Participants with baseline and last follow-up MMSE scores were included from ORIGIN (n = 11,691 mean age, 63.5 years, 65% male, 38% higher education, mean follow-up, 5.9 years) and TRANSCEND (n = 5815 mean age, 66.9 years, 57% male, 34% higher education, mean follow-up, 4.6 years) (Table 1). Participants with baseline and last follow-up MoCA scores were included from COMPASS (n = 17,864, mean age 68.2 years, 78% male, 47% higher education, mean follow-up, 1.9 years) and NAGIVATE-ESUS (n = 7016, mean age, 66.9 years, 62% male, 46% higher education, mean follow-up, 1 year). The mean baseline and follow-up MMSE scores for ORIGIN and TRANSCEND were 27.9 (SD = 2.9) vs. 27.3 (SD = 3.5) and 27.5 (SD = 3.1) vs. 27.4 (SD = 3.5), respectively, while for COMPASS and NAVIGATE-ESUS, the mean baseline and follow-up MoCA scores were 24.5 (SD = 3.9) vs. 24.4 (SD = 4.0) and 23.1 (SD = 5.5) vs. 23.5 (SD = 5.6), respectively (Tables 1 and 2). There was no significant difference in scores between baseline and follow-up in any of the four studies.

The follow-up MMSE distributions in the ORIGIN and TRANSCEND studies were similarly bounded, with most participants having a maximum score of 30 (Fig. 1). The distribution of change scores from baseline, since the models were adjusted for baseline, exhibited a lack of normality as well (Supplemental Fig. 1). However, the absolute skewness greater than 2 and the absolute kurtosis greater than 8 (Table 2) indicated that the distributions were substantially non-normal. Even after country-standardization of MMSE scores, the distributions still showed large absolute skewness and kurtosis values. Applying a square root or log transformation substantially reduced the skewness and kurtosis values. In contrast, the follow-up MoCA distributions in the COMPASS and NAVIGATE-ESUS studies were left-skewed but still considered normal, with most participants having scores close to the maximum (Fig. 1). The absolute skewness and kurtosis were high but still within the range expected for a normal distribution. Applying a square root or log transformation further improved the skewness of the distribution (Table 2).

Histograms for the MMSE of ORIGIN and TRANSCEND studies and the MoCA of COMPASS and NAVIGATE-ESUS studies

In terms of fitting the MLR with transformations, we found that the model performance significantly improved for the MMSE scores, and slightly improved for the MoCA scores when we applied either the square root or log transformation as compared to the untransformed MLR (Table 2). For instance, in ORIGIN with MMSE scores, the AIC decreased from 58,013 for the untransformed MLR to 50,317 with the square root transformation, and to 47,162 with the log transformation. In COMPASS with MoCA scores, the AIC decreased from 84,925 for the untransformed MLR to 82,322 with the square root transformation but increased to 85,861 with the log transformation. The histograms of residuals (Supplemental Figs. 2 and 3) became more normally distributed by reducing the initially high skewness and kurtosis values to a more acceptable range after applying square root or log transformations. This was more evident in the Q-Q plots, where most points aligned closely to the straight line for the cognitive measures with a log transformation (Supplemental Figs. 4 and 5).

SQRT: \(\sqrt+1}-\sqrt+1)-\text}\)

LOG: \(\text\left(\text+1\right)-\text((\text+1)-\text)\)

AIC for Beta-binomial were 46,342, 46,342, 22,387, 81,848, and 34,288 for ORIGIN, TRANSCEND, COMPASS, and NAVIGATE-ESUS, respectively.

AIC for Tobit were 49,095, 23,588, 84,112, and 35,641 for ORIGIN, TRANSCEND, COMPASS, and NAVIGATE-ESUS, respectively.

Next, we compared the MLR with two other regression models not only the model performance but also the treatment effects with the three models had the same unit of measure for the cognitive scores.

In the ORIGIN study, the beta-binomial regression model showed a lysignificant higher MMSE score of 0.38 units (95% CI, 0.11 to 0.66; CI width = 0.55, p = 0.006; AIC = 46,342) in the treatment group as compared to the control group. In contrast, both the untransformed MLR (0.049; 95% CI, − 0.06 to 0.15; CI width = 0.21; p = 0.36; AIC = 58,013) and the Tobit regression model (0.13; 95% CI, − 0.02 to 0.28; CI width = 0.30; p = 0.08; AIC = 49,095) showed no significant treatment effect (Fig. 2). A substantial AIC reduction was found in comparing between the MLR untransformed and the beta-binomial. The intracluster correlation coefficients obtained for the beta-binomial models were 0.090, 0.078, 0.022, and 0.044 for ORIGIN, TRANSCEND, COMPASS, and NAVIGATE-ESUS, respectively. Although treatment effects were insignificant across all models for the other studies, a consistent trend was observed in which all three approaches had similar treatment effect estimates. Moreover, the beta-binomial models consistently exhibited slightly lower AIC values, even when compared to the MLR with transformations.

Between-group mean difference and model performance given by AIC values for the selected methods

Similar results were found in the ORIGIN study with repeated measures, where the beta-binomial and MMRM showed similar treatment effect differences. However, the beta-binomial resulted in a narrower confidence interval width and a smaller AIC (Fig. 3).

Between-group mean difference and model performance given by AIC values for the ORIGIN study with repeated measures

The baseline cognitive score, baseline age, sex, and education level were mostly significantly associated with the dependent variable across the MLR untransformed, the beta-binomial, and the Tobit regression. Except in the NAVIGATE-ESUS study, the association between sex and cognitive scores was only found significant in the beta-binomial, and Tobit regression, but became not significant in the untransformed MLR. The degree of the associations between the covariates and the outcomes varied with the different approaches but the direction of the association remained consistent (Supplemental Table 1). The assumption of linearity was met, with points mostly hovering around the horizontal line for the generalized linear regression model, beta-binomial, and Tobit regression (Supplemental Figs. 6–7).