The conventional formulation of Partial Credit Model (PCM) is as follows:
where P_{k}(λ) is the probability of a student with ability λ obtaining the score k on an item with minimum mark equal to 0 and maximum mark equal to m, and {tj} are the non-centralized thresholds (= centralized threshold + item difficulty).
It can be observed that when the maximum mark of an item equals to m, the number of parameters that need to be estimated is also exactly equal to m. When m is very large (e.g., 20 or 30) which may not be uncommon for a non-multiple-choice item, the number of parameters subject to estimation may cause great problems or even a breakdown of commonly available software. In this brief note, a re-parameterization of PCM is proposed in order to cater for items with large values of maximum marks.
Re-parameterization of PCM
We re-formulate PCM using 4 parameters (S,q, d, c), at most, for items with any values of maximum marks. The idea of the re-formulation is shown in the following diagram:
S: The start point
θ: The first interval (i.e., θ1=θ and θi+1=θi+di where i ≥ 1)
d: The change applied to the first interval to obtain the second one (i.e., d1=d)
c: The variation of the change as compared with only the previous change (i.e. di+1=di+c where i ≥ 1)
Note that an approximation (i.e., assuming that the same c is applied for different di) is adopted here. By using the approximation, 4 parameters are enough to generate all the thresholds. The derivations of some thresholds are shown below:
t1 = S
t2 = S + θ1 = S + θ
t3 = S + θ1 + θ2 = (S + θ) + (θ 1+ d1) = S + 2θ + d
t4 = S + θ1 + θ2 + θ3 = (S + 2θ + d) + (θ2+d2) = (S + 2θ + d) + (θ 1+ d1+d1+c) =(S + 2θ + d) + (θ + 2d + c) = S + 3θ + 3d + c
It can be shown (by the use of mathematical induction) that the general form is:
tj = S + (j-1)θ + (j-1)(j-2) d /2 + (j-1)(j-2)(j-3)c /6
Based on the general form, we can immediately derive the following:
= kS + k(k-1)θ /2 + k(k-1)(k-2)d/6 + k(k-1)(k-2)(k-3)c/24 |
Therefore PCM can now be re-formulated using only the 4 parameters: S,θ, d, c.
Parameter Estimation using WinBugs
The estimation of the re-formulated parameters can be achieved directly using the freeware WinBugs. As an illustration, the relevant code to set up the probability model for an item with maximum mark equal to 18 in WinBugs is shown in Figure 2.
In addition, non-informative prior distributions are set up for the parameters concerned. The corresponding WinBugs code is shown in Figure 3.
Then the parameter estimation can be conducted using the built-in Markov Chain Monte Carlo (MCMC) method in WinBugs. The student ability and item parameters (from which all tj are derived) can be obtained.
We have applied this novel formulation of PCM to model responses of two items (one is Short Q and the other is Long Q) in the trial run of a test attempted by some 200 students. Short Q has maximum mark equal to 18 and Long Q has maximum mark equal to 20. The outcomes shown in Figure 1 are satisfactory.
All the values of the standard PCM thresholds tj derived from these re-formulated parameters (S, θ, d, c) are provided in Figure 4; together with frequency counts for different response categories.
Summary
In this brief note, we have proposed a novel re-parameterization for PCM in order to handle items with large values of maximum marks. The parameter estimation could be conducted using the freeware, WinBugs. We have applied the re-parameterization to model item responses in some real-life data. The outcomes of the estimations are satisfactory.
Dr. Fung Tze-ho
Manager-Assessment Technology & Research,
Hong Kong Examinations and Assessment Authority
Question | S | q | d | c | Infit Mean-square | OutfitMean-square |
---|---|---|---|---|---|---|
Short Q | -1.499 | 0.0347 | 0.0160 | 0.0001 | 0.96 | 1.00 |
Long Q | -1.434 | 0.0988 | 0.0028 | 0.0003 | 0.76 | 0.80 |
Figure 1. WinBugs parameter estimates and fit statistics.
for (i in 1:N) { # N Studentsnum.p1[i, 1]<-1
for (j in 1:18){ # An item with max mark = 18
fac11[i,j] <-j*base[1] # base[1]:S parameter
fac12[i,j]<-j*(j-1)*int[1]/2 # int[1]:q parameter
fac13[i,j]<-j*(j-1)*(j-2)*dev[1]/6 # dev[1]:d parameter
fac14[i,j]<-j*(j-1)*(j-2)*(j-3)*chg.dev[1]/24 # chg.dev[1]:c parameter
num.p1[i, j+1] <- exp(j*lambda[i] - (fac11[i,j] +fac12[i,j] + fac13[i,j] + fac14[i,j] ))}
den.p1[i] <- sum(num.p1[i,1:19] )
for (j in 1:19){p1[i,j] <- num.p1[i,j]/ den.p1[i] } } # normalization
for (i in 1:N)
{r[i, 1] ~ dcat(p1[i, 1:19 ])} # define a categorical distribution
# r[i,1] is the response of student i to the item
Figure 2. WinBugs code for reparameterized PCM with categories 0-18.
for (i in 1:N){lambda[i] ~ dnorm(0, tau.lambda)}for (j in 1:T){
base[t] ~ dnorm (mu.Base, tau.Base)
int[t] ~dnorm(mu.Int, tau.Int )I(0,)
dev[t]~dnorm(mu.Dev,tau.Dev)
chg.dev[t]~dnorm(mu.Chg, tau.Chg)
}
tau.lambda ~ dgamma(0.001, 0.001)
mu.Base ~ dnorm(0, 1.E-6)
tau.Base ~ dgamma(0.001, 0.001)
mu.Int ~ dnorm(0, 1.E-6)
tau.Int ~ dgamma(0.001, 0.001)
mu.Dev ~ dnorm(0, 1.E-6)
tau.Dev ~ dgamma(0.001, 0.001)
mu.Chg ~ dnorm(0, 1.E-6)
tau.Chg ~ dgamma(0.001, 0.001)
Figure 3. WinBugs code for non-informative priors.
Response Category | Frequency Count for Short Q | Frequency Count for Long Q | Derived PCM Threshold | Short Q | Long Q |
---|---|---|---|---|---|
0 | 0 | 0 | - | - | - |
1 | 1 | 3 | t1 | -1.50 | -1.43 |
2 | 1 | 1 | t2 | -1.46 | -1.34 |
3 | 1 | 3 | t3 | -1.41 | -1.23 |
4 | 0 | 4 | t4 | -1.35 | -1.13 |
5 | 5 | 3 | t5 | -1.26 | -1.02 |
6 | 5 | 5 | t6 | -1.16 | -0.91 |
7 | 5 | 9 | t7 | -1.05 | -0.79 |
8 | 9 | 15 | t8 | -0.92 | -0.67 |
9 | 15 | 11 | t9 | -0.77 | -0.55 |
10 | 17 | 22 | t10 | -0.60 | -0.42 |
11 | 24 | 19 | t11 | -0.42 | -0.29 |
12 | 40 | 29 | t12 | -0.22 | -0.15 |
13 | 26 | 19 | t13 | 0.00 | 0.00 |
14 | 27 | 18 | t14 | 0.23 | 0.15 |
15 | 20 | 17 | t15 | 0.48 | 0.31 |
16 | 25 | 17 | t16 | 0.75 | 0.47 |
17 | 5 | 12 | t17 | 1.03 | 0.65 |
18 | 3 | 16 | t18 | 1.34 | 0.82 |
19 | - | 3 | t19 | - | 1.01 |
20 | - | 3 | t20 | - | 1.20 |
Total | 229 | 229 | Average | -0.46 | -0.27 |
Figure 4. PCM thresholds for 19 categories (Short Q) and 21 categories (Long Q).
Note that category 0 is not observed, and nor is category 4 for Short Q. The reparameterized estimation is robust against unobserved categories.
Re-Parameterization of the Partial Credit Model for Estimating Items with Large Values of Maximum Marks, Fung Tze-ho ... Rasch Measurement Transactions, 2011, 24:4, 1301-2
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