1HGEN619 2006Thanks to Fruhling RijsdijkThreshold ModelsCategorical Datan Measuring instrument is able to only discriminate between two or a few ordered categories : e.g. absence or presence of a diseasen Data therefore take the form of counts, i.e. the number of individuals within each categoryn Underlying normal distribution of liability withone or more thresholds (cut-offs) is assumed2Standard Normal Distributionn Liability is a latent variable, the scale is arbitrary, distribution is, therefore, assumed to be a (SND)Standard Normal Distribution or z-distribution:¨ mean (µ) = 0 and SD (σ) = 1¨ z-values are number of SD away from mean¨ area under curve translates directly to probabilities > Normal Probability Density function (Φ)-33-10 12-268%Z0Area0 .50 50%.2 .42 42%.4 .35 35%.6 .27 27%.8 .21 21%1 .16 16%1.2 .12 12%1.4 .08 8%1.6 .06 6%1.8 .036 3.6%2 .023 2.3%2.2 .014 1.4%2.4 .008 .8%2.6 .005 .5%2.8 .003 .3%2.9 .002 .2%-33z0Area=P(z ≥ z0)Cumulative ProbabilityStandard Normal Cumulative Probability in right-hand tail(For negative z values, areas are found by symmetry)3Univariate Normal Distributionn For one variable it is possible to find a z-value(threshold) on SND, so that proportion exactly matches observed proportion of samplen i.e if from sample of 1000 individuals, 150 have met a criteria for a disorder (15%): the z-value is 1.04-331.04Two Categorical Traitsn With two categorical traits, data are represented in a Contingency Table, containing cell counts that can be translated into proportionsTrait2Trait10 1000 01 110 11 0=absent 1=presentTrait2Trait10 10545(.76)75(.11) 156(.08) 40(.05)4Categorical Data for Twinsn With dichotomous measured trait i.e. disorder either present or absent in unselected sample¨ cell a: number of pairs concordant for unaffected¨ cell d: number of pairs concordant for affected¨ cell b/c: number of pairs discordant for disorder0 = unaffected1 = affectedTwin2Twin10 1000 a01 b110 c11 dJoint Liability Model for Twinsn Assumed to follow a bivariate normal distributionn Shape of bivariate normal distribution is determined by correlation between traitsn Expected proportions under distribution can be calculated by numerical integration with mathematical subroutinesR=.00 R=.905Liab 2Liab 10 10 .87 .05 1.05 .03 R=0.6Th1=1.4Th2=1.4Expected Proportions for BNCorrelated Dimensionsn Correlation (shape) and two thresholds determine relative proportions of observations in 4 cells of CTn Conversely, sample proportions in 4 cells can be used to estimate correlation and thresholdsTwin2Twin10 1000 a01 b110 c11 dadbcacbd6ACE Liability Twin Modeln Variance decomposition (A, C, E) can be applied to liability, where correlations in liability are determined by path modeln This leads to an estimate of heritability of liability111Twin 1CEALCAELTwin 2Unaf¯AffUnaf¯Aff1/.5Fit to Ordinal Data in Mxn Summary Statistics: Contingency Tables¨ Built-in function for maximum likelihood analysis of 2 way contingency tables¨ Limited to two variablesn Raw Data¨ Built-in function for raw data ML¨ More flexible: multivariate, missing data, moderator variables¨ Frequency Data7Model Fitting to CTn Fit function is twice the log-likelihood of the observed frequency data calculated as:¨ Where nij is the observed frequency in cell ij¨ And pij is the expected proportion in cell ijn Expected proportions calculated by numerical integrationof the bivariate normal over two dimensions: the liabilities for twin1 and twin2Expected Proportionsn Probability that both twins are affected:¨ where F is bivariatenormal probability density function, L1 and L2 are liabilities of twin1 and twin2, with means 0, and Σ is correlation matrix of two liabilities (proportion d)¨ Example: for correlation of .9 and thresholds of 1, d is around .12n Probability that both twins are below threshold:¨ is given by integral function with reversed boundaries (proportion a) ¨ a is around .80 in this example2121),;,( dLdLLLT T∫∫∞ ∞Φ S02121),;,( dLdLLLT T∫ ∫∞− ∞−Φ S0dBBL2L1a8Chi-square Statisticn log-likelihood of the data under the model subtracted fromn log-likelihood of the observed frequencies themselvesn The model’s failure to predict the observed data i.e. a bad fitting model, is reflected in a significant ?²=∑∑==totijcjijrinnnL ln2ln211Model Fitting to Raw Dataordinal ordinalZyg response1 response21 0 01 0 01 0 12 1 02 0 01 1 12 . 12 0 .2 0 19Expected Proportionsn Likelihood of a vector of ordinal responses is computed by Expected Proportion in corresponding cell of Multivariate normal Distribution (MN), e.g.¨ where Φ is MN pdf, which is function of Σ, correlation matrix of variablesn Expected proportion are calculated by numerical integration of MN over n dimensions. In this example two, the liabilities for twin1 and twin2.n By maximizing the likelihood of data under a MN distribution, MLestimate of correlation matrix and thresholds are obtained2121),( dxdxxxT T∫ ∫∞− ∞−Φ2121),( dxdxxxTT∫∫∞−∞Φ2121),( dxdxxxTT∫ ∫∞∞−Φ2121),( dxdxxxT T∫ ∫∞− ∞−Φ2121),( dxdxxxT T∫∫∞ ∞Φ(0 1)(1 0)(0 0)(1 1)Numerical
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