Assignment 3 MESA
Andi Xu
2024-10-21
# Load necessary libraries
library(ltm)## 载入需要的程辑包：MASS## 载入需要的程辑包：msm## 载入需要的程辑包：polycorlibrary(mirt)## 载入需要的程辑包：stats4## 载入需要的程辑包：lattice##
## 载入程辑包：'mirt'## The following object is masked from 'package:ltm':
##
## Science# Load the data (assuming the .dat file is tab-separated; modify if needed)
pisa_data <- read.table("C:/Users/徐安迪/Downloads/PISA2009.dat", header=TRUE, sep="\t")
# Select the first column as character data
responses <- as.character(pisa_data[, 1])
# Remove the first 8 characters from each row and retain the remaining part
responses <- substr(responses, 9, nchar(responses))
# Verify that the length of the string is 23 characters
nchar(responses[1]) # The result should be 23 to ensure the data is correct## [1] 23# Split each string into 23 items (columns)
item_data <- do.call(rbind, strsplit(responses, ""))
# Confirm that the number of columns after splitting is 23
ncol(item_data)## [1] 23# Convert the resulting data into a data frame and ensure column names are unique
item_data <- as.data.frame(item_data)
# Assign unique column names to each column
colnames(item_data) <- paste0("Item_", 1:ncol(item_data))
# Check the first few rows of the data and the column names
head(item_data)## Item_1 Item_2 Item_3 Item_4 Item_5 Item_6 Item_7 Item_8 Item_9 Item_10
## 1 0 0 0 0 0 0 0 0 0 0
## 2 0 1 0 0 0 0 1 1 0 0
## 3 0 0 0 1 0 0 0 0 1 0
## 4 8 0 0 8 8 8 8 1 1 8
## 5 0 0 0 0 8 0 8 1 0 8
## 6 1 0 0 1 0 0 1 0 1 0
## Item_11 Item_12 Item_13 Item_14 Item_15 Item_16 Item_17 Item_18 Item_19
## 1 1 0 0 0 0 0 0 0 0
## 2 0 1 0 0 0 0 1 1 1
## 3 0 0 0 0 0 1 1 0 1
## 4 0 8 8 1 8 0 1 8 0
## 5 0 8 8 0 8 0 0 8 0
## 6 1 1 0 1 1 1 1 1 1
## Item_20 Item_21 Item_22 Item_23
## 1 1 0 0 0
## 2 0 0 1 0
## 3 0 0 1 0
## 4 1 0 8 8
## 5 1 0 8 8
## 6 1 1 1 0colnames(item_data)## [1] "Item_1" "Item_2" "Item_3" "Item_4" "Item_5" "Item_6" "Item_7"
## [8] "Item_8" "Item_9" "Item_10" "Item_11" "Item_12" "Item_13" "Item_14"
## [15] "Item_15" "Item_16" "Item_17" "Item_18" "Item_19" "Item_20" "Item_21"
## [22] "Item_22" "Item_23"# Convert the split item data to numeric
item_data <- apply(item_data, 2, as.numeric)
# Ensure the data frame is numeric and check data types
item_data <- as.data.frame(item_data)
# Review the structure of the data to confirm all columns are numeric
str(item_data)## 'data.frame': 2904 obs. of 23 variables:
## $ Item_1 : num 0 0 0 8 0 1 0 0 0 0 ...
## $ Item_2 : num 0 1 0 0 0 0 0 0 0 0 ...
## $ Item_3 : num 0 0 0 0 0 0 0 0 0 0 ...
## $ Item_4 : num 0 0 1 8 0 1 1 0 0 1 ...
## $ Item_5 : num 0 0 0 8 8 0 0 0 0 0 ...
## $ Item_6 : num 0 0 0 8 0 0 0 0 0 0 ...
## $ Item_7 : num 0 1 0 8 8 1 1 0 0 0 ...
## $ Item_8 : num 0 1 0 1 1 0 1 0 1 1 ...
## $ Item_9 : num 0 0 1 1 0 1 0 0 0 0 ...
## $ Item_10: num 0 0 0 8 8 0 1 1 0 1 ...
## $ Item_11: num 1 0 0 0 0 1 0 0 0 0 ...
## $ Item_12: num 0 1 0 8 8 1 1 0 0 1 ...
## $ Item_13: num 0 0 0 8 8 0 0 0 0 0 ...
## $ Item_14: num 0 0 0 1 0 1 1 0 0 1 ...
## $ Item_15: num 0 0 0 8 8 1 0 0 0 1 ...
## $ Item_16: num 0 0 1 0 0 1 1 0 0 1 ...
## $ Item_17: num 0 1 1 1 0 1 1 1 0 1 ...
## $ Item_18: num 0 1 0 8 8 1 1 0 1 0 ...
## $ Item_19: num 0 1 1 0 0 1 1 0 0 1 ...
## $ Item_20: num 1 0 0 1 1 1 0 0 0 1 ...
## $ Item_21: num 0 0 0 0 0 1 1 0 1 1 ...
## $ Item_22: num 0 1 1 8 8 1 1 0 0 1 ...
## $ Item_23: num 0 0 0 8 8 0 0 0 0 0 ...# Convert any values greater than 1 to 1 for each item
item_data <- apply(item_data, 2, function(x) ifelse(x > 1, 1, x))
# Ensure the data is numeric and convert back to a data frame
item_data <- as.data.frame(item_data)
# Check the first few rows of data to confirm changes
head(item_data)## Item_1 Item_2 Item_3 Item_4 Item_5 Item_6 Item_7 Item_8 Item_9 Item_10
## 1 0 0 0 0 0 0 0 0 0 0
## 2 0 1 0 0 0 0 1 1 0 0
## 3 0 0 0 1 0 0 0 0 1 0
## 4 1 0 0 1 1 1 1 1 1 1
## 5 0 0 0 0 1 0 1 1 0 1
## 6 1 0 0 1 0 0 1 0 1 0
## Item_11 Item_12 Item_13 Item_14 Item_15 Item_16 Item_17 Item_18 Item_19
## 1 1 0 0 0 0 0 0 0 0
## 2 0 1 0 0 0 0 1 1 1
## 3 0 0 0 0 0 1 1 0 1
## 4 0 1 1 1 1 0 1 1 0
## 5 0 1 1 0 1 0 0 1 0
## 6 1 1 0 1 1 1 1 1 1
## Item_20 Item_21 Item_22 Item_23
## 1 1 0 0 0
## 2 0 0 1 0
## 3 0 0 1 0
## 4 1 0 1 1
## 5 1 0 1 1
## 6 1 1 1 0# Perform a 3PL analysis on the 23-item data
model_3pl <- mirt(item_data, 1, itemtype = '3PL')## Iteration: 1, Log-Lik: -37370.625, Max-Change: 8.36211Iteration: 2, Log-Lik: -35132.719, Max-Change: 3.96630Iteration: 3, Log-Lik: -34492.076, Max-Change: 3.54052Iteration: 4, Log-Lik: -34093.726, Max-Change: 1.64753Iteration: 5, Log-Lik: -33945.238, Max-Change: 1.56847Iteration: 6, Log-Lik: -33854.724, Max-Change: 1.27794Iteration: 7, Log-Lik: -33803.122, Max-Change: 0.67287Iteration: 8, Log-Lik: -33778.038, Max-Change: 0.72729Iteration: 9, Log-Lik: -33766.604, Max-Change: 0.78857Iteration: 10, Log-Lik: -33759.181, Max-Change: 0.76641Iteration: 11, Log-Lik: -33754.113, Max-Change: 0.53015Iteration: 12, Log-Lik: -33750.427, Max-Change: 0.66618Iteration: 13, Log-Lik: -33743.585, Max-Change: 0.27706Iteration: 14, Log-Lik: -33740.185, Max-Change: 0.27565Iteration: 15, Log-Lik: -33738.585, Max-Change: 0.26376Iteration: 16, Log-Lik: -33735.224, Max-Change: 0.32328Iteration: 17, Log-Lik: -33733.000, Max-Change: 0.49897Iteration: 18, Log-Lik: -33731.928, Max-Change: 0.75076Iteration: 19, Log-Lik: -33730.172, Max-Change: 0.24293Iteration: 20, Log-Lik: -33729.147, Max-Change: 0.29684Iteration: 21, Log-Lik: -33728.494, Max-Change: 0.91503Iteration: 22, Log-Lik: -33727.513, Max-Change: 0.25384Iteration: 23, Log-Lik: -33727.045, Max-Change: 0.87502Iteration: 24, Log-Lik: -33726.550, Max-Change: 0.62937Iteration: 25, Log-Lik: -33726.167, Max-Change: 0.47747Iteration: 26, Log-Lik: -33725.543, Max-Change: 0.08838Iteration: 27, Log-Lik: -33725.367, Max-Change: 0.98817Iteration: 28, Log-Lik: -33725.013, Max-Change: 0.09265Iteration: 29, Log-Lik: -33724.881, Max-Change: 0.84638Iteration: 30, Log-Lik: -33724.649, Max-Change: 0.38752Iteration: 31, Log-Lik: -33724.522, Max-Change: 0.18245Iteration: 32, Log-Lik: -33724.426, Max-Change: 0.05138Iteration: 33, Log-Lik: -33724.371, Max-Change: 0.01110Iteration: 34, Log-Lik: -33724.359, Max-Change: 0.32082Iteration: 35, Log-Lik: -33724.272, Max-Change: 0.00958Iteration: 36, Log-Lik: 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0.00045Iteration: 499, Log-Lik: -33723.901, Max-Change: 0.00031Iteration: 500, Log-Lik: -33723.901, Max-Change: 0.00056## Warning in EM.group(pars = pars, constrain = constrain, Ls = Ls, PrepList =
## PrepList, : EM cycles terminated after 500 iterations.# Output the summary of the model
summary(model_3pl)## F1 h2
## Item_1 0.826 0.6819
## Item_2 0.204 0.0415
## Item_3 0.954 0.9096
## Item_4 0.548 0.3008
## Item_5 0.983 0.9660
## Item_6 0.832 0.6918
## Item_7 0.859 0.7380
## Item_8 0.118 0.0138
## Item_9 0.191 0.0366
## Item_10 0.955 0.9121
## Item_11 0.175 0.0307
## Item_12 0.768 0.5891
## Item_13 0.998 0.9955
## Item_14 0.226 0.0511
## Item_15 0.972 0.9457
## Item_16 0.227 0.0516
## Item_17 0.171 0.0292
## Item_18 0.906 0.8204
## Item_19 0.207 0.0429
## Item_20 0.251 0.0632
## Item_21 0.311 0.0969
## Item_22 0.846 0.7154
## Item_23 0.970 0.9401
##
## SS loadings: 10.664
## Proportion Var: 0.464
##
## Factor correlations:
##
## F1
## F1 1# Extract item coefficients from the 3PL model
item_stats <- coef(model_3pl, IRTpars=TRUE, simplify=TRUE)
# View the item statistics table
item_stats## $items
## a b g u
## Item_1 2.492 0.831 0.266 1
## Item_2 0.354 5.704 0.000 1
## Item_3 5.398 3.021 0.036 1
## Item_4 1.116 -0.210 0.001 1
## Item_5 9.067 0.617 0.041 1
## Item_6 2.550 0.715 0.058 1
## Item_7 2.856 0.640 0.144 1
## Item_8 0.202 -7.628 0.005 1
## Item_9 0.332 2.236 0.001 1
## Item_10 5.483 0.344 0.226 1
## Item_11 0.303 5.393 0.000 1
## Item_12 2.038 -0.044 0.000 1
## Item_13 25.185 0.781 0.008 1
## Item_14 0.395 1.694 0.000 1
## Item_15 7.104 0.519 0.026 1
## Item_16 0.397 1.171 0.000 1
## Item_17 0.295 -0.695 0.001 1
## Item_18 3.637 0.316 0.170 1
## Item_19 0.360 1.385 0.002 1
## Item_20 0.442 1.708 0.000 1
## Item_21 0.557 1.793 0.000 1
## Item_22 2.699 -0.085 0.663 1
## Item_23 6.744 0.510 0.060 1
##
## $means
## F1
## 0
##
## $cov
## F1
## F1 1# Create the data frame from the provided item statistics
item_stats <- data.frame(
item = paste0("Item_", 1:23),
a = c(2.492, 0.354, 5.398, 1.116, 9.066, 2.550, 2.856, 0.202, 0.332, 5.483,
0.303, 2.038, 25.185, 0.395, 7.104, 0.397, 0.295, 3.637, 0.360, 0.442,
0.557, 2.699, 6.744),
b = c(0.831, 5.704, 3.021, -0.210, 0.617, 0.715, 0.640, -7.629, 2.236, 0.344,
5.393, -0.045, 0.781, 1.694, 0.519, 1.171, -0.695, 0.316, 1.385, 1.708,
1.793, -0.085, 0.510),
g = c(0.266, 0, 0.036, 0.001, 0.041, 0.058, 0.144, 0.005, 0.001, 0.226,
0, 0, 0.008, 0, 0.026, 0, 0.001, 0.170, 0.002, 0, 0, 0.663, 0.060)
)
# Find the easiest item (lowest b value)
easiest_item <- item_stats[which.min(item_stats$b),]
# Find the most difficult item (highest b value)
most_difficult_item <- item_stats[which.max(item_stats$b),]
# Output the easiest and most difficult items
easiest_item## item a b g
## 8 Item_8 0.202 -7.629 0.005most_difficult_item## item a b g
## 2 Item_2 0.354 5.704 0# Plot the ICC for Item 2 using the mirt package.
cat("Plotting ICC for Item 2\n")## Plotting ICC for Item 2plot(model_3pl, type = "trace", which.items = 2)
# Plot the ICC for Item 8 using the mirt package.
cat("Plotting ICC for Item 8\n")## Plotting ICC for Item 8plot(model_3pl, type = "trace", which.items = 8)
# Obtain item parameter estimates from the 3PL model output
item_params <- coef(model_3pl, IRTpars=TRUE, simplify=TRUE)
# Convert the list of item parameters to a data frame for easier manipulation
item_params_df <- as.data.frame(item_params)
# Rename the columns clearly for discrimination, difficulty, and guessing parameters
colnames(item_params_df) <- c("discrimination", "difficulty", "guessing")
# Output the full item parameter estimates to review
item_params_df## discrimination difficulty guessing NA NA NA
## Item_1 2.4922223 0.83095722 0.2659943422 1 0 1
## Item_2 0.3543205 5.70357768 0.0003126850 1 0 1
## Item_3 5.3982546 3.02091959 0.0362084624 1 0 1
## Item_4 1.1164291 -0.21018070 0.0008037071 1 0 1
## Item_5 9.0665403 0.61685183 0.0409710976 1 0 1
## Item_6 2.5497818 0.71522453 0.0579314698 1 0 1
## Item_7 2.8564377 0.63997273 0.1441641991 1 0 1
## Item_8 0.2015628 -7.62810821 0.0047470592 1 0 1
## Item_9 0.3316297 2.23605642 0.0006417787 1 0 1
## Item_10 5.4832626 0.34446911 0.2255452890 1 0 1
## Item_11 0.3026939 5.39324989 0.0004724840 1 0 1
## Item_12 2.0379277 -0.04446747 0.0003560075 1 0 1
## Item_13 25.1847412 0.78061500 0.0081552884 1 0 1
## Item_14 0.3950907 1.69416520 0.0004645945 1 0 1
## Item_15 7.1042370 0.51866307 0.0262039360 1 0 1
## Item_16 0.3968793 1.17123768 0.0004225091 1 0 1
## Item_17 0.2951926 -0.69509842 0.0011984018 1 0 1
## Item_18 3.6370865 0.31634713 0.1699981483 1 0 1
## Item_19 0.3603339 1.38526002 0.0018273478 1 0 1
## Item_20 0.4420488 1.70805805 0.0004531190 1 0 1
## Item_21 0.5574628 1.79315783 0.0001501595 1 0 1
## Item_22 2.6987073 -0.08458097 0.6628333281 1 0 1
## Item_23 6.7442218 0.51017663 0.0600386775 1 0 1# Plot the test information function (TIF)
cat("Plotting the Test Information Function\n")## Plotting the Test Information Functioninfo_plot <- plot(model_3pl, type = "info", main = "Test Information Function")
# Calculate the standard error function from the information
# SE(θ) = 1 / sqrt(I(θ))
theta_vals <- seq(-4, 4, by=0.1)
info_vals <- testinfo(model_3pl, Theta = theta_vals)
se_vals <- 1 / sqrt(info_vals)
# Plot the standard error function
cat("Plotting the Standard Error Function\n")## Plotting the Standard Error Functionplot(theta_vals, se_vals, type = "l", col = "blue", lwd = 2,
xlab = "Ability (θ)", ylab = "Standard Error",
main = "Standard Error Function")
# Add a grid for better visualization
grid()
# Display both plots
info_plot
# Count the number of correct answers for each examinee
correct_counts <- rowSums(item_data)
# Find two examinees who answered the same number of items correctly
# Get the indices of two examinees with the same score
same_score_examinees <- which(correct_counts == correct_counts[1])[1:2]
# Extract the item response patterns for these two examinees
examinee_1 <- item_data[same_score_examinees[1], ]
examinee_2 <- item_data[same_score_examinees[2], ]
# Check their raw scores (number of correct answers)
cat("Examinee 1 Correct Answers: ", sum(examinee_1), "\n")## Examinee 1 Correct Answers: 2cat("Examinee 2 Correct Answers: ", sum(examinee_2), "\n")## Examinee 2 Correct Answers: 2# Estimate abilities for all examinees
abilities <- fscores(model_3pl)
# Retrieve and compare the ability estimates for these two examinees
cat("Examinee 1 Ability Estimate: ", abilities[same_score_examinees[1], ], "\n")## Examinee 1 Ability Estimate: -1.230896cat("Examinee 2 Ability Estimate: ", abilities[same_score_examinees[2], ], "\n")## Examinee 2 Ability Estimate: -1.393466```