Analyzing Ordinal Data With Linear Models

1

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: If two units are represented by same numbers, they have the sameproperty; if two units are represented by different numbers, they havedifferent properties.

Ordinal scale: 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.

Metric scale: If one unit is represented by a larger number than a second unit, thanthe difference of the numbers represent how much more of a propertythe first unit has compared with the second unit.

It depends on reality and theory about reality which measurement level is given.

Analyzing Ordinal Data With Linear Models

2

Problems of Regression Based on Ordinal Data Case 1: Depending variable is metric,predictor variable is ordinal..

0

1

2

3

4

5

6

7

8

9

10

1 2 3

Regression using rank numbers of X

X

Y

0

1

2

3

4

5

6

7

8

9

10

1 2 3 4

Regression using true distances of X

True distance between category 1 and category 2

True distance between category 2 and category 3

Y

X*

Then: Regression on X concealsthe exact functional formof a monotonic relationship.

Consequences:(a) Ignoring ordinality can be

captured by non-linearregression functions.

(b) If predictors are ordinal,there is no meaningful difference between linearand non-linear monotonicrelations.

3

Regression based on ranks of Y

-5-4-3-2-101234567

1 2 3

Y

Case 2: Depending variable is ordinal, predictor variable is metric or ordinal.

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.

True distancebetween firstand second category

Rank distancebetween firstand second category

-5-4-3-2-101234567

1 2 3

Y*

Regression based on true values

4

2211

YX00

-1-1

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 xi of X will be observed if and only if a realization of X* is in the interval between the thresholdsτi-1 and τi. In the same way any category yj of Y will be observed if and only if a realization of Y*

is in the interval between the thresholds τj-1 and τj.

Then:The probability of any cell in the contingencytable is given as the volume under the bivariatedensity in the regions defined by the thresholds.

Coping with Ordinality in Linear Models: Polychoric Correlation:

*i 1 iProb(X i) Prob( X )−= = τ < ≤ τ

*j 1 jProb(Y j) Prob( Y )−= = τ < ≤ τand

Mathematically:

The polychoric correlation is an estimate of theproduct moment correlation between X* and Y*

5

The Estimation of Polychoric Correlations and Covariances

*k 1 kProb(Y k) Prob( Y )−= = τ < ≤ τFrom it follows, that Y is related on Y* by

a step function:

Y=1

Y=3

Y=k

Y=2

τ1 τ2 τk-1Y*

Y

If the thresholds are unknown, not only the polychoric correlation but also the thresholds have to be estimated.

6

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2,.5 3.0 3.5 4.0

Y=1 Y=2 Y=3

f(Y*)

τ1 τ2

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:

* 2(Y )1k 2 2*k

k

1Prob(Y ) e dY*

2

( )

−µ

στ − ⋅

−∞≤ τ = ⋅∫

σ πτ − µ

= Φσ

The formula shows that the probabilities are given by the areas under the standard normal curve:

The Estimation of Polychoric Correlations and Covariances

7

f(Y*)

τ1 τ2

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

-4.0 -3.0 -2.0 -1.0 4.03.02.01.00.0

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.

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).

Indetermination of Mean and Variance

8

whereΦ-1(y) denotes the inverse of the 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.

In standard parametrization the thresholds can be estimated by the inverse of the cumulative standard normal distribution.

If the sample size is n and nk is the number of cases in category k the thresholds are estimated by:

Example data: (crosstab.dat)

X1 2 3 ∑

1 25 20 10 55Y 2 20 50 20 90

3 10 20 25 55∑ 55 90 55 200

1 11

1 12

55(0.275) 0.598

200145

(0.725) 0.598200

− −

− −

τ = Φ = Φ = − τ = Φ = Φ = +

both for X* and Y*.

( )

( )

10

1 11

1 1 22

1 1 2 k 1K 1

1K

0

nnn n

n ...

n n +...+nn

1

−

−

−

− −−

−

τ = Φ = −∞

τ = Φ

+ τ = Φ

+ τ = Φ

τ = Φ = +∞

Estimation of thresholds:

9

To estimate the correlation ρY*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:

( )( )j * * * *i

i 1 j 1 * *

* * * * * *

* *

*2 *2 * * 212 Y X Y X * *

2Y X

2 i j 2 i 1 j 2 i j 1Y X Y X Y X2 i 1 j 1 Y X

exp Y X 2 Y X /(1 )Prob(Y i,X j) Y X

2 1

( , , ) ( , , ) ( , , )( , , )

− −

ττ

τ τ

− −

− −

− + + ρ ⋅ ⋅ − ρ= = = ⋅∂ ⋅∂∫ ∫

π − ρ

= Φ τ τ ρ − Φ τ τ ρ − Φ τ τ ρ+Φ τ τ ρ

where Φ2(y,x,ρ) denotes the probability that a bivariate standard normal distribution withcorrelation ρ is smaller or equal X=x and Y=y.

Given the bivariate frequencies the polychoric correlation of Y and X can be estimated bymaximum-likelihood (ML). The ML-estimation is that correlation, that maximizes theprobability that the observed data could be realized by a bivariate normal distribution.

Estimation of polychoric correlations

10

For the example data and the threshold estimated before, the log-likelihood is

( )( )( )

22 2

22 22 2 2 2

ln L 25 ln ( .598, .598, )20 ln ( .598,.598, ) ( .598, .598, )10 ln ( .598) ( .598,.598, )20 ln (.598, .598, ) ( .598, .598, )50 ln (.598,.598, ) ( .598,.598, ) (.598, .598, ) ( .59

= ⋅ Φ − − ρ+ ⋅ Φ − ρ − Φ − − ρ+ ⋅ Φ − − Φ − ρ+ ⋅ Φ − ρ − Φ − − ρ+ ⋅ Φ ρ − Φ − ρ − Φ − ρ + Φ −( )

( )( )( )( )

2 22

2 22

8, .598, )20 ln (.598) ( .598) (.598,.598, ) ( .598,.598, )10 ln ( .598) (.598, .598, )20 ln (.598) ( .598) (.598,.598, ) (.598, .598, )25 ln 1 (.598) (.598) (.598,.598, )

− ρ+ ⋅ Φ − Φ − − Φ ρ + Φ − ρ+ ⋅ Φ − − Φ − ρ+ ⋅ Φ − Φ − − Φ ρ + Φ − ρ+ ⋅ − Φ − Φ + Φ ρ

where Φ2(∞,x,ρ)=Φ(x), Φ2(y,∞,ρ)=Φ(y), and Φ2(-∞,x,ρ)=Φ2(y,-∞,ρ)=0.

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=0.341.

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 2 i j 2 i j 1 2 i 1 jY X Y X Y XijY X i 1 j 1 2 i 1 j 1 Y X

( , , ) ( , , ) ( , , )ln L( ) n ln

( , , )− −

= = − −

Φ τ τ ρ − Φ τ τ ρ − Φ τ τ ρ ρ = ⋅∑ ∑ +Φ τ τ ρ

For each case in the sample the probability of its realization is given by the equation above.

11

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 alternat-ive parametrization from the standard parametrization.

The mean and the standard deviations can be computed from the first two estimated thresholds by:

−τµ = σ =

τ − τ τ − τ1

2 1 2 1

1 ;

The transformed thresholds are than:τ − τ

τ =τ − τ

* i 1i

2 1

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:

Σ = ⋅ ⋅

=

2

2

0.8360.341 0.836 0.836 0.836

0.7000.238 0.700

Estimation of covariances, standard-deviations and means

b) alternative parametrizationτ1 = 0, τ2 = 1,µ =-(-0.598)/(0.598+0.598)

= 0.5;σ=1/(0.598+0.598) = 0.836

a) standard parametrizationτ1 = −0.598, τ2 = +0.598,µ = 0, σ=1, ρ=0.341

12

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 variancesand covariances should be estimated simultaneously by ML-estimation based on amultidimensional contingency table. Because the effort would be to high, estimationsare done stepwise. But even then the computations are very tedious and differentcomputer programs use different algorithms resulting sometimes in slightly differentestimations.

(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 randomsampling with replacement. Then for each subsample the polychoric correlations arecomputed. The estimated variances and covariances of the polychoric correlations aregiven by the empirical variances and covariances of the correlations across thebootstrap-samples.

Estimation of Standard errors and covariances between estimations of polychoric correlations

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 datasamples 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, inparametric bootstrapping new data are generated by simulation.

Remarks:

Multivariate causal analysis of ordinal data using polychoric correlations is based on a prioriassumptions:

(1) It is assumed that all ordinal variables are crude measures of underlying continuousvariables.If ordinal variables are thought of as measures where distances are not only unknown butnot 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 notcontinuously, the logic of polychoric correlations does not fit.

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 evenif the threshold are random variables varying across units.

(3) Without an assumption on the the distribution of the underlying continuous variables itis impossible to estimate thresholds and polychoric correlations. Usually it is assumedthat all underlying continuous variables are normal distributed and that each pair ofvariables are binormal distributed. If this assumption does not hold, one can try to collapscategories 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 ispossible to estimate linear models. But it is not possible to estimate non-linearrelationships between the underlying continuous variables. Theory who stated non-linear relations cannot be tested empirically using this approach.

15

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 correlationsand their (co-) variances for Allbus subsetDA NI=7LAPolint1 Polint2 Impact Election Politicn Governm Leader/ RA=allb96s3.dat FO(7F1.0)OR ALLOU MA=PM PM=allb96s1.pm ac=allb96s1.acp BT

An example:

16

Total Sample Size = 1882

Univariate Marginal ParametersVariable Mean St. Dev. Thresholds-------- ---- -------- ----------Polint1 0.000 1.000 -1.677 -0.704 0.524 1.319Polint2 0.000 1.000 -1.711 -0.733 0.541 1.386Impact 0.000 1.000 -0.980 0.192 0.860 2.039

Election 0.000 1.000 -1.784 -1.012 -0.539 0.811Politicn 0.000 1.000 -0.944 0.113 0.825 2.018Governm 0.000 1.000 -1.185 -0.138 0.652 2.173Leader 0.000 1.000 -0.306 0.097 0.453 0.895 1.291

1.651....Bivariate Distributions for Ordinal Variables (Frequencies)

Polint2 Impact ----------------------- -----------------------

Polint1 1 2 3 4 5 1 2 3 4 5 ----------------------- -----------------------

1 63 15 9 1 0 34 28 19 5 2 2 15 292 57 1 0 61 174 68 53 9 3 4 42 783 34 1 125 362 221 141 15 4 0 3 36 338 12 50 154 87 89 9 5 0 2 7 24 143 38 58 36 40 4

Selected output:

17

(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. Polint1 0.905 (PC) 482.780 15 0.000 0.129 0.000W_A_R_N_I_N_G: Underlying bivariate normality may not hold, see BTS-file

Impact vs. Polint1 0.109 (PC) 46.156 15 0.000 0.033 1.000Impact vs. Polint2 0.136 (PC) 46.583 15 0.000 0.033 1.000

Election vs. Polint1 0.158 (PC) 39.318 15 0.001 0.029 1.000Election vs. Polint2 0.172 (PC) 40.990 15 0.000 0.030 1.000Election vs. Impact 0.312 (PC) 81.802 15 0.000 0.049 1.000Politicn vs. Polint1 0.050 (PC) 36.880 15 0.001 0.028 1.000Politicn vs. Polint2 0.062 (PC) 40.366 15 0.000 0.030 1.000Politicn vs. Impact 0.317 (PC) 129.032 15 0.000 0.064 1.000Politicn vs. Election 0.230 (PC) 125.406 15 0.000 0.063 1.000Governm vs. Polint1 0.044 (PC) 37.295 15 0.001 0.028 1.000Governm vs. Polint2 0.062 (PC) 50.939 15 0.000 0.036 1.000Governm vs. Impact 0.275 (PC) 91.635 15 0.000 0.052 1.000Governm vs. Election 0.269 (PC) 100.322 15 0.000 0.055 1.000Governm vs. Politicn 0.563 (PC) 232.222 15 0.000 0.088 0.977Leader vs. Polint1 -0.047 (PC) 58.742 23 0.000 0.029 1.000Leader vs. Polint2 -0.051 (PC) 61.720 23 0.000 0.030 1.000Leader vs. Impact 0.120 (PC) 45.630 23 0.003 0.023 1.000Leader vs. Election 0.159 (PC) 53.734 23 0.000 0.027 1.000Leader vs. Politicn 0.177 (PC) 19.567 23 0.668 0.000 1.000Leader vs. Governm 0.245 (PC) 40.806 23 0.012 0.020 1.000

Correlations and Test Statistics

18

Bivariate Table for Polint1 vs Polint2

Observed Frequencies 1 2 3 4 5 | Rowsum

----------------------------------------------------------------1 | 63 15 9 1 0 | 882 | 15 292 57 1 0 | 3653 | 4 42 783 34 1 | 8644 | 0 3 36 338 12 | 3895 | 0 2 7 24 143 | 176

----------------------------------------------------------------Colsum | 82 354 892 398 156 | 1882

Expected Frequencies 1 2 3 4 5 | Rowsum

----------------------------------------------------------------1 | 54.3 32.9 0.8 0.0 0.0 | 88.02 | 27.1 228.8 108.8 0.2 0.0 | 365.03 | 0.6 92.1 663.1 106.4 1.9 | 864.04 | 0.0 0.2 116.1 232.0 40.7 | 389.05 | 0.0 0.0 3.1 59.4 113.5 | 176.0

----------------------------------------------------------------Colsum | 82.0 354.0 892.0 398.0 156.0 | 1882.0

BTS-File:

19

1 2 3 4 5 | Rowsum ----------------------------------------------------------------

1 | 18.7 -23.5 42.7 27.6 0.0 | 65.52 | -17.8 142.4 -73.7 3.1 0.0 | 54.03 | 15.8 -66.0 260.3 -77.6 -1.2 | 131.44 | 0.0 16.6 -84.3 254.3 -29.3 | 157.35 | 0.0 40.6 11.3 -43.5 66.1 | 74.6

----------------------------------------------------------------Colsum | 16.8 110.1 156.2 164.1 35.6 | 482.8

GF Contributions 1 2 3 4 5 | Rowsum

----------------------------------------------------------------1 | 1.4 9.7 79.2******** 0.0 |********2 | 5.4 17.4 24.7 3.0 0.0 | 50.63 | 21.5 27.3 21.7 49.2 0.4 | 120.14 | 0.0 42.0 55.3 48.4 20.2 | 165.95 | 0.0 51817.2 4.8 21.1 7.7 | 51850.8

----------------------------------------------------------------Colsum | 28.3 51913.7 185.7******** 28.3 |********

LR Contributions

20

1 2 3 4 5 | Rowsum ----------------------------------------------------------------

1 | 2.1 -5.3 9.4 1004.1 0.0 | 1010.22 | -3.8 11.6 -9.0 1.8 0.0 | 0.53 | 4.8 -9.2 29.3 -12.4 -0.7 | 11.94 | 0.0 6.6 -13.2 17.1 -6.9 | 3.75 | 0.0 227.6 2.4 -7.5 5.9 | 228.5

----------------------------------------------------------------Colsum | 3.1 231.3 19.0 1003.1 -1.7 | 1254.9

Standardized Residuals

21

Correlation Matrix Polint1 Polint2 Impact Election Politicn Governm-------- -------- -------- -------- -------- --------

Polint1 1.000Polint2 0.905 1.000Impact 0.109 0.136 1.000

Election 0.158 0.172 0.312 1.000Politicn 0.050 0.062 0.317 0.230 1.000Governm 0.044 0.062 0.275 0.269 0.563 1.000Leader -0.047 -0.051 0.120 0.159 0.177 0.245

Leader--------

Leader 1.000

MeansPolint1 Polint2 Impact Election Politicn Governm-------- -------- -------- -------- -------- --------

0.000 0.000 0.000 0.000 0.000 0.000

Leader--------

0.000

Back to the PRELIS Output-file:

22

Polint1 Polint2 Impact Election Politicn Governm-------- -------- -------- -------- -------- --------

1.000 1.000 1.000 1.000 1.000 1.000

Leader--------

1.000

Standard Deviations

Alternative Parametrization can be realized by specifying MA=CM or AP on the output-command:Read asci raw data and compute polychoric covariancesand their (co-) variances for Allbus subsetDA NI=7LAPolint1 Polint2 Impact Election Politicn Governm Leader/ RA=allb96s3.dat FO(7F1.0)OR ALLOU MA=CM CM=allb96s3.cm ac=allb96s3.acc

23

-------- ---- -------- ----------Polint1 1.724 1.028 0.000 1.000 2.262 3.079Polint2 1.750 1.023 0.000 1.000 2.304 3.168Impact 0.836 0.854 0.000 1.000 1.570 2.577

Election 2.312 1.296 0.000 1.000 1.613 3.362Politicn 0.893 0.946 0.000 1.000 1.674 2.802Governm 1.131 0.955 0.000 1.000 1.754 3.206Leader 0.758 2.481 0.000 1.000 1.883 2.978 3.962 4.853

Covariance Matrix




Leader--------

Leader 6.155

Variable Mean St. Dev. Thresholds

The covariance matrix can be computed from the polychoric correlation matrix by multiplying the vector of standard deviations from left and from right.

24

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

WLS ij,lm ij ij lm lm iii 1 j 1 l 1 m 1 i 1

ˆ ˆ ˆF w (r ) (r ) (1 )− −

= = = = =

= ⋅ − ρ ⋅ − ρ + − ρ∑∑∑∑ ∑

The weights wij,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 areones.

Analyzing Correlations or Covariances ?

25

In a covariance matrix, the diagonal elements are sample variances. These elements areestimations 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:

ξ1

X1 X2 X3

λ λ λ

δ1 δ2 δ3

2 1

3 1 3 2

211

2 2X X 22

2 2 2X X X X 33

211

11

1

ˆ ˆ1

λ + θ ρ = λ λ + θ

ρ ρ λ λ λ + θ

⇒ θ = − λ

The diagonal elements are used in the estimation because formally a LISREL model is a covariance structure.

Correlation Structures and Covariance Structures

26

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 specifytau-equivalent measures. If loadings are restricted to be equal, measurement error varianceswill 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.

In a covariance structure variances and covariances are functions of parameters;in a correlation structure correlations are functions of parameters.

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: MLDA NI=7 NO=1882 MA=CMCM=allb96s1.pm! ac=allb96s1.acpLA Polint1 Polint2 Impact Election Politicn Governm Leader /MO NY=7 NE=4 LY=FU,FI BE=FU,FI PS=SY TE=DI,FR LEPOLINT 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 1.00 LY(1,1) LY(2,1) LY(3,2) LY(4,2) LY(5,3) LY(6,3) LY(7,4) PDOU AD=Off SO RS MI SS

28

POLINT0.91

EFFICACY0.16

TRUST0.51

LEADER1.00

Polint1 0.10

Polint2 0.09

Impact 0.68

Election 0.69

Politicn 0.45

Governm 0.43

Leader 0.00

Chi-Square=41.57, df=12, P-value=0.00004, RMSEA=0.036

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.13

0.45

0.05

0.07

0.22

-0.05

29

Model for Efficacy based on polychoric correlations: WLSDA NI=7 NO=1882 MA=PMpmallb96s1.pmac=allb96s1.acpLA Polint1 Polint2 Impact Election Politicn Governm Leader /MO NY=7 NE=4 LY=FU,FI BE=FU,FI PS=SY TE=DI,FR LEPOLINT 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 1.00 LY(1,1) LY(2,1) LY(3,2) LY(4,2) LY(5,3) LY(6,3) LY(7,4) PDOU AD=Off SO RS MI SS WL

Estimating method: WLS:

30

POLINT0.89

EFFICACY0.17

TRUST0.52

LEADER1.00

Polint1 0.11

Polint2 0.11

Impact 0.68

Election 0.68

Politicn 0.43

Governm 0.43

Leader 0.00


1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.14

0.45

0.06

0.08

0.22

-0.05

31

Model for Efficacy based on polychoric covariancesDA NI=7 NO=1882 MA=CMCM=allb96s3.cmac=allb96s3.accLA Polint1 Polint2 Impact Election Politicn Governm Leader /MO NY=7 NE=4 LY=FU,FI BE=FU,FI PS=SY TE=DI,FR LEPOLINT 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 1.00 LY(1,1) LY(2,1) LY(3,2) LY(4,2) LY(5,3) LY(6,3) LY(7,4) PDOU AD=Off SO RS MI SS WL

If polychoric covariances are analyzed again a different result occur:

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POLINT0.94

EFFICACY0.19

TRUST0.47

LEADER6.15

Polint1 0.11

Polint2 0.11

Impact 0.39

Election 1.34

Politicn 0.38

Governm 0.40

Leader 0.00


1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.12

0.47

0.02

0.07

0.08

-0.15

33

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=7LAPolint1 Polint2 Impact Election Politicn Governm Leader/ RA=allb96s3.dat FO(7F1.0)OR ALLET Polint1 Polint2ET Impact Election Politicn GovernmOU MA=PM

Restrictions on Thresholds

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Variable Mean St. Dev. Thresholds-------- ---- -------- ----------Polint1 0.005 1.016 -1.700 -0.711 0.537 1.346Polint2 -0.005 0.983 -1.700 -0.711 0.537 1.346Impact -0.360 0.985 -1.324 -0.171 0.487 1.648

Election 0.871 1.127 -1.324 -0.171 0.487 1.648Politicn -0.343 0.998 -1.324 -0.171 0.487 1.648Governm -0.169 0.874 -1.324 -0.171 0.487 1.648Leader 0.000 1.000 -0.306 0.097 0.453 0.895 1.291 1.651

Correlation Matrix equal thresholds




free thresholdsPolint1 Polint2 Impact Election Politicn Governm-------- -------- -------- -------- -------- --------



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Example with equal thresholds DA NI=7LAPolint1 Polint2 Impact Election Politicn Governm Leader/ RA=allb96s3.dat FO(7F1.0)OR ALLET Polint1 Polint2ET Impact Election Politicn GovernmOU MA=CM CM=a96s1et.cm AC=a96s1et.acc

Computation of polychoric covariances with equal thresholds

Variable Mean St. Dev. Thresholds-------- ---- -------- ----------Polint1 1.724 1.028 0.000 1.000 2.262 3.079Polint2 1.714 0.994 0.000 1.000 2.262 3.079Impact 0.836 0.854 0.000 1.000 1.570 2.577

Election 1.904 0.977 0.000 1.000 1.570 2.577Politicn 0.851 0.866 0.000 1.000 1.570 2.577Governm 1.002 0.758 0.000 1.000 1.570 2.577Leader 0.758 2.481 0.000 1.000 1.883 2.978 3.962 4.853

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Covariance Matrix equal thresholds




Leader--------

Leader 6.155

Covariance Matrix free thresholds




Leader--------

Leader 6.155

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POLINT0.92

EFFICACY0.15

TRUST0.33

LEADER6.15

Polint1 0.14

Polint2 0.07

Impact 0.45

Election 0.67

Politicn 0.39

Governm 0.22

Leader 0.00


1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.13

0.52

0.02

0.06

0.07

-0.13

38

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.

Fixed thresholds

Example: Computing thresholds for group comparionsDA NI=7LAPolint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.rawSD GroupOR ALLOU MA=PM TH=allb96s0.th

Univariate Marginal Parameters

Variable Mean St. Dev. Thresholds-------- ---- -------- ----------Polint1 0.000 1.000 -1.537 -0.606 0.569 1.374Polint2 0.000 1.000 -1.571 -0.647 0.579 1.426Impact 0.000 1.000 -0.827 0.280 0.929 1.978

Election 0.000 1.000 -1.682 -0.922 -0.438 0.818Politicn 0.000 1.000 -0.883 0.178 0.878 1.984Governm 0.000 1.000 -1.031 -0.044 0.739 2.192

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Computing pol. correlations based on given thresholds: WestDA NI=7LAPolint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.raw RESD Group=0OR ALLFT=allb96s0.th Polint1FT Polint2 ; FT Impact ; FT Election ; FT Politicn ; FT GovernmOU MA=PM Computing pm bsed on given thresholds: East DA NI=7LAPolint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.raw RESD Group=1OR ALLFT=allb96s0.th Polint1FT Polint2 ; FT Impact ; FT Election ; FT Politicn ; FT GovernmOU MA=PM

40

Computing cm and acc based on constant thresholds: West DA NI=7LAPolint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.raw RESD Group=0OR ALLFT=allb96s0.th Polint1FT Polint2 ; FT Impact ; FT Election ; FT Politicn ; FT GovernmOU MA=CM CM=a96s1pc.cm me=a96s1pc.me ac=a96s1pc.accComputing cm and acc based on constant thresholds: East DA NI=7LAPolint1 Polint2 Impact Election Politicn Governm Group/ RA FI=allb96s0.raw RESD Group=1OR ALLFT=allb96s0.th Polint1FT Polint2 ; FT Impact , FT Election ; FT Politicn ; FT GovernmOU MA=CM CM=a96s2pc.cm me=a96s2pc.me ac=a96s2pc.acc

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Group: West

Univariate Marginal Parameters


Election 2.306 1.276 0.000 1.000 1.637 3.289Politicn 0.895 0.926 0.000 1.000 1.660 2.702Governm 1.162 0.986 0.000 1.000 1.793 3.265

Covariance Matrix



Election 0.212 0.237 0.329 1.629Politicn 0.036 0.047 0.248 0.271 0.858Governm 0.038 0.061 0.230 0.339 0.516 0.973

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Group: East


Election 2.011 1.373 0.000 1.000 1.637 3.289Politicn 0.692 0.971 0.000 1.000 1.660 2.702Governm 0.794 1.037 0.000 1.000 1.793 3.265

Covariance Matrix

Polint1 Polint2 Impact Election Politicn GovernmPolint1 Polint2 Impact Election Politicn Governm

-------- -------- -------- -------- -------- --------Polint1 1.203Polint2 1.026 1.247Impact 0.115 0.143 1.018

Election 0.170 0.228 0.426 1.884Politicn 0.092 0.091 0.363 0.489 0.944Governm 0.099 0.080 0.388 0.433 0.688 1.075

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Group A: West Group B: East

Results of the estimation


Polint10.10

Polint20.10

Impact0.43

Election1.26

Politicn0.34

Governm0.45

POLINT 1.02

EFFICACY 0.34

TRUST 0.57

1.00

1.00

1.00

1.00

1.00

1.00

0.14

0.06

0.30

Polint10.18

Polint20.21

Impact0.64

Election1.42

Politicn0.27

Governm0.40

POLINT 1.02

EFFICACY 0.34

TRUST 0.57

1.00

1.00

1.00

1.00

1.00

1.00

0.14

0.06

0.30

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If it is possible to divide the variables in ordinal dependent variables and (fixed) metricexogenous variables, the underlying continuous variables can be regressed on the independentvariables. 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.

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:

K K

k k i k k i 1k=1 k=1

Prob(Y i) Prob Y*- X Prob Y*- X − = = β ⋅ < τ − β ⋅ < τ

∑ ∑

/*

1

ˆ ˆ ˆˆ ˆ;ˆK

X Y k kkX X

xµ β=

Β ⋅Σ ⋅Β + ΡΣ = = ⋅∑ Β ⋅Σ Σ

Probit Regression to Estimate Partial Polychoric Correlations

Analyzing Ordinal Data With Linear Models

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