Experimental design

Subject: RESEARCH METHODOLOGY,
BIOSTATISTICS AND IPR
Subject Code: MRM301T
Prepared by: Dolly Sadrani
M. Pharma
IIIrd SEM
Department of Pharmaceutics
Guided By: Dr. Sachin Narkhede
Department of Pharmacognosy
EXPERIMENTAL
DESIGN

 Optimization techniques and various method of optimization
 Full Factorial Design
 Introduction to Contour Plots
 Introduction of Response Surface Design
Classification
Characteristics of Design
Matrix and analysis of design with case study
Equation of Full and Reduced mathematical models in
experimental designs
 Central Composite designs
 Taguchi and mixtures designs
 Application of experimental designs in pharmacology for reduction
of animal
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 The term optimize is defined as to make perfect, effective or as
functional as possible.
 This is the process of finding the best way of using the existing
resources while taking in to the account of all the factors that
influences decisions in any experiment.
 Traditionally, optimization in pharmaceuticals refers to changing
one variable at a time, so to obtain solution of a problematic
formulation.
 Modern pharmaceutical optimization involves systematic design of
experiments (DoE) to improve formulation irregularities.
 In the other word we can say that quantitate a formulation that has
been qualitatively determined. It is not a screening technique.

 Primary objective may not be optimize absolutely but to
compromise effectively and thereby produce the best formulation
under a given set of restriction.

 Factor: It is an assigned variable such as concentration,
Temperature etc..,
Quantitative: Numerical factor assigned to it
Ex; Concentration 1 %, 2 %, 3% etc
Qualitative: Which are not numerical
Ex; Polymer grade, humidity condition etc
 LEVELS: Levels of a factor are the values or designations assigned
to the factor
5
Factors Levels
Concentration 1 %, 2%
Temperature 30oc, 50oc
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 RESPONSE: It is an outcome of the experiment.
It is the effect to evaluate.
Ex: Disintegration time etc..,
 EFFECT: It is the change in response caused by varying the levels
It gives the relationship between various factors & level
 INTERACTION: It gives the overall effect of two or more
variables
Ex: Combined effect of lubricant and glidant

 Yield the “BEST SOLUTION” within the domain of study.
 Require fewer experiments to achieve an optimum formulation.
 Can trace and rectify problem in a remarkably easier manner.

Unconstrained:
 In unconstrained optimization problems there are no restrictions.
 The making of the hardest tablet is the unconstrained optimization
problem.
Constrained:
 The constrained problem involved in it, is to make the hardest tablet
possible, but it must disintegrate in less than 15 minutes.

Independent variables: The independent variables are under the
control of the formulator. These might include the compression
force or the die cavity filling or the mixing time.
Example: Diluent ratio, Compressional force, Disintegrate level,
Binder level, Lubricant level
Dependent variables: The dependent variables are the responses or
the characteristics that are developed due to the independent
variables. The more the variables that are present in the system the
more the complications that are involved in the optimization.
Example: Disintegration time, Hardness, Friability, Weight
uniformity, Dissolution.

 Classical optimization is done by using the calculus to basic
problem to find the maximum and the minimum of a function.
 The curve in the fig represents the relationship between the
response Y and the single independent variable X and we can obtain
the maximum and the minimum.
 By using the calculus the graphical represented can be avoided. If
the relationship, the equation for Y as a function of X, is available.
Y = f(X)

 Limited applications.
 Problems that is too complex.
 They do not involve more than two variables. For more than two
variables graphical representation is impossible.

 EVOP method
 Simplex Lattice method
 Lagrangian method
 Search method

 Make very small changes in formulation repeatedly.
 The result of changes is statistically analyzed.
 If there is improvement, the same step is repeated until further
change doesn’t improve the product.

 Generates information on product development.
 Predict the direction of improvement.
 Help formulator to decide optimum conditions for the formulation
and process.

 More repetition is required
 Time consuming
 Not efficient to finding true optimum
 Expensive to use.

Tablet
By changing the conc. of binder how we can get Hardness?
Hardness
Response

 A simplex is a geometric figure, defined by no. of points or vertices
equal to one more than no. of factors examined.
 Once the shape of a simplex has been determined, the method can
employ a simplex of fixed size or of variable sizes that are
determined by comparing the magnitudes of the responses after
each successive calculation.
 It is of two types:
A. Basic Simplex Method
B. Modified Simplex Method.

 It is an experimental techniques & mostly used in analytical rather
than formulation & processing. Simplex is a geometric figure that
has one more point than the number of  factors.
Example: If 2 independent variables then simplex are
represented as triangle.
 The strategy is to move towards a better response by moving away
from worst response.
 Applied to optimize CAPSULES, DIRECT COMPRESSION
TABLET), liquid systems (physical stability).
 It is also called as Downhill Simplex / Nelder-Mead Method.

The simplex method is especially appropriate when:
 Process performance is changing over time.
 More than three control variables are to be changed.
 The process requires a fresh optimization with each new lot of
material.
 The simplex method is based on an initial design of k+1,
where k is the number of variables.
 A k+1 geometric figure in a k-dimensional space is called a
simplex.
 The corners of this figure are called vertices.

21
In simplex lattice, the response may be plotted as 2D
(contour plotted) or 3D plots (response surface methodology)

22
 This method will find the true optimum of a response with fewer
trials than the non-systematic approaches or the one variable-at-a-
time method.

 There are sets of rules for the selection of the sequential vertices in
the procedure.
 Requires mathematical knowledge.

 It represents mathematical techniques.
 It is an extension of classic method and applied to a pharmaceutical
formulation and processing.
 This technique follows the second type of statistical design.
 This technique require that the experimentation be completed
before optimization so that the mathematical models can be
generates

 Determine constraints. Determine objective formulation.
 Change inequality constraints to equality constraints.
 Form the Lagrange function F.
 Partially differentiate the lagrangian function for each variable &
set derivatives equal to zero.
 Solve the set of simultaneous equations.
 Substitute the resulting values in objective functions.

 Unlike the Lagrangian method, do not require differentiability of
the objective function.
 It is defined by appropriate equations.
 Used for more than two independent variables.
 The response surface is searched by various methods to find the
combination of independent variables yielding an optimum.
 It takes five independent variables into account and is computer
assisted.
 Persons unfamiliar with mathematics of optimization & with no
previous computer experience could carry out an optimization
study.

 Takes five independent variables in to account
 Person unfamiliar with the mathematics of optimization and with no
previous computer experience could carry out an optimization
study.
 It do not require continuity and differentiability of function.

 One possible disadvantage of the procedure as it is set up is that not
all pharmaceutical responses will fit a second-order regression
model.

 Factorial experiment is an experiment whose design consist of two
or more factor each with different possible values or “levels”.
 FD technique introduced by “Fisher” in 1926.
 Factorial design applied in optimization techniques.

 Factors can be “Quantitative” (numerical number) or they are
qualitative. They may be names rather than numbers like Method 1,
site B, or present or absent.
 Factorial design depends on independent variables for development
of new formulation.
 Factorial design also depends on Levels as well as Coding.
There are three types of levels :
1) LOW
2) INTERMEDIATE
3) HIGH
Simultaneously CODING takes place for Levels:
1) For LOW = (-1)
2) For intermediate = (0)
3) For HIGH = (+1)

 Full FD
 Fractional FD

 A design in which every setting of every factor appears with setting
of every other factor is full factorial design.
 If there is a K factor, each at Z level, a Full FD is Zk.
(Levels)Factors (ZK)
Factorial Design:
22, 23, 32, 33
 22 FD: 2 factors, 2 levels = 4 runs
 23 FD: 3 factors, 2 levels =8 runs

 2 factors: X1 and X2 (Independent variables)
 2 levels: Low and High
 Coding: (-1), (+1)

 In three level factorial designs, three levels are use,
1) Low (-1)
2) Intermediate (0)
3) High (+1)

 In Full FD, as a number of factor or level increases, the number of
experiment required exceeds to unmanageable levels.
 In such cases, the number of experiments can be reduced
systemically and resulting design is called as Fractional factorial
design (FFD).
 Applied if no. of factor is more than 5.
 Levels combinations are chosen to provide sufficient information to
determine the factor effect.

 Homogenous fractional
 Mixed level fractional
 Box-Hunter
 Plackett-Burman
 Taguchi
 Latin square

Homogenous fractional: Useful when large number of factors must
be screened.
Mixed level Fractional : Useful when variety of factors needs to be
evaluated for main effects and higher level interactions can be
assumed to be negligible.
Plackett- Burman:
 It is a popular class of screening design.
 These designs are very efficient screening designs when only the
main effects are of interest.
 These are useful for detecting large main effects economically,
assuming all interactions are negligible when compared with
important main effects.

 Factorial designs are most efficient for the experiments involve the
study of the effects of two or more factors.
 By a factorial design, we mean that in each complete trial or
replication of the experiment all possible combination of the levels
of the factors is investigated.
 When Factors are arranged in a factorial design, they are often said
to be crossed.

In chromatographic condition responses can be
 Efficiency.
 Retention Factor.
 Asymmetry.
 Retention Time.
 Resolution.

 Experiments for a 23 factorial design:

 Data Analysis for 23

 It’s easier to study the combined effect of two or more factors
simultaneously and analyze their interrelationships.
 It has a wide range of factor combination are used.
 It saves time.
 It permits the evaluation of interaction effects.

 It’s complex when several factors are involved simultaneously.
 Wasting of time and experimental material.
 Increase in factor size leads to increase in block size which increase
the chance of error.

 Formulation and processing
 Clinical Chemistry
 Medicinal Chemistry
 High performance liquid chromatographic analysis
 Formulation of culture medium in virological studies
 Study of pharmacokinetic parameters.

 It's a graphical representation of results.
 A contour plot is a graphic representation of the relationship among
three numeric variables in two dimensions.
 Contour plots are the geometric illustration of responses, obtained
by plotting one independent variable versus another while holding
the magnitude of response level and other variable is constant.
 From full mode equation eliminate insignificant terms gives refined
equation or reduced equation.
 This refined equation or full equation is transferred in form of
graphs. That is known as contour plot or response surface
methodology plot.
 Two variables are X and Y axes and a third variable Z is for contour
levels.
 The contour levels are plotted as curves: the area between curves
can be colour coded to indicate interpolated values.

The contour plot is formed by:
 Horizontal axis: Independent variable 1.
 Vertical axis: Independent variable 2.
 Lines: Iso-response values.
 An additional variable may be required to specify the Z values
drawing the iso-lines.
 If the data (or function) do not form a regular grid, you typically
need to perform a 2-D interpolation to form a regular grid.

 A contour plot is a graphical technique for representing a 3-
dimensional surface by plotting constant 2 slices called contour on a
2-dimensional format.
 That is given a value for z lines are drawn for connecting the (x, y)
coordinates where that z value occur.
 The contour plot is an alternative to a 3-D surface plot.
 The DEX contour plot is specialized contour plot used in the design
of experiments. In particular, it is useful for full and fractional
design.
 The contour plot is used to answer the question, How does Z change
as a function of X and Y?

 This contour plot shows that the surface is symmetric and peaks in
the centre.

1. Gridded data stored in a matrix.
2. XYZ triplets.
1.Gridded Data
 Gridded data is stored in a 2D wave or "matrix wave". By itself, the
matrix wave defines a regular XY grid.
 The X and Y coordinates for the grid lines are set by the matrix
wave's row X scaling and column Y scaling.
2. XYZ DATA
 XYZ triplets may be stored in a matrix wave of three columns or in
three separate 1D waves each supplying X, Y or Z values.

 Sample Maximum or Minimum.
 Stationary Ridge
 Rising Ridge

 Contour plot helps in visualizing the response surface.
 Contour plots are useful for establishing desirable response values
and operating conditions.
 This plot shows how a response variable relates to two factors based
on a model equation.
 Points that have the same response are connected to produce
contour lines of constant responses.

32 Full Factorial Designs
 In this factorial design conc. of Gellan gum and conc. of sodium
alginate is selected as independent variable.
 % Entrapment efficiency (Y1), Swelling ratio (Y2) and T90 (Y3)
was selected as dependent variable.
 In this we want to formulation showing maximum entrapment
efficiency, maximum swelling and about 10-11 hours for 90 % drug
released.

The polynomial equations for three responses are:
Y1 = 96.84 + 1.63X1 + 2.75 X2 – 0.83 X1X2
Y2 = 6.15 +0.83 X1 + 0.50 X2
Y3 = 7.50 + 1.98 X1 0.96 X2 + 0.83 X22

 RSM is a statistical technique that is useful for developing, improving and
optimizing process.
 It is an extension of other statistical approaches including Regression and
DoE.
 It should be applied after initial phases of experimentation, where the vital
few factors have been identified.
 RSM can be defined as a statistical method that uses quantitative data from
appropriate experiments to determine and simultaneously solve
multivariate equations.
 A response surface is a graph of a response variable as a function of the
various factors.
 In statistics, RSM explores the relationships between several explanatory
variables and one more response variables.
 RSM is useful for the modelling and analysis of programs in which a
response of interest is influenced by several variables and the objective is
to optimize this response.
Example: finds the level of temperature (x1) and pressure (x2) to
maximize the yield (y) of a process.

 The first step to RSM is that you must have or generate a large
number of data points.
 The second step is to assume a response equation. The most
common equations are linear and quadratic. Interaction effects can
also be included.
 The objective is to generate a map of response, either in form of
contours, or as a 3- dimensional graph.
 Take the given equation and data and solve for the regression
coefficient.
 Calculate the R2 and adjusted R2 values.
 Quadratic equation was fitted to the data points. If R2 value and
adjusted R2 value is not near to 1 then add other effect like linear to
quadratic interaction.

1. 3 D response surfaces
Sample maximum and minimum
Stationary Ridge
Rising Ridge
Saddle or Minimax
2. Contour response surface
Sample maximum and minimum
Stationary Ridge

 Models are simple and polynomial.
 Include terms for interaction and curvature.
 Coefficients are usually established by regression analysis with a
computer program.
 Insignificant terms are discarded.

Y = 𝛽0 Constant
+ 𝛽1 X1 + 𝛽2 X2 main effect
+ 𝛽3 X1
2 + 𝛽4 X2
2 curvature
+ 𝛽5X1 X2 interaction
ε error

Y = 𝛽0 Constant
+ 𝛽1 X1 + 𝛽2 X2 + 𝛽3 X3 main effect
+ 𝛽11 X1
2 + 𝛽22 X2
2 + 𝛽33 X3
2 curvature
+ 𝛽12 X1 X2 + 𝛽13 X1 X3 + 𝛽23 X2 X3 interaction
ε error

Designs for fitting the first order model
 The orthogonal first order design
 X’X is a diagonal matrix
 2k factorial and fraction of the 2kseries in which main effects are not
aliased with each other.
 Besides factorial designs, include several observations at the center.
 Simplex design.

Designs for fitting the second order model
 Central composite design
 Sequential experimentation
 The variance of the predicted response at X
 Rotatable design
 Box behnken design
 Cubical region
 Rotatable central composite design

 To determine the factor levels that will simultaneously satisfy a set
of design specifications.
 To determine the optimum combination of factors that yields a
desired response and describes the response near the optimum.
 To determine how a specific response is affected by changes in the
level of the factors over the specified levels of interest.
 To achieve a quantitative understanding of the system behaviour
over the region tested.
 To product product properties throughout the region even at factor
combination not actually run.
 To find condition for process stability = insensitive spot.

 Large variation in the factors can be misleading (errors, bias, no
replication)
 Critical factors may not be correctly defined or specified.
 Range of levels of factors to narrow or to wide optimum cannot be
defined.
 Lack of use of good statistical principles.
 Over reliance of computer make sure the results make good sense.

Full model equation:
Y = 𝛽0 + 𝛽1 X1 + 𝛽2 X2 + 𝛽12X1 X2 + 𝛽11 X1
2 + 𝛽22 X2
2 + E
Where, 𝛽0 = constant
𝛽1 and 𝛽2 = coefficient of X1 and X2 variable
𝛽12 = coefficient of interaction
𝛽11, 𝛽22 = coefficient of quadratic terms = non linearity
X1 and X2 = variables
E = error
Full model equation = Main effect + Interaction effect + Quadratic
effect
 Prediction power is good for full model equation compare to reduce
model.
Example: Y = 7.84 + 1.98X1 + 0.96X2 - 0.1X1X2 – 0.51 X1
2 + 0.83X2
2

 From the full model equation less significant terms or insignificant
term is removed will gives reduced model or refined model.
Example: Y = 7.84 + 1.98X1 + 0.96X2 + 0.83X2
2

 Central composite design (CCD) is one of the most commonly used
optimization technique in response surface designs.
 CCD commonly called ‘Box-Wilson Central Composite Design’
because it was first developed by Box and Wilson.
 A CCD can be run sequentially. It can be naturally partitioned into
two subsets of points; the first subset estimates linear and two-
factor interaction effects while the second subset estimates
curvature effects. The second subset need not be run when analysis
of the data from the first subset points indicates the absence of
significant curvature effects.
 CCDs are very efficient, providing much information on experiment
variable effects and overall experimental error in a minimum
number of required runs.
 CCDs are very flexible. The availability of several varieties of
CCDs enables their use under different experimental regions of
interest and operability.

The experimenter must first understand the differences between
these varieties in terms of the experimental region of interest
and region of operability
 Region of interest --a geometric region defined by lower and upper
limits on study-variable level setting combinations that are of
interest to the experimenter
 Region of operability --a geometric region defined by lower and
upper limits on study-variable level setting combinations that can be
operationally achieved with acceptable safety and that will output a
testable product.

 The design consists of three different sets of experimental runs:
 A factorial (or fractional) design, each having two levels.
 A set of centre points, whose values of each factor are the medians
of the values used in the factorial portion.
 A set of axial points (star points), whose values of each factor are
below and above the values of the two factorial levels.
 Design gives us the box, and adding the axial points (in green)
outside of the box gives us a spherical design (Figure A)
 This design in k = 3 dimensions can also be referred to as a central
composite design, chosen so that the design is spherical. (Figure B)

 Central Composite Design (CCD) is generally used for fitting a
second-order response surface model.
 CCD contains an imbedded factorial or fractional factorial design
with center points that is augmented with a group of `star points'
that allow estimation of curvature.
 If the distance from the center of the design space to a factorial
point is ±1 unit for each factor, the distance from the center of the
design space to a star point is ±α with |α| > 1.

 The design matrix consists of three distinct types of matrix:
 The matrix F obtained from the factorial experiment with 2^k rows.
The factor levels are scaled and coded as +1 and −1.
 The matrix C from the centre points, denoted in coded variables as
(0,0,0,...,0).
 A matrix E from the axial points, with 2k rows. Each factor is
sequentially placed at ±α and all other factors are at zero.
 Generation of a central composite design for two factors

 Three types of CCD designs, which depend on where the star points
are placed.
 Three main varieties of CCD are available in most statistical
software programs:
1. Rotatable (The circumscribed) CCD [CCCD]
2. Face-centered CCD [CCFD]
3. The Inscribed CCD [CCID]

 CCCD are the original form of the central composite design.
 This design has circular, spherical, or hyperspherical symmetry and
requires 5levels for each factor.
 CCCD provide high quality predictions over the entire design
space, it require factor settings outside the range of the factors in the
factorial part.
 The star points are at some distance from the center based on the
properties desired for the design and the number of factors in the
design.

 Rotatable CCD for time and temperature using defined lower and
upper variable bounds.
 the rotatable CCD uses an a value of 1.4 to describe a circular
design geometry (a sphere for three variables, a hypersphere for
four or more variables, etc.).

 In this design the star points are at the center of each face of the
factorial space, so α±1.
 CCF designs provide relatively high quality predictions over the
entire design space and do not require using points out side the
original factor rang.
 Requires 3 levels for each factor.

 Face-centered CCD for two study variables: time and temperature.
Both coded and natural variable level settings for time and
temperature are shown in the figure.
 The design consists of a center point, four factorial points (the
intersection points of the 1 coded variable bounds) and four axial
points (points parallel to each variable axis on a circle of radius
equal to 1.0 and origin at the center point).
 The dots in show Figure identify the variable level setting
combinations that constitute the nine design points (experiment
runs).

 Do not provide the same high quality prediction over the entire
space compared to the CCC.
 This design also requires 5 levels of each factor.
 CCI designs use only points within the factor ranges originally
specified.

 The value of a is chosen to maintain rotatability
 To maintain rotatability, the value of α depends on the number of
experimental runs in the factorial portion of the central composite
design:
 α = [number of factorial run]1/4
 If the factorial is a full factorial, then α = [2k]1/4
 If the factorial is a fractional factorial, then α = [2k-f]1/4

Selection of models for CCD
 Models are simple polynomials
 Include terms for interaction and curvature
 Coefficients are usually established by regression analysis with a
computer program
 Insignificant terms are discarded

The most common empirical models fit to the CCD experimental
data take as a full quadratic second order model.
Model for two factors X1 and X2 and measured response Y
Y = βo constant
+ β1X1 + β2X2 main effects
+ β3X11 + β4X22 curvature
+ β5X1X2 interaction
+ ε error

Model for two factors X1, X2 and X3 and measured response Y
Y = βo constant
+ β1X1 + β2X2 + β3X3 main effects
+ β11X11 + β22X22 + β33X33 curvature
+ β12X1X2 + β13X1X3 + β23X2X3 interaction
+ ε error

 It is the extension of 2 level factorial and fractional FD.
 To estimate non linearity of response in the given data set.
 Used to estimate curvature in a continuous response.
 Maximum information in minimum experimental trial.
 Reduction in no. of trail required to estimate the square terms in
second order model.
 Widely used in RSM and optimization.

 Star point outside the hypercube, so the no. of levels that have to be
adjusted for every factor so sometimes it is difficult to achieve the
adjusted values of factors.
 Fewer degrees of freedom for estimation of error terms in model.
 Inability to estimate certain interaction terms.

 Taguchi method is a statistical method developed by Taguchi and
Konishi.
 Initially it was developed for improving the quality of goods
manufactured, later it was expanded to many other fields.
 Fields such as Engineering, Biotechnology, Marketing and
Advertising.
 Sometimes called robust design methods.

 The most common goals are minimizing cost, maximizing
throughout, and/or efficiency.
 This is one of the major quantitative tools in industrial decision
making.

 Taguchi method contains system design, parameter design, and
tolerance design procedures to achieve a robust process and result
for the best product quality.
 Taguchi designs provide a powerful and efficient method for
designing processes that operate consistently and optimally over a
variety of conditions.
 Experimental design methods were developed in the early years of
20th century but they were not easy to use.
 Taguchi's approach is easy to be adopted and applied for users with
limited knowledge of statistics.
 Hence it has gained a wide popularity in the engineering and
scientific community.

Taguchi specified three situations:
 Larger the better (for example, agricultural yield)
 Smaller the better (for example, carbon dioxide emissions)
 On-target, minimum-variation (for example, a mating part in an
assembly)

 A specific loss function
 The philosophy of off-line quality control
 Taguchi rule for manufacturing.

L (y) = K (y – m) 2
 The loss due to performance variation is proportional to the square
of the deviation of the performance characteristics from nominal
value.

Loss at a point: L(x) = k*(x-t) ^2
Where, k = loss coefficient
x = measured value
t = target value
Average Loss of a sample set: L = k*(s^2 + (pm - t) ^2)
Where, s = standard deviation of sample
pm = process mean
Total Loss = Avg. Loss * number of samples

A specific loss function:
 Used to measure financial loss to society resulting from poor
quality.
The philosophy of off-line quality control:
 Taguchi proposed a standard 8-step procedure for applying his
method for optimizing any process.

 Identify the main function and its side effects.
 Identify the testing condition and quality characteristics.
 Identify the objective function to be optimized.
 Identify the control factors and their levels.
 Select a suitable Orthogonal Array and construct the Matrix
 Conduct the Matrix experiment.
 Examine the data; predict the optimum control factor levels and its
performance.
 Conduct the verification experiment.
Sadranidolly/192650820008/Pharmaceutics/BNBSPC 97

The process has three stages:
 System design
 Parameter (measure) design
 Tolerance design

System design: Involving creativity and innovation.
Parameter (measure) design:
 Detail design phase.
 The parameters to be chosen so as to minimize the effects.
 This is sometimes called robustification.
Tolerance design:
 Resources to be focused on reducing and controlling variation in the
critical few dimensions.

 Dr Taguchi's Signal-to-Noise ratios (S/N), which are log functions
is based on “ORTHOGONAL ARRAY” experiments which gives
much reduced “variance” for the experiment with “optimum
settings “of control parameters.
 "Orthogonal Arrays" (OA) provide a set of well balanced desired
output, serve as objective functions for optimization, help in data
analysis and prediction of optimum results.

 The parameters affecting a process that can be controlled have been
determined, the levels at which these parameters should be varied
must be determined.
 The Taguchi method is a powerful tool for designing high quality
systems.
 If the difference between the minimum and maximum is large, the
values being tested further.

 Design and Communicate the Objective
 Define the Process
 Select a Response and Measurement System
 Ensure that the Measurement System is Adequate
 Select Factors to be studied
 Select the Experimental Design
 Set Factor Levels
 Final Design Considerations

 Determine the parameters signification (ANOVA)-Analysis of
variance.
 Conduct a main effect plot analysis to determine the optimal level
of the control factors.
 Execute a factor contribution rate analysis.
 Confirm experiment and plan future application.

WHY FACTORIAL METHODS DON'T WORK WELL FOR
MIXTURE?
 Here we take an example of lemonade.

 Here we can see that in run1 (both factors low) and run4 (both
factors high) taste the same.
 It makes more sense to look at taste as a function of proportion of
lemons to water, not the amount.
 Mixture design accounts for the dependence of response on
proportionality of ingredients.

 In a mixture experiment, the independent factors are proportions of
different components of a blend. The fact that the proportions must
sum up to a constant (usually 1 or 100%) makes this type of
experiment a class on its own.
 Response surfaces and optimal regions for formulation
characteristics are frequently obtained from the application of
simplex lattice design. This class of design is particularly
appropriate in formulation optimization procedures where the total
quantity of the different ingredients under consideration must be
constant. Therefore, these are also called as "mixture design".

 For example, suppose that in a liquid formulation, the active
ingredient and solvent compose 90% of the product. The remaining
10% of the formulation consists of preservatives, colouring agents
and a stabilizer. We wish to prepare a formulation with a certain
optimal attributes which is dependent on the relative concentration
of preservative, color and stabilizer. In order to determine optimal
regions we vary the concentration of these three ingredients is
10%.In this example the total amount of the varying components is
maintained at 10%.Given the concentration of two of the
ingredients the third ingredient is fixed where in this example C
10%-A-B.

 To find a mathematical model EQ for forecasting the values of the
response variable Q in term of its components Xi.
 They are typically first or second degree polynomials

To model the blending surface with some form of mathematical
equation so that:
 Prediction of the response for any mixture or combination of
ingredients can be made empirically.
 Some measure of influence on the response of each component
singly and in combination with other components can be obtained.

 Like fractional experiments it is assumed that the errors are
independent and identically distributed with zero mean and
common variance.
 Another assumption is that the true underlying response surface is
continuous over the region being studied.
 The response is assumed to be only dependent on the ingredients
proportions and not on the amount of the mixture itself.

1. Standard mixture design
 The mixture components which are subjected to the constraint that
they must sum to one.
Example: simple lattice and simplex centroid design.
2. Constrained mixture design
 When the mixture components are subjected to additional constraint
such as maximum and minimum value for each component.
 Also known as extreme vertices design.
3. Mixture experiments and independent variable experiments
 Mixture experiments and independent variable experiments is that
with former input variables or components are non-negative
proportionate amount of mixture and if expressed as fraction of
mixture, they must sum to one.

 The linear model is used when the effect of the components in
mixture are additive and the response variable can be defined
as a linear combination of their fractions.
 The quadratic model considers antagonistic or synergistic
interactions between the pairs of components of mixture.
 The special cubic model considers interactions among the
three components.
 Due to the restriction x1+x2+...+xq= 1,the form of the
regression function that is to fit the data from a mixture
experiment is somewhat different from the traditional
polynomial fit and is often referred to as CANONICAL
polynomial.

Linear: E (Q) = B1X1 + B2X2 + B3X3
Quadratic: E (Q) = B1X1 + B2X2 + B3X3 +B12X12 +B13X13+B23X23
Special cubic: E (Q) = B1X1 + B2X2 + B3X3 +B12X12 +B13X13+B23X23
+ B123X123
Where X1 + X2 + X3 = 1

 To study possible interaction between non barbiturate anaesthetics
and ethanol.
 To study a strain difference in the response of mice to a compound
causing bladder toxicity.
 To study the effect of compound which is not strongly toxic on
reproductive and behavioural parameter in mice.
Sadranidolly/192650820008/Pharmaceutics/BNBSPC 114

 Pharmaceutics statistics by Sanford Bolton, Charles Bon.
 Pharmaceutical experimental design by Gareth Lewis and Didier
Mathieu.
 Experimental Design and Patent by Vipul P. Patel and Hardik D.
Shihora.
 Jain N. K., Pharmaceutical product development, CBS publishers
and distributors, 1st edition, 297-302, 2006.
 Cooper L. and Steinberg D., Introduction to methods of
optimization, W.B. Saunders, Philadelphia, 1970, 1st Edition, 301-
305.
 Former D.E., Mathematical optimization techniques in drug product
design and process analysis, Journal of pharmaceutical sciences. ,
vol-59 (11), 1587-1195, November 1970.

Experimental design

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