February 2013
Volume 54, Issue 2
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Eye Movements, Strabismus, Amblyopia and Neuro-ophthalmology  |   February 2013
Estimation of Spontaneous Blinking Main Sequence in Normal Subjects and Patients with Graves' Upper Eyelid Retraction
Author Affiliations & Notes
  • Denny M. Garcia
    From the Department of Ophthalmology, Otorhinolaryngology, and Head and Neck Surgery of the School of Medicine of Ribeirão Preto, University of São Paulo, São Paulo, Brazil; and the
  • José Carlos Barbosa
    Department of Exact Sciences, UNESP, Universidade Estadual Paulista, São Paulo, Brazil.
  • Carolina T. Pinto
    From the Department of Ophthalmology, Otorhinolaryngology, and Head and Neck Surgery of the School of Medicine of Ribeirão Preto, University of São Paulo, São Paulo, Brazil; and the
  • Antonio Augusto V. Cruz
    From the Department of Ophthalmology, Otorhinolaryngology, and Head and Neck Surgery of the School of Medicine of Ribeirão Preto, University of São Paulo, São Paulo, Brazil; and the
  • Corresponding author: Antonio Augusto V. Cruz, Department of Ophthalmology, Otorhinolaryngology, and Head and Neck Surgery, School of Medicine of Ribeirão Preto, University of São Paulo – Brazil, Av. Bandeirantes, 3900, 14049-900 – Ribeirão Preto, SP – Brazil; aavecruz@fmrp.usp.br
Investigative Ophthalmology & Visual Science February 2013, Vol.54, 1434-1442. doi:10.1167/iovs.12-11452
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      Denny M. Garcia, José Carlos Barbosa, Carolina T. Pinto, Antonio Augusto V. Cruz; Estimation of Spontaneous Blinking Main Sequence in Normal Subjects and Patients with Graves' Upper Eyelid Retraction. Invest. Ophthalmol. Vis. Sci. 2013;54(2):1434-1442. doi: 10.1167/iovs.12-11452.

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Abstract

Purpose.: We quantified the main sequence of spontaneous blinks in normal subjects and Graves' disease patients with upper eyelid retraction using a nonlinear and two linear models, and examined the variability of the main sequence estimated with standard linear regression for 10-minute periods of time.

Methods.: A total of 20 normal subjects and 12 patients had their spontaneous blinking measured with the magnetic search coil technique when watching a video during one hour. The main sequence was estimated with a power-law function, and with standard and trough the origin linear regressions. Repeated measurements ANOVA was used to test the mean sequence stability of 10-minute bins measured with standard linear regression.

Results.: In 95% of the sample the correlation coefficients of the main sequence ranged from 0.60 to 0.94. Homoscedasticity of the peak velocity was not verified in 20% of the subjects and 25% of the patients. The power-law function provided the best main sequence fitting for subjects and patients. The mean sequence of 10-minute bins measured with standard linear regression did not differ from the one-hour period value. For the entire period of observation and the slope obtained by standard linear regression, the main sequence of the patients was reduced significantly compared to the normal subjects.

Conclusions.: Standard linear regression is a valid and stable approximation for estimating the main sequence of spontaneous blinking. However, the basic assumptions of the linear regression model should be examined on an individual basis. The maximum velocity of large blinks is slower in Graves' disease patients than in normal subjects.

Introduction
The determination of the relationship between amplitude and maximum velocity, also known as main sequence, is accepted widely as an important parameter for the characterization of the neural control of eye saccadic movements. 1 Evinger probably was the first to apply the same concept to the study of eyelid movements, such as vertical lid saccades and blinks. 2,3  
Spontaneous blinking main sequence usually is assumed to be indicated by the slope of a simple linear regression line between the closing phase amplitude of the lid and its maximum or peak velocity. 4 However, the validity of using a simple linear regression line for this purpose depends on several assumptions that have not yet been discussed to our knowledge. The type of model chosen to fit the regression line also requires additional validation. 
In our study, we used two different linear and one nonlinear regression model to determine the goodness of fit of the main sequence of spontaneous blinks in normal subjects and patients with upper eyelid retraction caused by Graves' orbitopathy, and examined the common assumptions that the linear regression models require. 
Methods
This research adhered to the tenets of the Declaration of Helsinki. 
Subjects
A total of 20 normal controls and 12 patients with lid retraction resulting from Graves' orbitopathy had their spontaneous blinking activity measured when watching a one-hour video. The sample of normal subjects consisted of seven men and 13 women aged 28 to 64 years (41.2 ± 11.3 years, mean ± SD). None of the subjects had any history of eye disease or ocular symptoms. Their mid pupil upper lid margin distance ranged from 2.5 to 4.5 mm (3.5 ± 0.5 mm, mean ± SD).The patients were 3 men and 9 women aged 34 to 61 years (47.6 ± 8.2 years, mean ± SD). At the time of testing, 9 patients were euthyroid, 2 were using antithyroid drugs, and 1 had hypothyroidism. All presented with upper eyelid retraction. The upper mid pupil lid margin distances ranged from 4.6 to 9.0 mm (6.1 ± 1.2 mm, mean ± SD). 
Blink Measurement
A magnetic search coil was used to register the upper eyelid movements continuously during the experiment. 5 The subjects were seated comfortably with the head stabilized on a chin rest in a weak magnetic field. A small coil (3.8 mm diameter, 30 turns, 30 mg, copper wire 0.102 mm in diameter) was taped to the center of the pretarsal area of the upper eyelid. The coil did not impair lid movement, and subjects became unaware of the coil shortly after its application. During blinking, as the eyelid slid over the curved surface of the eye, the coil produced a current proportional to the angle between the coil and the magnetic field, and consequently proportional to lid angular position. The recordings were low-pass filtered at 10 kHz, amplified 20,000 times, digitized with 12-bit precision, and sampled at 200 Hz with a computer driven system (Remel Laboratories, San Antonio, TX), that resolved spatial changes in lid rotations as small as 0.1° (equivalent to a linear lid motion of 0.02 mm) every 5 ms. The magnetic search coil output current and lid angular position were calibrated by measuring the angle of lid rotation with a protractor, while a fine wire was placed perpendicular to the eyelid margin at the site of the coil. The experimenter centered the protractor over the point that was the projected center of rotation of the wire. Spontaneous blinks then were recorded continuously for one hour while the subjects watched the same video (a Brazilian commercial movie). 
Data Analysis
A specific program developed in C# language was used to analyze the data. An algorithm based on the derivatives of the blink signal detected the blinks automatically. In addition, the data retrieved were checked manually to avoid mistakes. For each blink, registered during the one-hour period of observation the software calculated the amplitude (degrees) and its respective maximum velocity (degree/s) of the closing phase of the movements. We also divided the 60 minutes of data into six 10-minute bins and analyzed amplitude–maximum velocity relationship for each bin. 
The slope of the regression lines describing the relationship between amplitude and maximum velocity (main sequence) was calculated with 3 different models: (1) a nonlinear function through the origin (power-law) (y = cxd ), (2) standard linear regression (y = a + bx), and (3) linear regression through the origin (y = bx). Group comparisons (main sequence for one-hour period, mean amplitude, and velocity) were performed using the Mann-Whitney nonparametric test. Two-way repeated measures ANOVA with post-hoc Tukey's test (group versus 10-minute bins) was used to assess the stability of the main sequence over time. Homogeneity of the dependent variables (homoscedasticity) was examined using standardized residuals plots. All statistical procedures were done using Microcal Origin 8.0 Pro (OriginLab, Northampton, MA) and JMP SAS 10.0 (SAS Institute, Inc., Cary, NC). 
Results
Overall, a total of 22,758 blinks was recorded for the subjects and 12,173 for the patients. Table 1 lists the number of blinks, range and mean blink amplitude, and maximum velocity obtained during the entire time of observation. We also tabulated the percentage of movements near the origin (0°–10° of amplitude). As shown in Table 1, the amplitude range was quite variable, and the number of movements with amplitudes between 0° and 10° was quite small. 
Table 1. 
 
Number of Blinks, Amplitude and Maximum Velocity Values Obtained during One Hour of a Video Observation
Table 1. 
 
Number of Blinks, Amplitude and Maximum Velocity Values Obtained during One Hour of a Video Observation
Group ID N Percentage of Blinks with Amplitude up to 10°
Amplitude (°)
Maximum Velocity (°/s)
Min Max Mean Min Max Mean
Subjects 1 1183 0.34 7.2 60.3 26.3 174.6 1449.9 550.6
2 830 6.02 6.7 49.0 18.9 117.8 2191.6 408.9
3 515 0.39 6.5 52.7 31.3 140.5 1694.0 674.2
4 1023 1.76 7.5 72.4 31.1 191.2 1957.2 694.3
5 2538 0.43 8.7 49.8 22.8 221.9 1305.3 566.0
6 2048 0.00 10.6 58.8 37.6 224.0 1713.5 766.5
7 945 0.11 9.5 59.9 41.4 213.1 1304.0 708.0
8 463 1.94 6.1 52.0 27.4 120.1 1177.5 476.1
9 877 0.00 10.8 52.5 32.2 296.5 1263.1 746.2
10 501 0.20 7.9 66.2 34.9 169.3 2045.4 700.8
11 572 0.52 6.4 69.5 27.5 129.7 2097.6 706.5
12 2463 0.04 9.6 64.6 36.3 166.3 1356.2 455.1
13 1393 0.00 11.8 53.1 39.0 280.3 1604.9 1101.5
14 1767 1.13 5.5 43.5 20.5 162.1 1133.2 417.2
15 357 2.24 7.0 38.6 21.5 108.8 730.7 401.7
16 1511 2.05 6.9 43.8 16.2 155.1 770.2 378.8
17 1474 0.41 6.0 40.1 31.4 180.6 929.5 586.3
18 1180 0.51 9.1 46.5 26.4 118.1 1183.4 492.3
19 669 0.30 8.7 42.5 23.4 172.4 1142.8 564.1
20 449 3.79 6.4 53.7 22.9 85.9 830.7 327.9
Patients 1 2258 0.18 9.0 43.9 26.3 169.2 1023.0 599.3
2 663 4.22 5.3 28.9 17.0 123.1 660.1 341.3
3 521 5.57 5.7 30.5 16.7 76.1 683.7 227.1
4 1143 0.00 15.2 52.8 30.9 186.1 1079.5 613.9
5 655 0.00 14.7 42.8 33.0 201.2 742.9 521.7
6 236 0.85 8.4 90.2 52.1 158.0 1741.5 684.7
7 666 2.40 6.8 32.1 18.3 113.5 466.0 264.1
8 1298 1.77 5.8 38.3 18.8 153.6 730.5 325.2
9 2189 0.09 8.7 44.3 19.8 168.1 839.7 345.9
10 174 1.15 8.4 49.3 27.9 208.8 932.8 556.9
11 1070 0.00 13.5 49.2 31.3 168.9 862.6 443.8
12 1300 2.85 5.4 39.8 19.9 173.7 772.1 387.1
Most amplitude and maximum velocity distributions were symmetric with skewness values between −1 and +1 (Fig. 1). There was no significant difference in median amplitude between subjects (27.5°) and patients (23.1°, U = 163, P = 0.1793). However, the median maximum velocity of the subjects (565.0°/s) was higher than that of the patients (415.5°/s, U = 141, P = 0.0279). 
Figure 1. 
 
Distribution of amplitude (left) and maximum velocity (right) of a subject.
Figure 1. 
 
Distribution of amplitude (left) and maximum velocity (right) of a subject.
The relationship between amplitude and peak velocity was linear in almost all subjects, including patients. In both groups the correlation coefficients ranged from 0.60 to 0.94. Linearity was poor in only one individual of each group (r = 0.35 and 0.40). Both had a small blink amplitude range, with the data tending to assume a cluster configuration (Fig. 2). Analysis of the residuals from the linear model showed that the assumption of homoscedasticity of the peak velocity was violated in 20% of the subjects and 25% of the patients. The standardized residuals of these regression lines were not distributed evenly around the zero level (Fig. 3). 
Figure 2. 
 
Maximum velocity and amplitude relationship. (AD) Normal subjects. Two patients are shown in (E, F). No linear relationship was observed in only one subject ([D], r = 0.35) and one patient ([F], r = 0.4).
Figure 2. 
 
Maximum velocity and amplitude relationship. (AD) Normal subjects. Two patients are shown in (E, F). No linear relationship was observed in only one subject ([D], r = 0.35) and one patient ([F], r = 0.4).
Figure 3. 
 
Maximum velocity and amplitude relationship. (A, B) Subjects 9 and 12. (C, D) Patients 4 and 6. Left: maximum velocity and amplitude relationship. Right: standardized residuals plot. Subject 12 and patient 6 had their homoscedasticity assumption violated.
Figure 3. 
 
Maximum velocity and amplitude relationship. (A, B) Subjects 9 and 12. (C, D) Patients 4 and 6. Left: maximum velocity and amplitude relationship. Right: standardized residuals plot. Subject 12 and patient 6 had their homoscedasticity assumption violated.
When the data were fitted to each of the three different models, it was apparent that the nonlinear power function provided a better fit for all subjects and patients, especially those whose maximum velocity variance increased with blinking amplitude (Table 2). In these cases, the linear fitting through the origin consistently underestimated the slope of the main sequence. For the eight (40%) subjects whose data points showed good linearity, each of the three models was adequate for data fitting. The same occurred in 58% of patients (Fig. 4). 
Figure 4. 
 
Maximum velocity and amplitude relationship fitted by the three models. No difference between the models in ([A, B]; normal subjects) and ([E], patient). In subjects (C, D) and patient (F), the power-law allowed a better fitting followed by the linear model with intercept.
Figure 4. 
 
Maximum velocity and amplitude relationship fitted by the three models. No difference between the models in ([A, B]; normal subjects) and ([E], patient). In subjects (C, D) and patient (F), the power-law allowed a better fitting followed by the linear model with intercept.
Table 2. 
 
Parameters of Three Different Models Used to Fit the Relationship between Amplitude and Maximum Velocity
Table 2. 
 
Parameters of Three Different Models Used to Fit the Relationship between Amplitude and Maximum Velocity
Group ID Model 1: y = a + b*x Model 2: y = b*x Model 3: y = c*xd
R M S R M S R M S
Subjects 1 0.92 −89.1 24.3 93.3 0.850 21.3 97.7 0.836 12.6 1.151 93.2 0.851
2 0.77 −3.3 21.8 112.2 0.587 21.7 112.2 0.757 21.1 1.005 74.7 0.757
3 0.89 −137.0 25.9 114.0 0.787 21.8 119.4 0.767 9.1 1.248 113.4 0.789
4 0.93 27.1 21.5 119.1 0.856 22.2 119.6 0.855 26.1 0.956 119.1 0.857
5 0.82 10.6 24.4 70.9 0.677 24.8 70.9 0.677 24.5 1.003 71.0 0.677
6 0.81 −296.9 28.2 125.4 0.662 20.6 134.4 0.612 3.4 1.491 122.4 0.678
7 0.82 −272.8 23.7 121.9 0.665 17.3 130.6 0.615 3.2 1.447 121.2 0.669
8 0.86 −29.6 18.5 88.5 0.731 17.5 88.8 0.729 11.5 1.123 87.8 0.735
9 0.91 86.6 20.5 72.5 0.832 23.0 75.3 0.819 34.8 0.883 72.4 0.833
10 0.78 −265.7 27.7 207.7 0.607 20.6 218.6 0.565 3.1 1.516 200.7 0.633
11 0.94 −147.4 31.0 115.8 0.878 26.3 126.2 0.855 12.2 1.219 112.9 0.884
12 0.81 −54.2 14.0 105.9 0.657 12.6 106.9 0.650 5.5 1.222 104.1 0.669
13 0.90 −193.4 33.2 91.9 0.805 28.4 95.9 0.787 13.0 1.211 91.5 0.806
14 0.69 90.7 15.9 103.4 0.480 20.0 106.7 0.447 38.7 0.789 103.2 0.482
15 0.86 59.2 15.9 63.1 0.744 18.5 65.4 0.725 30.7 0.840 62.6 0.748
16 0.35 237.3 8.7 87.7 0.123* 22.6 102.8 −0.206* 128.2 0.391 87.7 0.123
17 0.76 −72.4 21.0 69.6 0.575 18.7 70.2 0.568 8.9 1.214 69.1 0.582
18 0.83 −179.0 25.4 111.1 0.686 19.0 118.9 0.640 4.0 1.461 108.6 0.699
19 0.88 −45.6 26.1 82.9 0.772 24.2 83.6 0.769 17.9 1.093 82.8 0.773
20 0.61 109.5 9.5 99.8 0.367 13.8 106.0 0.286 43.5 0.650 98.3 0.386
Patients 1 0.85 17.5 22.1 68.0 0.717 22.7 68.0 0.717 25.9 0.961 67.9 0.718
2 0.81 20.9 18.8 57.0 0.653 20.0 57.2 0.651 23.5 0.943 57.1 0.652
3 0.76 −94.8 19.2 65.5 0.582 13.9 69.1 0.535 2.6 1.580 63.3 0.611
4 0.74 −36.2 21.1 104.7 0.548 19.9 104.8 0.547 17.1 1.044 104.7 0.547
5 0.40 245.3 8.4 87.4 0.160 15.7 93.6 0.037 90.0 0.504 87.0 0.168
6 0.84 −54.3 14.2 195.0 0.705 13.3 195.7 0.703 6.7 1.166 193.3 0.711
7 0.84 56.5 11.3 32.8 0.706 14.2 35.5 0.657 27.9 0.775 32.7 0.709
8 0.76 65.6 13.8 56.2 0.570 17.1 58.4 0.536 30.4 0.810 56.5 0.565
9 0.84 −33.1 19.2 48.0 0.712 17.5 48.5 0.706 14.2 1.070 48.2 0.710
10 0.83 190.1 13.2 70.5 0.689 19.5 87.6 0.519 66.3 0.642 69.9 0.695
11 0.76 −165.6 19.5 90.0 0.583 14.4 94.4 0.541 3.3 1.419 89.6 0.586
12 0.61 157.2 11.5 87.8 0.375* 18.8 98.5 0.214* 69.3 0.579 86.9 0.387
Taking into consideration only the slopes estimated by standard linear regression, the main sequence of the patients (median 16.5/s) was reduced significantly compared to the normal subjects (median 22.8/s, U = 131, P = 0.0056). 
In both groups the coefficient of variation of the main sequence distribution was around 30.0%. As expected, this interindividual variability was highly significant (subjects F = 230.5, P <0.0001; patients F = 169.1, P < 0.0001). On the other hand, the intraindividual variability (main sequence calculated for 10-minute periods) was not significant, showing that in normal subjects and patients the main sequence measurement was stable even with short time samples (F = 0.3039, P = 0.9101, Fig. 5). 
Figure 5. 
 
Slopes of the main sequence obtained over a period of one hour (solid line) was quite variable between subjects. The slopes estimated in the 10-minute bins (circles and squares) showed little variation. Circles: normal subjects 1 to 12. Squares: patients 1 to 8.
Figure 5. 
 
Slopes of the main sequence obtained over a period of one hour (solid line) was quite variable between subjects. The slopes estimated in the 10-minute bins (circles and squares) showed little variation. Circles: normal subjects 1 to 12. Squares: patients 1 to 8.
Discussion
Quantifying the slope of relationship between the maximum velocity and amplitude of the closing phase of spontaneous blinks has relevance, since it is assumed it is indicative of the aggregate motor neuron activity of the orbicularis oculi muscle. 6 The main sequence of spontaneous blinking is supposed to be linear and usually is expressed as the slope of a least-squared linear regression line. 7 However, a careful analysis of the literature on spontaneous blinking shows that the validity of basic assumptions underlying the use of least-squares linear regression has not been examined to our knowledge. 
In some reports, regression lines were determined for a pool of data from different subjects. 3,4,69 As the range of spontaneous blink amplitude varies across subjects, the calculated regression line relevance is open to question, especially regarding the variance in peak velocity for different levels of amplitude. In our sample, standardized residual plot analysis indicated that the homoscedasticity of the peak velocity could not be satisfied in 20% of our subjects. Although the lack of homoscedasticity of the peak velocity has not been addressed specifically in the literature, this feature clearly is apparent in some published data. 3,811  
Our findings indicated that a more meaningful approach is to analyze the range of blink amplitude and variance of peak velocity on an individual rather than a pooled basis. As it is not possible to control spontaneous blinking amplitude regression and subjects display distinct rates of blinks, regression with pooled data is quite inaccurate. Furthermore, assuming a linear relationship between peak velocity and amplitude warrants validation. The amplitude of spontaneous blink movements varies randomly within and across subjects. Two of our subjects did not perform large blinks (Figs. 2D, 2F), which accounts for why the linearity of their data was poor (r = 0.35 and r = 0.40). For the majority of subjects and patients the coefficients of correlation concurred well with the literature, with values ranging from 0.60 to 0.94. 12 As unequal variances increase the inaccuracy of the slope determination, a log transformation may be required to linearize the data. A residual plot or at least a graphic analysis is necessary to identify which data are suitable for straight regression. 
An interesting question in main sequence quantification using any type of regression model is whether or not an intercept is needed. From a pure biologic standpoint, it is meaningless to form an association between maximum velocity of a movement with zero amplitude. Regression instead through the origin, 13 thus, appears to be a meaningful approach to main sequence quantification. One problem in dropping the constant term is the low density of points near the origin. In fact, as shown in Table 1, few spontaneous blinks have less than 10° of amplitude. 
The curvilinear model through the origin, which has been used to fit eye saccadic main sequence, 14 was completely appropriate for fitting the data of all subjects. However, the biologic interpretation of curvilinear models is more complex than that of linear models. When the residuals of the linear models with and without intercept were compared to those resulting from the curvilinear function fitting, it became apparent that dropping the intercept imposed a restriction that diminished the ability of the model to fit the data. 
It is difficult to compare our results with the literature because the few normative data that have been published were reported as mean values 15,16 or single slopes derived from a pool of different subjects. In our sample, the slopes estimated by standard linear regression ranged from 8.7/s to 33.2/s. Two-way ANOVA with repeated measures indicated that the interindividual differences were significant and the slopes estimated within 10-minute bins were not different from the main sequence within one hour. The mean (21.9/s) concurs well with the figures reported by Huffman et al. (22.6/s) 7 and Stava et al. (20.9/s), 8 and they were slightly lower than those reported by Sun et al. 15 These investigators measured five age classes with eight subjects each, obtaining mean values ranging from 24.7 to 37.8/s. 
The constraints regarding the use of linear regression also were verified in the Graves' patients. The range of amplitude was similar, and the standardized residual plots indicated that the same degree of lack of homoscedasticity and linearity was poor for just one patient. The curvilinear model through the origin also was appropriate for fitting all patient data. 
The median main sequence of the patients was reduced significantly compared to the value obtained in the normal subjects. In our opinion, this relevant finding does not reflect a central anomaly of orbicularis oculi muscle recruitment. Graves' orbitopathy is an autoimmune disease involving the orbit and lids. Although many theories have been proposed to account for lid retraction, all are related to the involvement of levator palpebrae superioris and/or Müller's muscles in the inflammatory process induced by activated T lymphocytes. 17 We believe that the lower slope of the relationship between amplitude and maximum velocity in Graves' patients derives from the inability of the elastic elements of the lid to relax during spontaneous blinks. Maximum lid velocity, thus, is impaired for large movements. 
The results of both groups suggested that spontaneous blinks are not stereotyped movements and that there is a considerable interindividual variability in the range of amplitude of these movements. We do not know why some subjects display a small amplitude range of movements. This might reflect an intrinsic individual characteristic related to the attentional process involved in the visual display task or may be the result of differences in afferent inputs from the ocular surface. 18 This question deserves further investigation. The relationship between amplitude and velocity also varies across subjects and is not always linear. For a subset of subjects with large blink amplitudes the maximum velocity increased as a power function. This finding probably is related to the differences in the potential energy stored in the passive elements of the lid that is released in the closing phase of the movement. 19  
The fact that the patients with Graves' disease showed the same characteristics as the normal subjects indicated that these findings should be considered when spontaneous blinks are investigated in lid pathology. 
In summary, our analysis suggests that using classic linear regression is a valid approximation for estimating the slope of the relationship between amplitude and maximum velocity of spontaneous blinking. However, this model should be used with caution provided the homoscedasticity of the peak velocity and degree of linearity are satisfactory. 
Appendix
The coefficient of determination R 2 is used to assess the goodness fit of any model. The variety of R 2 statistics and its inappropriate use may produce misleading results when a linear no-intercept model is used. 20  
For a given dataset (xi , yi ), i = 1, 2,..., n, where X is the independent variable and Y is the dependent variable, linear regression fits the data to a model of the following form, ŷ = a + bx + ϵ, and residuals can be obtained by resi = yi − ŷ. In the most packages of statistics software R 2 is calculated by followed expression:  where is the mean of variable Y.  
For the linear with intercept model, the sum of residuals is equal to zero. Unfortunately, on regression through the origin that assumption is not warranted. In this case, Equation 1 may produce R 2 out of the 0 to 1 range. Alternatively, as Eisenhauer 13 recommends, R 2 can be obtained by Equation 2, a strictly non-negative coefficient of determination,  However, R 2 in Equation 2 may be higher than the expected and could not be used properly. Any such comparison between the fits of alternative models, however, is valid only if comparable measures of fit are used; that is, the same R 2 statistic must be used for the models being compared. 20  
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Footnotes
 Supported by Grant #301865/2009-4 from the Brazilian Research Council (CNPq).
Footnotes
 Disclosure: D.M. Garcia, None; J.C. Barbosa, None; C.T. Pinto, None; A.A.V. Cruz, None
Figure 1. 
 
Distribution of amplitude (left) and maximum velocity (right) of a subject.
Figure 1. 
 
Distribution of amplitude (left) and maximum velocity (right) of a subject.
Figure 2. 
 
Maximum velocity and amplitude relationship. (AD) Normal subjects. Two patients are shown in (E, F). No linear relationship was observed in only one subject ([D], r = 0.35) and one patient ([F], r = 0.4).
Figure 2. 
 
Maximum velocity and amplitude relationship. (AD) Normal subjects. Two patients are shown in (E, F). No linear relationship was observed in only one subject ([D], r = 0.35) and one patient ([F], r = 0.4).
Figure 3. 
 
Maximum velocity and amplitude relationship. (A, B) Subjects 9 and 12. (C, D) Patients 4 and 6. Left: maximum velocity and amplitude relationship. Right: standardized residuals plot. Subject 12 and patient 6 had their homoscedasticity assumption violated.
Figure 3. 
 
Maximum velocity and amplitude relationship. (A, B) Subjects 9 and 12. (C, D) Patients 4 and 6. Left: maximum velocity and amplitude relationship. Right: standardized residuals plot. Subject 12 and patient 6 had their homoscedasticity assumption violated.
Figure 4. 
 
Maximum velocity and amplitude relationship fitted by the three models. No difference between the models in ([A, B]; normal subjects) and ([E], patient). In subjects (C, D) and patient (F), the power-law allowed a better fitting followed by the linear model with intercept.
Figure 4. 
 
Maximum velocity and amplitude relationship fitted by the three models. No difference between the models in ([A, B]; normal subjects) and ([E], patient). In subjects (C, D) and patient (F), the power-law allowed a better fitting followed by the linear model with intercept.
Figure 5. 
 
Slopes of the main sequence obtained over a period of one hour (solid line) was quite variable between subjects. The slopes estimated in the 10-minute bins (circles and squares) showed little variation. Circles: normal subjects 1 to 12. Squares: patients 1 to 8.
Figure 5. 
 
Slopes of the main sequence obtained over a period of one hour (solid line) was quite variable between subjects. The slopes estimated in the 10-minute bins (circles and squares) showed little variation. Circles: normal subjects 1 to 12. Squares: patients 1 to 8.
Table 1. 
 
Number of Blinks, Amplitude and Maximum Velocity Values Obtained during One Hour of a Video Observation
Table 1. 
 
Number of Blinks, Amplitude and Maximum Velocity Values Obtained during One Hour of a Video Observation
Group ID N Percentage of Blinks with Amplitude up to 10°
Amplitude (°)
Maximum Velocity (°/s)
Min Max Mean Min Max Mean
Subjects 1 1183 0.34 7.2 60.3 26.3 174.6 1449.9 550.6
2 830 6.02 6.7 49.0 18.9 117.8 2191.6 408.9
3 515 0.39 6.5 52.7 31.3 140.5 1694.0 674.2
4 1023 1.76 7.5 72.4 31.1 191.2 1957.2 694.3
5 2538 0.43 8.7 49.8 22.8 221.9 1305.3 566.0
6 2048 0.00 10.6 58.8 37.6 224.0 1713.5 766.5
7 945 0.11 9.5 59.9 41.4 213.1 1304.0 708.0
8 463 1.94 6.1 52.0 27.4 120.1 1177.5 476.1
9 877 0.00 10.8 52.5 32.2 296.5 1263.1 746.2
10 501 0.20 7.9 66.2 34.9 169.3 2045.4 700.8
11 572 0.52 6.4 69.5 27.5 129.7 2097.6 706.5
12 2463 0.04 9.6 64.6 36.3 166.3 1356.2 455.1
13 1393 0.00 11.8 53.1 39.0 280.3 1604.9 1101.5
14 1767 1.13 5.5 43.5 20.5 162.1 1133.2 417.2
15 357 2.24 7.0 38.6 21.5 108.8 730.7 401.7
16 1511 2.05 6.9 43.8 16.2 155.1 770.2 378.8
17 1474 0.41 6.0 40.1 31.4 180.6 929.5 586.3
18 1180 0.51 9.1 46.5 26.4 118.1 1183.4 492.3
19 669 0.30 8.7 42.5 23.4 172.4 1142.8 564.1
20 449 3.79 6.4 53.7 22.9 85.9 830.7 327.9
Patients 1 2258 0.18 9.0 43.9 26.3 169.2 1023.0 599.3
2 663 4.22 5.3 28.9 17.0 123.1 660.1 341.3
3 521 5.57 5.7 30.5 16.7 76.1 683.7 227.1
4 1143 0.00 15.2 52.8 30.9 186.1 1079.5 613.9
5 655 0.00 14.7 42.8 33.0 201.2 742.9 521.7
6 236 0.85 8.4 90.2 52.1 158.0 1741.5 684.7
7 666 2.40 6.8 32.1 18.3 113.5 466.0 264.1
8 1298 1.77 5.8 38.3 18.8 153.6 730.5 325.2
9 2189 0.09 8.7 44.3 19.8 168.1 839.7 345.9
10 174 1.15 8.4 49.3 27.9 208.8 932.8 556.9
11 1070 0.00 13.5 49.2 31.3 168.9 862.6 443.8
12 1300 2.85 5.4 39.8 19.9 173.7 772.1 387.1
Table 2. 
 
Parameters of Three Different Models Used to Fit the Relationship between Amplitude and Maximum Velocity
Table 2. 
 
Parameters of Three Different Models Used to Fit the Relationship between Amplitude and Maximum Velocity
Group ID Model 1: y = a + b*x Model 2: y = b*x Model 3: y = c*xd
R M S R M S R M S
Subjects 1 0.92 −89.1 24.3 93.3 0.850 21.3 97.7 0.836 12.6 1.151 93.2 0.851
2 0.77 −3.3 21.8 112.2 0.587 21.7 112.2 0.757 21.1 1.005 74.7 0.757
3 0.89 −137.0 25.9 114.0 0.787 21.8 119.4 0.767 9.1 1.248 113.4 0.789
4 0.93 27.1 21.5 119.1 0.856 22.2 119.6 0.855 26.1 0.956 119.1 0.857
5 0.82 10.6 24.4 70.9 0.677 24.8 70.9 0.677 24.5 1.003 71.0 0.677
6 0.81 −296.9 28.2 125.4 0.662 20.6 134.4 0.612 3.4 1.491 122.4 0.678
7 0.82 −272.8 23.7 121.9 0.665 17.3 130.6 0.615 3.2 1.447 121.2 0.669
8 0.86 −29.6 18.5 88.5 0.731 17.5 88.8 0.729 11.5 1.123 87.8 0.735
9 0.91 86.6 20.5 72.5 0.832 23.0 75.3 0.819 34.8 0.883 72.4 0.833
10 0.78 −265.7 27.7 207.7 0.607 20.6 218.6 0.565 3.1 1.516 200.7 0.633
11 0.94 −147.4 31.0 115.8 0.878 26.3 126.2 0.855 12.2 1.219 112.9 0.884
12 0.81 −54.2 14.0 105.9 0.657 12.6 106.9 0.650 5.5 1.222 104.1 0.669
13 0.90 −193.4 33.2 91.9 0.805 28.4 95.9 0.787 13.0 1.211 91.5 0.806
14 0.69 90.7 15.9 103.4 0.480 20.0 106.7 0.447 38.7 0.789 103.2 0.482
15 0.86 59.2 15.9 63.1 0.744 18.5 65.4 0.725 30.7 0.840 62.6 0.748
16 0.35 237.3 8.7 87.7 0.123* 22.6 102.8 −0.206* 128.2 0.391 87.7 0.123
17 0.76 −72.4 21.0 69.6 0.575 18.7 70.2 0.568 8.9 1.214 69.1 0.582
18 0.83 −179.0 25.4 111.1 0.686 19.0 118.9 0.640 4.0 1.461 108.6 0.699
19 0.88 −45.6 26.1 82.9 0.772 24.2 83.6 0.769 17.9 1.093 82.8 0.773
20 0.61 109.5 9.5 99.8 0.367 13.8 106.0 0.286 43.5 0.650 98.3 0.386
Patients 1 0.85 17.5 22.1 68.0 0.717 22.7 68.0 0.717 25.9 0.961 67.9 0.718
2 0.81 20.9 18.8 57.0 0.653 20.0 57.2 0.651 23.5 0.943 57.1 0.652
3 0.76 −94.8 19.2 65.5 0.582 13.9 69.1 0.535 2.6 1.580 63.3 0.611
4 0.74 −36.2 21.1 104.7 0.548 19.9 104.8 0.547 17.1 1.044 104.7 0.547
5 0.40 245.3 8.4 87.4 0.160 15.7 93.6 0.037 90.0 0.504 87.0 0.168
6 0.84 −54.3 14.2 195.0 0.705 13.3 195.7 0.703 6.7 1.166 193.3 0.711
7 0.84 56.5 11.3 32.8 0.706 14.2 35.5 0.657 27.9 0.775 32.7 0.709
8 0.76 65.6 13.8 56.2 0.570 17.1 58.4 0.536 30.4 0.810 56.5 0.565
9 0.84 −33.1 19.2 48.0 0.712 17.5 48.5 0.706 14.2 1.070 48.2 0.710
10 0.83 190.1 13.2 70.5 0.689 19.5 87.6 0.519 66.3 0.642 69.9 0.695
11 0.76 −165.6 19.5 90.0 0.583 14.4 94.4 0.541 3.3 1.419 89.6 0.586
12 0.61 157.2 11.5 87.8 0.375* 18.8 98.5 0.214* 69.3 0.579 86.9 0.387
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