Design of Experiments: Factors and Responses

Method Development

• Method development is the process of setting up an analytical procedure that is suitable for analyzing a particular sample.
• It involves selecting and optimizing analytical test parameters to ensure that the test method meets the required standards.

Importance of Method Development

• Method development can improve the performance of testing by employing method development principles to standard test methods, especially for new products or materials.
• It enables the identification of the optimum test parameters to ensure the applicability and reliability of the data.

Method Development Steps

• Choice of analytical technique:
  • GC/HPLC is suitable for separating volatile substances.
  • Provides the best resolution, shortest analysis time, and highest sensitivity.
• Choice of phase system:
  • Based on the interactive character of the compound to be analyzed (polar, non-polar, or ionic).
• Detector selection:
  • Provides the required sensitivity, linearity, and specificity.

Optimization Design

• Traditional approach:
  • One-factor-at-a-time (OFAT) approach.
  • Incremental changes to each factor to determine the optimal conditions.
• Statistical design approach:
  • Design of Experiments (DoE) uses statistical tools to study multiple factors simultaneously.
  • Requires fewer experiments and provides more information about the interactions between factors.

Phases of DoE Process

• Planning:
  • Determine the objective of the study.
  • Identify the factors, factor ranges, and types of factors.
  • Select the design of the experiment.
• Designing:
  • Identify the design by selecting the level and range for each process variable.
  • Provide low to high input/factor levels.
• Conducting:
  • Run the experiment according to the design.
• Analyzing:
  • Analyze the results using ANOVA.
  • Identify the significant factors and their interactions.

Goal of DoE

• To optimize the costs by conducting as few experiments as possible.
• To learn how the inputs (factors) affect the output (responses).

Requirements of DoE

• Two types of variables: factors and responses.
• Factors: input parameters that can be changed.
• Responses: output parameters that provide information about the quality of the process.

Examples of Factors and Responses

• Growing tomatoes: fertilizer, soil pH, seed hybrid, water (factors); yield (response).
• Chemical reaction: pressure, temperature, catalyst concentration (factors); percent yield (response).

How to Select DoE

• Choose based on the research objectives:
  • Comparative objective: to compare the effects of different factors.
  • Screening objective: to select the most important factors.
  • Response Surface Method (RSM) objective: to estimate interaction and optimize the process.
  • Mixture design objective: to optimize the proportions of a mixture.
  • Regression design objective: to estimate a precise model.

Types of DoE

• Full factorial design: all possible combinations of variables and levels.
• Fractional factorial design: a subset of the full factorial design.
• Central composite design: suitable for fitting second and higher-order surfaces.
• Plackett-Burman design: a type of fractional factorial design.
• Box-Behnken design: a type of response surface design.

Analyzing the Design

• Multiple regression analysis.
• Analysis of Variance (ANOVA).
• Response surface plots.

Significance of Model Coefficients

• The p-value helps determine whether the relationships observed in the sample also exist in the population.
• A low p-value (< 0.05) indicates that the null hypothesis can be rejected.
• The relationships between the independent variables and the dependent variable are statistically significant.

Example of DoE Application

• Experimental design approach for the optimization of solid-phase microextraction (SPME) for the extraction of benzene and toluene from water.
• Central composite design was used to optimize the extraction time, extraction temperature, and desorption time.
• ANOVA and RSM were used to analyze the results and optimize the process.### Significance Test
• At the 95% confidence level (p < 0.05), two parameters, extraction time and desorption time, are significant in the Quadratic Model.
• The p-value determines the significance of a variable, with a value less than 0.05 indicating significance.

Response Surface 3-D Plot

• The plot illustrates the relationship between extraction time, desorption time, and toluene recovery.
• The optimal conditions for roasting Arabica coffee beans are temperature at 167.68°C for 22.50 minutes.

Method Validation

• Method validation is the process of proving that an analytical method is acceptable for its intended purpose.
• It involves verifying that a method is fit-for-purpose, i.e., suitable for solving a particular analytical problem.
• Method validation is necessary in analytical laboratories to ensure that reliable analytical procedures are used under defined conditions.

Importance of Method Validation

• Method validation increases the value of test results.
• It justifies customer trust.
• It helps to trace criminal activities.
• It proves the accuracy of claims.

Examples of Method Validation

• Valuing goods for trade purposes.
• Supporting healthcare.
• Checking the quality of drinking water.

Compliance to Standards

• FDA (Food and Drug Administration): regulates food safety, tobacco products, and pharmaceuticals.
• cGMP (current good manufacturing practice): provides systems that assure proper design, monitoring, and control of manufacturing processes and facilities.
• GLP (good laboratory practice): sets principles to assure the quality and integrity of non-clinical laboratory studies.

When to Validate a Method

• Before introducing a new method into routine use.
• When conditions change, such as using a method in a different laboratory or with a different analyst.
• When revising an established method or instrument.
• When comparing methods.

Method Validation Data

• Accuracy.
• Precision.
• Limit of Detection (LOD).
• Limit of Quantitation (LOQ).
• Specificity.
• Linearity and Range.
• Ruggedness/Robustness.
• System Suitability.

Accuracy

• Accuracy is the measure of the closeness of agreement between the measured value and true value.
• The true value for accuracy assessment can be obtained from a certified reference material or by spiking a sample with a known concentration.

Precision

• Precision is the degree of agreement among individual test results when the procedure is applied repeatedly to multiple samplings of a homogeneous sample.
• Precision is usually expressed in terms of standard deviation or Relative Standard Deviation percent (RSD).
• Repeatability and reproducibility are two types of precision measures.

Repeatability and Reproducibility

• Repeatability is the precision of a method when the same method is used on identical test items in the same laboratory, by the same operator, using the same equipment, within short time intervals.

• Reproducibility is the precision of a method when the same method is used on identical test items in different laboratories, with different operators, using different equipment.### Calculating Mean and Standard Deviation

• The standard deviation is a measure of how precise the average is.

• The standard deviation formula is: s =

• The relative standard deviation (RSD) is often more convenient and is always expressed as a percentage (%).

• The RSD formula is: RSD (%) =

Analyte Concentration vs. Precision

• The lower the concentration of analyte, the higher the %RSD obtained.
• Analyte concentration vs. precision within or between days:
  • 100% analyte: RSD = 1.3%
  • 10% analyte: RSD = 2.8%
  • 1% analyte: RSD = 2.7%
  • 0.1% analyte: RSD = 3.7%
  • 0.01% analyte: RSD = 5.3%
  • 0.001% analyte: RSD = 7.3%
  • 0.0001% analyte: RSD = 11%
  • 0.00001% analyte: RSD = 15%
  • 0.000001% analyte: RSD = 21%
  • 0.0000001% analyte: RSD = 30%

Accuracy vs. Precision

• “If you are playing soccer and you always hit the left goal post instead of scoring, then you are not accurate, but you are precise.”

Limit of Detection (LOD) and Limit of Quantification (LOQ)

• LOD is the lowest concentration of an analyte in a sample that can be detected, but not necessarily quantified, under the stated conditions of the test.

• LOD may be defined as the concentration of analyte which gives rise to a signal that is significantly different from the negative control or blank.

• The LOD is the lowest concentration of analyte that can be distinguished from background.

• The results obtained at the LOD are not necessarily precise or accurate or quantitated.

• LOD is usually expressed as the concentration of analyte generating an instrument response equivalent to three times the noise (S/N = 3).

• LOQ is the lowest concentration of an analyte in a sample that can be determined with acceptable precision and accuracy under the stated conditions of the test.

• A typical acceptable signal-to-noise ratio is 1:10 (S/N=10).

• The LOQ is calculated by plotting the RSD% of the precision against the analyte amount.

Linear Regression and Signal-to-Noise Ratio (SNR)

• The signal-to-noise ratio (SNR) is calculated by dividing the signal by the noise.
• The noise value is calculated based on the peak height of the blank (solvent) around the retention time of each analyte.
• Procedure to calculate LOD and LOQ:
  1. Run blank and the lowest concentration of analyte.
  2. Run 10 replicates independently.
  3. Measure the peak height values (noise magnitude was measured by auto-integrator).
  4. Calculate LOD: S/N = 3, and LOQ: S/N = 10.

Limit of Detection (LOD), Limit of Quantification (LOQ), and Signal-to-Noise Ratio (SNR)

• LOD ≅ xbi + 3sbi
• LOQ ≅ xbi + 10sbi
• xbi = mean concentration of the blank
• sbi = standard deviation of the blank

Linear Regression

• Based on linear calibration curve: y = a + bx
• LOD ≅ 3sa / b
• LOQ ≅ 10sa / b
• y: instrument responses
• x: analyte concentration
• a: y-intercept
• b: slope
• sa: standard deviation of the blank

Specificity and Selectivity

• Specificity: degree to which the measured response is due to the analyte of interest only and not to other substances expected to be present in the sample matrix.
• Selectivity: ability of an analytical method to differentiate various substances in a sample.

Linearity and Range

• Linearity: the ability of a method to provide measurement results that are directly proportional to the concentration of the analyte, or are directly proportional after mathematical transformation.
• Range: the area between the lower and the upper limits of quantitation that is also linear.
• How to determine linearity and range:
  1. Prepare five concentration levels.
  2. Analyze the response.
  3. Plot the graph analyte concentration vs. responses.
  4. The line generated should be submitted, together with slope, intercept, and correlation coefficient data.
  5. The measured slope should demonstrate a clear correlation between response and analyte concentration.
  6. Acceptability of linearity data is often judged by examining the correlation coefficient r2 and y-intercept of the linear regression line for the response versus concentration plot.

Robustness and Ruggedness

• Robustness: measure of its capability to remain unaffected by small, but deliberate variations in method parameters and provides an indication of its reliability during normal usage.
• Ruggedness: the degree of reproducibility of test results obtained by the analysis of same samples under a variety of conditions.

System Suitability

• System suitability: checking of a system to ensure system performance before or during the analysis of unknowns.
• To check system (equipment, electronics, samples, technique) is working properly before any samples are analyzed.

Types of Error

• Gross errors: errors that are so serious that there is no real alternative to abandoning the experiment.
• Random errors: cause replicate results to differ from one to another, so that the individual results fall on both sides of the average value.
• Systematic errors: cause all the results to be in error in the same sense (whether too high or too low from the true value).