# Estimation of Ability Parameters

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### Description

`estimate.theta.mle` estimates thetas with known item parameters using maximum likelihood.

`estimate.theta.map` estimates thetas with known item parameters using maximum a posterior.

`estimate.theta.eap` estimates thetas with known item parameters using expected a posterior.

### Usage

 ```1 2 3 4 5 6 7``` ```estimate.theta.mle(u, a, b, c, init = NULL, iteration = 15, delta = 0.005, bound = 3.5) estimate.theta.map(u, a, b, c, init = NULL, prior.mu = 0, prior.sig = 1, iteration = 15, delta = 0.005, bound = 3.5) estimate.theta.eap(u, a, b, c) ```

### Arguments

 `u` a response matrix, with people in rows and items in columns `a` a vector of item discrimination parameters `b` a vector of item difficulty parameters `c` a vector of item pseudo-guesing parameters `init` a vectoro of initial value for theta parmaeters `iteration` the maximum iterations of Newton-Raphson procedure `delta` convergence criterion used to terminate Newton-Raphson procedure `bound` the maximum absolute values of estimated theta parameters `prior.mu` the mean of the prior theta distribuiton `prior.sig` the standard deviation of the prior theta distribution

### Details

For the maximum likelihood estimation, refer to Baker and Kim (2004), pp. 66-69.

For the maximum a posteriori estimation, refer to Baker and Kim (2004), pp. 192.

For the expected a posteriori, refer to Baker and Kim (2004), pp. 193.

### Value

a vector of estimated thetas

Other estimation: `estimate.item.jmle`
Other estimation: `estimate.item.jmle`
Other estimation: `estimate.item.jmle`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```# MLE x <- gen.rsp(gen.irt(100, 20)) y <- estimate.theta.mle(x\$rsp, x\$items\$a, x\$items\$b, x\$items\$c) cor(x\$thetas, y) plot(x\$thetas, y, xlim=c(-4, 4), ylim=c(-4, 4), col=rgb(.8,.2,.2,.5), pch=16) abline(a=0, b=1) # MAP x <- gen.rsp(gen.irt(100, 20)) y <- estimate.theta.map(x\$rsp, x\$items\$a, x\$items\$b, x\$items\$c) cor(x\$thetas, y) plot(x\$thetas, y, xlim=c(-3, 3), ylim=c(-3, 3), col=rgb(.8,.2,.2,.5), pch=16) abline(a=0, b=1) # EAP x <- gen.rsp(gen.irt(100, 20)) y <- estimate.theta.eap(x\$rsp, x\$items\$a, x\$items\$b, x\$items\$c) cor(x\$thetas, y) plot(x\$thetas, y, xlim=c(-3, 3), ylim=c(-3, 3), col=rgb(.8,.2,.2,.5), pch=16) abline(a=0, b=1) ```