Design and Analysis of Experiments | DoE

Explore the design and analysis of single-factor experiments with multiple levels, using cotton content to show how randomization and replicates enable analysis of variance to compare level means.

Explore the one-way anova with the effects model, describing observations as an overall mean plus treatment effects and random error, and examine tau_i's null and alternative hypotheses.

Explore how the total sum of squares in analysis of variance decomposes into the sum of squares of treatments and error, linking variance, treatment means, and overall mean.

Use the anova table to decompose the total sum of squares into treatments and error, compute the mean squares, and apply the f-test using alpha and excel for calculations.

Select the data range and run the one-way anova in excel using the data analysis toolpak, with rows representing cotton percentages and replicates of tensile strength tests.

Analyze an Excel anova of cotton percentage treatments on tensile strength, showing f of 14.76 and p-value 9.01e-6. Conclude that at least one cotton percentage affects tensile strength.

Compare treatment means after analysis of variance to identify which cotton percentage changes tensile strength, using a t-distribution visualization to assess overlaps among 15, 20, 25, and 30 percent.

Apply Tukey’s test to compare treatment means in pairs using the error estimate, calculating T alpha from q alpha of p and f, with replicates in an ANOVA context.

Explain how Tukey's test uses superscript letters to mark significant differences among treatment means, illustrating the procedure with a carotenoid extraction table across temperatures and pre-treatment conditions.

Present results with tables and graphs showing means, standard deviations, and Tukey's test, noting 30% cotton yields highest tensile strength with no difference from 25%, and plan factorial designs.

Explore factorial designs to analyze several factors simultaneously by examining all possible level combinations in each trial, including crossed designs with ab treatments.

Explore main effects and interactions in a two-factor factorial experiment using commute time example, where leaving time and route influence commute time and reveal when effects depend on each other.

Explore how factorial designs reveal interactions between factors, including temperature and pH, avoid misleading conclusions from one-factor-at-a-time experiments, and efficiently maximize yield by analyzing these factors.

Analyze a two-factor factorial design with material type and temperature, nine treatments, and four replicates in a completely randomized design to assess main effects and the interaction via ANOVA.

Learn to solve a two-factor factorial design in Excel by performing a two-way analysis of variance with replication, analyzing material type and temperature, and interpreting the interaction and error terms.

Analyze how temperature, material, and their interaction affect battery life using an analysis of variance, identifying significant effects via F-zero, F-critical, and p-values.

Apply Tukey's test after anova to compare treatment means of battery life; conclude -10°C shows no differences, 20°C: materials 2 and 3 differ from 1, 50°C: none differ.

Explore 2^k factorial designs, where each of k factors is tested at two levels—numeric or categorical—and use to screen factors before optimization.

Explore the 2^2 factorial design by assessing substrate concentration (15 vs 25 g/L) and enzyme concentration (1 vs 2 g/L) on yield, with -1 and +1 coded, 12 randomized runs.

Represent a 2^2 factorial design on a square with substrate and enzyme as axes, labeling treatments 1, a, b, ab, and adding a totals column; lowercase letters denote high level.

Define and compute contrasts for main effects and the AB interaction in a factorial design, using high minus low level sums to obtain A, B, and AB contrasts.

Calculate the effects and sum of squares for substrate, enzyme, and their interaction using contrasts, and build the ANOVA table. Assess factor significance and interaction effects on yield.

Fit a regression model to a 2^2 factorial design, plot contour lines, and interpret coded versus natural variables to predict yield based on substrate and enzyme concentration.

Visualize yield with a contour plot of substrate and enzyme concentrations and assess model adequacy via regression factors, R squared (0.877), and residual analysis.

Present final results with contour plots, a reduced model, and a table of effects. Show substrate and enzyme influence on yield and introduce R Studio for p values.

Master how to analyze a 2^2 factorial design in R by ANOVA and regression on a CSV dataset, building contour plots and residual checks with the RSM package.

Explore a two in the power of three factorial design to evaluate temperature, substrate concentration, and enzyme type on enzymatic yield, with 16 randomized runs analyzed by ANOVA.

Import the csv data, define factor_T, factor_S, and factor_E for temperature, substrate, and enzyme, and run a three-factor ANOVA to study yield with main effects and interactions.

Explore building and comparing regression models in R for a 2^3 factorial design, including coded vs natural variables, and assess temperature, substrate, and enzyme interactions with r-squared and residual analysis.

Build and interpret contour plots for a 2^3 factorial (temperature, substrate, enzyme) using rsm in R-Studio, highlighting temperature-driven yield differences between E1 and E2, and substrate's smaller effect.

Explore a 2^k factorial design that analyzes k factors at two levels, built from a 2^2 design by doubling for each factor, with ANOVA testing main effects to k-factor interactions.

Analyze any 2^k design using an ANOVA table, run regression models for coded and natural variables, and present results as contour plots or an effects table.

Explore single replicate designs in a 2^k factorial framework, noting how the number of treatments grows with factors and how sparsity of effects helps estimate error when replicates are absent.

Explore how a single-replicate 2^4 factorial design evaluates temperature, pressure, concentration, and stirring rate on filtration rate, using a 16-run experimental matrix with coded and natural variables analyzed in R-Studio.

analyze a single-replicate design using an a n o v a to test main effects and all interactions, apply backwards elimination, and build a regression model for temperature, concentration, and stirring rate.

Analyze the regression model for coded and natural variables and interpret interaction effects with contour plots. Temperature and stirrer speed show strong effects; concentration and pressure have limited impact.

Explore how blocking a replicated design controls nuisance factors in a 2^2 design, and analyze the blocks with R-Studio to improve experimental analysis.

Load the data for a 2^2 design with replicates and blocks, define substrate and enzyme as factors, and perform ANOVA to evaluate blocks, main effects, and their interaction.

Explore confounding in a 2^2 block design using the plus/minus sign table to split a complete replicate into two blocks, confounding blocks with the interaction AB rather than main effects.

Confound a 2^3 design in two blocks using a table of plus and minus signs, and align the ABC interaction with block contrasts to split treatments by sign.

Explore how a 2^4 design can be confounded in two blocks by aligning the block contrasts with the ABCD interaction, then analyze the results in R.

Analyze a 2^4 design in blocks using ANOVA with temperature, pressure, concentration, and stirring rate as factors, including block effects and interactions, to identify significant factors and assess model adequacy.

Explore a 2^4 design in blocks, comparing coded regression with the analysis of variance; examine how block effects lower r-squared and influence contour plot versus bar graph choices.

Build bar graphs in r using ggplot to visualize temperature-stirring rate and temperature-concentration interactions, with means, standard errors, same y-axis, and faceted by stirring rate.

Assess how temperature, concentration, and stirring rate impact filtration rate. High temperature and high stirring rate boost filtration rate, while concentration's effect varies with temperature and pressure has no effect.

Blocking reduces variability from batch differences in a 2^4 factorial design by withdrawing raw-material variability from the error, improving detection of stirring rate and temperature–concentration interactions.

Address non-homogeneity in raw materials by mixing before runs or using blocks when unavoidable, and always randomize or split experiments into blocks to prevent ill planned designs.

Explore fractional factorial designs for screening experiments in design of experiments. Learn 2^k-p notation, sparsity of effects, projection property, and sequential experimentation to identify key factors and interactions.

Explore one-half fraction designs for three factors, selecting four treatments via the ABC generator and the defining relation. Understand contrasts and aliasing among main effects and interactions for estimating effects.

Determine aliases in a 2^(3-1) design using the defining relation ABC, with A aliasing BC, B aliasing AC, C aliasing AB, and identify the principal and complementary fractions.

Explore how design aliases define resolution in fractional designs, from three to five, revealing how main effects and two-factor interactions are aliased. Higher resolution reduces restrictive assumptions.

Build fractional designs by starting from a full 2 to the k minus 1 design and deriving the last column with the defining relation, using identity ABC.

Design and analyze a half-fraction factorial with four factors using a defining relation, alias structure, and resolution four, then build the experimental matrix and analyze in R.

Explore a 2^4−1 design by converting temperature, pressure, concentration, and stirring rate into factors, perform ANOVA and regression, and observe limited degrees of freedom and an R-squared of 1.

Use regression results to build a table of effects, untangling aliases to identify that temperature, concentration, and stirring rate are main drivers, with significant interactions: temperature–concentration and temperature–stirring rate.

Explore how sequential fractional designs, including the principal and complementary fractions of a 2^4 design, assess assumptions about two-factor interactions and build larger designs through sequential experimentation.

Explore one-quarter fraction designs for two-level factorial experiments, building 2^(5-2) designs using generators from ABC, D, and E, and understanding resolutions from III to V for main effects and interactions.

Use the FrF2 package in R to build and analyze fractional 2^k designs, explore resolution III and IV designs with generators and alias structures.

Explore a 2^(8-4) fractional design assessing eight wine-making factors on Pinot Noir quality, with 16 runs and expert rankings, and learn to analyze the data in R.

Analyze a highly fractionated design using the FrF2 package to identify significant main effects and two-factor interactions, assess aliasing with half-normal plots, and refine a screening model.

Create the final presentation of the results using dplyr and ggplot2. Compare eight treatments across yeast (Champagne, Montrachet), temperature (low, high), and stems with bar plots and error bars.

Choose the next design by discarding non-significant factors, focusing on Montrachet yeast. Use a 2^(5−1) resolution V fractional design for five factors, 16 runs, to study temperature and stems.

Apply simple, iterative experimental designs to plan and analyze experiments, balancing practical and statistical differences. Learn to select factors, levels, and replicates through careful pre-experiment planning and interpretation.

Explore insights for designing and analyzing your experiments, and share feedback to improve this design of experiments course.