Pharmaceutical Experimental Design and Interpretation

Transcription

1 Pharmaceutical Experimental Design and Interpretation N. ANTHONY ARMSTRONG, B. Pharm., Ph.D., F.R.Pharm.S., MCPP. KENNETH C. JAMES, M. Pharm., Ph.D., D.Sc, FRSC, F.R.Pharm.S., C.Chem. Welsh School of Pharmacy, University ol Wales, Cardiff, UK Tancis Publishers since 1798

2 1 Introduction to experimental design The experimental process Computers and experimental design Overview of experimental design and interpretation 4 2 Comparison of mean values Comparison of two means when the variance of the whole population is known Comparison of two means when the variance of the whole population is not known Comparison of means among more than two groups of data Analysis of variance Least significant difference Two-way analysis of variance 17 3 Non-parametric treatments Non-parametric tests for paired data The sign test The Wilcoxon signed rank test Non-parametric tests for unpaired data The Wilcoxon two-sample test 25 4 Correlation and regression Introduction Correlation Linear correlation Constitutional properties Resultant properties Linear regression The number of pairs of variables (n) The correlation coefficient (r) 33

3 4.3.3 The standard error of the estimate (s) The standard error of the coefficient The F value or variance ratio Inverse regression analysis Multiple regression analysis Correlation coefficients Standard error of the estimate Standard errors of the coefficients and the intercept F value Interaction between independent variables Stepwise regression Categorical data Curve fitting of non-linear relationships The power series Quadratic relationships Cubic equations Curve fitting with models Curve fitting without models Exponential plots Geometric plots Hyperbolic plots Rectangular hyperbolic plots Extrapolation Free-Wilson analysis 57 5 Multivariate methods Introduction Distance matrix Covariance matrix Correlation matrix Eigenvalues and eigenvectors 68 6 Cluster and discrimination analysis Cluster analysis Cartesian plots Andrews'plots Dendrograms Hierarchic or agglomerative methods Partitioning methods Discrimination 83 7 Principal components and factor analysis Principal components analysis Factor analysis Rotation 96 8 Sequential analysis Introduction Wald plots 107 vi

4 8.2.1 The sign test The sequential procedure Construction of barrier lines Bross plots Construction of barrier lines using the binomial theorem Confidence levels Prior distribution Triangular plots Calculation of barriers for triangular plots Truncation procedures Truncation using a vertical barrier Truncation using angled stopping lines Changing the confidence limits Truncation procedure for triangular plots Factorial design of experiments Two-factor, two-level experimental designs Notation in factorially designed experiments Factorial designs with interaction between factors Factorial designs with three factors Factorial designs and ANOVA Yates' treatment Linear regression Factorial designs with replication Factorial designs with three levels Three-factor, three-level factorial designs Blocks and fractional designs Blocked designs Fractional factorial designs Plackett-Burman designs , Central composite and other designs General comments on factorial design Model-dependent optimization and response surface methodology Optimization Model-dependent optimization Validation of the design and the regression equations Optimization when interaction occurs between the independent variables Use of coded data Second-order relationships between independent and dependent variables Optimization with three or more independent variables Optimization using the Pareto-optimality technique Model-independent optimization Optimization by simplex search Comparison of model-independent and model-dependent methods 199 vii

5 12 Experimental designs for mixtures Three component systems Mixtures with more than three components Optimization in experiments with mixtures Model-dependent methods Linear relationships between composition and response Higher order relationships between composition and response Derivation of contour plots Pareto-optimality and mixtures Process variables in mixture experiments 220 A1 Statistical tables 225 A 1.1 Cumulative normal distribution (Gaussian distribution) 225 A 1.2 Student's t distribution 225 A 1.3 Analysis of variance 226 A2 Computer programs in BASIC and MINITAB commands 229 A2.1 Calculation of mean, standard deviation etc. 229 A2.1.1 Insertion of data and instructions 231 A2.1.2 Calculation of mean etc. 232 A2.1.3 Standardization of data 232 A2.2 Linear regression 233 A2.2.1 Insertion of data and instructions 236 A2.2.2 Calculation of the regression equation etc. 236 A2.3 Parabolic curve fit 237 A2.3.1 Insertion of data and instructions 240 A2.3.2 Calculation of the regression equation etc. 241 A2.4 Three-variable regression 241 A2.4.1 Insertion of data and instructions 245 A2.4.2 Calculation of the regression equation etc. 246 A2.5 The determinant of a (3 x 3) matrix 246 A2.6 The determinant of a (4 x 4) matrix 248 A2.7 Determination of matrix parameters using MINITAB 249 A2.7.1 Insertion of data and instructions 250 A2.7.2 Standardization of data 250 A2.7.3 Calculation of covariance matrix 251 A2.7.4 Calculation of correlation matrix 251 A2.7.5 Calculation of eigenvalues and eigenvectors 251 A2.8 Three-factor, two-level factorial design 252 A2.8.1 Insertion of data and instructions 255 A2.8.2 Analysis of variance 256 A3 Sequential analysis grids 257 A3.1 A Wald grid for a probability level of IP = A3.2 A Wald grid for a probability level of IP = A3.3 A Bross grid for a probability level of IP = A3.4 A Bross grid for a probability level of IP = viii

6 A4 Matrices 261 A4.1 Introduction 261 A4.2 Addition and subtraction 263 A4.3 Multiplication 264 A4.3.1 Multiplying a matrix by a constant 264 A4.3.2 Multiplying a matrix by a column vector 264 A4.3.3 Multiplication of one matrix by another 264 A4.3.4 Multiplication by a unit matrix 265 A4.3.5 Multiplication by a null matrix 266 A4.4 Determinants 266 IX