2-factor Model

## `summarise()` ungrouping output (override with `.groups` argument)
## `summarise()` ungrouping output (override with `.groups` argument)
## `summarise()` ungrouping output (override with `.groups` argument)

Let’s plot F1 vs. F2 for the 2-factor model and label the extremes.

3-factor Model

Let’s plot F2 vs. F3 for the 3-factor model and label the extremes.

Next we will attempt to understand the 7-factor exploratory model through clustering analyses.

Clustering the Items

We attempt to understand the factors by clustering items’ factor loadings, and then look at acquisition curves (or item difficulties?) for each cluster. We’ll first use mclust’s Gaussian finite mixture model and t-SNE to plot the solution, and then move on to k-means and hierarchical clustering.

Mclust finds 4 clusters. Below are the number of words of each CDI category per cluster (1-4).

Cluster 1 has many (early-learned?) nouns (food & drink, animals, games/routines, sounds, vehicles), with the rest picked up in cluster 3 (furniture/rooms, household, places, body parts, toys, clothing).

Do the words in these clusters vary systematically in their difficulty? Shown below, the items in cluster 1 are much easier, followed by cluster 3 – the rest of the nouns. Words in clusters 2 (grammatical) and 4 (verbs/adjectives) tend to be much harder.

Clustered Items Compared to CDI Categories

We take the 7-factor loadings of each item and conduct a k-means clustering with $k = 22$, the same as the number of CDI categories (e.g., quantifiers, locations, toys, clothing, etc.). We use adjusted Rand Index to compare the clusters to the item category assignment (0 = chance agreement, 1 = perfect agreement), and find a value of 0.011. Next we will use gap statistics to choose the best $k$, plot the solution, and again examine adjusted Rand Index vs. the CDI categories.

##   cluster size ave.sil.width
## 1       1   93          0.19
## 2       2   82          0.14
## 3       3   83          0.19
## 4       4   59          0.29
## 5       5   90          0.39
## 6       6   92          0.21
## 7       7  106          0.19
## 8       8   75          0.28

According to the gap statistics, the optimal number of clusters $k$ = 8. The adjusted Rand Index of this clustering solution compared to the 22 CDI categories is 0.011.

Hierarchical Clustering

Bifactor Models

Since factors are loading on different lexical classes and CDI categories, and >6 factors are justified, let’s try bifactor models that load on 1) each lexical class (nouns, verbs, adjectives, function words, other) and 2) on each CDI category (22 levels, e.g. quantifiers, locations, animals, people, sounds, etc.).

Comparing the two bifactor models, the category model is preferred by AIC and BIC, but only fits almost as well (compare log-likelihoods) as the 2-factor exploratory model.

Further analysis is needed to understand the multidimensional structure of the CDI data, but it is intriguing that nouns, which represent the bulk of the items on the CDI, seem to hang together, and separately from other parts of speech.