, but slightly different, estimates for the main effects.  In either case, we would have used half the number of runs that the full factorial requires.  The half fraction we used is a new design written as 23-1. Note that 23-1 = 23/2 = 22 = 4, which is the number of runs in this half-fraction design
  • 94. 94 DoE : number of experiments D O E : Number of experiments
  • 95. 95  With n experiments, you can calculate the coefficients for n-1 factors and interactions ◦ For 2 factors, Factorial design requires 22 = 4 experiments, you can calculate the coefficients for 3 factors and interactions : A, B and interaction AB (green color), it’s not interesting to erase 1 experiment and loose informations on possible interaction between A and B.
  • 96. 96  With n experiments, you can calculate the coefficients for n-1 factors and interactions ◦ For 3 factors, Factorial design requires 23 = 8 experiments, you can calculate the coefficients for 7 factors and interactions : A, B, C and interactions AB, AC, BC and ABC. Using semi Factorial design (22 = 4 experiments) informations on possible interaction are also lost (red color).
  • 97. 97  With n experiments, you can calculate the coefficients for n-1 factors and interactions ◦ For more than 3 factors, the number of experiments should be limited to the number of factors tested + the number of single interactions. I never found (up to now) significant triple interactions (ABC), Why loose time, money for a large number of such interactions ABC…F, ABD…F, ACD…F, AED…F…(yellow color)
  • 98. 98 Trial A B C 1 Lo Lo Lo 2 Lo Lo Hi 3 Lo Hi Lo 4 Lo Hi Hi 5 Hi Lo Lo 6 Hi Lo Hi 7 Hi Hi Lo 8 Hi Hi Hi www.drugragulations.org
  • 99.  Replication ◦ Independent Repeat run of each factor combination ◦ Permits estimation of experimental error ◦ Estimation of Error : Determines if observed differences in data are statistically different. ◦ Permits more precise estimates of the sample statistics Replication is not repeated measurements. www.drugragulations.org 99
  • 100.  Replication ◦ Replication is not repeated measurements.  Case Study Formulation : Acetriptan Tablets ◦ In this example we are studying the effect of lubrication time on blend uniformity by assaying the blend. ◦ Blend analysis at lubrication stage after 3 and 5 minutes of mixing ◦ Samples are taken at each time interval from 6 different locations ◦ At each location 2 samples are taken ◦ For replicating this experiment you need have another experiment in which another bend is mixed for 5 minutes , samples are withdrawn at 3 and 5 minutes time interval for analysis. www.drugragulations.org 10 0
  • 101.  Randomization ◦ Corner stone of statistical experiments by design. ◦ Insures that observations or errors are more likely to be independent ◦ Helps “average out” effects of extraneous factors ◦ Special designs when complete randomization not feasible www.drugragulations.org 10 1
  • 102.  Randomization means ◦ Allocation of experimental material and ◦ The order in which the individual experimental runs are performed ◦ ARE RNADOMLY DETERMIED.  Case Study Formulation ◦ For the formulation development if a 23 experiment is being done in replicate , this will involve 8 experiments to be done in replicate i.e 16 experiments. ◦ Randomization means randomizing the experiments as well as randomizing the material used for experiments i.e. different batches of actives and excipient materials. ( If different batches are used) www.drugragulations.org 10 2
  • 103.  Randomization ◦ Computer programs are used in selecting and constructing experimental designs. ◦ These programs present the experimental run in random order. ◦ This is created by using a random number generator ◦ Even in such cases it is essential to assign  Experimental material  Operators  Gauges  Measurement devices ◦ This allocation also needs to be randomized if each can not be kept constant. ◦ If complete randomization is not possible there are statistical methods to deal with restriction on randomization. www.drugragulations.org 10 3
  • 104.  A good experimental design must ◦ Avoid systematic error ◦ Be precise ◦ Allow estimation of error ◦ Have broad validity. www.drugragulations.org 10 4
  • 105.  Comparative experiments estimate differences in response between treatments.  If experiment has systematic error, then comparisons will be biased, no matter how precise our measurements are or how many experimental units we use. www.drugragulations.org 10 5
  • 106.  Case Study Example : Consider following situations  The samples for blend uniformity are analyzed as follows: www.drugragulations.org 10 6 Case Study Number All Experiments with 3 min mixing time All Experiments with 5 min mixing time 1 Spectrophotome ter Perkin Elmer Model X : Unit 1 Perkin Elmer Model X : Unit 2 2 Analyst Analyst A Analyst B 3 Brand of filter used during filtration of final solution Type X Brand A Type X Brand B
  • 107.  We will not know if there is any difference in blend uniformity whether it is due to different mixing times ( 3 & 5 min) , or the difference is it is due to ◦ Case study 1 : Use of different Spectrophotometers  Differences in calibration ◦ Case study 2 : Use of different analyst  Expertise level of analyst A and Analyst B could be different ◦ Case study 3: Use of different brand of filters  Extraction of “leachables” may be different for brand A and brand B www.drugragulations.org 10 7
  • 108.  Experimental Error is the random variation present in all experimental results.  Different experimental units will give different responses to the same treatment.  It is often true that applying the same treatment over and over again to the same unit will result in different responses in different trials.  Experimental error does not refer to conducting the wrong experiment or dropping test tubes. www.drugragulations.org 10 8
  • 109.  Even without systematic error, there will be random error in the responses.  This will lead to random error in the treatment comparisons.  Experiments designed to increase precision are precise when this random error in “treatment comparisons” is small.  Precision depends on the ◦ Size of the random errors in the “responses” ( ie assay value ◦ The number of experimental units used, and ◦ The experimental design used.  There are several designs to improve precision. www.drugragulations.org 10 9
  • 110.  Experiments must be designed so that we have an estimate of the size of the random error.  This permits statistical inference: ◦ Confidence intervals or ◦ Tests of significance.  We cannot do inference without an estimate error. www.drugragulations.org 11 0
  • 111.  An experiment may have several randomized features in addition to the assignment of treatments to units.  Randomization is one of the most important elements of a well-designed experiment. www.drugragulations.org 11 1
  • 112.  We defined confounding as occurring when the effect of one factor or treatment cannot be distinguished from that of another factor or treatment.  Randomization helps prevent confounding.  Let’s start by looking at the trouble that can happen when we don’t randomize. www.drugragulations.org 11 2
  • 113.  Consider a new drug treatment for coronary artery disease.  We wish to compare this drug treatment with bypass surgery, which is costly and invasive.  We have 100 patients in our pool of volunteers that have agreed via informed consent to participate in our study;  They need to be assigned to the two treatments.  We then measure five-year survival as a response www.drugragulations.org 11 3
  • 114.  What sort of trouble can happen if we fail to randomize?  Bypass surgery is a major operation, and patients with severe disease may not be strong enough to survive the operation.  It might thus be tempting to assign the stronger patients to surgery and the weaker patients to the drug therapy.  This confounds strength of the patient with treatment differences ( Surgery or New Drug)  The drug therapy would likely have a lower survival rate because it is getting the weakest patients, even if the drug therapy is every bit as good as the surgery. www.drugragulations.org 11 4
  • 115.  Alternatively, perhaps only small quantities of the drug are available early in the experiment, so that we assign more of the early patients to surgery, and more of the later patients to drug therapy.  There will be a problem if the early patients are somehow different from the later patients. For example, the earlier patients might be from your own practice, and the later patients might be recruited from other doctors and hospitals.  The patients could differ by age, socioeconomic status, and other factors that are known to be associated with survival. www.drugragulations.org 11 5
  • 116.  Here is how randomization has helps us.  No matter which features of the population of experimental units are associated with our response, our randomizations put approximately half the patients with these features in each treatment group. ◦ Approximately half the men get the drug; ◦ Approximately half the older patients get the drug; ◦ Approximately half the stronger patients get the drug; and so on.  These are not exactly 50/50 splits, but the deviation from an even split follows rules of probability that we can use when making inference about the treatments. www.drugragulations.org 11 6
  • 117.  This example is, of course, an oversimplification.  A real experimental design would include considerations for age, gender, health status, and so on.  The beauty of randomization is that it helps prevent confounding, even for factors that we do not know are important. www.drugragulations.org 11 7
  • 118.  We have taken a very simplistic view of experiments;  “assign treatments to units and then measure responses” hides a multitude of potential steps and choices that will need to be made.  Many of these additional steps can be randomized, as they could also lead to confounding. www.drugragulations.org 11 8
  • 119.  If the experimental units are not used simultaneously, you can randomize the order in which they are used.  If the experimental units are not used at the same location, you can randomize the locations at which they are used.  If you use more than one measuring instrument for determining response, you can randomize which units are measured on which instruments. www.drugragulations.org 11 9
  • 120. 12 0  Factor A – a input levels  Factor B – b input levels  n measurements for each input combination  abn total measurements
  • 121. Copyright 2004 David J. Lilja 121 Factor A 1 2 … j … a FactorB 1 … … … … … … 2 … … … … … … … … … … … … … i … … … yijk … … … … … … … … … b … … … … … … n replications
  • 122. 122  Each individual measurement is composition of ◦ Overall mean ◦ Effect of alternatives ◦ Measurement errors errortmeasuremen Atodueeffect meanoverall.. ..     ij i ijiij e y eyy  
  • 123. 123  Each individual measurement is composition of ◦ Overall mean ◦ Effects ◦ Interactions ◦ Measurement errors errortmeasuremen BandAofninteractiotodueeffect Btodueeffect Atodueeffect meanoverall... ...       ijk ij j i ijkijjiijk e y eyy    
  • 124. 12 4  As before, use sum-of-squares identity SST = SSA + SSB + SSAB + SSE  Degrees of freedom ◦ df(SSA) = a – 1 ◦ df(SSB) = b – 1 ◦ df(SSAB) = (a – 1)(b – 1) ◦ df(SSE) = ab(n – 1) ◦ df(SST) = abn - 1
  • 126. 12 6  If n=1 ◦ Only one measurement of each configuration  Can then be shown that ◦ SSAB = SST – SSA – SSB  Since ◦ SSE = SST – SSA – SSB – SSAB  We have ◦ SSE = 0
  • 127. 12 7  Thus, when n=1 ◦ SSE = 0 ◦ → No information about measurement errors  Cannot separate effect due to interactions from measurement noise  Must replicate each experiment at least twice
  • 128. 12 8  Replication ◦ Completely re-run experiment with same input levels ◦ Used to determine impact of measurement error
  • 129.  Recognition and statement of the problem in non statistical language ◦ Selection of factors, levels, ranges ◦ Selection of response variables ◦ Choice of experimental design ◦ Performance of the experiment ◦ Statistical analysis of the data ◦ Conclusions and recommendations www.drugragulations.org 12 9
  • 130.  Use team’s non-statistical knowledge of the problem to: ◦ Choose factors ◦ Determine proper levels ◦ Decide number of replications ◦ Interpret results  Keep the design and analysis as simple as  possible  Recognize the difference between practical and statistical significance  Be prepared to iterate – commit no more than 25% of available resources to first series www.drugragulations.org 13 0