In general, a complementary part of the studies reporting the regression polynomials that relate AA with age is their comparison with the Hofstetter’s equations. However, studies aimed at the inter-comparison of these regression polynomials are limited. Such inter-comparison is necessary to analyze the trends of AA decrease with age in different populations, age groups, measurement techniques, etc. Accordingly, this study was designed to extensively collect, analyze and compare the reported regression polynomials connecting AA with age and deduce an overall trend. The results, as presented below, revealed that the published studies have reported a linear decrease of AA with age, with few exceptions [6, 52]. However, large variations exist in the rate of annual decrease of AA with age.

The data (except AA values) extracted from the eligible studies have been summarized in Supplementary Table 1 [1, 2, 13, 15–17, 19–21, 27, 28, 35, 4, 36–38, 42–48, 5, 49–53, 55–59, 6, 60–69, 7, 70–79, 9, 84, 87–89, 10–12]. A total of 101 data sets of AA values from 50 studies has been collected. The overall data set consisted of 5433 subjects, with age ranging from 3 to 86 years. In the selected studies, different techniques have been utilized to measure the AA values; among these, the push-up method was most commonly used, followed by the push-down, minus lens to blur, dynamic retinoscopy and autorefractometry. Moreover, the collected dataset comprised of subject from almost all ethnic groups, such as White population (i.e., studies from USA, Australia, Canada, Germany), Black population (i.e., studies from Nigeria, Ghana), East Asian (i.e., studies from China, Japan, Korea) and Indians. Further, subjects from both genders were included and the AA values were measured for each of monocular site individually and for binocular sites. Based on the extensive cohort of the collected data, we postulate that the influence of these factors (i.e., the measurement technique and age, ethnicity and gender of the subjects), which are presumably responsible for the variations in the measured values of AA, may have been normalized.

Figure 2 presents a scatter plot for the AA values as a function of age for 5433 subjects; these data were collected from all included studies. Several characteristic features of AA can be appreciated from Fig. 2. To specify, the AA data shows an overall decreasing trend with age. The values of AA are negligibly small for subjects older than 50 years. Wide variations in the values of AA can be noted for subjects of every age group; such variations are strongly age-dependent, being particularly prominent at the young age. Moreover, a linear regression fitting, illustrated by red pseudocolor, was applied to the complete data set of AA. The slope and intercept of the linear regression equation were − 0.24 and 14.9, respectively; these values closely match to the corresponding values for Hofstetter equation for minimum AA. It is also imperative to highlight that a large number of younger subjects of age below 10 years were present in the data cohort, as represented by the cluster of data points in Fig. 2; Hofstetter equations does not account for such young subjects.

A comparison of the three linear regression lines, representation the trends in the data shown in Fig. 2, with Hofstetter equations [40, 41] has been depicted in Fig. 3. The three trend lines correspond to the minimum, mean and maximum values of AA. It is obvious that the AA values predicted by all three Hofstetter equations are systematically higher than the corresponding values predicted by the regression equations derived in the current study. Perhaps, the most prominent feature of this objective comparison is the nearly perfect match between the minimum of Hofstetter equation and mean regression relation from the present study, where the values for the slope and intercept were − 0.25 vs. -0.24 and 15 vs. 14.9, respectively. For a one-to-one comparison, the Hofstetter equations and regression expressions from the present study are summarized in Supplementary Table 1. The quantitative analyses demonstrated that both the slope and intercept- the two important metrics- of all three Hofstetter equations were higher than that of the relations derived herein. Moreover, the regression relations of the present study cover much wider variations in the values of AA as compared to the Hofstetter equations. Collectively, all these features, summarized in Fig. 3 and Table 2, demonstrated the large differences in the two set of linear equations.

Table 2

Comparison of the linear regression relations derived from the trends in Fig. 3 in the present study with the Hofstetter equations

| Equation from Hofstetter | Equation from this study |

Minimum AA | \({A}_{mini} =15-0.25Age\) | \({A}_{mini} =9-0.15Age\) |

Mean AA | \({A}_{mean}=18.5-0.3Age\) | \({A}_{mean}=14.9-0.24Age\) |

Maximum AA | \({A}_{maxi}=25-0.4Age\) | \({A}_{maxi}=22.4-0.35Age\) |

A comparison of the residual AA for the Hofstetter equation and the regression relation from the present study has been shown in Fig. 4. It may be mentioned that, for a particular age of the subject, the residual AA was defined as the difference between the value of AA predicted by the regression equation and the observed/ measured value of AA. Analyzing the two residual plots, the data in Fig. 4a (regression equation from this study) more symmetric about the origin (i.e., residual AA = 0) as compared to Fig. 4c (regression equation from Hofstetter). In other words, for the Hofstetter equation, a higher number of data points are present below the horizontal line compared to the regression relations of the present study. The data in Fig. 4a and 4c have also been presented in the form of histograms. From Fig. 4b and 4d, the Hofstetter equations generated large number of residual AA with negative values (as highlighted by black arrows for one *bin*), indicating the overestimation of the measured AA. Also, the relatively higher number of data points for the residual AA of ± 2 Diopters indicates better agreement of the regression equation derived in the present study.