5 Test Characteristic Curves, Information Functions, and SEM

Author

Derek C. Briggs and Claude Code (Opus 4.6 & 4.7)

5.1 Introduction

This document covers three important concepts in IRT:

Test Characteristic Curves (TCC): The expected test score as a function of ability
Information Functions: How much “information” items and tests provide about ability
Standard Error of Measurement (SEM): The precision of ability estimates

5.2 Part 1: Test Characteristic Curves

5.2.1 Expected Item Response

For a dichotomously scored item, the expected value of a respondent’s answer is:

\[E(X_{ip} | \theta_p) = P(X_{ip} = 1 | \theta_p) = P_i(\theta)\]

Because $X_{ip}$ is a binary variable (0 or 1), the mean equals the probability.

This means: of all people with ability $\theta_p$, we expect $P_i(\theta_p)$ proportion to get the item right.

5.2.2 Expected Test Score: The TCC

The expected observed (number correct) score for a person can be calculated by summing the ICCs across items:

\[TCC(\theta_p) = \sum_{i=1}^{I} P_i(\theta_p)\]

This Test Characteristic Curve tells us the expected number of correct answers for each level of theta.

We can think of this as an estimate of the true score associated with a particular set of items.

5.2.3 Example: Three-Item Test

Code

# 3PL probability function
calc_prob <- function(theta, a, b, c) {
  c + (1 - c) * exp(a * (theta - b)) / (1 + exp(a * (theta - b)))
}

# Item parameters (from three_item_test.xlsx)
items <- data.frame(
  item = 1:3,
  a = c(1, 1, 2),
  b = c(-1.5, 0, 0.2),
  c = c(0.2, 0.2, 0.3)
)

knitr::kable(items, caption = "Item Parameters")

Item Parameters
item	a	b	c
1	1	-1.5	0.2
2	1	0.0	0.2
3	2	0.2	0.3

Code

theta <- seq(-4, 4, 0.1)

# Calculate individual ICCs
p1 <- calc_prob(theta, items$a[1], items$b[1], items$c[1])
p2 <- calc_prob(theta, items$a[2], items$b[2], items$c[2])
p3 <- calc_prob(theta, items$a[3], items$b[3], items$c[3])

# Calculate TCC (sum of probabilities)
tcc <- p1 + p2 + p3

par(mfrow = c(1, 2))

# Plot individual ICCs
plot(theta, p1, type = "l", lwd = 2, col = "blue",
     xlab = expression(theta), ylab = expression(P[i](theta)),
     main = "Individual ICCs", ylim = c(0, 1))
lines(theta, p2, lwd = 2, col = "red")
lines(theta, p3, lwd = 2, col = "darkgreen")
legend("bottomright", legend = paste("Item", 1:3),
       col = c("blue", "red", "darkgreen"), lwd = 2)

# Plot TCC
plot(theta, tcc, type = "l", lwd = 3, col = "purple",
     xlab = expression(theta), ylab = "Expected Score",
     main = "Test Characteristic Curve (TCC)")
abline(h = c(0, 1, 2, 3), lty = 2, col = "gray")

Code

par(mfrow = c(1, 1))

Interpretation:

At $\theta = -4$: Expected score $\approx$ 0.78 (mostly guessing)
At $\theta = 0$: Expected score $\approx$ 2.03
At $\theta = 4$: Expected score $\approx$ 2.98 (nearly all correct)

5.3 Part 2: Item Information Functions

5.3.1 What is “Information”?

Each item provides a certain amount of “information” about respondents’ ability levels. The item information function is:

\[I_i(\theta) = \frac{[P'_i(\theta)]^2}{P_i(\theta) Q_i(\theta)}\]

where:

$P_i(\theta) = P(X_{ip} = 1 | \theta_p)$
$Q_i(\theta) = 1 - P_i(\theta)$
$P'_i(\theta)$ is the first derivative of $P_i(\theta)$

Key insight: Items provide different amounts of information at different points on the theta scale.

5.3.2 Model-Specific Information Functions

Model	Item Information Formula	Maximum At
1PL	$P_i(\theta) Q_i(\theta)$	$b_i$
2PL	$a_i^2 P_i(\theta) Q_i(\theta)$	$b_i$
3PL	$\frac{a_i^2 Q_i(\theta)}{P_i(\theta)} \cdot \frac{[P_i(\theta) - c_i]^2}{(1-c_i)^2}$	Slightly above $b_i$

5.3.3 Computing Item Information

Code

# 3PL item information function
calc_info <- function(theta, a, b, c) {
  # Calculate P and Q
  L <- 1 / (1 + exp(-a * (theta - b)))
  P <- c + (1 - c) * L
  Q <- 1 - P

  # Calculate derivative of P
  dP <- (1 - c) * a * L * (1 - L)

  # Information
  info <- (dP^2) / (P * Q)
  return(info)
}

Code

# Calculate item information for each item
info1 <- calc_info(theta, items$a[1], items$b[1], items$c[1])
info2 <- calc_info(theta, items$a[2], items$b[2], items$c[2])
info3 <- calc_info(theta, items$a[3], items$b[3], items$c[3])

par(mfrow = c(1, 2))

# Plot ICCs
plot(theta, p1, type = "l", lwd = 2, col = "blue",
     xlab = expression(theta), ylab = expression(P[i](theta)),
     main = "ICCs", ylim = c(0, 1))
lines(theta, p2, lwd = 2, col = "red")
lines(theta, p3, lwd = 2, col = "darkgreen")
legend("bottomright", legend = paste("Item", 1:3),
       col = c("blue", "red", "darkgreen"), lwd = 2, cex = 0.8)

# Plot Item Information
plot(theta, info1, type = "l", lwd = 2, col = "blue",
     xlab = expression(theta), ylab = expression(I[i](theta)),
     main = "Item Information Functions",
     ylim = c(0, max(c(info1, info2, info3)) * 1.1))
lines(theta, info2, lwd = 2, col = "red")
lines(theta, info3, lwd = 2, col = "darkgreen")
legend("topright", legend = paste("Item", 1:3),
       col = c("blue", "red", "darkgreen"), lwd = 2, cex = 0.8)

Code

par(mfrow = c(1, 1))

Observations:

Item 3 (green) has the highest peak information because it has the highest discrimination ($a = 2$)
Information is maximized near each item’s difficulty parameter ($b$)
Items with higher discrimination provide more information but over a narrower range

5.4 Part 3: Test Information Function

5.4.1 Summing Item Information

The test information function is calculated by summing the item information functions:

\[I(\theta) = \sum_{i=1}^{I} I_i(\theta)\]

5.4.2 Relationship to SEM

The standard error of measurement for theta is:

\[SEM(\theta) = \frac{1}{\sqrt{I(\theta)}}\]

Key relationships:

As information increases, SEM decreases
$SEM(\theta)$ varies across the theta distribution
Tests provide more precise estimates where they have more information

Code

# Calculate test information
test_info <- info1 + info2 + info3

# Calculate SEM
sem <- 1 / sqrt(test_info)

par(mfrow = c(1, 2))

# Plot Test Information
plot(theta, test_info, type = "l", lwd = 3, col = "purple",
     xlab = expression(theta), ylab = expression(I(theta)),
     main = "Test Information Function")

# Also show individual item contributions
lines(theta, info1, lwd = 1, col = "blue", lty = 2)
lines(theta, info2, lwd = 1, col = "red", lty = 2)
lines(theta, info3, lwd = 1, col = "darkgreen", lty = 2)
legend("topright",
       legend = c("Test Info", "Item 1", "Item 2", "Item 3"),
       col = c("purple", "blue", "red", "darkgreen"),
       lwd = c(3, 1, 1, 1), lty = c(1, 2, 2, 2), cex = 0.8)

# Plot SEM
plot(theta, sem, type = "l", lwd = 3, col = "darkred",
     xlab = expression(theta), ylab = expression(SEM(theta)),
     main = "Standard Error of Measurement",
     ylim = c(0, max(sem[is.finite(sem)]) * 1.1))

Code

par(mfrow = c(1, 1))

5.4.3 Combined Plot: Information and SEM

Code

library(ggplot2)

# Create data frame
plot_df <- data.frame(
  theta = theta,
  info = test_info,
  sem = sem
)

# Scale factor for dual axis
scale_factor <- max(plot_df$info, na.rm = TRUE) / max(plot_df$sem[is.finite(plot_df$sem)], na.rm = TRUE)
plot_df$sem_scaled <- plot_df$sem * scale_factor

# Dual-axis plot
ggplot(plot_df, aes(x = theta)) +
  geom_line(aes(y = info), linewidth = 1.2, color = "blue") +
  geom_line(aes(y = sem_scaled), linewidth = 1.2, linetype = "dashed", color = "red") +
  scale_y_continuous(
    name = "Test Information",
    sec.axis = sec_axis(~ . / scale_factor, name = "SEM (logits)")
  ) +
  labs(
    x = expression(theta),
    title = "Three-Item Test: Information and SEM Functions",
    subtitle = "Solid blue = Information | Dashed red = SEM"
  ) +
  theme_minimal(base_size = 14) +
  theme(
    axis.title.y.left = element_text(color = "blue"),
    axis.title.y.right = element_text(color = "red")
  )

Interpretation:

The test provides maximum information (minimum SEM) around $\theta \approx 0$
At the extremes ($\theta < -2$ or $\theta > 2$), information drops and SEM increases
This three-item test measures best in the middle of the ability range

5.5 Part 4: Practice with Real Data Using mirt

Now let’s apply these concepts to real data using the mirt package.

Code

library(mirt)

5.5.1 Load Data and Fit Model

Code

# Load data
forma <- read.csv("../Data/pset1_formA.csv")
forma <- forma[, 1:15]

# Fit 2PL model
mirt_2pl <- mirt(forma, model = 1, itemtype = "2PL", method = "EM")

5.5.2 Test Characteristic Curve

Code

plot(mirt_2pl, main = "Test Characteristic Curve")

The TCC shows the expected number of correct answers (out of 15) at each ability level.

5.5.3 Item Information Functions

Code

# Show item information for first 3 items
plot(mirt_2pl, type = 'infotrace', which.items = 1:3)

5.5.4 Test Information Function

Code

plot(mirt_2pl, type = 'info', main = "Test Information Function")

5.5.5 Test Information and SEM

Code

plot(mirt_2pl, type = 'infoSE', theta_lim = c(-3, 3),
     main = "Test Information and Standard Error")

5.6 Practical Applications

5.6.1 Using Information for Test Construction

Item and test information are especially useful for:

Constructing tests: Select items that provide information where it’s needed
Predicting precision: Know the SEM of theta scores before administering
Targeted measurement: Increase information at specific points in the theta distribution

5.6.2 Example: Comparing Two Tests

Code

# Simulate two different tests
theta <- seq(-4, 4, 0.1)

# Test A: Items clustered around theta = 0
test_a_info <- calc_info(theta, 1.5, -0.5, 0) +
               calc_info(theta, 1.5, 0, 0) +
               calc_info(theta, 1.5, 0.5, 0)

# Test B: Items spread across theta range
test_b_info <- calc_info(theta, 1.5, -2, 0) +
               calc_info(theta, 1.5, 0, 0) +
               calc_info(theta, 1.5, 2, 0)

par(mfrow = c(1, 2))

# Information comparison
plot(theta, test_a_info, type = "l", lwd = 2, col = "blue",
     xlab = expression(theta), ylab = "Test Information",
     main = "Information Comparison", ylim = c(0, max(c(test_a_info, test_b_info))))
lines(theta, test_b_info, lwd = 2, col = "red")
legend("topright", legend = c("Test A (clustered)", "Test B (spread)"),
       col = c("blue", "red"), lwd = 2)

# SEM comparison
sem_a <- 1 / sqrt(test_a_info)
sem_b <- 1 / sqrt(test_b_info)

plot(theta, sem_a, type = "l", lwd = 2, col = "blue",
     xlab = expression(theta), ylab = "SEM",
     main = "SEM Comparison", ylim = c(0, 2))
lines(theta, sem_b, lwd = 2, col = "red")
legend("topright", legend = c("Test A (clustered)", "Test B (spread)"),
       col = c("blue", "red"), lwd = 2)

Code

par(mfrow = c(1, 1))

Conclusion:

Test A (clustered items): High precision around $\theta = 0$, poor at extremes
Test B (spread items): More uniform precision across the ability range

The choice depends on your measurement goals!

5.7 Summary

Concept	Formula	Key Points
TCC	$\sum P_i(\theta)$	Expected score at each ability level
Item Info	$\frac{[P'_i(\theta)]^2}{P_i(\theta)Q_i(\theta)}$	Varies by item and ability
Test Info	$\sum I_i(\theta)$	Sum of item information
SEM	$1/\sqrt{I(\theta)}$	Inverse relationship with information

Key takeaways:

The TCC links ability ($\theta$) to expected test score
Items provide maximum information near their difficulty parameter
Higher discrimination = more information (but narrower range)
Test information determines measurement precision
SEM varies across the ability distribution

--- title: "Test Characteristic Curves, Information Functions, and SEM" author: "Derek C. Briggs and Claude Code (Opus 4.6 & 4.7)" output: html_document: toc: true toc_float: true code_folding: show pdf_document: toc: true latex_engine: xelatex --- ```{r inject-rootdir, include=FALSE} knitr::opts_knit$set(root.dir = "/Users/briggsd/Library/CloudStorage/Dropbox/Github/Measurement and Psychometrics/IRT Models for Dichotomously Scored Items/R Markdown Tutorials") ``` ```{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE) ``` ## Introduction This document covers three important concepts in IRT: 1. **Test Characteristic Curves (TCC)**: The expected test score as a function of ability 2. **Information Functions**: How much "information" items and tests provide about ability 3. **Standard Error of Measurement (SEM)**: The precision of ability estimates --- ## Part 1: Test Characteristic Curves ### Expected Item Response For a dichotomously scored item, the expected value of a respondent's answer is: $$E(X_{ip} | \theta_p) = P(X_{ip} = 1 | \theta_p) = P_i(\theta)$$ Because $X_{ip}$ is a binary variable (0 or 1), the mean equals the probability. This means: of all people with ability $\theta_p$, we expect $P_i(\theta_p)$ proportion to get the item right. ### Expected Test Score: The TCC The expected observed (number correct) score for a person can be calculated by summing the ICCs across items: $$TCC(\theta_p) = \sum_{i=1}^{I} P_i(\theta_p)$$ This **Test Characteristic Curve** tells us the expected number of correct answers for each level of theta. We can think of this as an estimate of the true score associated with a particular set of items. ### Example: Three-Item Test ```{r tcc-functions} # 3PL probability function calc_prob <- function(theta, a, b, c) { c + (1 - c) * exp(a * (theta - b)) / (1 + exp(a * (theta - b))) } # Item parameters (from three_item_test.xlsx) items <- data.frame( item = 1:3, a = c(1, 1, 2), b = c(-1.5, 0, 0.2), c = c(0.2, 0.2, 0.3) ) knitr::kable(items, caption = "Item Parameters") ``` ```{r tcc-plot, fig.width=10, fig.height=5} theta <- seq(-4, 4, 0.1) # Calculate individual ICCs p1 <- calc_prob(theta, items$a[1], items$b[1], items$c[1]) p2 <- calc_prob(theta, items$a[2], items$b[2], items$c[2]) p3 <- calc_prob(theta, items$a[3], items$b[3], items$c[3]) # Calculate TCC (sum of probabilities) tcc <- p1 + p2 + p3 par(mfrow = c(1, 2)) # Plot individual ICCs plot(theta, p1, type = "l", lwd = 2, col = "blue", xlab = expression(theta), ylab = expression(P[i](theta)), main = "Individual ICCs", ylim = c(0, 1)) lines(theta, p2, lwd = 2, col = "red") lines(theta, p3, lwd = 2, col = "darkgreen") legend("bottomright", legend = paste("Item", 1:3), col = c("blue", "red", "darkgreen"), lwd = 2) # Plot TCC plot(theta, tcc, type = "l", lwd = 3, col = "purple", xlab = expression(theta), ylab = "Expected Score", main = "Test Characteristic Curve (TCC)") abline(h = c(0, 1, 2, 3), lty = 2, col = "gray") par(mfrow = c(1, 1)) ``` **Interpretation:** - At $\theta = -4$: Expected score $\approx$ `r round(tcc[theta == -4], 2)` (mostly guessing) - At $\theta = 0$: Expected score $\approx$ `r round(tcc[theta == 0], 2)` - At $\theta = 4$: Expected score $\approx$ `r round(tcc[theta == 4], 2)` (nearly all correct) --- ## Part 2: Item Information Functions ### What is "Information"? Each item provides a certain amount of "information" about respondents' ability levels. The item information function is: $$I_i(\theta) = \frac{[P'_i(\theta)]^2}{P_i(\theta) Q_i(\theta)}$$ where: - $P_i(\theta) = P(X_{ip} = 1 | \theta_p)$ - $Q_i(\theta) = 1 - P_i(\theta)$ - $P'_i(\theta)$ is the first derivative of $P_i(\theta)$ **Key insight:** Items provide different amounts of information at different points on the theta scale. ### Model-Specific Information Functions | Model | Item Information Formula | Maximum At | |:------|:------------------------|:-----------| | 1PL | $P_i(\theta) Q_i(\theta)$ | $b_i$ | | 2PL | $a_i^2 P_i(\theta) Q_i(\theta)$ | $b_i$ | | 3PL | $\frac{a_i^2 Q_i(\theta)}{P_i(\theta)} \cdot \frac{[P_i(\theta) - c_i]^2}{(1-c_i)^2}$ | Slightly above $b_i$ | ### Computing Item Information ```{r item-info-function} # 3PL item information function calc_info <- function(theta, a, b, c) { # Calculate P and Q L <- 1 / (1 + exp(-a * (theta - b))) P <- c + (1 - c) * L Q <- 1 - P # Calculate derivative of P dP <- (1 - c) * a * L * (1 - L) # Information info <- (dP^2) / (P * Q) return(info) } ``` ```{r item-info-plot, fig.width=10, fig.height=5} # Calculate item information for each item info1 <- calc_info(theta, items$a[1], items$b[1], items$c[1]) info2 <- calc_info(theta, items$a[2], items$b[2], items$c[2]) info3 <- calc_info(theta, items$a[3], items$b[3], items$c[3]) par(mfrow = c(1, 2)) # Plot ICCs plot(theta, p1, type = "l", lwd = 2, col = "blue", xlab = expression(theta), ylab = expression(P[i](theta)), main = "ICCs", ylim = c(0, 1)) lines(theta, p2, lwd = 2, col = "red") lines(theta, p3, lwd = 2, col = "darkgreen") legend("bottomright", legend = paste("Item", 1:3), col = c("blue", "red", "darkgreen"), lwd = 2, cex = 0.8) # Plot Item Information plot(theta, info1, type = "l", lwd = 2, col = "blue", xlab = expression(theta), ylab = expression(I[i](theta)), main = "Item Information Functions", ylim = c(0, max(c(info1, info2, info3)) * 1.1)) lines(theta, info2, lwd = 2, col = "red") lines(theta, info3, lwd = 2, col = "darkgreen") legend("topright", legend = paste("Item", 1:3), col = c("blue", "red", "darkgreen"), lwd = 2, cex = 0.8) par(mfrow = c(1, 1)) ``` **Observations:** - Item 3 (green) has the highest peak information because it has the highest discrimination ($a = 2$) - Information is maximized near each item's difficulty parameter ($b$) - Items with higher discrimination provide more information but over a narrower range --- ## Part 3: Test Information Function ### Summing Item Information The test information function is calculated by summing the item information functions: $$I(\theta) = \sum_{i=1}^{I} I_i(\theta)$$ ### Relationship to SEM The standard error of measurement for theta is: $$SEM(\theta) = \frac{1}{\sqrt{I(\theta)}}$$ **Key relationships:** - As information **increases**, SEM **decreases** - $SEM(\theta)$ **varies** across the theta distribution - Tests provide more precise estimates where they have more information ```{r test-info, fig.width=10, fig.height=5} # Calculate test information test_info <- info1 + info2 + info3 # Calculate SEM sem <- 1 / sqrt(test_info) par(mfrow = c(1, 2)) # Plot Test Information plot(theta, test_info, type = "l", lwd = 3, col = "purple", xlab = expression(theta), ylab = expression(I(theta)), main = "Test Information Function") # Also show individual item contributions lines(theta, info1, lwd = 1, col = "blue", lty = 2) lines(theta, info2, lwd = 1, col = "red", lty = 2) lines(theta, info3, lwd = 1, col = "darkgreen", lty = 2) legend("topright", legend = c("Test Info", "Item 1", "Item 2", "Item 3"), col = c("purple", "blue", "red", "darkgreen"), lwd = c(3, 1, 1, 1), lty = c(1, 2, 2, 2), cex = 0.8) # Plot SEM plot(theta, sem, type = "l", lwd = 3, col = "darkred", xlab = expression(theta), ylab = expression(SEM(theta)), main = "Standard Error of Measurement", ylim = c(0, max(sem[is.finite(sem)]) * 1.1)) par(mfrow = c(1, 1)) ``` ### Combined Plot: Information and SEM ```{r info-sem-dual, fig.width=10, fig.height=6} library(ggplot2) # Create data frame plot_df <- data.frame( theta = theta, info = test_info, sem = sem ) # Scale factor for dual axis scale_factor <- max(plot_df$info, na.rm = TRUE) / max(plot_df$sem[is.finite(plot_df$sem)], na.rm = TRUE) plot_df$sem_scaled <- plot_df$sem * scale_factor # Dual-axis plot ggplot(plot_df, aes(x = theta)) + geom_line(aes(y = info), linewidth = 1.2, color = "blue") + geom_line(aes(y = sem_scaled), linewidth = 1.2, linetype = "dashed", color = "red") + scale_y_continuous( name = "Test Information", sec.axis = sec_axis(~ . / scale_factor, name = "SEM (logits)") ) + labs( x = expression(theta), title = "Three-Item Test: Information and SEM Functions", subtitle = "Solid blue = Information | Dashed red = SEM" ) + theme_minimal(base_size = 14) + theme( axis.title.y.left = element_text(color = "blue"), axis.title.y.right = element_text(color = "red") ) ``` **Interpretation:** - The test provides maximum information (minimum SEM) around $\theta \approx 0$ - At the extremes ($\theta < -2$ or $\theta > 2$), information drops and SEM increases - This three-item test measures best in the middle of the ability range --- ## Part 4: Practice with Real Data Using mirt Now let's apply these concepts to real data using the `mirt` package. ```{r load-packages} library(mirt) ``` ### Load Data and Fit Model ```{r fit-model, results='hide'} # Load data forma <- read.csv("../Data/pset1_formA.csv") forma <- forma[, 1:15] # Fit 2PL model mirt_2pl <- mirt(forma, model = 1, itemtype = "2PL", method = "EM") ``` ### Test Characteristic Curve ```{r mirt-tcc, fig.width=8, fig.height=5} plot(mirt_2pl, main = "Test Characteristic Curve") ``` The TCC shows the expected number of correct answers (out of 15) at each ability level. ### Item Information Functions ```{r mirt-item-info, fig.width=10, fig.height=4} # Show item information for first 3 items plot(mirt_2pl, type = 'infotrace', which.items = 1:3) ``` ### Test Information Function ```{r mirt-test-info, fig.width=8, fig.height=5} plot(mirt_2pl, type = 'info', main = "Test Information Function") ``` ### Test Information and SEM ```{r mirt-info-sem, fig.width=8, fig.height=5} plot(mirt_2pl, type = 'infoSE', theta_lim = c(-3, 3), main = "Test Information and Standard Error") ``` --- ## Practical Applications ### Using Information for Test Construction Item and test information are especially useful for: 1. **Constructing tests**: Select items that provide information where it's needed 2. **Predicting precision**: Know the SEM of theta scores before administering 3. **Targeted measurement**: Increase information at specific points in the theta distribution ### Example: Comparing Two Tests ```{r test-comparison, fig.width=10, fig.height=5} # Simulate two different tests theta <- seq(-4, 4, 0.1) # Test A: Items clustered around theta = 0 test_a_info <- calc_info(theta, 1.5, -0.5, 0) + calc_info(theta, 1.5, 0, 0) + calc_info(theta, 1.5, 0.5, 0) # Test B: Items spread across theta range test_b_info <- calc_info(theta, 1.5, -2, 0) + calc_info(theta, 1.5, 0, 0) + calc_info(theta, 1.5, 2, 0) par(mfrow = c(1, 2)) # Information comparison plot(theta, test_a_info, type = "l", lwd = 2, col = "blue", xlab = expression(theta), ylab = "Test Information", main = "Information Comparison", ylim = c(0, max(c(test_a_info, test_b_info)))) lines(theta, test_b_info, lwd = 2, col = "red") legend("topright", legend = c("Test A (clustered)", "Test B (spread)"), col = c("blue", "red"), lwd = 2) # SEM comparison sem_a <- 1 / sqrt(test_a_info) sem_b <- 1 / sqrt(test_b_info) plot(theta, sem_a, type = "l", lwd = 2, col = "blue", xlab = expression(theta), ylab = "SEM", main = "SEM Comparison", ylim = c(0, 2)) lines(theta, sem_b, lwd = 2, col = "red") legend("topright", legend = c("Test A (clustered)", "Test B (spread)"), col = c("blue", "red"), lwd = 2) par(mfrow = c(1, 1)) ``` **Conclusion:** - **Test A** (clustered items): High precision around $\theta = 0$, poor at extremes - **Test B** (spread items): More uniform precision across the ability range The choice depends on your measurement goals! --- ## Summary | Concept | Formula | Key Points | |:--------|:--------|:-----------| | TCC | $\sum P_i(\theta)$ | Expected score at each ability level | | Item Info | $\frac{[P'_i(\theta)]^2}{P_i(\theta)Q_i(\theta)}$ | Varies by item and ability | | Test Info | $\sum I_i(\theta)$ | Sum of item information | | SEM | $1/\sqrt{I(\theta)}$ | Inverse relationship with information | **Key takeaways:** 1. The TCC links ability ($\theta$) to expected test score 2. Items provide maximum information near their difficulty parameter 3. Higher discrimination = more information (but narrower range) 4. Test information determines measurement precision 5. SEM varies across the ability distribution