Reevaluating statistical methods for proportion-based laboratory outcomes in assisted reproduction

Abstract

Background

Proportion-based laboratory outcomes in assisted reproductive technology (ART), including two-pronuclear fertilization rate, good-quality embryo rate, and blastocyst formation rate, are widely used to evaluate treatment effectiveness and laboratory performance. These outcomes are bounded between 0 and 1, frequently skewed, and arise from hierarchical data structures in which oocytes or embryos are nested within cycles and patients. Inappropriate statistical handling of such data can lead to biased inference and misinterpretation of treatment effects. This narrative review aimed to identify common methodological misuses in the analysis of proportion-based ART laboratory outcomes and to summarize appropriate statistical approaches.

Methods

This narrative review examined methodological and applied ART studies published between 2000 and 2025. Statistical practices were extracted and categorized according to data reporting (numerator–denominator structure), modeling strategies, and handling of clustering and boundary values. Recommended analytical frameworks were synthesized for binary outcomes reported either as a proportion or a count.

Results

Common analytical issues included pooling embryo- or oocyte-level observations and applying chi-square tests, reporting absolute counts without denominators, describing skewed proportion outcomes using mean ± standard deviation and linear models, and failing to account for intra-patient or intra-cycle clustering. Appropriate approaches depended on data structure. Beta regression was suitable for continuous proportions within the open interval (0,1), while zero–one-inflated beta models accommodated observed boundary values. When numerator and denominator counts were available, binomial generalized linear mixed models (GLMMs) and generalized estimating equations (GEEs) preserved denominator information and accounted for hierarchical clustering.

Conclusions

Misapplication of statistical methods to proportion-based ART laboratory outcomes remains common and may compromise reproducibility and clinical interpretation. Adoption of model–data alignment and transparent reporting practices will improve the validity of ART laboratory research and support evidence-based decision-making in reproductive medicine.

Introduction

In assisted reproductive technology (ART) research, proportion-based laboratory outcomes, such as 2PN fertilization rate, good-quality embryo rate, and blastocyst formation rate, are widely used to evaluate oocyte fertilization capacity, embryo developmental potential, and the effectiveness of interventions. These indicators serve not only as important intermediate outcomes for predicting clinical pregnancy and live birth [1, 2], but also directly inform clinical decisions regarding ovarian stimulation protocol optimization, embryo culture conditions, and the adoption of new technologies. Ensuring that such proportion-based outcomes are analyzed with appropriate statistical methods is therefore essential for generating robust research conclusions and for translating findings into clinical practice.

Unlike clinical endpoints such as live birth or miscarriage rates, which are typically analyzed at the cycle level, laboratory outcomes possess an inherent hierarchical structure: multiple oocytes or embryos may derive from a single cycle, and multiple cycles may belong to the same patient. These oocytes or embryos share the same maternal genetic background, hormonal environment, and stimulation protocol, leading to substantial intra-class correlation (ICC). Moreover, proportion data are often skewed and may include boundary values (0 or 1), making them unsuitable for traditional statistical methods that rely on normality assumptions. Ignoring these features risks biased statistical inference, thereby undermining the validity of study findings.

The risks of overlooking clustering effects have been well documented across disciplines. In dental epidemiology, Masood et al. highlighted the hierarchical structure of tooth surface data, where surfaces are nested within teeth, and teeth within individuals. Correlated observations within the same oral cavity yielded significant ICC, and treating them as independent artificially inflated sample size, underestimated standard errors, and exaggerated estimated intervention effects, ultimately increasing the risk of false-positive findings (type I error) [3]. Similarly, in the field of reproductive medicine, Mumusoglu et al. observed significant within-patient consistency in embryo developmental timing in a study using time-lapse morphokinetics to predict euploidy, with patient-level ICCs accounting for 16–47% of the total variance [4]. Failure to account for this clustering overstated the predictive value of morphokinetic parameters, leading to potentially biased conclusions.

Despite growing awareness, proportion-based laboratory outcomes in ART research remain subject to three common methodological misuses. First, clustered data are often aggregated into overall proportions and analyzed using chi-square tests, disregarding the non-independence of samples within the same patient. Second, absolute counts are reported without the corresponding proportions, conflating “quantity effects” with “efficiency effects.” Third, even when proportions are calculated at the patient level, analyses frequently rely on mean ± standard deviation and linear models based on normal distribution assumptions, overlooking the boundary constraints and skewness of proportion data. Such practices can exaggerate statistical significance, inflate the risk of false positives, misattribute intervention effects, and ultimately compromise reproducibility, generalizability, and clinical interpretation.

Although more suitable approaches, such as beta regression, zero-one inflated Beta (ZOIB) regression, generalized linear mixed models (GLMMs), and generalized estimating equations (GEEs) have been introduced, their adoption remains limited in ART research. Therefore, the present study evaluates current analytical practices for proportion-based laboratory outcomes in ART, illustrates common statistical misuses and their potential consequences using representative case examples, and discusses the applicability and limitations of more robust alternatives. The overall logic of this review is summarized in Fig. 1. Our goal is to provide a rigorous statistical framework and practical guidance for reproductive medicine research, thereby improving the reliability of study conclusions and their value for clinical decision-making.

Fig. 1.

Conceptual overview of statistical approaches for analyzing proportion-based laboratory outcomes in assisted reproduction. This figure illustrates the hierarchical nature of ART laboratory data, where oocytes and embryos are nested within treatment cycles and patients. It contrasts commonly misused methods, such as chi-square tests and linear regression, which assume data independence and normality, with more appropriate modeling frameworks that account for the bounded and clustered characteristics of proportion outcomes. Beta and zero-one-inflated beta (ZOIB) regressions are suitable for proportions within the (0,1) interval, while generalized linear mixed models (GLMMs) and generalized estimating equations (GEEs) effectively handle binomial numerator–denominator data and intra-class correlation. The schematic highlights how selecting the correct model improves statistical validity, reduces bias, and enhances the interpretability and reproducibility of ART research findings

Common statistical misuses

Ignoring the nested data structure: use of overall proportions with chi-square test

A frequent misuse in the analysis of ART laboratory outcomes is treating oocyte- or embryo-level data as if they were independent observations. Researchers often calculate an overall proportion across the entire cohort (e.g., total number of 2PN embryos divided by the total number of retrieved oocytes) and then perform a chi-square test. This approach completely disregards the hierarchical structure of the data (oocyte or embryo → cycle → patient). Such handling introduces four major drawbacks. First, violation of the independence assumption. A fundamental prerequisite of the chi-square test is that observations are independent. However, oocytes or embryos derived from the same patient share maternal genetic, endocrine, and clinical environments and are therefore correlated. Treating them as independent artificially inflates the sample size, underestimates the standard error, and exaggerates statistical significance, which increases the risk of type I error [5]. Second, neglect of inter-patient heterogeneity. The overall proportion functions as a “weighted average” across all oocytes, disproportionately influenced by patients with many retrieved oocytes. Patients contributing only a few oocytes are nearly diluted out, while “large-sample” patients dominate the results. This obscures the relationship between patient variability and undermines the representativeness of the outcome at the population level. Clinically, however, the relevant measure is the fertilization rate per patient, not the pooled oocyte-level rate. Third, the inability to adequately control confounders. Chi-square tests and aggregated group-level proportions cannot incorporate covariates such as female age. Even when logistic regression is attempted, all oocytes from a single patient share identical covariate values, producing duplicated entries, misrepresenting correlation among observations, and failing to achieve true adjustment for confounders. Fourth, loss of information and reduced statistical efficiency. The chi-square test analyzes a binary outcome at the level of individual oocytes (fertilized vs. not fertilized), implicitly assuming independence among all observations. However, the clinically relevant unit of analysis is the patient-level proportion (e.g., fertilization rate per patient). By failing to account for this hierarchical structure and bypassing patient-level aggregation, the approach neglects within-patient correlation and variability, thereby reducing statistical efficiency and potentially biasing estimation of the true patient-level effect. Importantly, clustering is not limited to oocytes or embryos within a cycle. When patients undergo multiple treatment cycles, outcomes across cycles remain correlated. Treating these repeated observations as independent introduces additional bias.

A representative example is provided by Zhou et al. [6], who examined the association between sperm DNA fragmentation and high-quality embryo rates. The authors reported 72.61% high-quality embryos in the normal sperm DNA group, versus 57.83% in the damaged group, with a statistically significant difference (P < 0.01) based on a chi-square test. However, this analysis treated 544 embryos from 93 patients as independent observations, ignoring embryo-level ICC. Moreover, despite claiming that baseline characteristics were “well matched,” the damaged-DNA group showed significantly lower female BMI (20.57 vs. 21.76 kg/m², P < 0.01) and borderline differences in serum AMH levels (1.89 vs. 3.01 ng/mL, P = 0.06) and sperm concentration (39 × 10⁶/mL vs. 58 × 10⁶/mL, P = 0.05). Because AMH levels are a well-established determinant of embryo quality [7], inadequate adjustment for these and other confounders risks over-attributing differences to sperm DNA damage [8]. This example highlights the necessity of rigorous confounding control in observational studies [9]. Unfortunately, similar methodological flaws are prevalent throughout the ART literature [10–21].

Reporting absolute numbers only: confusing the “Denominator Effect” with the “Efficiency Effect”

Another frequent misuse in ART research is reporting absolute counts of laboratory outcomes, such as the number of 2PN oocytes or high-quality embryos (often expressed as median or mean ± SD), as outcome indicators without simultaneously presenting the corresponding proportions [22–27]. This practice risks conflating differences in oocyte yield with differences in fertilization or developmental efficiency. Taking 2PN fertilization as an example, reporting only the absolute number of 2PN oocytes, without calculating the 2PN rate, can substantially bias interpretation. Fanton et al. [28], in a large-scale analysis of 402,411 IVF cycles, demonstrated a strong positive correlation between the number of retrieved oocytes and the absolute number of 2PN embryos (r = 0.86, P < 0.01). In other words, more retrieved oocytes almost inevitably lead to more fertilized embryos. Thus, when study groups differ in oocyte yield, comparing absolute 2PN counts alone may misleadingly attribute an apparent advantage to the intervention under study, when in reality the difference reflects stimulation intensity rather than an actual improvement in fertilization efficiency. For this reason, reporting 2PN rate in parallel, or applying models adjusted for oocyte number, is essential.

Gallardo et al. [29] further highlighted this paradoxical issue in a study of thawed oocytes. When oocyte survival fell sharply from ≥ 95% to < 50%, the absolute number of 2PN oocytes declined significantly (8.0 ± 2.8 → 5.5 ± 2.2). However, the 2PN fertilization rate remained essentially stable (76.8% → 75.5%, a difference of only 1.3% points). Reporting absolute numbers alone would have suggested impaired fertilization capacity. However, closer inspection revealed that the reduction in 2PN counts was driven by a smaller available oocyte pool (mean MII oocytes decreased from 10.48 ± 2.9 to 7.31 ± 2.6). In contrast, the fertilization potential of surviving oocytes was preserved.

Ultimately, the decline in cumulative live birth rate (CLBR) was therefore attributable to reduced embryo pool size, not diminished fertilization efficiency. Together, these findings reinforce the risk of distorted interpretations when absolute counts are reported in isolation. By conflating denominator effects with efficiency effects, studies may misattribute outcomes to interventions rather than to differences in oocyte yield. Proportion-based indicators, reported alongside absolute numbers, are thus indispensable for accurate interpretation and the avoidance of misleading conclusions in both research and clinical practice.

Ignoring distributional characteristics of proportion data: misuse of mean ± standard deviation and linear regression

Even when studies correctly calculate patient-level proportions, inappropriate descriptive and inferential methods are frequently applied, undermining the validity of their conclusions [30–34].

Problems with mean ± standard deviation

A common issue arises when means and standard deviations (SDs) are reported for skewed proportion data. For instance, in a study examining the timing of luteal-phase ovarian stimulation (LPOS) initiation [30], several laboratory outcomes were described using mean ± SD, including the mature oocyte rate (0.78 ± 0.74) and the 2PN fertilization rate (0.58 ± 0.68). In both cases, the SD approached or even exceeded the mean, indicating substantial skewness and the presence of extreme values. Using the Empirical Rule, one would expect approximately 68% of 2PN rates to lie within [− 0.10, 1.26] (square brackets denote closed intervals, including the endpoints). This range extends beyond the logical bounds of proportion data [0,1], producing negative values and rates above 100%, which is statistically nonsensical. Using the conventional 95% range (± 2 SD) further expands the interval to [− 0.78, 1.94]. Even Chebyshev’s inequality, which makes no distributional assumptions, yields similarly implausible intervals. These results highlight the inadequacy of mean ± SD for proportion data. More appropriate descriptive measures include the median with interquartile range (IQR) or the median absolute deviation (MAD) [35]. Indeed, both of these two measurements have already been adopted in several ART studies [36–38].

Misuse of linear regression

Despite clear evidence of skewed distributions with high variability, Tan et al. [30] employed linear regression models based on the assumption of normally distributed residuals, for example, analyzing the effect of follicle diameter on oocyte retrieval rates. While confounder adjustment was attempted, the reliance on linear models for skewed, bounded outcomes raise serious concerns about both statistical validity and biological interpretability [39].

Taken together, these methodological limitations undermine the study’s conclusion that “smaller follicle diameter leads to higher efficiency.” The combination of inappropriate descriptive statistics and misuse of linear regression illustrate the broader risks of applying conventional methods to proportion-based outcomes. More robust alternatives, such as generalized linear models, beta regression, or mixed-effects frameworks, are required to ensure valid inference and biologically meaningful interpretation.

Recommended methods

To overcome the limitations outlined above, we focus on four complementary modeling families that align with the characteristics of proportion-based laboratory outcomes in ART.

Beta regression

Model concept

Beta regression is a type of generalized linear model specifically designed for analyzing proportion or percentage data that falls strictly within the open interval (0,1) (i.e., values strictly greater than 0 and less than 1). Unlike traditional linear regression, which assumes a normal distribution, beta regression assumes that the dependent variable follows a flexible beta distribution, which means the model can handle data of various shapes (symmetric, skewed, or U-shaped) without requiring transformation. The core model can be written as:

In the equation, is the expected mean proportion for the 𝑖-th observation (e.g., the predicted fertilization rate for a patient with certain characteristics), represents the predictor variables (e.g., treatment, age, BMI), 𝛽 represents the vector of regression coefficients, and is the linear predictor (representing the combined effect of all clinical factors on the outcome). The function g (⋅) is the link function. When the most commonly used logit link function g(µ) = log[µ/(1 − µ)] is employed, the regression coefficient 𝛽_k can be interpreted as follows: with other variables held constant, for each one-unit increase in the independent variable xₖ, the log-odds of the expected value of the dependent variable will increase by 𝛽_k units, and the corresponding odds ratio (OR) is exp (𝛽_k). In other words, this interpretation is analogous to that of logistic regression but applied to proportions.

Applications in ART research

Several studies have demonstrated the practicality and interpretability of Beta regression for laboratory outcomes in ART. For example, Xu et al. [40] used Beta regression to evaluate the association between follicular fluid exposure to perfluoroalkyl and polyfluoroalkyl substances (PFAS) and the rate of high-quality embryo, adjusting for key confounders such as age and BMI. Similarly, Butler et al. applied Beta regression to examine the relationship between plasma microRNAs and fertilization rate in IVF patients, identifying several miRNAs that were significantly associated with fertilization capacity while simultaneously controlling for confounders [41]. These examples illustrate that Beta regression accommodates proportional outcomes while flexibly incorporating multiple covariates, making it a powerful alternative to conventional approaches.

Advantages over traditional methods

A significant advantage of Beta regression is that it eliminates the need for arbitrary data transformations (e.g., logarithmic, square-root, or arcsine) often employed to improve residual normality before applying linear regression. Although transformations can stabilize variance, they shift parameters away from the original scale of measurement, complicating biological interpretation [42]. In contrast, Beta regression directly models the raw proportional data. This approach not only effectively addressing the skewness and heteroscedasticity commonly observed when proportion data are clustered near 0 or 1, but also ensures that parameter estimates remain on the intuitive original scale [42]. Consequently, it provides a clear biological interpretation of the results, significantly improving interpretability and stability. Well-established R packages such as betareg and Bayesianbetareg support Beta regression and allow covariate adjustment.

It is important to note that although Beta regression offers advantages in clinical interpretability, the traditional arcsine transformation remains well established in statistical practice. From a theoretical perspective, the arcsine transformation is highly effective for variance stabilization and reducing heteroscedasticity [43]. More recently, Laurencelle and Cousineau [44] further introduced the Analysis of Proportions Using Arcsine Transform (ANOPA) framework, which extends the classical approach to accommodate complex experimental designs, including nested and within-subject structures. ANOPA also facilitates the estimation of effect sizes and confidence intervals, with computations implemented in the R package ANOPA. Accordingly, the arcsine transformation and Beta regression should be viewed as complementary rather than competing approaches: the former emphasizes variance stabilization and compatibility with ANOVA-based inference, whereas the latter directly models bounded proportional outcomes within a flexible regression framework.

Limitations and boundary values

One important limitation is that the Beta distribution is defined on the open interval (0,1). Datasets containing rare boundary values (e.g., a patient’s 2PN rate equal to exactly 0% or 100%) require special handling. For rare boundary values, Smithson & Verkuilen [42] proposed a simple linear shrinkage method to rescale observations into the (0,1) interval:

where n is the sample size. This approach preserves the proportional meaning of the variable while satisfying the distributional assumptions of Beta regression. However, when boundary values of 0 or 1 occur frequently, simple shrinkage may underestimate their biological relevance or distort estimates, thereby masking important clinical implications. In such cases, a zero-one inflated beta (ZOIB) model is preferable, as it explicitly models 0 and 1 as discrete point masses while using the Beta distribution for intermediate values, thereby providing more accurate and interpretable parameter estimates.

Zero-One Inflated Beta (ZOIB) regression

When proportion outcomes are distributed within the interval (0,1) but also contain frequent boundary values at 0 or 1 (or both), standard Beta regression or simple linear shrinkage is inadequate. In such cases, Zero-One Inflated Beta regression (ZOIB) provides a more flexible and data-appropriate alternative.

Model concept

The ZOIB model is essentially a segmented model comprising three parts: two discrete components representing the point masses at 0 and 1, and a continuous component for observations within the interval (0,1). Think of the ZOIB model as a three-step decision process for each clinical observation (e.g., a patient’s fertilization rate). The model simultaneously estimates three separate equations:

The three equations above correspond respectively to: (1) the zero-inflation component, modeling the probability of total failure (), (2) the one-inflation component, modeling the probability of perfect success ( ); and (3) the Beta regression component, modeling the varying degrees of partial success within the (0,1) range. , , and are the intercepts, while , , and are the regression coefficients representing the influence of clinical predictors (such as age or treatment) on each specific component; , , and denote the values of the i-th sample on the k-th predictor variable, respectively.

The ZOIB model is highly flexible. If boundary inflation occurs only at one side, the model can be simplified to a Zero-Inflated Beta (ZIB) or a One-Inflated Beta (OIB) model by setting the point mass probability at the other side to zero. In the absence of boundary values, ZOIB reduces naturally to standard Beta regression. This flexibility enables ZOIB to handle outcomes defined on (0,1), [0,1), (0,1], and [0,1], allowing researchers to flexibly tailor the model to the actual data distribution.

Practical implementation

To facilitate practical implementation of this mixture framework, a study group developed the zoib R package, which implements ZOIB regression within a Bayesian framework using Markov Chain Monte Carlo (MCMC) sampling. The package provides a unified platform for ZOIB, ZIB, OIB, and standard beta regression, where zero- and one-inflation components can be activated or disabled via simple parameter settings (zero.inflation, one.inflation).

Advantages for ART research

In assisted reproduction research, when boundary value piling occurs in laboratory outcomes (e.g., a high proportion of patients with a 0% high-quality embryo rate), ignoring these extreme observations risks either statistical bias (if discarded) or distorted estimates (if shrunk). ZOIB regression directly models these biologically meaningful extremes, while appropriately handling the distribution of intermediate values. This approach not only yields more robust and reliable parameter estimates but also enhances biological interpretability, facilitating the identification and investigation of unique factors contributing to “complete failure” or “complete success”.

Generalized Linear Mixed Models (GLMMs)

Introduction to mixed effects models

As previously discussed, laboratory outcomes in ART studies exhibit an inherent clustering or hierarchical structure (oocytes/embryos → cycles → patients), implying that observations from the same source (e.g., the same patient) are not independent. Ignoring such a correlation and employing traditional statistical methods (e.g., ordinary linear regression) can inflate the risk of false positives. Mixed effects models are specifically designed to address this issue. In brief, a mixed effects model comprises two components: fixed effects, representing the average impact of the independent variables of interest (e.g., interventions) and confounding factors to be controlled for (e.g., age) on the outcome; and random effects, primarily used to characterize and control for the non-independence arising from the data structure (e.g., ovarian stimulation cycles nested within patients).

Introduction to GLMMs

Early mixed effects models primarily referred to linear mixed effects models, which assume that the outcome variable is continuous and normally distributed. However, this assumption is often violated for proportion-based outcomes (e.g., fertilization rate per patient), which are restricted between 0 and 1 and do not follow a normal distribution. GLMM represents a significant extension of the linear mixed effects framework. It integrates the capability of generalized linear models to handle non-normally distributed data with the ability of mixed effects models to address data non-independence. The general form of GLMMs can be expressed as:

In plain language, the model separates the influences on the outcome into two parts: fixed effects and random effects. The fixed-effects function, like standard regression coefficients, estimates the average effect of predictors (e.g., treatment, age) across all patients. The random effects account for the fact that repeated observations from the same patient (e.g., multiple cycles) are more alike than observations from different patients. More specifically, represents the conditional probability of the outcome event given the random effects . Indices 𝑖, j, and k represent the clustering individuals (e.g., patients), repeated measures (e.g., treatment cycles), and predictor variables (e.g., interventions), respectively. represents the fixed effects coefficients, and exp (β) can be interpreted as the conditional OR under the logit link. ​ is the design matrix for random effects, and represents the random effects for the 𝑖-th higher-level unit (e.g., patient). The choice of link function g (⋅) depends on the distribution of the dependent variable: the logit link is most commonly used for binary outcomes (e.g., whether an oocyte is fertilized or not), whereas the log link is appropriate for count data (e.g., the number of oocytes retrieved).

Extended applications of mixed effects models

The mixed modeling framework represented by GLMM, which combines fixed and random effects, offers considerable flexibility. It can be integrated with other complex distributional assumptions, such as zero-inflated binomial mixed models and mixed-effects ZOIB regression models. In these models, researchers can introduce fixed effects in both the inflation and Beta distribution components, as well as incorporate patient-level random effects. This approach simultaneously addresses the challenges of boundary value accumulation, confounding factor control, and data hierarchy.

Application to proportion outcomes

In ART research, when a single patient contributes multiple ovarian stimulation cycles, the GLMM can effectively model proportion-based laboratory outcomes. This model incorporates fixed effect covariates to control for confounding factors, while explicitly modeling and estimating the intra-class correlation (ICC) between cycles within the same patient through the inclusion of patient-level random effects. Moreover, it allows for in-depth investigation of individual heterogeneity in outcomes. For example, to analyze the impact of an intervention on the high-quality embryo rate, let 𝑌_𝑖𝑗 represent the number of high-quality embryos for the 𝑗-th cycle of the 𝑖-th patient, and 𝑛_𝑖𝑗 represent the total number of embryos in that cycle. The model can then be formulated as follows:

where the fixed effects component, , represents the intervention, and represents the patient’s age (a representative potential confounder), with and as the regression coefficients for the intervention and age, respectively. The random effects component, , is the patient-level random intercept, reflecting the deviation of the high-quality embryo rate of the 𝑖-th patient from the overall average due to individual characteristics (such as diminished ovarian reserve, DOR). The term represents the patient-level random slope, capturing the deviation of the 𝑖-th patient’s response to the intervention from the average treatment effect. If the variance of the random slope () is found to be significant, it indicates substantial individual differences in the treatment effect. Based on this finding, researchers can further investigate which patient characteristics (e.g., the presence of polycystic ovary syndrome (PCOS), DOR, etc.) may explain such heterogeneity in treatment response. For instance, this can be examined by incorporating a “Treatment × PCOS” interaction term into the model, thereby providing statistical evidence for stratified therapy and precision medicine.

This model can directly incorporate the number of successes and total attempts (i.e., the number of high-quality embryos and the total number of embryos) to model the success probability (high-quality embryo rate ) based on the binomial distribution. This approach not only retains the original count information but also accommodates extreme probabilities of 0% or 100%. Furthermore, the GLMM can simultaneously provide both subject-specific (conditional) effects and population-averaged (marginal) effects, thereby enhancing the interpretability and stability of parameter estimates.

Example in ART research

Fouks et al. applied GLMMs to examine the association between weight change and IVF laboratory outcomes using a total of 1,922 cycles from 961 patients [45]. By including patient ID as a random effect and adjusting for confounders such as age, AMH, and gonadotropin dosage, their model accounted for repeated cycles within patients and produced valid, interpretable estimates. This example demonstrates how GLMMs can effectively address clustering while incorporating clinically relevant covariates.

Practical implementation

GLMMs are widely supported in statistical software. In R, common packages include lme4 (function glmer), glmmTMB, and brms, all of which allow flexible specification of link functions, error distributions, and random effects structures. By modeling laboratory outcomes at the count level and explicitly handling multi-level clustering, GLMMs directly address the methodological flaws highlighted in Sect.  2. They provide robust estimates that respect the hierarchical structure of ART data, making them one of the most versatile and reliable approaches for proportion-based outcomes when multiple cycles per patient are included.

Generalized Estimating Equations (GEE)

Model concept

Generalized estimating equations (GEE) are a method employed in the analysis of repeated measures or clustered data (e.g., multiple ovarian stimulation cycles from the same patient). The primary advantage of GEE lies in its ability to describe within-subject correlations by specifying a working correlation structure and to prevent the underestimation of standard errors caused by clustering without relying on strict distributional assumptions. Even if the chosen working correlation structure is not perfectly specified, GEE can still provide valid population-averaged effect estimates, as long as the mean model is correct.

The mean model for GEE is specified as:

where represents the value of the k-th predictor (e.g., treatment) for the j-th observation (e.g., ovarian stimulation cycle) from the 𝑖-th cluster (e.g., patient), and represents the corresponding marginal regression coefficient which describes how the average outcome changes with each predictor across the entire population. Based on this, the model incorporates an appropriate working correlation structure (e.g., assuming all cycles from the same patient are equally correlated) to adjust the parameter estimation process, thereby enhancing the reliability of the inferences.

Application to proportion outcomes

Similar to GLMM, GEE can directly model outcomes as “number of successes/total trials” under a binomial framework, avoiding the need to convert them into pre-calculated rates. This approach retains denominator information, preserves statistical efficiency, and accommodates extreme values of 0% and 100%.

Examples in ART research

Recent studies have demonstrated the application of GEE in the field of assisted reproduction. For example, Fouks et al. analyzed 1,922 IVF cycles from 961 patients using both GLMM and GEE, clustering by patient ID to account for multiple cycles. Their results demonstrated that GEE provides a valid and interpretable alternative for modeling clustered ART outcomes [45]. Le et al. adopted a different approach by aggregating embryo-level data into cycle-level proportions (e.g., blastocyst formation rate) and using these as inputs for GEE [46]. However, this strategy discards information on denominator size, which may bias estimates. Whenever both numerator and denominator counts are available, modeling outcomes directly under a binomial distribution is preferred.

Practical implementation

In R, GEE can be implemented using packages such as geepack or gee, both of which support a variety of correlation structures. The method is relatively straightforward, computationally efficient, and well-suited for large-sample settings in ART studies. By producing robust population-level estimates while accommodating clustering, GEE represents an accessible and practical approach for analyzing proportion-based laboratory outcomes. It is particularly useful in large-scale ART datasets where patient- or cycle-level clustering must be addressed, but the primary interest lies in population-averaged inference.

Other Strategies

While advanced statistical models such as beta regression, ZOIB, GLMMs, and GEEs provide powerful solutions for handling distributional characteristics, hierarchical structures, and confounder adjustment, robust inference in ART research can also be achieved through rigorous study design combined with appropriate basic statistical methods. The study by Moutos et al., which compared laboratory outcomes in non-obstructive azoospermia (NOA) patients using fresh versus cryopreserved oocytes, provides an instructive example [47].

Example of a self-paired design

A study employed a paired design in which fresh mTESE sperm from the same male were used to inseminate both fresh and cryopreserved oocytes from the same female partner on the same day. This approach minimized individual-level confounding, effectively isolating “oocyte status” as the primary explanatory variable [47].

Additional strategies to ensure validity

The investigators further strengthened their study by implementing supplementary analyses: (1) In the overall cohort analysis, they restricted each patient to contributing only the cycle with the youngest oocyte age, ensuring data independence. (2) They applied 1:2 age matching to control maternal age, a critical confounder, thereby improving comparability across groups.

Data reporting and statistical methods

Results were reported at both the absolute count and patient-level proportion scales, with outcomes summarized using medians and interquartile ranges (IQRs). Statistical comparisons were made using non-parametric tests appropriate for the data structure: the Wilcoxon signed-rank test for paired observations and the Wilcoxon rank-sum test for unpaired comparisons. Their analyses showed that cryopreserved oocytes produced fewer good-quality embryos in absolute terms (median 1 vs. 2, P = 0.042), but the good-quality embryo rate did not differ significantly (61% vs. 50%, P = 0.69). This distinction highlighted that cryopreservation primarily reduces oocyte quantity rather than fertilization efficiency or embryo quality.

Implications

Although the study did not employ multi-level models such as GLMM or GEE, its thoughtful design and appropriate use of non-parametric methods effectively addressed issues of clustering and confounding. This case underscores that study design can be as important as, if not more important than, complex post hoc modeling. A prospective design mindset, combined with judicious application of basic statistical tools, represents an alternative methodological strategy that can yield results as reliable, and sometimes more interpretable than those obtained from sophisticated statistical models.

Summary

In summary, Beta regression, ZOIB, GLMM, and GEE each provide viable approaches for analyzing proportion-based laboratory outcomes in assisted reproduction, such as 2PN fertilization rate, good-quality embryo rate, and blastocyst formation rate. Each method carries distinct assumptions, input format, and interpretive strengths (Table 1).

Table 1.

Comparison of Different Models for Analyzing Proportion-Based Laboratory Outcomes

Characteristic	Beta regression	ZOIB	GLMM	GEE
Input format	Aggregated proportion values	Proportion values + separate modeling of 0/1 extremes	Number of successes + total attempts	Number of successes + total attempts
Distribution assumption	Beta (0,1)	Zero-One Inflated Beta	Binomial	Binomial
Denominator information	Not retained	Not retained (extremes modeled separately)	Retained	Retained
Clustering handling	Not supported (unless extended version)	Not supported (unless extended version)	Random effects	Working correlation matrix
Effect interpretation	Population-average effect	Population-average + extreme probabilities	Subject-specific + population-average	Population-average effect
Extreme values	Requires shrinkage	Directly accommodates 0/1	Naturally supported	Naturally supported
Typical scenarios	Single-level proportions	Proportions including 0/1	Repeated measures or hierarchical data	Population-level inference

Beta regression and ZOIB models are typically applied to aggregated proportion data, assuming that the outcome follows a Beta distribution or a ZOIB mixture distribution, respectively. In contrast, GLMMs and GEEs directly model the binomial data defined by the number of successes over total trials, thereby retaining the original denominator information and effectively accounting for data clustering. Regarding effect interpretation, GLMMs incorporate random effects to estimate both fixed and subject-specific effects, making them suitable for analyzing within-subject variability and between-patient heterogeneity. GEEs, on the other hand, focus on population-averaged effects and are more appropriate for evaluating overall intervention efficacy rather than individual-level prediction (Fig. 2).

Fig. 2.

Decision framework for selecting appropriate statistical models for proportion-based laboratory outcomes in assisted reproduction. This flowchart outlines the decision process for choosing suitable analytical models based on data format, boundary characteristics, and hierarchical structure. When data are expressed as success or attempt counts, generalized estimating equations (GEEs) or generalized linear mixed models (GLMMs) are recommended, depending on whether population-averaged or subject-specific effects are of interest. For rate values strictly within the (0,1) interval, standard beta regression is appropriate, whereas data with hierarchical structure may require extensions of beta regression. In cases where observed zeros or ones are infrequent, boundary values may be adjusted using data shrinkage transformations for standard beta models. However, when significant boundary value piling is present, zero-one-inflated beta (ZOIB) regression provides a robust alternative. This framework emphasizes aligning model choice with data properties to ensure valid, interpretable, and reproducible analysis of ART laboratory outcomes

When dealing with boundary values (0/1), Beta regression requires transformation of the boundary observations to fall within the open interval (0,1). GLMMs and GEEs can naturally accommodate observations equal to 0 or 1, but in the presence of significant boundary value piling, the ZOIB model provides a more appropriate framework. Therefore, in practical analyses, researchers should carefully select the most suitable modeling approach based on the study objective, data structure, boundary distribution, and clustering characteristics. Conducting sensitivity analyses across multiple models is also recommended to enhance the reliability of conclusions.

Finally, it is important to emphasize that rigorous study design, careful control of confounders, and transparent reporting practices remain as critical as advanced statistical modeling. Well-designed studies, even when paired with simpler methods, can yield conclusions that are as valid and clinically informative as those obtained from complex models.

Conclusion

This review evaluated current analytical practices for proportion-based laboratory outcomes in ART, identified common methodological pitfalls, and introduced more robust alternatives. As reproductive medicine continues to advance, the statistical methods used to analyze laboratory outcomes must also evolve. Future studies should carefully select and transparently report statistical methods that align with their research objectives, data structures, and outcome characteristics. Importantly, no single approach is universally optimal. Leveraging the complementarity of different models and incorporating cross-validation can enhance the robustness of findings and promote more standardized practices across the field. However, the use of multiple analytical approaches requires careful consideration of multiplicity and the potential for p-hacking. To maintain transparency and methodological rigor, a primary analysis should be pre-specified, with additional models clearly designated as sensitivity analyses. Equally essential is the recognition that rigorous study design and appropriate statistical analysis are interdependent. Thoughtful design choices can reduce reliance on post hoc modeling, while sophisticated models can maximize the value of well-collected data. Together, these refinements will ensure that ART laboratory outcomes are interpreted with accuracy, reproducibility, and clinical relevance, ultimately strengthening their translation into evidence-based decision-making and improved patient care.

Abbreviations

2PN

Two pronuclear (normal fertilization)

AFC

Antral follicle count

AMH

Anti-Müllerian hormone

ART

Assisted reproductive technology

ASRM

American Society for Reproductive Medicine

BMI

Body mass index

CLBR

Cumulative live birth rate

COS

Controlled ovarian stimulation

DOR

Diminished ovarian reserve

GEE

Generalized estimating equations

GLMM

Generalized linear mixed models

ICC

Intra-class correlation (intraclass correlation coefficient)

IQR

Interquartile range

IVF

In vitro fertilization

LPOS

Luteal-phase ovarian stimulation

MAD

Median absolute deviation

MCMC

Markov Chain Monte Carlo

MII

Metaphase II oocyte

mTESE

Microdissection testicular sperm extraction

NGS

Next-generation sequencing

NOA

Non-obstructive azoospermia

OIB

One-inflated beta (regression)

OR

Odds ratio

PCOS

Polycystic ovary syndrome

PFAS

Perfluoroalkyl and polyfluoroalkyl substances

PGT-A

Preimplantation genetic testing for aneuploidy

R

Statistical software (R language)

SD

Standard deviation

VIF

Variance inflation factor

WGA

Whole genome amplification

ZIB

Zero-inflated beta (regression)

ZOIB

Zero-one inflated beta (regression)

Authors’ contributions

C.W. and H-M.C. contributed to the study’s conception and conducted the systematic search. All authors contributed to data analysis and interpretation; drafted the article and revised it critically for content; approved the final version; and agreed to be accountable for all aspects of the work.

Funding

This work is supported by the Canadian Institutes of Health Research (143317). Additionally, this work is supported by the National Science and Technology Council (NSTC 113-2314-B-039-055), the China Medical University Hospital Research Foundation (CMUH114-REC2-164), National Natural Science Foundation of China (No.82474560), and the Program of China Scholarship Council (Grant No.202408370230).

Data availability

No new data were generated or analyzed in support of this research.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.