nag_binary_factor (g11sac) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_binary_factor (g11sac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_binary_factor (g11sac) fits a latent variable model (with a single factor) to data consisting of a set of measurements on individuals in the form of binary-valued sequences (generally referred to as score patterns). Various measures of goodness-of-fit are calculated along with the factor (theta) scores.

2  Specification

#include <nag.h>
#include <nagg11.h>
void  nag_binary_factor (Nag_OrderType order, Integer p, Integer n, Nag_Boolean gprob, Integer ns, Nag_Boolean x[], Integer pdx, Integer irl[], double a[], double c[], Integer iprint, const char *outfile, double cgetol, Integer maxit, Nag_Boolean chisqr, Integer *niter, double alpha[], double pigam[], double cm[], Integer pdcm, double g[], double expp[], Integer pde, double obs[], double exf[], double y[], Integer iob[], double *rlogl, double *chi, Integer *idf, double *siglev, NagError *fail)

3  Description

Given a set of p dichotomous variables x~=x1,x2,,xp, where  denotes vector or matrix transpose, the objective is to investigate whether the association between them can be adequately explained by a latent variable model of the form (see Bartholomew (1980) and Bartholomew (1987))
Gπiθ=αi0+αi1θ. (1)
The xi are called item responses and take the value 0 or 1. θ denotes the latent variable assumed to have a standard Normal distribution over a population of individuals to be tested on p items. Call πiθ=Pxi=1θ the item response function: it represents the probability that an individual with latent ability θ will produce a positive response (1) to item i. αi0 and αi1 are item parameters which can assume any real values. The set of parameters, αi1, for i=1,2,,p, being coefficients of the unobserved variable θ, can be interpreted as ‘factor loadings’.
G is a function selected by you as either Φ-1 or logit, mapping the interval 0,1 onto the whole real line. Data from a random sample of n individuals takes the form of the matrices X and R defined below:
Xs×p= x11 x12 x1p x21 x22 x2p xs1 xs2 xsp = x~1 x~2 x~s ,  Rs×1= r1 r2 rs
where x~l = xl1,xl2,,xlp  denotes the lth score pattern in the sample, rl the frequency with which x~l occurs and s the number of different score patterns observed. (Thus l=1srl=n). It can be shown that the log-likelihood function is proportional to
l=1 s rl logPl ,
where
Pl = P x~ = x~l = - P x~ = x~l θ ϕθ dθ (2)
(ϕθ being the probability density function of a standard Normal random variable).
Pl denotes the unconditional probability of observing score pattern x~l. The integral in (2) is approximated using Gauss–Hermite quadrature. If we take Gz=logitz=logz1-z in (1) and reparameterise as follows,
αi = αi1, πi = logit-1αi0,
then (1) reduces to the logit model (see Bartholomew (1980))
πiθ = πi πi + 1-πi exp - αi θ .
If we take Gz=Φ-1z (where Φ is the cumulative distribution function of a standard Normal random variable) and reparameterise as follows,
αi = αi11+αi12 γi = -αi01+αi12 ,
then (1) reduces to the probit model (see Bock and Aitkin (1981))
πiθ=ϕ αiθ-γi 1-αi2 .
An E-M algorithm (see Bock and Aitkin (1981)) is used to maximize the log-likelihood function. The number of quadrature points used is set initially to 10 and once convergence is attained increased to 20.
The theta score of an individual responding in score pattern x~l is computed as the posterior mean, i.e., Eθx~l. For the logit model the component score Xl=j=1pαjxlj is also calculated. (Note that in calculating the theta scores and measures of goodness-of-fit nag_binary_factor (g11sac) automatically reverses the coding on item j if αj<0; it is assumed in the model that a response at the one level is showing a higher measure of latent ability than a response at the zero level.)
The frequency distribution of score patterns is required as input data. If your data is in the form of individual score patterns (uncounted), then nag_binary_factor_service (g11sbc) may be used to calculate the frequency distribution.

4  References

Bartholomew D J (1980) Factor analysis for categorical data (with Discussion) J. Roy. Statist. Soc. Ser. B 42 293–321
Bartholomew D J (1987) Latent Variable Models and Factor Analysis Griffin
Bock R D and Aitkin M (1981) Marginal maximum likelihood estimation of item parameters: Application of an E-M algorithm Psychometrika 46 443–459

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     pIntegerInput
On entry: p, the number of dichotomous variables.
Constraint: p3.
3:     nIntegerInput
On entry: n, the number of individuals in the sample.
Constraint: n7.
4:     gprobNag_BooleanInput
On entry: must be set equal to Nag_TRUE if Gz=Φ-1z and Nag_FALSE if Gz=logitz.
5:     nsIntegerInput
On entry: ns must be set equal to the number of different score patterns in the sample, s.
Constraint: 2×p<nsmin2p,n.
6:     x[dim]Nag_BooleanInput/Output
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×p when order=Nag_ColMajor;
  • max1,ns×pdx when order=Nag_RowMajor.
Where Xl,j appears in this document, it refers to the array element
  • x[j-1×pdx+l-1] when order=Nag_ColMajor;
  • x[l-1×pdx+j-1] when order=Nag_RowMajor.
On entry: the first s rows of x must contain the s different score patterns. The lth row of x must contain the lth score pattern with Xl,j set equal to Nag_TRUE if xlj=1 and Nag_FALSE if xlj=0. All rows of x must be distinct.
On exit: given a valid parameter set then the first s rows of x still contain the s different score patterns. However, the following points should be noted:
(i) If the estimated factor loading for the jth item is negative then that item is re-coded, i.e., 0s and 1s (or Nag_TRUE and Nag_FALSE) in the jth column of x are interchanged.
(ii) The rows of x will be reordered so that the theta scores corresponding to rows of x are in increasing order of magnitude.
7:     pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxns;
  • if order=Nag_RowMajor, pdxp.
8:     irl[ns]IntegerInput/Output
On entry: the ith component of irl must be set equal to the frequency with which the ith row of x occurs.
Constraints:
  • irl[i-1]0, for i=1,2,,s;
  • i=0 s-1irl[i-1]=n.
On exit: given a valid parameter set then the first s components of irl are reordered as are the rows of x.
9:     a[p]doubleInput/Output
On entry: a[j-1] must be set equal to an initial estimate of αj1. In order to avoid divergence problems with the E-M algorithm you are strongly advised to set all the a[j-1] to 0.5.
On exit: a[j-1] contains the latest estimate of αj1, for j=1,2,,p. (Because of possible recoding all elements of a will be positive.)
10:   c[p]doubleInput/Output
On entry: c[j-1] must be set equal to an initial estimate of αj0. In order to avoid divergence problems with the E-M algorithm you are strongly advised to set all the c[j-1] to 0.0.
On exit: c[j-1] contains the latest estimate of αj0, for j=1,2,,p.
11:   iprintIntegerInput
On entry: the frequency with which the maximum likelihood search function is to be monitored.
iprint>0
The search is monitored once every iprint iterations, and when the number of quadrature points is increased, and again at the final solution point.
iprint=0
The search is monitored once at the final point.
iprint<0
The search is not monitored at all.
iprint should normally be set to a small positive number.
Suggested value: iprint=1.
12:   outfileconst char *Input
On entry: the name of a file to which diagnostic output will be directed. If outfile is NULL the diagnostic output will be directed to standard output.
13:   cgetoldoubleInput
On entry: the accuracy to which the solution is required.
If cgetol is set to 10-l and on exit fail.code= NE_NOERROR or NE_ZERO_DF, then all elements of the gradient vector will be smaller than 10-l in absolute value. For most practical purposes the value 10-4 should suffice. You should be wary of setting cgetol too small since the convergence criterion may then have become too strict for the machine to handle.
If cgetol has been set to a value which is less than the square root of the machine precision, ε, then nag_binary_factor (g11sac) will use the value ε instead.
14:   maxitIntegerInput
On entry: the maximum number of iterations to be made in the maximum likelihood search. There will be an error exit (see Section 6) if the search function has not converged in maxit iterations.
Suggested value: maxit=1000.
Constraint: maxit1.
15:   chisqrNag_BooleanInput
On entry: if chisqr is set equal to Nag_TRUE, then a likelihood ratio statistic will be calculated (see chi).
If chisqr is set equal to Nag_FALSE, no such statistic will be calculated.
16:   niterInteger *Output
On exit: given a valid parameter set then niter contains the number of iterations performed by the maximum likelihood search function.
17:   alpha[p]doubleOutput
On exit: given a valid parameter set then alpha[j-1] contains the latest estimate of αj. (Because of possible recoding all elements of alpha will be positive.)
18:   pigam[p]doubleOutput
On exit: given a valid parameter set then pigam[j-1] contains the latest estimate of either πj if gprob=Nag_FALSE (logit model) or γj if gprob=Nag_TRUE (probit model).
19:   cm[dim]doubleOutput
Note: the dimension, dim, of the array cm must be at least pdcm×2×p.
Where CMi,j appears in this document, it refers to the array element
  • if order=Nag_ColMajor, cm[j-1×pdcm+i-1];
  • if order=Nag_RowMajor, cm[i-1×pdcm+j-1].
On exit: given a valid parameter set then the strict lower triangle of cm contains the correlation matrix of the parameter estimates held in alpha and pigam on exit. The diagonal elements of cm contain the standard errors. Thus:
CM2×i-1,2×i-1 = standard error alpha[i-1]
CM2×i,2×i = standard error pigam[i-1]
CM2×i,2×i-1 = correlation pigam[i-1],alpha[i-1],
for i=1,2,,p;
CM2×i-1,2×j-1 = correlation alpha[i-1],alpha[j-1]
CM2×i,2×j = correlation pigam[i-1],pigam[j-1]
CM2×i-1,2×j = correlation alpha[i-1],pigam[j-1]
CM2×i,2×j-1 = correlation alpha[j-1],pigam[i-1],
for j=1,2,,i-1.
If the second derivative matrix cannot be computed then all the elements of cm are returned as zero.
20:   pdcmIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix C in the array cm.
Constraint: pdcm2×p.
21:   g[2×p]doubleOutput
On exit: given a valid parameter set then g contains the estimated gradient vector corresponding to the final point held in the arrays alpha and pigam. g[2×j-2] contains the derivative of the log-likelihood with respect to alpha[j-1], for j=1,2,,p. g[2×j-1] contains the derivative of the log-likelihood with respect to pigam[j-1], for j=1,2,,p.
22:   expp[dim]doubleOutput
Note: the dimension, dim, of the array expp must be at least pde×p.
Where EXPPi,j appears in this document, it refers to the array element
  • if order=Nag_ColMajor, expp[j-1×pde+i-1];
  • if order=Nag_RowMajor, expp[i-1×pde+j-1].
On exit: given a valid parameter set then EXPPi,j contains the expected percentage of individuals in the sample who respond positively to items i and j (ji), corresponding to the final point held in the arrays alpha and pigam.
23:   pdeIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix E in the array expp.
Constraint: pdep.
24:   obs[dim]doubleOutput
Note: the dimension, dim, of the array obs must be at least pde×p.
Where OBSi,j appears in this document, it refers to the array element
  • if order=Nag_ColMajor, obs[j-1×pde+i-1];
  • if order=Nag_RowMajor, obs[i-1×pde+j-1].
On exit: given a valid parameter set then OBSi,j contains the observed percentage of individuals in the sample who responded positively to items i and j (ji).
25:   exf[ns]doubleOutput
On exit: given a valid parameter set then exf[l-1] contains the expected frequency of the lth score pattern (lth row of x), corresponding to the final point held in the arrays alpha and pigam.
26:   y[ns]doubleOutput
On exit: given a valid parameter set then y[l-1] contains the estimated theta score corresponding to the lth row of x, for the final point held in the arrays alpha and pigam.
27:   iob[ns]IntegerOutput
On exit: given a valid parameter set then iob[l-1] contains the number of items in the lth row of x for which the response was positive (Nag_TRUE).
28:   rlogldouble *Output
On exit: given a valid parameter set then rlogl contains the value of the log-likelihood kernel corresponding to the final point held in the arrays alpha and pigam, namely
l=0 s-1irl[l]×logexf[l]/n.
29:   chidouble *Output
On exit: if chisqr was set equal to Nag_TRUE on entry, then given a valid parameter set, chi will contain the value of the likelihood ratio statistic corresponding to the final parameter estimates held in the arrays alpha and pigam, namely
2×l=0 s-1irl[l]×logexf[l]/irl[l].
The summation is over those elements of irl which are positive. If exf[l-1] is less than 5.0, then adjacent score patterns are pooled (the score patterns in x being first put in order of increasing theta score).
If chisqr has been set equal to Nag_FALSE, then chi is not used.
30:   idfInteger *Output
On exit: if chisqr was set equal to Nag_TRUE on entry, then given a valid parameter set, idf will contain the degrees of freedom associated with the likelihood ratio statistic, chi.
idf=s0-2×p if s0<2p;
idf=s0-2×p-1 if s0=2p,
where s0 denotes the number of terms summed to calculate chi (s0=s only if there is no pooling).
If chisqr has been set equal to Nag_FALSE, then idf is not used.
31:   siglevdouble *Output
On exit: if chisqr was set equal to Nag_TRUE on entry, then given a valid parameter set, siglev will contain the significance level of chi based on idf degrees of freedom. If idf is zero or negative then siglev is set to zero.
If chisqr was set equal to Nag_FALSE, then siglev is not used.
32:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, maxit=value.
Constraint: maxit1.
On entry, n=value.
Constraint: n7.
On entry, p=value.
Constraint: p3.
On entry, pdcm=value.
Constraint: pdcm>0.
On entry, pde=value.
Constraint: pde>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, i=value and irl[i-1]=value.
Constraint: irl[i-1]0.
On entry, irl[0]++irl[ns-1]=value and n=value.
Constraint: irl[0]++irl[ns-1]=n.
On entry, ns=value and n=value.
Constraint: nsn.
On entry, ns=value and p=value.
Constraint: ns>2×p.
On entry, ns=value and p=value.
Constraint: ns2p.
On entry, pdcm=value and p=value.
Constraint: pdcm2×p.
On entry, pde=value and p=value.
Constraint: pdep 
On entry, pdx=value and ns=value.
Constraint: pdxns.
On entry, pdx=value and p=value.
Constraint: pdxp.
On entry, rows i and j of x are identical: i=value and j=value.
NE_INT_3
On entry, p=value, n=value and ns=value.
Constraint: 2×p<nsmin2p,n.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_MAT_INV
Failure to invert Hessian matrix and maxit iterations made: maxit=value.
Failure to invert Hessian matrix plus Heywood case encountered.
NE_NOT_CLOSE_FILE
Cannot close file value.
NE_NOT_WRITE_FILE
Cannot open file value for writing.
NE_REAL_ARRAY_ELEM_CONS
One of the elements of a has exceeded 10 in absolute value (Heywood case).
NE_RESPONSE_LEVEL
For at least one of the p items the responses are all at the same level.
NE_TOO_MANY_ITER
maxit iterations have been performed: maxit=value.
NE_ZERO_DF
Chi-squared statistic has idf degrees of freedom: idf=value.

7  Accuracy

On exit from nag_binary_factor (g11sac) if fail.code= NE_NOERROR or NE_ZERO_DF then the following condition will be satisfied:
max 0i2×p-1 g[i] < cgetol .
If fail.code= NE_MAT_INV or NE_TOO_MANY_ITER on exit (i.e., maxit iterations have been performed but the above condition does not hold), then the elements in a, c, alpha and pigam may still be good approximations to the maximum likelihood estimates. You are advised to inspect the elements of g to see whether this is confirmed.

8  Further Comments

8.1  Timing

The number of iterations required in the maximum likelihood search depends upon the number of observed variables, p, and the distance of the starting point you supplied from the solution. The number of multiplications and divisions performed in an iteration is proportional to p.

8.2  Initial Estimates

You are strongly advised to use the recommended starting values for the elements of a and c. Divergence may result from values you supplied even if they are very close to the solution. Divergence may also occur when an item has nearly all its responses at one level.

8.3  Heywood Cases

As in normal factor analysis, Heywood cases can often occur, particularly when p is small and n not very big. To overcome this difficulty the maximum likelihood search function is terminated when the absolute value of one of the αj1 exceeds 10.0. You have the option of deciding whether to exit from nag_binary_factor (g11sac) (by setting fail.print=NAGERR_DEFAULT on entry) or to permit nag_binary_factor (g11sac) to proceed onwards as if it had exited normally from the maximum likelihood search function (see fail.print=Nag_TRUE or Nag_FALSE on entry). The elements in a, c, alpha and pigam may still be good approximations to the maximum likelihood estimates. You are advised to inspect the elements g to see whether this is confirmed.

8.4  Goodness of Fit Statistic

When n is not very large compared to s a goodness-of-fit statistic should not be calculated as many of the expected frequencies will then be less than 5.

8.5  First and Second Order Margins

The observed and expected percentages of sample members responding to individual and pairs of items held in the arrays obs and expp on exit can be converted to observed and expected numbers by multiplying all elements of these two arrays by n/100.0.

9  Example

A program to fit the logit latent variable model to the following data:
Index Score Pattern Observed Frequency
01 0000 0154
02 1000 0011
03 0001 0042
04 0100 0049
05 1001 0002
06 1100 0010
07 0101 0027
08 0010 0084
09 1101 0010
10 1010 0025
11 0011 0075
12 0110 0129
13 1011 0030
14 1110 0050
15 0111 0181
16 1111 0121
    ––––
Total   1000

9.1  Program Text

Program Text (g11sace.c)

9.2  Program Data

Program Data (g11sace.d)

9.3  Program Results

Program Results (g11sace.r)


nag_binary_factor (g11sac) (PDF version)
g11 Chapter Contents
g11 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012