purpose. It is often difficult to estimate parameters from individual clinical data because of noisy or incomplete measurements. Nonlinear mixed-effects (NLME) modeling provides a statistical framework for analyzing population parameters and the associated variations, even when individual data sets are incomplete. The authors demonstrate the application of NLME by analyzing data from the MNREAD, a continuous-text reading-acuity chart.

methods. The authors analyzed MNREAD data (measurements of reading speed vs. print size) for two groups: 42 adult observers with normal vision and 14 patients with age-related macular degeneration (AMD). Truncated sets of MNREAD data were generated from the individual observers with normal vision. The MNREAD data were fitted with a two-limb function and an exponential-decay function using an individual curve-fitting approach and an NLME modeling approach.

results. The exponential-decay function provided slightly better fits than the two-limb function. When the parameter estimates from the truncated data sets were used to predict the missing data, NLME modeling gave better predictions than individual fitting. NLME modeling gave reasonable parameter estimates for AMD patients even when individual fitting returned unrealistic estimates.

conclusions. These analyses showed that (1) an exponential-decay function fits MNREAD data very well, (2) NLME modeling provides a statistical framework for analyzing MNREAD data, and (3) NLME analysis provides a way of estimating MNREAD parameters even for incomplete data sets. The present results demonstrate the potential value of NLME modeling for clinical vision data.

^{ 1 }

^{ 2 }

^{ 3 }

^{ 4 }Here we present nonlinear mixed-effects (NLME) modeling

^{ 5 }

^{ 6 }

^{ 7 }of MNREAD data sets as a robust method of analyzing and summarizing reading speed data, even when the data are incomplete. Thereby, we demonstrate the potential value of NLME modeling for clinical vision data.

^{ 8 }Precise measurements of reading performance provide valuable information for assessment

^{ 9 }

^{ 10 }

^{ 11 }

^{ 12 }and rehabilitation evaluation.

^{ 13 }

^{ 14 }

^{ 15 }Reading speed is measured as a function of print size in the MNREAD test.

^{ 16 }

^{ 17 }

^{ 18 }

^{ 19 }

^{ 20 }Research has shown that reading speed increases sharply with print size at small print, plateaus at mid-range print, and decreases gradually at large print (>1.4 logMAR or 2° of visual angle; see Mansfield and Legge

^{ 21 }for details on print size definitions and conversions).

^{ 22 }

^{ 23 }

^{ 24 }

^{ 17 }

^{ 18 }

^{ 20 }It has 19 sentences at print sizes from −0.5 to 1.3 logMAR in 0.1-log steps at a standard viewing distance of 40 cm, capturing the sharp rising part of the reading speed versus print size curve at small print and the asymptote at the mid-range print (Fig. 1) . When the MNREAD chart is used at a closer viewing distance, the measured reading speed curve may also exhibit a decline at large print size.

^{ 22 }Three key parameters are used to summarize the function

^{ 17 }

^{ 20 }: maximum reading speed (MRS), critical print size (CPS, the smallest print size at which MRS can be attained), and reading acuity (RA, the smallest print size that can be resolved). These parameters have revealed important differences in reading performance among patients with age-related macular degeneration (AMD),

^{ 19 }retinitis pigmentosa,

^{ 25 }and dyslexia

^{ 26 }compared with people with normal vision. Effects of children’s grade level and age on these MNREAD parameters have also been found

^{ 12 }(Cheung S-H, et al.

*IOVS*2006;47:ARVO E-Abstract 5830). Robust methods for estimating and analyzing the MNREAD curve provide the tools for answering important research questions about reading performance.

^{ 19 }pointed out that because of the rapid deterioration in reading speed at small print and noisier reading speed measurements near the acuity limit, the estimation of the rising slope can be poor and, thus, the accuracy and precision of the CPS estimates can be compromised. Moreover, as shown in Figure 1A , the elbow of the two-limb function fit is usually to the left of the observed data, tending to underestimate the CPS. In this study, we introduce an exponential-decay function to provide smooth fits for the reading speed versus print size data (Fig. 1B) . Similar smooth nonlinear functions have also been used by other researchers

^{ 27 }(Massof RW.

*IOVS*2003;44:ARVO E-Abstract 1284).

^{ 28 }parameter estimation may be difficult, or even impossible, when one person’s data set is noisy or incomplete. For instance, acuity reduction or inadequate testing time may result in reading speed measurements for a compressed range of print sizes. Appropriate grouping of MNREAD data, for example by diagnosis, and NLME modeling of the MNREAD curves from appropriately defined groups of patients can be a method for estimating parameters if data sets are incomplete. In clinical settings, when incomplete MNREAD data are collected, the new data set can be grouped with existing data sets from patients of similar conditions, and NLME modeling can be applied. We will illustrate NLME modeling of an incomplete data set produced by truncating complete sets of normal data and a data set from a group of patients with AMD.

*IOVS*2005;46:ARVO E-Abstract 4784). Their ages ranged from 19 to 65 years (mean, 33 ± 14 years). Best-corrected binocular visual acuity measured with the Lighthouse distance acuity chart (Optelec, US Inc., Vista, CA) was better than 20/20 for all observers in this group. They were all fluent English speakers. The second group consisted of 14 patients with a primary ocular diagnosis of AMD; their ages ranged from 65 to 87 years (mean, 78.6 ± 5.6 years). They were recruited from the Low Vision Center of the University of Minnesota and Vision Loss Resources (Minneapolis, MN), for another study in the laboratory.

^{ 29 }All observers in this second group were native English speakers. Our AMD participants were considerably older than our normal-vision participants, but this study did not include any direct comparison between the two groups. Informed consent was obtained from each observer before testing. The protocol of this study followed the tenets of the Declaration of Helsinki and was approved by the Institutional Review Board of the University of Minnesota.

*two-limb*function, composed of two straight lines:

*x*represents the print size in logMAR,

*f*(

*x*) the corresponding reading speed in log words per minute (logWPM), θ

_{1}the maximum reading speed (MRS) in logWPM, exp(θ

_{2}) the slope of the first limb (the slope of the second limb is 0), and θ

_{3}the CPS

_{TL}. θ

_{3}is the intersection of the two straight lines. The slope of the first limb was parametrized as exp(θ

_{2}) to ensure positivity. The three parameters (θ

_{1}, θ

_{2}, and θ

_{3}) were estimated using optimization procedures either through individual curve fitting or nonlinear mixed-effects (NLME) modeling.

^{ 5 }

^{ 6 }

^{ 7 }

*exponential-decay*function

*x*represents the print size in logMAR,

*g*(

*x*) the corresponding reading speed in logWPM, φ

_{1}the MRS in logWPM, exp(φ

_{2}) the rate of change in reading speed as a function of print size, φ

_{3}the print size at which reading speed is 0 logWPM (i.e., 1 WPM). The rate of change in the function was parametrized as exp(φ

_{2}) to ensure positivity. The three parameters (φ

_{1}, φ

_{2}, and φ

_{3}) were estimated using optimization procedures through either individual curve fitting or NLME modeling.

^{ 5 }

^{ 6 }

^{ 7 }The CPS

_{ED}was defined as the print size that yielded a criterion percentage of the MRS. Linear regressions were calculated for the CPS

_{ED}in the exponential-decay function at five different criterion reading speeds (75%–95% of the MRS) against the CPS

_{TL}in the two-limb function.

^{ 5 }

^{ 6 }

^{ 7 }Fitting of NLME models involves an iterative process, in which the means of the different groups (fixed effects) and the variances within the groups (random effects) are estimated. Parameters for each individual data set are then estimated given the estimated means and variances. Details of the NLME models used in this study are included in the Appendix.

^{ 30 }

^{ 31 }in most of our statistical analyses with 10,000 resampling. All reported intervals (in parentheses) were 95% bootstrap confidence intervals (CI

_{bootstrap}) estimated using the bias-corrected and accelerated percentile method (BC

_{a}).

^{ 32 }

*x*-axis) and the exponential-decay function (

*y*-axis) from the data set of the 42 observers with normal vision. RMS error from the two functions clustered around the equality line, with more data points below and to the right of the equality line, indicating that the exponential-decay function provided better fits (smaller errors). The average differences (95% CI

_{bootstrap}in parentheses) between the RMS errors from the two functions were 0.0079 (0.0026, 0.0138) logWPM and 0.0061 (0.0007, 0.0121) logWPM for regular contrast polarity (i.e., black on white) and reversed contrast polarity respectively. Although the differences were small, they were statistically significant (

*ps*

_{bootstrap}≤ 0.05).

_{TL}instead of the CPS

_{ED}. As shown in Figure 3A , residuals from TL fits tended to be negative around the CPS

_{TL}, indicating that the data tended to lie below the fitted curve. The two right panels (B and D) of Figure 3show histograms of the residuals for print sizes within 0.1 logMAR (i.e., one line on the MNREAD chart) of the CPS

_{TL}in the normal-vision data set. The means of the residuals around the CPS

_{TL}were −0.054 (−0.064, −0.045) logWPM and 0.016 (0.006, 0.027) logWPM for TL fits and ED fits, respectively. In other words, the TL fits overestimated the reading speed, whereas the ED fits underestimated it, but to a lesser extent, near the critical print size.

_{ED}) versus Critical Print Size from TL Fits (CPS

_{TL})

*R*

^{2}for linear regressions of CPS

_{ED}for different reading speed criteria (75%–95% of the MRS) versus CPS

_{TL}for regular polarity. The linear regression results for the reverse polarity data set were similar to those for the regular polarity data set and are not shown here. Not surprisingly, the correlations between CPS

_{ED}for different criteria and CPS

_{TL}were high (adjusted

*R*

^{2}varies from 0.673 to 0.853 for regular polarity and from 0.653 to 0.765 for reverse polarity). Although regression with a CPS

_{ED}criterion of 75% gave the highest adjusted

*R*

^{2}values, CPS

_{ED}criterion of 85% yielded slopes that were closest to 1. A slope of ∼1 makes the conversion between CPS

_{ED}and CPS

_{TL}straightforward as a mere offset (the

*y*-intercept). For instance, given that the slope is close to 1 for CPS

_{ED}criteria of 80% and 85%, the

*y*-intercept is approximately the average difference between the CPS

_{ED}and CPS

_{TL}. A CPS

_{ED}criterion of 80% is a good compromise for high adjusted

*R*

^{2}and slope ∼1. Figure 4shows the data and the regression line for CPS

_{ED}with a criterion level of 80% versus CPS

_{TL}for the regular polarity data set.

_{NLME}) against the RMS errors from individual curve fitting (RMS

_{IND}) with the exponential-decay function. RMS errors fall almost perfectly on the equality line. The mean differences of RMS

_{NLME}− RMS

_{IND}were 0.0025 (0.0017, 0.0040) logWPM for regular polarity and 0.0020 (0.0016, 0.0029) logWPM for reverse polarity.

_{NLME}− RMS

_{IND}were 0.0053 (0.0035, 0.0091) logWPM for regular polarity and 0.0084 (0.0058, 0.0121) logWPM for reverse polarity. The fitted curves for the regular polarity data sets are shown in Figure 6 . Given a relatively complete data set, individual curve fitting will often have better fits because estimation of individual parameter sets from NLME modeling is constrained by the estimated means (fixed effects) and variances (random effects) of the groups.

_{IND}>6 logWPM, are not shown on the plot. These extreme values were usually a consequence of problematic predictions of the missing data on the rising part of the reading speed curve.) We used median values to summarize the RMS prediction errors to limit the biases introduced by the extreme values. Median differences of RMS

_{NLME}− RMS

_{IND}were −0.09 (−0.16, −0.03) logWPM for regular polarity and −0.06 (−0.10, −0.04) logWPM for reverse polarity.

_{ED}(80% of MRS) from individual curve fitting were 7.21 logWPM and 19.82 logMAR, respectively, whereas those from NLME modeling were 2.15 logWPM and 2.20 logMAR. Table 2shows the estimated MRS and CPS

_{ED}(80% of MRS) from individual curve fitting and NLME modeling for each of the 14 AMD patients.

_{ED}of 80% of MRS yielded print sizes slightly larger (0.07 logMAR) than the CPS

_{TL}values from the two-limb fit. Adopting a more conservative CPS

_{ED}criterion of 90% MRS produced print sizes averaging 0.15 logMAR larger than the CPS

_{TL}. When a clinician prescribes a reading magnifier for a patient with visual impairment, the goal is usually for the magnified print to be at least as large as the CPS for the patient to achieve MRS.

^{ 33 }

^{ 34 }Whittaker and Lovie-Kitchin defined “acuity reserve” as the ratio of the print size intended for reading to the acuity print size.

^{ 35 }If the acuity reserve is insufficient, reading speed will be compromised. The CPS

_{TL}, estimated from the two-limb fit, often underestimates the print size required for reading at the MRS. As a result, magnifier prescription based on the CPS

_{TL}estimate may underestimate the magnification required for best reading performance. Magnifier prescription based on the CPS

_{ED}, with the criterion of either 80% or 90% of MRS, is more likely to yield adequate reading performance than magnifier prescription based on the CPS

_{TL}from the two-limb fit.

_{ED}criterion, depending on circumstances. For example, a higher criterion could be used for fluent or maximum reading, whereas a lower criterion could be used for spot reading.

^{ 35 }A lower criterion would correspond to a smaller print size, lower magnification, and perhaps slower reading. A higher criterion would correspond to higher magnification and closer approximation to maximum reading speed.

^{ 36 }suggested a fixed-acuity-reserve (0.3 log unit) method to determine the magnification needed by low-vision patients. Table 3shows the CPS

_{ED}(80% of MRS) along with the suggested print size using the fixed-acuity-reserve method (PS

_{reserve}) and their corresponding reading speeds. The CPS

_{ED}and the PS

_{reserve}are also plotted in Figure 7 . In general, the CPS

_{ED}suggests a larger print size compared with the fixed-acuity-reserve method. It should be noted that for some patients, the CPS

_{ED}is larger than 1.5 logMAR and might already have reached the declining portion of the reading-speed curve for very large print sizes. However, it is unclear whether AMD patients experience the same downturn as normally sighted people or whether it sets in at the same logMAR values. In most cases, magnifier prescription based on the CPS

_{ED}may result in better reading performance than the fixed-acuity reserve method. Magnification based on the CPS

_{ED}provides a good starting point in the performance trial process of magnifier prescription. Thus, it helps reduce the consulting time and frustration for trial and error in the magnifier prescription process.

*IOVS*2005;46:ARVO E-Abstract 4589). In its current form, NLME modeling may not be easily accessible to many clinical practitioners. However, it is possible to make NLME modeling through a Web page interface. A Web-based version of the statistical software used is available (http://rweb.stat.umn.edu/Rweb/), which means model fitting can be performed on any regular computer with Web access.

^{ 37 }or for different observers from different populations and different patient groups (Kallie CS.

*IOVS*2005;46:ARVO E-Abstract 4589), in psychophysical studies.

^{ 5 }

^{ 6 }:

*y*

_{ ij }is the reading speed in logWPM at the

*j*th print size

*x*

_{ ij },

*f*(

*x*

_{ ij },

**Φ**

_{ i }) and is either the

*two-limb*function or the

*exponential-decay*function with parameter vector

**Φ**

_{ i }for the

*i*th observer. ε

_{ ij }is the residual error, which follows the normal distribution with mean zero and variance σ

^{2}. The parameter vector

**Φ**

_{ i }determines the reading speed curve for each observer, and is modeled as:

**β**is the mean parameter vector and b

_{i}is the vector for the random effects for the

*i*th observer. b

_{i}is normally distributed with a zero mean vector and variance-covariance matrix

**Ψ**.

**Φ**

_{i}can be formulated by replacing the mean parameter vector

**β**with any fixed-effect structure of interest and the random-effect vector b

_{i}with the corresponding more complex random-effect structure. In our first analysis, we had fixed effects of contrast polarity and random effects of both contrast polarity and individual variations. The same model was also fitted to the truncated data sets. Another NLME model with a mean parameter vector and random effects of individual variations was used to analyze the data sets from our patients with AMD. After fitting an NLME model, the parameter vector

**Φ**

_{i}for each observer was estimated with best linear unbiased predictor, or BLUP.

^{ 38 }

^{ 39 }

^{ 40 }with the NLME library.

^{ 7 }The Nelder-Mead simplex algorithm

^{ 41 }was used to find the parameter combinations that minimize the residual sum of squares in individual curve fitting. R scripts for fitting NLME models to our data sets are available for download at http://vision.psych.umn.edu/∼gellab/mnread/.

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

**Figure 3.**

**Figure 3.**

Threshold (% of MRS) | y-Intercept | Slope | Adjusted R ^{2} |
---|---|---|---|

75 | 0.0374 (0.0233, 0.0505) | 0.956 (0.857, 1.075) | 0.853 (0.729, 0.914) |

80 | 0.0670 (0.0518, 0.0816) | 0.996 (0.882, 1.131) | 0.836 (0.704, 0.903) |

85 | 0.1041 (0.0867, 0.1213) | 1.047 (0.905, 1.200) | 0.806 (0.655, 0.886) |

90 | 0.1547 (0.1336, 0.1760) | 1.116 (0.930, 1.299) | 0.757 (0.576, 0.858) |

95 | 0.2387 (0.2108, 0.2679) | 1.230 (0.968, 1.469) | 0.673 (0.445, 0.812) |

**Figure 4.**

**Figure 4.**

**Figure 5.**

**Figure 5.**

**Figure 6.**

**Figure 6.**

**Figure 7.**

**Figure 7.**

Individual Curve Fitting | NLME Modeling | |||||
---|---|---|---|---|---|---|

MRS (logWPM) | CPS_{ED} (logMAR) | MRS (logWPM) | CPS_{ED} (logMAR) | |||

AMD1 | 2.07 | 0.52 | 2.08 | 0.54 | ||

AMD2 | 2.32 | 1.27 | 2.26 | 1.18 | ||

AMD3 | 1.68 | 0.25 | 1.71 | 0.32 | ||

AMD4 | 3.19 | 2.59 | 2.36 | 1.43 | ||

AMD5 | 2.09 | 0.73 | 2.10 | 0.75 | ||

AMD6 | 5.25 | 8.35 | 2.17 | 2.09 | ||

AMD7 | 2.24 | 0.79 | 2.20 | 0.66 | ||

AMD8 | 2.28 | 0.52 | 2.28 | 0.54 | ||

AMD9 | 3.16 | 5.57 | 2.09 | 2.77 | ||

AMD10 | 7.21 | 19.82 | 2.15 | 2.20 | ||

AMD11 | 2.02 | 0.79 | 2.03 | 0.81 | ||

AMD12 | 29.82 | 531.70 | 1.97 | 2.93 | ||

AMD13 | 2.19 | 1.26 | 2.17 | 1.23 | ||

AMD14 | 1.64 | 2.00 | 2.08 | 2.48 |

NLME Modeling | Fixed-Acuity-Reserve Method | |||||
---|---|---|---|---|---|---|

CPS_{ED} (logMAR) | Reading Speed (logWPM) | PS_{reserve} (logMAR) | Reading Speed (logWPM) | |||

AMD1 | 0.54 | 1.98 | 0.68 | 2.06 | ||

AMD2 | 1.18 | 2.16 | 0.93 | 2.00 | ||

AMD3 | 0.32 | 1.61 | 0.58 | 1.71 | ||

AMD4 | 1.43 | 2.26 | 1.13 | 2.04 | ||

AMD5 | 0.75 | 2.00 | 0.72 | 1.98 | ||

AMD6 | 2.09 | 2.07 | 1.52 | 1.74 | ||

AMD7 | 0.66 | 2.10 | 0.40 | 1.95 | ||

AMD8 | 0.54 | 2.18 | 0.60 | 2.22 | ||

AMD9 | 2.77 | 2.00 | 1.36 | 1.40 | ||

AMD10 | 2.20 | 2.05 | 1.22 | 1.61 | ||

AMD11 | 0.81 | 1.94 | 0.68 | 1.85 | ||

AMD12 | 2.93 | 1.88 | 1.42 | 1.46 | ||

AMD13 | 1.23 | 2.07 | 0.92 | 1.86 | ||

AMD14 | 2.48 | 1.98 | 1.71 | 1.46 |