About MEAJO | Editorial board | Search | Ahead of print | Current Issue | Archives | Instructions to authors | Online submission | Subscribe | Advertise | Contact | Login 
Middle East African Journal of Ophthalmology Middle East African Journal of Ophthalmology
Users Online: 769   Home Print this page Email this page Small font sizeDefault font sizeIncrease font size

  Table of Contents 
Year : 2012  |  Volume : 19  |  Issue : 4  |  Page : 397-401  

Analysis of familial aggregation in total, against-the-rule, with-the-rule, and oblique astigmatism by conditional and marginal models in the Tehran eye study

1 Department of Epidemiology and Biostatistics, School of Public Health, Tehran University of Medical Sciences, Tehran, Iran
2 Noor Ophthalmology Research Center, Noor Eye Hospital; Department of Ophthalmology, Farabi Eye Hospital, School of Medicine, Tehran University of Medical Sciences, Tehran, Iran

Date of Web Publication20-Oct-2012

Correspondence Address:
Akbar Fotouhi
Department of Epidemiology and Biostatistics, School of Public Health, Tehran University of Medical Sciences, Tehran
Login to access the Email id

Source of Support: This project was partly supported by a grant from the Iranian National Research Center for Medical Sciences, and the random sampling for the study was performed by the Iranian Statistics Center according to the 1996 national census. The study was supported by Tehran University of Medical Sciences as a PhD thesis., Conflict of Interest: None

DOI: 10.4103/0974-9233.102746

Rights and Permissions

Purpose: The purpose was to determine the familial aggregation of the total, against-the-rule (ATR), with-the-rule (WTR), and oblique astigmatism by conditional and marginal models in the Tehran Eye Study.
Materials and Methods: Total, ATR, WTR, and oblique astigmatism were studied in 3806 participants older than 5 years from August 2002 to December 2002 in the Tehran Eye Study. Astigmatism was defined as a cylinder worse than or equal to −0.5 D. WTR astigmatism was defined as 0 ± 19°, ATR astigmatism was defined as 90 ± 19°, and oblique when the axes were 20-70° and 110-160°. The familial aggregation was investigated with a conditional model (quadratic exponential) and marginal model (alternating logistic regression) after controlling for confounders.
Results: Using the conditional model, the conditional familial aggregation odds ratios (OR) (95% confidence interval) for the total, WTR, ATRs, and oblique astigmatism were 1.49 (1.43-1.72), 1.91 (1.65-2.20), 2.00 (1.70-2.30), and 1.86 (1.37-2.54), respectively. In the marginal model, the marginal OR of the parent-offspring and sib-sib in the total astigmatism were 1.35 (1.13-1.63) and 1.54 (1.13-2.11), respectively; WTR 1.53 (1.06-2.20) and 1.94 (1.21-3.13) and; ATR 2.13 (1.01-4.50) and 2.23 (1.52-3.30). The model was statistically significant in sib-sib relationship only for oblique astigmatism with OR of 3.00 (1.25-7.20).
Conclusion: The results indicate familial aggregation of astigmatism in the population in Tehran adjusted for age, gender, cataract, duration of education, and body mass index, so that the addition of a new family member affected with astigmatism, as well as having a sibling or parents with astigmatism, significantly increases the odds of exposure to the disease for all four phenotypes. This aggregation can be due to genetic and/or environmental factors. Dividing astigmatism into three phenotypes increased the odds ratios.

Keywords: Astigmatism, Conditional and Marginal models, Familial Aggregation

How to cite this article:
Rakhshani MH, Mohammad K, Zeraati H, Nourijelyani K, Hashemi H, Fotouhi A. Analysis of familial aggregation in total, against-the-rule, with-the-rule, and oblique astigmatism by conditional and marginal models in the Tehran eye study. Middle East Afr J Ophthalmol 2012;19:397-401

How to cite this URL:
Rakhshani MH, Mohammad K, Zeraati H, Nourijelyani K, Hashemi H, Fotouhi A. Analysis of familial aggregation in total, against-the-rule, with-the-rule, and oblique astigmatism by conditional and marginal models in the Tehran eye study. Middle East Afr J Ophthalmol [serial online] 2012 [cited 2018 May 27];19:397-401. Available from: http://www.meajo.org/text.asp?2012/19/4/397/102746

   Introduction Top

Determinants of astigmatism including age, race, genetics, gender, other refractive errors, ophthalmological surgeries, and eyelid pressure have been reviewed. [1] One of the probable factors is genetics. Study types and their results on the effects of genetics on astigmatism vary widely. Some studies on twins, [2],[3] a proband case control study, [4] a cross-sectional, [5] and an epidemiological study [6] have emphasized the role of genetics and/or a common environment on inducing astigmatism. However, other studies on twins, [7],[8] a cross-sectional population-based study, [9] a cohort population-based study, [10] and a study by Cagigrigoriu et al. [11] have reported opposite outcomes. However, the drawback of these includes bias in some studies or a small sample size. Additionally, the type of study, lack of division of astigmatism into oblique, against-the-rule (ATR), and with-the-rule (WTR) phenotypes, and merely relying on marginal models can be considered controversial in such studies.

In nonlinear models such as the logistic regression, interpretation of parameters in conditional models is distinct from marginal models, and accordingly the former has been the focus of attention by researchers. However, due to their sensitivity to family size, they were not used in familial aggregation analysis until recently. [12] One study that managed to solve this problem was conducted by Matthews et al., [13] who employed a modified quadratic exponential model. Simple interpretation of the parameters by usual epidemiological terms (e.g., odds ratio (OR)) and avoiding the use of terms such as recurrence risk ratio which are associated with some bias, [14] as well as easy fitting by current software programs in probability and nonprobability sampling are among the advantages of the model.

In this article, we attempt to provide results of the conditional model and those of the marginal model so that the interpretation of the data can be possible in both models. Also, we have provided more reliable results by investigating oblique, ATR, and WTR astigmatism together with the total astigmatism in a population-based study with an appropriate sample size.

   Materials and Methods Top

The Tehran Eye Study is a population-based cross-sectional study the details of which have been previously reported. [15] In short, based on the block enumeration of the 1996 census of the Iranian population which announced the Tehran population to be 1,660,219 households with the mean family size of 3.6, 160 clusters in 22 municipal districts of Tehran were selected by a stratified random cluster sampling approach. The enumeration was continued for 10 neighboring households by proceeding systematically in a clockwise direction from the initial household within the cluster. Finally, all household members were invited for a complete eye examination at Noor Vision Correction Center in Tehran, Iran. This study adhered to the tenets of the Helsinki Declaration and has been approved by the Ethics Committee of the National Research Center for Medical Sciences of Iran.


Visual acuity was determined with tumbling E letters at a distance of 4 m. Refraction was measured by optometrists for all participants over 5 years of age using a Topcon automated refractometer (Topcon KR 8000, Topcon Corporation, Tokyo, Japan). Results from autorefraction were used as a starting point for full manifest refraction. In order to increase the precision of subjective refraction examinations, a red-green test was used. If, in the ophthalmologist's judgment, there was no contraindication, cycloplegic refraction was performed. For cycloplegia, 2 drops of 1% cyclopentolate were instilled 30 and 25 minutes prior to refraction.


Astigmatism was defined as a cylinder worse than or equal to -0.5 D. WTR astigmatism was defined as 0 ± 19°, ATR astigmatism was defined as 90 ± 19° and axes between 20-70° and 110-160° were considered oblique astigmatism. The degree of lens opacity was graded according to the lens opacities classification system (LOCS III) grading guidelines. A gradable cataract was defined as LOCS III grade 3 or more in C and/or N and/or a grade or more posterior subcapsular cataract.

Statistical methods

Four phenotypes of astigmatism (total, oblique, ATR, and WTR) were defined as no/yes variables, and multiple logistic regression was used for calculating familial aggregation. This model can be fitted into conditional and marginal modes with some differences in parameter interpretation. For example, for a binary covariate Xij, a marginal regression coefficient would be interpreted as a log odds for disease for those in the population with Xij = 1 covariate relative to those with Xij = 0.Conversely, a conditional regression coefficient would be interpreted as log odds for disease in a family member j with Xij = 1 relative to another family member within the same family i with Xij = 0.

The conditional modeling approach is preferable if one wants to specify a mechanism that could generate positive association among clustered observation, estimate cluster-specific effects, and estimate their variability. Given a conditional model, one can recover information about marginal model; that is, a conditional model implies a marginal model but a marginal model does not itself imply a conditional model. [16] In many surveys or epidemiological studies, the goal is to compare the relative frequency of occurrence of some outcome for different groups in population. Then the quantities of primary interest include between-group OR among marginal probabilities for different groups.

For fitting the marginal model, the alternating logistic regression (ALR) was used. [17] This is the extended version of the second-order generalized estimated equation (GEE2), where the inference of correlation and regression parameters is done simultaneously. However, in the conditional model, the quadratic exponential model (QEM) was used. This was introduced for the first time by Zhao and Prentice [18] in 1990 and was developed in 1996 by Betensky and Whittemore [19] for studies on familial aggregation. QEM can be considered for a binary outcome of an n-member family with a simple conversion as follows:

where y-i is the vector of all outcomes within a family excluding the ith member.

This is a simple logistic regression model and for its fitting, data on the status of the disorder of a certain individual family member can be considered as the outcome and number of remaining affected family members can be considered as covariates.

As with the above relation, α is the logarithm of the odds of the disorder of the ith family member, provided that there is no other affected member in the family; and γ is the logarithm of conditional OR (the parameter representing familial aggregation) in the family. This is the simplest form of the model, and confounders can be added to it where appropriate. In the present study, in order to control the confounders in both models, methods of confounder control were used in logistic regression. [20]

The QEM is especially useful for the family studies dealing with disease aggregation. In particular, its conditioning is related to family history of disease; in other words, it is precisely related to individuals wishing to figure out their own risk of disease. Matthews et al. [13] generalized this model for the families with different sizes, and we used their marginal approach in the present study.

An alternative modeling approach, for the conditional model, in handling the correlated binary data is the introduction of random effects. [21],[22],[23],[24],[25] However, its major disadvantage is its computational complexity, even in the case of a single random effect for each family.

In order to fit ALR, the GENMOD procedure was used in SAS, where familial relation makes up the model matrix as a separate variable, and the procedure is able to estimate OR of pairs like sib-sib or parent-offspring. However, in the conditional model the conditions are different and data related to each family (e.g., with 2 or 3 members) should be separately organized in separate files. The disease status of family members and their familial relations are entered in one row.

   Results Top

Data from 1595 male (41.9%) and 2211 female (58.1%) participants, 5 years to 96 years old, were analyzed. The prevalence and 95% confidence intervals of the four phenotypes of total, WTR, ATR, and oblique astigmatism are presented by gender in [Table 1]. After screening astigmatism, the highest prevalence was associated with ATR. The models were fitted after adjusting for the potential confounders such as age, gender, body mass index (BMI), duration of education, occupation, and cataract.
Table 1: The prevalence of four phenotypes of astigmatism by gender (Tehran Eye Study, Iran, 2002)

Click here to view

In the marginal model, we investigated three relations: sib-sib, parent-offspring, and spouse. The OR of the relation of spouse with all phenotypes and that of parent-offspring with oblique astigmatism did not turn out to be significant [Table 2]. In all phenotypes, the OR associated with siblings had greater point estimation than the parent-offspring OR. Based on the results from the model (for instance in WTR), odds of a participant with affected sibling is 94% more than a participant with healthy sibling. Irrespective of the OR of sibling in the oblique phenotype with a greater value and wider range, the strongest relationship is associated with ATR.
Table 2: Estimation of familial aggregation based on the marginal model (Tehran Eye Study, Iran, 2002)

Click here to view

Then we fitted the conditional model. The estimation of familial aggregation is represented in [Table 3]. Therefore, the addition of an affected member into the family increases the odds of developing all four phenotypes of the disorder [Table 3]. For instance, addition of a member with WTR to the family can increase the odds of the disorder for a member of that family by 91%. Dividing the total astigmatism into three distinct phenotypes caused an increase in the OR in all cases with the strongest relation associated with the ATR.
Table 3: Estimation of familial aggregation based on the conditional model (Tehran Eye Study, Iran, 2002)

Click here to view

   Discussion Top

Two strategies were applied in the present study, which distinguish it from other studies: division of astigmatism into three phenotypes and application of conditional models together with conventional marginal models. The results of the present study suggest that division of astigmatism into three phenotypes (i.e., ATR, WTR, and oblique), in both marginal and conditional models, can play an effective part in confirming the role of genetics and/or environment in developing astigmatism. [Table 2] and [Table 3] indicate that in cases where the OR is significant, dividing the astigmatism into three phenotypes (WTR, ATR, and oblique) has increased the OR in both conditional and marginal models. This seems to be natural due to the invariant number of participants, reduced positive cases, and increased negative cases because of the division and considering the OR formula. The OR increase has been significant only in the marginal model and for the parent-offspring relation of the oblique phenotype. Based on the conditional model, addition of an affected member to the family increases the odds of developing total astigmatism by 50%, while the increase for the other three phenotypes is almost double. This increase for the marginal model and in relation with siblings is more than double on average; and for the parent-offspring relation, it is approximately double. However, the figures are 54% and 35% for total astigmatism respectively. These findings may contradict previous research where astigmatism was not divided into three phenotypes as we did in the present study.

On studying twins, Teikari and colleagues, after adjusting for effects of environmental factors on astigmatism, showed that genetic factors have no effects on inducing astigmatism. [7] Valluri et al. also found no significant mean differences between the monozygotic and dizygotic twins in the magnitude or axis of astigmatic vector and horizontal or vertical meridians. [8] However, the results of the study by Hammond et al. [2] and Dirani et al. [3] contradicted the findings of earlier studies. In a study by Teikari et al., [7] in addition to the small sample size, the age range of participants was very limited. In another study by Dirani et al., [3] the correlation between data from twins has been calculated by the Pearson correlation coefficient. Agresti [26] stated that model-based estimations are far better than sample-based results, except when the sample is quite large. Therefore, calculating a correlation coefficient cannot guarantee valid results by itself. Twin studies have some advantages but the major disadvantage is the small sample size. Previous studies with [2],[7] small sample size can affect the test power and lead to tests with lower powers in differentiating significant differences. Clementi et al. [4] studied 125 probands and 351 first-degree relatives and found that when the disease was defined as affected/healthy and either eyes astigmatism α1 D, the no-familial-transmission hypothesis was rejected. Also, no significant differences were observed between the two groups with regard to the degree of astigmatism in a study by Cagigrigoriu et al. [11] On the other hand, case-control studies are associated with some degrees of bias. [27],[28] Furthermore, a detailed discussion is presented on how to determine the proband when there is more than one proband in each group by Matthews et al. [29] This is a point which is barely considered in earlier studies (e.g., in Clementi et al. or Cagigrigoriu et al.). [4],[11] In a cross-sectional study, Hashemi et al. [5] stressed familial aggregation; in another epidemiological study Grijbosky et al. [6] emphasized the effects of genetics on inducing astigmatism. However, in a cross-sectional population-based study, Jenny et al. [9] found that the degree of astigmatism was similar among children between 6 years and 12 years old with both healthy and affected parents. Additionally in a cohort population-based study, Lee et al. [10] did not confirm the role of genetics. [11] In a study of 6- to12-year-old children and their parents by Jenny et al., [9] only parents with contact lenses were included, which led to some bias. Other examples include Grjiobovski et al.'s [6] study that was based on the response rate (65%) for self-reports and postal surveys; and Hammond et al. who focused only on female patients. [2] Hence, the results of various studies on the effects of genetic and environmental factors on astigmatism are controversial. In other words, the study type can make a difference in obtaining different results.

Another problem in earlier research is the exclusive application of marginal models. Due to the higher homogeneity of the environmental and genetic factors within families, we believe that obtaining the risk of exposure to a certain disease for a person based on the intrafamily conditions can provide us with more valid results than what we obtain in marginal models, where the risk calculated for a pair (e.g., sib-sib) is calculated across the community.

Another advantage of the conditional model used in the present study is that it allows the researcher to determine the coaggregation besides determining the familial aggregation. [30],[31] Therefore, we fitted the conditional model in the first step which is one of the most common procedures in the epidemiology of diseases. The application of the conditional model QEM as the primary model, which did not have the problems of the early conditional models, division of astigmatism into three phenotypes and a cross-sectional population-based study, and an appropriate sample size have all contributed to solving the aforementioned problems to a great extent, and to making the results more reliable in the relevant community. The present search suggests that different phenotypes of astigmatism have not been investigated with conditional models.

   Conclusion Top

The results of the present study suggested that a relatively average familial aggregation adjusted for variables such as age, gender, cataract, duration of education, occupation, and BMI in the population of Tehran, Iran. Addition of an affected member into the family, having an affected sibling and/or parents significantly increased the odds of developing the disorder. Also, division of astigmatism into three phenotypes (WTR, ATR, and oblique) strengthened the significance of relations.

   Acknowledgments Top

This project was partly supported by a grant from the Iranian National Research Center for Medical Sciences, and the random sampling for the study was performed by the Iranian Statistics Center according to the 1996 national census. The study was supported by Tehran University of Medical Sciences as a PhD thesis.

   References Top

1.Read SA, Collins MJ, Carney LG. A review of astigmatism and its possible genesis. Clin Exp Optom 2007;90:5-19.  Back to cited text no. 1
2.Hammond CJ, Snieder H, Gilbert CE, Spector TD. Genes and environment in refractive error: The twin eye study. Invest Ophthalmol Vis Sci 2001;42:1232-6.  Back to cited text no. 2
3.Dirani M, Islam A, Sbekar SR, Baird PN. Dominant genetic effects on corneal astigmatism: The genes in myopia (GEM) twin study. Invest Ophthalmol Vis Sci 2008;49:1339-44.  Back to cited text no. 3
4.Clementi M, Angi M, Forabosco P, Di Gianantonio E, Tenconi R. Inheritance of astigmatism: Evidence for a major autosomal dominant locus. Am J Hum Genet 1998;63:825-30.  Back to cited text no. 4
5.Hashemi H, Hatef E, Fotouhi A, Mohammad K. Astigmatism and its determinants in the Tehran population: The Tehran eye study. Ophthalmic Epidemiol 2005;12:373-81.  Back to cited text no. 5
6.Grjibovski AM, Magnus P, Midelfart A, Harris JR. Epidemiology and heritability of astigmatism in Norwegian twins: An analysis of self-reported data. Ophthalmic Epidemiol 2006;13:245-52.  Back to cited text no. 6
7.Teikari J, O'Donnell J, Kaprio J, Koskenvuo M. Genetic and environmental effects on oculometric traits. Optom Vis Sci 1980;66:594-9.  Back to cited text no. 7
8.Valluri S, Minkovitz JB, Budak K, Essary LR, Walker RS, Chansue E, et al. Comparative corneal topography and refractive variables in monozygotic and dizygotic twins. Am J Ophthalmol 1999;127:158-63.  Back to cited text no. 8
9.Ip JM, Kifley A, Rose KA, Mitchell P. Refractive findings in children with astigmatic parents: The Sydney Myopia Study. Am J Ophthalmol 2007;144:304-6.  Back to cited text no. 9
10.Lee KE, Klein BE, Klein R, Fine JP. Aggregation of refractive error and 5-year changes in refractive error among families in the Beaver Dam Eye Study. Arch Ophthalmol 2001;119:1679-85.  Back to cited text no. 10
11.Cagigrigoriu A, Gregori D, Cortassa F, Catena F, Marra A. Heritability of corneal curvature and astigmatism: A videokeratographic child-parent comparison study. Br J Ophthalmol 2001;85:1470-6.  Back to cited text no. 11
12.Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data. 2 [nd] ed. Oxford: Oxford University Press; 2002.  Back to cited text no. 12
13.Matthews AG, Finkelstein DM, Betensky RA. Analysis of familial aggregation in the presence of varying family size. Appl Stat 2005;54:847-62.  Back to cited text no. 13
14.Guo SW. Inflation of sibling recurrence-risk ratio, due to ascertainment bias and/or overreporting. Am J Hum Genet 1998;63:252-8.  Back to cited text no. 14
15.Hashemi H, Fotouhi A, Mohammad K. The Tehran Eye Study: Research design and eye examination protocol. BMC Ophthalmol 2003;3:8.  Back to cited text no. 15
16.Lee Y, Nelder JA. Conditional and marginal models: Another view. Stat Sci 2004;19:219-38.  Back to cited text no. 16
17.Carey V, Zeger SL, Diggle P. Modeling multivariate binary data with alternating logistic regressions. Biometrika 1993;80:517-26.  Back to cited text no. 17
18.Zhao LP, Prentice RL. Correlated binary regression using a quadratic exponential model. Biometrika 1990;77:642-8.  Back to cited text no. 18
19.Betensky RA, Whittemore AS. An analysis of correlated multivariate binary data: Application to familial cancers of the ovary and breast. Appl Stat 1996;45:411-29.  Back to cited text no. 19
20.Jewell NP. Statistics for epidemiology. Boca Raton: Chapman & Hall/CRC; 2004.  Back to cited text no. 20
21.Betensky RA, Williams PL, Lederman HM. A comparison of models for clustered binary outcomes: Analysis of a designed immunology experiment. Appl Stat 2001;50:43-61.  Back to cited text no. 21
22.Anderson DA, Aitkin M. Variance component models with binary response. J R Stat Soc Ser B 1985;47:203-10.  Back to cited text no. 22
23.Commenges D, Jacqmin H, Letenneur L, Van Duijn CM. Score test for familial aggregation in probands studies: Application to Alzheimer's disease. Biometrics 1995;51:542-51.  Back to cited text no. 23
24.Korn EL, Whittemore AS. Methods of analyzing panel studies of acute health effects of air pollution. Biometrics 1979;35:745-802.  Back to cited text no. 24
25.Prentice RL. Correlated binary regression with covariate specific to each binary observation. Biometrics 1988;44:1033-48.  Back to cited text no. 25
26.Agresti A. Categorical data analysis, 2 [nd] ed. New York: John Wiley & Sons, Inc; 2002.  Back to cited text no. 26
27.Khoury MJ, Flanders WD. Bias in using family history as a risk factor in case-control studies of disease. Epidemiology 1995;6:511-9.  Back to cited text no. 27
28.Zimmerman R, Pal DK, Tin A, Ahsan H, Greenberg DA. Method of assessing familial aggregation: Family history measures and confounding in the standard cohort, reconstructed cohort and case-control designs. Hum Hered 2009;68:201-8.  Back to cited text no. 28
29.Matthews AG, Finkelstein DM, Betensky RA. Analysis of familial aggregation studies with complex ascertainment schemes. Stat Med 2008;27:5076-92.  Back to cited text no. 29
30.Hudson JI, Laird NM, Betensky RA. Multivariate logistic regression for familial aggregation of two disorders: I. Development of models and methods. Am J Epidemiol 2001;153:500-5.  Back to cited text no. 30
31.Hudson JI, Laird NM, Betensky RA. Multivariate logistic regression for familial aggregation of two disorders: II. Analysis of studies of eating and mood disorders. Am J Epidemiol 2001;153:506-14.  Back to cited text no. 31


  [Table 1], [Table 2], [Table 3]


    Similar in PUBMED
   Search Pubmed for
   Search in Google Scholar for
 Related articles
    Access Statistics
    Email Alert *
    Add to My List *
* Registration required (free)  

  In this article
    Materials and Me...
    Article Tables

 Article Access Statistics
    PDF Downloaded91    
    Comments [Add]    

Recommend this journal