Diagnosing Singular Fits in Linear Mixed Models: Practical Examples, Code, and Alternatives

 

Introduction

This post is the practical follow‑up to When Linear Mixed Models Don’t Converge, where I discussed the theoretical reasons why convergence issues often arise in linear mixed models. Here, we move from theory to practice: simulated examples, diagnostic tools, interpretation of singular fits, and practical modeling alternatives.

If you haven’t read the first part yet, you may want to start there for the conceptual foundations.

Figure. Conceptual map showing how model complexity and available information interact

Legend of Figure 1. Green = identifiable models; yellow = risk of singularity; red = non‑identifiable models.

A Concrete Example Using R: When a Random-Slope Model Becomes Singular

Let’s simulate a simple dataset with only 8 groups and a very small random‑slope variance. This is a classic scenario where the model converges numerically but is statistically non‑identifiable.

```r

set.seed(123) n_groups <- 8 n_per_group <- 20 group <- factor(rep(1:n_groups, each = n_per_group)) x <- rnorm(n_groups * n_per_group) beta0 <- 2 beta1 <- 1 u0 <- rnorm(n_groups, 0, 1) u1 <- rnorm(n_groups, 0, 0.05) y <- beta0 + u0[group] + (beta1 + u1[group]) * x + rnorm(n_groups * n_per_group, 0, 1) df <- data.frame(y, x, group) library(lme4) m <- lmer(y ~ x + (x | group), data = df) summary(m) isSingular(m)

```

Typical output:

• Random‑slope variance ≈ 0.05

• Correlation between intercept and slope = 1.00

• Message: boundary (singular) fit

• isSingular(m) returns TRUE

This is not a numerical failure. It is a mathematical signal that the model is too complex for the available information.

How to Interpret a Singular Fit

A singular fit occurs when the estimated random‑effects variance–covariance matrix is not of full rank. This typically means:

• A random‑effect variance is estimated as zero

• A correlation hits ±1

• The Hessian is not invertible

• The model lies on the boundary of the parameter space

In practice, this means the data do not support the random‑effects structure you specified.

Diagnostic Checklist

Red flags

• Variance of a random effect is exactly zero

• Correlation between random effects is ±1

• isSingular(model) returns TRUE

• Large gradients or warnings about the Hessian

• Estimates change dramatically when switching optimizers

Useful tools

• lme4::isSingular()

• lmerControl(optimizer = ...) for diagnosis only

• Likelihood‑ratio tests between nested models

• Checking the number of levels per random effect

Practical Alternatives When the Model Is Too Complex

1. Simplify the random‑effects structure

Often, the most appropriate solution.

Examples:

• Remove the random slope

• Remove correlations using (1 | group) + (0 + x | group)

• Fit a random‑intercept model first and evaluate necessity of slopes

2. Bayesian models with weakly informative priors

Bayesian estimation stabilizes variance estimates near zero.

Example:

```r

brm(y ~ x + (x | group), data = df, prior = prior(exponential(2), class = sd))

```

3. Penalized mixed models (e.g., glmmTMB)

Penalization shrinks unstable variance components.

4. Marginal models (GEE)

When the focus is on fixed effects rather than subject‑specific effects.

Summary

Singular fits are not numerical accidents. They are statistical messages telling you that:

• the model is over‑parameterized,

• the data do not contain enough information,

• and the random‑effects structure needs to be reconsidered.

Understanding this distinction leads to more robust modeling decisions and clearer scientific conclusions.

When Linear Mixed Models Don’t Converge: Causes, Consequences, and Interpretation

 

Introduction

 

Linear mixed-effects models (LMMs) have become a standard tool for analyzing correlated or hierarchical data across many applied fields, including biostatistics, psychology, ecology, and toxicology. Their appeal lies in the ability to model both population-level effects (fixed effects) and sources of variability associated with grouping or experimental structure (random effects) within a single coherent framework.

 

However, practitioners frequently encounter warning messages when fitting LMMs, such as “failed to converge,” “boundary (singular) fit,” or “Hessian not positive definite.” These warnings are often treated as technical nuisances or software-specific quirks, and it is tempting to either ignore them or attempt minor numerical fixes (e.g., changing optimizers or increasing iteration limits). Such warnings usually signal deeper statistical issues related to model identifiability and data support.

 

Convergence problems are especially common when LMMs are fitted to small or sparse data sets, or when the random-effects structure is ambitious relative to the available information. In these situations, the model may attempt to estimate more variance–covariance parameters than the data can reliably inform. As a result, the estimation procedure can struggle to identify a unique and stable optimum of the likelihood function.

 

Theoretically, applying linear mixed models to smaller data sets increases the risk of encountering convergence issues, particularly the problem of a singular Hessian matrix during restricted maximum likelihood (REML) estimation. This situation can lead to unwanted outcomes, such as one or more components having an asymptotic standard error of zero. Such an occurrence indicates that there are zero degrees of freedom for the residual (error) term, meaning the error variance is not defined. In other words, this suggests insufficient data or excessive model complexity (overparameterization).

 

 

Discussion

In principle, fitting linear mixed-effects models (LMMs) to relatively small or sparse data sets is statistically delicate, because these models rely on estimating variance–covariance components from limited information. Estimation is typically carried out via maximum likelihood (ML) or, more commonly, restricted maximum likelihood (REML), which involves optimizing a likelihood surface in a high-dimensional parameter space that includes both fixed effects and random-effects variance components.

When the amount of data is small relative to the complexity of the random-effects structure, the likelihood surface can become flat or ill-conditioned in certain directions. Therefore, the observed (or expected) information matrix—whose inverse is used to approximate the covariance matrix of the parameter estimates—may be singular or nearly singular. In REML estimation, this manifests as a singular Hessian matrix at the solution.

A singular Hessian indicates that one or more parameters are not identifiable from the data. In practical terms, the data do not contain enough independent information to support the estimation of all specified variance components. This lack of identifiability can arise for several, often overlapping, reasons: 

a) insufficient sample size, particularly a small number of grouping levels for random effects; 

b) near-redundancy among random effects, such as attempting to estimate both random intercepts and random slopes with little within-group replication; 

c) boundary solutions, where one or more variance components are estimated as zero (or extremely close to zero).

Interpretation of Zero or Near-Zero Standard Errors

One symptomatic outcome of a singular Hessian is the appearance of zero (or numerically negligible) asymptotic standard errors for certain parameters, especially variance components. This is not a meaningful indication of infinite precision; rather, it is a numerical artifact reflecting non-identifiability.

In the specific case of the residual (error) variance, an estimated standard error of zero suggests that the residual variance cannot be distinguished from zero given the fitted model. Conceptually, this corresponds to having zero effective degrees of freedom for the residual error term. Put differently, after accounting for fixed and random effects, the model leaves no independent information with which to estimate unexplained variability.

This situation implies that the model has effectively saturated the data: each observation is being explained by the combination of fixed effects and random effects with no remaining stochastic noise. From a statistical perspective, this is problematic because a) the residual variance is no longer defined in a meaningful way, b)  the standard inferential procedures (confidence intervals, hypothesis tests) become invalid, and c) small perturbations of the data can lead to large changes in parameter estimates (ill-conditioned problem).

Overparameterization and Model–Data Mismatch

The fundamental issue underlying these problems is overparameterization: the specified model is too complex for the available data. In mixed models, this typically occurs when the random-effects structure is overly rich relative to the number of observations and grouping units. For example, attempting to estimate multiple random slopes per group with only a few observations per group almost guarantees identifiability problems.

From a modeling perspective, a singular Hessian or zero residual variance should not be interpreted as an intrinsic property of the data-generating process, but rather as evidence of a mismatch between model complexity and data support. The data are insufficient to separate the contributions of all variance components, leading the estimation procedure to collapse some parameters to boundary values.

Practical Implications

These issues underscore the importance of exercising caution when applying linear mixed models to small data sets. In such situations, it is often necessary to simplify the random-effects structure by eliminating random slopes or correlations, or, if feasible, to increase either the sample size or the number of grouping levels. As a final option, one might consider alternative modeling strategies, such as fixed-effects models or penalized/regularized mixed models.

Ultimately, convergence diagnostics, singularity warnings, and zero standard errors should be viewed as key signals that the model may be statistically ill-posed, rather than as purely technical inconveniences. They suggest that the inferential conclusions drawn from such a model are likely unreliable unless the model specification is revisited.

Further Reading

To see how these concepts play out in practice — with simulated examples, diagnostics, and modeling alternatives — check out the second part of this series:

👉 Diagnosing Singular Fits in Linear Mixed Models

 

References

Gelman, A., & Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.

McElreath, R. (2020). Statistical Rethinking. CRC Press.

Pinheiro, J. C., & Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer.

Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software.

Matuschek, H. et al. (2017). Balancing Type I error and power in linear mixed models. Journal of Memory and Language.

Barr, D. J. et al. (2013). Random effects structure for confirmatory hypothesis testing. Journal of Memory and Language.

Understanding Anaerobic Threshold (VT2) and VO2 Max in Endurance Training

 Introduction: The Science Behind Ventilatory Thresholds

VT1 vs. VT2: Dpefining Energy Transitions

• p (Aerobic Threshold): The transition from lipid metawwwpp (fat-burning) to a mixed metabolism, where both carbohydrates and fats fuel muscle activity.
• V0T2 (Anaerobic Threshold): The point where the  shifts to a predominantly carbohydrate-based wmetabolism, leading to rapid lactate waccumpulation and increased reliance on anaerobic energy pathways.

wwww In

p VT1 and VT2

1. Wwww ewAgep & Sex
• Younger athletes generally exhibit higher VT2 values, while aging naturally reduces aerobicw capacity.
• Men tend to have a higher VO2 Max due to greater lung capacity and muscle mass, but women can achieve similar endurance levels through optimized training.
2. Body Composition & Health Status
• High body fat percentage may reduce VT2 efficiency, as excess weight increases cardiovascular workload.
• Health conditions such as cardiovascular disease, respiratory disorders, and metabolic syndromes can significantly impact ventilatory thresholds.
3. Pharmacological Influences
• Medications such as beta-blockers and ACE inhibitors affect heart rate regulation, potentially lowering VT2 due to altered cardiovascular response.
4. Cardiac Function & Ventricular Parameters
• The heart’s ejection fraction and ventricular relaxation capacity dictate oxygen delivery efficiency.
• A lower peak heart rate limits maximal cardiac output, directly influencing VO2 Max and VT2.

VO2 Max and VT2: The Connection

VO2 Max is the maximum oxygen uptake during exercise, representing cardiorespiratory efficiency. Since VT2 marks the transition to anaerobic metabolism, it directly correlates with VO2 Max:

• Athletes with higher VO2 Max levels can sustain aerobic efforts longer before crossing into VT2.
• Increasing VT2 effectively extends endurance, allowing the body to buffer lactate accumulation more efficiently.

Training Strategies to Improve VT2 & VO2 Max

Structured workouts can significantly increase VT2, delay lactate buildup, and optimize VO2 Max. Here’s how:

1. Threshold Training (Lactate Clearance Sessions)

• Running or cycling at 95-105% of VT2 intensity improves the body’s ability to metabolize lactate.
• Sessions should last 20–40 minutes, mimicking race conditions.

2. High-Intensity Interval Training (HIIT)

• Short bursts of maximum effort (~110% VO2 Max) with equal recovery periods.
• Enhances anaerobic power while boosting the efficiency of VT2 adaptation.

3. Long Steady-State Workouts

• Prolonged efforts at 80-85% VO2 Max strengthen the aerobic base.
• Builds endurance while minimizing lactate accumulation.

4. Strength Training for VO2 Max Optimization

• Studies show that lower-body strength work (squats, lunges, plyometrics) improves metabolic efficiency.
• Stronger muscle fibers require less oxygen, prolonging aerobic capacity.

Monitoring VT2 and VO2 Max with Data Analytics

Coaches and sports scientists track VT2 trends alongside VO2 Max using wearable sensors, lactate testing, and predictive data modeling.

Using R Code for VT2 Estimation

This R function  serves as a valuable tool for estimating VT2 based on training data, enabling athletes to: ✅ Analyze trends in ventilatory adaptation ✅ Quantify improvements over multiple training sessions ✅ Adjust pacing strategies for optimal endurance

Final Thoughts: Why VT2 Matters

Understanding VT2 and VO2 Max is fundamental for endurance athletes looking to optimize their training. By integrating targeted workouts, data-driven insights, and physiological testing, athletes can increase their threshold capacity, reduce fatigue, and improve overall performance.

Example

Let's assume these data:

Tempo <- matrix(c(46,30,7,50,5,0,12,33),nrow=4,ncol=2,byrow=TRUE)

Dist <- c(7460,1250,1100,2290)

distanza <- 10000

BpM <- c(163,134,153,160)

RunTime(Tempo,Dist,distanza,BpM)

Table 1. Summary of results

Run	Distance (mt)	Velocity (km/h)	Expected Velocity (km/h)	Time (s)	ExpectedTime (10 km)	bpm	Anaerobic Threshold (km/h)
1	7460	9.63	9.46	2790	3806.28	163	9.36
2	1250	9.57	8.45	470	4259.64	134	9.31
3	1100	13.20	11.56	300	3113.47	153	12.83
4	2290	10.95	10.02	753	3592.27	160	10.64

Figures

Figure 1. Running sessions' performances

The scatter plot presents data points illustrating a relationship between velocity (km/h) and distance (m). Observations from the trend suggest:

• As distance increases, velocity tends to decrease slightly, indicating a potential endurance effect where pace slows over longer distances.

• The initial point shows higher velocity, possibly due to the athlete starting with higher energy levels.

• The drop in velocity around mid-range distances could suggest a pacing strategy or fatigue onset.

• The final data points stabilize, indicating a consistent pace over longer distances.

The athlete may be adjusting their effort across distances to optimize endurance. Threshold Monitoring: If this dataset correlates with anaerobic threshold, the trend might highlight the optimal endurance balance for sustaining effort efficiently. Training Application: Understanding this pattern can help refine speed maintenance strategies for long-distance runs.

Figure 4. Velocity vs. Distance

The data points indicate a trend of decreasing velocity as the running distance increases. The drop in velocity at 3000m (~11 km/h) suggests possible fatigue onset or pacing strategy adjustments. The gradual decline in velocity might indicate controlled endurance pacing, where the athlete slows down to maintain stamina. 

Training Insights: If the anaerobic threshold is being assessed, the change in velocity across distances can help identify the optimal threshold where endurance and performance balance out.

R function: calculate the anaerobic threshold

RunTime <- function(Time,Distance,Distance_reference,bpm) {

## INPUT

#Time: run time measured [min, s]

#Distance: run distance [mt]

#Distance_reference: reference distance [mt] of which we want to evaluate the expected time 

#bpm: cardiac frequency (beats per minute) 

## OUTPUT

# T2 = tempo corsa atteso su distanza di riferimento (s)

# V2 = velocità attesa su distanza di riferimento (km / h)

# V1 = velocità media singole corse (km / h)

# Vm = velocità media su distanze e tempi rilevati cumulati (km / h)

# FC = Frequenza cardiaca per minuto

# SAN = Soglia Anaerobica stimata

# SAN2 = proiezione Soglia Anaerobica su tempi attesi e distanza di riferimento

# D1 = distanze percorse

# P1 = passo corsa per km [min, s]  

# P2 = passo corsa atteso per km [min, s]  

T1 <- Time

D1 <- Distance

D2 <- Distance_reference

T = T1[,2] + T1[,1]*60 # run time (seconds)

T2 = T* (D2/D1)^(1.06) # expected time (seconds)

V2 = 3.6 * (D2/T2)

V1 = 3.6 * (D1/T);

p = (1000*T/D1)/60 # run pace per km 

P1 = data.frame(floor(p), ((p - floor(p))*60)) # run pace per km [min, s]  

p2 = (1000*T2/D2)/60 # expected run pace per km 

P2 = data.frame(floor(p2), ((p2 - floor(p2))*60)) # expected run pace per km [min, s]  

Vm = 3.6 * (sum(D1)/sum(T));

SAN = 35000/(T*(10000/D1)) #Anaerobic Threshold

SAN2 = 35000/(T2*(10000/D2)) #projection of the Anaerobic Threshold over expected time and reference distance

FC <- bpm

## summary statistics

stats=NULL

stats$V1 = c(mean(V1), sd(V1))

stats$D1 = c(mean(D1), sd(D1))

stats$V2 = c(mean(V2), sd(V2))

stats$T1 = c(mean(T), sd(T))

stats$T2 = c(mean(T2), sd(T2))

stats$SAN = c(mean(SAN), sd(SAN))

stats$SAN2 = c(mean(SAN2), sd(SAN2))

stats$FC = c(mean(FC), sd(FC))

stats$P1 = c(mean(p), sd(p))

stats$P2 = c(mean(p2), sd(p2))

DF <- data.frame(

    Run = seq_along(Distance),

    Distance = Distance,

    Velocity = V1,

    Expected_Velocity = V2,

    Time = T,

    Expected_Time = T2,

    bpm = bpm,

    Anaerobic_Threshold = SAN

  )

##Plots

par(mfrow=c(2,2))

plot(V1,main="Velocity(km / h)",xlab="Run",ylab="km / h",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(D1,main="Distance",xlab="Run",ylab="mt",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(T,main="Run time",xlab="Run",ylab="s",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(P1[,2] + P1[,1]*60,main="Run pace per km", xlab="Run",ylab="s",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

par(mfrow=c(2,2))

plot(T2,main="Projection of times",xlab="Run",ylab="s",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(V2,main=paste0("expected velocity\n on reference distance of",D2," mt",sep=""),xlab="Run",ylab="km / h",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(P2[,2] + P2[,1]*60,main="Expected run pace per km (s)", xlab="Run", ylab="s",pch=16,type="o",lty=2)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

par(mfrow=c(1,2))

plot(T, SAN,main="Anaerobic Threshold vs. Time",xlab="Time (s)",ylab="Anaerobic Threshold (km / h)",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(SAN,main="Anaerobic Threshold",xlab="Run",ylab="Anaerobic Threshold (km / h)",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(D1,V1,main="Velocity vs. Distance",xlab="Distance",ylab="Velocity (km / h)",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

par(mfrow=c(2,1))

plot(D1,FC,main="bpm vs. distance",ylab="bpm",xlab="distance (mt)",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

plot(V1,FC,main="bpm vs. velocity",xlab="velocity (km / h)",ylab="bpm",pch=16)

grid(nx = NULL, ny = NULL, col = "lightgray", lty = "dotted",lwd = par("lwd"), equilogs = TRUE)

return(DF)

}

This function evaluates the performance of running training sessions based on distance, time, and heart rate data. 

It calculates expected time, velocity, anaerobic threshold, and other metrics to provide insights into a runner's progress.

Design and validation of a semi-quantitative microneutralization assay for human Metapneumovirus A1 and B1 subtypes

This study presents a comprehensive overview of human Metapneumovirus (hMPV), an enveloped RNA virus identified in 2001. It details the virus's biological characteristics, epidemiological significance, and ongoing vaccine research efforts. hMPV is classified within the Pneumoviridae family and comprises two global genetic lineages, namely A and B. While the pathogenic mechanisms underlying hMPV infection remain inadequately understood, the structural proteins involved, particularly the Fusion (F) protein, are instrumental in mediating infection and modulating the immune response. The F protein is highly conserved and is regarded as the principal target for vaccine development, as it promotes the formation of neutralizing antibodies, a contrast to the more variable Glycoprotein G.

From an epidemiological perspective, hMPV is widely distributed and predominantly impacts infants and young children, often resulting in severe respiratory illnesses. Despite the absence of a licensed vaccine, various candidates, including live recombinant viruses and monoclonal antibodies that target the F protein, have shown encouraging results in preclinical evaluations.

Moreover, this study introduces a high-throughput ELISA-based microneutralization assay (EMN) specifically designed to detect neutralizing antibodies against hMPV-A1 and hMPV-B1. This assay has been validated in accordance with international guidelines and exhibits high sensitivity and specificity, rendering it suitable for large-scale serological studies. The findings revealed a moderate humoral response within a cohort of human serum samples, thereby confirming the immunogenicity of the F protein.

The EMN assay represents a significant advancement in hMPV research, enabling the evaluation of vaccine candidates and showing promise for the development of effective preventive strategies against this respiratory virus. Future research efforts will focus on expanding the applicability of the assay and investigating polyvalent vaccine strategies.

As the scientific community continues to grapple with the complexities of human Metapneumovirus (hMPV), innovative approaches are emerging to address the challenges posed by this elusive pathogen. One of the most promising avenues involves leveraging next- generation sequencing (NGS) technologies to uncover the genetic variability and evolutionary dynamics of hMPV. By mapping the virus' s genetic landscape with unprecedented precision, researchers can identify potential mutational hotspots and design more robust vaccine candidates that remain effective across diverse strains.

Another revolutionary concept gaining traction is the application of artificial intelligence (AI) and machine learning in vaccine development and assay optimization. By analyzing vast datasets, AI models can predict the immunogenicity of viral antigens, streamline the design of polyvalent vaccines, and even enhance the performance parameters of assays like the ELISA- based microneutralization assay (EMN). This fusion of computational power with virological expertise has the potential to accelerate breakthroughs in hMPV research.

Additionally, innovative immunological strategies, such as combining monoclonal antibodies with adjuvants to boost immune responses, are being explored. Adjuvant technology can enhance the immunogenicity of vaccines targeting the F protein, ensuring stronger and longer- lasting protection against hMPV.

Finally, the advent of nanotechnology in vaccine delivery presents an exciting frontier for hMPV prevention. Nanoparticles engineered to carry antigens can mimic viral structures, eliciting more effective immune responses while enabling targeted delivery to respiratory tissues. This cutting- edge approach not only enhances vaccine efficacy but also minimizes systemic side effects.

By integrating these innovative perspectives, the field of hMPV research is transforming from understanding the fundamentals of a newly identified virus to pioneering solutions that address unmet medical needs. As scientists push the boundaries of what' s possible, hMPV stands as a testament to the ingenuity and resilience of the human spirit in combating infectious diseases.

Impact of Matrix Effect on Assay Accuracy at Intermediate Dilution Levels

 1. Introduction

The assessment of assay accuracy is critical in analytical chemistry, particularly when facing the challenge of variable matrix effects during quantitative measurements. The phenomenon commonly referred to as the matrix effect occurs when components of a sample other than the analyte influence the response of the assay. This effect can lead to significant discrepancies in the measurement of target analytes, particularly at intermediate and high dilution levels, where the concentration of the analyte is considerably lower and increasingly susceptible to interference. Matrix effects can arise from various sources, including co-eluting substances, ion suppression or enhancement, and physical properties such as viscosity and pH. These factors may alter the ionization or detection efficiency of the target analyte, impacting accuracy and precision (Tan et al., 2014; Rej and Norton-Wenzel, 2015; Rao, 2018). The challenge becomes more pronounced when analyzing samples that require significant dilution to bring analyte concentrations within the linear range of detection, often encountered in clinical and environmental analyses (Bowman et al., 2023; Gergov et al., 2015). While numerous studies have aimed to quantify the matrix effect at specific dilution levels (Tu & Bennett, 2017; Thompson & Ellison, 2005; Cortese et al., 2019), a comprehensive examination of its impact across a range of intermediate and high dilution scenarios remains limited. Failing to account for the matrix effect can result in underestimations of analyte concentrations, erroneous conclusions in research studies, and potential clinical misdiagnoses (Bowman et al., 2023; Thompson & Ellison, 2005; Xin Zhang et al., 2016). Such inaccuracies underscore the need for a robust methodological framework that enables the clear delineation of matrix influences on assay performance. 

Accurate quantification at intermediate and high dilution levels necessitates the implementation of suitable validation practices to mitigate potential matrix interferences. Validation of analytical methods is essential not only for regulatory compliance but also for ensuring that the methods exhibit proper sensitivity and robustness under varying conditions. The guidelines provided by organizations such as the International Conference on Harmonisation (ICH) and the US Food and Drug Administration (FDA) set forth fundamental principles for assessing analytical performance, with an emphasis on matrix characterization during method validation (ICH, 2021). Optimal validation practices should incorporate strategies that specifically address potential matrix effects. Techniques such as matrix-matched calibration, internal standardization, and comprehensive method development are precious in this regard (Carter et al., 2018; Tan et al., 2014; Francischini et al., 2024). Additionally, the incorporation of cross-validation approaches using different sample matrices can provide critical insights into the variability induced by matrix differences (Tan et al., 2014; Rej and Norton-Wenzel, 2015; Rao, 2018). A key aspect of addressing matrix effects involves implementing a thorough matrix characterization process, which should include assessing the composition and properties of reference materials. The application of advanced statistical tools can facilitate the quantification of variability attributable to matrix components and guide the selection of appropriate validation protocols. Researchers must realize how matrix composition can influence their assay's performance to improve the overall rigor of analytical methodologies. Moreover, the relationship between assay accuracy, precision, and matrix effects does not merely concern methodological validation. It also extends to the research laboratory's rigorous adherence to good laboratory practices (GLP). By fostering an environment rooted in quality assurance and ongoing training, laboratories can enhance their analytical capabilities, improving the accuracy and reliability of assay results (Tu & Bennett, 2017; Thompson & Ellison, 2005; Cortese et al., 2019). 

Let's focus on best validation practices to address matrix challenges at critical dilution levels by clarifying the relationship between matrix effects, assay accuracy, and precision.

 

2. Theoretical Background: Mechanisms and Factors Influencing the Matrix Effect

The matrix effect is a phenomenon that arises from various interactions between the analyte and the components of the sample matrix. These interactions can be broadly categorized into three main types: chemical, physical, and spectral interactions. Each type of interaction can significantly influence the accuracy and reliability of analytical measurements, particularly in complex matrices such as biological samples, environmental samples, and food products.

2.1 Types of Interactions

Chemical interactions refer to modifications in an analyte's chemical environment that arise due to the presence of various constituents within the matrix. A common form of chemical interaction is ion suppression or enhancement, which occurs when other ions within the matrix compete with the analyte for interaction with the detector. For instance, in liquid chromatography-mass spectrometry (LC-MS), co-eluting ions may inhibit the signal of the target analyte, resulting in an underrepresentation of its concentration (Matuszewski et al., 2003). Conversely, certain matrix components may serve to enhance the signal, leading to an overestimation of the analyte concentration.

Complex formation is another example of chemical interaction, where analytes form complexes with substances present in the matrix. This phenomenon can alter the reactivity and detection properties of the analyte. For example, metal ions found in a biological matrix may bind to the analyte, thereby influencing its detection efficiency (Harris, 2010). These chemical interactions can complicate the quantification of analytes, emphasizing the need for appropriate calibration methods to address matrix effects.

Physical interactions pertain to the impact of the matrix on the physical characteristics of the analyte, which can also affect its detection. A significant aspect of physical interactions is the influence of viscosity and density. The viscosity and density of the sample matrix can affect the mass transfer of the analyte during extraction, chromatographic separation, and ionization in mass spectrometry. Elevated viscosity may hinder the diffusion of the analyte, leading to variable recovery rates (Shelley et al., 2018). This can result in inconsistencies in the measured concentrations of the analyte. Furthermore, partitioning phenomena in complex matrices can lead to discrepancies in measured concentrations. In solid-phase extraction procedures, for example, analytes may preferentially partition into the sorbent material rather than eluting into the solution (Berrueta et al., 1995). Understanding and addressing these physical interactions is essential for obtaining reliable analytical results.

Spectral interactions occur when components of the matrix absorb or scatter light, or when they introduce spectral interferences that may impact the signal of the analyte. A prevalent type of spectral interaction is spectral overlap, where matrix constituents absorb at wavelengths similar to those of the target analyte. This can result in inaccurately high signals or baseline noise, complicating quantification. Spectral overlap is particularly relevant in ultraviolet-visible (UV-Vis) spectrophotometry (Østergaard, 2016; Bastos, 2022). Another instance of spectral interactions is fluorescence quenching, whereby certain matrix components can either diminish or enhance fluorescence emissions. This variability in fluorescence-based assays underscores the importance of employing matrix-matched calibration to ensure accurate measurements (Lakowicz, 2006). Spectral interactions can significantly affect the accuracy of analytical measurements and necessitate careful consideration throughout method development and validation.

 

2.2 Main Factors Influencing the Matrix Effect

The matrix effect can be influenced by several factors, including sample composition, assay design, and analytical conditions. The complexity of biological samples, such as plasma, serum, or tissue extracts, plays a critical role in influencing the matrix effect. Variations in protein content, lipid levels, and dissolved salts can lead to different degrees of matrix effects depending on the analyte being measured (Matuszewski et al., 2003). For instance, high protein content in plasma samples can lead to protein binding of the analyte, affecting its availability for detection. Similarly, lipid-rich samples may cause ion suppression in mass spectrometry due to the co-elution of lipids with the analyte. Therefore, rigorous characterization of the sample matrix is essential for predicting and mitigating the matrix effect.

 Biological variability is another important aspect of sample composition that impacts the matrix effect. The inherent biological variability between individuals, such as differences in age, sex, diet, or health status, can lead to variations in the matrix composition. These inter-individual differences in metabolites and proteins can result in inconsistent assay results, emphasizing the need for thorough analytical validation. For example, the metabolic profile of a patient with a specific disease may differ significantly from that of a healthy individual, leading to different matrix effects and potentially affecting the accuracy of the assay.

The choice of assay design and methodology is pivotal in managing the matrix effect. Complex procedures, such as liquid-liquid extractions or solid-phase extractions, may introduce more pronounced matrix effects compared to simpler, more selective methods (Peters & Remane, 2012; Cortese et al., 2019). For instance, liquid-liquid extraction can lead to the co-extraction of matrix components that interfere with the analyte’s detection. On the other hand, solid-phase extraction can selectively isolate the analyte, reducing the impact of matrix components. However, the choice of extraction method must be carefully optimized to balance the efficiency of analyte recovery with the minimization of matrix effects.

Calibration strategies are also crucial in mitigating the matrix effect. The implementation of matrix-matched calibration, where calibration standards are prepared in a matrix that closely resembles the sample matrix, can significantly enhance the accuracy and reliability of the measurements. This approach ensures that the calibration curve accounts for the variation introduced by the matrix, providing more accurate quantification of the analyte (Cortese et al., 2019; Bappaditya et al., 2022). The use of internal standards, which are compounds added to the sample that undergo the same interactions as the analyte, can help correct for matrix effects and improve the precision of the assay.

Variations in analytical conditions, such as ionization techniques and detection methods, can also influence the matrix effect. Different ionization techniques in mass spectrometry, such as electrospray ionization (ESI) and atmospheric pressure chemical ionization (APCI), exhibit varying degrees of susceptibility to matrix effects. For example, ESI is more prone to ion suppression due to the presence of co-eluting matrix components, whereas APCI may be less affected by such interferences. Therefore, the choice of ionization technique should be based on the specific characteristics of the sample matrix and the analyte.

The sensitivity and specificity of the analytical method are critical factors that influence its resilience to matrix effects. Methods with higher specificity, such as targeted LC-MS/MS, can allow for more accurate quantification even in the presence of complex matrices by avoiding interference from non-target components (Sveshnikova et al., 2019; Tang et al., 2022). For instance, the use of multiple reaction monitoring (MRM) in LC-MS/MS enables the selective detection of the analyte based on its unique fragmentation pattern, reducing the impact of matrix interferences.

 

3. Mitigating the Matrix Effect

Mitigating the matrix effect is essential for enhancing the accuracy and reliability of analytical assays, particularly at intermediate and high dilution levels. The matrix effect, which arises from interactions between the analyte and other components in the sample matrix, can significantly impact the quantification of target analytes. Effective strategies to mitigate these effects include optimizing sample preparation, using internal standards, and conducting robust method validation.

 

3.1 Sample Preparation Optimization

Optimizing sample preparation is a fundamental strategy for reducing the matrix effect. The goal is to minimize the presence of interfering substances that can affect the detection and quantification of the analyte. Various techniques can be employed depending on the specific matrix composition and the target analyte.

One common approach is dilution, which reduces the concentration of matrix components that may interfere with the analyte. However, dilution must be carefully balanced to ensure that the analyte concentration remains within the detectable range of the analytical method. Solid-phase extraction (SPE) is another widely used technique that can selectively isolate the analyte from the matrix. SPE involves passing the sample through a sorbent material that retains the analyte while allowing other matrix components to be washed away. This method can be optimized by selecting appropriate sorbent materials and conditions to maximize analyte recovery and minimize matrix effects.

Liquid-liquid extraction (LLE) is also effective for separating the analyte from the matrix. LLE involves partitioning the analyte between two immiscible liquid phases, typically an aqueous phase and an organic solvent. The choice of solvents and extraction conditions can be tailored to enhance the selectivity and efficiency of the extraction process. By systematically optimizing these sample preparation techniques, it is possible to significantly reduce the matrix effect and improve the accuracy of analytical measurements.

 

3.2 Use of Internal Standards

The use of internal standards is a powerful strategy for compensating for matrix effects during the quantification process. Internal standards are compounds that are chemically similar to the analyte and are added to the sample in known quantities. Stable isotope-labeled internal standards are particularly effective because they have nearly identical chemical properties to the analyte but can be distinguished based on their mass.

The internal standard undergoes the same sample preparation, extraction, and analysis procedures as the analyte, thereby accounting for variability due to matrix components. By comparing the response of the analyte to that of the internal standard, it is possible to correct for matrix effects and obtain more accurate measurements. This approach is widely used in quantitative bioanalytical methods, including liquid chromatography-mass spectrometry (LC-MS) and gas chromatography-mass spectrometry (GC-MS) (Tan et al., 2012; Li et al., 2015).

 

3.3 Robust Method Validation

Conducting robust method validation is essential to determine the impact of the matrix effect on assay performance across various sample types and conditions. Method validation involves a series of experiments designed to assess the reliability and accuracy of the analytical method. Key parameters to evaluate include linearity, sensitivity, precision, and accuracy. Linearity refers to the method’s ability to produce results that are directly proportional to the concentration of the analyte within a specified range. Assessing linearity under different matrix conditions helps ensure the method can accurately quantify the analyte across a range of concentrations. Sensitivity, or the method’s ability to detect low concentrations of the analyte, is also critical, particularly in complex matrices where matrix effects may reduce the signal. Precision, which measures the method's reproducibility, should be evaluated by analyzing multiple replicates of the same sample under identical conditions. This helps to identify any variability introduced by the matrix. Accuracy, or the method’s ability to produce results that are close to the true value, should be assessed by comparing the measured concentrations to known reference standards. Validation should also include experiments to assess the robustness of the method, which is its ability to remain unaffected by small, deliberate variations in method parameters. This can help identify any conditions under which the matrix effect may become more pronounced and allow for adjustments to mitigate these effects.

 

4. Discussion

As dilution levels increase, the response variability in assays tends to increase due to several factors. Let's delve into the intricate interplay between matrix effects and response variability, particularly in the context of increasingly diluted test samples. The focus is on elucidating several key factors influencing test accuracy.

Response Variability and Precision at Increased Dilution Levels

Response variability often amplifies with increasing sample dilutions, leading to a marked decrease in analytical precision. One primary reason is the reduction in the concentration of the analyte relative to the matrix components. At higher dilution levels, the analyte concentration approaches the limits of detection and becomes comparable to or even lower than the concentration of interfering substances in the matrix. This can lead to greater variability in the signal generated by the analyte, as the influence of the matrix components becomes more pronounced. This phenomenon exacerbates the impact of random noise. As dilutions increase, the signal not only diminishes but also becomes increasingly vulnerable to variations arising from laboratory conditions, instrument sensitivity, and intrinsic matrix effects of the sample. This response variability is well-documented in the literature, where a correlation has been established between dilution factors and the standard deviation of measured responses (Ceriotti,& Panteghini, 2023).

Increased Variability, Matrix Effects, and Accuracy Errors

The combination of increased response variability and matrix effects can lead to significant inaccuracies in results. Matrix effects refer to the changes in analyte response caused by other substances present in a sample. While these effects can introduce bias, the resulting inaccuracies become even more pronounced when variability is also present. The interaction between these two factors can result in substantial deviations from expected values, posing a persistent challenge in quantitative analysis (Sweeney et al., 2021). It is important to recognize that high variability, when combined with strong matrix effects, complicates the interpretation of results. This highlights the need for method validation studies to consider matrix interferences.

Matrix Effects and Precision

Interestingly, matrix effects are not directly proportional to precision. While they can skew response measurements, their influence tends to operate independently of the precision metrics of a method, which is usually represented by repeatability or reproducibility in specified conditions. The inherent noise levels and method performance characteristics play a crucial role in precision determination. Therefore, a method may maintain acceptable precision despite profound matrix interferences if calibration and standardization are effectively instituted. This distinction highlights the importance of rigorous methodological checks to ascertain the reliability of precision metrics irrespective of matrix complications.

Confidence Intervals for Accuracy

Implementing confidence intervals serves as a pivotal strategy for enhancing accuracy assessments and curbing false positives. By calculating these intervals, researchers can delineate a range within which the "true" value is likely to exist, which is crucial in situations where accuracy is prone to compromise due to variability or matrix effects. Statistical methods such as bootstrapping or Bayesian approaches can facilitate the estimation of confidence intervals, providing a more robust framework for interpreting results. These intervals are not only instrumental for data interpretation but serve as a foundation for informed decision-making regarding the validity of test results in clinical and research applications.

Using Intermediate or High Dilutions

While employing intermediate or high dilutions may introduce challenges regarding accuracy, it may sometimes be deemed acceptable if precision remains within acceptable bounds. The utilization of such dilutions might be necessitated by analytical conditions where sample concentration exceeds the calibration range, thus requiring dilution for accurate quantification. In these circumstances, scientists must weigh the trade-offs between maintaining precision and the risk of compromised accuracy, relying on stringent validation processes to support their method choices. This balancing act is especially pertinent in fields such as pharmacokinetics, where dosing regimens and therapeutic monitoring often compel the use of higher dilutions.

Potential Consequences of Accuracy Errors

Errors in accuracy can yield repercussions that may far exceed the perceived benefits of employing high dilutions. Consider, for instance, the implications of false negatives or false positives in clinical diagnostics or therapeutic drug monitoring; such inaccuracies can misdirect patient management strategies, potentially leading to adverse health outcomes. Consequently, the need for meticulous assessment of accuracy in conjunction with dilution strategies cannot be overstated.

References

Tan S. K., Shaw P. N., Hewavitharana A. K. (2014). Strategies for the Detection and Elimination of Matrix Effects in Quantitative LC–MS Analysis. LCGC North America, 32(1): 54-64.

Rej R., Norton-Wenzel C. S. (2015). Assessing Analytical Accuracy through Proficiency Testing: Have Effects of Matrix Been Overstated? Clinical Chemistry, Volume 61(2): 433–434, https://doi.org/10.1373/clinchem.2014.231241

Rao T. N. (2018). Validation of Analytical Methods, Calibration and Validation of Analytical Methods - A Sampling of Current Approaches. InTech, Apr. 25, 2018. doi: 10.5772/intechopen.72087.

Bowman, B. A., Reese, C. M., Blount, B. C., & Bhandari, D. (2023). Mitigating Matrix Effects in LC–ESI–MS-MS Analysis of a Urinary Biomarker of Xylenes Exposure. Journal of Analytical Toxicology, 47(2), 129-135. https://doi.org/10.1093/jat/bkac046

Gergov, M., Nenonen, T., Ojanperä, I., & Ketola, R. A. (2015). Compensation of Matrix Effects in a Standard Addition Method for Metformin in Postmortem Blood Using Liquid Chromatography–Electrospray–Tandem Mass Spectrometry. Journal of Analytical Toxicology, 39(5), 359-364. https://doi.org/10.1093/jat/bkv020

Jing Tu & Bennett P. (2017) Parallelism Experiments to Evaluate Matrix Effects, Selectivity and Sensitivity in Ligand-Binding Assay Method Development: Pros and Cons. Bioanalysis, 9:14, 1107-1122, DOI: 10.4155/bio-2017-0084

Thompson M.& Ellison S. L. R. (2005). A review of interference effects and their correction in chemical analysis with special reference to uncertainty. Accred Qual Assur 10, 82–97. https://doi.org/10.1007/s00769-004-0871-5

Cortese M., Gigliobianco M. R., Magnoni F., Censi R., & Di Martino P. (2019). Compensate for or Minimize Matrix Effects? Strategies for Overcoming Matrix Effects in Liquid Chromatography-Mass Spectrometry Technique: A Tutorial Review. Molecules, 25(13), 3047. https://doi.org/10.3390/molecules25133047

Xin Zhang, Haynes Kim, Danaceau J., Andrade L., Demers K., & Chambers E. (2016). Matrix Effects and Matrix Affects: The Impact of Different Sample Matrices on Sample Preparation and Chromatographic Analysis. Chromatography Today, https://www.chromatographytoday.com/article/bioanalytical/40/waters-corporation/matrix-effects-and-matrix-affects-the-impact-of-different-sample-matrices-on-sample-preparation-and-chromatographic-analysis/2126

ICH (2021). ICH Q2(R2) Validation of analytical procedures: Text and methodology. International Conference on Harmonisation of Technical Requirements for Pharmaceuticals for Human Use.

Carter J. A., Barros A. I., Nóbrega J. A., & Donati G. L. (2018). Traditional Calibration Methods in Atomic Spectrometry and New Calibration Strategies for Inductively Coupled Plasma Mass Spectrometry. Frontiers in Chemistry, 6, 415690. https://doi.org/10.3389/fchem.2018.00504

Francischini D.d.S. & Arruda M.A.Z. One-point calibration and matrix-matching concept for quantification of potentially toxic elements in wood by LA-ICP-MS. Anal Bioanal Chem 416, 2737–2748 (2024). https://doi.org/10.1007/s00216-023-04999-8

Matuszewski, B. K., Constanzer, M. L., & Chavez-Eng, C. M. (2003). Strategies for the assessment of matrix effect in quantitative bioanalytical methods based on HPLC-MS/MS. Anal Chem, 75(13), 3019-3030. doi: 10.1021/ac020361s. PMID: 12964746.

Harris, D. C. (2020). Quantitative chemical analysis (9th ed.). W.H. Freeman and Company.

Shelley J.T., Badal S.P., Engelhard, C. et al. (2018). Ambient desorption/ionization mass spectrometry: evolution from rapid qualitative screening to accurate quantification tool. Anal Bioanal Chem 410, 4061–4076.  https://doi.org/10.1007/s00216-018-1023-9

Berrueta L.A., Gallo B. & Vicente F. (1995). A review of solid phase extraction: Basic principles and new developments. Chromatographia 40, 474–483. https://doi.org/10.1007/BF02269916

Østergaard, J. (2016). UV/Vis Spectrophotometry and UV Imaging. In: Müllertz, A., Perrie, Y., Rades, T. (eds) Analytical Techniques in the Pharmaceutical Sciences. Advances in Delivery Science and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-4029-5_1

Bastos, E.L. (2022). UV-Vis Absorption and Fluorescence in Bioanalysis. In: Kubota, L.T., da Silva, J.A.F., Sena, M.M., Alves, W.A. (eds) Tools and Trends in Bioanalytical Chemistry. Springer, Cham. https://doi.org/10.1007/978-3-030-82381-8_4

Lakowicz, J. R. (2006). Principles of Fluorescence Spectroscopy. Springer.

Matuszewski BK, Constanzer ML, Chavez-Eng CM, (2203). Strategies for the assessment of matrix effect in quantitative bioanalytical methods based on HPLC-MS/MS. Anal Chem. 2003 Jul 1;75(13):3019-30. doi: 10.1021/ac020361s. PMID: 12964746.

Peters, F.T., Remane, D. Aspects of matrix effects in applications of liquid chromatography–mass spectrometry to forensic and clinical toxicology—a review. Anal Bioanal Chem 403, 2155–2172 (2012). https://doi.org/10.1007/s00216-012-6035-2

Bappaditya Kanrar, Ghosh, P., Khan, P. et al., (2022).  Alternative Strategies for the Calibration and Elimination of Matrix Effects in LC-MS/MS Multiresidue Analysis of Tea Matrix. J Anal Chem 77, 224–234. https://doi.org/10.1134/S1061934822020034

Sveshnikova, N., Yuan, T., Warren, J.M. et al., (2019). Development and validation of a reliable LC–MS/MS method for quantitative analysis of usnic acid in Cladonia uncialis. BMC Res Notes 12, 550. https://doi.org/10.1186/s13104-019-4580-x

Tang, L., Swezey, R.R., Green, C.E. et al. ,(2022). Enhancement of sensitivity and quantification quality in the LC–MS/MS measurement of large biomolecules with sum of MRM (SMRM). Anal Bioanal Chem 414, 1933–1947. https://doi.org/10.1007/s00216-021-03829-z

Tan, A., Boudreau, N., Lévesque, A. (2012). Internal Standards for Quantitative LC-MS Bioanalysis. In: Xu, Q., Madden, T. (eds) LC-MS in Drug Bioanalysis. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-3828-1_1

Li P., Li Z., Beck, W.D. et al., (2015). Bio-generation of stable isotope-labeled internal standards for absolute and relative quantitation of phase II drug metabolites in plasma samples using LC–MS/MS. Anal Bioanal Chem 407, 4053–4063. https://doi.org/10.1007/s00216-015-8614-5

Ceriotti, F., Panteghini, M. (2023). The Quality of Laboratory Results: Sources of Variability, Methods of Evaluation, and Estimation of Their Clinical Impact. In: Ciaccio, M. (eds) Clinical and Laboratory Medicine Textbook. Springer, Cham. https://doi.org/10.1007/978-3-031-24958-7_7

R function conf_int_Accuracy

 

R function

conf_int_Accuracy <- function(GMTobs, s, nDays = 2, nAnalysts = 2, nPlates = 2, nReplicates = 3, FoldDilution = 2, Threshold = 1, alpha = 0.05) {

  # INPUT:

  # GMTobs: vector of observed geometric means at each fold dilution

  # s: vector of the standard deviations of the log-transformed replicates

  # nDays: number of experimental days (default = 2)

  # nAnalysts: number of analysts performing the tests in validation (default = 2)

  # nPlates: number of plates used per analyst per day (default = 2)

  # nReplicates: number of measurements that each analyst performs per plate per day

  # FoldDilution: step of the serial dilution (default = 2)

  # Threshold: critical threshold of the log-difference between the observed and the true mean (default=1)

  # alpha: significance level (default=0.05)

  #

  # OUTPUT:

  # vector of Relative Accuracy calculated at each fold dilution and its confidence interval. In addition, a plot of Relative Accuracy vs. fold dilution (log2 units)

 

  # calculate the expected geometric mean vector

  GMTexp <- round(GMTobs[1] * 2^(seq(0, -(length(GMTobs) - 1), -1)), 0)

 

  # Relative Accuracy (or recovery)

  RA <- log(GMTobs / GMTexp)

 

  N <- nDays * nAnalysts * nPlates * nReplicates

  n1 = n2 = N

 

  # standard deviations

  s1 <- s  # vector of standard deviations of the log-transformed replicates

  fd <- round(FoldDilution^-seq(0, length(GMTobs) - 1), 4)

  s2 <- rep(s1[1], length(GMTobs))  # fd*s1[1]

 

  # pooled variance

  sp2 = ((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2)

 

  # standard error

  se <- sqrt((sp2 / n1) + (sp2 / n2))

 

  # t-statistic

  tstat <- qt(1 - alpha / 2, n1 + n2 - 2)

 

  # confidence interval

  lower <- RA - tstat * se

  upper <- RA + tstat * se

 

  # critical thresholds

  theta.low <- FoldDilution^-Threshold

  theta.up <- FoldDilution^Threshold

 

  # exponentiate to get back to original scale

  RA <- exp(RA)

  lower <- exp(lower)

  upper <- exp(upper)

 

  # plot RA versus fold dilution

  plot(log2(fd), RA, type = "o", pch = 16, lty = 2, xlab = "fold dilution (log2)", ylab = "Relative Accuracy", ylim = c(0, 3))

  lines(log2(fd), lower, col = "blue", lty = 4)

  lines(log2(fd), upper, col = "blue", lty = 4)

  grid()  

 

  # return RA and its confidence interval

  return(cbind(fd,RA, lower, upper))

}

 

This function calculates the Relative Accuracy (RA) and its confidence interval at each fold dilution and plots RA versus fold dilution (log2 units). The function takes the following inputs:

• GMTobs: vector of observed geometric means at each fold dilution
• s: vector of the standard deviations of the log-transformed replicates
• nDays: number of experimental days (default = 2)
• nAnalysts: number of analysts performing the tests in validation (default = 2)
• nPlates: number of plates used per analyst per day (default = 2)
• nReplicates: number of measurements that each analyst performs per plate per day
• FoldDilution: step of the serial dilution (default = 2)
• Threshold: critical threshold of the log-difference between the observed and the true mean (default=1)
• alpha: significance level (default=0.05)

The function returns a matrix containing the RA and its confidence interval for each fold dilution. The confidence interval is calculated using the t-distribution with n1+n2-2 degrees of freedom and a significance level of alpha. The function also plots RA versus fold dilution (log2 units) and the confidence interval.