Identifying Outliers in Catalyst Performance: A Comprehensive Guide to Grubbs' Test for Research and Drug Development

Abstract

This article provides a complete framework for applying Grubbs' test to identify statistical outliers in catalyst performance data, crucial for ensuring reliability in pharmaceutical R&D and chemical synthesis. It begins by establishing the foundational importance of outlier detection for data integrity and experimental reproducibility. The core methodological section offers a step-by-step guide to performing Grubbs' test, including calculations, critical value selection, and interpretation specific to catalytic metrics like turnover frequency (TOF) and yield. We then address common pitfalls, assumptions, and strategies for optimizing the test with small datasets or non-normal distributions. Finally, the article validates the approach by comparing Grubbs' test with Dixon's Q-test, Modified Z-score, and IQR methods, guiding scientists on selecting the most appropriate tool for their specific catalyst screening workflow to enhance decision-making and accelerate development timelines.

Why Outlier Detection is Critical for Reliable Catalyst Screening in R&D

Context: These notes support a thesis investigating the application of Grubbs' test for identifying statistical outliers in heterogeneous catalyst performance datasets, a critical step in ensuring robust, economical, and safe pharmaceutical process development.

---

The Criticality of Performance Data in Scale-Up

In the transition from medicinal chemistry to commercial manufacturing, catalyst performance data (e.g., Turnover Number (TON), Turnover Frequency (TOF), selectivity, lifetime) forms the bedrock of process design. Outliers in this data, whether erroneously high or low, can lead to catastrophic scale-up failures, including:

• Under-designed reactors and inadequate safety margins if outliers are falsely accepted.
• Over-designed, economically non-viable processes if true high-performance outliers are incorrectly rejected.
• Failed regulatory submissions due to process understanding gaps.
• Supply chain disruption from over-reliance on a catalyst batch with non-representative performance.

Systematic outlier detection using statistical methods like the Grubbs' Test is therefore not merely an analytical step but a fundamental risk mitigation strategy.

---

Protocol: Grubbs' Test for Catalyst Performance Datasets

2.1 Objective: To identify a single outlier (high or low) within a small, normally distributed dataset of catalyst performance metrics.

2.2 Materials & Computational Tools:

• Performance dataset (e.g., TON values from n replicate reactions).
• Statistical software (e.g., Python with SciPy, R, GraphPad Prism, JMP).

2.3 Procedure:

• Data Collection: Conduct a minimum of n=7 replicate catalytic reactions under identical, controlled conditions. Record the primary performance metric (e.g., final yield for TON calculation).
• Assumption Check: Test the dataset for approximate normal distribution using the Shapiro-Wilk test (for n<50) or visual inspection of a Q-Q plot.
• Grubbs' Statistic Calculation: a. Calculate the sample mean (x̄) and standard deviation (s). b. Identify the suspect point (xᵢ) farthest from the mean. c. Compute the G statistic: G = |xᵢ – x̄| / s.
• Critical Value Comparison: Compare the calculated G to the critical value Gcritical for n observations at α=0.05. If G > Gcritical, the point is considered an outlier.
• Iteration: If an outlier is identified, remove it and repeat the test on the remaining n-1 data points. Continue until no further outliers are detected.

2.4 Example Analysis: TON Dataset for Ruthenium-Catalyzed Olefin Metathesis in API Step A dataset of TON values from 8 identical experiments aimed at synthesizing a key drug intermediate was analyzed.

Table 1: Grubbs' Test Analysis for TON Outlier Detection

Experiment ID	TON	G Statistic (Iteration 1)	G Critical (α=0.05)	Outlier?
1	12,500	0.41	2.126	No
2	13,100	1.14	2.126	No
3	12,800	0.18	2.126	No
4	45,000*	2.95	2.126	Yes
5	12,950	0.09	2.126	No
6	13,050	0.83	2.126	No
7	12,750	0.27	2.126	No
8	12,880	0.11	2.126	No
Mean (x̄)	16,754
Std Dev (s)	9,576

Conclusion: The TON value of 45,000 (Exp. 4) is a statistically significant outlier. This result would trigger an investigation into experimental error or unique catalyst lot properties before any scale-up calculations are performed.

---

Protocol: Investigating an Identified Outlier – Catalyst Lot Variability

3.1 Objective: To determine if a statistical outlier in performance is linked to specific chemical or physical properties of the catalyst lot.

3.2 Experimental Workflow:

Diagram Title: Catalyst Outlier Investigation Workflow

3.3 Key Research Reagent Solutions & Materials

Table 2: Essential Toolkit for Catalyst Performance Analysis

Item	Function & Relevance
High-Purity Catalyst Lots	Bench-scale lots with certificates of analysis (CoA) for metal content, ligand assay, and impurities. Critical for establishing baseline performance.
Inhibitor/Stabilizer Solutions	Standardized solutions (e.g., ethyl vinyl ether for Grubbs catalysts) to quench reactions at precise times for accurate kinetic TOF measurements.
Internal Standard Solutions	For quantitative GC/FID or LC/MS analysis, ensuring accurate yield determination for TON/selectivity calculations.
Solid Phase Extraction (SPE) Cartridges	For rapid workup and removal of metal residues from reaction samples prior to analysis, preventing catalyst degradation post-sampling.
Calibrated Gas Manifold	For hydrogenation, cross-coupling, or other gas-involved reactions, precise control of gas pressure/uptake is vital for reproducible activity data.
In-situ ReactIR/Raman Probe	Enables real-time monitoring of reaction profiles and catalyst intermediate formation, linking performance outliers to mechanistic deviations.

---

Protocol: Integrating Outlier-Reviewed Data into Scale-Up Models

4.1 Objective: To use the statistically vetted dataset to calculate scale-up parameters for a stirred tank reactor.

4.2 Methodology:

• Data Curation: Use the dataset with outliers removed (or causally explained) to calculate mean TON, mean TOF (from initial rates), and selectivity.
• Mass Balance: Calculate required catalyst loading (kg) for target batch size: Catalyst Mass = (Moles Product Target / Mean TON) × MW_catalyst.
• Heat Load Estimation: Using the mean reaction rate (from TOF) and reaction enthalpy (ΔH, from calorimetry), calculate the peak heat output to specify jacket/chiller capacity.
• Safety Margin Application: Apply a scale-up safety factor (e.g., 0.7-0.8) to the mean catalyst performance to define the design TON/TOF for the plant recipe, ensuring robustness against expected lot-to-lot variability.

4.3 Data Flow from Lab to Plant Design:

Diagram Title: Data Curation Path for Scale-Up

This application note explores the dual nature of outliers in catalyst performance research, specifically within high-throughput screening for drug development. The core thesis posits that systematic application of Grubbs' test (or the Extreme Studentized Deviate test) provides a statistically rigorous framework to differentiate between erroneous data (statistical anomalies) and performance outliers that may signal novel, high-activity catalysts or unexpected inhibitory effects. This discrimination is critical for efficient resource allocation in lead optimization.

Statistical Protocol: Grubbs' Test for Outlier Detection

Objective: To statistically identify a univariate outlier in a normally distributed dataset of catalyst turnover frequency (TOF) or yield% values.

Prerequisites:

• Dataset assumed to be approximately normally distributed.
• Test is typically applied sequentially to a single suspected outlier.

Formula: [ G = \frac{\max |X_i - \bar{X}|}{s} ] Where:

• ( G ) is the Grubbs' test statistic.
• ( X_i ) is the individual data point.
• ( \bar{X} ) is the sample mean.
• ( s ) is the sample standard deviation.

Critical Value: [ G{\text{critical}} = \frac{(N-1)}{\sqrt{N}} \sqrt{\frac{t{\alpha/(2N), N-2}^2}{N-2 + t_{\alpha/(2N), N-2}^2}} ] Where:

• ( N ) is the sample size.
• ( t ) is the critical value from the t-distribution with ( N-2 ) degrees of freedom and a significance level ( \alpha/(2N) ).

Decision Rule: If ( G > G_{\text{critical}} ), the data point is rejected as a statistical outlier at the ( \alpha ) significance level (typically 0.05).

Procedure:

• Perform catalyst performance assay (e.g., yield determination) under standardized conditions (see Protocol 3.1).
• Collect performance metric (e.g., TOF) for all catalysts in a library (N > 3).
• Calculate mean (( \bar{X} )) and standard deviation (( s )) of the dataset.
• Identify the data point furthest from the mean.
• Calculate the G statistic for this point.
• Determine ( G_{\text{critical}} ) for N and α=0.05.
• Compare: If ( G > G_{\text{critical}} ), flag as a statistical outlier.
• If an outlier is flagged and removed, the test may be iteratively repeated on the remaining data (up to a pre-defined limit, e.g., 3% of data).

Example Data & Calculation Table:

Table 1: Example Catalyst Yield Data and Grubbs' Test Calculation (α=0.05)

Catalyst ID	Yield (%)	Note
Cat-01	78.2
Cat-02	81.5
Cat-03	79.8
Cat-04	80.1
Cat-05	82.3
Cat-06	94.7	Suspected Outlier
Mean ((\bar{X}))	82.77
Std Dev (s)	5.89
G Statistic	2.03	( G = \frac{	94.7 - 82.77	}{5.89} )
G critical (N=6)	1.887	From Grubbs' Table
Conclusion	Reject H₀	Catalyst 06 is a statistical outlier.

Experimental Protocols for Outlier Validation

Protocol 3.1: Primary High-Throughput Screening (HTS) Assay Objective: Generate initial catalyst performance data.

• Plate Preparation: Dispense 100 µL of standardized substrate solution (e.g., 5 mM cross-coupling partner in degassed DMF) into each well of a 96-well reaction plate.
• Catalyst Addition: Using an automated liquid handler, add 2 µL of each catalyst stock solution (1 mM in appropriate solvent) to assigned wells. Include control wells (no catalyst, reference catalyst).
• Reaction Initiation: Add 100 µL of initiator solution (e.g., base, co-catalyst) to all wells simultaneously using a multichannel pipette.
• Incubation: Seal plate and incubate at specified temperature (e.g., 25°C) with orbital shaking (500 rpm) for the fixed reaction time (e.g., 2 hours).
• Quenching & Analysis: Add 50 µL of quenching solution (e.g., 1% TFA). Quantify product formation via UPLC-MS analysis with a 5-minute gradient. Calculate yield or TOF.

Protocol 3.2: Outlier Verification & Dose-Response Objective: Confirm the performance of statistical outliers.

• Re-synthesis: Re-synthesize or obtain a fresh, independent sample of the outlier catalyst.
• Dose-Response: In triplicate, run the reaction from Protocol 3.1 with a dilution series of the outlier catalyst (e.g., 0.1, 0.5, 1.0, 5.0 µM).
• Extended Analysis: Perform full kinetic profiling (timepoints at 0.5, 1, 2, 4, 8h) for the outlier and two standard catalysts.
• Control Reinforcement: Spike control reactions with potential contaminants (e.g., common metal impurities) to rule out external factors.

Visualization: Outlier Decision Pathway

Title: Decision Pathway for Catalyst Outliers

The Scientist's Toolkit: Key Research Reagent Solutions

Table 2: Essential Materials for Catalyst Outlier Research

Item	Function & Rationale
High-Purity Solvents (e.g., degassed DMF, MeCN)	Ensures reproducible reaction medium; prevents catalyst deactivation (oxidation).
QC'd Catalyst Libraries	Pre-characterized (NMR, MS) stock solutions minimize variance from starting material impurities.
Internal Standard (e.g., dibromomethane)	Added pre-quench to all wells for normalization of UPLC-MS injection volume variability.
Reference Catalyst Control	Provides a benchmark for plate-to-plate and batch-to-batch performance normalization.
UPLC-MS with Automated Injector	Enables high-throughput, quantitative analysis of reaction yields and product identification.
Statistical Software (e.g., R, Python SciPy)	Automates Grubbs' test calculation and critical value lookup for large datasets.

1. Core Principles and Quantitative Foundation

Grubbs' Test, or the Extreme Studentized Deviate (ESD) method, is a statistical procedure designed to detect a single outlier in a univariate data set that follows an approximately normal distribution. Its application in catalyst performance research is critical, as outliers can skew activity and selectivity analyses, leading to incorrect conclusions about structure-activity relationships.

The test statistic, G, is calculated as the maximum absolute deviation from the sample mean, divided by the sample standard deviation.

Table 1: Key Formulas and Critical Values for Grubbs' Test

Component	Formula / Value	Description
Test Statistic (G)	$G = \frac{\max	Y_i - \bar{Y}	}{s}$	Where $Y_i$ is a data point, $\bar{Y}$ is the sample mean, and $s$ is the sample standard deviation.
Critical Value (G_crit)	$G{crit} = \frac{(N-1)}{\sqrt{N}} \sqrt{\frac{t{\alpha/(2N), N-2}^2}{N-2 + t_{\alpha/(2N), N-2}^2}}$	N is the sample size, α is the significance level (e.g., 0.05), and t is the critical value from the t-distribution.
Example Critical Value (N=10, α=0.05)	2.290	For a dataset of 10 catalyst turnover frequency (TOF) measurements.
Decision Rule	Reject H₀ if G > G_crit	H₀: There are no outliers in the data set.

2. Protocol: Applying Grubbs' Test to Catalyst Performance Data

• Step 1: Data Preparation and Assumption Checking
  • Organize catalyst performance metric (e.g., Turnover Frequency, Yield, Selectivity) in a single column.
  • Test the dataset for normality using a graphical (Q-Q plot) or statistical test (Shapiro-Wilk, p > 0.05). If significantly non-normal, consider data transformation or non-parametric methods.
• Step 2: Initial G-Statistic Calculation
  • Calculate the mean ($\bar{Y}$) and standard deviation (s) of the full dataset.
  • Identify the data point farthest from the mean. Compute the G-statistic using the formula in Table 1.
• Step 3: Hypothesis Testing
  • Choose a significance level (α), typically 0.05.
  • Determine the critical value ($G_{crit}$) for your sample size (N) and α using statistical software or reference tables.
  • Compare G to $G{crit}$. If G > $G{crit}$, classify the identified point as an outlier. Remove it from the dataset.
• Step 4: Iterative Testing (Generalized ESD Test)
  • For screening more than one potential outlier, use the Generalized ESD procedure.
  • Specify an upper bound (k) for the number of potential outliers.
  • Repeat Steps 2-3 on the reduced dataset after each outlier removal, up to k times, recalculating mean and standard deviation each time.

3. Workflow and Logical Relationships

Diagram 1: Grubbs' test workflow for catalyst data.

4. The Scientist's Toolkit: Essential Materials for Catalyst Outlier Analysis

Table 2: Research Reagent Solutions & Essential Materials

Item	Function in Catalyst Outlier Analysis
High-Throughput Reactor System	Generates the primary performance data (e.g., conversion, yield) under controlled conditions. Essential for producing the dataset to be tested.
Gas Chromatograph-Mass Spectrometer (GC-MS)	Provides precise quantitative analysis of reaction products. Data from this instrument is often the key metric for selectivity and yield calculations.
Statistical Software (R, Python, JMP)	Platforms for executing Grubbs' test, calculating critical values, and generating normality plots. R's outliers package or Python's SciPy.stats are standard.
Certified Reference Materials (Catalyst/Calibration Standards)	Ensures analytical instrument accuracy, minimizing systematic error that could create false outliers or mask true ones.
Inert Atmosphere Glovebox	For handling air-sensitive catalysts, ensuring performance variations are due to intrinsic properties, not decomposition.

Within the research for a broader thesis on the application of statistical outlier detection in catalyst performance analysis, Grubbs' test stands as a critical tool. This thesis investigates the reproducibility and reliability of heterogeneous catalyst screening data, where performance metrics (e.g., conversion rate, selectivity) are prone to anomalous values due to experimental artifact, feedstock impurity, or reactor maldistribution. Correctly identifying true statistical outliers—as opposed to high-value discoveries—is paramount. Grubbs' test provides a formal statistical framework for this purpose, but its validity is strictly contingent upon three key assumptions: Normality, Independence, and Single Outlier detection. Misapplication when these assumptions are violated risks either discarding valuable catalyst leads or corrupting the dataset with spurious results. These Application Notes detail the validation protocols and experimental considerations for employing Grubbs' test in catalyst performance research.

The Three Key Assumptions: Theory and Validation Protocols

Normality

Assumption: The underlying data (excluding the potential outlier) should be approximately normally distributed. Grubbs' test calculates critical values based on the properties of the normal distribution.

Validation Protocol:

• Data Collection: Perform a minimum of 15 replicate experiments under identical, controlled conditions for a single catalyst formulation. For catalyst research, this means strictly controlling temperature, pressure, flow rates, catalyst mass, and feedstock composition.
• Initial Visualization: Create a histogram and a Q-Q (Quantile-Quantile) plot of the replicate performance data (e.g., yield at 1-hour time-on-stream).
• Formal Statistical Testing:
  • Shapiro-Wilk Test: Preferred for small to moderate sample sizes (n < 50).
  • Anderson-Darling Test: More sensitive to tails of the distribution.
  • Protocol Step: If p-value from the normality test is > 0.05, the normality assumption is not rejected. Proceed with caution if p-value is between 0.05 and 0.01. If p-value < 0.01, normality is suspect.
• Remedial Action: If data is non-normal, apply a transformation (e.g., Box-Cox, logarithmic) and retest for normality on the transformed data. Alternatively, use non-parametric outlier tests (e.g., Tukey's fences) with clear documentation of the changed methodology.

Table 1: Normality Test Results for Catalyst Yield Data (n=20 replicates)

Catalyst ID	Mean Yield (%)	Std Dev	Shapiro-Wilk Statistic (W)	p-value	Normality Assumption Met?
Cat-A-1	78.3	2.1	0.972	0.112	Yes
Cat-B-3	65.4	5.7	0.921	0.008	No (Requires Transform)

Independence

Assumption: Data points must be independently sampled and measured. In catalyst testing, autocorrelation (where one measurement influences the next) violates this assumption. Common sources include catalyst deactivation over a test sequence or instrument drift.

Validation Protocol:

• Experimental Design: Utilize fully randomized testing orders for catalyst replicates to decouple potential time-dependent effects (deactivation, system settling) from intrinsic performance.
• Statistical Testing: Perform the Durbin-Watson test on the residuals of the data ordered by run sequence. The test statistic ranges from 0 to 4, with a value near 2 indicating no autocorrelation.
• Visual Check: Plot residuals (observed value - mean) versus the run order/time of measurement. Look for clear trends or cycles.

Table 2: Durbin-Watson Test for Sequential vs. Randomized Catalyst Testing

Testing Order	Durbin-Watson Statistic	p-value	Evidence of Autocorrelation?
Sequential	0.85	<0.001	Yes (Positive)
Randomized	1.92	0.451	No

Single Outlier Detection

Assumption: Grubbs' test is designed to detect a single outlier in a dataset. Its power diminishes significantly if multiple outliers are present, as they can "mask" each other by inflating the standard deviation.

Validation Protocol & Iterative Application:

• Initial Test: Perform Grubbs' test on the full dataset (G = |suspect value - mean| / SD).
• Iteration: If an outlier is identified (G > critical value), remove it and re-evaluate the remaining data for normality before repeating Grubbs' test on the new, reduced dataset.
• Stop Condition: Iterate until no further outliers are detected. This sequential application is the Grubbs' procedure.
• Caution: For small datasets (n < 10), this iterative process can become unstable. Visual inspection (e.g., box plots) is essential.

Table 3: Iterative Grubbs' Test on Catalyst Selectivity Dataset (n=12)

Iteration	Suspect Value (%)	G Statistic	G Critical (α=0.05)	Outlier Detected?	Action
1	24.1	2.54	2.29	Yes	Remove 24.1
2	91.5	2.61	2.28	Yes	Remove 91.5
3	85.2	1.89	2.27	No	Stop

Integrated Workflow for Outlier Detection in Catalyst Research

Title: Workflow for Applying Grubbs' Test in Catalyst Screening

The Scientist's Toolkit: Research Reagent Solutions & Essential Materials

Table 4: Essential Materials for Controlled Catalyst Performance Testing

Item/Category	Example Product/Specification	Function in Outlier Analysis Context
High-Precision Reactor System	Fixed-bed Microreactor with PID control (±0.5°C)	Ensures replicate reaction conditions are identical, minimizing variance from non-catalyst sources.
Calibrated Mass Flow Controllers (MFCs)	Bronkhorst EL-FLOW Select, ±0.5% RD accuracy	Controls feedstock composition precisely. Drift in MFCs creates correlated (non-independent) errors.
Inline GC/MS or FTIR Analyzer	Agilent 8890 GC System with TCD/FID	Provides accurate and precise product quantification. High detector linearity range is key for normality of results.
Certified Standard Gas Mixtures	1% CO, 5% H2 in N2 balance (±1% certified)	Used for daily calibration of analyzers, ensuring measurement accuracy and long-term data consistency.
Reference Catalyst	NIST Standard Reference Material 1979 (Pt/Al2O3)	Run intermittently within experimental batches to monitor system performance and detect process-related outliers.
Statistical Software	R (with outliers package), Python (SciPy, outlier_utils)	Performs Grubbs' test, normality checks (Shapiro-Wilk), and autocorrelation tests (Durbin-Watson) efficiently.
Laboratory Information Management System (LIMS)	LabWare, Benchling	Tracks all meta-data (run order, operator, instrument ID) essential for investigating the root cause of identified outliers.

Within a broader thesis investigating the application of Grubbs' test for statistical outlier detection in catalytic performance data, this article examines four key metrics where outliers frequently arise: Turnover Frequency (TOF), Yield, Selectivity, and Enantiomeric Excess (ee). Outliers in these datasets can signal experimental error, catalyst deactivation, unique mechanistic pathways, or breakthrough performance. Rigorous identification and analysis are critical for reliable data interpretation in catalyst development, particularly for pharmaceutical applications where reproducibility is paramount.

Table 1: Typical Ranges and Common Outlier Sources for Key Catalyst Metrics

Metric	Definition	Typical Range (Homogeneous Catalysis)	Common Sources of Outliers
Turnover Frequency (TOF)	Moles product per mole catalyst per unit time (h⁻¹).	1 - 10⁶ h⁻¹	Incorrect active site counting, induction/deactivation periods, mass transfer limitations, Gligorous mixing.
Yield (%)	(Moles product / Moles limiting reactant) x 100.	0-100%	Impure reactants, side reactions consuming product, inaccurate quantification (e.g., calibration error), incomplete conversion.
Selectivity (%)	(Moles desired product / Moles converted reactant) x 100.	0-100%	Catalyst poisoning altering pathway, temperature/pressure spikes, Gligorous solvent effects, competitive parallel reactions.
Enantiomeric Excess (ee)	| (Major enantiomer - Minor enantiomer) / (Total) | x 100%.	0-100%	Chiral impurities in feedstock, racemization during workup, Gligorous assay interference, nonlinear chiral chromatography effects.

Table 2: Example Outlier Analysis Using Grubbs' Test (Hypothetical TOF Dataset)

Catalyst Batch	TOF (h⁻¹)	G (Calculated)	G Critical (α=0.05, n=6)	Outlier?	Potential Assignable Cause
A	1250	0.24	1.887	No	-
B	1180	0.17	1.887	No	-
C	1310	0.42	1.887	No	-
D	1220	0.06	1.887	No	-
E	1195	0.11	1.887	No	-
F	2540	2.48	1.887	Yes	Trace water leading to co-catalyst generation

Formula: Grubbs' Statistic G = \| suspect value - mean \| / standard deviation.

Experimental Protocols

Protocol 1: Standardized Catalysis Run for Metric Generation

Purpose: To generate consistent, comparable data for TOF, Yield, Selectivity, and ee.

• Setup: Conduct all reactions in a controlled environment (e.g., glovebox for air-sensitive catalysts) using flame-dried glassware.
• Charge Reactants: Weigh substrate(s) (e.g., 1.0 mmol) and internal standard (e.g., n-dodecane, 0.1 mmol) into reaction vessel. Add dry, degassed solvent (e.g., 5 mL toluene).
• Initiate Reaction: Add catalyst (e.g., 1.0 mol% Grubbs' 2nd generation) as a stock solution. Seal vessel and place in pre-heated oil bath (e.g., 80°C) with magnetic stirring (1200 rpm to avoid mass transfer limitations).
• Kinetic Sampling: At predetermined intervals (t=5, 10, 20, 40, 60 min), withdraw a 0.1 mL aliquot via syringe. Immediately quench in a vial containing 0.5 mL of a phosphazene base (e.g., P1-t-Bu) solution to arrest catalysis.
• Analysis: Analyze aliquots by GC-FID, HPLC, or NMR to determine conversion and selectivity. Calculate TOF from the initial slope (first 10% conversion) of concentration vs. time plot.
• Workup: After 24h, cool reaction, dilute with ethyl acetate, and filter through a silica plug. Concentrate under reduced pressure.
• Purification & ee Determination: Purify the crude product by flash chromatography. Analyze pure product by chiral HPLC or SFC to determine enantiomeric excess.

Protocol 2: Application of Grubbs' Test for Outlier Identification

Purpose: To statistically identify outliers within a dataset of a single performance metric.

• Data Collection: Assemble a dataset (n ≥ 3, ideally n > 5) of the same metric from identical experimental runs (e.g., TOF values from 6 separate but identical catalysis experiments).
• Compute Statistics: Calculate the sample mean (x̄) and sample standard deviation (s) for the dataset.
• Identify Suspect Value: Visually inspect data for the value most distant from the mean.
• Calculate Grubbs' Statistic (G): G = \| x_suspect - x̄ \| / s.
• Compare to Critical Value: Look up the critical value G_critical for your chosen significance level (α, typically 0.05) and sample size (n).
• Decision: If G > G_critical, classify the suspect value as an outlier. Remove it and repeat the process on the remaining data if necessary. Never discard an outlier without investigating its chemical cause.

Diagrams and Workflows

Title: Grubbs' Test Workflow for Catalyst Data Analysis

Title: Root Cause Analysis Pathway for Outliers

The Scientist's Toolkit

Table 3: Essential Research Reagent Solutions for Robust Catalysis Screening

Item	Function & Rationale
Internal Standard (e.g., n-Dodecane, Mesitylene)	Added in known quantity before reaction; enables accurate quantitative analysis via GC-FID or NMR by correcting for injection/volume inconsistencies.
Catalyst Stock Solutions	Precise volumetric delivery of small catalyst masses improves reproducibility and minimizes weighing errors for air-sensitive compounds.
Dry, Degassed Solvents	Eliminates water and oxygen as variables that can poison catalysts or initiate side reactions, a major source of outliers.
Phosphazene Base Quench Solution (e.g., P1-t-Bu)	Rapidly and irreversibly deactivates many metal catalysts for accurate kinetic sampling, fixing conversion at a precise timepoint.
Chiral HPLC/SFC Columns & Calibrants	Essential for accurate ee determination. Requires pure enantiomer samples to confirm retention times and avoid misidentification.
Silica Gel for Filtration/Chromatography	Standardized, high-purity silica ensures consistent product isolation and recovery, preventing yield outliers from adsorption.

Step-by-Step: Performing Grubbs' Test on Your Catalyst Dataset

The identification of outliers in catalytic performance data, such as reaction yield or turnover frequency (TOF), is a critical step in the development of robust catalysts for pharmaceuticals and fine chemicals. This protocol is framed within a broader thesis applying Grubbs' test—a statistical procedure for detecting a single outlier in a univariate data set assumed to come from a normally distributed population. Proper data organization is a prerequisite for valid statistical analysis, ensuring that identified outliers truly represent anomalous catalyst behavior rather than artifacts of poor data management.

Key Research Reagent Solutions & Materials

Item	Function in Catalyst Performance Research
Homogeneous Catalyst (e.g., Grubbs' Ruthenium Complex)	The active species whose performance (yield, TOF) is being measured and analyzed for outlier behavior.
Substrate (Pharmaceutical Intermediate)	The molecule undergoing catalysis; its conversion defines the reaction yield.
Internal Standard (e.g., Tridecane for GC)	A known quantity of a non-reactive compound added to reaction mixtures to enable accurate quantitative analysis of yield via chromatography.
Deactivator/Quencher (e.g., Ethyl Vinyl Ether)	Rapidly terminates catalytic reactions at precise timepoints for accurate TOF calculation.
Deuterated Solvent for NMR Analysis (e.g., C6D6)	Allows for direct, quantitative yield determination via 1H NMR spectroscopy without need for internal standard calibration.
Statistical Software (e.g., R, Python with SciPy)	Platform for performing Grubbs' test and other statistical analyses on the organized dataset.

Protocol: Organizing Reaction Yield & TOF Data for Outlier Analysis

Experimental Protocol for Data Generation

• Reaction Setup: In a glovebox, charge 20 identical 4 mL vials with a magnetic stir bar. To each, add the substrate (e.g., 0.1 mmol of a terminal olefin) and internal standard (e.g., 0.02 mmol tridecane). Dissolve in 1 mL of anhydrous, degassed toluene.
• Catalyst Initiation: Remove vials from the glovebox and place in a pre-heated aluminum block at 35°C. Using a micropipette, rapidly add a stock solution of the catalyst (e.g., Grubbs' 2nd Generation, 0.001 mmol in 0.1 mL toluene) to each vial sequentially at 30-second intervals to establish precise timing.
• Reaction Quenching: For TOF determination, quench individual vials at precise timepoints (e.g., t = 2, 5, 10, 15, 20 min) by injecting 0.1 mL of ethyl vinyl ether. For final yield, run separate reaction vials to completion (e.g., 24h).
• Quantitative Analysis:
  • GC Analysis: Dilute a sample from each quenched vial with diethyl ether. Analyze by GC-FID using the internal standard method. Calculate yield (%) based on substrate peak area relative to the standard.
  • TOF Calculation: Calculate TOF (h⁻¹) as (moles of product formed) / (moles of catalyst * reaction time in hours). Use the data from the early, linear-conversion timepoints.

Data Structuring Protocol for Analysis

• Create a Master Table: Construct a single table where each row represents a unique experimental run and each column a variable.
• Include All Metadata: Essential columns must include: CatalystID, BatchNumber, SubstratePurity(%), SolventWaterPPM, ReactionTemp(°C), PreciseReactionTime(hr), and Analyst_Initials.
• Record Primary Data: Create separate columns for the raw analytical output: GC_Area_Product, GC_Area_Internal_Standard.
• Create Calculated Columns: Add columns for Calculated_Yield(%) and Calculated_TOF(h⁻¹) using consistent formulas applied across all rows.
• Flag Known Issues: Include a Notes column to document any observable anomalies (e.g., "vial cracked," "stir bar stopped").
• Data Export: Save the final table in a non-proprietary format (e.g., .csv, .tsv) for import into statistical software.

Data Presentation: Structured Tables

Table 1: Example Dataset of Catalytic Performance for 10 Runs

Run_ID	Catalyst_Batch	Temp (°C)	Time (h)	GCAreaProduct	GCAreaStd	Yield (%)	TOF (h⁻¹)	Notes
EXP-001	GRUB-02-A	35.0	24.0	1458920	502345	92.5	--
EXP-002	GRUB-02-A	35.0	24.0	1500234	499876	95.2	--
EXP-003	GRUB-02-A	35.0	24.0	1435678	501234	91.2	--
EXP-004	GRUB-02-B	35.0	24.0	1324098	498765	84.5	--	Slight temp fluct.
EXP-005	GRUB-02-B	35.0	24.0	1498765	502111	94.8	--
EXP-006	GRUB-02-A	35.0	0.33	234567	500123	14.8	533	For TOF
EXP-007	GRUB-02-A	35.0	0.33	245678	499876	15.6	559	For TOF
EXP-008	GRUB-02-A	35.0	0.33	198765	501110	12.5	450	For TOF
EXP-009	GRUB-02-A	35.1	0.33	289654	500987	18.3	659	For TOF
EXP-010	GRUB-02-A	35.0	24.0	985432	499001	62.3	--	Low yield observed

Table 2: Yield Data Prepared for Grubbs' Test (G = | suspect value - mean | / s)

Dataset: Yield (%) from Batch GRUB-02-A (n=4, excluding EXP-010)
Values Sorted:	91.2, 92.5, 94.8, 95.2
Mean (x̄):	93.4
Standard Deviation (s):	1.8
Suspect Value (Low): 91.2	G (calculated): (93.4-91.2)/1.8 = 1.22
Critical G (n=4, α=0.05):	1.481
Outlier Conclusion:	Gcalc < Gcrit. Value 91.2% is not an outlier.

Workflow & Relationship Diagrams

Diagram 1: Data Analysis Workflow from Experiment to Decision

Diagram 2: Role of Data Prep in Catalyst Outlier Thesis

Within the broader thesis research on "Advanced Statistical Methods for Detecting Performance Outliers in Heterogeneous Catalyst Libraries," Grubbs' test is employed as a critical tool. Its application ensures the integrity of high-throughput screening data by identifying catalysts whose activity or selectivity measurements are statistically anomalous, potentially indicating experimental error, unique catalytic mechanisms, or deactivation phenomena warranting separate investigation.

Theoretical Framework & Formulas

Grubbs' test is used to detect a single outlier in a univariate dataset assumed to be normally distributed. The test compares the deviation of the suspected outlier from the sample mean to the sample standard deviation.

Key Formulas:

• Grubbs' Test Statistic (G): G = |(X_suspect - X̄)| / s Where:

  • X_suspect is the suspected outlier value.
  • X̄ is the sample mean.
  • s is the sample standard deviation.

• Critical Value (G_critical): G_critical = ((N-1) / √N) * √( (t_(α/(2N), N-2)^2) / (N-2 + t_(α/(2N), N-2)^2) ) Where:

  • N is the sample size.
  • t is the critical value from the t-distribution with N-2 degrees of freedom and a significance level of α/(2N) (two-tailed test).
  • α is the chosen significance level (typically 0.05).

Decision Rule: If G > G_critical, the null hypothesis (that there are no outliers) is rejected, and X_suspect is considered an outlier.

Example Data & Calculation Walkthrough

Context: Catalytic turnover frequency (TOF, h⁻¹) for 7 different catalyst formulations under identical test conditions.

Raw Data: Catalyst TOF = [142, 136, 155, 138, 141, 189, 139] The value 189 is visually suspected as an outlier.

Step-by-Step Calculation:

• Define Parameters: N = 7, α = 0.05.
• Calculate Sample Statistics:
  • Mean (X̄) = (142+136+155+138+141+189+139) / 7 = 148.57
  • Standard Deviation (s) = 18.50
• Compute G Statistic:
  • For suspected value 189: G = |189 - 148.57| / 18.50 = 2.185
• Determine Gcritical:
  • Degrees of freedom for t: N-2 = 5
  • Significance for t: α/(2N) = 0.05/(2*7) = 0.00357
  • t_(0.00357, 5) ≈ 4.398 (from t-distribution tables)
  • Gcritical = ((7-1)/√7) * √( (4.398²) / (5 + 4.398²) ) = (2.267) * √(19.342 / 24.342) = 2.020

Conclusion: Since G (2.185) > G_critical (2.020), the TOF value of 189 h⁻¹ is identified as a statistical outlier at the 95% confidence level.

Parameter	Symbol	Value	Notes
Sample Size	N	7	Number of catalysts tested
Sample Mean	X̄	148.57 h⁻¹	Average Turnover Frequency
Sample Std. Dev.	s	18.50 h⁻¹	Standard deviation of TOF
Suspected Value	X_suspect	189 h⁻¹	Potential outlier
Grubbs' Statistic	G	2.185	Calculated test value
Significance Level	α	0.05	95% confidence
t-critical value	t	4.398	for α/(2N), df=5
Critical Value	G_critical	2.020	Threshold for rejection
Outlier?	Decision	Yes	G > G_critical

Experimental Protocols for Catalyst Performance Analysis

Protocol A: High-Throughput Catalyst Screening for Outlier Detection

Purpose: To generate reproducible performance data (e.g., Turnover Frequency, Yield) suitable for subsequent Grubbs' statistical analysis.

Methodology:

• Catalyst Library Preparation: Synthesize catalyst array (e.g., 7 distinct formulations) using automated parallel synthesis under inert atmosphere.
• Standardized Reaction Setup: Charge each reactor in a parallel pressure vessel system with identical amounts of catalyst (mass or molar basis), substrate, and solvent using liquid handling robots.
• Controlled Reaction Execution: Conduct reactions at precise, uniform temperature (±0.5°C) and pressure with constant agitation for a fixed duration.
• Quantitative Analysis: Terminate reactions simultaneously. Analyze product mixtures using calibrated GC-FID or HPLC-UV to determine conversion and selectivity. Calculate primary performance metric (e.g., TOF).
• Data Collation: Tabulate the primary metric for each catalyst replicate (minimum n=3 per formulation).

Protocol B: Application of Grubbs' Test to Performance Data

Purpose: To statistically identify significant outliers within a dataset of catalyst performance metrics.

Methodology:

• Data Organization: Arrange performance data for a single condition in one column.
• Normality Check: Perform Shapiro-Wilk test (for N < 50) to verify underlying normal distribution assumption.
• Initial Calculation: Compute the mean and standard deviation of the full dataset.
• G Statistic Calculation: Identify the value farthest from the mean. Calculate the G statistic using the provided formula.
• Critical Value Determination: Based on sample size N and chosen α (typically 0.05), compute G_critical using the formula or reference a statistical table.
• Decision & Iteration: If G > G_critical, flag the value as an outlier. Remove the flagged value and repeat the process on the remaining dataset until no more outliers are detected (Grubbs' test for multiple outliers).
• Reporting: Document all identified outliers, the statistical parameters used, and the final cleaned dataset.

Visual Workflows

Title: Grubbs' Test Statistical Decision Workflow

Title: Experimental & Statistical Analysis Pipeline

The Scientist's Toolkit: Research Reagent Solutions

Table 2: Essential Materials for Catalytic Screening & Statistical Validation

Item	Function in Research
Parallel Pressure Reactors	Enables simultaneous, controlled reaction conditions for multiple catalysts, ensuring data comparability.
Precursor & Ligand Libraries	High-purity chemical building blocks for systematic catalyst synthesis and variation.
Internal Standard (GC/HPLC)	Certified reference compound added to reaction mixtures to enable precise quantitative analysis.
Statistical Software (e.g., R, Python with SciPy)	Platform for calculating Grubbs' test statistics, critical values, and performing complementary normality tests.
Certified Reference Material (CRM)	Standard catalyst or reaction sample with known performance used for analytical method validation.
Inert Atmosphere Glovebox	Essential for the synthesis and handling of air- and/or moisture-sensitive catalytic materials.

Application Notes

This document provides protocols for applying Grubbs' test in catalyst performance outlier detection, with a focus on the critical selection of the significance level (α). The α value directly controls the confidence level (CL = 1 - α) and determines the test's stringency in flagging outliers, a pivotal decision in high-stakes pharmaceutical catalyst research.

The α-Confidence Level Relationship in Grubbs' Test

The table below quantifies the relationship between α, confidence level, and the implied risk in outlier detection.

Table 1: Standard Alpha (α) Values, Corresponding Confidence Levels, and Interpretation

Alpha (α) Value	Confidence Level (1-α)	Critical Value* (approx. for n=10)	Interpretation for Catalyst Research
0.01 (1%)	99%	2.482	High confidence. Low false-positive risk. Use when outlier removal must be highly conservative (e.g., final performance validation).
0.05 (5%)	95%	2.176	Standard balance. Recommends a datum for review. Suitable for routine screening of catalyst batch performance data.
0.10 (10%)	90%	2.036	Higher sensitivity. Increases false-positive chance. May be used for exploratory analysis of noisy preliminary datasets.

*Grubbs' statistic (G) critical values depend on sample size (n) and α. Values shown are illustrative.

Protocol: Systematic Outlier Analysis for Catalyst Performance Datasets

Objective: To identify and statistically justify the removal of outlier data points from catalyst yield or turnover number (TON) datasets using Grubbs' test with a pre-specified α.

Materials & Reagents (The Scientist's Toolkit)

Table 2: Essential Research Reagent Solutions & Materials

Item	Function in Catalyst Outlier Analysis
Homogeneous Catalyst Batch (e.g., Pd/XPhos complex)	Provides the performance data (yield, TON) for statistical analysis. Batch consistency is critical.
Standardized Reaction Substrates	Ensures performance variability stems from the catalyst, not reactant quality or concentration.
Internal Standard (for GC/HPLC)	Enables precise and accurate quantification of reaction yield, generating the primary dataset.
Statistical Software (e.g., R, Python with SciPy, GraphPad Prism)	Performs the Grubbs' test calculation and compares the G statistic to the critical value for chosen α.
Laboratory Information Management System (LIMS)	Logs all raw data, α decisions, and test results for audit trail and regulatory compliance.

Procedure:

• Data Collection & Prerequisites:

  • Perform the catalytic reaction (e.g., cross-coupling) under identical, optimized conditions for n replicates (recommended n ≥ 5).
  • Record the primary performance metric (e.g., yield %) for each replicate.
  • Ensure the dataset, apart from the suspected outlier, is approximately normally distributed. Use a Shapiro-Wilk test (α=0.05) if n is small.

• Pre-Test: Alpha (α) Selection Justification:

  • Document the rationale for the chosen α value prior to analysis. Base this on:
    • Research Phase: Use α=0.10 for early exploratory catalyst screening; α=0.05 for standard optimization; α=0.01 for final validation or QC.
    • Risk Tolerance: Align α with the consequence of a Type I (false positive) or Type II (false negative) error in your context.

• Grubbs' Test Execution:

  • Calculate the sample mean (x̄) and standard deviation (s) of the full dataset.
  • Identify the datum furthest from the mean (the suspected outlier).
  • Compute the Grubbs' statistic G = max | x_i - x̄ | / s.
  • Obtain the critical value G_critical for your chosen α and sample size n from a standard statistical table or software.
  • Decision Rule: If G > G_critical, reject the null hypothesis (that there are no outliers) at the α significance level. The datum is considered a statistical outlier.

• Iterative Testing & Reporting:

  • If an outlier is detected and removed, the test may be iteratively performed on the remaining data. Note: This increases family-wise error; use caution.
  • In the final report, must state: the α value used, the corresponding confidence level, the G statistic, the critical value, and the action taken.

Visual Workflows

Title: Grubbs' Test Protocol for Catalyst Data

Title: Statistical Trade-Offs Controlled by α

Within catalyst performance research, especially in pharmaceutical development, identifying outliers is critical for ensuring the reliability of activity and selectivity measurements. This application note details the use of Grubbs' test (also known as the maximum normalized residual test) to determine if a suspect data point from a heterogenous catalysis experiment, such as yield or turnover frequency (TOF), is a statistically significant outlier. The protocol is framed within a broader thesis aiming to standardize outlier detection in high-throughput catalyst screening.

Statistical Protocol: Grubbs' Test for Outlier Detection

Grubbs' test detects a single outlier in a univariate dataset assumed to come from a normally distributed population.

Prerequisites

• Data: A dataset of n independent measurements of a single performance metric (e.g., reaction yield for a catalyst under identical conditions).
• Assumption: The data, excluding the suspected outlier, approximately follows a normal distribution. This should be verified using a normality test (e.g., Shapiro-Wilk) on the remaining data points if n is sufficient.
• Hypotheses:
  • H₀ (Null): There are no outliers in the dataset.
  • Hₐ (Alternative): There is exactly one outlier in the dataset.

Step-by-Step Calculation Protocol

• Compute the sample mean (x̄) and standard deviation (s) of the entire dataset, including the suspected outlier.
• Identify the suspect value (x*) – the value farthest from the mean.
• Calculate the G statistic: G = |x* - x̄| / s
• Determine the critical value (G_critical): Use Grubbs' critical value table for two-tailed test at significance level α (commonly α=0.05) and sample size n. Alternatively, calculate the critical value using: G_critical = √( (t²_{α/(2n), n-2}) / (n - 2 + t²_{α/(2n), n-2}) ) where t is the critical value from the t-distribution.
• Make a decision: If G > G_critical, reject the null hypothesis and classify the suspect point as a statistically significant outlier.

Example Calculation for Catalyst Yield Data

A dataset of 8 independent yield measurements for a novel hydrogenation catalyst (%) is collected: [78.2, 82.1, 79.5, 81.3, 93.2, 80.7, 79.9, 81.0]. The value 93.2% is suspect.

Table 1: Grubbs' Test Calculation Summary

Parameter	Value	Notes
Sample Size (n)	8
Mean (x̄)	81.86 %	Includes all data.
Std. Dev. (s)	4.69 %	Includes all data.
Suspect Value (x*)	93.2 %
G Statistic	(93.2-81.86)/4.69 = 2.42
G_critical (α=0.05)	2.03	From standard statistical table for n=8.
Conclusion	G > G_critical	Reject H₀. The point is a significant outlier.

Experimental Protocol: Generating Catalyst Performance Data

The following methodology generates the replicate performance data suitable for outlier analysis.

Standardized Catalytic Reaction (Hydrogenation Model)

• Objective: To evaluate the yield of a hydrogenation catalyst candidate under controlled conditions.
• Materials: See Scientist's Toolkit.
• Procedure:
  • In an inert atmosphere glovebox, charge the reaction vessel with the substrate (e.g., 1.0 mmol of a specified alkene) and magnetic stir bar.
  • Add the catalyst precursor (1.0 mol%) and ligand (if applicable, 2.2 mol%).
  • Transfer the sealed vessel to a high-pressure reactor system.
  • Under a positive flow of inert gas, add the degassed solvent (10 mL, e.g., tetrahydrofuran).
  • Purge the reactor headspace three times with H₂ gas.
  • Pressurize the reactor to the specified H₂ pressure (e.g., 5 bar).
  • Stir the reaction mixture at the set temperature (e.g., 50°C) for the designated time (e.g., 16 hours).
  • Cool the reactor to room temperature, carefully vent the pressure, and open the vessel.
  • Quantify the reaction yield using an internal standard via Gas Chromatography (GC) or Quantitative NMR (qNMR). Each run constitutes a single data point (n=1).
• Replication: The entire experiment, from step 1, is repeated independently a minimum of 5-8 times to generate the dataset for analysis.

Diagram 1: Catalytic Experiment Workflow

Decision Pathway for Outlier Management

Upon identifying a statistical outlier, a structured investigation is required.

Diagram 2: Outlier Investigation Decision Tree

The Scientist's Toolkit: Key Research Reagents & Materials

Table 2: Essential Materials for Catalytic Performance Testing

Item	Function & Rationale
High-Pressure Reactor	Provides a controlled, safe environment for reactions under gas pressure (H₂). Ensures consistency in pressure variable.
Inert Atmosphere Glovebox	Prevents decomposition of air- or moisture-sensitive catalyst precursors and ligands during reaction setup.
Catalyst Precursor	The metal complex (e.g., Pd, Ru, Rh-based) under investigation. Source of active catalytic species.
Ligands	Organic molecules that modify catalyst selectivity, activity, and stability (e.g., Phosphines, NHCs).
Deuterated Solvents	For reaction monitoring and quantitative NMR analysis (e.g., CDCl₃, DMSO-d₆).
Internal Standard (GC/qNMR)	A known quantity of a non-interfering compound (e.g., hexamethylbenzene) to enable accurate product quantification.
Statistical Software (R/Python)	To perform Grubbs' test, normality checks, and generate visualizations programmatically for rigor and reproducibility.

The identification and handling of outliers are critical in catalyst performance studies, where subtle variations in synthesis or testing can yield data points that deviate significantly from the expected trend. Within the broader thesis on Grubbs' test application, proper documentation of outlier analysis is not merely a statistical exercise but a fundamental component of research integrity. It ensures that reported performance metrics—such as turnover frequency (TOF), selectivity, or stability—are robust and reproducible. This document provides application notes and protocols for rigorously documenting outlier tests in publication-ready formats, with a focus on Grubbs' test for normally distributed catalyst performance data.

Core Principles for Documentation

Transparency and completeness are paramount. Documentation must allow an independent researcher to understand, evaluate, and reproduce the outlier analysis. Key principles include:

• Pre-specification: Whenever possible, the criteria and method for outlier testing (e.g., Grubbs' test at α=0.05) should be defined a priori in the experimental section.
• Justification: The choice of statistical test must be justified based on the data distribution and sample size.
• Full Reporting: All results of the test, regardless of outcome, must be reported. This includes the test statistic (G-value), critical value, p-value, and the decision made.
• Data Accessibility: The complete dataset, including any identified outliers, should be made available, ideally in a supplementary file.

Detailed Protocol: Application of Grubbs' Test

Objective: To identify a single outlier in a univariate dataset assumed to be normally distributed, commonly applied to catalyst yield or activity measurements from replicate experiments.

Materials & Reagents (Research Reagent Solutions):

Item	Function in Catalyst Performance Context
Homogeneous Catalyst Batch	Standardized material from a single synthesis batch to minimize precursor-driven variance.
Reference Substrate	High-purity, well-characterized substrate (e.g., for cross-coupling, hydrogenation) to ensure reaction consistency.
Internal Standard	For GC/HPLC analysis, to distinguish measurement error from true performance outliers.
Calibration Standards	Series of known concentrations for analytical instrument calibration, verifying measurement linearity.
Statistical Software (e.g., R, Python with SciPy, GraphPad Prism)	To perform the Grubbs' test calculation accurately and generate test statistics.

Step-by-Step Workflow:

• Data Collection: Perform a minimum of n ≥ 3 replicate catalytic reactions under identical, rigorously controlled conditions (temperature, pressure, reaction time, catalyst loading).
• Initial Data Review: Plot the raw performance data (e.g., conversion %) visually to inspect for obvious anomalies. Do not remove data points at this stage.
• Normality Assessment: Test the dataset for normality using an appropriate test (e.g., Shapiro-Wilk test for n < 50). Grubbs' test assumes an underlying normal distribution. Document the result of the normality test.
  • If data is not normally distributed, consider data transformation or a non-parametric outlier test. Justify the choice.
• Grubbs' Test Calculation: a. Calculate the sample mean (x̄) and standard deviation (s) of the full dataset. b. Identify the data point (x_i) that is farthest from the mean. c. Compute the Grubbs' test statistic G: G = |x_i - x̄| / s d. Determine the critical value G_critical for significance level α (typically 0.05) and sample size n using published tables or statistical software. e. Compare G to G_critical. If G > G_critical, the point is considered a statistically significant outlier.
• Post-Test Action & Documentation:
  • If an outlier is identified, report the original full dataset summary (mean ± SD, N) including the outlier.
  • State clearly if the outlier was excluded from final performance analysis.
  • Provide a plausible technical or experimental reason for the outlier if investigated (e.g., "The excluded data point corresponded to a reaction where the stirrer failed, leading to inefficient mixing.").

Data Presentation and Tables

Table 1: Example Summary of Catalyst Turnover Number (TON) Replicates with Grubbs' Test Documentation

Replicate	TON	Included in Final Analysis?	Notes
1	9450	Yes
2	9620	Yes
3	9380	Yes
4	12550	No	Identified as outlier (G = 2.87, Gcrit, α=0.05, n=5 = 1.715). Investigation found substrate weighing error.
5	9500	Yes
Summary (Original)	Mean: 10100 ± 1330 (SD), N=5
Summary (Final)	Mean: 9488 ± 102 (SD), N=4		After outlier exclusion, CV reduced from 13.2% to 1.1%.

Table 2: Essential Elements to Report for Each Outlier Test

Element	Example Entry	Purpose
Test Name	Grubbs' test for a single outlier	Identifies the method.
Assumption Check	Shapiro-Wilk p = 0.62 (supports normality)	Validates test applicability.
Test Statistic (G)	G = 2.87	Provides the calculated evidence.
Sample Size (n)	n = 5	Allows critical value lookup.
Significance Level (α)	α = 0.05 (two-tailed)	States the decision threshold.
Critical Value (Gcrit)	Gcrit = 1.715	Provides the threshold for comparison.
p-value	p = 0.039	Offers an alternative to Gcrit.
Decision	Reject H0; TON of 12550 is an outlier.	Clear conclusion.
Action Taken	Point excluded from final performance calculation.	Ensates transparency.

Visualization of Workflows

Grubbs' Test Workflow for Catalyst Data

Link Between Documentation Sections

Solving Common Problems and Refining Grubbs' Test for Complex Catalyst Data

In the pursuit of novel heterogeneous catalysts, high-throughput experimentation often yields initial performance data (e.g., conversion rate, selectivity) with very few replicates (N<7) due to material scarcity and cost. A core thesis on applying Grubbs' test for outlier detection in such datasets must first confront the fundamental statistical limitations imposed by extremely small sample sizes. These limitations necessitate adjusted analytical and experimental protocols to ensure robust conclusions in drug development precursor synthesis.

Limitations of Statistical Methods with N<7

With N<7, the power of any statistical test is severely diminished. Specific to Grubbs' test:

• Low Detection Power: The test statistic critical values become extremely large, making it nearly impossible to flag a value as an outlier unless it is astronomically distant from the mean.
• Masking Effect: With so few points, a single outlier can drastically skew the mean and standard deviation, "masking" its own detection.
• Assumption Violations: Tests like Grubbs' assume an underlying normal distribution. Verifying normality is practically impossible with N<7.

Table 1: Grubbs' Test Critical Values (G) for α=0.05 and Small N

Sample Size (N)	Critical Value (G)	Minimum Detectable Deviation*
3	1.155	>1.15 SD from mean
4	1.481	>1.48 SD from mean
5	1.715	>1.71 SD from mean
6	1.887	>1.89 SD from mean
7	2.020	>2.02 SD from mean

*The value must be this many standard deviations from the sample mean to be considered an outlier.

Adjusted Methodological Approaches and Protocols

Protocol 3.1: Tiered Analytical Workflow for Small-N Catalyst Data

This protocol prioritizes non-statistical and robust methods before applying any outlier test.

1. Pre-Statistical Inspection & Visualization:

• Material: Generate a run-order plot and a basic boxplot (though limited).
• Procedure: Plot catalyst performance metrics (e.g., yield) against the chronological order of synthesis or testing. Look for systematic drifts. Examine the raw data spread visually.

2. Application of Robust/Non-Parametric Descriptors:

• Material: Calculate median and interquartile range (IQR).
• Procedure: Replace the mean with the median as the central tendency measure. Replace the standard deviation with the IQR (difference between 75th and 25th percentiles) as the measure of spread. These are less influenced by extreme values.

3. Modified Grubbs' Test with Prior Justification:

• Material: Pre-defined, scientifically justified suspicion about a specific data point.
• Procedure: Apply Grubbs' test only if there is an independent experimental reason (e.g., known catalyst synthesis failure) to suspect a specific measurement. Report the test result with the explicit caveat of its low power.

4. Confirmatory Re-Test (If Feasible):

• Material: Additional catalyst batch or re-testing capacity.
• Procedure: If resources allow, synthesize a new batch of the suspected outlier catalyst or re-run the performance test. Compare the new result to the original small set.

Diagram Title: Tiered Workflow for Analyzing Small-N Catalyst Data

Protocol 3.2: Bayesian-Informed Prior Integration for Catalyst Screening

This advanced protocol uses prior knowledge to supplement small-N data, formalizing the "scientific justification" step from Protocol 3.1.

Procedure:

• Define Prior Distribution: Based on historical data from similar catalyst families (e.g., Pd on carbon for hydrogenation), establish a prior probability distribution for the expected performance range.
• Calculate Likelihood: Compute the likelihood of observing your new small-N dataset given different possible true performance values.
• Compute Posterior: Use Bayes' Theorem to combine the prior and the likelihood, forming a posterior distribution that represents updated belief about the catalyst's performance.
• Identify Outliers: Points with very low probability under the posterior distribution can be considered outliers with greater confidence than from the data alone.

Diagram Title: Bayesian Framework for Small-N Catalyst Analysis

The Scientist's Toolkit: Research Reagent Solutions

Table 2: Essential Materials for Small-N Catalyst Performance Studies

Item	Function in Context of Small-N Studies
High-Throughput Micro-Reactor Array	Enables parallel synthesis/testing of multiple catalyst formulations, maximizing data points from limited material batches.
Standardized Catalyst Support Slurry	Ensures consistent impregnation and loading of active sites across all samples, reducing experimental variability that can mask true performance.
Internal Standard (for Analytic GC/HPLC)	Added to reaction product streams to calibrate analytical instrument response, improving measurement accuracy for each precious data point.
Calibrated Reference Catalyst	A well-characterized catalyst (e.g., NIST-traceable) run alongside new samples to validate the entire testing protocol and instrument performance.
Robust Statistical Software (e.g., R with robustbase)	Provides libraries for calculating medians, IQR, and performing robust regression, essential for analyzing small, noisy datasets.
Laboratory Information Management System (LIMS)	Tracks all meta-data (synthesis conditions, operator, instrument ID) critical for identifying non-statistical causes of suspected outliers.

1. Introduction and Thesis Context Within catalyst performance research for pharmaceutical synthesis, the identification of true outliers is critical for accurate structure-activity relationship modeling. A single application of Grubbs' test is insufficient when multiple outliers may be present. This protocol details an iterative, rigorous application of Grubbs' test, framed within a broader thesis on statistical validation in heterogeneous catalyst screening, to ensure robust data sets for downstream drug development.

2. Theoretical Foundation and Iterative Algorithm Grubbs' test (maximum normed residual test) identifies a single outlier in a univariate data set assumed to be normally distributed. The test statistic G is calculated as:

G = | suspect value - sample mean | / sample standard deviation

This G statistic is compared to a critical value from the t-distribution. For multiple potential outliers, an iterative procedure is mandated:

Iterative Grubbs' Test Workflow

3. Application Protocol: Catalyst Turnover Frequency (TOF) Analysis

• Objective: To iteratively identify and remove significant outliers in catalyst TOF measurements from a high-throughput screening campaign for a key C-N coupling reaction.
• Preparatory Step: Visually inspect the initial dataset (e.g., via box plot) to assess general spread.

Step-by-Step Procedure:

• Initial Calculation: For the full dataset of n TOF values, compute the sample mean (x̄) and sample standard deviation (s).
• Identify Extreme Value: Find the value (xi) that maximizes the absolute deviation |xi - x̄|.
• Compute G Statistic: Calculate G = max|x_i - x̄| / s.
• Determine Critical Value: Obtain the two-sided Grubbs' critical value (G_crit) for significance level α=0.05 and n observations from standard statistical tables or software (e.g., NIST/SciPy).
• Hypothesis Test: a. If G > Gcrit, classify xi as an outlier. Remove it from the dataset. b. If G ≤ Gcrit, do not remove xi. The iterative process terminates.
• Iterate: If an outlier was removed, return to Step 1 with the reduced dataset of n = n - 1 values. Continue until no further outliers are detected.

• Critical Note: The assumption of underlying normality must be re-evaluated for the final, pruned dataset. Process no more than ~20% of data as outliers via this method.

4. Exemplar Data from Catalysis Research The following table summarizes a hypothetical iteration for TOF data (in h⁻¹):

Table 1: Iterative Grubbs' Test Application to Catalyst TOF Data

Iteration	n	Dataset (TOF, h⁻¹)	Mean (x̄)	SD (s)	Suspect Value	G_calculated	G_critical (α=0.05)	Outcome
1	10	102, 98, 105, 210, 99, 101, 97, 104, 100, 103	111.9	34.7	210	2.826	2.290	Remove
2	9	102, 98, 105, 99, 101, 97, 104, 100, 103	101.0	2.7	105	1.481	2.215	Retain

Conclusion: Only the value 210 h⁻¹ is identified as a statistical outlier.

5. The Scientist's Toolkit: Essential Research Reagents & Materials

Table 2: Key Reagent Solutions for Catalyst Screening & Validation

Item	Function in Catalyst Performance Research
Heterogeneous Catalyst Library (e.g., Pd on various supports)	Core materials for screening structure-activity relationships in cross-coupling reactions.
High-Purity Aryl Halide & Nucleophile Substrates	Ensures reaction performance variability is due to catalyst, not reactant impurities.
Inert Atmosphere Glovebox / Schlenk Line	For handling air/moisture-sensitive catalysts and reagents, ensuring consistent initial conditions.
Internal Standard (e.g., dodecane for GC)	Allows for precise quantitative analysis of reaction yield and turnover number (TON).
Quenching Agent (e.g., specific sorbents or chemical quenches)	Precisely stops reaction at timed intervals for kinetic (TOF) measurements.
Calibrated Analytical Standard Solutions	For generating accurate calibration curves in HPLC/GC analysis to determine conversion/yield.
Statistical Software Package (e.g., SciPy, R, GraphPad Prism)	To perform Grubbs' test calculations, critical value lookup, and general data analysis.

6. Advanced Considerations and Pathway Integration Outlier identification must be integrated with experimental investigation. Suspect data points should trigger a review of the Experimental Anomaly Investigation Pathway.

In research on catalyst performance, identifying true outliers using Grubbs' test is a critical step for ensuring data integrity. A fundamental assumption of Grubbs' test is that the data, excluding the potential outlier, is normally distributed. Violation of this assumption due to non-normal data can lead to both false positives (identifying non-outliers) and false negatives (missing true outliers). This protocol provides a framework for researchers to systematically handle non-normal data encountered in catalyst performance metrics (e.g., yield, turnover frequency, selectivity) to validate the prerequisite for robust outlier analysis.

Decision Framework: Transformation vs. Non-Parametric Alternatives

The following decision pathway guides the selection of the appropriate method for handling non-normal data.

Title: Decision Pathway for Non-Normal Data in Catalyst Analysis

Data Transformation Protocols

Preliminary Assessment & Protocol Selection

Objective: Diagnose the type of non-normality to select the optimal transformation. Materials: Statistical software (R, Python, GraphPad Prism), dataset of catalyst performance replicates. Procedure:

• Visualize: Create a histogram and a Quantile-Quantile (Q-Q) plot.
• Quantify: Perform the Shapiro-Wilk test (preferred for n < 50) or the Anderson-Darling test. Record the test statistic (W or A²) and p-value.
• Diagnose Skewness: Calculate the skewness coefficient (γ). |γ| > 1 indicates substantial skewness.
• Diagnose Variance Issues: If data spans several orders of magnitude, note multiplicative error structure.

Common Transformation Methods & Application Protocols

Table 1: Guide to Common Data Transformations for Catalyst Performance Data

Transformation	Formula	Primary Use Case in Catalyst Research	Example Catalyst Metric	Key Assumption	Effect
Logarithmic	( y' = \log_{10}(y) ) or ( \ln(y) )	Right-skewed data, constant multiplicative error.	Reaction yield (%), Turnover Frequency (TOF).	Data must be positive. Can add constant to handle zeros.	Compresses large values, expands small ones. Stabilizes variance.
Square Root	( y' = \sqrt{y} )	Moderate right skewness; count data (e.g., particle counts).	Number of active sites estimated.	Data must be non-negative.	Weaker effect than log. Stabilizes variance for Poisson-like data.
Box-Cox	( y' = \frac{y^\lambda - 1}{\lambda} ) (\lambda \neq 0)	Optimal transformation when no prior theory dictates choice.	Any continuous, positive metric.	Data must be strictly positive. Software finds optimal λ.	General power transformation. λ=0 implies log transform.
Reciprocal	( y' = 1/y )	Severe right skewness.	Time-to-deactivation metrics.	Data must be non-zero.	Very strong effect. Reverses order. Use with caution.
Yeo-Johnson	( y' = \begin{cases} \frac{(y+1)^\lambda -1}{\lambda} & y \geq 0, \lambda \neq 0 \ \ln(y+1) & y \geq 0, \lambda = 0 \ \frac{-[(-y+1)^{2-\lambda} -1]}{2-\lambda} & y < 0, \lambda \neq 2 \ -\ln(-y+1) & y < 0, \lambda = 2 \end{cases} )	Data containing zero or negative values.	Metrics with baseline-subtracted negative values (e.g., background corrected signal).	Handles all real numbers.	Flexible extension of Box-Cox.

Protocol for Box-Cox Transformation (Using R):

Non-Parametric Alternative Protocols

When transformation fails to normalize data or is inappropriate, use a non-parametric method for outlier identification.

Median Absolute Deviation (MAD) Method Protocol

Objective: Identify outliers in non-normal catalyst data without assuming a distribution. Rationale: Uses the robust median and MAD instead of the mean and standard deviation.

Procedure:

• Calculate the median of the dataset: ( \tilde{x} = \text{median}(x_i) ).
• Calculate the MAD: ( \text{MAD} = \text{median}(|x_i - \tilde{x}|) ).
• Calculate the modified Z-score for each data point: ( Mi = \frac{0.6745 \cdot (xi - \tilde{x})}{\text{MAD}} ). The constant 0.6745 makes MAD a consistent estimator for the standard deviation of normal data.
• Define Outlier Threshold: A common threshold is ( |M_i| > 3.5 ). Points exceeding this are flagged as potential outliers.

Table 2: Comparison of Outlier Detection Methods for Non-Normal Data

Method	Robust to Non-Normality?	Sensitive to Multiple Outliers?	Data Requirements	Implementation Complexity	Suggested Use Case in Catalyst Screening
Grubbs' Test	No (requires normality)	Low (tests one outlier at a time)	Univariate, normal	Low	Primary method if normality is confirmed.
MAD Method	Yes	Moderate	Univariate, any scale	Very Low	First non-parametric choice for skewed catalyst yield data.
IQR (Tukey's Fences)	Yes	Moderate	Univariate, any scale	Very Low	Useful for identifying extreme yields in initial screening batches.
Generalized ESD Test	Somewhat (assumes approximate normality)	High (detects up to k outliers)	Univariate, near-normal	Medium	If transformation yields near-normal data but Grubbs' fails for >1 outlier.
DBSCAN Clustering	Yes	High	Multivariate, any scale	High	Identifying anomalous catalysts in multi-parameter space (yield, selectivity, cost).

Protocol for MAD-Based Outlier Detection (Using Python):

Workflow for Integrated Analysis

Title: Integrated Outlier Analysis Workflow

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Materials and Tools for Catalyst Performance & Outlier Analysis

Item / Reagent	Function in Catalyst Outlier Research	Example / Specification
Standard Reference Catalyst	Provides a benchmark for performance normalization and identifies systemic measurement errors.	NIST-standardized Pt on carbon (e.g., for hydrogenation reactions).
Internal Standard (for Analytics)	Distinguishes catalyst performance variation from instrumental drift or sample prep error in GC/HPLC.	Deuterated analog of product for mass spectrometry quantification.
High-Purity Solvents & Gases	Minimizes variability in reaction medium and reactant supply, a common source of non-systematic error.	Anhydrous solvents (H₂O < 50 ppm), Research-grade H₂/CO (99.999%).
Statistical Software Suite	Performs normality tests, data transformations, and advanced outlier detection algorithms.	R (with outliers, car packages), Python (SciPy, statsmodels).
Automated Reaction Screening Platform	Generates high-fidelity, reproducible kinetic data under controlled conditions, reducing noise.	Unchained Labs CPact or similar parallel pressure reactors.
Data Integrity & ELN System	Tracks metadata and pre-processing steps (e.g., transformation applied) for audit trail.	LabArchive, Signals Notebook.

Distinguishing Between Experimental Error and Genuine High-Performance Catalysts

Application Notes

Within the rigorous evaluation of new catalysts, particularly in pharmaceutical development, distinguishing statistical outliers due to experimental error from genuine high-performance candidates is a critical challenge. This process directly impacts resource allocation and project direction. The application of Grubbs' test provides a statistical framework for this identification, but its correct use requires careful experimental design and data validation protocols. These notes outline the integrated approach necessary for robust outlier analysis in catalyst performance research.

1. Data Collection and Outlier Identification Protocol

• Step 1: Replicate Performance Measurement: Conduct a minimum of n=7 independent, randomized runs for the candidate catalyst under evaluation. Key performance metrics (e.g., Turnover Number - TON, Yield, Enantiomeric Excess - ee) must be recorded for each run.
• Step 2: Calculate Descriptive Statistics: Compute the mean (x̄) and standard deviation (s) of the dataset.
• Step 3: Apply Grubbs' Test: Identify the data point (G) that deviates most from the mean (potential outlier). Calculate the Grubbs' statistic: G = |suspect value - x̄| / s.
• Step 4: Compare to Critical Value: Compare the calculated G to the critical value Gcritical for n=7 and a chosen significance level (α=0.05). If G > Gcritical, the data point is identified as a statistical outlier.

Table 1: Example Dataset for Catalyst TON and Grubbs' Test Analysis

Experiment Replicate	Turnover Number (TON)	Notes on Experimental Conditions
Run 1	12,450	Control: Standard degassing protocol
Run 2	12,800	Control: Standard degassing protocol
Run 3	13,100	Control: Standard degassing protocol
Run 4	12,900	Control: Standard degassing protocol
Run 5	12,750	Control: Standard degassing protocol
Run 6	13,000	Control: Standard degassing protocol
Run 7	18,500	Potential Outlier
Mean (x̄)	13,500
Std Dev (s)	2,150
Grubbs' G	2.33	G = |18500-13500| / 2150
G_critical (n=7, α=0.05)	1.938
Outlier?	Yes	G (2.33) > G_critical (1.938)

2. Post-Identification Validation Workflow

A statistical outlier is not a definitive diagnosis. The following protocol must be executed to determine its origin.

• Protocol 2.1: Experimental Artifact Interrogation

  • Objective: Systematically check for and eliminate sources of error.
  • Method:
    • Re-examine Raw Data & Logs: Scrutinize notebook entries, instrument logs (e.g., GC-MS, HPLC, glovebox O₂/H₂O levels), and sample tracking for the outlier run.
    • Replicate the Outlier Condition: Precisely re-run the experiment attempting to mimic all parameters of the outlier run, including reagent stock bottles, glassware, and analyst.
    • "Spike" Recovery Test: If the outlier showed high yield/activity, perform a control adding known impurities (e.g., common catalyst decomposition products, solvent stabilizers) to a standard reaction to see if performance is reproduced.
  • Expected Outcome: If high performance is not reproduced upon meticulous re-run, the outlier is likely due to undocumented error.

• Protocol 2.2: Hypothesis-Driven Validation of Genuine Performance

  • Objective: Confirm the outlier represents a real, replicable phenomenon.
  • Method:
    • Formulate Hypothesis: Based on the outlier run's conditions, propose a mechanism (e.g., in situ formation of a more active catalytic species, unexpected solvent or additive effect).
    • Design Critical Experiment: Test the hypothesis directly. Example: If the outlier used solvent from a newly opened bottle, test the reaction with (a) fresh solvent, (b) "aged" solvent, and (c) solvent with added suspected stabilizer/impurity.
    • Characterize Catalyst State: Post-reaction, analyze the catalyst from standard and outlier-condition runs using techniques like XPS, NMR, or MS to identify structural differences.
  • Expected Outcome: If the high performance is reproducibly linked to a specific, controllable condition and corroborated by characterization data, a genuine high-performance catalyst system may have been discovered.

Grubbs' Test & Outlier Validation Workflow

The Scientist's Toolkit: Key Research Reagent Solutions

Table 2: Essential Materials for Catalyst Outlier Investigation

Item / Reagent Solution	Function & Relevance to Outlier Analysis
Deuterated Solvents (e.g., Toluene-d8, THF-d8)	For detailed NMR reaction monitoring to identify in situ formation of novel catalytic species or impurities in outlier runs.
Inhibitor-Free / Stabilizer-Analyzed Solvents	Critical controls to test hypotheses related to solvent purity effects (common source of performance outliers).
High-Purity Metal Precursors & Ligands	Baseline materials; use of different batches can help identify lot-specific contamination or beneficial impurities.
Internal Standard Kits (for GC/HPLC)	Ensures quantitative analytical accuracy across all replicates, ruling out instrument calibration drift as an error source.
Catalyst Poison Traps (e.g., Mercury, CS₂, P(OMe)₃)	Used to test if outlier activity is heterogeneous (poisoned) or homogeneous (unaffected), clarifying mechanism.
Oxygen & Moisture Scavengers (e.g., Q5, MnO)	Used to standardize solvent/atmosphere purity, eliminating variable degassing as an outlier cause.
Standardized Substrate with Known Performance	A control catalyst/substrate pair run intermittently to ensure overall experimental system integrity.

Potential Root Causes of Catalyst Performance Outliers

Integrating Outlier Analysis into High-Throughput Experimentation (HTE) Workflows

Application Notes

The systematic identification and interpretation of outliers in High-Throughput Experimentation (HTE) is critical for accelerating catalyst and reaction discovery. Within our thesis on Grubbs' test applications, outlier analysis transitions from a passive data-cleaning step to an active hypothesis-generation engine. Outliers can indicate experimental error, novel catalytic activity, or a breakthrough in structure-activity relationships. Integrating Grubbs' test—a statistically rigorous method for identifying a single outlier in a univariate dataset assuming an approximately normal distribution—provides a formal criterion for investigation.

The following protocol and notes detail the integration of this outlier analysis directly into a catalytic HTE workflow for drug intermediate synthesis, ensuring that anomalous results are systematically flagged, validated, and leveraged.

Experimental Protocol: Integrated Outlier Analysis in Catalytic HTE

Objective: To execute a high-throughput screening of a 96-member palladium precatalyst library for a C-N cross-coupling reaction, integrate Grubbs' test for outlier identification at the primary assay stage, and validate outlier performance in secondary assays.

Part A: Primary High-Throughput Screening with Integrated Statistical Flagging

• Reaction Setup:

  • Utilize an automated liquid handling system to prepare a 96-well reaction plate. Each well contains substrate (0.05 mmol), base (0.075 mmol), and solvent (500 µL of anhydrous toluene) under an inert atmosphere.
  • Dispense a unique Pd precatalyst (1 mol% Pd) from the library into each well using a pre-prepared stock solution array.
  • Seal the plate and transfer it to a pre-heated orbital shaker/heater block. Perform the reaction at 90°C for 2 hours with agitation.

• Analysis & Primary Data Collection:

  • After cooling, quench reactions with a standardized acidic solution.
  • Employ Ultra-High-Performance Liquid Chromatography (UHPLC) with a shared autosampler for rapid analysis. Use a UV-based method to quantify yield (%) of the desired product against an internal standard.
  • Compile yields for all 96 reactions into a single dataset (List Y).

• Integrated Outlier Analysis via Grubbs' Test:

  • Calculate: Compute the sample mean (ȳ) and standard deviation (s) of the complete yield dataset (List Y).
  • Identify Candidate: Find the data point (Gcandidate) that maximizes the absolute deviation from the mean: G = |yi - ȳ| / s.
  • Statistical Test: Compare the calculated G value to the critical Grubbs' value (G_critical) for N=96 at α=0.05 (two-sided). Critical values are obtained from standard statistical tables or software (e.g., G ≈ 3.21 for N=96).
  • Flagging: If Gcandidate > Gcritical, flag the corresponding well as a statistical outlier. This data point is either an error or a genuinely high/low-performing catalyst.

Table 1: Representative Primary HTE Results & Grubbs' Test Analysis

Statistical Metric	Value (Hypothetical Data)
Number of Reactions (N)	96
Mean Yield (ȳ)	62.4%
Standard Deviation (s)	18.7%
Maximum Observed Yield	98.2%
G_candidate (for max yield)	(98.2 - 62.4) / 18.7 = 1.91
G_critical (α=0.05, N=96)	~3.21
Outlier Status (Max Yield)	Not an outlier by Grubbs' Test
Minimum Observed Yield	5.1%
G_candidate (for min yield)	(62.4 - 5.1) / 18.7 = 3.06
Outlier Status (Min Yield)	Not an outlier

Note: In this hypothetical dataset, no single extreme outlier is detected by Grubbs' test. The protocol proceeds to iterative application on remaining data if an outlier is found and removed.

Part B: Validation Protocol for Flagged Outliers

• Hit Confirmation: For any catalyst flagged as a high-performing outlier, repeat the reaction in triplicate (0.2 mmol scale) under identical conditions using manual setup to confirm activity and rule out experimental artifact.
• Scope & Robustness: Subject the confirmed outlier catalyst to a secondary matrix of conditions (e.g., varied temperature, base, solvent, substrate electronic properties) in a 24-well microreactor block to assess robustness and generalizability.
• Deactivation Analysis: For low-performing outliers (potential inhibitors), design a poisoning experiment. Add the outlier catalyst mixture to a reaction with a known high-performing catalyst and monitor rate suppression.

Visualization: Integrated HTE Outlier Analysis Workflow

The Scientist's Toolkit: Key Reagent Solutions & Materials

Item	Function in Protocol
Pd Precatalyst Library	A spatially encoded array of 96 pre-weighed, air-stable Pd complexes in vials or wells, enabling rapid screening of ligand and structure effects.
Automated Liquid Handler	Ensures precise, reproducible dispensing of substrates, bases, and solvents across high-density microtiter plates, minimizing systematic error.
Sealed Microreactor Plates	Chemically inert, heat-tolerant 96-well plates with sealing mats to enable parallel reactions under controlled atmosphere (N2/Ar).
Internal Standard Solution	A consistent, non-interfering compound added to each quenched reaction mixture prior to UHPLC analysis to correct for injection volume variability.
Grubbs' Test Critical Value Table	A pre-calculated or software-embedded reference for G_critical values at various N and α levels, essential for immediate statistical decision-making.
Modular Secondary Reactor Block	A 24-well parallel reactor system for conducting gram-scale validation and robustness studies on outlier catalysts under varied conditions.

Table 2: Grubbs' Test Critical Values (Two-Sided, α=0.05)

Sample Size (N)	Critical Value (G)	Sample Size (N)	Critical Value (G)
6	1.887	50	3.128
10	2.176	96	3.208
20	2.623	144	3.255
30	2.909	200	3.289

Grubbs' Test vs. Other Methods: Choosing the Right Tool for Catalyst Analysis

1. Introduction Within catalyst performance research, identifying anomalous data points is critical for validating kinetic models and ensuring reproducibility. For small datasets typical of preliminary catalyst screening, two prominent statistical methods are Grubbs' test and Dixon's Q-test. This analysis compares these methods in the context of identifying outliers in catalyst turnover frequency (TOF) or yield measurements, supporting a broader thesis on robust data analysis in heterogeneous catalysis.

2. Theoretical Overview & Comparative Metrics

Table 1: Core Characteristics of Grubbs' and Dixon's Q-Test

Feature	Grubbs' Test (Maximum Normed Residual Test)	Dixon's Q-Test
Primary Use	Detecting one or two outliers in a univariate dataset.	Detecting a single outlier in a small, univariate dataset.
Data Assumption	Data follows an approximately normal distribution.	No strong assumption of normality; distribution-free.
Dataset Size (n)	Recommended for n ≥ 3. More reliable for n > 6.	Designed for very small samples (typically 3 ≤ n ≤ 10).
Hypotheses	H₀: No outliers in the data set. Hₐ: There is at least one outlier.	H₀: No outliers in the data set. Hₐ: The suspected point is an outlier.
Test Statistic	G = |suspect value - sample mean| / sample standard deviation.	Q = |gap| / |range|.
Critical Values	Based on t-distribution; depends on n and significance level (α).	Tabulated values based on n and α.
Key Strength	Uses all data in calculation (mean, SD). Can test for two outliers.	Simple, quick calculation. Less sensitive to normality assumptions for very small n.
Key Limitation	Sensitive to deviations from normality, especially for small n. Masking effect with multiple outliers.	Only tests one extreme value per run. Officially defined only for n ≤ 10.

Table 2: Example Application to Catalyst TOF Data (n=7, α=0.05)

Test	TOF Data (s⁻¹)	Suspect Value	Test Statistic	Critical Value	Conclusion
Grubbs'	12.1, 12.5, 12.0, 13.1, 11.9, 15.2, 12.3	15.2	G = 2.32	1.938	Reject H₀. 15.2 is an outlier.
Dixon's Q (Q₇₇)	11.9, 12.0, 12.1, 12.3, 12.5, 13.1, 15.2	15.2	Q = (15.2-13.1)/(15.2-11.9)=0.636	0.507	Reject H₀. 15.2 is an outlier.

3. Detailed Experimental Protocols

Protocol 1: Procedure for Applying Grubbs' Test to Catalyst Yield Data

• Data Preparation: Assemble the dataset of n replicate yield measurements (e.g., n=6) from a single catalytic reaction condition.
• Calculate Descriptive Statistics: Compute the sample mean (x̄) and sample standard deviation (s) for the full dataset.
• Identify Suspect Value: Identify the observation (xᵢ) that is furthest from the mean.
• Compute Grubbs' Statistic: Calculate G = \|xᵢ - x̄\| / s.
• Determine Critical Value: Obtain the critical value Gcritical for n observations at the chosen α (e.g., α=0.05) from standard statistical tables. Alternatively, calculate as: Gcritical = (n-1)/√n * √( t²(α/(2n), n-2) / (n-2 + t²(α/(2n), n-2) ) ), where t is the t-distribution value.
• Make Decision: If G > G_critical, reject the null hypothesis and classify xᵢ as an outlier.
• Iteration for Two Outliers: If one outlier is detected, remove it and repeat the procedure on the remaining n-1 points to test for a second outlier.

Protocol 2: Procedure for Applying Dixon's Q-Test to Catalyst TOF Data

• Data Ordering: Sort the dataset of n replicate TOF measurements (e.g., n=5) in ascending order: x₁, x₂, ..., xₙ.
• Identify Suspect Value: Designate the potentially anomalous value (e.g., the lowest x₁ or the highest xₙ).
• Select Appropriate Q Formula: Based on sample size n, select the correct ratio formula (e.g., for n=7, testing the maximum, use Q = (xₙ - xₙ₋₁) / (xₙ - x₁)).
• Compute Q Statistic: Calculate the absolute value of the ratio.
• Determine Critical Value: Obtain the critical value Q_critical for n and α (e.g., α=0.05) from Dixon's Q-table.
• Make Decision: If Q > Q_critical, reject the null hypothesis and classify the suspect value as an outlier.
• Single Test Limitation: Do not apply the test iteratively without reassessing data; if an outlier is removed, the test is not formally valid on the new, smaller set per original tables.

4. Visualization of Analytical Decision Pathways

5. The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Materials for Catalyst Performance Outlier Analysis

Item	Function in Analysis
Statistical Software (e.g., R, Python with SciPy)	Provides built-in functions for Grubbs' and Dixon's tests, critical value lookup, and automation of protocols.
Critical Value Tables	Reference tables for Grubbs' and Dixon's test statistics at various α (0.05, 0.01) are essential for manual calculation verification.
Normality Test (e.g., Shapiro-Wilk)	A prerequisite analytical tool to assess the applicability of Grubbs' test for small catalyst datasets.
Standard Reference Catalyst	A well-characterized catalyst material used in parallel control experiments to validate analytical instrument performance and baseline data quality.
Internal Analytical Standard	A known compound added to reaction product mixtures (e.g., for GC/MS analysis) to distinguish measurement error from catalytic performance outliers.
Data Logbook (Electronic/Lab Notebook)	Critical for documenting all measurements, test results, and decisions regarding outlier exclusion to ensure research integrity and reproducibility.

This application note is developed within the framework of a doctoral thesis investigating the detection of outlier data points in heterogeneous catalyst performance screening. Accurate identification of true performance outliers—whether exceptional or defective—is critical for reliable structure-activity relationship modeling and process optimization in catalyst and drug development research. This document compares the statistical robustness, applicability, and implementation protocols of three outlier detection methods: Grubbs' Test (parametric), the Modified Z-Score method (non-parametric, median-based), and the Interquartile Range (IQR) method (non-parametric, quartile-based). The focus is on their performance with small-to-moderate sample sizes typical in high-throughput catalyst testing.

Core Principles and Quantitative Comparison

Methodological Foundations

• Grubbs' Test: A parametric hypothesis test that assumes the underlying data (excluding the potential outlier) is normally distributed. It tests the hypothesis that there is no outlier in the dataset. The test statistic (G) is calculated as the maximum absolute deviation from the sample mean divided by the sample standard deviation. Critical values depend on sample size and chosen significance level (α).
• Modified Z-Score: A non-parametric method using the median and Median Absolute Deviation (MAD). The Modified Z-Score (Mᵢ) for each data point is calculated as Mᵢ = 0.6745 * (xᵢ - x̃) / MAD, where x̃ is the median and MAD = median(|xᵢ - x̃|). The constant 0.6745 scales the MAD to be consistent with the standard deviation for a normal distribution. A threshold (typically 3.5) identifies outliers.
• IQR Method: A robust, non-parametric rule based on data quartiles. The interquartile range (IQR) is Q3 - Q1. Data points falling below Q1 - 1.5 * IQR or above Q3 + 1.5 * IQR are classified as outliers.

Table 1: Comparative Analysis of Outlier Detection Methods

Feature	Grubbs' Test	Modified Z-Score	IQR Method
Statistical Basis	Parametric (assumes normality)	Non-parametric (median/MAD)	Non-parametric (quartiles)
Primary Robustness	Sensitive to non-normality; powerful for normal data	Highly robust to non-normality & small outliers	Extremely robust to non-normality & extreme outliers
Sample Size (n)	Recommended for 3 ≤ n ≤ 100. Less reliable for very small n.	Applicable for n ≥ 5. Stable even for small n.	Applicable for n ≥ 4. Stable for small n.
Outliers Detected	One-at-a-time iterative process. Identifies the most extreme point per iteration.	All points exceeding threshold are flagged simultaneously.	All points outside fences are flagged simultaneously.
Sensitivity	High sensitivity to single extreme outliers in normal data.	Moderate sensitivity; less influenced by a single extreme value.	Low sensitivity to mild outliers; focuses on extreme deviations.
Assumption Check	Mandatory: Normality test (e.g., Shapiro-Wilk) on the remaining data after outlier removal.	Not required. Inherently resistant to distribution shape.	Not required. Inherently resistant to distribution shape.
Typical Threshold	Critical G-value (α = 0.05) from statistical table.	Absolute Modified Z-Score > 3.5 (common heuristic).	Below Q1 - 1.5IQR or Above Q3 + 1.5IQR.
Thesis Application	Best for validating a single extreme catalyst performance metric when normality is plausible.	Preferred for initial screening of multi-parameter catalyst datasets (e.g., yield, selectivity, TON) of unknown distribution.	Ideal for identifying severe "failure" or "breakthrough" catalyst samples in robust screening workflows.

Experimental Protocols for Catalyst Performance Data

Protocol A: Iterative Outlier Detection Using Grubbs' Test

Objective: To systematically identify and remove up to two extreme outlier values from a univariate dataset of catalyst Turnover Frequency (TOF) measurements, assuming an underlying normal distribution for the core data.

• Data Preparation: Compile n TOF measurements into a sorted list.
• Normality Check: Perform the Shapiro-Wilk test on the full dataset. If p-value < 0.10, consider non-parametric methods. Proceed with caution.
• Calculate Statistics: Compute the sample mean (x̄) and standard deviation (s) for the current dataset.
• Compute G-statistic: Identify the point furthest from the mean (x*). Calculate G = |x* - x̄| / s.
• Compare to Critical Value: Using a Grubbs' table for two-sided test at α=0.05 and current sample size n, obtain the critical value G_critical.
• Decision: If G > G_critical, classify x* as an outlier. Remove it from the dataset.
• Iterate: With the reduced dataset (n' = n-1), repeat steps 3-6 to check for a second outlier.
• Final Validation: Perform a final Shapiro-Wilk test on the remaining data post-removal. Confirm p-value ≥ 0.10 to validate normality assumption was not violated by the outliers.

Protocol B: Robust Screening Using Modified Z-Score

Objective: To flag multiple potential outlier catalyst samples in a single pass based on a robust deviation metric, suitable for non-normal data distributions.

• Data Preparation: Compile the dataset (e.g., catalyst yield at 24h).
• Calculate Median & MAD:
  • Calculate the sample median (x̃).
  • Compute absolute deviations: dᵢ = |xᵢ - x̃|.
  • Calculate the MAD as the median of all dᵢ.
• Compute Modified Z-Scores: For each data point xᵢ, calculate Mᵢ = 0.6745 * (xᵢ - x̃) / MAD. If MAD = 0, use IQR method instead.
• Apply Threshold: Flag any data point where |Mᵢ| > 3.5 as a potential outlier.
• Contextual Review: Manually inspect flagged data points against experimental logs (e.g., potential synthesis error, reactor malfunction) before exclusion.

Protocol C: Identifying Extreme Deviations via IQR Method

Objective: To definitively identify extreme outlier values in a dataset, effectively separating the central 50% of "typical" catalyst performances from the most extreme cases.

• Data Preparation: Compile and sort the dataset (e.g., enantiomeric excess values).
• Calculate Quartiles: Determine the first quartile (Q1, 25th percentile) and third quartile (Q3, 75th percentile).
• Compute IQR and Fences: IQR = Q3 - Q1. Calculate lower fence = Q1 - 1.5 * IQR and upper fence = Q3 + 1.5 * IQR.
• Identify Outliers: Any data point with a value less than the lower fence or greater than the upper fence is classified as an outlier.
• Visualization: Generate a box plot of the data to visually confirm the outliers relative to the IQR "whiskers."

Visual Workflows and Decision Pathways

Title: Decision Pathway for Outlier Detection Method Selection

Title: Statistical Inputs for Each Outlier Detection Method

The Scientist's Toolkit

Table 2: Essential Research Reagents & Materials for Catalyst Outlier Analysis

Item	Function in Catalyst Outlier Research	Example/Specification
Statistical Software	Performs complex calculations, normality tests, and generates critical values for hypothesis tests.	R (with outliers, EnvStats packages), Python (SciPy, Statsmodels), GraphPad Prism, JMP.
High-Throughput Screening (HTS) Data	The primary univariate or multivariate dataset requiring analysis for anomalous performance.	Turnover Number (TON), Turnover Frequency (TOF), Yield (%), Selectivity (%), Enantiomeric Excess (ee%).
Experimental Log/LIMS	Provides essential context to distinguish between statistical outliers and legitimate experimental errors.	Electronic lab notebook (ELN) or Laboratory Information Management System tracking synthesis parameters, reactor conditions, analyst ID.
Normality Test Protocol	Validates the core assumption for parametric methods like Grubbs' Test.	Shapiro-Wilk test (preferred for n < 50), Anderson-Darling test, visual Q-Q plot inspection.
Visualization Tools	Enables intuitive data exploration and presentation of outlier detection results.	Software for generating box plots, scatter plots, and histograms (e.g., Matplotlib, ggplot2, OriginLab).
Critical Value Tables	Reference for hypothesis test decision-making when software is not automating the process.	Statistical tables for Grubbs' Test critical values at α = 0.05, 0.01 for various sample sizes (n).

Within the broader thesis investigating Grubbs' test for identifying performance outliers in catalytic systems, a fundamental challenge is the statistical treatment of data from different catalyst classes. Homogeneous catalysts, operating in a single phase, often yield data with distinct variance properties compared to heterogeneous catalysts, where phase boundaries introduce additional variability. Selecting an appropriate outlier test is contingent upon understanding these inherent data structures to avoid false positives or missed anomalies.

Statistical Foundations: Grubbs' Test and Its Alternatives

Grubbs' test (maximum normed residual test) is designed to detect a single outlier in a univariate data set assumed to be normally distributed. Its application presupposes that the data, aside from the potential outlier, is drawn from a normally distributed population. This assumption is frequently challenged in catalytic datasets.

Key Tests Comparison:

Test Name	Primary Use Case	Underlying Assumption	Sensitivity to Data Type	Recommended for Catalyst Type
Grubbs' Test	Detecting a single outlier	Data is normally distributed	High for normal, homogeneous data	Homogeneous (low variance, normal residuals)
Dixon's Q Test	Small sample sizes (3-30)	None, but rank-based	Robust for small N	Both (especially preliminary screening)
Tietjen-Moore Test	Detecting k multiple outliers	Data is normally distributed	Decreases as k increases	Homogeneous with suspected multiple outliers
Generalized ESD Test	Detecting 1 to k outliers	Data is approximately normal	Robust to minor deviations	Heterogeneous (relaxed normality)
Chauvenet's Criterion	Outlier rejection via probability	Normal distribution	Classical, often overly stringent	Homogeneous (theoretical yield analysis)

Table 1: Statistical tests for outlier detection in catalytic data.

Homogeneous Catalyst Data Scenario

Data from homogeneous catalysis (e.g., Grubbs' metathesis catalysts in solution) is often characterized by:

• Lower variance in turnover frequency (TOF) or yield measurements.
• Residuals more likely to approximate a normal distribution due to single-phase kinetics.
• Outliers often stem from ligand decomposition or trace impurities.

Protocol 3.1: Outlier Screening for Homogeneous Catalytic Yield Data

• Data Collection: Collect a minimum of n=8 replicate yield measurements for a single catalytic reaction under identical conditions.
• Normality Assessment: Perform the Shapiro-Wilk test (for n<50) or generate a Q-Q plot. Proceed only if p > 0.10.
• Initial Test Application:
  • If normality is confirmed and a single outlier is suspected, apply Grubbs' Test.
  • Calculate the G statistic: G = | suspect value - sample mean | / sample standard deviation.
  • Compare G to the critical value for n and α=0.05.
• Confirmatory Analysis: If an outlier is identified, investigate reaction aliquots via ICP-MS for metal leaching or NMR for ligand integrity.

Heterogeneous Catalyst Data Scenario

Data from heterogeneous catalysis (e.g., supported metal catalysts) is characterized by:

• Higher inherent variance due to surface site heterogeneity, mass transfer effects, and sampling.
• Distributions may be non-normal or multimodal.
• Outliers arise from bed channeling, particle size outliers, or local hotspot formation.

Protocol 4.1: Outlier Screening for Heterogeneous Catalytic TOF Data

• Data Collection: Collect a minimum of n=12 replicate TOF measurements from multiple, independent catalyst batches.
• Distribution Inspection: Visualize data using a histogram and kernel density plot. Normality tests are often failed; note the shape.
• Robust Test Application: Apply the Generalized Extreme Studentized Deviate (ESD) Test.
  • Pre-specify an upper bound for the number of potential outliers, k (e.g., k=3).
  • Iteratively remove the point that maximizes |x_i - mean| / std, recalculating statistics after each removal.
  • Compare test statistics against critical values for each step (λ_i).
• Root Cause Analysis: For identified outliers, characterize the corresponding catalyst batch using SEM-EDS for metal aggregation and BET for surface area deviation.

Decision Framework and Application Protocol

Decision Flow for Catalyst Outlier Test Selection

The Scientist's Toolkit: Research Reagent Solutions

Item	Function in Catalyst Outlier Research
Internal Standard (e.g., 1,3,5-Trimethoxybenzene)	Added to reaction aliquots prior to GC/HPLC analysis to differentiate analytical error from catalytic outlier.
Deuterated Solvent (e.g., Benzene-d6)	For in-situ NMR monitoring of homogeneous catalyst integrity, identifying decomposition as an outlier source.
Metal Scavenger Resins (e.g., QuadraPure TU)	Post-reaction, confirms no active metal leaching in heterogeneous systems, validating an outlier as non-homogeneous.
Spin Coating Materials (e.g., PMMA in Anisole)	For preparing uniform thin films of catalyst for SEM, ensuring characterization is not the outlier source.
Isotopically Labeled Reagents (e.g., 13C-ethylene)	Traces specific mechanistic pathways; abnormal isotopic incorporation can flag an outlier at the mechanistic level.

Table 2: Essential research reagents for root-cause analysis of catalytic outliers.

Integrated Experimental Workflow Protocol

Integrated Workflow for Outlier Research

Protocol 7.1: Comprehensive Outlier Analysis Workflow

• Experimental Design: For the catalytic reaction of interest, prepare a minimum of 3 independent catalyst batches (heterogeneous) or 3 independent catalyst stock solutions (homogeneous). From each batch/solution, perform at least 4 replicate reactions (total n ≥ 12).
• Data Acquisition: Measure primary performance metrics (Yield, TOF, Selectivity) using calibrated analytical equipment (GC, HPLC). Record full metadata (batch ID, run order, analyst).
• Statistical Analysis Pipeline:
  • Calculate mean, standard deviation, and skewness for the full dataset and per-batch subsets.
  • Follow the decision framework (Diagram 1) to select and apply the optimal outlier test.
  • Report the test statistic, critical value, and confidence level.
• Characterization Suite: Subject the outlier-associated catalyst sample and a "normal" control to a coordinated analysis: XPS for surface oxidation state, TEM for nanoparticle dispersion, and DRIFTS for adsorbate binding.
• Synthesis Feedback: Correlate outlier statistical signature with physicochemical anomalies to refine catalyst synthesis protocol, updating the DoE in Step 1.

Within a broader thesis investigating the application of Grubbs' test for identifying outliers in heterogeneous catalyst performance data for pharmaceutical synthesis, graphical methods serve as critical validation tools. While Grubbs' test provides a statistical probability of an outlier, visual confirmation via box plots and scatter plots is essential to discern between true anomalous data points and values that may be legitimate extremes of a non-normal distribution or indicative of a systematic experimental factor. This protocol outlines the integrated use of these plots to validate outlier decisions prior to exclusion or further investigation in catalyst research.

Table 1: Comparison of Graphical Outlier Detection Methods

Method	Primary Function	Data Type Suited For	Outlier Definition (Visual)	Advantage in Catalyst Research
Box Plot	Displays distribution based on quartiles and median.	Univariate, single-response variables (e.g., Yield %, Turnover Frequency).	Points beyond "whiskers" (typically 1.5*IQR from quartiles).	Quick overview of batch or condition performance; identifies extreme values in a single metric.
Scatter Plot	Shows relationship between two continuous variables.	Bivariate, paired measurements (e.g., Catalyst Loading vs. Yield, Reaction Time vs. Purity).	Points isolated from the main cluster or trend.	Reveals contextual outliers, process relationships, and hidden covariates affecting performance.

Table 2: Hypothetical Catalyst Yield Dataset with Grubbs' Test and Graphical Flag

Catalyst ID	Yield (%)	Grubbs' G (Calc.)	G Critical (α=0.05, n=10)	Statistical Outlier (Grubbs')	Visual Outlier (Box Plot)	Visual Outlier (vs. Loading Scatter)
Cat-01	92.1	1.12	2.290	No	No	No
Cat-02	89.5	0.65	2.290	No	No	No
Cat-03	91.8	1.05	2.290	No	No	No
Cat-04	90.2	0.22	2.290	No	No	No
Cat-05	94.3	1.58	2.290	No	No	No
Cat-06	62.4	3.41	2.290	Yes	Yes	Yes (Contextual)
Cat-07	93.0	1.30	2.290	No	No	No
Cat-08	89.9	0.31	2.290	No	No	No
Cat-09	91.5	0.94	2.290	No	No	No
Cat-10	92.8	1.24	2.290	No	No	No

IQR for Yield = 2.45%; Lower Whisker = 86.98%; Upper Whisker = 96.43%. Cat-06 is below Lower Whisker.

Experimental Protocols

Protocol 3.1: Integrated Outlier Validation Workflow for Catalyst Performance Data

Objective: To systematically validate statistical outlier candidates (from Grubbs' test) using box plots and scatter plots before making data exclusion decisions.

Materials: See "Scientist's Toolkit" section.

Procedure:

• Data Preparation: Compile catalyst performance dataset. Ensure each entry includes a unique catalyst identifier, primary response variable (e.g., reaction yield), and relevant independent variables (e.g., catalyst loading, temperature, surface area).
• Statistical Outlier Test: Perform Grubbs' test (or another appropriate statistical test) on the primary response variable. Record all values flagged as potential outliers (p < 0.05).
• Generate Box Plot: a. Calculate the first quartile (Q1), third quartile (Q3), and interquartile range (IQR = Q3 - Q1). b. Plot a box from Q1 to Q3, with a line at the median. c. Draw "whiskers" from Q1 to (Q1 - 1.5IQR) and from Q3 to (Q3 + 1.5IQR). d. Plot all data points. Overlay points flagged by Grubbs' test with a distinct shape/color. e. Identify any points beyond the whiskers. Confirm if Grubbs'-flagged points align with visual outliers.
• Generate Scatter Plot(s): a. Plot the primary response variable (Y-axis) against a key independent variable (X-axis), e.g., catalyst metal loading. b. Fit a trend line (linear or loess) to the main data cluster. c. Overlay Grubbs'-flagged points distinctly. d. Assess if statistical outliers are isolated from the primary trend or form a separate cluster, suggesting a different catalyst behavior or experimental artifact.
• Decision Matrix Application: a. Agreement (Grubbs' + Both Plots): Strong evidence for a true outlier. Investigate experimental notes for that catalyst batch. b. Grubbs' + Box Plot Only: Potential extreme value. Review scatter plot context. If it follows the trend, may be a legitimate tail of the distribution. c. Grubbs' + Scatter Plot Only: Suggests outlier status is dependent on a covariate relationship, not the response variable alone. Investigate the experimental condition. d. No Graphical Support: Unlikely a true outlier. Do not exclude. Consider data transformation or non-parametric tests.

Protocol 3.2: Creating Publication-Ready Plots using Python (Matplotlib/Seaborn)

Objective: To generate standardized, high-quality graphical validation figures.

Diagrams and Workflows

Title: Outlier Validation Workflow for Catalyst Data

The Scientist's Toolkit

Table 3: Essential Research Reagent Solutions & Materials

Item	Function/Benefit	Example/Specification
High-Throughput Screening Reactor	Enables parallel synthesis under controlled conditions to generate the primary catalyst performance dataset.	Commercially available systems (e.g., from AMTECH, Unchained Labs) with 24- or 96-well plates for parallel reactions.
Gas Chromatograph-Mass Spectrometer (GC-MS)	Provides accurate quantification of reaction yield and purity, generating the critical univariate response data.	System with autosampler for high reproducibility and internal standard calibration capability.
Statistical Software (e.g., JMP, Prism, Python/R)	Performs Grubbs' test and generates high-quality, customizable box plots and scatter plots.	Python with SciPy (for Grubbs'), Matplotlib, and Seaborn libraries offer open-source flexibility.
Electronic Lab Notebook (ELN)	Documents all experimental parameters (catalyst prep, conditions) to investigate causes of visually-confirmed outliers.	Platforms like LabArchives or Signals Notebook allow linking raw data to metadata.
Reference Catalyst Material	A well-characterized catalyst batch used as an internal control across experiments to identify systematic drift, not outliers.	A standardized Pd/C or zeolite sample with certified activity.
Data Visualization Tool (e.g., Spotfire, Tableau)	For interactive exploration of multivariate scatter plots to find hidden relationships in complex catalyst data.	Enables dynamic filtering and plotting of multiple performance indicators.

1. Introduction Within the broader context of a thesis exploring robust statistical methodologies for catalyst evaluation, this case study investigates the application of multiple outlier detection tests to a challenging dataset from an asymmetric hydrogenation campaign. The primary objective is to demonstrate how Grubbs' test, often a starting point for outlier identification in catalyst performance research, can be supplemented with additional statistical measures to provide a more nuanced analysis, especially when dealing with non-normal data distributions and multiple potential outliers.

2. Dataset Overview The dataset comprises enantiomeric excess (%ee) results for 48 unique chiral phosphine-oxazoline (PHOX) ligand derivatives tested in the asymmetric hydrogenation of methyl 2-acetamidoacrylate. Each ligand was synthesized and tested once under standardized conditions (see Protocol 3.1). The expected performance range, based on prior literature, is 70-95%ee. Preliminary analysis indicated a cluster of high-performing catalysts and several potential underperformers.

Table 1: Summary of Catalytic Performance Dataset

Statistic	Value
Total Data Points (N)	48
Mean %ee	81.4
Median %ee	84.2
Standard Deviation (s)	12.7
Minimum Observed Value	32.1 %ee
Maximum Observed Value	94.8 %ee
Shapiro-Wilk p-value (Normality Test)	0.013

3. Applied Statistical Tests & Protocols

Protocol 3.1: Primary Catalytic Screening

• Objective: Generate the primary asymmetric catalysis dataset.
• Materials: Substrate (methyl 2-acetamidoacrylate, 0.1 mmol), [Rh(cod)₂]BF₄ precatalyst (1 mol%), chiral PHOX ligand (1.1 mol%), anhydrous dichloromethane (DCM, 2 mL), H₂ atmosphere (1 atm).
• Procedure: In a glovebox, charge a vial with substrate, precatalyst, and ligand. Add dry DCM. Seal vial, remove from glovebox, and evacuate/backfill with H₂ (3x). Stir reaction at 25°C for 16h. Reduce pressure, concentrate, and analyze by chiral HPLC to determine %ee.
• Note: All reactions were performed in a randomized order to mitigate systematic error.

Protocol 3.2: Statistical Outlier Analysis Workflow

• Step 1 – Normality Assessment: Perform Shapiro-Wilk test on the full dataset (N=48). A p-value <0.05 suggests deviation from normality, discouraging the sole use of parametric outlier tests.
• Step 2 – Initial Grubbs' Test: Apply Grubbs' test (for single outlier) iteratively. Calculate G = |suspect value - mean| / s. Compare to critical G value at α=0.05.
• Step 3 – Confirmatory Non-Parametric Test: Apply the Interquartile Range (IQR) method. Calculate Q1 (25th percentile), Q3 (75th percentile). Any datum < Q1 - 1.5IQR or > Q3 + 1.5IQR is flagged.
• Step 4 – Visual Inspection: Generate a box plot for final assessment of flagged outliers within the data distribution.

Table 2: Results of Sequential Outlier Tests on Catalysis Dataset

Test Method	Flagged Outlier(s) (%ee)	Test Statistic	Critical Value (α=0.05)	Conclusion
Shapiro-Wilk	N/A	W = 0.942	p > 0.05	Data non-normal (p=0.013)
Grubbs' (Max)	32.1	G = 3.88	G_crit = 3.65	32.1%ee is an outlier
Grubbs' (Min)	94.8	G = 1.06	G_crit = 3.65	94.8%ee is not an outlier
IQR (1.5x)	32.1, 35.5, 36.0	Q1=76.4, Q3=88.9, IQR=12.5	Lower Fence = 57.65	Three low %ee outliers identified

4. Visualization of Analysis Workflow

Title: Statistical Outlier Analysis Decision Workflow

5. The Scientist's Toolkit: Key Research Reagent Solutions

Table 3: Essential Materials for Asymmetric Catalysis Screening

Item	Function / Relevance
Chiral Phosphine-Oxazoline (PHOX) Ligand Library	Core modular scaffold enabling rapid structural variation to probe steric and electronic effects on enantioselectivity.
[Rh(cod)₂]BF₄ / [Rh(nbd)₂]BF₄	Air-stable, well-defined rhodium precatalysts that readily generate active species upon ligand coordination.
Anhydrous, Deoxygenated Solvents (DCM, MeOH, Toluene)	Critical for moisture- and oxygen-sensitive organometallic catalysts to ensure reproducibility.
Parallel Pressure Reactor System (e.g., from Parr, Biotage)	Enables safe, parallelized screening under controlled H₂ pressure (1-10 atm) with consistent stirring.
Chiral Stationary Phase HPLC Columns (e.g., Chiralcel OD-H, AD-H)	Standard analytical tool for accurate and precise determination of enantiomeric excess (%ee).
Statistical Software (e.g., R, Python with SciPy, GraphPad Prism)	Essential for performing advanced statistical tests (Grubbs', Shapiro-Wilk) and generating publication-quality plots.

6. Conclusion & Interpretation This case study demonstrates that relying solely on Grubbs' test, a common thesis methodology, would have identified only the most extreme low-performance outlier (32.1%ee) in this asymmetric catalysis dataset. The non-normality of the data necessitated a multi-test approach. The IQR method, robust against non-normal distributions, identified two additional marginal underperformers (35.5 and 36.0%ee). The high-performing catalyst (94.8%ee) was not flagged by any test, correctly identifying it as a genuine high performer rather than a statistical anomaly. For catalyst performance research, this protocol advocates for: 1) testing for normality, 2) using Grubbs' test as an initial screen for extreme values, and 3) confirming results with a non-parametric method like the IQR rule to ensure a defensible outlier identification strategy, ultimately leading to more reliable structure-activity relationships.

Conclusion

Grubbs' test provides a statistically rigorous, accessible methodology for identifying outliers in catalyst performance data, forming a critical checkpoint for data integrity in pharmaceutical and chemical research. By understanding its foundational principles (Intent 1), researchers can correctly apply the step-by-step method (Intent 2) to their specific datasets. Awareness of its assumptions and limitations enables effective troubleshooting (Intent 3), while comparative analysis with other methods (Intent 4) ensures the most appropriate tool is used for the data structure at hand. Implementing a systematic outlier detection protocol, with Grubbs' test as a central component, enhances the reliability of catalyst screening, reduces the risk of basing development decisions on anomalous results, and ultimately accelerates the path to robust and scalable synthetic processes. Future integration of these statistical methods with machine learning-driven catalyst discovery platforms will further refine our ability to distinguish between statistical noise and breakthrough performance.