18.2: Correlation coefficient: How to calculate the reliability coefficient

In the previous section, we define the reliability coefficient ${ RC} [{\mathsf M}_{{\mathsf O}_\tau}^\otimes] := \frac{{ Var} [{\mathsf M}_{{\mathsf O}_E}^\otimes]}{{ Var} [{\mathsf M}_{{\mathsf O}_\tau}^\otimes]}$. However, from the measured data $(x_1,x_2,\dots,x_n) \;(\in X_{\mathbb R}^n)$, we can not get the variance of mathematical intelligences of $n$ students ${ Var}[{\mathsf M}_{{\mathsf O}_E}^\otimes]$ directly (though we can calculate the ${ Var}[{\mathsf M}_{{\mathsf O}_\tau}^\otimes]$). Thus, we focus on the problem how to estimate the reliability coefficient. Here we consider one typical method, say the split-half method.

[Split-half method:]

 $\quad$ This method is appropriate where the testing procedure may in some fashion be divided into two halves and two scores obtained. These may be correlated. With psychological tests a common procedure is to obtain scores on the odd and even items.

Now we introduce the measurement theoretical characterizations of the split-half method.

Definition 18.9 [Group simultaneous test] Let $\Theta := \{\theta_1,\theta_2,\dots,\theta_n\}$, $X_{\mathbb R} = \Omega_{\mathbb R} = {\mathbb R}$ and $\Phi_\ast : L_{+1}^1(\Theta,\nu_c) \to L_{+1}^1(\Omega_{\mathbb R},d\omega)$ be as in Example 18.1Example}. Let ${\mathsf O}_{\tau_1} := (X_{\mathbb R},{\cal F}_{X_{\mathbb R}},F_{\tau_1})$ and ${\mathsf O}_{\tau_2} := (X_{\mathbb R},{\cal F}_{X_{\mathbb R}},F_{\tau_2})$ be test observables in $L^\infty(\Omega_{\mathbb R},d\omega)$. The measurement

$$\otimes_{\theta_i \in \Theta} {\mathsf M}_{L^\infty(\Omega_{\mathbb R},d\omega)} ({\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}, S_{[\ast]}(\Phi_\ast(1_{\theta_i}))),$$

is called a group simultaneous test of ${\mathsf O}_{\tau_1}$ and ${\mathsf O}_{\tau_2}$ and it is symbolized by ${\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes$ for short.

Axiom${}^{{ (m)}}$ 1 ($\S$9.1) says that
 $(A):$ $\;$ the probability that the score $((x_1^1,x_1^2),(x_2^1. x_2^2),\dots,(x_n^1,x_n^2)) \;(\in X_{\mathbb R}^{2n})$ obtained by the group simultaneous test $\otimes_{\theta_i \in \Theta} {\mathsf M}_{L^\infty(\Omega_{\mathbb R},d\omega)} ({\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}, S_{[\ast]}(\Phi_\ast(1_{\theta_i})))$ (or in short, ${\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes$) belongs to the set $\times_{i=1}^n (\Xi_i^1 \times \Xi_i^2) \;(\in {\cal F}_{X_{\mathbb R}^{2n}})$ is given by \begin{align} {\Large{\times}}_{\theta_i \in \Theta} {}_{L^1(\Omega_{\mathbb R},d\omega)} \langle {\Phi_\ast(1_{\theta_i})}, (F_{\tau_1} \times F_{\tau_2})(\Xi_i^1 \times \Xi_i^2) \rangle_{L^\infty(\Omega_{\mathbb R},d\omega)} \Big(=: {\widehat P}_2(\times_{i=1}^n (\Xi_i^1 \times \Xi_i^2)) \Big). \tag{18.14} \end{align}

Here note that $(X_{\mathbb R}^{2n}, {\cal F}_{X_{\mathbb R}^{2n}}, {\widehat P}_2)$ is a sample probability space. Let $W_2: X_{\mathbb R}^{2n} \to {\mathbb R}$ be a statistics (i.e., measurable function). Then, ${\cal E}_{{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes}[W_2]$, the expectation of $W_2$, is defined by

$${\cal E}_{{ M}^\otimes_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}}[W_2] = \int_{X_{\mathbb R}^n} W(x_1^1,x_1^2,x_2^1,x_2^2,\dots,x_n^1,x_n^2) \, {\widehat P}_2(dx_1^1 \, dx_1^2 \, dx_2^1 \, dx_2^2 \cdots dx_n^1 \, dx_n^2).$$

We use the following notations: \begin{align} &\mbox{(i)} \;\; { Av}^{(k)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes] := \mathcal{E}_{{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes} \Big[ \frac1n \sum_{i=1}^n x_i^k \Big] \qquad (k=1,2), \\ &\mbox{(ii)} \;\; { Var}^{(k)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes] := {\cal E}_{{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes} \Big[ \frac1n \sum_{i=1}^n (x_i^k- { Av}^{(k)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes])^2 \Big] \qquad (k=1,2), \\ &\mbox{(iii)} \;\; { Cov}[{ M}^\otimes_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}] := {\cal E}_{{ M}^\otimes_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}} \Big[ \frac1n \sum_{i=1}^n (x_i^1-{ Av}^{(1)}[{ M}^\otimes_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}])(x_i^2-{ Av}^{(2)}[{ M}^\otimes_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}]) \Big]. \end{align} It is clear that ${ Av}^{(k)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes] = { Av}[{\mathsf M}_{{\mathsf O}_{\tau_k}}^\otimes] = { Av}[{\mathsf M}_{{\mathsf O}_E}^\otimes]$ ($k=1,2$).

Definition 18.10 [Equivalency of test observables] We call that test observables ${\mathsf O}_{\tau_1} := (X_{\mathbb R},{\cal F}_{X_{\mathbb R}},F_{\tau_1})$ and ${\mathsf O}_{\tau_2} := (X_{\mathbb R},{\cal F}_{X_{\mathbb R}},F_{\tau_2})$ in $L^\infty(\Omega_{\mathbb R},d\omega)$ are equivalent if it holds

\begin{align} \Delta_\omega^{(1)} = \Delta_{\omega}^{(2)} \quad (\forall \omega \in \Omega_{\mathbb R}), \tag{18.15} \end{align}

where $\Delta_\omega^{(k)} := ( \int_{X_{\mathbb R}} (x-\omega)^2 \, [F_{\tau_k}(dx)](\omega) )^{1/2}$ (see (18.9)). In case that test observables ${\mathsf O}_{\tau_1} := (X_{\mathbb R},{\cal F}_{X_{\mathbb R}},F_{\tau_1})$ and ${\mathsf O}_{\tau_2} := (X_{\mathbb R},{\cal F}_{X_{\mathbb R}},F_{\tau_2})$ in $L^\infty(\Omega_{\mathbb R},d\omega)$ are equivalent and ${\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}$ is a product test observable in $L^\infty(\Omega_{\mathbb R},d\omega)$, it holds that

\begin{align} { Var}[{\mathsf M}_{{\mathsf O}_{\tau_1}}^\otimes] = { Var}^{(1)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes] = { Var}^{(2)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes] = { Var}[{\mathsf M}_{{\mathsf O}_{\tau_2}}^\otimes]. \tag{18.16} \end{align}

In consequence of these properties, we introduce the correlation coefficient of the measured values $(x_1^1,x_2^1,\dots,x_n^1) \;(\in X_{\mathbb R}^n)$ and $(x_1^2,x_2^2,\dots,x_n^2) \;(\in X_{\mathbb R}^n)$ which are obtained by the group simultaneous test ${\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes$.

Theorem 18.11 [The reliability coefficient and the correlation coefficient in group simultaneous tests] Let ${\mathsf O}_{\tau_1}$ and ${\mathsf O}_{\tau_2}$ be equivalent test observables in $L^\infty(\Omega_{\mathbb R},d\omega)$. And let ${\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}$ be a product test observable in $L^\infty(\Omega_{\mathbb R},d\omega)$. Let ${\mathsf M}_{{\mathsf O}_{\tau_k}}^\otimes := \otimes_{\theta_i \in \Theta} {\mathsf M}_{L^\infty(\Omega_{\mathbb R},d\omega)}({\mathsf O}_{\tau_k}, S_{[\ast]}(\Phi_\ast(1_{\theta_i})))$ ($k=1,2$) and ${\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes := \otimes_{\theta_i \in \Theta} { M}({\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}, S_{[\ast]}(\Phi_\ast(1_{\theta_i})))$ be group tests as above notations. Then we see that

\begin{align} { RC}[{ M}^\otimes_{{\mathsf O}_{\tau_1}}] = { RC}[{ M}^\otimes_{{\mathsf O}_{\tau_2}}] = \frac{{ Cov}[{ M}^\otimes_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}]}{\sqrt{{ Var}[{ M}^\otimes_{{\mathsf O}_{\tau_1}}]} \cdot \sqrt{{ Var}[{ M}^\otimes_{{\mathsf O}_{\tau_2}}]}}. \tag{18.17} \end{align}

Proof From the (18.3), we get the following:

\begin{align} &{ Cov}[{ M}^\otimes_{{\mathsf O}_{\tau_1} \times{\mathsf O}_{\tau_2}}] := {\cal E}_{{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes} \Big[ \frac1n \sum_{i=1}^n (x_i^1 - { Av}^{(1)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes])(x_i^2 - { Av}^{(2)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes])\Big] \nonumber \\ &= \int_{\Omega_{\mathbb R}} \cdots \int_{\Omega_{\mathbb R}} \Big( \int_{X_{\mathbb R}} \cdots \int_{X_{\mathbb R}} \frac1n \sum_{i=1}^n (x_i^1-{ Av}^{(1)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes])(x_i^2-{ Av}^{(2)}[{\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes]) \nonumber \\ &\quad \times \times_{i=1}^n [F_{\tau_1}(dx_i^1) \, F_{\tau_2}(dx_i^2)](\omega_i) \Big) \times_{i=1}^n [\Phi_\ast(1_{\theta_i})](\omega_i) \, d\omega_i \nonumber \\ &= \frac1n \sum_{i=1}^n \Big( \int_{\Omega_{\mathbb R}} \Big( \int_{X_{\mathbb R}} \int_{X_{\mathbb R}} (x_i^1-{ Av}[{\mathsf M}_{{\mathsf O}_E}^\otimes])(x_i^2-{ Av}[{\mathsf M}_{{\mathsf O}_E}^\otimes]) \, [F_{\tau_1}(dx_i^1)](\omega) \, [F_{\tau_2}(dx_i^2)](\omega) \Big) \nonumber \\ &\quad \times [\Phi_\ast(1_{\theta_i})](\omega) \, d\omega \Big) \nonumber \\ &= \frac1n \sum_{i=1}^n \Big( \int_{\Omega_{\mathbb R}} \Big( \int_{X_{\mathbb R}} (x_i^1-{ Av}[{\mathsf M}_{{\mathsf O}_E}^\otimes]) \, [F_{\tau_1}(dx_i^1)](\omega) \cdot \int_{X_{\mathbb R}} (x_i^2-{ Av}[{\mathsf M}_{{\mathsf O}_E}^\otimes]) \, [F_{\tau_2}(dx_i^2)](\omega) \Big) \nonumber \\ &\quad \times [\Phi_\ast(1_{\theta_i})](\omega) \, d\omega \Big) \nonumber \\ &= \frac1n \sum_{i=1}^n \int_{\Omega_{\mathbb R}} (\omega-{ Av}[{\mathsf M}_{{\mathsf O}_E}^\otimes])^2 \, [\Phi_\ast(1_{\theta_i})](\omega) \, d\omega = { Var}[{\mathsf M}_{{\mathsf O}_E}^\otimes]. \tag{18.18} \end{align}

Then, we see that

\begin{align} &\frac{{ Cov}[{ M}^\otimes_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}]}{\sqrt{{ Var}[{ M}^\otimes_{{\mathsf O}_{\tau_1}}]} \cdot \sqrt{{ Var}[{ M}^\otimes_{{\mathsf O}_{\tau_2}}]}} = \frac{{ Var}[{\mathsf M}_{{\mathsf O}_E}^\otimes]}{{ Var}^{(1)}[{ M}^\otimes_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}]} = \frac{{ Var}[{\mathsf M}_{{\mathsf O}_E}^\otimes]}{{ Var}^{(2)}[{ M}^\otimes_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}]}. \tag{18.19} \end{align}
$\square \quad$
18.3: Conclusions

In this chapter, we introduce the measurement theoretical understanding of psychological test and the split-half method which estimate reliability. Measurement theoretical approach show the following correspondences: \begin{align} {\mbox{split-half method }} \longleftrightarrow \underset{\displaystyle {\mathsf M}_{{\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}}^\otimes := \otimes_{\theta_i \in \Theta} {\mathsf M}_{L^\infty(\Omega_{\mathbb R},d\omega)}({\mathsf O}_{\tau_1} \times {\mathsf O}_{\tau_2}, S_{[\ast]}(\Phi_\ast(1_{\theta_i})))}{{{\mbox{ group simultaneous test.}}}} \end{align} And further, we show the well-known theorem: $$\mbox{ "reliability coefficient" = "correlation coefficient" }$$ in Theorem 18.11.