Threshold ModelsCategorical DataStandard Normal DistributionCumulative ProbabilityUnivariate Normal DistributionTwo Categorical TraitsCategorical Data for TwinsJoint Liability Model for TwinsExpected Proportions for BNCorrelated DimensionsACE Liability Twin ModelFit to Ordinal Data in MxModel Fitting to CTExpected ProportionsChi-square StatisticModel Fitting to Raw DataSlide 17Numerical IntegrationHGEN619 2006Thanks to Fruhling Rijsdijk Threshold ModelsCategorical DataMeasuring instrument is able to only discriminate between two or a few ordered categories : e.g. absence or presence of a diseaseData therefore take the form of counts, i.e. the number of individuals within each categoryUnderlying normal distribution of liability with one or more thresholds (cut-offs) is assumedStandard Normal DistributionLiability is a latent variable, the scale is arbitrary, distribution is, therefore, assumed to be a (SND) Standard Normal Distribution or z-distribution:mean () = 0 and SD () = 1z-values are number of SD away from meanarea under curve translates directly to probabilities > Normal Probability Density function ()-33-1 012-2 68%Z0Area 0 .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)Univariate Normal DistributionFor one variable it is possible to find a z-value (threshold) on SND, so that proportion exactly matches observed proportion of samplei.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 TraitsWith two categorical traits, data are represented in a Contingency Table, containing cell counts that can be translated into proportions Trait2Trait10 1000 01 110 11 0=absent 1=present Trait2Trait10 10 545 (.76) 75(.11) 1 56(.08) 40(.05)Categorical Data for TwinsWith dichotomous measured trait i.e. disorder either present or absent in unselected samplecell a: number of pairs concordant for unaffectedcell d: number of pairs concordant for affectedcell b/c: number of pairs discordant for disorder0 = unaffected1 = affected Twin2Twin10 1000 a01 b110 c11 dJoint Liability Model for TwinsAssumed to follow a bivariate normal distributionShape of bivariate normal distribution is determined by correlation between traitsExpected proportions under distribution can be calculated by numerical integration with mathematical subroutinesR=.00 R=.90Liab 2Liab 10 10 .87 .05 1 .05 .03 R=0.6Th1=1.4Th2=1.4Expected Proportions for BNCorrelated DimensionsCorrelation (shape) and two thresholds determine relative proportions of observations in 4 cells of CTConversely, sample proportions in 4 cells can be used to estimate correlation and thresholds Twin2Twin10 1000 a01 b110 c11 dadbcacbdACE Liability Twin ModelVariance decomposition (A, C, E) can be applied to liability, where correlations in liability are determined by path modelThis leads to an estimate of heritability of liability111Twin 1 C EA L C AE LTwin 2Unaf¯AfUnaf¯Af1/.5Fit to Ordinal Data in MxSummary Statistics: Contingency TablesBuilt-in function for maximum likelihood analysis of 2 way contingency tablesLimited to two variablesRaw DataBuilt-in function for raw data MLMore flexible: multivariate, missing data, moderator variablesFrequency DataModel Fitting to CTFit function is twice the log-likelihood of the observed frequency data calculated as:Where nij is the observed frequency in cell ijAnd pij is the expected proportion in cell ij Expected proportions calculated by numerical integration of the bivariate normal over two dimensions: the liabilities for twin1 and twin2Expected ProportionsProbability that both twins are affected:where Φ is bivariate normal probability density function, L1 and L2 are liabilities of twin1 and twin2, with means 0, and is correlation matrix of two liabilities (proportion d)Example: for correlation of .9 and thresholds of 1, d is around .12Probability 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 Σ02121),;,( dLdLLLT T Σ0dBBL2L1aChi-square Statisticlog-likelihood of the data under the model subtracted fromlog-likelihood of the observed frequencies themselvesThe model’s failure to predict the observed data i.e. a bad fitting model, is reflected in a significant χ²totijcjijrinnnL ln2ln211Model Fitting to Raw Dataordinal ordinalZyg response1 response21 0 01 0 01 0 12 1 02 0 01 1 12 . 12 0 .2 0 1Expected ProportionsLikelihood of a vector of ordinal responses is computed by Expected Proportion in corresponding cell of Multivariate normal Distribution (MN), e.g.where is MN pdf, which is function of , correlation matrix of variablesExpected proportion are calculated by numerical integration of MN over n dimensions. In this example two, the liabilities for twin1 and twin2.By maximizing the likelihood of data under a MN distribution, ML estimate of correlation matrix and thresholds are obtained2121),( dxdxxxT T 2121),( dxdxxxTT2121),( dxdxxxTT2121),( dxdxxxT T 2121),( dxdxxxT T (0 1)(1 0)(0 0)(1 1)Numerical
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