Graduation Rates

In this post, I want to run an example of absorbing states in stochastic processes. This example is based on Example 3.29 in Introduction to Stochastic Processes in R by Robert Dobrow.

Data

The data I have assembled is based on IRDS reports from my own University of California at Merced.

# weights
weights <- c(20, 70, 0, 0, 10, 0, 
       0, 5, 89, 0, 6, 0, 
       0, 0, 3, 94, 3, 0, 
       0, 0, 0, 24, 1, 76, 
       0, 0, 0, 0, 100, 0, 
       0, 0, 0, 0, 0, 100)
sparse_weights <- weights[weights > 0]

# transition matrix (row stochastic)
P <- matrix(weights, nrow = 6, byrow = TRUE)*0.01

# states
rownames(P) <- c("Freshman", "Sophomore", "Junior", "Senior", "Dropped", "Graduated")
colnames(P) <- c("Freshman", "Sophomore", "Junior", "Senior", "Dropped", "Graduated")
P
##           Freshman Sophomore Junior Senior Dropped Graduated
## Freshman       0.2      0.70   0.00   0.00    0.10      0.00
## Sophomore      0.0      0.05   0.89   0.00    0.06      0.00
## Junior         0.0      0.00   0.03   0.94    0.03      0.00
## Senior         0.0      0.00   0.00   0.24    0.01      0.76
## Dropped        0.0      0.00   0.00   0.00    1.00      0.00
## Graduated      0.0      0.00   0.00   0.00    0.00      1.00

This formulation assumes that about 10 percent of students drop after the first year, and about 24 percent of seniors continue to take classes into the fifth year or more.

Network Graph

library(igraph)
network <- graph_from_adjacency_matrix(P, mode = "directed", weighted = TRUE)
plot(network,
     edge.arrow.size = 0.5,
     edge.color = "#D4AE00",
     edge.label = sparse_weights,
     layout = layout_in_circle(network))

Absorbing States

# canonical form
Q <- P[1:4, 1:4]
R <- P[1:4, 5:6]

# fundamental matrix (inverse of (I - Q))
fundamental_matrix <- solve(diag(4) - Q)

# absorption
absorption <- fundamental_matrix %*% R
round(100*absorption)
##           Dropped Graduated
## Freshman       22        79
## Sophomore      10        91
## Junior          4        97
## Senior          1       100

This model predicts that, starting with incoming students, about 79 percent of those students will eventually graduate while about 21 percent will eventually drop from UC Merced enrollment.

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