11  Nonparametric Item Response Theory and Mokken Scaling

Item Response Theory (IRT) models describe how responses to items depend on a person’s position on a latent trait. Many familiar IRT models are parametric: they specify a particular mathematical form for the item response function and estimate parameters within that form.

Mokken Scale Analysis (MSA) takes a different approach. It belongs to nonparametric IRT because it places order restrictions on item response functions without requiring a specific logistic or normal-ogive shape (Mokken, 1971; Sijtsma & Molenaar, 2002).

ImportantNonparametric does not mean assumption-free

Mokken models still make strong psychometric assumptions. The difference is that they do not prescribe the exact functional shape of the item response curves. Instead, they emphasize properties such as unidimensionality, local independence, monotonicity, and—in the stronger model—invariant item ordering.

NoteChapter map

This chapter proceeds from theory to analysis:

1. distinguish parametric and nonparametric IRT;
2. define the Monotone Homogeneity and Double Monotonicity models;
3. introduce Loevinger’s scalability coefficients;
4. use the Automatic Item Selection Procedure to explore scales;
5. test monotonicity and local independence; and
6. examine nonintersection/invariant item ordering in R.

11.1 Parametric and Nonparametric IRT

A parametric IRT model specifies a response function such as a logistic curve. Once the model form is selected, item parameters determine features such as location, discrimination, or guessing.

A nonparametric IRT model is less restrictive about the shape of that function. This flexibility is useful when the substantive requirement is mainly ordinal: respondents with higher levels of the latent trait should have at least as high a probability of endorsing or answering an item correctly as respondents with lower levels.

The distinction can be summarized as follows:

Parametric IRT	Nonparametric IRT / Mokken
Specifies a particular item-response function	Does not specify an exact parametric curve
Parameters describe the curve’s form	Order restrictions describe how the curve may behave
Stronger assumptions can yield efficient parameter estimates	Weaker functional-form assumptions provide greater flexibility
Misfit can arise from an incorrect curve form	Misfit is evaluated through violations of monotonicity, scalability, ordering, and related assumptions

Franco et al. (2022) provide a detailed psychometric introduction to Mokken Scale Analysis and its use in psychology.

12 The Mokken Models

Mokken Scale Analysis is built around two nested models:

1. the Monotone Homogeneity Model (MHM); and
2. the Double Monotonicity Model (DMM).

The DMM adds a stronger item-ordering requirement to the MHM (Mokken, 1971; Sijtsma & Molenaar, 2002).

12.1 Assumption 1: Unidimensionality

For a unidimensional Mokken scale, the responses are organized primarily by one latent variable, usually denoted

\[ \theta. \]

The assumption does not mean that every human response is literally caused by only one psychological process. It means that the item set is modeled as measuring one dominant ordering variable.

12.2 Assumption 2: Local Independence

Let

\[ \mathbf{X}=(X_1,\ldots,X_k) \]

be the vector of item responses and let \(\mathbf{x}=(x_1,\ldots,x_k)\) be one response pattern. Local independence states that, conditional on \(\theta\),

\[ P(\mathbf{X}=\mathbf{x}\mid\theta) = \prod_{i=1}^{k} P(X_i=x_i\mid\theta). \]

Thus, once people are compared at the same trait level, knowing their response to one item should not provide additional information about their response to another item.

A related implication is

\[ \operatorname{Cov}(X_i,X_j\mid\theta)=0. \]

Observed items may still correlate in the population because respondents differ in \(\theta\). Local independence says that this covariance should be accounted for by the latent trait rather than by additional direct item dependence.

12.3 Assumption 3: Latent Monotonicity

For an item \(i\), let

\[ P_i(\theta) \]

denote the probability of endorsing the item or obtaining a higher score. Monotonicity requires this function to be nondecreasing:

\[ \theta_a \leq \theta_b \quad\Rightarrow\quad P_i(\theta_a)\leq P_i(\theta_b). \]

A person with more of the latent trait should therefore never have a systematically lower probability of obtaining the higher item score.

NoteWhat monotonicity does not require

The curve does not have to be logistic, linear, or symmetric. It can take many shapes as long as it does not systematically decrease as \(\theta\) increases.

12.4 Assumption 4: Nonintersection / Invariant Item Ordering

The Double Monotonicity Model adds a stronger requirement: item response functions should preserve their order across the latent trait.

For appropriately ordered items,

\[ P_1(\theta) \leq P_2(\theta) \leq \cdots \leq P_k(\theta) \qquad\text{for all }\theta. \]

If Item 1 is more difficult than Item 2, it should remain more difficult throughout the trait continuum. The item-response functions should not cross in a way that reverses the item ordering.

12.5 MHM versus DMM

Model	Assumptions	Main ordering implication
Monotone Homogeneity Model	Unidimensionality + local independence + monotonicity	Respondents can be ordered on the latent trait
Double Monotonicity Model	MHM + invariant item ordering	Respondents and items can both be ordered

The MHM is therefore sufficient when the aim is an ordinal ordering of persons. The DMM is useful when the researcher also wants a stable ordering of items (Sijtsma & Molenaar, 2002).

13 Loevinger’s Scalability Coefficients

A central quantity in Mokken Scale Analysis is Loevinger’s scalability coefficient \(H\) (Loevinger, 1948). The coefficient evaluates how well observed item responses approach the ordering expected under a Guttman-type scale while allowing probabilistic errors.

There are three levels of scalability.

13.1 Pairwise scalability: (H_{ij})

For items \(i\) and \(j\),

\[ H_{ij} = \frac{ \operatorname{Cov}(X_i,X_j) }{ \operatorname{Cov}(X_i,X_j)^{\max} }. \]

The denominator is the maximum covariance attainable under the marginal item-score distributions without Guttman errors.

13.2 Item scalability: (H_i)

For item \(i\),

\[ H_i = \frac{ \operatorname{Cov}(X_i,R_{-i}) }{ \operatorname{Cov}(X_i,R_{-i})^{\max} }, \]

where \(R_{-i}\) is the rest score: the total score on the remaining items after item \(i\) is removed.

13.3 Scale scalability: (H)

At the complete-scale level,

\[ H = \frac{ \sum_i \operatorname{Cov}(X_i,R_{-i}) }{ \sum_i \operatorname{Cov}(X_i,R_{-i})^{\max} }. \]

Under a properly oriented Mokken scale, the pair, item, and scale coefficients are expected to be nonnegative. Negative coefficients are a strong signal that an item’s ordering is inconsistent with the remaining scale (Sijtsma & Molenaar, 2002).

A commonly used descriptive convention is:

Scale (H)	Descriptive interpretation
(H < .30)	Insufficient scalability for a Mokken scale
(.30 H < .40)	Weak
(.40 H < .50)	Medium
(H > .50)	Strong

These values are useful descriptive references, not substitutes for assumption checking or substantive interpretation.

ImportantH is not a reliability coefficient

A high scalability coefficient indicates that the item ordering is close to the pattern expected by the Mokken model. It does not by itself establish unidimensionality, local independence, monotonicity, validity, or score reliability.

14 Mokken Scale Analysis in R

The mokken package implements Mokken Scale Analysis in R (Ark, 2007; Ark, 2012).

14.1 Install and load the package

install.packages("mokken")

library(mokken)

Carregando pacotes exigidos: poLCA

Carregando pacotes exigidos: scatterplot3d

Carregando pacotes exigidos: MASS

14.2 Example data: transitive reasoning

We use the transreas dataset included with the package. The data contain responses from 425 Dutch children to a set of dichotomous transitive reasoning tasks (Verweij et al., 1996).

data(transreas)

items <- transreas[, 2:ncol(transreas)]

dim(items)

[1] 425  12

The item set contains substantive transitive-reasoning tasks as well as pseudo-items that are useful for illustrating how Mokken procedures separate scalable from nonscalable indicators.

15 Step 1: Explore Dimensionality with AISP

Mokken Scale Analysis does not require factor-retention procedures such as parallel analysis. Instead, the Automatic Item Selection Procedure (AISP) searches for subsets of items that satisfy a chosen lower bound for scalability (Mokken, 1971).

The procedure begins with strongly scalable items, builds a scale by adding compatible items, and then attempts to form additional scales from items that remain unassigned.

Genetic-algorithm search can improve the item-selection procedure in complex solutions (Ark, 2012; Straat et al., 2013).

AISP <- mokken::aisp(
  items,
  search = "ga",
  lowerbound = seq(.30, .80, by = .05)
)

print(AISP)

     0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
T09L   1    1   1    1   1    1   2    1   2    2   1
T12P   0    0   0    0   0    0   0    0   0    0   0
T10W   1    1   1    1   1    2   0    2   0    0   0
T11P   0    0   0    0   0    0   0    0   0    0   0
T04W   2    2   0    0   0    0   0    0   0    0   0
T05W   0    0   0    0   0    0   0    0   0    0   0
T02L   2    2   0    0   0    0   0    0   0    0   0
T07L   1    1   1    1   1    1   1    0   1    1   0
T03W   1    1   1    1   1    2   0    0   0    0   0
T01L   1    1   1    1   0    0   0    0   0    0   0
T08W   1    1   1    1   1    1   2    1   2    2   1
T06A   1    1   1    1   1    1   1    2   1    1   0

Each column corresponds to a lower-bound value. The entries indicate which scale an item is assigned to:

• 0 = the item is not assigned to a scale at that lower bound;
• 1 = Scale 1;
• 2 = Scale 2; and so forth.

15.1 How to interpret the AISP table

Do not select a lower bound only because it produces the preferred number of scales. Look for a solution that is stable across nearby lower bounds and substantively interpretable.

For the example used in the original chapter, the first scale remains relatively stable through a lower bound of approximately .45, whereas a second scale is less consistent across lower-bound choices. This makes the first grouping a reasonable candidate for subsequent assumption checking.

scale1 <- items[
  ,
  colnames(items)[which(AISP[, "0.45"] == 1)],
  drop = FALSE
]

colnames(scale1)

[1] "T09L" "T10W" "T07L" "T03W" "T01L" "T08W" "T06A"

WarningAISP is exploratory

AISP helps identify scalable item subsets. It does not prove that the resulting scale is unidimensional or that every retained item satisfies all Mokken assumptions. The candidate scale still needs to be checked.

16 Step 2: Check Scalability Coefficients

Before moving to more detailed assumption checks, inspect the pairwise, item, and total scalability coefficients:

H_results <- mokken::coefH(scale1)

$Hij
     T09L    se      T10W    se      T07L    se      T03W    se      T01L   
T09L                  0.463  (0.072)  0.497  (0.140)  0.526  (0.161)  0.336 
T10W  0.463  (0.072)                  0.534  (0.096)  0.568  (0.111)  0.538 
T07L  0.497  (0.140)  0.534  (0.096)                  0.444  (0.080)  0.479 
T03W  0.526  (0.161)  0.568  (0.111)  0.444  (0.080)                  0.548 
T01L  0.336  (0.260)  0.538  (0.161)  0.479  (0.114)  0.548  (0.108)        
T08W  1.000  (0.000)  0.588  (0.209)  0.493  (0.154)  0.677  (0.135)  0.317 
T06A  0.698  (0.286)  0.650  (0.222)  0.785  (0.137)  0.589  (0.162)  0.420 
     se      T08W    se      T06A    se     
T09L (0.260)  1.000  (0.000)  0.698  (0.286)
T10W (0.161)  0.588  (0.209)  0.650  (0.222)
T07L (0.114)  0.493  (0.154)  0.785  (0.137)
T03W (0.108)  0.677  (0.135)  0.589  (0.162)
T01L          0.317  (0.133)  0.420  (0.157)
T08W (0.133)                  0.530  (0.153)
T06A (0.157)  0.530  (0.153)                

$Hi
     Item H  se     
T09L   0.496 (0.070)
T10W   0.518 (0.057)
T07L   0.507 (0.060)
T03W   0.531 (0.064)
T01L   0.462 (0.085)
T08W   0.546 (0.103)
T06A   0.592 (0.098)

$H
Scale H      se 
  0.515 (0.052) 

H_results

$Hij
     T09L    se      T10W    se      T07L    se      T03W    se      T01L   
T09L                  0.463  (0.072)  0.497  (0.140)  0.526  (0.161)  0.336 
T10W  0.463  (0.072)                  0.534  (0.096)  0.568  (0.111)  0.538 
T07L  0.497  (0.140)  0.534  (0.096)                  0.444  (0.080)  0.479 
T03W  0.526  (0.161)  0.568  (0.111)  0.444  (0.080)                  0.548 
T01L  0.336  (0.260)  0.538  (0.161)  0.479  (0.114)  0.548  (0.108)        
T08W  1.000  (0.000)  0.588  (0.209)  0.493  (0.154)  0.677  (0.135)  0.317 
T06A  0.698  (0.286)  0.650  (0.222)  0.785  (0.137)  0.589  (0.162)  0.420 
     se      T08W    se      T06A    se     
T09L (0.260)  1.000  (0.000)  0.698  (0.286)
T10W (0.161)  0.588  (0.209)  0.650  (0.222)
T07L (0.114)  0.493  (0.154)  0.785  (0.137)
T03W (0.108)  0.677  (0.135)  0.589  (0.162)
T01L          0.317  (0.133)  0.420  (0.157)
T08W (0.133)                  0.530  (0.153)
T06A (0.157)  0.530  (0.153)                

$Hi
     Item H  se     
T09L   0.496 (0.070)
T10W   0.518 (0.057)
T07L   0.507 (0.060)
T03W   0.531 (0.064)
T01L   0.462 (0.085)
T08W   0.546 (0.103)
T06A   0.592 (0.098)

$H
Scale H      se 
  0.515 (0.052) 

$covHij
               [,1]          [,2]          [,3]          [,4]           [,5]
 [1,]  5.124444e-03  2.291763e-03  5.188406e-04  1.470397e-03   5.215660e-85
 [2,]  2.291763e-03  1.950048e-02  7.257806e-03  1.335455e-02   6.127697e-84
 [3,]  5.188406e-04  7.257806e-03  2.576480e-02  7.412712e-03   1.236029e-83
 [4,]  1.470397e-03  1.335455e-02  7.412712e-03  6.745195e-02   1.754834e-83
 [5,]  5.215660e-85  6.127697e-84  1.236029e-83  1.754834e-83  4.060058e-163
 [6,] -6.777996e-04  1.213038e-02  6.654589e-04  3.048915e-02   2.503084e-83
 [7,]  2.991652e-04  2.197929e-03  1.032205e-03  4.798878e-03   8.304260e-84
 [8,]  2.631726e-04  1.027956e-03  8.371341e-03 -6.215615e-04   1.204504e-83
 [9,] -9.204136e-06  4.688150e-03 -8.377601e-04  1.747996e-02   1.265807e-83
[10,]  1.075011e-03  1.855521e-03  2.612667e-03  3.738671e-03  -3.693808e-85
[11,]  3.418055e-04  7.533212e-03  1.423006e-03  1.768830e-02  -2.713458e-85
[12,] -2.495751e-04  5.282013e-04  1.087163e-03  4.762026e-04   9.461434e-84
[13,]  1.879366e-04 -8.275895e-04 -2.020509e-04 -1.095805e-03   1.665723e-83
[14,]  3.922079e-04  2.264756e-03  1.854325e-03  4.923104e-03  -1.136134e-84
[15,]  2.669003e-04  4.581488e-04  3.495675e-04  1.481677e-04  -1.669361e-83
[16,] -8.381994e-05  2.340116e-03  1.100019e-03  1.126950e-02   1.210758e-83
[17,]  4.254962e-04  1.392247e-03  2.257697e-03  2.870105e-03  -1.096703e-84
[18,]  3.174172e-05  3.227922e-03  2.321078e-03  1.033224e-02   1.796487e-83
[19,]  2.023778e-04  1.332078e-03  9.332798e-04  6.502099e-03  -1.128107e-84
[20,]  1.880219e-04 -1.695662e-03  8.616335e-04  2.923469e-04   4.481609e-84
[21,] -2.055487e-04  1.675852e-03  7.926000e-04  7.098194e-03   3.994796e-83
               [,6]          [,7]          [,8]          [,9]         [,10]
 [1,] -6.777996e-04  2.991652e-04  2.631726e-04 -9.204136e-06  1.075011e-03
 [2,]  1.213038e-02  2.197929e-03  1.027956e-03  4.688150e-03  1.855521e-03
 [3,]  6.654589e-04  1.032205e-03  8.371341e-03 -8.377601e-04  2.612667e-03
 [4,]  3.048915e-02  4.798878e-03 -6.215615e-04  1.747996e-02  3.738671e-03
 [5,]  2.503084e-83  8.304260e-84  1.204504e-83  1.265807e-83 -3.693808e-85
 [6,]  8.203175e-02  7.120081e-03  1.023608e-03  1.736636e-02  7.414980e-04
 [7,]  7.120081e-03  9.272834e-03  2.487859e-03  4.295681e-03  7.390190e-04
 [8,]  1.023608e-03  2.487859e-03  1.223636e-02  4.721542e-03  1.722962e-03
 [9,]  1.736636e-02  4.295681e-03  4.721542e-03  2.597981e-02  1.799901e-03
[10,]  7.414980e-04  7.390190e-04  1.722962e-03  1.799901e-03  4.363708e-02
[11,]  4.255383e-02  4.145425e-03  1.199686e-03  9.907663e-03  1.938324e-02
[12,]  1.595661e-03  1.006671e-03  2.438972e-03  1.343312e-03  1.476339e-03
[13,] -3.294696e-03  2.592726e-04  1.164842e-03  1.291066e-03  2.742581e-03
[14,]  1.436415e-03  1.796721e-03  1.312949e-03  3.057440e-03  1.935734e-02
[15,] -5.813420e-03  5.525154e-04  7.952012e-05  1.643483e-04  8.967135e-03
[16,]  9.006840e-03  1.652664e-03  8.632106e-04  5.301939e-03  2.205295e-03
[17,]  1.433212e-03  8.821499e-04  1.718701e-03  1.688280e-03  2.321261e-02
[18,]  1.939368e-02  1.755570e-03  2.079917e-03  6.207543e-03  1.202063e-02
[19,]  2.076368e-03  9.828260e-04  7.280852e-04  4.358679e-03  1.078883e-02
[20,] -1.553164e-02 -8.915929e-04  6.554929e-04  8.500721e-04  6.628352e-03
[21,]  1.517961e-02  8.180671e-04  8.709281e-04  4.304422e-03  1.670189e-03
              [,11]         [,12]         [,13]         [,14]         [,15]
 [1,]  3.418055e-04 -2.495751e-04  1.879366e-04  3.922079e-04  2.669003e-04
 [2,]  7.533212e-03  5.282013e-04 -8.275895e-04  2.264756e-03  4.581488e-04
 [3,]  1.423006e-03  1.087163e-03 -2.020509e-04  1.854325e-03  3.495675e-04
 [4,]  1.768830e-02  4.762026e-04 -1.095805e-03  4.923104e-03  1.481677e-04
 [5,] -2.713458e-85  9.461434e-84  1.665723e-83 -1.136134e-84 -1.669361e-83
 [6,]  4.255383e-02  1.595661e-03 -3.294696e-03  1.436415e-03 -5.813420e-03
 [7,]  4.145425e-03  1.006671e-03  2.592726e-04  1.796721e-03  5.525154e-04
 [8,]  1.199686e-03  2.438972e-03  1.164842e-03  1.312949e-03  7.952012e-05
 [9,]  9.907663e-03  1.343312e-03  1.291066e-03  3.057440e-03  1.643483e-04
[10,]  1.938324e-02  1.476339e-03  2.742581e-03  1.935734e-02  8.967135e-03
[11,]  4.937387e-02  1.519570e-03 -4.774828e-04  9.637893e-03  1.181554e-02
[12,]  1.519570e-03  6.344432e-03  3.204835e-03  3.783223e-03  2.062627e-03
[13,] -4.774828e-04  3.204835e-03  1.308394e-02  3.056845e-03  1.656607e-03
[14,]  9.637893e-03  3.783223e-03  3.056845e-03  2.380768e-02  1.097068e-02
[15,]  1.181554e-02  2.062627e-03  1.656607e-03  1.097068e-02  1.876210e-02
[16,]  5.825290e-03  1.180921e-03  5.280061e-03  2.909738e-03  7.921922e-04
[17,]  1.147000e-02  1.240152e-03  2.052567e-03  1.520042e-02  4.725970e-03
[18,]  2.243866e-02  2.051200e-03  1.605642e-03  6.548249e-03  2.979217e-03
[19,]  6.723704e-03  2.188778e-03  4.871780e-03  1.323700e-02  6.852792e-03
[20,] -1.482173e-03  1.737301e-03  5.382593e-03  8.013988e-03  9.264682e-03
[21,]  1.470099e-03  2.957008e-04  5.487076e-04  2.194908e-03 -9.030244e-03
              [,16]         [,17]        [,18]         [,19]         [,20]
 [1,] -8.381994e-05  4.254962e-04 3.174172e-05  2.023778e-04  1.880219e-04
 [2,]  2.340116e-03  1.392247e-03 3.227922e-03  1.332078e-03 -1.695662e-03
 [3,]  1.100019e-03  2.257697e-03 2.321078e-03  9.332798e-04  8.616335e-04
 [4,]  1.126950e-02  2.870105e-03 1.033224e-02  6.502099e-03  2.923469e-04
 [5,]  1.210758e-83 -1.096703e-84 1.796487e-83 -1.128107e-84  4.481609e-84
 [6,]  9.006840e-03  1.433212e-03 1.939368e-02  2.076368e-03 -1.553164e-02
 [7,]  1.652664e-03  8.821499e-04 1.755570e-03  9.828260e-04 -8.915929e-04
 [8,]  8.632106e-04  1.718701e-03 2.079917e-03  7.280852e-04  6.554929e-04
 [9,]  5.301939e-03  1.688280e-03 6.207543e-03  4.358679e-03  8.500721e-04
[10,]  2.205295e-03  2.321261e-02 1.202063e-02  1.078883e-02  6.628352e-03
[11,]  5.825290e-03  1.147000e-02 2.243866e-02  6.723704e-03 -1.482173e-03
[12,]  1.180921e-03  1.240152e-03 2.051200e-03  2.188778e-03  1.737301e-03
[13,]  5.280061e-03  2.052567e-03 1.605642e-03  4.871780e-03  5.382593e-03
[14,]  2.909738e-03  1.520042e-02 6.548249e-03  1.323700e-02  8.013988e-03
[15,]  7.921922e-04  4.725970e-03 2.979217e-03  6.852792e-03  9.264682e-03
[16,]  1.167544e-02  1.620302e-03 3.922872e-03  4.306217e-03  2.308159e-03
[17,]  1.620302e-03  1.825994e-02 7.644636e-03  8.481083e-03  3.932522e-03
[18,]  3.922872e-03  7.644636e-03 2.635703e-02  4.976398e-03  7.931710e-03
[19,]  4.306217e-03  8.481083e-03 4.976398e-03  1.774891e-02  1.053639e-02
[20,]  2.308159e-03  3.932522e-03 7.931710e-03  1.053639e-02  2.473096e-02
[21,]  2.556516e-03  2.377019e-03 1.116614e-02  2.744000e-03  2.416039e-03
              [,21]
 [1,] -2.055487e-04
 [2,]  1.675852e-03
 [3,]  7.926000e-04
 [4,]  7.098194e-03
 [5,]  3.994796e-83
 [6,]  1.517961e-02
 [7,]  8.180671e-04
 [8,]  8.709281e-04
 [9,]  4.304422e-03
[10,]  1.670189e-03
[11,]  1.470099e-03
[12,]  2.957008e-04
[13,]  5.487076e-04
[14,]  2.194908e-03
[15,] -9.030244e-03
[16,]  2.556516e-03
[17,]  2.377019e-03
[18,]  1.116614e-02
[19,]  2.744000e-03
[20,]  2.416039e-03
[21,]  2.353244e-02

$covHi
            [,1]        [,2]        [,3]        [,4]        [,5]        [,6]
[1,] 0.004875122 0.002521933 0.001537349 0.001674098 0.001611928 0.001088075
[2,] 0.002521933 0.003198798 0.001843146 0.001938512 0.001784295 0.001997937
[3,] 0.001537349 0.001843146 0.003564549 0.002587688 0.002769382 0.002814759
[4,] 0.001674098 0.001938512 0.002587688 0.004139013 0.002837914 0.003183693
[5,] 0.001611928 0.001784295 0.002769382 0.002837914 0.007192337 0.004400494
[6,] 0.001088075 0.001997937 0.002814759 0.003183693 0.004400494 0.010555713
[7,] 0.001596821 0.001958767 0.002247952 0.002983221 0.004387283 0.006076041
            [,7]
[1,] 0.001596821
[2,] 0.001958767
[3,] 0.002247952
[4,] 0.002983221
[5,] 0.004387283
[6,] 0.006076041
[7,] 0.009647901

$covH
            [,1]
[1,] 0.002693066

Pay particular attention to:

• negative or very small \(H_{ij}\) values;
• items with \(H_i\) below the intended lower bound; and
• the overall \(H\) of the candidate scale.

The coefficients should be interpreted together with the AISP solution, not as independent pass/fail tests.

17 Step 3: Check Latent Monotonicity

MonLat <- mokken::check.monotonicity(scale1)

summary(MonLat)

     ItemH #ac #vi #vi/#ac maxvi sum sum/#ac zmax #zsig crit
T09L  0.50   1   0       0     0   0       0    0     0    0
T10W  0.52   3   0       0     0   0       0    0     0    0
T07L  0.51   3   0       0     0   0       0    0     0    0
T03W  0.53   3   0       0     0   0       0    0     0    0
T01L  0.46   3   0       0     0   0       0    0     0    0
T08W  0.55   1   0       0     0   0       0    0     0    0
T06A  0.59   0   0     NaN     0   0     NaN    0     0    0

The monotonicity output evaluates whether item response functions show substantial decreases across increasing rest-score groups.

Important quantities typically include:

Output	Meaning
#ac / active comparisons	Number of comparisons available for assessing monotonicity
#vi	Number of observed monotonicity violations
maxvi	Magnitude of the largest violation
zmax	Standardized statistic for the largest violation
#zsig	Number of statistically significant violations

Statistical significance alone should not decide whether an item is removed. The number, magnitude, location, and substantive importance of violations all matter.

In the example, Item T06A may provide little useful variation and can be removed before continuing:

scale2 <- scale1[
  ,
  setdiff(colnames(scale1), "T06A"),
  drop = FALSE
]

If the item name differs in your installed version of the dataset, inspect colnames(scale1) and remove the corresponding item explicitly.

18 Step 4: Check Local Independence

The conditional-association procedure can be used to investigate local dependence:

CA <- mokken::check.ca(scale2)

print(CA)

[[1]]
[1] TRUE TRUE TRUE TRUE TRUE TRUE

Evidence of conditional association between item pairs suggests that their relationship is not fully accounted for by the latent ordering variable. Possible reasons include highly similar wording, shared stimulus material, testlets, or additional substantive dimensions.

TipLocal dependence is substantive information

If two items remain dependent after conditioning on the scale, do not automatically delete one. First ask why the dependence exists. A pair of nearly duplicated items implies a different measurement problem than a pair that shares a theoretically meaningful secondary process.

19 Step 5: Check Invariant Item Ordering

For the Double Monotonicity Model, we also examine whether item response functions preserve their ordering.

NI <- mokken::check.pmatrix(scale2)

summary(NI)

     ItemH #ac #vi #vi/#ac maxvi sum sum/#ac zmax #zsig crit
T09L  0.49  20   0       0     0   0       0    0     0    0
T10W  0.51  20   0       0     0   0       0    0     0    0
T07L  0.49  20   0       0     0   0       0    0     0    0
T03W  0.53  20   0       0     0   0       0    0     0    0
T01L  0.47  20   0       0     0   0       0    0     0    0
T08W  0.55  20   0       0     0   0       0    0     0    0

A solution without important violations provides evidence consistent with invariant item ordering. If item curves cross substantially, the weaker Monotone Homogeneity Model may still be useful even when the Double Monotonicity Model is not supported.

20 A Practical Mokken Workflow

Stage	Question
AISP	Which groups of items form stable scalable subsets?
(H_{ij}, H_i, H)	How strongly do item pairs, items, and the full scale satisfy the scalability ordering?
Monotonicity	Do higher trait/rest-score levels correspond to nondecreasing item endorsement?
Local independence	Are important item relations left after conditioning on the latent ordering?
Invariant item ordering	Does the relative difficulty/order of items remain stable across respondents?
Substantive review	Do the retained items still represent the intended construct adequately?

21 Reporting Mokken Scale Analysis

A transparent report should include:

• the item response format;
• the AISP search method and lower-bound values examined;
• the selected scale solution and how it was chosen;
• pairwise, item, and scale-level \(H\) coefficients;
• monotonicity diagnostics;
• the method used to inspect local independence;
• evidence for or against invariant item ordering;
• any items removed and the reason for removal; and
• the substantive interpretation of the final scale.

A concise description could be:

Mokken Scale Analysis was conducted using the mokken package. Candidate scales were explored with the automatic item selection procedure using a genetic-algorithm search across scalability lower bounds from .30 to .80. The retained solution was then evaluated using Loevinger’s pairwise, item, and total scalability coefficients, monotonicity checks, conditional-association diagnostics for local independence, and invariant-item-ordering analyses.

Report the actual coefficients and violation statistics from your analysis rather than relying only on qualitative labels such as weak or strong.

22 Concluding Remarks

Mokken Scale Analysis is useful when the primary measurement goal is ordinal and the researcher does not want to impose a specific parametric shape on item response functions. This flexibility does not eliminate psychometric assumptions; it shifts attention toward order restrictions that can be evaluated directly.

The Monotone Homogeneity Model supports an ordinal ordering of persons when unidimensionality, local independence, and monotonicity are reasonable. The stronger Double Monotonicity Model additionally supports an invariant ordering of items. Loevinger’s \(H\), AISP, monotonicity checks, conditional association, and invariant-item-ordering diagnostics address different parts of this framework.

The important lesson is therefore not simply that Mokken scaling is “nonparametric.” Its value lies in making a different set of measurement claims—particularly monotonic and ordinal claims—explicit and empirically examinable.

Ark, L. A. van der. (2007). Mokken scale analysis in r. Journal of Statistical Software, 20(11), 1–19. https://doi.org/10.18637/jss.v020.i11

Ark, L. A. van der. (2012). New developments in mokken scale analysis in r. Journal of Statistical Software, 48(5), 1–27. https://doi.org/10.18637/jss.v048.i05

Franco, V. R., Laros, J. A., & Bastos, R. V. S. (2022). Theoretical and practical foundations of mokken scale analysis in psychology. Paidéia (Ribeirão Preto), 32, e3223. https://doi.org/10.1590/1982-4327e3223

Loevinger, J. (1948). The technique of homogenous tests compared with some aspects of “scale analysis” and factor analysis. Psychological Bulletin, 45(6), 507–530. https://doi.org/10.1037/h0055827

Mokken, R. J. (1971). A theory and procedure of scale analysis. De Gruyter. https://doi.org/10.1515/9783110813203

Sijtsma, K., & Molenaar, I. W. (2002). Introduction to nonparametric item response theory. Sage. https://doi.org/10.4135/9781412984676

Straat, J. H., Ark, L. A. van der, & Sijtsma, K. (2013). Comparing optimization algorithms for item selection in mokken scale analysis. Journal of Classification, 30, 72–99. https://doi.org/10.1007/s00357-013-9122-y

Verweij, A. C., Sijtsma, K., & Koops, W. (1996). A mokken scale for transitive reasoning suited for longitudinal research. International Journal of Behavioral Development, 19(1), 219–238. https://doi.org/10.1177/016502549601900115