# Weighted pairwise likelihood and limited information goodness-of-fit tests for binary factor models

Adolphe Quetelet Seminar Series, Ghent University

Haziq Jamil

Assistant Professor in Statistics, Universiti Brunei Darussalam
Visiting Fellow, London School of Economics and Political Science

April 15, 2024

H Jamil, I Moustaki, C Skinner. 2023+. Pairwise likelihood estimation and limited information goodness-of-fit test statistics for binary factor analysis models under complex survey sampling. Manuscript under revision.

# Introduction

## Introduction

Context

Employ latent variable models (factor models) to analyse binary data $y_1,\dots,y_p$ collected via simple random or complex sampling.

## Introduction (cont.)

$i$ $y_1$ $y_2$ $y_3$ $y_4$ $y_5$
1 1 0 0 1 1
2 1 1 1 1 1
3 1 1 1 0 1
$\vdots$ $\vdots$
$n$ 1 0 0 1 1
$i$ $y_1$ $y_2$ $y_3$ $y_4$ $y_5$ Pattern
1 1 0 0 1 1 10011
2 1 1 1 1 1 11111
3 1 1 1 0 1 11101
$\vdots$ $\vdots$ $\vdots$
$n$ 1 0 0 1 1 10011
$r$ $y_1$ $y_2$ $y_3$ $y_4$ $y_5$ Pattern Obs. freq
1 1 1 1 1 1 11111 343
2 1 1 0 1 1 11011 153
3 1 0 1 1 1 10111 71
$\vdots$ $\vdots$ $\vdots$ $\vdots$
$R$ 0 1 1 1 0 01110 1

$R = 2^p$

$r$ Pattern Obs. freq
1 11111 343
2 11011 153
3 10111 71
$\vdots$ $\vdots$ $\vdots$
32 01110 1
$r$ Pattern Obs. freq Exp. freq
1 11111 343 342.1
2 11011 153 151.3
3 10111 71 62.81
$\vdots$ $\vdots$ $\vdots$ $\vdots$
32 01110 1 0.948
$r$ Pattern Obs. freq Exp. freq
1 11111 343 342.1
2 11011 153 151.3
3 10111 71 62.81
$\vdots$ $\vdots$ $\vdots$ $\vdots$
28 01000 1 1.831
29 01010 1 3.276
30 01100 1 0.948
31 01101 0 0.013
32 01110 0 0.009
$r$ Pattern Obs. freq Exp. freq
1 11111 343 342.1
2 11011 153 151.3
3 10111 71 62.81
$\vdots$ $\vdots$ $\vdots$ $\vdots$
28 01000 1 1.831
29 01010 1 3.276
30 01100 1 0.948
31 01101 0 0.013
32 01110 0 0.009
$r$ Pattern Obs. freq Exp. freq
1 11111 360.9 342.1
2 11011 181.2 151.3
3 10111 68.05 62.81
$\vdots$ $\vdots$ $\vdots$ $\vdots$
28 01000 1.716 1.831
29 01010 1.120 3.276
30 01100 0.591 0.948
31 01101 0 0.013
32 01110 0 0.009
• Sparsity affects reliability of goodness-of-fit tests.
• Limited information tests
• Computational burden of likelihood-based models.
• Pairwise likelihood
• Unequal probability sampling (e.g. due to a complex design)
• Incorporate design weights

## Definitions

• Let ${\mathbf y}= (y_1, \ldots, y_p)^\top \in\{0,1\}^p$ be a vector of Bernoulli random variables.
• The joint probability of observing a response pattern ${\mathbf c}_r=(c_{r1},\dots,c_{rp})^\top$, for each $r=1,\dots,R:=2^p$, is given by $\pi_r = \operatorname{P}({\mathbf y}= {\mathbf c}_r) = \operatorname{P}(y_1=c_{r1},\dots,y_p=c_{rp}), \tag{1}$ with $\sum_{r=1}^R \pi_r = 1$.

• Suppose observations ${\mathcal Y}:= \big\{{\mathbf y}^{(h)}\big\}_{h=1}^n$ are obtained, where each unit $h$ has a probability of selection $1/w_h$.

• Out of convenience, sampling weights are (typically) normalised so that $\sum_{h=1}^n w_h = n$.

• Simple random sampling (SRS): $w_h=1, \forall h$.
• Stratified sampling: $w_h = N_s(h)/N$, the stratum fractions.
• Etc.

## Multinomial distribution

• Let $p_r = \hat n_r \big/ \sum_h w_h$ be the the $r$th entry of the $R$-vector of proportions ${\mathbf p}$, with $\hspace{1.4cm}$

$\hat n_r = \sum_{h=1}^n w_h [{\mathbf y}^{(h)} = {\mathbf c}_r]. \tag{2}$

• The random vector $\hat{\mathbf n}= (\hat n_1,\dots,\hat n_R)^\top$ follows a multinomial distribution with parameters $n$, $R$, and ${\boldsymbol\pi}:=(\pi_1,\dots,\pi_R)^\top$, with

$\mathop{\mathrm{E}}(\hat{\mathbf n}) = n{\boldsymbol\pi}\hspace{2em}\text{and}\hspace{2em} \mathop{\mathrm{Var}}(\hat{\mathbf n}) = n\big( \ {\color{lightgray}\overbrace{\color{black}\mathop{\mathrm{diag}}({\boldsymbol\pi}) - {\boldsymbol\pi}{\boldsymbol\pi}^\top}^{{\boldsymbol\Sigma}}} \ \big).$

• It is widely known that for IID samples that $\sqrt n ({\mathbf p}- {\boldsymbol\pi}) \xrightarrow{\text D} {\mathop{\mathrm{N}}}_R({\mathbf 0}, {\boldsymbol\Sigma}) \tag{3}$ as $n\to\infty$. This also works for complex sampling , but ${\boldsymbol\Sigma}$ need not take a multinomial form.

## Parametric models

• E.g. binary factor model with underlying variable approach (s.t. constraints) $\begin{gathered} y_i = \begin{cases} 1 & y_i^* > \tau_i \\ 0 & y_i^* \leq \tau_i \end{cases} \\ {\mathbf y}^* = {\boldsymbol\Lambda}{\boldsymbol\eta}+ {\boldsymbol\epsilon}\\ {\boldsymbol\eta}\sim {\mathop{\mathrm{N}}}_q({\mathbf 0}, {\boldsymbol\Psi}), \hspace{3pt} {\boldsymbol\epsilon}\sim {\mathop{\mathrm{N}}}_p({\mathbf 0}, {\boldsymbol\Theta}_{\epsilon}) \end{gathered} \tag{4}$
• The log-likelihood for ${\boldsymbol\theta}^\top = ($${\boldsymbol\lambda}$$,\,$${\boldsymbol\rho}$$,\,$${\boldsymbol\tau}$$)\in\mathbb{R}^m$ is $\log L({\boldsymbol\theta}\mid {\mathcal Y}) = \sum_{r=1}^R \hat n_r \log \pi_r({\boldsymbol\theta}) \tag{5}$ where $\pi_r({\boldsymbol\theta}) = \int \phi_p({\mathbf y}^* \mid {\mathbf 0}, {\boldsymbol\Lambda}{\boldsymbol\Psi}{\boldsymbol\Lambda}^\top + {\boldsymbol\Theta}_\epsilon) \mathop{\mathrm{d}}\hspace{0.5pt}\!{\mathbf y}^*$.
• FIML may be difficult (high-dimensional integral; perfect separation).

## Composite likelihood

• Terminology: Pseudo-likelihood, quasi-likelihood (à la Wedderburn (1974) or misspecified models), limited information methods.
• Let $\{{\mathcal A}_1,\dots,{\mathcal A}_K\}$ be a set of marginal or conditional events (partitioning the variable space). The composite likelihood is defined as ${\mathcal L}({\boldsymbol\theta}\mid {\mathbf y}) = \prod_{k=1}^K L({\boldsymbol\theta}\mid {\mathbf y}\in {\mathcal A}_k)^{\textcolor{lightgray}{\omega_k}}$
• Component likelihoods $L({\boldsymbol\theta}\mid {\mathbf y}\in {\mathcal A}_k)$ are either conditional or marginal densities.

• Composite likelihood enjoys nice features : relatively efficient, robust, and easier to maximise (smoother surface).

## An analogy

One may enjoy the approximate picture despite
not being able to see every blade of grass.

## Pairwise likelihood estimation

• For pairs of variables $y_i$ and $y_j$, $i,j=1,\dots,p$, and $i<j$, define $\hspace{5cm}$ $\pi_{cc'}^{(ij)}({\boldsymbol\theta}) = \operatorname{P}_{{\boldsymbol\theta}}(y_i = c, y_i = c'), \hspace{2em} c,c'\in\{0,1\}. \tag{6}$ There are $\tilde R = 4 \times \binom{p}{2}$ such probabilities, with $\sum_{c,c'} \pi_{cc'}^{(ij)}({\boldsymbol\theta}) = 1$.
• The pairwise log-likelihood takes the form $\log \operatorname{\mathcal L_{\text P}}({\boldsymbol\theta}\mid {\mathcal Y}) = \sum_{i<j} \sum_{c}\sum_{c'} \hat n_{cc'}^{(ij)} \log \pi_{y_iy_j}^{(ij)}({\boldsymbol\theta}), \tag{7}$ where $\hat n_{cc'}^{(ij)} = \sum_h w_h [{\mathbf y}^{(h)}_i = c, {\mathbf y}^{(h)}_j = c']$.

• The evaluation of Equation 7 now involves only bivariate normal integrals! $\pi_{cc'}^{(ij)}({\boldsymbol\theta}) = \iint \phi_2\big({\mathbf y}^*_{ij} \mid {\mathbf 0}, {\boldsymbol\Sigma}_{y^*}^{(ij)} ({\boldsymbol\theta})\big) \mathop{\mathrm{d}}\hspace{0.5pt}\!{\mathbf y}^*_{ij}$

## MPLE properties

• Let $\hat{\boldsymbol\theta}_{\text{PL}}= \mathop{\mathrm{argmax}}_{{\boldsymbol\theta}} \operatorname{\mathcal L_{\text P}}({\boldsymbol\theta}\mid {\mathcal Y})$. Under certain regularity conditions , as $n\to\infty$, $\sqrt n (\hat{\boldsymbol\theta}_{\text{PL}}- {\boldsymbol\theta}) \xrightarrow{\text D} {\mathop{\mathrm{N}}}_m \left( {\mathbf 0}, \left\{ {\mathcal H}({\boldsymbol\theta}){\mathcal J}({\boldsymbol\theta})^{-1}{\mathcal H}({\boldsymbol\theta}) \right\}^{-1} \right),\hspace{1em} \text{where} \tag{8}$
• ${\mathcal H}({\boldsymbol\theta})=-\mathop{\mathrm{E}}\nabla^2\log \operatorname{\mathcal L_{\text P}}({\boldsymbol\theta}\mid {\mathbf y}^{(h)})$ is the sensitivity matrix; and
• ${\mathcal J}({\boldsymbol\theta})=\mathop{\mathrm{Var}}\big(\nabla\log\operatorname{\mathcal L_{\text P}}({\boldsymbol\theta}\mid {\mathbf y}^{(h)})\big)$ is the variability matrix.
• Estimators of these matrices are given by

\begin{aligned} \hat{\mathbf H}&= - \frac{1}{\sum_h w_h} \sum_h \nabla^2\log \operatorname{\mathcal L_{\text P}}({\boldsymbol\theta}\mid {\mathbf y}^{(h)}) \Bigg|_{{\boldsymbol\theta}= \hat{\boldsymbol\theta}_{\text{PL}}} \hspace{2em}\text{and} \\ \hat{\mathbf J}&= \frac{1}{\sum_h w_h} \sum_h \nabla\log \operatorname{\mathcal L_{\text P}}({\boldsymbol\theta}\mid {\mathbf y}^{(h)}) \nabla\log \operatorname{\mathcal L_{\text P}}({\boldsymbol\theta}\mid {\mathbf y}^{(h)})^\top \Bigg|_{{\boldsymbol\theta}= \hat{\boldsymbol\theta}_{\text{PL}}}. \end{aligned} \tag{9}

# Limited information GOF tests

## Goodness-of-fit

• GOF tests are usually constructed by inspecting the fit of the joint probabilities $\hat\pi_r := \pi_r(\hat{\boldsymbol\theta})$.
• Most common tests are

• LR: $X^2 = 2n\sum_r p_r\log( p_r/\hat\pi_r)$;
• Pearson: $X^2 = n\sum_r ( p_r - \hat\pi_r)^2 / \hat\pi_r$.

These tests are asymptotically distributed as chi square.

• Likely to face sparsity issues (small or zero cell counts) which distort the approximation to the chi square.

## Limited information goodness-of-fit (LIGOF)

Consider instead the fit of the lower order marginals.

• Univariate: $\ \dot\pi_i := \operatorname{P}(y_i = 1)$

• Bivariate: $\ \dot\pi_{ij} := \operatorname{P}(y_i = 1, y_j=1)$

• Collectively ${\boldsymbol\pi}_2 = \begin{pmatrix} \dot{\boldsymbol\pi}_1 \\ \dot{\boldsymbol\pi}_2 \\ \end{pmatrix} = \begin{pmatrix} (\dot\pi_1, \dots, \dot\pi_p)^\top \\ \big(\dot\pi_{ij}\big)_{i<j} \\ \end{pmatrix}$ This is of dimensions $S=p + p(p-1)/2 \ll R.$

## Transformation matrix

Define ${\mathbf T}_2: \mathbb{R}^R \to \mathbb{R}^S$ such that ${\boldsymbol\pi}\mapsto {\boldsymbol\pi}_2$. To illustrate, consider $p=3$ so that $R=2^3=8$ and $S=3+3=6$.

${\color{lightgray}\overbrace{\color{black} \left( \begin{array}{c} \dot\pi_1 \\ \dot\pi_2 \\ \dot\pi_3 \\ \hdashline \dot\pi_{12} \\ \dot\pi_{13} \\ \dot\pi_{23} \\ \end{array} \right) \vphantom{ \begin{array}{c} \pi_{000} \\ \pi_{100} \\ \pi_{010} \\ \pi_{001} \\ \pi_{110} \\ \pi_{101} \\ \pi_{011} \\ \pi_{111} \\ \end{array} } }^{{\boldsymbol\pi}_2}} = {\color{lightgray}\overbrace{\color{black} \left( \begin{array}{cccccccc} 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ \hdashline 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ \end{array} \right) \vphantom{ \begin{array}{c} \pi_{000} \\ \pi_{100} \\ \pi_{010} \\ \pi_{001} \\ \pi_{110} \\ \pi_{101} \\ \pi_{011} \\ \pi_{111} \\ \end{array} } }^{{\mathbf T}_2}} \ {\color{lightgray}\overbrace{\color{black} \left( \begin{array}{c} \pi_{000} \\ \pi_{100} \\ \pi_{010} \\ \pi_{001} \\ \pi_{110} \\ \pi_{101} \\ \pi_{011} \\ \pi_{111} \\ \end{array} \right) }^{{\boldsymbol\pi}}}$

## Asymptotic distribution of residuals

Theorem

Consider the lower order residuals $\hat{\mathbf e}_2 = {\mathbf p}_2 - {\boldsymbol\pi}_2(\hat{\boldsymbol\theta}_{\text{PL}})$. Then as $n\to\infty$, $\sqrt n \, \hat{\mathbf e}_2 \xrightarrow{\text D} {\mathop{\mathrm{N}}}_S\left({\mathbf 0}, {\boldsymbol\Omega}_2\right)$ where ${\boldsymbol\Omega}_2 = \left( {\mathbf I}- {\boldsymbol\Delta}_2{\mathcal H}({\boldsymbol\theta})^{-1} {\mathbf B}({\boldsymbol\theta}) \right) {\boldsymbol\Sigma}_2 \left( {\mathbf I}- {\boldsymbol\Delta}_2{\mathcal H}({\boldsymbol\theta})^{-1} {\mathbf B}({\boldsymbol\theta}) \right)^\top$,

• ${\boldsymbol\Sigma}_2 = {\mathbf T}_2{\boldsymbol\Sigma}{\mathbf T}_2^\top$ (uni & bivariate multinomial matrix);
• ${\boldsymbol\Delta}_2 = {\mathbf T}_2 \big(\partial\pi_r({\boldsymbol\theta}) / \partial\theta_k \big)_{r,k}$ (uni & bivariate derivatives);
• ${\mathcal H}({\boldsymbol\theta})$ (sensitivity matrix); and
• ${\mathbf B}({\boldsymbol\theta})$ (some transformation matrix dependent on ${\boldsymbol\theta}$).

To use ${\boldsymbol\Omega}_2$ in practice, use “hat versions” of relevant matrices.

## Distribution of test statistics

LIGOF test statistics generally take the quadratic form $X^2 = n \hat{\mathbf e}_2^\top \hat{\boldsymbol\Xi}\hat{\mathbf e}_2,$ where ${\boldsymbol\Xi}(\hat{\boldsymbol\theta}) =: \hat{\boldsymbol\Xi}\xrightarrow{\text P} {\boldsymbol\Xi}$ is some $S\times S$ weight matrix. Generally, $X^2$ is reffered to a chi square distribution under $H_0$, because $X^2 \xrightarrow{\text D} \sum_{s=1}^S \delta_s\chi^2_1 \quad \text{as} \quad n\to\infty,$ where the $\delta_s$ are the eigenvalues of ${\mathbf M}= {\boldsymbol\Omega}_2{\boldsymbol\Xi}$. Two cases:

1. If ${\mathbf M}$ is idempotent, then the chi square is exact.
2. Otherwise, it is a sum of scaled chi squares. Can be approximated by a chi square with degrees of freedom needing estimation .

## Test statistics used

$\begin{gathered}X^2 = n \hat{\mathbf e}_2^\top \hat{\boldsymbol\Xi}\hat{\mathbf e}_2 \\\sqrt n \hat{\mathbf e}_2 \approx {\mathop{\mathrm{N}}}_S ({\mathbf 0}, \hat{\boldsymbol\Omega}_2)\end{gathered}$

Name $\hat{{\boldsymbol\Xi}}$ D.f.
1 Wald $\hat{{\boldsymbol\Omega}}^+_2$ $S-m$
2 Wald (VCF) ${\boldsymbol\Xi}\hat{{\boldsymbol\Omega}}_2{\boldsymbol\Xi}$ $S-m$
3 Wald (Diag.) $\mathop{\mathrm{diag}}(\hat{{\boldsymbol\Omega}}_2)^{-1}$ est.
4 Pearson $\mathop{\mathrm{diag}}(\hat{{\boldsymbol\pi}}_2)^{-1}$ est.
5 RSS $\mathbf I$ est.
6 Multinomial $\hat{{\boldsymbol\Sigma}}_2^{-1}$ est.

# Simulation results

## Factor models

Loadings
5 x 1 sparse Matrix of class "dgCMatrix"

[1,] 0.80
[2,] 0.70
[3,] 0.47
[4,] 0.38
[5,] 0.34

Thresholds
[1] -1.43 -0.55 -0.13 -0.72
[5] -1.13

Factor correlations
[1] 1
Loadings
8 x 1 sparse Matrix of class "dgCMatrix"

[1,] 0.80
[2,] 0.70
[3,] 0.47
[4,] 0.38
[5,] 0.34
[6,] 0.80
[7,] 0.70
[8,] 0.47

Thresholds
[1] -1.43 -0.55 -0.13 -0.72
[5] -1.13 -1.43 -0.55 -0.13

Factor correlations
[1] 1
Loadings
15 x 1 sparse Matrix of class "dgCMatrix"

[1,] 0.80
[2,] 0.70
[3,] 0.47
[4,] 0.38
[5,] 0.34
[6,] 0.80
[7,] 0.70
[8,] 0.47
[9,] 0.38
[10,] 0.34
[11,] 0.80
[12,] 0.70
[13,] 0.47
[14,] 0.38
[15,] 0.34

Thresholds
[1] -1.43 -0.55 -0.13 -0.72
[5] -1.13 -1.43 -0.55 -0.13
[9] -0.72 -1.13 -1.43 -0.55
[13] -0.13 -0.72 -1.13

Factor correlations
[1] 1
Loadings
10 x 2 sparse Matrix of class "dgCMatrix"

[1,] 0.80 .
[2,] 0.70 .
[3,] 0.47 .
[4,] 0.38 .
[5,] 0.34 .
[6,] .    0.80
[7,] .    0.70
[8,] .    0.47
[9,] .    0.38
[10,] .    0.34

Thresholds
[1] -1.43 -0.55 -0.13 -0.72
[5] -1.13 -1.43 -0.55 -0.13
[9] -0.72 -1.13

Factor correlations
[,1] [,2]
[1,]  1.0  0.3
[2,]  0.3  1.0
Loadings
15 x 3 sparse Matrix of class "dgCMatrix"

[1,] 0.80 .    .
[2,] 0.70 .    .
[3,] 0.47 .    .
[4,] 0.38 .    .
[5,] 0.34 .    .
[6,] .    0.80 .
[7,] .    0.70 .
[8,] .    0.47 .
[9,] .    0.38 .
[10,] .    0.34 .
[11,] .    .    0.80
[12,] .    .    0.70
[13,] .    .    0.47
[14,] .    .    0.38
[15,] .    .    0.34

Thresholds
[1] -1.43 -0.55 -0.13 -0.72
[5] -1.13 -1.43 -0.55 -0.13
[9] -0.72 -1.13 -1.43 -0.55
[13] -0.13 -0.72 -1.13

Factor correlations
[,1] [,2] [,3]
[1,]  1.0  0.2  0.3
[2,]  0.2  1.0  0.4
[3,]  0.3  0.4  1.0

## Informative sampling

• Using M1: 1F5V, generate a fixed population size of $N$. Then, assign each unit $h$ a probability of selection as follows: $w_h^{-1} = \frac{1}{1 + \exp(y_1^*)}.$ Larger values of $y_1^*$ result in smaller probabilities of selection.

• Sample $n\in\{500, 1000, 5000\}$ units from a population of size $N=n/0.01$ which ensures no need for FPC factor .

• In repeated sampling ($B=1000$), interested in performance of PMLE vis-à-vis

• Bias
• Coverage for 95% CI
• SD/SE ratio

## Informative sampling (results)

### Bias

True values $n = 500$ $n = 1000$ $n = 5000$
PML PMLW PML PMLW PML PMLW

$\lambda_1$

0.80 −0.03 0.00 −0.03 −0.01 −0.02 0.00

$\lambda_2$

0.70 −0.03 −0.01 −0.02 0.00 −0.03 −0.01

$\lambda_3$

0.47 −0.02 0.00 −0.02 0.00 −0.02 0.00

$\lambda_4$

0.38 −0.02 0.00 −0.02 0.00 −0.02 0.00

$\lambda_5$

0.34 0.00 0.01 −0.02 0.00 −0.02 −0.01
Thresholds

$\tau_1$

-1.43 0.31 0.00 0.31 0.00 0.30 −0.01

$\tau_2$

-0.55 0.22 −0.01 0.22 0.00 0.21 −0.01

$\tau_3$

-0.13 0.15 −0.01 0.16 0.01 0.15 0.00

$\tau_4$

-0.72 0.12 0.00 0.12 0.00 0.12 0.00

$\tau_5$

-1.13 0.11 0.01 0.11 0.01 0.11 0.00

## Informative sampling (results)

### Coverage and SD/SE ratio

Coverage SD/SE Coverage SD/SE Coverage SD/SE
$n = 500$ $n = 1000$ $n = 5000$
PML PMLW PML PMLW PML PMLW PML PMLW PML PMLW PML PMLW

$\lambda_1$

0.95 0.96 1.01 0.99 0.94 0.95 1.04 1.01 0.89 0.95 1.22 1.01

$\lambda_2$

0.95 0.95 1.03 0.99 0.93 0.95 1.09 1.01 0.85 0.95 1.31 0.99

$\lambda_3$

0.95 0.96 1.02 0.97 0.94 0.95 1.06 0.97 0.85 0.95 1.35 0.99

$\lambda_4$

0.94 0.94 1.04 1.03 0.94 0.95 1.05 1.00 0.88 0.95 1.25 1.01

$\lambda_5$

0.95 0.95 0.99 0.98 0.95 0.96 1.02 1.00 0.90 0.95 1.19 1.01
Thresholds

$\tau_1$

0.01 0.96 4.38 0.98 0.00 0.96 6.24 0.97 0.00 0.96 13.81 0.97

$\tau_2$

0.04 0.95 3.91 1.05 0.00 0.94 5.47 1.01 0.00 0.94 12.03 1.06

$\tau_3$

0.22 0.96 2.93 1.02 0.03 0.94 4.03 1.02 0.00 0.96 8.76 0.96

$\tau_4$

0.49 0.94 2.22 1.03 0.20 0.95 2.97 1.03 0.00 0.95 6.40 1.04

$\tau_5$

0.61 0.94 1.88 1.02 0.42 0.95 2.39 1.01 0.00 0.95 5.13 1.04

## Educational survey

Simulate a population of $1e6$ students clustered within classrooms and stratified by school type (correlating with abilities).

Type $N$ Classes Avg. class size
A 400 33.0 15.2
B 1000 19.6 25.6
C 600 24.9 20.4

ICC for test items range between 0.05 and 0.60.

• Cluster sample: Sample $n_C$ schools using PPS, then sample 1 classroom via SRS, then select all students in classroom.
• Stratified cluster sample: For each stratum, sample $n_S$ schools using SRS, then sample 1 classroom via SRS, then select all students in classroom.

# Summary

## Conclusions

• Pairwise likelihood estimation alleviates some issues associated with the UV approach for binary factor models.

• Sampling weights are easily incorporated in the PML estimation, and found to perform well in (limited) simulation studies.

• Sparsity impairs the dependability of GOF tests but are circumvented by considering lower order statistics. Wald, Pearson, and others are investigated across a variety of scenarios.

• Generally all tests have acceptable Type I errors, except WaldDiag test.
• Wald test needs ${\boldsymbol\Omega}_2^{-1}$, but WaldVCF does not.
• Pearson test has more power than the Wald-type test in more complicated scenarios.
• Added variability due to complex designs (expected), but overall power curves are not drastically different.

## Software

R software implementation in {lavaan} 0.6-17 (PML weights) and a separate package {lavaan.bingof} for the LIGOF tests.

fit <- lavaan::sem(
model = "eta1 =~ y1 + y2 + y3 + y4 + y5",
data = lavaan.bingof::gen_data_bin_wt(n = 1000),
std.lv = TRUE,
estimator = "PML",
sampling.weights = "wt"
)
lavaan.bingof::all_tests(fit)
# A tibble: 6 × 6
X2    df name          pval Xi_rank     S
<dbl> <dbl> <chr>        <dbl>   <int> <int>
1  6.39  5    Wald         0.270      13    15
2  6.25  5    WaldVCF      0.283       5    15
3  3.65  3.29 WaldDiag,MM3 0.345      15    15
4  5.24  3.14 Pearson,MM3  0.169      15    15
5  5.86  3.82 RSS,MM3      0.191      15    15
6  5.25  3.14 Multn,MM3    0.168      15    15

# Thanks!

https://haziqj.ml/plgof-gent

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