A latent class model for individual differences in the interpretation of conditionals

Abstract.

We investigated the hypothesis that there are three levels of performance associated with conditional reasoning: (1) Unsophisticated reasoners solve a modus tollens by accepting the invited inferences, treating the conditional as if it were a biconditional. (2) Reasoners of an intermediate level can resist the invited inferences, but cannot find the line of reasoning needed to endorse modus tollens. (3) Sophisticated reasoners do not draw the invited inferences either, but they do master the strategy to solve a modus tollens.

On a first set of six problems, solved by 214 adolescents, an unrestricted latent class analysis revealed the existence of a large subgroup of reasoners with a biconditional interpretation of the conditional, and a smaller subgroup with a conditional interpretation.

On a second set of 24 problems, solved by the same participants, a restricted latent class model corroborated the existence of a large subgroup of unsophisticated reasoners and a smaller subgroup of reasoners of an intermediate level. No evidence was found for the existence of a subgroup of sophisticated reasoners.

As expected, the class of biconditional reasoners was associated with the class of unsophisticated reasoners, and the class of conditional reasoners was associated with the class of reasoners of an intermediate level. Furthermore, the former showed a biconditonal response pattern on truth table tasks, whereas the latter showed a conditional response pattern.

Appendix

EM algorithm for constrained latent class analysis.

Our use of the EM algorithm for the discrete mixture model of Equation 9 resembles the use of the EM algorithm as proposed by Mislevy and Verhelst (1990), and Mislevy and Wilson (1996) for these kind of models.

The marginal likelihood function for the model of Equation 9 is

$$ L\left( {\varvec{\delta} ,{\bf p},\sigma ^2 \left| {\bf Y} \right.} \right) = \prod\limits_{n = 1}^N {\sum\limits_{t = 1}^T {p_t \left\{ {\int\limits_{\theta _{nt} } {\left( {\prod\limits_{i = 1}^I {{{\exp \left[ {y_{ni} \left( {{\bf x}_{nit}^{\bf '} \varvec{\delta} + \theta _{nt} } \right)} \right]} \over {1 + \exp \left( {{\bf x}_{nit}^{\bf '} \varvec{\delta} + \theta _{nt} } \right)}}} } \right)} N\left( {\theta _{nt} \left| {0,\sigma ^2 } \right.} \right)d\theta _{nt} } \right\}} ,} $$

(10)

where δ is a parameter vector containing both the class specific logistic regression weights and the logistic regression weights constant across classes; ${\bf x}_{nit}^{'}$ is a class specific covariate vector, defined such that ${\bf x}_{nit}^{\prime} \varvec{\delta} = {\bf x}_{ni}^{\prime}\varvec{\beta} _t$ . p=(p ₁ , ..., p _T ) with $p_T = 1 - \sum\limits_{t = 1}^{T - 1} {p_t } ;$ and Y is the N×I datamatrix.

Using ${\bf x}_{nit}^{'}$ and δ instead of ${\bf x}_{ni}^{'}$ and β _t is done for computational convenience. It allows to treat all logistic regression weights as logistic regression weights that are constant across classes. If a logistic regression weight β _tj is class specific for a particular class t, this can be taken into account by setting x _nit'j to zero for all t' ≠ t.

The idea behind the EM-algorithm is that maximizing (10) with respect to the parameters would be easier if the ς_n and θ _nt were known. In this case, the complete data (consisting of the observed data Y and the "missing data" ς_n and θ _nt) likelihood L _c reduces to:

$$ L_c \left({\delta} , {p},\sigma ^2 | {\bf Y},\sigma {\bf ,}{\theta} _\varsigma \right) = \prod\limits_{n = 1}^N {p_t} \left( {\prod\limits_{i = 1}^I {{{\exp \left[ {y_{ni} \left( {{\bf x}_{nit}^{\bf '} {\delta} + \theta _{nt} } \right)} \right]} \over {1 + \exp \left( {{\bf x}_{nit}^{\bf '} {\delta} + \theta _{nt} } \right)}}} } \right)N\left( {\theta _{nt} \left| {0,\sigma ^2 } \right.} \right) $$

(11)

Where: ς and θ _ς are N×1 vectors, containing for each person n the class t he/she belongs to and the ability θ _nt of that person within the class t he/she belongs to.

The missing data being unknown (by definition), the EM algorithm iterates as follows: at iteration k + 1,

• 1. Compute the expected complete data loglikelihood, given the observed data and provisional parameter estimates obtained in iteration k.

• 2. Maximize the expected complete data loglikelihood with respect to the parameters of interest, resulting in new provisional parameter estimates for iteration k + 1.

Iterate until some convergence criterion is met. For the first iteration, initial parameter estimates can be chosen at random or based on some preliminary analysis. Dempster et al. (1977) show that, under regularity conditions, the likelihood is not decreased after an iteration.

The expected complete data loglikelihood, given the observed data and the provisional parameter estimates is:

$$ \matrix{ {Q\left( {\varvec{\alpha} \left| {\varvec{\alpha} ^{(k)} } \right.} \right)} \hfill & = \hfill & {E_{\varvec{\alpha} ^{(k)} } \left[ {\log L_c \left( {\varvec{\alpha} \left| {{\bf Y}{\bf ,}\varvec{\sigma} ,\varvec{\theta} _\sigma } \right.} \right)\left| {\bf Y} \right.} \right]} \hfill \cr {} \hfill & = \hfill & {\sum\limits_{n = 1}^N {\sum\limits_{t = 1}^T {p\left( {\varsigma _n = t\left| {{\bf y}_n ,\varvec{\alpha} ^{\left( k \right)} } \right.} \right)\int\limits_{\theta _{nt} } {\log g\left( {{\bf y}_n {\bf ,}\varsigma _n = t,\theta _{nt} \left|\varvec{\alpha} \right.} \right)f\left( {\theta _{nt} \left| {\varsigma _n = t,{\bf y}_n ,\varvec{\alpha} ^{(k)} } \right.} \right)d\theta _{nt} } } } } \hfill \cr } $$

(12)

where

α is the vector of all parameters (δ , p , σ ²),

α ^(k) is the vector of provisional parameter estimates obtained in iteration k, and

$E_{\varvec{\alpha} ^{(k)} }$denotes the expectation using parameter vector α ^(k).

Differentiating Q(α|α ^{( k )}) with respect to the parameters α results in the following score equations:

$${{\partial Q\left( {\varvec{\alpha} \left| {\varvec{\alpha} ^{(k)} } \right.} \right)} \over {\partial p_t }}=\sum\limits_{n=1}^N {\left[ {p\left( {\varsigma _n =t\left| {{\bf y}_n ,\varvec{\alpha} ^{\left( k \right)} } \right.} \right)/p_t - p\left( {\varsigma _n =T\left| {{\bf y}_n ,\varvec{\alpha} ^{\left( k \right)} } \right.} \right)/\left( {p_T } \right)} \right]} $$

(13)

for t = 1,..., T-1

$$ {{\partial Q\left( {\varvec{\alpha} \left| {\varvec{\alpha} ^{(k)} } \right.} \right)} \over {\partial \sigma ^2 }} = \sum\limits_{n = 1}^N {\sum\limits_{t = 1}^T {p\left( {\varsigma _n = t\left| {{\bf y}_n ,\varvec{\alpha} ^{\left( k \right)} } \right.} \right)\int\limits_{\theta _{nt} } {\left[ { - 1/2\sigma ^2 + \theta _{nt}^2 /2\sigma ^4 } \right]} f\left( {\theta _{nt} \left| {\varsigma _n = t,{\bf y}_n ,\varvec{\alpha} ^{(k)} } \right.} \right)d\theta _{nt} } } , $$

(14)

and

$$ {{\partial Q\left( {\varvec{\alpha} \left| {\varvec{\alpha} ^{(k)} } \right.} \right)} \over {\partial \delta _j }} = \sum\limits_{n = 1}^N {\sum\limits_{t = 1}^T {p\left( {\varsigma _n = t\left| {{\bf y}_n ,\varvec{\alpha} ^{\left( k \right)} } \right.} \right)\left\{ {\sum\limits_{i = 1}^I {y_{ni} x_{nitj} } - \int\limits_{\theta _{nt} } {\sum\limits_{i = 1}^I {{{\exp \left( {\theta _{nt} + {\bf x}_{nit}^{\prime} \varvec{\delta} } \right)x_{nitj} } \over {1 + \exp \left( {\theta _{nt} + {\bf x}_{nit}^{\prime} \varvec{\delta} } \right)}}} } f\left( {\theta _{nt} \left| {\varsigma _n = t,{\bf y}_n ,\varvec{\alpha} ^{(k)} } \right.} \right)d\theta _{nt} } \right\}} } . $$

(15)

By equating the partial derivatives to zero, closed form solutions can be found for p ₁ ,...p _{T- 1}, and σ ² that maximize Q(α|α ^{( k )}), serving as provisional parameter estimates for the E-step of iteration k+2 :

$$ \matrix{ {p_t^{(k + 1)} = {{\sum\limits_{n = 1}^N {p\left( {\varsigma _n = t\left| {{\bf y}_n ,\varvec{\alpha} ^{\left( k \right)} } \right.} \right)} } \over N}} & {{\rm for}\;t\; = \;1,...,\;T - 1} \cr } , $$

(16)

and

$$ \sigma ^{2(k + 1)} = {{\sum\limits_{n = 1}^N {\sum\limits_{t = 1}^T {p\left( {\varsigma _n = t\left| {{\bf y}_n ,\varvec{\alpha} ^{\left( k \right)} } \right.} \right)\int\limits_{\theta _{nt} } {\theta _{nt}^2 } f\left( {\theta _{nt} \left| {\varsigma _n = t,{\bf y}_n ,\varvec{\alpha} ^{(k)} } \right.} \right)d\theta _{nt} } } } \over N}. $$

(17)

For the logistic regression weights δ _j, no closed form solution exists. Instead, the parameter estimates are updated using one Newton-Raphson iteration. Hence actually, the algorithm we implemented is a Generalised EM algorithm (Dempster et al., 1977). The second derivatives needed for the Newton-Raphson algorithm are

$$ {{\partial ^2 Q\left( {\varvec{\alpha} \left| {\varvec{\alpha} ^{(k)} } \right.} \right)} \over {\partial \delta _j \partial \delta _{j'} }} = - \sum\limits_{n = 1}^N {\sum\limits_{t = 1}^T {p\left( {\varsigma _n = t\left| {{\bf y}_n ,\varvec{\alpha} ^{\left( k \right)} } \right.} \right)\int\limits_{\theta _{nt} } {\left\{ {\sum\limits_{i = 1}^I {{{\exp \left( {\theta _{nt} + {\bf x}_{nit}^{\prime} \varvec{\delta} } \right)x_{nitj} x_{nitj'} } \over {\left[ {1 + \exp \left( {\theta _{nt} + {\bf x}_{nit}^{\prime} \varvec{\delta} } \right)} \right]^2 }}} } \right\}} f\left( {\theta _{nt} \left| {\varsigma _n = t,{\bf y}_n ,\varvec{\alpha} ^{(k)} } \right.} \right)d\theta _{nt} } } . $$

(18)

The posterior densities ƒ(θ_nt|ς _n = t, y_n, α ^{( k )})and the posterior probabilities p(ς _n = t|y_n, α ^{( k )}) are computed using Bayes' theorem:

$$ f\left( {\theta _{nt} \left| {\varsigma _n = t,{\bf{y}}_n ,\varvec{\alpha} ^{(k)} } \right.} \right) = {{p\left( {{\bf{y}}_n \left| {\varsigma _n = t,\theta _{nt} ,\varvec{\delta} } \right.} \right)N\left( {\theta _{nt} \left| {0,\sigma ^2 } \right.} \right)} \over {\int\limits_{\theta _{nt} } {p\left( {{\bf{y}}_n \left| {\varsigma _n = t,\theta _{nt} ,\varvec{\delta} } \right.} \right)N\left( {\theta _{nt} \left| {0,\sigma ^2 } \right.} \right)d\theta _{nt} } }} $$

(19)

$$ p\left( {\varsigma _n = t\left| {{\bf y}_n ,\varvec{\alpha} ^{\left( k \right)} } \right.} \right) = {{p_t \int\limits_{\theta _{nt} } {p\left( {{\bf y}_n \left| {\varsigma _n = t,\theta _{nt} ,\varvec{\delta} } \right.} \right)N\left( {\theta _{nt} \left| {0,\sigma ^2 } \right.} \right)d\theta _{nt} } } \over {\sum\limits_{t = 1}^T {p_t \int\limits_{\theta _{nt} } {p\left( {{\bf y}_n \left| {\varsigma _n = t,\theta _{nt} ,\varvec{\delta} } \right.} \right)N\left( {\theta _{nt} \left| {0,\sigma ^2 } \right.} \right)d\theta _{nt} } } }} $$

(20)

The integrals over θ _nt are approximated with Gaussian quadrature.

After convergence, the observed information matrix for the maximum likelihood estimates $\hat {\alpha}$ can be approximated by

$$ I\left( {\hat {\alpha} \left| {\bf Y} \right.} \right) \approx \sum\limits_{n = 1}^N {{\bf s}\left( {{\bf y}_n \left| {\hat {\alpha} } \right.} \right){\bf s}^{\prime} \left( {{\bf y}_n \left| {\hat {\alpha} } \right.} \right)} $$

(21)

where $s\left( {{\bf y}_n \left| {\hat {\alpha} } \right.} \right) = \partial \log L_n \left( {{\delta} ,{\bf p},\sigma ^2 \left| {{\bf y}_n } \right.} \right)/\partial {\alpha}$ , the score function based on the single observation y _n. The latter can be computed using the complete data loglikelihood as follows (McLachlan & Krishnan, 1997):

$${\bf{s}}\left( {{\bf{y}}_n \left| {\bf{\varvec{\alpha} }} \right.} \right) = E_{\bf{\varvec{\alpha} }} \left[ {\partial \log L_{cn} \left( {{\bf{\varvec{\alpha} }}\left| {{\bf{y}}_n {\bf{,}}\varsigma _n ,\theta _{nt} } \right.} \right)/\partial {\bf{\varvec{\alpha} }}\left| {{\bf{y}}_n } \right.} \right]$$

(22)

where L_cn(α|y _n, ς _n, θ _nt)is the complete data loglikelihood formed from the single observation y _n.