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Design of Experiments

Presentation prepared by Drug Regulations – a not for
profit organization. Visit www.drugregulations.org for
             the latest in Pharmaceuticals.




                               www.drugragulations.org   1
Product Profile      Quality Target Product Profile (QTPP)


     CQA’s            Determine ―potential‖ critical quality attributes (CQAs)


Risk Assessments      Link raw material attributes and process parameters to
                       CQAs and perform risk assessment
  Design Space        Develop a design space (optional and not required)


Control Strategy      Design and implement a control strategy

    Continual         Manage product lifecycle, including continual
  Improvement
                       improvement


                                              www.drugragulations.org             2
Product Profile



     CQA’s
                      This presentation Part VI of the
Risk Assessments       series ―QbD for Beginners‖ covers
                       basic aspects of
  Design Space
                       ◦ Design of Experiments
Control Strategy


    Continual
  Improvement




                                         www.drugragulations.org   3
   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.
   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.



                             www.drugragulations.org   4
   Discovery
    ◦ Designed to generate new ideas or approaches
    ◦ Usually involve ―hands-on‖ activities
    ◦ May involve systems or processes that are not well
      understood or refined.
   Hypothesis
    ◦ Closer to the traditional academic approach
    ◦ Seek to falsify specific hypotheses
    ◦ Used often in the attempt to ―prove‖ a
      theory, idea, or approach


                                 www.drugragulations.org   5
   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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   Best Guess
    ◦ PRO: Works reasonably well when used by SMEs
      with solid foundational knowledge on known issues
    ◦ CONs: If it fails, need to guess again…and
      again…until….
    ◦ If get acceptable results first time, may stop without
      discovering ―better‖




                                  www.drugragulations.org      7
   One Factor at a Time
    ◦ PRO: Straight-forward, easily understood
    ◦ CONs: Impossible to address interactions
    ◦ Tends to ―over collect‖ data, not efficient sample
      sizes




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   Factorial
    ◦ PROs: Full evaluation of individual and interaction
      effects
    ◦ Most efficient design with respect to sample sizes
    ◦ CON: More complex to explain to untrained
      audiences




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   Replication
    ◦ Permits estimation of experimental error
    ◦ Permits more precise estimates of the sample
      statistics
    ◦ Not to be confused with repeated measures




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   Randomization
    ◦ 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




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   Blocking
    ◦ Designed to improve precision of comparisons
    ◦ Used to reduce or eliminate nuisance factors




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◦ 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, 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   14
   Treatments are the different procedures we
    want to compare.
   Experimental units are the things to which we
    apply the treatments.
   Responses are outcomes that we observe
    after applying a treatment to an experimental
    unit.




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   Randomization is the use of a
    known, understood probabilistic mechanism
    for the assignment of treatments to units.
   Other aspects of an experiment can also be
    randomized: for example,
    ◦ The order in which units are evaluated for their
      responses.




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   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.



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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.




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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)?
                             www.drugragulations.org   19
   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      20
   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.




                            www.drugragulations.org   22
   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   23
   Factors combine to form treatments.
   For example, the baking treatment for a cake
    involves a given time at a given temperature.
   The treatment is the combination of time and
    temperature, but we can vary the time and
    temperature separately.
   Thus we speak of a time factor and a
    temperature factor.
   Individual settings for each factor are called
    levels of the factor.



                             www.drugragulations.org   24
   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 planting wheat variety A in Punjab and
    Wheat variety B in M.P .
   In this experiment, we cannot distinguish
    location effects from variety effects—the variety
    factor and the location factor are confounded.

                              www.drugragulations.org   25
   A good experimental design must
    ◦   Avoid systematic error
    ◦   Be precise
    ◦   Allow estimation of error
    ◦   Have broad validity.




                                    www.drugragulations.org   26
   Comparative experiments estimate
    differences in response between treatments.
   If our experiment has systematic error, then
    our comparisons will be biased, no matter
    how precise our measurements are or how
    many experimental units we use.




                            www.drugragulations.org   27
   For example, if responses for units receiving
    treatment one are measured with instrument
    A, and responses for treatment two are
    measured with instrument B, then we don’t
    know if any observed differences are due to
    treatment effects or instrument Mis-
    calibrations.
   Randomization, is our main tool to combat
    systematic error


                             www.drugragulations.org   28
   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,
    ◦ The number of units used, and
    ◦ The experimental design used.
   There are several designs to improve precision.


                                     www.drugragulations.org   29
   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   30
   We characterize an experiment by the
    treatments and experimental units to be
    used, the way we assign the treatments to
    units, and the responses we measure.
   An experiment is randomized if the method
    for assigning treatments to units involves a
    known, well-understood probabilistic
    scheme. The probabilistic scheme is called a
    randomization.



                            www.drugragulations.org   31
   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   32
   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   33
   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   34
   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.
   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        35
   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       36
   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      37
   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   38
   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   39
   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    40
   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   41
   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   42
   Goal:
    ◦ Compare two or more means; variances; probabilities
    ◦ Compare A versus B: [better or worse] – paired
      comparison is a special case of randomized block design
   Major Considerations
    ◦ Sample size
    ◦ Distributional knowledge: Normal, χ2, F …..etc.
    ◦ Structure of the statistical hypothesis
      One-tailed




                                   www.drugragulations.org      43
   One Factor – Multiple Levels
    ◦   ―One-level-at-a-time‖ analysis isn’t efficient
    ◦   Consider one factor with five levels
    ◦   Pair-wise comparison requires 10 pairs [ 5C2 = 10 ]
    ◦   If each comparison has α = 0.05, then
    ◦   Probability(correct assessment) = (1-α)10 = 0.60




                                   www.drugragulations.org    44
   Technique of Choice – ANOVA
    ◦ Tests hypothesis H0: μ1 = μ2 = μ3 = … μn
   Assumptions
    ◦ Error term is Normal (0,σ2) ⇒ test residuals to confirm
    ◦ Conditions properly randomized
    ◦ Results are independent; errors are independent
   If reject H0 (i.e., failed the test) then use
    Newman-Keuls Range Test or Duncan’s Multiple
    Range Test to determine the specifics
   Note – there are non-parametric tests in lieu of
    ANOVA if assumptions are not met (e.g. Kruskal-
    Wallis Test)


                                    www.drugragulations.org     45
   Single Factor – Unit Days of Supply
   Levels – 5, 10, 15, 20, 25
                              www.drugragulations.org   46
    Latin Square: An arrangement of conditions
    such that each combination occurs only once
     in each row and column of the test matrix.

     A   B   C   D
     B   C   D   A                                   Orthogonal
                                                     Latin
                                                     Square
     C   D   A   B
     D   A   B   C

Graeco-Latin Square: The superposition of two Latin
 Squares such that each paired-combination occurs
        only once in each row and column.

                           www.drugragulations.org                47
   Definition: An experiment in which for each
    completed trial or replication of the experiment
    all possible combinations of the levels of the
    factors are investigated.
   Design Notation
    ◦ General Notation for 2-level experiment ⇒ 2k where k
      = number of factors
    ◦ 3 factors 2 levels each = 23 design
    ◦ Factors and Levels ⇒ example for 3 factors, 2 levels
    ◦ Aa Bb Cc
    ◦ A+A- B+B- C+C-
    ◦ (1) a b c


                                www.drugragulations.org   48
   All combinations are examined
    ◦ Example 23 design = 8 experimental settings:
    ◦ A+B+C+, A-B+C+, A+B-C+, A+B+C-, A-B-C+, A-B+C-
      , A+B-C- ,A-B-C-

   Effects Evaluated
    ◦   Main effects of single factors: A, B, C
    ◦   Second Order (2-factor) interactions: AB, AC, BC
    ◦   Third Order (3-factor) interactions: ABC
    ◦   In general, a 2k design evaluates all 1, 2, …k-1, k-
        factor effects



                                    www.drugragulations.org    49
   Advantages over ―one-factor-at-a-time‖
    ◦ More efficient in time, resources, sample size
    ◦ Addresses interactions
    ◦ Provides insight over a range of experimental
      conditions




                                 www.drugragulations.org   50
bc
                     abc                            A        B   C
                                                  -      -       --
     c                           ac               +      -       -
                                                  -      +       -
                     ab                           +      +       -
         C           +
                          B                       -      -       +
b
     _                          _                 +      -       +
             _
         1       A            + a                 -      +       +
                                                  +      +       +




                               www.drugragulations.org                51
DoE : number of experiments
D O E : Number of experiments




                                52
   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.




                                              53
   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).




                                              54
   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)



                                              55
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        56
   ANOVA
   Standard deviation
   Coefficient of variation
   R2 Measures the proportion of total variability explained by the model
   Adjusted R2 - adjusted for the number of Factors .If non-significant terms
    are ―forced‖ into the model this can decrease

   PRESS = Prediction Error Sum of Squares -                                                        A
    measure of how well the model will ―predict‖ new data. Smaller is better but can only be used in a
    comparative sense

   Linear regression


                                                            www.drugragulations.org                      57
   Tests of Significance
    ◦ Overall model response
    ◦ Individual coefficients
   Diagnostic tests
    ◦ Residuals
    ◦ Outliers
    ◦ Lack of Fit




                                www.drugragulations.org   58
   A way to reduce a huge full factorial to
    something manageable
    Considerations
    ◦ Required time, resources
    ◦ Complexity of set-up for experiments
   Major use is in screening experiments where
    the knowledge of basic effects is not well
    known
   If 2k is very large, may need to run reduced
    experiment


                               www.drugragulations.org   59
   Justification
    ◦ Sparsity of Effects – in general, even complex
      systems are usually driven by a few main effects
      and low-level interactions
    ◦ Projection Property – fractional factorial designs can
      be ―projected‖ into larger designs in the subset of
      significant factors
    ◦ Sequential Experimentation – can combine runs of 2
      or more fractional designs into larger designs




                                  www.drugragulations.org      60
   Issue:
    ◦ Confounding of Effects (also called ―aliasing‖) ⇒ reduced
      experiments do not evaluate all levels of the factors and
      their interactions .
    ◦ Some mixture of effects is ―confounded‖ and not
      identifiable
   Challenge:
    ◦ To select the best combination of test elements that stands
      a reasonable chance of revealing the true effects
    ◦ Alias the (most likely) insignificant or unwanted factors
   Symbology
    ◦ 2k-p designs



                                     www.drugragulations.org        61
L
Factorial analysis : N


                                             (L-X)
Semi Factorial analysis : N



                               Loss of resolution (aliase)

                                                            Number of factor
                                   2          3       4            5            6           7      8
                                  Full        1/2
 Number of experiments




                           4
                           8                 Full     1/2          1/4          1/8        1/16
                          16                         Full          1/2          1/4         1/8   1/16
                          32                                      Full          1/2         1/4    1/8
                          64                                                   Full         1/2    1/4
                         128                                                               Full    1/2
                         256                                                                      Full




                                                                                      62
22   Factional Factorial

      No of       Factors
       exp
              A               B
       1      +               -
       2      -               -
       3      +               +
       4      -               +




                   www.drugragulations.org   63
22   Factional Factorial

No of exp       Factors                Interaction
            A             B                     AB
   1        +             -                     -
   2        -             -                     +
   3        +             +                     +
   4        -             +                     -




                      www.drugragulations.org        64
23-1 Factional   Factorial

No of          Factors
 exp                                          C=AB
         A       B       C=AB
 1       +       -        -            C is confounding with AB
 2       -       -        +
 3       +       +        +
 4       -       +        -




                          www.drugragulations.org                 65
23-1 Factional      Factorial

No of     Factors and interaction
 exp
          A       B=AC       C=AB                       C=AB
 1        +         -          -               C is confounding with AB
 2        -         -          +                      B=AC
 3        +         +          +                B is confounding with AC
 4        -         +          -




                              www.drugragulations.org               66
23-1 Factional      Factorial


No of     Factors and interaction
 exp
         A=BC     B=AC       C=AB                       C=AB
 1        +         -          -              C is confounding with AB
 2        -         -          +                      B=AC
 3        +         +          +              B is confounding with AC
                                                     A=BC
 4        -         +          -
                                              A is confounding with BC




                              www.drugragulations.org               67
A    B    C     D                   E    Yield %
-1   -1   -1    -1                   1    49.8
 1   -1   -1    -1                  -1    51.2
-1   1    -1    -1                  -1    50.4
 1   1    -1    -1                   1    52.4
-1   -1   1     -1                  -1    49.2
 1   -1   1     -1                   1    67.1
-1   1    1     -1                   1    59.6
 1   1    1     -1                  -1    67.9
-1   -1   -1    1                   -1    59.3
 1   -1   -1    1                    1    70.4
-1   1    -1    1                    1    69.6
 1   1    -1    1                   -1      64
-1   -1   1     1                    1    53.1
 1   -1   1     1                   -1    63.2
-1   1    1     1                   -1    58.4
 1   1    1     1                    1    64.3


               www.drugragulations.org             68
B                                                    B

            +1                                                  +1
                 50.40                      52.40                     69.60                   64.00




    59.60                           67.90               58.40                       64.30




            -1   49.80                      51.20   A           -1    59.30                   70.40   A
                           -1                                                  -1                     +1

                   -1                                                   -1


    49.20                           67.10               53.10                       63.20



C    +1                                             C    +1
                 Left Cube is D = LOW                                Right Cube is D = HIGH

                         E = HIGH                         www.drugragulations.org
                                                                         E = LOW                      69
   Resolution ⇒ a measure of confounding



   Resolution III     2K-p III
    ◦ No main effect aliased with any other main effect
    ◦ Main effects are aliased with 2-factor interactions
    ◦ 2-factor interactions may be aliased with each other




                                  www.drugragulations.org    70
   Resolution ⇒ a measure of confounding



   Resolution IV      2K-p IV
    ◦ No main effect aliased with any other main effect
    ◦ No main effect aliased with 2-factor interactions
    ◦ 2-factor interactions may be aliased with each other




                                 www.drugragulations.org     71
   Resolution ⇒ a measure of confounding



   Resolution V       2K-p V
    ◦ No main effect aliased with any other main effect
    ◦ No main effect aliased with 2-factor interactions
    ◦ No 2-factor interactions may be aliased with each
      other
    ◦ 2-factor interactions are aliased with 3-factor
      interactions



                                www.drugragulations.org   72
DoE : number of experiments
D O E Resolution Trade off’s




                               73
   Three levels for k-factors (3k) designs
   Fractional 3-level designs (3k-p)
   Adding Center runs to
    ◦ Get estimates of process variability
    ◦ Gain familiarity with the process
    ◦ Identify system performance limits
   Mixture Designs – where one or more factors are
    constrained to add to something
    ◦ Usually have constraints like: x1 + x2 + x3 +…+xp = 1
    ◦ Example: A mixture of contributing probabilities
   Nested and Split-Plot designs for experiments
    with random factors


                                    www.drugragulations.org   74
   Irregular Fraction
    ◦ Usually a Resolution V design for 4 to 9 factors
      where each factor is varied over only 2 levels.
    ◦ Two-factor interactions aliased with three-factor
      and higher
    ◦ Excellent to reduce number of runs and still get
      clean results
   General Factorial
    ◦ For 1 to 12 factors where each factor may have a
      different number of levels
    ◦ All factors treated as categorical


                                 www.drugragulations.org   75
   D-Optimal
    ◦ A special design for categorical factors based on a
      analyst specified model
    ◦ Design will be a subset of the possible
      combinations
    ◦ Generated to minimize the error associated with the
      model coefficients




                                www.drugragulations.org     76
   Plackett-Burman
    ◦ Specialized design for 2 to 31 factors where each
      factor is varied over only 2 levels
    ◦ Use only if you can reasonably assume NO two-
      factor interactions; otherwise, use fractional
      factorial designs.




                                 www.drugragulations.org   77
Matrix

X1   X2   X3   X4   X5    X6   X7   X8   X9   X10   X11

1    1    1    1    1     1    1    1    1     1     1

1    1    1    -1   -1    -1   1    -1   -1    1    -1

1    1    -1   -1   -1    1    -1   -1   1    -1     1

1    -1   1    1    1     -1   -1   -1   1    -1    -1

1    -1   -1   1    -1    1    1    1    -1   -1    -1                         y( 1) y( 1)
1    -1   -1   -1   1     -1   -1   1    -1    1     1           Ex
-1   1    1    1    -1    -1   -1   1    -1   -1     1                             N /2
-1   1    -1   1    1     1    -1   -1   -1    1    -1

-1   1    -1   -1   1     -1   1    1    1    -1    -1

-1   -1   1    -1   1     1    1    -1   -1   -1     1

-1   -1   1    -1   -1    1    -1   1    1     1    -1

-1   -1   -1   1    -1    -1   1    -1   1     1     1




                                                          www.drugragulations.org            78
   Taguchi OA
    ◦ Saturated orthogonal arrays – all main effects and
      NO interactions
    ◦ Special attention must be paid to the alias structure
      for proper interpretation at both the design phase
      (prior to runs) and during final analysis




                                  www.drugragulations.org     79
   Goal: develop a model that describes a
    continuous curve, or surface, that connects
    the measured data taken at strategically
    important places in the experimental window




                           www.drugragulations.org   80
   RSM uses a least-squares curve-fit
    (regression analysis) to:
    ◦ calculate a system model (what is the process?)
    ◦ test its validity (does it fit?)
    ◦ analyze the model (how does it behave?)

                    Bond = f(temperature, pressure, duration)
                    Y = a0 + a1 T + a 2 P + a3 D
                            + a11T2 + a22P2 + a33D2
                            + a12TP + a13TD + a23PD




                                 www.drugragulations.org        81
RSM characteristics

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
Model equation for 2 factors

Y = β0                      constant
    + β 1X1 + β2X2          main effects
    + β 3X12 + β4X22        curvature
    + β 5X1X2               interaction
    + ε                     error                     Model equation for 3 factors
                                     Y=       β0                              constant
    Higher order interaction terms        +   β1X1 + β 2X2 + β3X3             main effects
    are not included                      +   β11X12 + β22X22 + β33X32        curvature
                                          +   β12X1X2 + β13X1X3 + β23X2X3     interactions
                                          +    ε                               error
                                                    www.drugragulations.org                  82
Response surface methodology
Central composite design (CCD)
 eg. 2 factor
                       Central composite circumscribed (CCC)

                         5 Levels
                         α (star point) are beyond levels



                      Central composite face centered (CCF)

                         3 Levels
                         α (star point) are within levels (center)



                        Central composite inscribed (CCI)
                         5 Levels
                         α (star point) are within levels
                         Scale down of CCC
                                        www.drugragulations.org      83
Response surface methodology
Central composite design (CCD)

                               Central composite circumscribed (CCC)

                 3 factors
     Total exp:                    20
         Full factorial      8                      Pattern      X1       X2       X3

         Axial points        6
                                                     +++           1       1        1
                                                     ++−           1       1        -1
                                                     +−+
         Center points       6
                                                                   1       -1       1
                                                     +−−           1       -1       -1
                                                     −++          -1       1        1
                                                     −+−          -1       1        -1
                                                     −−+          -1       -1       1
         -                                           −−−          -1       -1       -1

         ++                      ++                    0
                                                       0
                                                                   0
                                                                   0
                                                                           0
                                                                           0
                                                                                    0
                                                                                    0
    --                    +-     +                     0           0       0        0
                                                       0           0       0        0
    +                     +                            0           0       0        0
                                                       0           0       0        0
           -                      +                  00a           0       0    -1.68179
           +-                     +-
                                                     00A
                                                     0a0
                                                                   0
                                                                   0
                                                                           0
                                                                       -1.68179
                                                                                1.681793
                                                                                    0
                                                     0A0           0   1.681793     0
     -                                               A00      1.681793     0        0
    --                    +-                         a00      -1.68179     0        0

                          -                   www.drugragulations.org                      84
Response surface methodology
Central composite design (CCD)

                            Central composite circumscribed (CCC)
  Randomization:

 To avoid effect of uncontrollable nuisance variables
                                                 Pattern      X1       X2       X3
                                                  000           0       0        0
                                                  A00      1.681793     0        0
                                                  ++−           1       1        -1
                                                  00a           0       0    -1.68179
                                                  −−+          -1       -1       1
                                                  000           0       0        0
                                                  0a0           0   -1.68179     0
         -                                        −−−          -1       -1       -1

         ++                  ++                   −+−
                                                  000
                                                               -1
                                                                0
                                                                        1
                                                                        0
                                                                                 -1
                                                                                 0
    --                 +-    +                    +−+           1       -1       1
                                                  000           0       0        0
    +                  +                          000           0       0        0
                                                  a00      -1.68179     0        0
          -                   +                   000           0       0        0
          +-                  +-
                                                  +−−
                                                  −++
                                                                1
                                                               -1
                                                                        -1
                                                                        1
                                                                                 -1
                                                                                 1
                                                  00A           0       0    1.681793
     -                                            0A0           0   1.681793     0
    --                 +-                         +++           1       1        1

                       -                   www.drugragulations.org                      85
Response surface methodology
Central composite design (CCD)

                            Central composite circumscribed (CCC)
  Blocking:

 To avoid effect of controllable nuisance variables
                                                  Pattern      X1       X2       X3      Block
                                                   −−−          -1       -1       -1       1
                                                   −++          -1       1        1        1
                                                   ++−           1       1        -1       1
                                                   +−+           1       -1       1        1
                                                   000           0       0        0        1
                                                   000           0       0        0        1
              -                ++                  −+−          -1       1        -1       2
                                                   000           0       0        0        2
              +                +                   −−+          -1       -1       1        2
              +                                    000           0       0        0        2
     --                +-                          +++           1       1        1        2
                                                   +−−
                       +
                                                                 1       -1       -1       2
     +                                             000           0       0        0        3
                                                   000           0       0        0        3
          -                    +                   a00      -1.63299     0        0        3
          +                    +-                  0a0
                                                   A00
                                                                 0
                                                            1.632993
                                                                     -1.63299
                                                                         0
                                                                                  0
                                                                                  0
                                                                                           3
                                                                                           3
          -
     --                +-                          0A0
                                                   00a
                                                                 0
                                                                 0
                                                                     1.632993
                                                                         0
                                                                                  0
                                                                              -1.63299
                                                                                           3
                                                                                           3
     -                 -                           00A           0       0    1.632993     3

                                            www.drugragulations.org                              86
Response surface methodology
Box Behnen

   It is portion of 3k Factorial
   3 levels of each factor is used
   Center points should be included
   It is possible to estimate main effects and second order terms
   Box-Behnken experiments are particularly useful if
  some       boundary areas of the design region are
  infeasible, such as the extremes of the experiment
  region




                                       eg. 3 factor




                                          www.drugragulations.org
                                                                    12 experiments   87
Response surface methodology
Comparison of RSM experiments




       * One third replicate is used for a 3 k factorial design and one-
       half replicate is used for a 2 k factorial design with the CCD for
       5, 6 and 7 factors.



                                         www.drugragulations.org            88
(1) Choose experimental design
    (e.g., full factorial, d-optimal)       (2) Conduct randomized
                                                 experiments
                                              Experiment    Factor A   Factor B   Factor C

                                                  1           +          -          -
      A
                                                  2           -          +          -
                                                  3           +          +          +
          B
                C                                 4           +          -          +
(3) Analyze data
                                           (4) Create multidimensional
                                                 surface model
                                               (for optimization or control)




                        www.minitab.com
                                          www.drugragulations.org
                                                                                             89
Product Profile      Quality Target Product Profile (QTPP)


     CQA’s            Determine ―potential‖ critical quality attributes (CQAs)


Risk Assessments      Link raw material attributes and process parameters to
                       CQAs and perform risk assessment
  Design Space        Develop a design space (optional and not required)


Control Strategy      Design and implement a control strategy

    Continual         Manage product lifecycle, including continual
  Improvement
                       improvement


                                              www.drugragulations.org             90

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Quality by Design : Design of experiments

  • 1. Design of Experiments Presentation prepared by Drug Regulations – a not for profit organization. Visit www.drugregulations.org for the latest in Pharmaceuticals. www.drugragulations.org 1
  • 2. Product Profile  Quality Target Product Profile (QTPP) CQA’s  Determine ―potential‖ critical quality attributes (CQAs) Risk Assessments  Link raw material attributes and process parameters to CQAs and perform risk assessment Design Space  Develop a design space (optional and not required) Control Strategy  Design and implement a control strategy Continual  Manage product lifecycle, including continual Improvement improvement www.drugragulations.org 2
  • 3. Product Profile CQA’s  This presentation Part VI of the Risk Assessments series ―QbD for Beginners‖ covers basic aspects of Design Space ◦ Design of Experiments Control Strategy Continual Improvement www.drugragulations.org 3
  • 4. 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.  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. www.drugragulations.org 4
  • 5. Discovery ◦ Designed to generate new ideas or approaches ◦ Usually involve ―hands-on‖ activities ◦ May involve systems or processes that are not well understood or refined.  Hypothesis ◦ Closer to the traditional academic approach ◦ Seek to falsify specific hypotheses ◦ Used often in the attempt to ―prove‖ a theory, idea, or approach www.drugragulations.org 5
  • 6. 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 6
  • 7. Best Guess ◦ PRO: Works reasonably well when used by SMEs with solid foundational knowledge on known issues ◦ CONs: If it fails, need to guess again…and again…until…. ◦ If get acceptable results first time, may stop without discovering ―better‖ www.drugragulations.org 7
  • 8. One Factor at a Time ◦ PRO: Straight-forward, easily understood ◦ CONs: Impossible to address interactions ◦ Tends to ―over collect‖ data, not efficient sample sizes www.drugragulations.org 8
  • 9. Factorial ◦ PROs: Full evaluation of individual and interaction effects ◦ Most efficient design with respect to sample sizes ◦ CON: More complex to explain to untrained audiences www.drugragulations.org 9
  • 10. Replication ◦ Permits estimation of experimental error ◦ Permits more precise estimates of the sample statistics ◦ Not to be confused with repeated measures www.drugragulations.org 10
  • 11. Randomization ◦ 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 11
  • 12. Blocking ◦ Designed to improve precision of comparisons ◦ Used to reduce or eliminate nuisance factors www.drugragulations.org 12
  • 13. ◦ 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 13
  • 14. 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 14
  • 15. Treatments are the different procedures we want to compare.  Experimental units are the things to which we apply the treatments.  Responses are outcomes that we observe after applying a treatment to an experimental unit. www.drugragulations.org 15
  • 16. Randomization is the use of a known, understood probabilistic mechanism for the assignment of treatments to units.  Other aspects of an experiment can also be randomized: for example, ◦ The order in which units are evaluated for their responses. www.drugragulations.org 16
  • 17. 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 17
  • 18. Measurement units (or response units) are the actual objects on which the response is measured.  These may differ from the experimental units. www.drugragulations.org 18
  • 19. 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 19
  • 20. 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 20
  • 21. 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 21
  • 22. 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 22
  • 23. 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 23
  • 24. Factors combine to form treatments.  For example, the baking treatment for a cake involves a given time at a given temperature.  The treatment is the combination of time and temperature, but we can vary the time and temperature separately.  Thus we speak of a time factor and a temperature factor.  Individual settings for each factor are called levels of the factor. www.drugragulations.org 24
  • 25. 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 planting wheat variety A in Punjab and Wheat variety B in M.P .  In this experiment, we cannot distinguish location effects from variety effects—the variety factor and the location factor are confounded. www.drugragulations.org 25
  • 26. A good experimental design must ◦ Avoid systematic error ◦ Be precise ◦ Allow estimation of error ◦ Have broad validity. www.drugragulations.org 26
  • 27. Comparative experiments estimate differences in response between treatments.  If our experiment has systematic error, then our comparisons will be biased, no matter how precise our measurements are or how many experimental units we use. www.drugragulations.org 27
  • 28. For example, if responses for units receiving treatment one are measured with instrument A, and responses for treatment two are measured with instrument B, then we don’t know if any observed differences are due to treatment effects or instrument Mis- calibrations.  Randomization, is our main tool to combat systematic error www.drugragulations.org 28
  • 29. 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, ◦ The number of units used, and ◦ The experimental design used.  There are several designs to improve precision. www.drugragulations.org 29
  • 30. 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 30
  • 31. We characterize an experiment by the treatments and experimental units to be used, the way we assign the treatments to units, and the responses we measure.  An experiment is randomized if the method for assigning treatments to units involves a known, well-understood probabilistic scheme. The probabilistic scheme is called a randomization. www.drugragulations.org 31
  • 32. 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 32
  • 33. 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 33
  • 34. 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 34
  • 35. 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.  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 35
  • 36. 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 36
  • 37. 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 37
  • 38. 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 38
  • 39. 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 39
  • 40. 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 40
  • 41. 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 41
  • 42. 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 42
  • 43. Goal: ◦ Compare two or more means; variances; probabilities ◦ Compare A versus B: [better or worse] – paired comparison is a special case of randomized block design  Major Considerations ◦ Sample size ◦ Distributional knowledge: Normal, χ2, F …..etc. ◦ Structure of the statistical hypothesis  One-tailed www.drugragulations.org 43
  • 44. One Factor – Multiple Levels ◦ ―One-level-at-a-time‖ analysis isn’t efficient ◦ Consider one factor with five levels ◦ Pair-wise comparison requires 10 pairs [ 5C2 = 10 ] ◦ If each comparison has α = 0.05, then ◦ Probability(correct assessment) = (1-α)10 = 0.60 www.drugragulations.org 44
  • 45. Technique of Choice – ANOVA ◦ Tests hypothesis H0: μ1 = μ2 = μ3 = … μn  Assumptions ◦ Error term is Normal (0,σ2) ⇒ test residuals to confirm ◦ Conditions properly randomized ◦ Results are independent; errors are independent  If reject H0 (i.e., failed the test) then use Newman-Keuls Range Test or Duncan’s Multiple Range Test to determine the specifics  Note – there are non-parametric tests in lieu of ANOVA if assumptions are not met (e.g. Kruskal- Wallis Test) www.drugragulations.org 45
  • 46. Single Factor – Unit Days of Supply  Levels – 5, 10, 15, 20, 25 www.drugragulations.org 46
  • 47. Latin Square: An arrangement of conditions such that each combination occurs only once in each row and column of the test matrix. A B C D B C D A Orthogonal Latin Square C D A B D A B C Graeco-Latin Square: The superposition of two Latin Squares such that each paired-combination occurs only once in each row and column. www.drugragulations.org 47
  • 48. Definition: An experiment in which for each completed trial or replication of the experiment all possible combinations of the levels of the factors are investigated.  Design Notation ◦ General Notation for 2-level experiment ⇒ 2k where k = number of factors ◦ 3 factors 2 levels each = 23 design ◦ Factors and Levels ⇒ example for 3 factors, 2 levels ◦ Aa Bb Cc ◦ A+A- B+B- C+C- ◦ (1) a b c www.drugragulations.org 48
  • 49. All combinations are examined ◦ Example 23 design = 8 experimental settings: ◦ A+B+C+, A-B+C+, A+B-C+, A+B+C-, A-B-C+, A-B+C- , A+B-C- ,A-B-C-  Effects Evaluated ◦ Main effects of single factors: A, B, C ◦ Second Order (2-factor) interactions: AB, AC, BC ◦ Third Order (3-factor) interactions: ABC ◦ In general, a 2k design evaluates all 1, 2, …k-1, k- factor effects www.drugragulations.org 49
  • 50. Advantages over ―one-factor-at-a-time‖ ◦ More efficient in time, resources, sample size ◦ Addresses interactions ◦ Provides insight over a range of experimental conditions www.drugragulations.org 50
  • 51. bc abc A B C - - -- c ac + - - - + - ab + + - C + B - - + b _ _ + - + _ 1 A + a - + + + + + www.drugragulations.org 51
  • 52. DoE : number of experiments D O E : Number of experiments 52
  • 53. 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. 53
  • 54. 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). 54
  • 55. 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) 55
  • 56. 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 56
  • 57. ANOVA  Standard deviation  Coefficient of variation  R2 Measures the proportion of total variability explained by the model  Adjusted R2 - adjusted for the number of Factors .If non-significant terms are ―forced‖ into the model this can decrease  PRESS = Prediction Error Sum of Squares - A measure of how well the model will ―predict‖ new data. Smaller is better but can only be used in a comparative sense  Linear regression www.drugragulations.org 57
  • 58. Tests of Significance ◦ Overall model response ◦ Individual coefficients  Diagnostic tests ◦ Residuals ◦ Outliers ◦ Lack of Fit www.drugragulations.org 58
  • 59. A way to reduce a huge full factorial to something manageable  Considerations ◦ Required time, resources ◦ Complexity of set-up for experiments  Major use is in screening experiments where the knowledge of basic effects is not well known  If 2k is very large, may need to run reduced experiment www.drugragulations.org 59
  • 60. Justification ◦ Sparsity of Effects – in general, even complex systems are usually driven by a few main effects and low-level interactions ◦ Projection Property – fractional factorial designs can be ―projected‖ into larger designs in the subset of significant factors ◦ Sequential Experimentation – can combine runs of 2 or more fractional designs into larger designs www.drugragulations.org 60
  • 61. Issue: ◦ Confounding of Effects (also called ―aliasing‖) ⇒ reduced experiments do not evaluate all levels of the factors and their interactions . ◦ Some mixture of effects is ―confounded‖ and not identifiable  Challenge: ◦ To select the best combination of test elements that stands a reasonable chance of revealing the true effects ◦ Alias the (most likely) insignificant or unwanted factors  Symbology ◦ 2k-p designs www.drugragulations.org 61
  • 62. L Factorial analysis : N (L-X) Semi Factorial analysis : N Loss of resolution (aliase) Number of factor 2 3 4 5 6 7 8 Full 1/2 Number of experiments 4 8 Full 1/2 1/4 1/8 1/16 16 Full 1/2 1/4 1/8 1/16 32 Full 1/2 1/4 1/8 64 Full 1/2 1/4 128 Full 1/2 256 Full 62
  • 63. 22 Factional Factorial No of Factors exp A B 1 + - 2 - - 3 + + 4 - + www.drugragulations.org 63
  • 64. 22 Factional Factorial No of exp Factors Interaction A B AB 1 + - - 2 - - + 3 + + + 4 - + - www.drugragulations.org 64
  • 65. 23-1 Factional Factorial No of Factors exp C=AB A B C=AB 1 + - - C is confounding with AB 2 - - + 3 + + + 4 - + - www.drugragulations.org 65
  • 66. 23-1 Factional Factorial No of Factors and interaction exp A B=AC C=AB C=AB 1 + - - C is confounding with AB 2 - - + B=AC 3 + + + B is confounding with AC 4 - + - www.drugragulations.org 66
  • 67. 23-1 Factional Factorial No of Factors and interaction exp A=BC B=AC C=AB C=AB 1 + - - C is confounding with AB 2 - - + B=AC 3 + + + B is confounding with AC A=BC 4 - + - A is confounding with BC www.drugragulations.org 67
  • 68. A B C D E Yield % -1 -1 -1 -1 1 49.8 1 -1 -1 -1 -1 51.2 -1 1 -1 -1 -1 50.4 1 1 -1 -1 1 52.4 -1 -1 1 -1 -1 49.2 1 -1 1 -1 1 67.1 -1 1 1 -1 1 59.6 1 1 1 -1 -1 67.9 -1 -1 -1 1 -1 59.3 1 -1 -1 1 1 70.4 -1 1 -1 1 1 69.6 1 1 -1 1 -1 64 -1 -1 1 1 1 53.1 1 -1 1 1 -1 63.2 -1 1 1 1 -1 58.4 1 1 1 1 1 64.3 www.drugragulations.org 68
  • 69. B B +1 +1 50.40 52.40 69.60 64.00 59.60 67.90 58.40 64.30 -1 49.80 51.20 A -1 59.30 70.40 A -1 -1 +1 -1 -1 49.20 67.10 53.10 63.20 C +1 C +1 Left Cube is D = LOW Right Cube is D = HIGH E = HIGH www.drugragulations.org E = LOW 69
  • 70. Resolution ⇒ a measure of confounding  Resolution III 2K-p III ◦ No main effect aliased with any other main effect ◦ Main effects are aliased with 2-factor interactions ◦ 2-factor interactions may be aliased with each other www.drugragulations.org 70
  • 71. Resolution ⇒ a measure of confounding  Resolution IV 2K-p IV ◦ No main effect aliased with any other main effect ◦ No main effect aliased with 2-factor interactions ◦ 2-factor interactions may be aliased with each other www.drugragulations.org 71
  • 72. Resolution ⇒ a measure of confounding  Resolution V 2K-p V ◦ No main effect aliased with any other main effect ◦ No main effect aliased with 2-factor interactions ◦ No 2-factor interactions may be aliased with each other ◦ 2-factor interactions are aliased with 3-factor interactions www.drugragulations.org 72
  • 73. DoE : number of experiments D O E Resolution Trade off’s 73
  • 74. Three levels for k-factors (3k) designs  Fractional 3-level designs (3k-p)  Adding Center runs to ◦ Get estimates of process variability ◦ Gain familiarity with the process ◦ Identify system performance limits  Mixture Designs – where one or more factors are constrained to add to something ◦ Usually have constraints like: x1 + x2 + x3 +…+xp = 1 ◦ Example: A mixture of contributing probabilities  Nested and Split-Plot designs for experiments with random factors www.drugragulations.org 74
  • 75. Irregular Fraction ◦ Usually a Resolution V design for 4 to 9 factors where each factor is varied over only 2 levels. ◦ Two-factor interactions aliased with three-factor and higher ◦ Excellent to reduce number of runs and still get clean results  General Factorial ◦ For 1 to 12 factors where each factor may have a different number of levels ◦ All factors treated as categorical www.drugragulations.org 75
  • 76. D-Optimal ◦ A special design for categorical factors based on a analyst specified model ◦ Design will be a subset of the possible combinations ◦ Generated to minimize the error associated with the model coefficients www.drugragulations.org 76
  • 77. Plackett-Burman ◦ Specialized design for 2 to 31 factors where each factor is varied over only 2 levels ◦ Use only if you can reasonably assume NO two- factor interactions; otherwise, use fractional factorial designs. www.drugragulations.org 77
  • 78. Matrix X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 y( 1) y( 1) 1 -1 -1 -1 1 -1 -1 1 -1 1 1 Ex -1 1 1 1 -1 -1 -1 1 -1 -1 1 N /2 -1 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 1 www.drugragulations.org 78
  • 79. Taguchi OA ◦ Saturated orthogonal arrays – all main effects and NO interactions ◦ Special attention must be paid to the alias structure for proper interpretation at both the design phase (prior to runs) and during final analysis www.drugragulations.org 79
  • 80. Goal: develop a model that describes a continuous curve, or surface, that connects the measured data taken at strategically important places in the experimental window www.drugragulations.org 80
  • 81. RSM uses a least-squares curve-fit (regression analysis) to: ◦ calculate a system model (what is the process?) ◦ test its validity (does it fit?) ◦ analyze the model (how does it behave?) Bond = f(temperature, pressure, duration) Y = a0 + a1 T + a 2 P + a3 D + a11T2 + a22P2 + a33D2 + a12TP + a13TD + a23PD www.drugragulations.org 81
  • 82. RSM characteristics 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 Model equation for 2 factors Y = β0 constant + β 1X1 + β2X2 main effects + β 3X12 + β4X22 curvature + β 5X1X2 interaction + ε error Model equation for 3 factors Y= β0 constant Higher order interaction terms + β1X1 + β 2X2 + β3X3 main effects are not included + β11X12 + β22X22 + β33X32 curvature + β12X1X2 + β13X1X3 + β23X2X3 interactions + ε error www.drugragulations.org 82
  • 83. Response surface methodology Central composite design (CCD) eg. 2 factor Central composite circumscribed (CCC) 5 Levels α (star point) are beyond levels Central composite face centered (CCF) 3 Levels α (star point) are within levels (center) Central composite inscribed (CCI) 5 Levels α (star point) are within levels Scale down of CCC www.drugragulations.org 83
  • 84. Response surface methodology Central composite design (CCD) Central composite circumscribed (CCC) 3 factors Total exp: 20 Full factorial 8 Pattern X1 X2 X3 Axial points 6 +++ 1 1 1 ++− 1 1 -1 +−+ Center points 6 1 -1 1 +−− 1 -1 -1 −++ -1 1 1 −+− -1 1 -1 −−+ -1 -1 1 - −−− -1 -1 -1 ++ ++ 0 0 0 0 0 0 0 0 -- +- + 0 0 0 0 0 0 0 0 + + 0 0 0 0 0 0 0 0 - + 00a 0 0 -1.68179 +- +- 00A 0a0 0 0 0 -1.68179 1.681793 0 0A0 0 1.681793 0 - A00 1.681793 0 0 -- +- a00 -1.68179 0 0 - www.drugragulations.org 84
  • 85. Response surface methodology Central composite design (CCD) Central composite circumscribed (CCC) Randomization: To avoid effect of uncontrollable nuisance variables Pattern X1 X2 X3 000 0 0 0 A00 1.681793 0 0 ++− 1 1 -1 00a 0 0 -1.68179 −−+ -1 -1 1 000 0 0 0 0a0 0 -1.68179 0 - −−− -1 -1 -1 ++ ++ −+− 000 -1 0 1 0 -1 0 -- +- + +−+ 1 -1 1 000 0 0 0 + + 000 0 0 0 a00 -1.68179 0 0 - + 000 0 0 0 +- +- +−− −++ 1 -1 -1 1 -1 1 00A 0 0 1.681793 - 0A0 0 1.681793 0 -- +- +++ 1 1 1 - www.drugragulations.org 85
  • 86. Response surface methodology Central composite design (CCD) Central composite circumscribed (CCC) Blocking: To avoid effect of controllable nuisance variables Pattern X1 X2 X3 Block −−− -1 -1 -1 1 −++ -1 1 1 1 ++− 1 1 -1 1 +−+ 1 -1 1 1 000 0 0 0 1 000 0 0 0 1 - ++ −+− -1 1 -1 2 000 0 0 0 2 + + −−+ -1 -1 1 2 + 000 0 0 0 2 -- +- +++ 1 1 1 2 +−− + 1 -1 -1 2 + 000 0 0 0 3 000 0 0 0 3 - + a00 -1.63299 0 0 3 + +- 0a0 A00 0 1.632993 -1.63299 0 0 0 3 3 - -- +- 0A0 00a 0 0 1.632993 0 0 -1.63299 3 3 - - 00A 0 0 1.632993 3 www.drugragulations.org 86
  • 87. Response surface methodology Box Behnen  It is portion of 3k Factorial  3 levels of each factor is used  Center points should be included  It is possible to estimate main effects and second order terms  Box-Behnken experiments are particularly useful if some boundary areas of the design region are infeasible, such as the extremes of the experiment region eg. 3 factor www.drugragulations.org 12 experiments 87
  • 88. Response surface methodology Comparison of RSM experiments * One third replicate is used for a 3 k factorial design and one- half replicate is used for a 2 k factorial design with the CCD for 5, 6 and 7 factors. www.drugragulations.org 88
  • 89. (1) Choose experimental design (e.g., full factorial, d-optimal) (2) Conduct randomized experiments Experiment Factor A Factor B Factor C 1 + - - A 2 - + - 3 + + + B C 4 + - + (3) Analyze data (4) Create multidimensional surface model (for optimization or control) www.minitab.com www.drugragulations.org 89
  • 90. Product Profile  Quality Target Product Profile (QTPP) CQA’s  Determine ―potential‖ critical quality attributes (CQAs) Risk Assessments  Link raw material attributes and process parameters to CQAs and perform risk assessment Design Space  Develop a design space (optional and not required) Control Strategy  Design and implement a control strategy Continual  Manage product lifecycle, including continual Improvement improvement www.drugragulations.org 90