Our purpose is to describe in detail a convenient procedure for
performing a new kind of item analysis. This new item analysis is
different in a vital way from that described in textbooks like
Gulliksen's *Theory of Mental Tests *and used in computing
programs like TSSA2. The difference is that (a) test calibrations
are independent of the sample of persons used to estimate item
parameters, and (b) person measurements, the transformation of test
scores into estimates of person ability, are independent of the
selection of items used to obtain test scores.

The procedure for sample-free item analysis is based on a very simple model (Rasch, 1960, 1966a, 1966b) for what happens when any person encounters any item. The model says that the outcome of such an encounter is governed by the product of the ability of the person and the easiness of the item and nothing more! The more able the person, the better his chances for success with any item. The more easy the item, the more likely any person is to solve it.

This means that variation in additional item characteristics, like guessing and discrimination, must be dealt with during the construction and selection of items for the final sample-free pool. The aim is to create a pool of items with similar discrimination and minimal guessing. Since the method for measuring person ability is quite robust with respect to departures from the assumption that the only characteristic on which items differ is easiness, this aim is not difficult to satisfy. The procedure to be described includes a statistical test for item fit which facilitates the identification of "bad" items which do not conform to the assumptions of the model. [BDW would later say "specifications of the model."]

The use of this simple model for mental measurement makes it possible to take into account whatever abilities persons in the calibration sample happen to have and to free the calibration of test items from the particulars of these abilities. As a result no assumptions need be made about the distribution of ability in the target population or in the calibration sample.

In its mathematical form this model for sample-free item
analysis says that the observed response *a _{ni } *of
person

Pr(*a _{ni } *= 1) =

and the probability of a wrong response is:

Pr(*a _{ni } *= 0) = 1 - Pr(

Taking advantage of the convention that *a _{ni } *=
1 means

Pr(*a _{ni }*) =
(

It is also convenient to express (2) in an alternative form in
which we write the model parameters *Z _{n } *and

Pr(*a _{ni }*) = exp
(

where *b _{n }* = log

An important consequence of this model is that the number of
correct responses to a given set of items is a sufficient statistic
for estimating person ability. This score is the *only
*information needed from the data to make the ability estimate.
Therefore, we need only estimate an ability for each possible
score. Any person who gets a certain score will be estimated to
have the ability associated with that score. All persons who get
the same score will be estimated to have the same ability.

This encourages us to rewrite (3) in terms of score groups.

Pr(*a _{ni }*) = exp
(

where *j *is the score obtained by person *n *and
all persons with a score *j *are estimated to have the same
probability governing their responses to item *i*.

There are two stages in the measurement of person ability. The
first stage, *item calibration, *consists in estimating the
item parameters *d _{i }* and their standard errors.
This is done by analyzing the responses of a sample of

The failure of an item to fit the model can be traced to two main sources. One is that the model is too simple. It takes account of only one item characteristic - item easiness. Other item parameters like item discrimination and guessing are neglected. As a matter of fact, parameters for discrimination and guessing can easily be included in a more general model. Unfortunately their inclusion makes the application of the model to actual measurement very complicated, if not impossible. The sample-free model assumes that all items have the same discrimination, and that the effect of guessing is negligible. Our experience with the analysis of real data suggests that the model is quite robust with respect to departures from these assumptions.

The other source of lack of fit of an item lies in the content of the item. The model assumes that all the items used are measuring the same trait. Items in a "test" may not fit together if the "test" is composed of items which measure different abilities. This includes the situation in which the item is so badly constructed or so mis-scored that what it measures is irrelevant to the rest of the "test."

If a given set of items fit the model this is the evidence that they refer to a unidimensional ability, that they form a conformable set. Fit to the model also implies that item discriminations are uniform and substantial, that there are no errors in item scoring and that guessing has had a negligible effect. Thus the criterion of fit to the model enables us to identify and delete "bad" items. Item calibration is concluded by reanalyzing the retained items to obtain the final estimates of their easinesses.

In the second stage, *person measurement, *some or all of
the calibrated items are used to obtain a test score. An estimate
of person ability and the standard error of this estimate are made
from the score and from the easinesses of the items used. The
flexibility of being able to use some or all of a set of items in
a "test" is an important advantage of this method of item analysis.
Meaningful comparisons of ability can be made even when the
particular items used to make the different measurements are not
the same. The number of items selected for any measurement can be
determined by the testing time available and the accuracy
required.

In this procedure the "reliability" of a test, a concept which depends upon the ability distribution of the sample, is replaced by the precision of measurement. The standard error of the ability estimate is a measure of the precision attained. This standard error depends primarily upon the number of items used. The range of item easiness with respect to the ability level being measured, also affects the standard error of the ability estimate. But in practice this effect is minor compared to the effect of test length. It is possible to reach any desired level of precision by varying the number of items used in the measurement, just providing that the range of item easiness is reasonably appropriate to the abilities being measured.

We shall describe two methods for the estimation of item and person parameters and their standard errors. Both methods are such that ability estimates are obtained at the same time as item estimates. The equations used for person measurement, given calibrated items, are similar to those used during item calibration. The difference being that during person measurement the items are assumed calibrated, and so item easinesses are no longer estimated but kept fixed. However, one is not usually interested in ability measurement at the stage of item calibration. Usually a pool of items are calibrated first and then later used selectively for measurement.

The first method of estimation uses unweighted least squares and will be referred to as LOG. The second method uses maximum likelihood and will be referred to as MAX [also known as UCON and JMLE]. In general MAX is preferable to LOG. MAX gives better estimates of the model parameters, and the standard errors of estimate are better approximated. However, when the calibration sample is large, and the ability range of the sample is wider than the easiness range of the item parameters, then the item estimates obtained by LOG are equivalent to the estimates obtained by MAX.

In general we recommend that MAX be used whenever possible. Our reason for describing LOG is that it is conceptually and computationally simple. If a small computer is unavailable, LOG can be used to obtain rough parameter estimates and their standard errors.

Despite the simplicity of LOG we would like to emphasize that MAX is not much more complicated. The characteristic which makes MAX more difficult to use is its system of implicit equations which must be solved by an iterative procedure. This iterative procedure is easy to perform on a small computer but tedious on a desk calculator.

*Methods*

*A. LOG Method:*

1. Description.

The log method of estimation is based on using the observed
proportion of successes *a _{ji }/r_{j }* within
a particular score group

*p _{ji } ~= a_{ji }/r_{i }
*

*p _{ji }* = exp (

where *b _{i }* is the ability associated with
score group

and *(r _{i } - a_{ji })/r_{i }
*~= 1/(1 + exp (

so *a _{ji }/(r_{i } - a_{ji })
*~= exp (

and *t _{ji } *= log

so t_{ji } = b_{i }^{*} +
d_{i }_{* } (7)

where d_{i }^{*} = estimate of d_{i }

and b_{j }^{*} = estimate of b_{j }.

This leads to the estimation equations

d_{i }^{*} - d_{. }^{*} = t_{.i }
- t_{.. } (8)

where d_{. }^{*} = (1/k) sum i=i to k
(d_{i }^{*})

t_{.i } = (1/(k - 1)) sum j=1 to k-1 t_{ji }

t.. = (1/k) sum i=1 to k t_{.i }

Since there is an indeterminacy in the scale of easiness we can
determine the scale so that d_{. }^{*} = 0 to give:

log E_{i }^{*} = d_{i }^{*} =
t_{.i } - t_{.. } (9)

as the basic equation for estimating item easiness.

We also obtain an estimation equation for ability:

log Z_{j }^{*} = b_{j }^{*} =
t_{j. } - t^{..} (10)

Equations (9) and (10) are the basic estimation equations for the log method.

To calculate standard errors of the estimates
b_{j }^{*} and d_{i }^{*} we need
expressions for the variance of t_{ji }. This is obtained
from the variance of a_{ji }. The number of successes
a_{ji } in the score group *j *has a binomial
distribution, and hence the variance of a_{ji }, will be
given by:

V(a_{ji }) = r_{i }p_{ji }(1 -
p_{ji })

where p_{ji } is the probability of obtaining a success.
The variance of t_{ji } can be approximated from:

V(t_{ji }) ~=
(dt_{ji }/da_{ji })^{2} V(a_{ji })

~= 1/r_{i }p_{ji }(1-
p_{ji })

or V^{*}(t_{ji }) =
1/r_{i }p_{ji }^{*}(1-p_{ji }*) (11)

where p_{ji }^{*} = exp (b_{j }^{*} +
d_{i }^{*})/(1 + exp (b_{j }^{*} +
d_{i }^{*}))

and (dt_{ji }/da_{ji }) is the partial derivative
of t_{ji } with respect to a_{ji } and equals
1/r_{i }p_{ji }^{*}(1-p_{ji }*)

From (9) we get for the variance of d_{i }^{*}:

V(d_{i }^{*}) = V(t_{.i } -
t_{.. }).

We know that the t_{ji }'s are independent with respect
to variation in *j, *that is for given _{i, }
t_{ji } and t_{li } are independent, because they come
from different groups of persons. However, there is a relationship
between t_{ji } and t_{jl }, for any score group *j
*because of the constraint sum i=1 to k a_{ji } -
jr_{i }. In fact, the actual covariances between
t_{ji } and t_{jl } are very small. For simplicity we
will assume that the t_{ji }'s are independent of each other
in both directions. Then for the variance of d_{i }^{*}
we get:

V(d_{i }^{*}) ~= (1 - 1/k)V(t_{.i }) < V(t_{.i })

so ~= V(t_{.i })

V^{*}(d_{i }^{*}) = (1/(k - 1)_{2 }) sum
from i=1 to k-1 V(t_{ji }). (12)

This approximation is conservative. The exact variances of estimates are smaller than those given by (12). The standard error of the ability estimate is approximated by:

V^{*}(b_{i }^{*}) = (1/k_{2 }) sum from
i=1 to k V(t_{ji }). (13)

*Procedure*

*A. Data Handling*

The observations consist of the responses of *N
*individuals to each of *k *items which compose the
test. The response to an item is coded 1 or 0, 1 if the response is
correct and 0 otherwise. (The procedure is restricted to
dichotomous items, i.e., to items that can be coded right or
wrong.)

A *k-*dimensional response vector *I *of 1's and
0's can represent the response of an individual to the test. Hence,
the data could be conceived of as an N x k matrix containing the
responses of all the N persons to the k items. However, for
estimation that matrix contains superfluous information because the
ability estimate of an individual is entirely dependent on his
score - the exact pattern of responses is immaterial. We do not
need to know the response of an individual to a particular item,
but only his total score to classify him according to estimated
ability.

The distribution of estimated ability for the whole sample can
be summarized in a score vector R of dimension k-1. The element
r_{j } of the vector R is set equal to the number of persons
with a score of *j.*

Scores of 0 and k are excluded because they do not contribute to the item calibration. They provide no differential information about the items. For these people all the items appear either equally hard or equally easy. In fact we cannot obtain point estimates of ability for such people. Items which everyone gets right or everyone gets wrong are also excluded. At the calibration stage we cannot obtain point estimates for them from the sample, and at the measurement stage at least among the calibrating sample they do not provide differential information about the ability of the individuals being measured.

Thus the original N x k data matrix can be collapsed into a (k
- 1) x k matrix *A, *such that an element a_{ji }
represents the number of persons with a score of j who get item i
correct. This *A *matrix contains all the information
bearing on test calibration.

The first step in the procedure then consists in computing
*A* and *R*. The total number of persons N'
(excluding those that get zero and maximum scores) can be counted
at the same time. The most convenient way of setting up the matrix
A and vector R is to read in one case (vector I) at a time. The
score j is calculated by summing over all the responses.

j = sum i=1 to k (I_{i }) (14)

or j_{n } = sum i=1 to k a_{ni }.

If j = 0 or k the case is disregarded and the next case is read in. When j is in the permissible range the appropriate accumulation is made to R and A. This is demonstrated below in terms of a FORTRAN program segment which can be used as a subroutine acting on each case:

[Obsolete source code omitted]

I = Response vector I

IA = Matrix A in fixed point

K = Number of items k in test

RN = N' number of persons with scores not 0 or K

R = Vector R of score group sizes.

It is assumed that IA, R and RN are zeroed before any cases are accumulated into them.

If any r_{j } is zero we disregard the score group j. An
empty score group does not contribute any information to the item
estimation or to the test for the item fit. Also in the case of the
log method we cannot obtain ability estimates directly for empty
score groups. Therefore, the number of useful score groups are
score groups which have one or more persons in them. We compute
*m, *the number of such useful score groups by scanning the
vector R,

m = sum i=1 to k-1 x_{i } (15)

where x_{i }=1 if r_{i }>0

x_{i } = 0 if r_{i } = 0.

The information from the data contained in R, A, N' and *m
*is enough to enable us to estimate the model parameters and
their standard errors.

*b. Estimation*

To get estimates by the log method we transform the data in A to
a matrix T where the element t_{ji } is given by

t_{ji }= log (a_{ji }/(r_{i }-
a_{ji })). (16)

We run into problems when a_{ji } = 0 or when
a_{ji } = r_{i }, because at these values
t_{ji } is infinite. To avoid this difficulty we modify T
such that:

t_{ji }= log ((a_{ji }+w)/((r_{i }-
a_{ji }+w)). (17)

where w = r_{i }/N'.

The advantage of this adjustment is that now when a_{ji }
= 0 or a_{ji } = r_{i } then t_{ji } =
±log (1 + N'). These limits for extreme values of
t_{ji } seem reasonable, because for N' persons log(1 + N')
is an outside limit on the magnitude that any cell in T can take.
Thus the matrix T is set up using the expression (17) for each
element of the matrix.

The estimates d_{i }^{*} are obtained from T using
(9)

(18)

In principle this is as far as we need proceed to obtain item
estimates by the log method, but the d_{i }^{*}'s
obtained above contain the extreme values for the empty and full
cells in A, i.e., when a_{ji } = 0 or a_{ji } =
r_{i }. We can improve the estimates by substituting values
for the unknown t_{ji }'s according to the model. To do this
we also need the ability estimates, which are obtained from T by
(10)

(19)

From the model the estimated value we get for the cell
t_{ji } is:

t_{ji }^{*} = d_{i }^{*} +
b_{j }^{*} + t_{.. } (20)

therefore for the extreme cells we substitute this value in place of ±log(1 + N').

With these new values for the unknown cells in T we again
compute d_{i }^{*} and b_{j }^{*} according
to (18) and (19). The results will differ from the previous values
depending upon the number of empty and full cells in the matrix
A.

The program steps in FORTRAN required for obtaining the
estimates d_{i }^{*}, b_{j }^{*} and the
matrix T are shown below.

[obsolete source code omitted]

B is the vector of ability estimates

D is the vector of item estimates.

*Methods*

*B. MAX Method:*

1 Description.

Maximum likelihood is a widely used method for estimating model parameters. The assumption involved in obtaining parameter estimates is that the observed data is the most likely occurrence. Parameters are estimated so that they maximize the probability (likelihood) of obtaining the sample of observations.

The equations obtained when the condition of a maximum likelihood is satisfied for the sample free model (3) in the introduction are:

i=1,2...k (21)

i=1,2...k-1 (22)

where a_{+i } = number of persons who get item i correct
(item score)

j = the score, an ability estimate is obtained for each score

r_{i } = number of persons in score group j,

and the log likelihood is

The method consists in computing d_{i }^{*} and
b_{j }^{*} from the implicit equations (21) and (22).
It should be noted that each of the equations (21) involves only
one item estimate, even though it does depend on all (k - 1)
ability estimates b_{j }^{*}. Similarly, each equation
in (22) involves only one ability estimate and of all the item
estimates d_{i }^{*}. We handle these equations as two
independent sets, and solve them accordingly.

When the item estimates are assumed known, (22) is the set of equations used for person measurement. From (22) we can obtain a scoring table, a table which will show the estimated ability corresponding to every score, for a given set of items. This scoring table involves only the item estimates. Therefore, a scoring table can be provided for any specific test, and the ability of an individual can be estimated by looking up his score in the scoring table. Once the scoring table is obtained no further computations are necessary. Thus computations are in general only necessary at the item calibration stage. They become necessary at the measurement stage only if one does not want to use a set of items for which a scoring table has been provided.

The approximation of a standard error for item estimates can be
approached in two ways. In equation (21) we can assume that the
variance of the item estimate is due primarily to the uncertainty
in the item score a_{+i }. To a first approximation this
gives:

which from (21) leads to:

(23)

An alternative is to approximate the standard error f m the asymptotic value of the variance of a maximum likelihood estimate. But this leads to the same equation (23).

To obtain estimates for the item parameters, we have to solve
the two sets of equations (21) and (22). Since these equations are
implicit in d_{i }^{*} and b_{j }^{*}, we
cannot solve them directly. In our analysis we use the
Newton-Raphson procedure to solve for the unknown parameter
estimates. This procedure is an iterative one. We start with an
initial estimate x_{0 }, and using the Newton-Raphson
equation obtain an improved estimate x_{1 }. Now using the
new value x_{1 } as the starting estimate, we repeat the
procedure until the estimates do not change appreciably. If f(x) =
0 is the implicit equation to be solved for x, the value of x at
the (n+1)th iteration is given by

x_{n+1 } = x_{n } -
(f(x_{n })/f'(x_{n })) (24)

where x_{n } = value of x at the nth iteration

f'(x) = df(x)/dx, the differential of f(x) with respect to x and
f(x)/f'(x) is evaluated at x = x_{n }.

Equation (24) is suitable for equations which are functions of only one unknown. This is adequate for our purposes because we can solve (21) and (22) as two independent sets of equations, in which each of the k equations in (21) and each of the (k - 1) equations in (22) are locally functions of only one unknown.

To facilitate a description of the procedure we write equations (21) and (22) in a form analogous to equation (24).

i = 1, 2 ... k

(25)

(26)

j = 1, 2 ... k-1

Also if

j = 1, 2 ... k-1

(27)

(28)

Since the method is iterative, we need some basis for termination. We employ two different criteria for judging whether convergence has been reached. An obvious consideration is to look at the average squared difference SD between the values of estimates obtained from two consecutive iterations. If SD is less than some criterion value SC, we stop the procedure, because insufficient improvement is obtained in the estimates by continuing the procedure further. An alternate criterion is to monitor the value of the likelihood function. This can be accomplished by computing the likelihood at each iteration and observing the rate of increase. If things are as they should be, the likelihood will increase rapidly at first, and then become approximately constant. The procedure can be stopped when the increase in the likelihood is less than some specified value CM.

*Procedure*

The first part of the procedure for MAX is the same as that described for LOG. The data is edited in exactly the same way, and the LOG procedure followed until initial item estimates are obtained. These item estimates are then used as the initial values for the iterative procedure described in MAX. The initial values for the ability estimates are taken to be zero.

Using the LOG item estimates and zero ability estimates as starting values, the iterative procedure, described by the Newton-Raphson equations (25) and (27), is continued until stable estimates are obtained both for the item and the ability estimates.

This is accomplished by solving (25) for the item estimates assuming that the abilities are zero. The obtained item estimates are substituted in (27) and these equations are solved for improved ability estimates. The improved ability estimates are then substituted in (25) and improved item estimates obtained. This procedure of alternately solving (25) and (27) using improved estimates at each stage is continued till the process converges.

Two criteria for convergence were described in the previous
section. We use both criteria. First we examine the average squared
deviation SD and then test the change in the likelihood ELD. If
either SD or ELD is less than the specified criterion value we stop
the procedure. The criterion values we use are 10_{-5 } for
SD and 10_{-2 } for ELD. We find that these cut-off values
ensure sufficient convergence. When the procedure is continued
further no appreciable change is observed in the estimates. The
FORTRAN programming steps required for implementing the successive
solutions for (25) and (27) are shown below:

[obsolete source code omitted]

The log likelihood EL is initialized at a negative value since
it is expected to increase. This is necessary to do in order to
compute the change in the likelihood for the first iteration. The
vector B, ability estimates, are initially set to zero, and the
vector D, item estimates, are those obtained from the LOG method.
From our experience we find that the maximum number of times we
might expect to go through this procedure is less than 20,
therefore we set the maximum index of the loop at 20. SC and CM are
the criterion values discussed above, e.g. SC = 10_{-5 } and
CM = 10_{-2 } and

K = number of items

NGK = K - 1, the number of score groups

R = vector of score group sizes

IA = data matrix in fixed point mode.

AP is the vector of item scores which can be computed from the data matrix as follows:

AP_{i } = sum i=1 to k-1 a_{ji }.

MAXLIK and LIKE are subroutines. MAXLIK performs the iterations for the individual sets of equations, i.e. for (25) and (27). LIKE computes the likelihood. The steps required for these subroutines are indicated below.

[obsolete source code omitted]

It should be noted that, as in the LOG method, here also the
item estimates are constrained so that they add to zero, i.e. sum
from i=1 to k d_{i }^{*} = 0. The iterations for the
Newton-Raphson method are performed in subroutine NEWT. It is a
general subroutine and is applicable to any equation of the
form:

where X = the unknown

C, and vectors A and Y are given constants.

The steps required for the programming are shown below:

[obsolete source code omitted]

Finally Subroutine LIKE is given below:

[obsolete source code omitted]

Once the item and ability estimates have been obtained, by the procedure described above, the standard error of item estimates is easily computed from equation (23). The vector SI of standard errors of the item estimates depends mainly upon the number of persons in the sample, i.e., the vector R of score group sizes. The larger the elements of this vector R, the smaller will be the standard errors. The program segment for computing SI is shown below:

[obsolete source code omitted]

*Methods *

*C. Person Measurement*

1. Ability Estimation:

This part of the procedure is especially important for test users. Ordinarily test users are not concerned with calibrating items. Given a pool of calibrated items, however, they want to estimate abilities for persons to whom sets of items have been administered.

As mentioned earlier, if a scoring table is provided with the items and all the items used to compute the scoring table are used in the test, there is no need to compute new ability estimates. They can be obtained immediately by referring to the scoring table. If only some of the items are used, however, one needs to compute the abilities and their standard errors for scores on this selection of items. That procedure is given in this section.

The equations to be solved have been discussed previously (22). The only way to solve these implicit equations (22) is by means of an iterative method. The Newton-Raphson procedure gives the relationship between two successive values of the estimates in terms of the functional form of the equation to be solved. This procedure was discussed previously (27), but we will restate the equations for the convenience of those interested in ability estimation only.

j=1,2,...,k- 1

j = the score, an estimated ability b_{j }^{*} is
associated with each score

d_{i } = the item estimates, assumed known from the
calibration of the item pool

k = number of items used for the test.

b_{n }^{*} = value of the estimate at the nth
iteration

b_{n+1 }^{*} = value of the estimate at the (n+1)th
iteration

g(b^{*})/g'(b^{*}) is evaluated at b^{*} =
b_{n }^{*}.

Since we are solving the equations by means of an iterative
method, we need some criterion for terminating the procedure. We
stop the iterations when SD, the square of the relative change in
the estimate, is less than some specified value SC. We find that no
appreciable change is observed in the estimates if the procedure is
carried on beyond the point when SD becomes less than 10^{-6}.
Therefore, we set SC = 10^{-5}.

The FORTRAN program segment for this procedure is given below:

[obsolete source code omitted]

Thus we obtain an ability estimate for each of the k-1 scores 1, 2 ... k-1. One advantage of using this metric for the abilities instead of the observed score is that the scale of this metric is an interval scale, whereas, in general the raw score scale is not. Another important consideration is that abilities in this metric, obtained from different sets of calibrated items, are comparable. In the case of the raw score there is no rigorous method of putting the score on a common scale.

2. Standard Error of Ability Estimate:

The accuracy [precision] of any ability measurement is an important consideration. Not only do we want to be able to measure the ability of a person, but we would also like to know how well we have been able to make the measurement. The major contribution to the error variance of the ability estimate comes from the variance in scores produced by a given individual. As we shall later see, this part of the error variance depends upon the number of items and their easiness range. Therefore, in designing a measurement, for example constructing a test, it will be the accuracy desired which will determine the number and easiness range of the items selected for the ability estimation.

A smaller number of items is needed to produce a given level of precision in the measurement when the difficulty level of the items is approximately equal to the ability of the person being measured. This is similar to choosing items at the fifty per cent level of difficulty in classical item analysis. For a given set of k items the standard errors of the ability estimates corresponding to raw scores around k/2 will be smaller than the standard errors for the more extreme scores near 1 and k-1. Hence, by choosing items with the appropriate difficulties it is possible to economize on the number of items administered.

Another component which makes a small contribution to the variance of ability estimates comes from the imprecision in item calibration. This effect can be made negligible by calibrating the items on large samples so that the standard errors of item estimates are very small.

An approximation of the variance of the ability estimate
b^{*} is given by:

(29)

where

V(d_{i }) is the variance of the item calibration
d_{i }.

The first term in the right hand side of the expression (29) is
due to the variance in the score and the second term is due to the
imprecision of item calibration. The first term is always larger
than the second. For example, if we assume that all
V(d_{i }) are one (usually V(d_{i }) is much less than
one) the second term is p(1-p) times the first. We know that the
maximum value of p(1-p) is 0.25, therefore, the second term will,
at the most, contribute one fourth as much variance as that due to
the uncertainty in the score, in other words, at most 20 per cent
of the total error variance. The magnitude of the first term
depends primarily on the number of items, and to a lesser degree on
the relationship between their easiness range and the ability being
measured.

Given ability estimates, item estimates and their variances we can compute the standard errors of the ability estimates by means of the following FORTRAN program segment:

[obsolete source code omitted]

SA = vector of standard errors of ability estimates

K = number of items

B = vector of ability estimates

D = vector of item estimates

SI = vector of standard errors of item estimates.

*D. Testing the Fit of the Item:*

During item calibration it is necessary to decide whether all the items that have been tried are to be retained for the final pool. We need a statistical criterion for deciding whether an item is good enough from the point of view of the model.

To make this decision we need to investigate how the elements
a_{ji } in the data matrix A depend upon the estimates
d_{i }^{*} and b_{j }^{*}. If we can derive
the expectation E (a_{ji }) of these elements in terms of the
obtained estimates we can form a standard deviate

(30)

and use this deviate as the basis for a test of item fit. If
item i fits the model, and the score group r_{j } is large
enough, then y_{ji } will have an approximately unit normal
distribution.

Now a_{ji } has a binomial distribution with parameters
p_{ji }, the probability of making a correct response, and
r_{j }, the number of persons with a score j. Therefore, the
expectation of a_{ji } is given by:

(31)

and its variance by

Since b_{j } and d_{i } are not known we use their
estimates and approximate the expectation and variance of
a_{ji } as

and

Examination of the matrix Y, with the standard deviates
y_{ji } as elements, will show us how well the items fit, and
indicate where there are signs of misfit.

From the matrix Y we can obtain statistics which will enable us to evaluate the fit of the model to the data as a whole, and we can also form approximate statistics which will help identify items which are bad, and hence need to be reconsidered. As discussed in the introduction, an item may not fit for a number of reasons. It may be badly constructed or incorrectly scored. Its discrimination may be very different from the discriminations of the other items. It could be measuring some ability other than that being measured by the rest of the items. In any case, the item will be detected so that it can be examined for deletion or revision.

The over-all statistic used in the procedure is a chi-square
statistic χ_{2 } which is obtained by summing the
squared unit normal deviates over the entire matrix Y

(32)

with degrees of freedom = (k-1)(m-1)

where m = number of score groups with
r_{i }><0.

The degrees of freedom are obtained from the number of
observations in the data matrix, taking account of the loss of
degrees of freedom due to constraints and parameter estimation.
There are k x m observations in the data matrix. There are m
constraints on the score margins since sum for i=1 to k
a_{ji } = jr_{j }. Finally (k-1) item parameters have
been estimated. Therefore the degrees of freedom for
χ^{2} are:

d.f.= km -m -(k-1) (33)

= (m-1)(k-1).

An approximate χ^{2} statistic can also be obtained
for each item by summing y_{ji }^{2} over the score
groups to give

(34)

with

d.f. = m-1.

Since (34) is an approximate χ_{i }^{2}, we do
not think it advisable to mechanically delete all items for which
the χ_{i }^{2} is significant at some level. We
prefer to examine in detail items for which
χ_{i }^{2} is large. This may mean evaluating the
possible effects of discrimination and guessing in these "bad"
items. Then when we have decided which of the "bad" items to
delete, we rerun the analysis to see how the remaining set of items
look.

A FORTRAN program segment which will implement the procedure in this section is given below:

[obsolete source code omitted]

CH = mean square for the entire data.

CHI = vector of item mean squares.

R = vector of score group sizes.

M = number of occupied score groups with r_{j }<>0.

IA = data matrix.

K = number of items.

D = vector of item estimates.

B = vector of ability estimates.

A (FORTRAN II) PROGRAM FOR SAMPLE-FREE ITEM ANALYSIS

This program estimates item and ability parameters from item analysis data according to the logistic response model:

[details of obsolete computer program omitted]

REFERENCES

Gulliksen, H. *Theory of Mental Tests. *New York: John
Wiley & Sons, 1950.

Rasch, G. *Probabilistic Models for Some Intelligence and
Attainment Tests. *Copenhagen: Danish Institute for Educational
Research, 1960. Chapters V-VII, X.

Rasch, G. An Individualistic Approach to Item Analysis.
*Readings in Mathematical Social Science. *Edited by
Lazarsfeld and Henry. Chicago: Science Research Associates Inc.
1966, 89-107. (a)

Rasch, G. An Item Analysis Which Takes Individual Differences
into Account. *British Journal of Mathematical and Statistical
Psychology. *London: 1966. Vol. 19, Part l, 49-57. (b)

Wright, B. D. Sample-Free Test Calibration and Person
Measurement. *Proceedings of the 1967 Invitational Conference on
Testing Problems. *Princeton: Educational Testing Service,
1968, 85-101.

This memo was published as: Wright, B. D., & Panchapakesan, N. (1969) A procedure for sample-free item analysis.Educational and Psychological Measurement, 29,23-48.

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