In taylordunn/gasr: Simulate and Analyze Goal Attainment Scaling Data

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.height = 3, fig.width = 5
)

This vignette is still under construction.

library(gasr)
library(dplyr)

A key metric in the theory of GAS, and in psychometrics in general, is internal consistency. This concept is discussed in Chapter 9 (p. 192) of @Kiresuk1994, where they cite the following definition:

Internal consistency refers to an estimate of test reliability based upon the average intercorrelation of items within the test [@Nunnally1978].

The average scale intercorrelation can be estimated most easily by the intraclass correlation obtained from a subjects-by-scales analysis of variance.

We've shown elsewhere that the ICC of continuous goal scores $y_{ij}$ and $y_{ij'}$ is a function of treatment group:

$$ \rho_g = \frac{\sigma_u^2}{\sigma_u^2 + g \sigma_B^2 + \sigma_{\epsilon}^2}. $$

To compute this empirically, @Kiresuk1994 give the following formula:

$$ \tag{9.7} \bar{r}_{ij} = \frac{\text{MS}_B - \text{MS}_W}{\text{MS}_B + (k - 1) \text{MS}_W}, $$

where $\bar{r}_{ij}$ is the average scale intercorrelation, $\text{MS}_B$ is the between-subjects mean square, $\text{MS}_W$ is the within-subjects mean square, and $k$ is the number of scales (goals) per subject.

Proof

For $n$ subjects with $k$ goals each and no treatment effect, the between-subjects mean square (on the latent score $y_{ij}$) is

$$ \text{MS}_B = k\sigma_u^2 + \sigma_e^2 $$

with degrees of freedom $df = n-1$, where $\sigma_u^2$ is the subject-level random effect variance, and $\sigma_{\epsilon}^2$ is the goal-level random effect variance.

The within-subjects mean square is

$$ \text{MS}W = \sigma{\epsilon}^2 $$

with degrees of freedom $df = n(k-1)$.

Rearranging $\text{MS}_B$ in terms of the subject-level variance:

$$ \sigma_u^2 = \frac{\text{MS}B - \sigma{\epsilon}^2}{k} = \frac{\text{MS}_B - \text{MS}_W}{k} $$

Then Pearson's correlation is calculated as:

$$ \begin{align} \bar{r}{ij} &= \frac{\sigma_u^2}{\sigma_u^2 + \sigma{\epsilon}^2} \ &= \frac{\frac{\text{MS}_B - \text{MS}_W}{k}}{\frac{\text{MS}_B - \text{MS}_W}{k} + \text{MS}_W} \ &= \frac{\text{MS}_B - \text{MS}_W}{\text{MS}_B + (k - 1) \text{MS}_W} \end{align} $$

For instance, simulate 20 subjects in a control group with three goals each:

gas_data <- sim_trial(n_subjects = 20, n_goals = 3,
                      group_allocation = list("control" = 1.0))

Manually compute the total between-subjects sum of squares (SS) and mean:

gas_data_between <- gas_data %>%
  mutate(grand_mean = mean(score_continuous)) %>%
  group_by(grand_mean, subject_id) %>%
  summarise(subject_mean = mean(score_continuous),
            .groups = "drop") %>%
  summarise(
    ss_b = sum((subject_mean - grand_mean)^2),
    df = n() - 1,
    ms_b = ss_b / df
  )
gas_data_between

And within-subjects:

gas_data_within <- gas_data %>%
  group_by(subject_id) %>%
  summarise(
    ss_subj = sum((score_continuous - mean(score_continuous))^2),
    .groups = "drop"
  ) %>%
  summarise(
    ss_w = sum(ss_subj),
    df = n() * (3 - 1), # n*(k-1)
    ms_w = ss_w / df
  )
gas_data_within

Then the inter-correlation is:

(gas_data_between$ms_b - gas_data_within$ms_w) /
  (gas_data_between$ms_b + (3 - 1) * gas_data_within$ms_w)

$$ \bar{r}_{ij} = \frac{\text{MS}_B - \text{MS}_W}{\text{MS}_B + (k-1) \text{MS}_W} $$

aov(score_continuous ~ subject_id, data = gas_data) %>%
  summary()

@Streiner2008

TBD

References

taylordunn/gasr documentation built on April 5, 2022, 1:37 a.m.