Analyzing Ordinal Data With Linear Models

Transcription

1 Analyzing Ordinal Data With Linear Models Consequences of Ignoring Ordinality In statistical analysis numbers represent properties of the observed units. Measurement level: Which features of numbers correspond to empirical meanings? Nominal scale: Ordinal scale: Metric scale: If two units are represented by same numbers, they have the same property; if two units are represented by different numbers, they have different properties. If one unit is represented by a larger number than a second unit, than the first unit has more of a property than the second unit. If one unit is represented by a larger number than a second unit, than the difference of the numbers represent how much more of a property the first unit has compared with the second unit. It depends on reality and theory about reality which measurement level is given. 1

2 Problems of Regression Based on Ordinal Data Case 1: Depending variable is metric, predictor variable is ordinal Y Regression using rank numbers of X Y Regression using true distances of X True distance between category 1 and category X True distance between category 2 and category 3 Then: Regression on X conceals the exact functional form of a monotonic relationship. Consequences: (a) Ignoring ordinality can be captured by non-linear regression functions. (b) If predictors are ordinal, there is no meaningful difference between linear and non-linear monotonic relations. X* 2

3 Case 2: Depending variable is ordinal, predictor variable is metric or ordinal Y Regression based on ranks of Y Rank distance between first and second category Y* Regression based on true values 1 2 True distance 3 between first and second category Because means utilize on distances, means and variances are not defined for ordinal variables. Consequences: It is not possible to infer from linear regression of Y on X whether a relationship is positive, negative, or non-monotonic, or whether there is any relationship at all. Ignoring ordinality of dependent variables can cause meaningless results. 3

4 Coping with Ordinality in Linear Models: Polychoric Correlation: To cope with ordinality in linear models it is assumed that the ordinal variables X and Y are crude measures of unobserved metric continuous variables X * and Y *. Then any category x i of X will be observed if and only if a realization of X * is in the interval between the thresholds τ i-1 and τ i. In the same way any category y j of Y will be observed if and only if a realization of Y * is in the interval between the thresholds τ j-1 and τ j. Mathematically: Prob(X= i) = Prob( τ < X τ) i 1 * i and Prob(Y= j) = Prob( τ < Y τ ) j1 * j Then: The probability of any cell in the contingency table is given as the volume under the bivariate density in the regions defined by the thresholds. The polychoric correlation is an estimate of the product moment correlation between X* and Y* X Y 1 2 4

5 The Estimation of Polychoric Correlations and Covariances From Prob(Y= k) = Prob( τ k 1< Y τk) a step function: Y * it follows, that Y is related on Y * by Y=3 Y=k Y=2 Y=1 τ 1 τ 2 τ k-1 Y * If the thresholds are unknown, not only the polychoric correlation but also the thresholds have to be estimated. 5

6 The Estimation of Polychoric Correlations and Covariances Because the realizations of an ordinal category depends on the probability distribution of the underlying continuous variable it is necessary to assume a distribution for the unobserved variables Y *. Usually it is assumed that the underlying continuous variable Y * is normal distributed. If Y * is normally distributed, the following relation holds: τ (Y * ) 2 1 µ 2 σ 2 k * 1 Prob(Y τ k ) = e dy* σ 2π τ ( k µ = Φ ) σ τ 1 τ 2 The formula shows that the probabilities are given by the areas under the standard normal curve: f(y*) Y=1 Y=2 Y= ,

7 Indetermination of Mean and Variance From the formula above or the figure it can be easily justified that the translation of the curve to the right or to the left may be compensated by corresponding shifts of all thresholds. Therefore, the thresholds and the mean of Y * cannot be estimated independently. Either one threshold or the mean must be fixed to an a priori number. τ 1 τ 2 f(y*) An increase in the variance of Y * will result in a flatter curve or equivalently in a rescaling of the horizontal axis. Again this can be compensated by the values of the thresholds. That means that also the variance of Y * cannot be estimated independent of the values of the thresholds. This is a reflection of the fact that an ordinal variable has no mean and no variance. Statistically, both the mean and the variance of the underlying metric variables are not identified. If there is no other information, often the mean will be fixed to zero and the variance to one (standard parametrization).an alternative is to fix the first two thresholds to 0 and 1(Alternative parametrization). 7

8 Estimation of thresholds: In standard parametrization the thresholds can be estimated by the inverse of the cumulative standard normal distribution. If the sample size is n and n k is the number of cases in category k the thresholds are estimated by: τ =Φ 0 1 ( 0) 1 τ 1 1 =Φ 1 τ =Φ 1 τ 1 2 k 1 K 1=Φ K... 1 n n n n ( 1) = + n n τ =Φ =+ + n +...+n n Example data: (crosstab.dat) X Y τ 1 =Φ =Φ (0.275) = τ 2 =Φ =Φ (0.725) = whereφ -1 (y) denotes the inverse of the both for X * and Y *. standard normal distribution function, that is the z-value for which the probability that a standard normal distribution is less or equal is given by the proportion in the argument of the function. 8

9 Estimation of polychoric correlations To estimate the correlation ρ Y*X* between two unobserved continuous variables Y* and X* it is assumed that Y* and X* have a standard bivariate normal distribution. Then the probabilities of any cell of the contingency table is given by: *2 *2 * * 2 ( ( + + ρ ) ρ ) 1 2 YX YX * * τ τ i j exp Y X 2 * * Y X /(1 * * ) Prob(Y= i,x= j) = Y X τ τ 2π 1 ρ 2 YX i 1 j1 * * = Φ ( τ, τ, ρ * *) Φ ( τ, τ, ρ * *) Φ ( τ, τ, ρ * *) +Φ τ τ ρ 2 i j YX 2 i 1 j YX 2 i j1 YX 2( i 1, j1, * *) YX where Φ 2 (y,x,ρ) denotes the probability that a bivariate standard normal distribution with correlation ρ is smaller or equal X=x and Y=y. Given the bivariate frequencies the polychoric correlation of Y and X can be estimated by maximum-likelihood (ML). The ML-estimation is that correlation, that maximizes the probability that the observed data could be realized by a bivariate normal distribution. 9

10 For each case in the sample the probability of its realization is given by the equation above. If the units are sampled independently, the log-likelihood-function, that is the logarithm of the probability of the whole sample given the correlation, is: I J Φ2( τi, τj, ρ * *) Φ2( τi, τj1 * * 2 i 1 j * * YX, ρ ) Φ ( τ YX, τ, ρ ) YX lnl( ρ * *) = n YX ij ln i= 1j1 = 2( i 1, j1, * *) +Φ τ τ ρ YX For the example data and the threshold estimated before, the log-likelihood is lnl = 25 ln Φ2(.598,.598, ρ) + 20 ln Φ (.598,.598, ρ) Φ (.598,.598, ρ) + 10 ln Φ (.598) Φ (.598,.598, ρ) + 20 ln Φ (.598,.598, ρ) Φ (.598,.598, ρ) + 50 ln Φ (.598,.598, ρ) Φ (.598,.598, ρ) Φ (.598,.598, ρ ) +Φ ( ln Φ(.598) Φ (.598) Φ (.598,.598, ρ ) +Φ (.598,.598, ρ) + 10 ln Φ (.598) Φ (.598,.598, ρ) + 20 ln Φ(.598) Φ (.598) Φ (.598,.598, ρ ) +Φ (.598,.598, ρ) + 25 ln 1 Φ(.598) Φ (.598) +Φ (.598,.598, ρ) ( 2 2 ) ( 2 ) ( 2 2 ) ( ,.598, ρ) ) ( 2 2 ) ( 2 ) ( 2 2 ) ( ) where Φ 2 (,x,ρ)=φ(x), Φ 2 (y,,ρ)=φ(y), and Φ 2 (-,x,ρ)=φ 2 (y,-,ρ)=0. 2 The ML-estimation is the value of ρ, where lnl reached a maximum or where -lnl reached a minimum. In the example the maximum is reached, when ρ=

11 Estimation of covariances, standard-deviations and means Using alternative parametrization the first two thresholds are fixed to 0 and 1. Then not only the polychoric correlations, but also the mean and the standard deviation can be estimated. Because reparametrizations can be transformed in each other, it is also possible to compute the alternative parametrization from the standard parametrization. The mean and the standard deviations can be computed from the first two estimated thresholds by: τ1 1 µ= ; σ= τ τ τ τ The transformed thresholds are than: τ τ τ i = τ τ * i From this transformation it can be easily seen that the values of the first two transformed thesholds are 0 and 1. For the example data the alternative parametrization for both indicators becomes: a) standard parametrization τ 1 = 0.598, τ 2 = , µ = 0, σ=1, ρ=0.341 b) alternative parametrization τ 1 = 0, τ 2 = 1, µ =-(-0.598)/( ) = 0.5; σ=1/( ) = Σ= =

12 Estimation of Standard errors and covariances between estimations of polychoric correlations For statistical inferences it is necessary to estimate not only the polychoric correlations, but also their standard errors and covariances between pairs of polychoric correlations. Using alternative parametrization the asymptotic covariances of the covariance matrix is needed. There are three methods to estimate these (asymptotic) variances and covariances of the polychoric correlations: (a) As the correlations their variances and covariances can be estimated from the data too. These estimation is based on multivariate crosstabulations of more than two variables. From a statistical point of view, thresholds, polychoric correlations and their variances and covariances should be estimated simultaneously by ML-estimation based on a multidimensional contingency table. Because the effort would be to high, estimations are done stepwise. But even then the computations are very tedious and different computer programs use different algorithms resulting sometimes in slightly different estimations. (b) The variances and covariances can be estimated by non-parametric bootstrapping. From the empirical data a lot of bootstrap-samples are drawn by simple random sampling with replacement. Then for each subsample the polychoric correlations are computed. The estimated variances and covariances of the polychoric correlations are given by the empirical variances and covariances of the correlations across the bootstrap-samples. 12

13 (c) The variances and covariances can be estimated by parametric bootstrapping or simulation. In the first step of parametric bootstrapping all parameters of a model are estimated. Here the parameters are the estimated thresholds and the polychoric correlations, or the thresholds and the parameters of a statistical model for the polychoric correlations. Next these parameters will be used to generate samples of simulated data. The variances and covariances of the estimated parameters across the simulated data samples can be used as estimation of the variances and covariances of the parameters. Whereas in usual bootstrap-samples the (same) empirical data are used repeatedly, in parametric bootstrapping new data are generated by simulation. Remarks: Multivariate causal analysis of ordinal data using polychoric correlations is based on a priori assumptions: (1) It is assumed that all ordinal variables are crude measures of underlying continuous variables. If ordinal variables are thought of as measures where distances are not only unknown but not defined, the logic of polychoric correlations does not fit. Further, if it makes no sense to assume that the property - one is interested in - is not continuously, the logic of polychoric correlations does not fit. 13

14 (2) It is assumed that the relation between the crude measures and the underlying continuous variable can be formalized by a threshold model. Note, that the model postulate that the threshold are parameters that are valid for all units of a population. Nevertheless, simulation give hints that the model is stable even if the threshold are random variables varying across units. (3) Without an assumption on the the distribution of the underlying continuous variables it is impossible to estimate thresholds and polychoric correlations. Usually it is assumed that all underlying continuous variables are normal distributed and that each pair of variables are binormal distributed. If this assumption does not hold, one can try to collaps categories or to reduce the number of variables. Further, simulation studies give hints that the true correlations can be estimated even if the assumption of binormality is false. That is, the estimation algorithm seems to be robust with respect to the distribution of the continuous variables. (4) Correlations are sufficient association measures for linear models. Therefore, it is possible to estimate linear models. But it is not possible to estimate non-linear relationships between the underlying continuous variables. Theory who stated nonlinear relations cannot be tested empirically using this approach. 14

15 An example: Estimating the polychoric correlations or polychoric covariances of the efficacy items The Allbus raw data of the 7 items to model political efficacy are stored in the file allb96s3.dat. The following PRELIS-command are used to estimate the polychoric correlations and their asymptotic covariances for the 7 variables: Read asci raw data and compute polychoric correlations and their (co-) variances for Allbus subset DA NI=7 LA Polint1 Polint2 Impact Election Politicn Governm Leader/ RA=allb96s3.dat FO (7F1.0) OR ALL OU MA=PM PM=allb96s1.pm ac=allb96s1.acp BT 15

16 Selected output: Total Sample Size = 1882 Univariate Marginal Parameters Variable Mean St. Dev. Thresholds Polint Polint Impact Election Politicn Governm Leader Bivariate Distributions for Ordinal Variables (Frequencies) Polint2 Impact Polint

17 Correlations and Test Statistics (PE=Pearson Product Moment, PC=Polychoric, PS=Polyserial) Test of Model Test of Close Fit Variable vs. Variable Correlation Chi-Squ. D.F. P-Value RMSEA P-Value Polint2 vs. Polint (PC) W_A_R_N_I_N_G: Underlying bivariate normality may not hold, see BTS-file Impact vs. Polint (PC) Impact vs. Polint (PC) Election vs. Polint (PC) Election vs. Polint (PC) Election vs. Impact (PC) Politicn vs. Polint (PC) Politicn vs. Polint (PC) Politicn vs. Impact (PC) Politicn vs. Election (PC) Governm vs. Polint (PC) Governm vs. Polint (PC) Governm vs. Impact (PC) Governm vs. Election (PC) Governm vs. Politicn (PC) Leader vs. Polint (PC) Leader vs. Polint (PC) Leader vs. Impact (PC) Leader vs. Election (PC) Leader vs. Politicn (PC) Leader vs. Governm (PC)

18 BTS-File: Bivariate Table for Polint1 vs Polint2 Observed Frequencies Rowsum Colsum Expected Frequencies Rowsum Colsum

19 LR Contributions Rowsum Colsum GF Contributions Rowsum ******** 0.0 ******** Colsum ******** 28.3 ******** 19

20 Standardized Residuals Rowsum Colsum

21 Back to the PRELIS Output-file: Correlation Matrix Polint1 Polint2 Impact Election Politicn Governm Polint1 0 Polint Impact Election Politicn Governm Leader Leader Leader 0 Means Polint1 Polint2 Impact Election Politicn Governm Leader

22 Standard Deviations Polint1 Polint2 Impact Election Politicn Governm Leader Alternative Parametrization can be realized by specifying MA=CM or AP on the outputcommand: Read asci raw data and compute polychoric covariances and their (co-) variances for Allbus subset DA NI=7 LA Polint1 Polint2 Impact Election Politicn Governm Leader/ RA=allb96s3.dat FO (7F1.0) OR ALL OU MA=CM CM=allb96s3.cm ac=allb96s3.acc 22

23 Variable Mean St. Dev. Thresholds Polint Polint Impact Election Politicn Governm Leader Covariance Matrix Polint1 Polint2 Impact Election Politicn Governm Polint Polint Impact Election Politicn Governm Leader Leader Leader The covariance matrix can be computed from the polychoric correlation matrix by multiplying the vector of standard deviations from left and from right. 23

24 Analyzing Correlations or Covariances? As a consequences of ordinal data, with standard parametrization correlations instead of variances and covariances will be analyzed. If a covariance structure model is specified it is not certain that all diagonal elements have the estimations one. Because the diagonal ones are fixed numbers they have no asymptotic variances and covariances. LISREL uses a different WLS fit-function if the matrix to be analyzed is a correlation matrix: k i 1 k l 1 k F = w (r ρˆ )(r ρ ˆ ) + (1 ρˆ ) WLS ij,lm ij ij lm lm ii i= 1 j= 1 l= 1 m= 1 i= 1 The weights w ij,lm are the asymptotic variances and covariances of the (polychoric, polyserial or product moment) correlations, estimated by PRELIS. Note, that this formula is an ad-hoc solution. It is not guaranteed that the fitted diagonals are ones. 24

25 Correlation Structures and Covariance Structures In a covariance matrix, the diagonal elements are sample variances. These elements are estimations of the population variances. In a correlation matrix, the diagonal elements are fixed numbers. Therefore, the diagonal elements contain no empirical information that can be used to estimate a linear model. On the other hand, we have seen, that even when analyzing a matrix of polychoric correlations the diagonal elements are used in the estimation of the model parameters. In the example of the 1-factor model the diagonal ones are used to estimate the error covariances: ξ 1 λ λ λ 2 1 λ +θ ρ XX 1 = λ λ +θ ρxx ρ 3 1 XX 1 λ λ λ +θ X 1 X 2 X 3 δ 1 δ 2 δ 3 θ ˆ 11 = 1 λˆ 2 The diagonal elements are used in the estimation because formally a LISREL model is a covariance structure. 25

26 In a covariance structure variances and covariances are functions of parameters; in a correlation structure correlations are functions of parameters. Does it matter if a covariance structure is estimated by a correlation matrix? There are three problems that can occur if a correlation matrix is analyzed as if it is a covariance matrix: it is not possible to estimate specific restrictions the chi-square goodness-of-fit statistic may be incorrect standard errors may be incorrect. An example for the first problem is, that analyzing correlations it is not possible to specify tau-equivalent measures. If loadings are restricted to be equal, measurement error variances will be equal too. Estimating a covariance structure using correlations the chi-square goodness-of-fit statistic is often correct, if two condition are given: the model have to be scale invariant the diagonal elements are fitted to one that is all diagonal residual elements are zero. Even if the chi-square is correct, the standard errors may be wrong. 26

27 Differences can be seen, if the efficacy model is estimated based on polychoric correlations by ML-method or by WLS-method: Model for Efficacy based on polychoric correlations: ML DA NI=7 NO=1882 MA=CM CM=allb96s1.pm! ac=allb96s1.acp LA Polint1 Polint2 Impact Election Politicn Governm Leader / MO NY=7 NE=4 LY=FU,FI BE=FU,FI PS=SY TE=DI,FR LE POLINT EFFICACY TRUST LEADER FI TE(7,7) FR PS(4,1) BE(2,1) BE(3,1) BE(2,3) BE(2,4) BE(3,4) VA LY(1,1) LY(2,1) LY(3,2) LY(4,2) LY(5,3) LY(6,3) LY(7,4) PD OU AD=Off SO RS MI SS 27

28 POLINT Polint Polint EFFICACY TRUST 0.22 Impact 0.68 Election 0.69 Politicn 0.45 LEADER Governm 0.43 Leader 0.00 Chi-Square=41.57, df=12, P-value= , RMSEA=

29 Estimating method: WLS: Model for Efficacy based on polychoric correlations: WLS DA NI=7 NO=1882 MA=PM pmallb96s1.pm ac=allb96s1.acp LA Polint1 Polint2 Impact Election Politicn Governm Leader / MO NY=7 NE=4 LY=FU,FI BE=FU,FI PS=SY TE=DI,FR LE POLINT EFFICACY TRUST LEADER FI TE(7,7) FR PS(4,1) BE(2,1) BE(3,1) BE(2,3) BE(2,4) BE(3,4) VA LY(1,1) LY(2,1) LY(3,2) LY(4,2) LY(5,3) LY(6,3) LY(7,4) PD OU AD=Off SO RS MI SS WL 29

30 POLINT Polint Polint EFFICACY TRUST 0.22 Impact 0.68 Election 0.68 Politicn 0.43 LEADER Governm 0.43 Leader 0.00 Chi-Square=27.13, df=12, P-value= , RMSEA=

31 If polychoric covariances are analyzed again a different result occur: Model for Efficacy based on polychoric covariances DA NI=7 NO=1882 MA=CM CM=allb96s3.cm ac=allb96s3.acc LA Polint1 Polint2 Impact Election Politicn Governm Leader / MO NY=7 NE=4 LY=FU,FI BE=FU,FI PS=SY TE=DI,FR LE POLINT EFFICACY TRUST LEADER FI TE(7,7) FR PS(4,1) BE(2,1) BE(3,1) BE(2,3) BE(2,4) BE(3,4) VA LY(1,1) LY(2,1) LY(3,2) LY(4,2) LY(5,3) LY(6,3) LY(7,4) PD OU AD=Off SO RS MI SS WL 31

32 POLINT Polint Polint EFFICACY TRUST 0.08 Impact 0.39 Election 1.34 Politicn LEADER Governm 0.40 Leader 0.00 Chi-Square=40.94, df=12, P-value= , RMSEA=

33 Restrictions on Thresholds Equal thresholds If the thresholds are known or if they can be restricted it is possible to estimate also variances and covariances In PRELIS it is possible to set the thresholds equal for different variables. Then not only the correlations but also the means and variances of this variables can be estimated. But note that the means of variables with equal thresholds will sum to zero and the variances will sum to one. Example: Equal thresholds for equal formats in the efficacy model: Example with equal thresholds DA NI=7 LA Polint1 Polint2 Impact Election Politicn Governm Leader/ RA=allb96s3.dat FO (7F1.0) OR ALL ET Polint1 Polint2 ET Impact Election Politicn Governm OU MA=PM 33

34 Variable Mean St. Dev. Thresholds Polint Polint Impact Election Politicn Governm Leader Correlation Matrix equal thresholds Polint1 Polint2 Impact Election Politicn Governm Polint1 0 Polint Impact Election Politicn Governm Leader free thresholds Polint1 Polint2 Impact Election Politicn Governm Polint1 0 Polint Impact Election Politicn Governm Leader

35 Computation of polychoric covariances with equal thresholds Example with equal thresholds DA NI=7 LA Polint1 Polint2 Impact Election Politicn Governm Leader/ RA=allb96s3.dat FO (7F1.0) OR ALL ET Polint1 Polint2 ET Impact Election Politicn Governm OU MA=CM CM=a96s1et.cm AC=a96s1et.acc Variable Mean St. Dev. Thresholds Polint Polint Impact Election Politicn Governm Leader

36 Covariance Matrix equal thresholds Polint1 Polint2 Impact Election Politicn Governm Polint Polint Impact Election Politicn Governm Leader Leader Leader Covariance Matrix free thresholds Polint1 Polint2 Impact Election Politicn Governm Polint Polint Impact Election Politicn Governm Leader Leader Leader

37 POLINT Polint Polint EFFICACY TRUST 0.07 Impact 0.45 Election 0.67 Politicn LEADER Governm 0.22 Leader 0.00 Chi-Square=27.79, df=12, P-value= , RMSEA=

38 Fixed thresholds Another possibility to identify means and variances is to fix threshold to given values. This is useful for the same variables in a group comparison. In a first step common thresholds are estimated and saved. These thresholds are used in a second step to compute polychoric covariances and means. Lastly a structured mean model can be estimated. Example: Computing thresholds for group comparions DA NI=7 LA Polint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.raw SD Group OR ALL OU MA=PM TH=a llb96s0. th Univariate Marginal Parameters Variable Mean St. Dev. Thresholds Polint Polint Impact Election Politicn Governm

39 Computing pol. correlations based on given thresholds: West DA NI=7 LA Polint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.raw RE SD Group=0 OR ALL FT=allb96s0.t h Polint1 FT Polint2 ; FT Impact ; FT Election ; FT Politicn ; FT Governm OU MA=PM Computing pm bsed on given thresholds: East DA NI=7 LA Polint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.raw RE SD Group=1 OR ALL FT=allb96s0.th Polint1 FT Polint2 ; FT Impact ; FT Election ; FT Politicn ; FT Governm OU MA=PM 39

40 Computing cm and acc based on constant thresholds: West DA NI=7 LA Polint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.raw RE SD Group=0 OR ALL FT=allb96s0.t h Polint1 FT Polint2 ; FT Impact ; FT Election ; FT Politicn ; FT Governm OU MA=CM CM=a96s1pc.cm me=a96s1pc.me ac=a96s1pc.acc Computing cm and acc based on constant thresholds: East DA NI=7 LA Polint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.raw RE SD Group=1 OR ALL FT=allb96s0.th Polint1 FT Polint2 ; FT Impact, FT Election ; FT Politicn ; FT Governm OU MA=CM CM=a96s2pc.cm me=a96s2pc.me ac=a96s2pc.acc 40

41 Group: West Univariate Marginal Parameters Variable Mean St. Dev. Thresholds Polint Polint Impact Election Politicn Governm Covariance Matrix Polint1 Polint2 Impact Election Politicn Governm Polint Polint Impact Election Politicn Governm

42 Group: East Variable Mean St. Dev. Thresholds Polint Polint Impact Election Politicn Governm Covariance Matrix Polint1 Polint2 Impact Election Politicn Governm Polint1 Polint2 Impact Election Politicn Governm Polint Polint Impact Election Politicn Governm

43 Results of the estimation Group A: West Group B: East Polint1 Polint2 POLINT Polint1 Polint2 POLINT Impact Election EFFICACY Impact Election EFFICACY Politicn TRUST Politicn TRUST Governm 0.40 Governm Chi-Square=80.07, df=27, P-value= , RMSEA=

44 Probit Regression to Estimate Partial Polychoric Correlations If it is possible to divide the variables in ordinal dependent variables and (fixed) metric exogenous variables, the underlying continuous variables can be regressed on the independent variables. Such regression is known as probit regression. Here, the distributional assumption is that the residuals of the variables given the exogenous variables will be standard multinormal distributed. K K k k i k k i 1 k=1 k=1 Prob(Y= i) = Prob Y*- β X <τ Prob Y*- β X <τ PRELIS computes not only the thresholds and the regression coefficients but also the conditional (polychoric) correlations of the residuals. Further given these residual correlation matrix, the regression coefficients, and the variances and covariances of the exogenous variables also the unconditional variances and covariances of all variables may be computed. Additionally the asymptotic variances and covariances of this variances and covariances can be estimated: ˆ ˆ / ˆ Σ= Β ΣX Β +Ρ ; µ ˆ β ˆ = Β ΣX ΣX K ˆ Y* k xk k= 1 44