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Pointwise univariate linear regression of perimetric sensitivity against follow-up time in glaucoma
Ophthalmology
Volume 105, Number 2, February •998
asma, in the second category, and that lesions in this category do not require histopathologic confirmation. In this study, 13 of 692 lesions that were thought to be benign turned out to be malignant. Ten of those 13 lesions were basal cell carcinomas. I wonder exactly how much morbidity is generated by excising an occasional undiagnosed basal cell carcinoma. Since these lesions are usually quite small at the time of excision, I suspect that a number of them would have been completely excised, and thus would not recur. Of those undiagnosed basal cell carcinomas that were not completely excised, one would expect only one third of them to recur. 2 Unless they occurred in a particularly treacherous location, I believe that recurrences could be dealt with quite satisfactorily with excision using frozen sections or the Mohs' technique. It is not clear that falling to diagnose malignancy in these ten lesions would have any significant consequences for the patients involved. I recently removed approximately 50 lesions that appeared to be papillomas from a patient's periocular area and face. Had I submitted these for histopathologic confirmation, the mere task of labeling each specimen and mapping exactly where it came from would have been daunting. Furthermore, using the authors' reported cost o~ $62.49 per lesion for examination and reporting by the pathologist, a pathology bill of over $3000 would have been generated, with dubious value to the patient. While this study demonstrates that some lesions that appear benign may in fact be malignant or premalignant, it does not support the sweeping generalization that all excised eyelid lesions require histopathologic confirmation. Clihical judgment, while not perfect, still has a role to play. LAWRENCE H. QUIST, MD
Minneapolis, Minnesota References
1. Kersten RC, Ewing-Chow D, Kulwin DR, Gallon M. Accuracy of clinical diagnosis of cutaneous eyelid lesions. Ophthalmology 1997; 104:479-84. 2. Gooding CA, White G, Yatsuhashi M. Significance of marginal extension in excised basal-cell carcinoma. N Engl J Med 1965;273:923-4. Author's reply Dear Editor: At the onset of this study I too believed that it would be possible to clinically identify those lesions that "appear benign and could not possibly be malignant." Dr. Quist suggests that xanthelasmas, and by inference, "papillomas" are lesions that fall into this category, and would not require histopathologic confirmation. Only a few xanthelasmas were included in our study, as we limited it to those lesions that were completely removed in the office chair. We did, however, remove over 300 "papillomas," all of which were thought to be benign, and yet 6 proved histopathologically to harbor malignancies. Of the 154 "epidermal inclusion cysts," all of which we thought could not possibly be malignant, 2 were cystic basal cell carcinomas. Eighty-seven "me|anocytic nevi," all of
204
which were presumed to be benign, harbored I basal cell carcinoma and 1 non-Hodgkin lymphoma, and 75 "hydrocystomas" also contained 2 unsuspected malignancies. Clinical medicine does require us to make cost-benefit decisions. I do agree with Dr. Quist that the morbidity arising from excising an occasional undiagnosed basal cell carcinoma may not be high. On the other hand, a "papilloma" could conceivably also represent an amelanotic melanoma. If one elects to depend on clinical judgement rather than histopathologic evaluation, one is obliged to thoroughly educate the patient about the potential for missing a malignancy and the need for assiduous follow-up should a "benign" lesion recur. ROBERT C. KERSTEN, MD
Cincinnati, Ohio Pointwise Univariate Linear Regression of Perimetric Sensitivity Against Follow-up Time in G l a u c o m a Dear Editor: The Appendix of the article by Wild et al, "Pointwise Univariate Linear Regression of Perimetric Sensitivity Against Follow-up Time in Glaucoma" (Ophthalmology 1997; 104:808-15) contains some statistical mistakes and misconceptions. 1. The formula given for standard error is wrong. In the formula as stated, the right-hand side is identically zero. 2. The t-statistic given for comparison of individual slopes with those derived from the normal database is not correct. The decline in sensitivity of a test location with age presumably varies between healthy individuals, yet this variability does not appear in the authors' statistic. Thus, the statistic provides no information about how significant an individual slope really is. For example, they might determine that an individual slope is almost certainly less than the population average, but we would expect a large percentage of nomaal slopes to fall into this category, the exact proportion depending on how certain we are of the individual slope as well as the magnitude of variation in the population. They might also say that their slope estimate is five or ten times as large in absolute value as the population average, but, again, a high percentage of healthy individuals could share this characteristic. Also, since the "normal slope" that they use is an estimate of the true (unknown) population average, it contributes an additional element of uncertainty that should be included in the formula. These additional variance terms should appear in the denominator of the test statistic. Thus, the statistic that they use is too small, and it is possible that far too many significant values will be found. Since most of the content of the paper depends on this formula, few of their results and conclusions are correct. 3. An additional source of error in their test statistics is the failure to account for temporal dependence between measurements. Let us assume that sensitivity at a location is measured on five different occasions and that the correlation between measurements varies with time in an autoregressive pattern, i.e., correlation decreases the further apart measurements are taken. We fit a regression line to the data using ordinary least squares, as do the authors.
Letters to the Editor We also lit a line to the data using weighted least squares, which takes into account the correlation between measurements. The two lines will have slopes that are close, but not identical? More importantly, the confidence intervals for the slopes will differ, since ordinary least squares will underestimate the variance.-" In other words, if there is temporal correlation between measurements, we will find more significant slopes than we should. Temporal correlation could be estimated by looking at the data from all the patients in the study and/or the normal database. 4. The authors state that they have modeled spatial correlation in a previous study? In fact, spatial location, not spatial correlation, was modeled in that paper, using an infinite (!?) number of parameters. TERRY A. COX, MD
Chapel Hill, North Carolina References
1. Diggle PJ, Liang K-Y, Zeger SL. Analysis of Longitudinal Data. Oxford: Oxford University Press, 1994;58-62. 2. Diggle PJ. Time Series: A Biostatistical Introduction. Oxford: Oxford University Press, 1990;87-92. 3. Wild JM, Hussey MK, Flanagan JG, Trope GE. Pointwise topographical and longitudinal modeling of the visual field in glaucoma. Invest Ophthalmol Vis Sci 1993;34:190716. Authors' reply Dear Editor: We thank Dr. Cox for initiating a discussion of our paper. The formula for the standard deviation, not the standard error as Dr. Cox points out, should read:
Y' (z~mzj) 2 S 2
~J
~
i=l
n-2
The squared term present on both sides of the equation became inadvertently omitted during the generation of the publication. The right-hand side of the equation specified in the paper defaults to zero, and had the equation been incorporated into the software, the computer program developed for the study would have reported a "divide by zero error." The two major criticisms of Dr. Cox (Items 2 and 3) involve subjective judgments about the level of detail required in our model, i.e., the lack of a variance term for the presumed, but unknown, between-individual variation in the decline of normal sensitivity with age and the lack of a variance term for within-individual differences in the interval between successive follow-ups. The mathematical modeling of any physical process involves abstracting from reality. The degree of abstraction depends on the quantity and the precision of the available data and on the purpose for which the model is being devel-
oped. The fundamental nature of the model described in our paper does not invalidate the rigor of the statistical procedures applied, and Dr. Cox is misguided in his assertion that " f e w of the results and conclusions are correct." Before employing more complex model formulations, it seemed sensible, both theoretically and clinically, to investigate the more fundamental model specifications that are present at the earliest stage of the abstraction hierarchy. The observed data were tested against a null hypothesis of a prespecified constant'slope of the decline in sensitivity with increase in age. Such a procedure is consistent with sound statistical practice. The slope of the decline in sensitivity with increase in age was that derived from cross-sectional data. No data are available as to the format of the slope based on longitudinal studies, and therefore the magnitude of the between-individual variation in normals is unknown. A term for the variability of the data around the slope derived from the cross-sectional data was considered; however, the inclusion of additional variance terms in any iteration of the model has to be considered in the context of the 2-dB final step size used for the measurement of sensitivity and the magnitude in glaucoma of the short-term fluctuation of at least 1.5 dB and the long-term fluctuation of approximately 2.5 dB. A similar argument can be applied as to the necessity for the inclusion of a term to account for the_ impact of the temporal correlation of sensitivities derived at separate examinations. It should also be recognized that the possibilities for the level of abstraction of the model are almost limitless and extend beyond those suggested by Dr. Cox: the model could theoretically be refined to include variance terms to account for such factors as the level of prior perimetric experience, the severity of field loss, the order of eye examined, and so forth. Dr. Cox argues that our model would produce too many false-positives. In reality, and as might be expected clinically from an a priori knowledge of the variability that exists both within- and between-visual field examinations, the opposite was found. Dr. Cox's final point for discussion (Item 4) addresses the question of semantics: the topographic component of the multiple regression of sensitivity against time to follow-up models the sensitivity at each stimulus location collectively, and thus uses the correlation in sensitivity that exists between locations. The identification, at the earliest possible opportunity, of progressive visual field loss remains one of the major problems in ophthalmology. Discussions such as these can only serve to stimulate the development of analytical techniques capable of satisfying such a need. JOHN M. WILD, PhD NATALIE HUTCHINGS,BSc MICHAEL K. HUSSEY, MSc JOHN G. FLANAGAN, PhD GRAHAM E. TROPE, FRCS(C)
Birmillgham, England
i
205
Volume 105, Number 2, February •998
asma, in the second category, and that lesions in this category do not require histopathologic confirmation. In this study, 13 of 692 lesions that were thought to be benign turned out to be malignant. Ten of those 13 lesions were basal cell carcinomas. I wonder exactly how much morbidity is generated by excising an occasional undiagnosed basal cell carcinoma. Since these lesions are usually quite small at the time of excision, I suspect that a number of them would have been completely excised, and thus would not recur. Of those undiagnosed basal cell carcinomas that were not completely excised, one would expect only one third of them to recur. 2 Unless they occurred in a particularly treacherous location, I believe that recurrences could be dealt with quite satisfactorily with excision using frozen sections or the Mohs' technique. It is not clear that falling to diagnose malignancy in these ten lesions would have any significant consequences for the patients involved. I recently removed approximately 50 lesions that appeared to be papillomas from a patient's periocular area and face. Had I submitted these for histopathologic confirmation, the mere task of labeling each specimen and mapping exactly where it came from would have been daunting. Furthermore, using the authors' reported cost o~ $62.49 per lesion for examination and reporting by the pathologist, a pathology bill of over $3000 would have been generated, with dubious value to the patient. While this study demonstrates that some lesions that appear benign may in fact be malignant or premalignant, it does not support the sweeping generalization that all excised eyelid lesions require histopathologic confirmation. Clihical judgment, while not perfect, still has a role to play. LAWRENCE H. QUIST, MD
Minneapolis, Minnesota References
1. Kersten RC, Ewing-Chow D, Kulwin DR, Gallon M. Accuracy of clinical diagnosis of cutaneous eyelid lesions. Ophthalmology 1997; 104:479-84. 2. Gooding CA, White G, Yatsuhashi M. Significance of marginal extension in excised basal-cell carcinoma. N Engl J Med 1965;273:923-4. Author's reply Dear Editor: At the onset of this study I too believed that it would be possible to clinically identify those lesions that "appear benign and could not possibly be malignant." Dr. Quist suggests that xanthelasmas, and by inference, "papillomas" are lesions that fall into this category, and would not require histopathologic confirmation. Only a few xanthelasmas were included in our study, as we limited it to those lesions that were completely removed in the office chair. We did, however, remove over 300 "papillomas," all of which were thought to be benign, and yet 6 proved histopathologically to harbor malignancies. Of the 154 "epidermal inclusion cysts," all of which we thought could not possibly be malignant, 2 were cystic basal cell carcinomas. Eighty-seven "me|anocytic nevi," all of
204
which were presumed to be benign, harbored I basal cell carcinoma and 1 non-Hodgkin lymphoma, and 75 "hydrocystomas" also contained 2 unsuspected malignancies. Clinical medicine does require us to make cost-benefit decisions. I do agree with Dr. Quist that the morbidity arising from excising an occasional undiagnosed basal cell carcinoma may not be high. On the other hand, a "papilloma" could conceivably also represent an amelanotic melanoma. If one elects to depend on clinical judgement rather than histopathologic evaluation, one is obliged to thoroughly educate the patient about the potential for missing a malignancy and the need for assiduous follow-up should a "benign" lesion recur. ROBERT C. KERSTEN, MD
Cincinnati, Ohio Pointwise Univariate Linear Regression of Perimetric Sensitivity Against Follow-up Time in G l a u c o m a Dear Editor: The Appendix of the article by Wild et al, "Pointwise Univariate Linear Regression of Perimetric Sensitivity Against Follow-up Time in Glaucoma" (Ophthalmology 1997; 104:808-15) contains some statistical mistakes and misconceptions. 1. The formula given for standard error is wrong. In the formula as stated, the right-hand side is identically zero. 2. The t-statistic given for comparison of individual slopes with those derived from the normal database is not correct. The decline in sensitivity of a test location with age presumably varies between healthy individuals, yet this variability does not appear in the authors' statistic. Thus, the statistic provides no information about how significant an individual slope really is. For example, they might determine that an individual slope is almost certainly less than the population average, but we would expect a large percentage of nomaal slopes to fall into this category, the exact proportion depending on how certain we are of the individual slope as well as the magnitude of variation in the population. They might also say that their slope estimate is five or ten times as large in absolute value as the population average, but, again, a high percentage of healthy individuals could share this characteristic. Also, since the "normal slope" that they use is an estimate of the true (unknown) population average, it contributes an additional element of uncertainty that should be included in the formula. These additional variance terms should appear in the denominator of the test statistic. Thus, the statistic that they use is too small, and it is possible that far too many significant values will be found. Since most of the content of the paper depends on this formula, few of their results and conclusions are correct. 3. An additional source of error in their test statistics is the failure to account for temporal dependence between measurements. Let us assume that sensitivity at a location is measured on five different occasions and that the correlation between measurements varies with time in an autoregressive pattern, i.e., correlation decreases the further apart measurements are taken. We fit a regression line to the data using ordinary least squares, as do the authors.
Letters to the Editor We also lit a line to the data using weighted least squares, which takes into account the correlation between measurements. The two lines will have slopes that are close, but not identical? More importantly, the confidence intervals for the slopes will differ, since ordinary least squares will underestimate the variance.-" In other words, if there is temporal correlation between measurements, we will find more significant slopes than we should. Temporal correlation could be estimated by looking at the data from all the patients in the study and/or the normal database. 4. The authors state that they have modeled spatial correlation in a previous study? In fact, spatial location, not spatial correlation, was modeled in that paper, using an infinite (!?) number of parameters. TERRY A. COX, MD
Chapel Hill, North Carolina References
1. Diggle PJ, Liang K-Y, Zeger SL. Analysis of Longitudinal Data. Oxford: Oxford University Press, 1994;58-62. 2. Diggle PJ. Time Series: A Biostatistical Introduction. Oxford: Oxford University Press, 1990;87-92. 3. Wild JM, Hussey MK, Flanagan JG, Trope GE. Pointwise topographical and longitudinal modeling of the visual field in glaucoma. Invest Ophthalmol Vis Sci 1993;34:190716. Authors' reply Dear Editor: We thank Dr. Cox for initiating a discussion of our paper. The formula for the standard deviation, not the standard error as Dr. Cox points out, should read:
Y' (z~mzj) 2 S 2
~J
~
i=l
n-2
The squared term present on both sides of the equation became inadvertently omitted during the generation of the publication. The right-hand side of the equation specified in the paper defaults to zero, and had the equation been incorporated into the software, the computer program developed for the study would have reported a "divide by zero error." The two major criticisms of Dr. Cox (Items 2 and 3) involve subjective judgments about the level of detail required in our model, i.e., the lack of a variance term for the presumed, but unknown, between-individual variation in the decline of normal sensitivity with age and the lack of a variance term for within-individual differences in the interval between successive follow-ups. The mathematical modeling of any physical process involves abstracting from reality. The degree of abstraction depends on the quantity and the precision of the available data and on the purpose for which the model is being devel-
oped. The fundamental nature of the model described in our paper does not invalidate the rigor of the statistical procedures applied, and Dr. Cox is misguided in his assertion that " f e w of the results and conclusions are correct." Before employing more complex model formulations, it seemed sensible, both theoretically and clinically, to investigate the more fundamental model specifications that are present at the earliest stage of the abstraction hierarchy. The observed data were tested against a null hypothesis of a prespecified constant'slope of the decline in sensitivity with increase in age. Such a procedure is consistent with sound statistical practice. The slope of the decline in sensitivity with increase in age was that derived from cross-sectional data. No data are available as to the format of the slope based on longitudinal studies, and therefore the magnitude of the between-individual variation in normals is unknown. A term for the variability of the data around the slope derived from the cross-sectional data was considered; however, the inclusion of additional variance terms in any iteration of the model has to be considered in the context of the 2-dB final step size used for the measurement of sensitivity and the magnitude in glaucoma of the short-term fluctuation of at least 1.5 dB and the long-term fluctuation of approximately 2.5 dB. A similar argument can be applied as to the necessity for the inclusion of a term to account for the_ impact of the temporal correlation of sensitivities derived at separate examinations. It should also be recognized that the possibilities for the level of abstraction of the model are almost limitless and extend beyond those suggested by Dr. Cox: the model could theoretically be refined to include variance terms to account for such factors as the level of prior perimetric experience, the severity of field loss, the order of eye examined, and so forth. Dr. Cox argues that our model would produce too many false-positives. In reality, and as might be expected clinically from an a priori knowledge of the variability that exists both within- and between-visual field examinations, the opposite was found. Dr. Cox's final point for discussion (Item 4) addresses the question of semantics: the topographic component of the multiple regression of sensitivity against time to follow-up models the sensitivity at each stimulus location collectively, and thus uses the correlation in sensitivity that exists between locations. The identification, at the earliest possible opportunity, of progressive visual field loss remains one of the major problems in ophthalmology. Discussions such as these can only serve to stimulate the development of analytical techniques capable of satisfying such a need. JOHN M. WILD, PhD NATALIE HUTCHINGS,BSc MICHAEL K. HUSSEY, MSc JOHN G. FLANAGAN, PhD GRAHAM E. TROPE, FRCS(C)
Birmillgham, England
i
205