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<h1>Estimate Person Parameters for the GPCM</h1>
<small class="dont-index">Source: <a href='https://github.com/jansteinfeld/PP/blob/master/R/PP_gpcm.R'><code>R/PP_gpcm.R</code></a></small>
<div class="hidden name"><code>PP_gpcm.Rd</code></div>
</div>
<div class="ref-description">
<p>Compute person parameters for the GPCM and choose between five common estimation techniques: ML, WL, MAP, EAP and a robust estimation. All item parameters are treated as fixed.</p>
</div>
<pre class="usage"><span class='fu'>PP_gpcm</span><span class='op'>(</span>
<span class='va'>respm</span>,
<span class='va'>thres</span>,
<span class='va'>slopes</span>,
theta_start <span class='op'>=</span> <span class='cn'>NULL</span>,
mu <span class='op'>=</span> <span class='cn'>NULL</span>,
sigma2 <span class='op'>=</span> <span class='cn'>NULL</span>,
type <span class='op'>=</span> <span class='st'>"wle"</span>,
maxsteps <span class='op'>=</span> <span class='fl'>100</span>,
exac <span class='op'>=</span> <span class='fl'>0.001</span>,
H <span class='op'>=</span> <span class='fl'>1</span>,
ctrl <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/list.html'>list</a></span><span class='op'>(</span><span class='op'>)</span>
<span class='op'>)</span></pre>
<h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a>Arguments</h2>
<table class="ref-arguments">
<colgroup><col class="name" /><col class="desc" /></colgroup>
<tr>
<th>respm</th>
<td><p>An integer matrix, which contains the examinees responses. A persons x items matrix is expected.</p></td>
</tr>
<tr>
<th>thres</th>
<td><p>A numeric matrix which contains the threshold parameter for each item. If the first row of the matrix is not set to zero (only zeroes in the first row) - then a row-vector with zeroes is added by default.</p></td>
</tr>
<tr>
<th>slopes</th>
<td><p>A numeric vector, which contains the slope parameters for each item - one parameter per item is expected.</p></td>
</tr>
<tr>
<th>theta_start</th>
<td><p>A vector which contains a starting value for each person. If NULL is submitted, the starting values are set automatically. If a scalar is submitted, this start value is used for each person.</p></td>
</tr>
<tr>
<th>mu</th>
<td><p>A numeric vector of location parameters for each person in case of MAP or EAP estimation. If nothing is submitted this is set to 0 for each person for MAP estimation.</p></td>
</tr>
<tr>
<th>sigma2</th>
<td><p>A numeric vector of variance parameters for each person in case of MAP or EAP estimation. If nothing is submitted this is set to 1 for each person for MAP estimation.</p></td>
</tr>
<tr>
<th>type</th>
<td><p>Which maximization should be applied? There are five valid entries possible: "mle", "wle", "map", "eap" and "robust". To choose between the methods, or just to get a deeper understanding the papers mentioned below are quite helpful. The default is <code>"wle"</code> which is a good choice in many cases.</p></td>
</tr>
<tr>
<th>maxsteps</th>
<td><p>The maximum number of steps the NR Algorithm will take. Default = 100.</p></td>
</tr>
<tr>
<th>exac</th>
<td><p>How accurate are the estimates supposed to be? Default is 0.001.</p></td>
</tr>
<tr>
<th>H</th>
<td><p>In case <code>type = "robust"</code> a Huber ability estimate is performed, and <code>H</code> modulates how fast the downweighting takes place (for more Details read Schuster & Yuan 2011).</p></td>
</tr>
<tr>
<th>ctrl</th>
<td><p>more controls</p><ul>
<li><p><code>killdupli</code>: Should duplicated response pattern be removed for estimation (estimation is faster)? This is especially resonable in case of a large number of examinees and a small number of items. Use this option with caution (for map and eap), because persons with different <code>mu</code> and <code>sigma2</code> will have different ability estimates despite they responded identically. Default value is <code>FALSE</code>.</p></li>
<li><p><code>skipcheck</code>: Default = FALSE. If TRUE data matrix and arguments are not checked - this saves time e.g. when you use this function for simulations.</p></li>
</ul></td>
</tr>
</table>
<h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>
<p>The function returns a list with the estimation results and pretty much everything which has been submitted to fit the model. The estimation results can be found in <code>OBJ$resPP</code>. The core result is a number_of_persons x 2 matrix, which contains the ability estimate and the SE for each submitted person.</p>
<h2 class="hasAnchor" id="details"><a class="anchor" href="#details"></a>Details</h2>
<p>Please note, that <code>robust</code> estimation with (Huber ability estimate) polytomous items is still experimental!</p>
<p>The probability choosing the k-th category is as follows:</p>
<p>$$P(x_{ij} = k | \hat \alpha_i, \hat\beta_{iv}, \theta_j) = \frac{exp(\sum_{v=0}^{(k-1)}\hat \alpha_{i}(\theta_j - \hat \beta_{iv}))}{\,\sum_{c=0}^{m_i - 1}exp(\sum_{v=0}^{c}\hat \alpha_{i}(\theta_j - \hat \beta_{iv})))}$$</p>
<p>In our case \(\theta\) is to be estimated. The item parameters are assumed as fixed (usually these are estimates of a former scaling procedure).</p>
<p>The model simplifies to the Partial Credit Model by setting \(\alpha_{i} = 1, \forall i\).</p>
<h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>
<p>Baker, Frank B., and Kim, Seock-Ho (2004). Item Response Theory - Parameter Estimation Techniques. CRC-Press.</p>
<p>Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.</p>
<p>Muraki, Eiji (1992). A Generalized Partial Credit Model: Application of an EM Algorithm. Applied Psychological Measurement, 16, 159-176.</p>
<p>Muraki, Eiji (1993). Information Functions of the Generalized Partial Credit Model. Applied Psychological Measurement, 17, 351-363.</p>
<p>Samejima, Fumiko (1993). The bias function of the maximum likelihood estimate of ability for the dichotomous response level. Psychometrika, 58, 195-209.</p>
<p>Samejima, Fumiko (1993). An approximation of the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete. Psychometrika, 58, 119-138.</p>
<p>Schuster, C., & Yuan, K. H. (2011). Robust estimation of latent ability in item response models. Journal of Educational and Behavioral Statistics, 36(6), 720-735.</p>
<p>Wang, S. and Wang, T. (2001). Precision of Warm's Weighted Likelihood Estimates for a Polytomous Model in Computerized Adaptive Testing. Applied Psychological Measurement, 25, 317-331.</p>
<p>Warm, Thomas A. (1989). Weighted Likelihood Estimation Of Ability In Item Response Theory. Psychometrika, 54, 427-450.</p>
<h2 class="hasAnchor" id="see-also"><a class="anchor" href="#see-also"></a>See also</h2>
<div class='dont-index'><p><a href='PPass.html'>PPass</a>, <a href='PPall.html'>PPall</a>, <a href='PP_4pl.html'>PP_4pl</a>, <a href='JKpp.html'>JKpp</a>, <a href='PV.html'>PV</a></p></div>
<h2 class="hasAnchor" id="author"><a class="anchor" href="#author"></a>Author</h2>
<p>Manuel Reif</p>
<h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
<pre class="examples"><div class='input'><span class='co'>################# GPCM ###########################################################################</span>
<span class='co'># some threshold parameters</span>
<span class='va'>THRES</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/matrix.html'>matrix</a></span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span><span class='op'>(</span><span class='op'>-</span><span class='fl'>2</span>,<span class='op'>-</span><span class='fl'>1.23</span>,<span class='fl'>1.11</span>,<span class='fl'>3.48</span>,<span class='fl'>1</span>
,<span class='fl'>2</span>,<span class='op'>-</span><span class='fl'>1</span>,<span class='op'>-</span><span class='fl'>0.2</span>,<span class='fl'>0.5</span>,<span class='fl'>1.3</span>,<span class='op'>-</span><span class='fl'>0.8</span>,<span class='fl'>1.5</span><span class='op'>)</span>,nrow<span class='op'>=</span><span class='fl'>2</span><span class='op'>)</span>
<span class='co'># slopes</span>
<span class='va'>sl</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span><span class='op'>(</span><span class='fl'>0.5</span>,<span class='fl'>1</span>,<span class='fl'>1.5</span>,<span class='fl'>1.1</span>,<span class='fl'>1</span>,<span class='fl'>0.98</span><span class='op'>)</span>
<span class='va'>awmatrix</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/matrix.html'>matrix</a></span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span><span class='op'>(</span><span class='fl'>1</span>,<span class='fl'>0</span>,<span class='fl'>2</span>,<span class='fl'>0</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>1</span>
,<span class='fl'>2</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>2</span>,<span class='fl'>2</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>1</span><span class='op'>)</span>,byrow<span class='op'>=</span><span class='cn'>TRUE</span>,nrow<span class='op'>=</span><span class='fl'>5</span><span class='op'>)</span>
<span class='co'>## GPCM model ##### </span>
<span class='co'># MLE</span>
<span class='va'>resgpcmlmle</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix</span>,thres <span class='op'>=</span> <span class='va'>THRES</span>, slopes <span class='op'>=</span> <span class='va'>sl</span>,type <span class='op'>=</span> <span class='st'>"mle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: GPCM ...
#> type = mle
#> Estimation finished!</div><div class='input'><span class='co'># WLE</span>
<span class='va'>resgpcmwle</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix</span>,thres <span class='op'>=</span> <span class='va'>THRES</span>, slopes <span class='op'>=</span> <span class='va'>sl</span>,type <span class='op'>=</span> <span class='st'>"wle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: GPCM ...
#> type = wle
#> Estimation finished!</div><div class='input'><span class='co'># MAP estimation</span>
<span class='va'>resgpcmmap</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix</span>,thres <span class='op'>=</span> <span class='va'>THRES</span>, slopes <span class='op'>=</span> <span class='va'>sl</span>,type <span class='op'>=</span> <span class='st'>"map"</span><span class='op'>)</span>
</div><div class='output co'>#> <span class='warning'>Warning: all mu's are set to 0! </span></div><div class='output co'>#> <span class='warning'>Warning: all sigma2's are set to 1! </span></div><div class='output co'>#> Estimating: GPCM ...
#> type = map
#> Estimation finished!</div><div class='input'><span class='co'># EAP estimation</span>
<span class='va'>resgpcmeap</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix</span>,thres <span class='op'>=</span> <span class='va'>THRES</span>, slopes <span class='op'>=</span> <span class='va'>sl</span>,type <span class='op'>=</span> <span class='st'>"eap"</span><span class='op'>)</span>
</div><div class='output co'>#> <span class='warning'>Warning: all mu's are set to 0! </span></div><div class='output co'>#> <span class='warning'>Warning: all sigma2's are set to 1! </span></div><div class='output co'>#> Estimating: GPCM ...
#> type = eap
#> Estimation finished!</div><div class='input'><span class='co'># robust estimation</span>
<span class='va'>resgpcmrob</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix</span>,thres <span class='op'>=</span> <span class='va'>THRES</span>, slopes <span class='op'>=</span> <span class='va'>sl</span>,type <span class='op'>=</span> <span class='st'>"robust"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: GPCM ...
#> type = robust </div><div class='output co'>#> <span class='warning'>Warning: Robust estimation for GPCM is still very experimental! </span></div><div class='output co'>#> Estimation finished!</div><div class='input'>
<span class='co'>## PCM model ##### </span>
<span class='co'># MLE</span>
<span class='va'>respcmlmle</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix</span>,thres <span class='op'>=</span> <span class='va'>THRES</span>, slopes <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/rep.html'>rep</a></span><span class='op'>(</span><span class='fl'>1</span>,<span class='fu'><a href='https://rdrr.io/r/base/nrow.html'>ncol</a></span><span class='op'>(</span><span class='va'>THRES</span><span class='op'>)</span><span class='op'>)</span>,type <span class='op'>=</span> <span class='st'>"mle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: GPCM ...
#> type = mle
#> Estimation finished!</div><div class='input'><span class='co'># WLE</span>
<span class='va'>respcmwle</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix</span>,thres <span class='op'>=</span> <span class='va'>THRES</span>, slopes <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/rep.html'>rep</a></span><span class='op'>(</span><span class='fl'>1</span>,<span class='fu'><a href='https://rdrr.io/r/base/nrow.html'>ncol</a></span><span class='op'>(</span><span class='va'>THRES</span><span class='op'>)</span><span class='op'>)</span>,type <span class='op'>=</span> <span class='st'>"wle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: GPCM ...
#> type = wle
#> Estimation finished!</div><div class='input'><span class='co'># MAP estimation</span>
<span class='va'>respcmmap</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix</span>,thres <span class='op'>=</span> <span class='va'>THRES</span>, slopes <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/rep.html'>rep</a></span><span class='op'>(</span><span class='fl'>1</span>,<span class='fu'><a href='https://rdrr.io/r/base/nrow.html'>ncol</a></span><span class='op'>(</span><span class='va'>THRES</span><span class='op'>)</span><span class='op'>)</span>,
type <span class='op'>=</span> <span class='st'>"map"</span><span class='op'>)</span>
</div><div class='output co'>#> <span class='warning'>Warning: all mu's are set to 0! </span></div><div class='output co'>#> <span class='warning'>Warning: all sigma2's are set to 1! </span></div><div class='output co'>#> Estimating: GPCM ...
#> type = map
#> Estimation finished!</div><div class='input'><span class='co'># EAP estimation</span>
<span class='va'>respcmeap</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix</span>,thres <span class='op'>=</span> <span class='va'>THRES</span>, slopes <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/rep.html'>rep</a></span><span class='op'>(</span><span class='fl'>1</span>,<span class='fu'><a href='https://rdrr.io/r/base/nrow.html'>ncol</a></span><span class='op'>(</span><span class='va'>THRES</span><span class='op'>)</span><span class='op'>)</span>,
type <span class='op'>=</span> <span class='st'>"eap"</span><span class='op'>)</span>
</div><div class='output co'>#> <span class='warning'>Warning: all mu's are set to 0! </span></div><div class='output co'>#> <span class='warning'>Warning: all sigma2's are set to 1! </span></div><div class='output co'>#> Estimating: GPCM ...
#> type = eap
#> Estimation finished!</div><div class='input'>
<span class='co'>#### with different number of categories ##</span>
<span class='va'>THRES</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/matrix.html'>matrix</a></span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span><span class='op'>(</span><span class='op'>-</span><span class='fl'>2</span>,<span class='op'>-</span><span class='fl'>1.23</span>,<span class='fl'>1.11</span>,<span class='fl'>3.48</span>,<span class='fl'>1</span>,<span class='fl'>2</span>,<span class='op'>-</span><span class='fl'>1</span>,<span class='op'>-</span><span class='fl'>0.2</span>,<span class='fl'>0.5</span>,<span class='fl'>1.3</span>,<span class='op'>-</span><span class='fl'>0.8</span>,<span class='fl'>1.5</span><span class='op'>)</span>,nrow<span class='op'>=</span><span class='fl'>2</span><span class='op'>)</span>
<span class='va'>THRES1</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/cbind.html'>rbind</a></span><span class='op'>(</span><span class='va'>THRES</span>,<span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span><span class='op'>(</span><span class='cn'>NA</span>,<span class='cn'>NA</span>,<span class='cn'>NA</span>,<span class='cn'>NA</span>,<span class='fl'>1.7</span>,<span class='fl'>1</span><span class='op'>)</span><span class='op'>)</span>
<span class='va'>awmatrix1</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/matrix.html'>matrix</a></span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span><span class='op'>(</span><span class='fl'>1</span>,<span class='fl'>0</span>,<span class='fl'>2</span>,<span class='fl'>0</span>,<span class='fl'>1</span>,<span class='fl'>3</span>,<span class='fl'>1</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>1</span>,<span class='fl'>3</span>,<span class='fl'>1</span>,<span class='fl'>0</span>,<span class='fl'>0</span>
,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>2</span>,<span class='fl'>2</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>1</span>,<span class='fl'>0</span>,<span class='fl'>0</span>,<span class='fl'>1</span><span class='op'>)</span>, byrow<span class='op'>=</span><span class='cn'>TRUE</span>, nrow<span class='op'>=</span><span class='fl'>5</span><span class='op'>)</span>
<span class='co'># MLE estimation</span>
<span class='va'>respcmlmle1</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix1</span>,thres <span class='op'>=</span> <span class='va'>THRES1</span>, slopes <span class='op'>=</span> <span class='va'>sl</span>,type <span class='op'>=</span> <span class='st'>"mle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: GPCM ...
#> type = mle
#> Estimation finished!</div><div class='input'><span class='co'># WLE estimation</span>
<span class='va'>respcmwle1</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix1</span>,thres <span class='op'>=</span> <span class='va'>THRES1</span>, slopes <span class='op'>=</span> <span class='va'>sl</span>,type <span class='op'>=</span> <span class='st'>"wle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: GPCM ...
#> type = wle
#> Estimation finished!</div><div class='input'><span class='co'># MAP estimation</span>
<span class='va'>respcmmap1</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix1</span>,thres <span class='op'>=</span> <span class='va'>THRES1</span>, slopes <span class='op'>=</span> <span class='va'>sl</span>,type <span class='op'>=</span> <span class='st'>"map"</span><span class='op'>)</span>
</div><div class='output co'>#> <span class='warning'>Warning: all mu's are set to 0! </span></div><div class='output co'>#> <span class='warning'>Warning: all sigma2's are set to 1! </span></div><div class='output co'>#> Estimating: GPCM ...
#> type = map
#> Estimation finished!</div><div class='input'><span class='co'># EAP estimation</span>
<span class='va'>respcmeap1</span> <span class='op'><-</span> <span class='fu'>PP_gpcm</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awmatrix1</span>, thres <span class='op'>=</span> <span class='va'>THRES1</span>, slopes <span class='op'>=</span> <span class='va'>sl</span>,type <span class='op'>=</span> <span class='st'>"eap"</span><span class='op'>)</span>
</div><div class='output co'>#> <span class='warning'>Warning: all mu's are set to 0! </span></div><div class='output co'>#> <span class='warning'>Warning: all sigma2's are set to 1! </span></div><div class='output co'>#> Estimating: GPCM ...
#> type = eap
#> Estimation finished!</div></pre>
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