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<h1>Estimate Person Parameters for the 4-PL model</h1>
<small class="dont-index">Source: <a href='https://github.com/jansteinfeld/PP/blob/master/R/PP_4pl.R'><code>R/PP_4pl.R</code></a></small>
<div class="hidden name"><code>PP_4pl.Rd</code></div>
</div>
<div class="ref-description">
<p>Compute Person Parameters for the 1/2/3/4-PL model 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_4pl</span><span class='op'>(</span>
<span class='va'>respm</span>,
<span class='va'>thres</span>,
slopes <span class='op'>=</span> <span class='cn'>NULL</span>,
lowerA <span class='op'>=</span> <span class='cn'>NULL</span>,
upperA <span class='op'>=</span> <span class='cn'>NULL</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'>40</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 vector or a numeric matrix which contains the threshold parameter (also known as difficulty parameter or beta parameter) for each item. If a matrix is submitted, the first row must contain only <b>zeroes</b>!</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>lowerA</th>
<td><p>A numeric vector, which contains the lower asymptote parameters (kind of guessing parameter) for each item.</p></td>
</tr>
<tr>
<th>upperA</th>
<td><p>numeric vector, which contains the upper asymptote parameters for each item.</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>With this function you can estimate:</p><ul>
<li><p><b>1-PL model</b> (Rasch model) by submitting: the data matrix, item difficulties and <b>nothing else</b>, since the 1-PL model is merely a 4-PL model with: any slope = 1, any lower asymptote = 0 and any upper asymptote = 1!</p></li>
<li><p><b>2-PL model</b> by submitting: the data matrix, item difficulties and slope parameters. Lower and upper asymptotes are automatically set to 0 und 1 respectively.</p></li>
<li><p><b>3-PL model</b> by submitting anything except the upper asymptote parameters</p></li>
<li><p><b>4-PL model</b> ---> submit all parameters ...</p></li>
</ul>
<p>The probability function of the 4-PL model is:
$$P(x_{ij} = 1 | \hat \alpha_i, \hat\beta_i, \hat\gamma_i, \hat\delta_i, \theta_j ) = \hat\gamma_i + (\hat\delta_i-\hat\gamma_i) \frac{exp(\hat \alpha_i (\theta_{j} - \hat\beta_{i}))}{\,1 + exp(\hat\alpha_i (\theta_{j} - \hat\beta_{i}))}$$</p>
<p>In our case \(\theta\) is to be estimated, and the four item parameters are assumed as fixed (usually these are estimates of a former scaling procedure).</p>
<p>The 3-PL model is the same, except that \(\delta_i = 1, \forall i\).</p>
<p>In the 2-PL model \(\delta_i = 1, \gamma_i = 0, \forall i\).</p>
<p>In the 1-PL model \(\delta_i = 1, \gamma_i = 0, \alpha_i = 1, \forall i\).</p>
<p>.</p>
<p>The <b>robust</b> estimation method, applies a Huber-type estimator (Schuster & Yuan, 2011), which downweights responses to items which provide little information for the ability estimation. First a residuum is estimated and on this basis, the weight for each observation is computed.</p>
<p>residuum:
$$r_i = \alpha_i(\theta - \beta_i)$$</p>
<p>weight:</p>
<p>$$w(r_i) = 1 \rightarrow if\, |r_i| \leq H$$
$$w(r_i) = H/|r| \rightarrow if\, |r_i| > H$$</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>Barton, M. A., & Lord, F. M. (1981). An Upper Asymptote for the Three-Parameter Logistic Item-Response Model.</p>
<p>Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F.M. & Novick, M.R. (Eds.), Statistical theories of mental test scores. Reading, MA: Addison-Wesley.</p>
<p>Magis, D. (2013). A note on the item information function of the four-parameter logistic model. Applied Psychological Measurement, 37(4), 304-315.</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>Warm, Thomas A. (1989). Weighted Likelihood Estimation Of Ability In Item Response Theory. Psychometrika, 54, 427-450.</p>
<p>Yen, Y.-C., Ho, R.-G., Liao, W.-W., Chen, L.-J., & Kuo, C.-C. (2012). An empirical evaluation of the slip correction in the four parameter logistic models with computerized adaptive testing. Applied Psychological Measurement, 36, 75-87.</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_gpcm.html'>PP_gpcm</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'>################# 4 PL #############################################################</span>
<span class='co'>### real data ##########</span>
<span class='fu'><a href='https://rdrr.io/r/utils/data.html'>data</a></span><span class='op'>(</span><span class='va'>pp_amt</span><span class='op'>)</span>
<span class='va'>d</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/matrix.html'>as.matrix</a></span><span class='op'>(</span><span class='va'>pp_amt</span><span class='op'>$</span><span class='va'>daten_amt</span><span class='op'>[</span>,<span class='op'>-</span><span class='op'>(</span><span class='fl'>1</span><span class='op'>:</span><span class='fl'>7</span><span class='op'>)</span><span class='op'>]</span><span class='op'>)</span>
<span class='va'>rd_res</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>d</span>, thres <span class='op'>=</span> <span class='va'>pp_amt</span><span class='op'>$</span><span class='va'>betas</span><span class='op'>[</span>,<span class='fl'>2</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: 1pl model ...
#> type = wle
#> Estimation finished!</div><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span><span class='op'>(</span><span class='va'>rd_res</span><span class='op'>)</span>
</div><div class='output co'>#> PP Version: 0.6.3.11
#>
#> Call: PP_4pl(respm = d, thres = pp_amt$betas[, 2], type = "wle")
#> - job started @ Mon May 24 13:27:52 2021
#>
#> Estimation type: wle
#>
#> Number of iterations: 4
#> -------------------------------------
#> estimate SE
#> [1,] 1.4099 0.4178
#> [2,] 0.3516 0.3729
#> [3,] 1.0745 0.3741
#> [4,] 0.3205 0.6900
#> [5,] 1.3879 0.4644
#> [6,] 0.9457 0.4185
#> [7,] 1.8277 0.4109
#> [8,] -0.2274 0.3964
#> [9,] 1.4426 0.4074
#> [10,] 1.6430 0.4033
#> [11,] -0.4802 0.4759
#> [12,] -0.3591 0.4086
#> [13,] -0.1911 0.4137
#> [14,] 0.5781 0.4177
#> [15,] -0.8898 0.4215
#> --------> output truncated <--------</div><div class='input'>
<span class='va'>rd_res1</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>d</span>, thres <span class='op'>=</span> <span class='va'>pp_amt</span><span class='op'>$</span><span class='va'>betas</span><span class='op'>[</span>,<span class='fl'>2</span><span class='op'>]</span>, theta_start <span class='op'>=</span> <span class='fl'>0</span>,type <span class='op'>=</span> <span class='st'>"wle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 1pl model ...
#> type = wle
#> Estimation finished!</div><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span><span class='op'>(</span><span class='va'>rd_res1</span><span class='op'>)</span>
</div><div class='output co'>#> PP Version: 0.6.3.11
#>
#> Call: PP_4pl(respm = d, thres = pp_amt$betas[, 2], theta_start = 0, type = "wle")
#> - job started @ Mon May 24 13:27:52 2021
#>
#> Estimation type: wle
#>
#> Number of iterations: 40
#> -------------------------------------
#> estimate SE
#> [1,] 1.4099 0.4178
#> [2,] 0.3516 0.3729
#> [3,] 1.0745 0.3741
#> [4,] 0.3205 0.6900
#> [5,] 1.3879 0.4644
#> [6,] 0.9457 0.4185
#> [7,] 1.8277 0.4109
#> [8,] -0.2274 0.3964
#> [9,] 1.4426 0.4074
#> [10,] 1.6430 0.4033
#> [11,] -0.4802 0.4759
#> [12,] -0.3591 0.4086
#> [13,] -0.1911 0.4137
#> [14,] 0.5781 0.4177
#> [15,] -0.8898 0.4215
#> --------> output truncated <--------</div><div class='input'>
<span class='co'>### fake data ##########</span>
<span class='co'># smaller ... faster</span>
<span class='fu'><a href='https://rdrr.io/r/base/Random.html'>set.seed</a></span><span class='op'>(</span><span class='fl'>1522</span><span class='op'>)</span>
<span class='co'># intercepts</span>
<span class='va'>diffpar</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span><span class='op'>(</span><span class='op'>-</span><span class='fl'>3</span>,<span class='fl'>3</span>,length<span class='op'>=</span><span class='fl'>12</span><span class='op'>)</span>
<span class='co'># slope parameters</span>
<span class='va'>sl</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/Round.html'>round</a></span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/stats/Uniform.html'>runif</a></span><span class='op'>(</span><span class='fl'>12</span>,<span class='fl'>0.5</span>,<span class='fl'>1.5</span><span class='op'>)</span>,<span class='fl'>2</span><span class='op'>)</span>
<span class='va'>la</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/Round.html'>round</a></span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/stats/Uniform.html'>runif</a></span><span class='op'>(</span><span class='fl'>12</span>,<span class='fl'>0</span>,<span class='fl'>0.25</span><span class='op'>)</span>,<span class='fl'>2</span><span class='op'>)</span>
<span class='va'>ua</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/Round.html'>round</a></span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/stats/Uniform.html'>runif</a></span><span class='op'>(</span><span class='fl'>12</span>,<span class='fl'>0.8</span>,<span class='fl'>1</span><span class='op'>)</span>,<span class='fl'>2</span><span class='op'>)</span>
<span class='co'># response matrix</span>
<span class='va'>awm</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/sample.html'>sample</a></span><span class='op'>(</span><span class='fl'>0</span><span class='op'>:</span><span class='fl'>1</span>,<span class='fl'>10</span><span class='op'>*</span><span class='fl'>12</span>,replace<span class='op'>=</span><span class='cn'>TRUE</span><span class='op'>)</span>,ncol<span class='op'>=</span><span class='fl'>12</span><span class='op'>)</span>
<span class='co'>## 1PL model ##### </span>
<span class='co'># MLE</span>
<span class='va'>res1plmle</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,type <span class='op'>=</span> <span class='st'>"mle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 1pl model ...
#> type = mle
#> Estimation finished!</div><div class='input'><span class='co'># WLE</span>
<span class='va'>res1plwle</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,type <span class='op'>=</span> <span class='st'>"wle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 1pl model ...
#> type = wle
#> Estimation finished!</div><div class='input'><span class='co'># MAP estimation</span>
<span class='va'>res1plmap</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</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: 1pl model ...
#> type = map
#> Estimation finished!</div><div class='input'><span class='co'># EAP estimation</span>
<span class='va'>res1pleap</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</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: 1pl model ...
#> type = eap
#> Estimation finished!</div><div class='input'><span class='co'># robust estimation</span>
<span class='va'>res1plrob</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,type <span class='op'>=</span> <span class='st'>"robust"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 1pl model ...
#> type = robust
#> Estimation finished!</div><div class='input'>
<span class='co'># summarize results</span>
<span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span><span class='op'>(</span><span class='va'>res1plmle</span><span class='op'>)</span>
</div><div class='output co'>#> PP Version: 0.6.3.11
#>
#> Call: PP_4pl(respm = awm, thres = diffpar, type = "mle")
#> - job started @ Mon May 24 13:27:53 2021
#>
#> Estimation type: mle
#>
#> Number of iterations: 5
#> -------------------------------------
#> estimate SE
#> [1,] 0.5903 0.7715
#> [2,] 3.5841 1.1379
#> [3,] 0.5903 0.7715
#> [4,] 0.5903 0.7715
#> [5,] 0.5903 0.7715
#> [6,] 0.0000 0.7668
#> [7,] 0.5903 0.7715
#> [8,] -1.1966 0.7880
#> [9,] 0.5903 0.7715
#> [10,] -1.1966 0.7880</div><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span><span class='op'>(</span><span class='va'>res1plwle</span><span class='op'>)</span>
</div><div class='output co'>#> PP Version: 0.6.3.11
#>
#> Call: PP_4pl(respm = awm, thres = diffpar, type = "wle")
#> - job started @ Mon May 24 13:27:53 2021
#>
#> Estimation type: wle
#>
#> Number of iterations: 4
#> -------------------------------------
#> estimate SE
#> [1,] 0.5778 0.7713
#> [2,] 3.2839 1.0488
#> [3,] 0.5778 0.7713
#> [4,] 0.5778 0.7713
#> [5,] 0.5778 0.7713
#> [6,] 0.0000 0.7668
#> [7,] 0.5778 0.7713
#> [8,] -1.1667 0.7869
#> [9,] 0.5778 0.7713
#> [10,] -1.1667 0.7869</div><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span><span class='op'>(</span><span class='va'>res1plmap</span><span class='op'>)</span>
</div><div class='output co'>#> PP Version: 0.6.3.11
#>
#> Call: PP_4pl(respm = awm, thres = diffpar, type = "map")
#> - job started @ Mon May 24 13:27:53 2021
#>
#> Estimation type: map
#>
#> Number of iterations: 3
#> -------------------------------------
#> estimate SE
#> [1,] 0.3706 0.7686
#> [2,] 1.9069 0.8297
#> [3,] 0.3706 0.7686
#> [4,] 0.3706 0.7686
#> [5,] 0.3706 0.7686
#> [6,] 0.0000 0.7668
#> [7,] 0.3706 0.7686
#> [8,] -0.7436 0.7745
#> [9,] 0.3706 0.7686
#> [10,] -0.7436 0.7745</div><div class='input'>
<span class='co'>## 2PL model ##### </span>
<span class='co'># MLE</span>
<span class='va'>res2plmle</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</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: 2pl model ...
#> type = mle
#> Estimation finished!</div><div class='input'><span class='co'># WLE</span>
<span class='va'>res2plwle</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</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: 2pl model ...
#> type = wle
#> Estimation finished!</div><div class='input'><span class='co'># MAP estimation</span>
<span class='va'>res2plmap</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</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: 2pl model ...
#> type = map
#> Estimation finished!</div><div class='input'><span class='co'># EAP estimation</span>
<span class='va'>res2pleap</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</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: 2pl model ...
#> type = eap
#> Estimation finished!</div><div class='input'><span class='co'># robust estimation</span>
<span class='va'>res2plrob</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</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: 2pl model ...
#> type = robust
#> Estimation finished!</div><div class='input'>
<span class='co'>## 3PL model ##### </span>
<span class='co'># MLE</span>
<span class='va'>res3plmle</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,
slopes <span class='op'>=</span> <span class='va'>sl</span>,lowerA <span class='op'>=</span> <span class='va'>la</span>,type <span class='op'>=</span> <span class='st'>"mle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 3pl model ...
#> type = mle
#> Estimation finished!</div><div class='input'><span class='co'># WLE</span>
<span class='va'>res3plwle</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,
slopes <span class='op'>=</span> <span class='va'>sl</span>,lowerA <span class='op'>=</span> <span class='va'>la</span>,type <span class='op'>=</span> <span class='st'>"wle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 3pl model ...
#> type = wle
#> Estimation finished!</div><div class='input'><span class='co'># MAP estimation</span>
<span class='va'>res3plmap</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,
slopes <span class='op'>=</span> <span class='va'>sl</span>,lowerA <span class='op'>=</span> <span class='va'>la</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: 3pl model ...
#> type = map
#> Estimation finished!</div><div class='input'><span class='co'># EAP estimation</span>
<span class='va'>res3pleap</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,
slopes <span class='op'>=</span> <span class='va'>sl</span>,lowerA <span class='op'>=</span> <span class='va'>la</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: 3pl model ...
#> type = eap
#> Estimation finished!</div><div class='input'>
<span class='co'>## 4PL model ##### </span>
<span class='co'># MLE</span>
<span class='va'>res4plmle</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,
slopes <span class='op'>=</span> <span class='va'>sl</span>,lowerA <span class='op'>=</span> <span class='va'>la</span>,upperA<span class='op'>=</span><span class='va'>ua</span>,type <span class='op'>=</span> <span class='st'>"mle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 4pl model ...
#> type = mle
#> Estimation finished!</div><div class='input'><span class='co'># WLE</span>
<span class='va'>res4plwle</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,
slopes <span class='op'>=</span> <span class='va'>sl</span>,lowerA <span class='op'>=</span> <span class='va'>la</span>,upperA<span class='op'>=</span><span class='va'>ua</span>,type <span class='op'>=</span> <span class='st'>"wle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 4pl model ...
#> type = wle
#> Estimation finished!</div><div class='input'><span class='co'># MAP estimation</span>
<span class='va'>res4plmap</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,
slopes <span class='op'>=</span> <span class='va'>sl</span>,lowerA <span class='op'>=</span> <span class='va'>la</span>,upperA<span class='op'>=</span><span class='va'>ua</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: 4pl model ...
#> type = map
#> Estimation finished!</div><div class='input'><span class='co'># EAP estimation</span>
<span class='va'>res4pleap</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>awm</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>,
slopes <span class='op'>=</span> <span class='va'>sl</span>,lowerA <span class='op'>=</span> <span class='va'>la</span>,upperA<span class='op'>=</span><span class='va'>ua</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: 4pl model ...
#> type = eap
#> Estimation finished!</div><div class='input'>
<span class='co'>## A special on robust estimation:</span>
<span class='co'># it reproduces the example given in Schuster & Ke-Hai 2011:</span>
<span class='va'>diffpar</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'>3</span>,<span class='op'>-</span><span class='fl'>2</span>,<span class='op'>-</span><span class='fl'>1</span>,<span class='fl'>0</span>,<span class='fl'>1</span>,<span class='fl'>2</span>,<span class='fl'>3</span><span class='op'>)</span>
<span class='va'>AWM</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='fl'>0</span>,<span class='fl'>7</span>,<span class='fl'>7</span><span class='op'>)</span>
<span class='fu'><a href='https://rdrr.io/r/base/diag.html'>diag</a></span><span class='op'>(</span><span class='va'>AWM</span><span class='op'>)</span> <span class='op'><-</span> <span class='fl'>1</span>
<span class='va'>res1plmle</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>AWM</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>, type <span class='op'>=</span> <span class='st'>"mle"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 1pl model ...
#> type = mle
#> Estimation finished!</div><div class='input'>
<span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span><span class='op'>(</span><span class='va'>res1plmle</span><span class='op'>)</span>
</div><div class='output co'>#> PP Version: 0.6.3.11
#>
#> Call: PP_4pl(respm = AWM, thres = diffpar, type = "mle")
#> - job started @ Mon May 24 13:27:53 2021
#>
#> Estimation type: mle
#>
#> Number of iterations: 4
#> -------------------------------------
#> estimate SE
#> [1,] -2.9409 1.2524
#> [2,] -2.9409 1.2524
#> [3,] -2.9409 1.2524
#> [4,] -2.9409 1.2524
#> [5,] -2.9409 1.2524
#> [6,] -2.9409 1.2524
#> [7,] -2.9409 1.2524</div><div class='input'>
<span class='va'>res1plrob</span> <span class='op'><-</span> <span class='fu'>PP_4pl</span><span class='op'>(</span>respm <span class='op'>=</span> <span class='va'>AWM</span>,thres <span class='op'>=</span> <span class='va'>diffpar</span>, type <span class='op'>=</span> <span class='st'>"robust"</span><span class='op'>)</span>
</div><div class='output co'>#> Estimating: 1pl model ...
#> type = robust
#> Estimation finished!</div><div class='input'>
<span class='fu'><a href='https://rdrr.io/r/base/summary.html'>summary</a></span><span class='op'>(</span><span class='va'>res1plrob</span><span class='op'>)</span>
</div><div class='output co'>#> PP Version: 0.6.3.11
#>
#> Call: PP_4pl(respm = AWM, thres = diffpar, type = "robust")
#> - job started @ Mon May 24 13:27:53 2021
#>
#> Estimation type: robust
#>
#> Number of iterations: 10
#> -------------------------------------
#> estimate SE
#> [1,] -2.7467 1.2114
#> [2,] -2.7467 1.2114
#> [3,] -4.0393 1.6545
#> [4,] -4.2548 1.7791
#> [5,] -4.4176 1.8855
#> [6,] -4.5492 1.9800
#> [7,] -4.6600 2.0658</div></pre>
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