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name learning-experiment description Educational psychology experiment design and analysis covering IRT, growth modeling, A/B testing, and learning curve estimation for research. tags ["education-research","item-response-theory","learning-analytics","experimental-design","growth-modeling"] version 1.0.0 authors ["@xjtulyc"] license MIT platforms ["claude-code","codex","gemini-cli","cursor"] dependencies {"python":["numpy>=1.24","pandas>=2.0","scipy>=1.11","statsmodels>=0.14","scikit-learn>=1.3","matplotlib>=3.7"]} last_updated 2026-03-17 status stable
Learning Experiment Design and Analysis
When to Use This Skill
Use this skill when you need to:
Design randomized controlled trials for educational interventions
Analyze item response data with IRT models (1PL/2PL/3PL)
Estimate student growth models and learning trajectories
Conduct A/B testing for educational technology features
Fit learning curves (power law, exponential decay of errors)
Measure knowledge retention (forgetting curves, spacing effects)
Compute test reliability (Cronbach's alpha, inter-rater reliability)
Trigger keywords : item response theory, Rasch model, 2PL, 3PL, student growth, learning curve, forgetting curve, spaced repetition, educational A/B test, mastery learning, knowledge tracing, Bayesian knowledge tracing, DIF analysis, NAEP, PISA, CAT, adaptive testing, effect size, pre-post design.
Background & Key Concepts
Item Response Theory (IRT)
The 3-parameter logistic model (3PL) gives the probability that student $\theta$ answers item $i$ correctly:
$$P(X_i=1|\theta) = c_i + (1-c_i) \cdot \frac{e^{a_i(\theta - b_i)}}{1 + e^{a_i(\theta - b_i)}}$$
$a_i$: discrimination parameter
$b_i$: difficulty parameter (item location)
$c_i$: guessing parameter (lower asymptote)
The 2PL sets $c_i=0$; the 1PL (Rasch) additionally fixes $a_i=1$.
Item Information Function
$$I_i(\theta) = a_i^2 \cdot \frac{(P_i(\theta) - c_i)^2}{(1-c_i)^2} \cdot \frac{(1-P_i(\theta))}{P_i(\theta)}$$
Total test information: $I(\theta) = \sum_i I_i(\theta)$. Standard error: $SE(\theta) = 1/\sqrt{I(\theta)}$.
Power Law of Learning (Newell & Rosenbloom 1981)
$$T_n = T_1 \cdot n^{-\alpha}$$
where $T_n$ is the time (or error rate) on trial $n$ and $\alpha$ is the learning rate. Log-linearized: $\ln T_n = \ln T_1 - \alpha \ln n$.
Bayesian Knowledge Tracing (BKT)
Hidden Markov model with states $K \in {0,1}$ (unlearned/learned):
$$P(K_{n+1}=1 | K_n=0) = p_T \quad \text{(transit)}$$
$$P(X=1 | K=1) = 1-p_S,\quad P(X=1|K=0) = p_G$$
Effect Size Measures
Cohen's $d$: $d = (\bar{x}_1 - \bar{x}_2)/s_p$ (pooled SD)
Glass's $\Delta$: $\Delta = (\bar{x}_1 - \bar{x}2)/s {\text{control}}$
Hedges' $g$: small-sample corrected version of $d$
Environment Setup
pip install numpy>=1.24 pandas>=2.0 scipy>=1.11 statsmodels>=0.14 \
scikit-learn>=1.3 matplotlib>=3.7
import
as
import
as
import
as
import
as
import
as
print
"Learning experiment analysis environment ready"
Core Workflow
Step 1: IRT Model Estimation (2PL) import numpy as np
import pandas as pd
import scipy.optimize as opt
import matplotlib.pyplot as plt
np.random.seed(42 )
n_students = 500
n_items = 20
theta_true = np.random.normal(0 , 1 , n_students)
a_true = np.random.uniform(0.5 , 2.5 , n_items)
b_true = np.random.linspace(-2.5 , 2.5 , n_items)
def p2pl (theta, a, b ):
"""2PL probability of correct response."""
return 1.0 / (1.0 + np.exp(-a * (theta[:, None ] - b[None , :])))
P_matrix = p2pl(theta_true, a_true, b_true)
responses = (np.random.uniform(size=P_matrix.shape) < P_matrix).astype(int )
print (f"Response matrix shape: {responses.shape} " )
print (f"Mean item difficulty (p-values): {responses.mean(axis=0 ).round (2 )} " )
print (f"Overall correct rate: {responses.mean():.3 f} " )
def neg_log_likelihood_2pl (params, responses ):
"""Compute -2LL for 2PL model with fixed quadrature points for theta.
Args:
params: vector of length 2*n_items (a1..an, b1..bn)
responses: (n_students, n_items) binary matrix
Returns:
negative log-likelihood
"""
n_i = responses.shape[1 ]
a = np.abs (params[:n_i]) + 0.2
b = params[n_i:]
n_quad = 15
quad_pts, quad_wts = np.polynomial.hermite.hermgauss(n_quad)
theta_q = quad_pts * np.sqrt(2 )
wts = quad_wts / np.sqrt(np.pi)
total_ll = 0.0
for s in range (len (responses)):
p_iq = 1.0 / (1.0 + np.exp(-a[None , :] * (theta_q[:, None ] - b[None , :])))
r = responses[s]
log_lik_q = (r * np.log(np.clip(p_iq, 1e-9 , 1 - 1e-9 )) +
(1 - r) * np.log(np.clip(1 - p_iq, 1e-9 , 1 - 1e-9 ))).sum (axis=1 )
lik_q = np.exp(log_lik_q)
marginal_lik = (lik_q * wts).sum ()
total_ll += np.log(max (marginal_lik, 1e-300 ))
return -total_ll
a_init = np.ones(n_items)
b_init = np.zeros(n_items)
params_init = np.concatenate([a_init, b_init])
print ("\nFitting 2PL model (simplified MML)..." )
subset_students = responses[:100 ]
result = opt.minimize(neg_log_likelihood_2pl, params_init,
args=(subset_students,),
method="L-BFGS-B" ,
options={"maxiter" : 50 , "ftol" : 1e-4 })
a_est = np.abs (result.x[:n_items]) + 0.2
b_est = result.x[n_items:]
print (f"Optimization converged: {result.success} " )
print (f"Correlation(a_true, a_est): {np.corrcoef(a_true, a_est)[0 ,1 ]:.3 f} " )
print (f"Correlation(b_true, b_est): {np.corrcoef(b_true, b_est)[0 ,1 ]:.3 f} " )
fig, axes = plt.subplots(1 , 3 , figsize=(15 , 5 ))
theta_plot = np.linspace(-4 , 4 , 200 )
colors = ["steelblue" , "orange" , "green" , "red" ]
for idx, item_idx in enumerate ([0 , 5 , 10 , 15 ]):
p_true = 1 / (1 + np.exp(-a_true[item_idx] * (theta_plot - b_true[item_idx])))
axes[0 ].plot(theta_plot, p_true, color=colors[idx],
label=f"Item {item_idx+1 } (b={b_true[item_idx]:.1 f} )" )
axes[0 ].set_xlabel("θ (Ability)" ); axes[0 ].set_ylabel("P(correct)" )
axes[0 ].set_title("Item Characteristic Curves" )
axes[0 ].legend(fontsize=8 )
axes[0 ].axhline(0.5 , color="gray" , ls="--" , lw=0.8 )
for idx, item_idx in enumerate ([0 , 5 , 10 , 15 ]):
a, b = a_true[item_idx], b_true[item_idx]
p = 1 / (1 + np.exp(-a * (theta_plot - b)))
info = a**2 * p * (1 - p)
axes[1 ].plot(theta_plot, info, color=colors[idx],
label=f"Item {item_idx+1 } (a={a:.2 f} )" )
axes[1 ].set_xlabel("θ" ); axes[1 ].set_ylabel("Information" )
axes[1 ].set_title("Item Information Functions" )
axes[1 ].legend(fontsize=8 )
P_all = 1 / (1 + np.exp(-a_true[:, None ] * (theta_plot[None , :] - b_true[:, None ])))
test_info = (a_true[:, None ]**2 * P_all * (1 - P_all)).sum (axis=0 )
se_theta = 1 / np.sqrt(np.maximum(test_info, 0.01 ))
axes[2 ].plot(theta_plot, test_info, "steelblue" , lw=2 , label="Test Information" )
ax2r = axes[2 ].twinx()
ax2r.plot(theta_plot, se_theta, "r--" , lw=1.5 , label="SE(θ)" )
ax2r.set_ylabel("Standard Error" , color="red" )
axes[2 ].set_xlabel("θ" ); axes[2 ].set_ylabel("Information" )
axes[2 ].set_title("Test Information & SE" )
axes[2 ].legend(loc="upper left" )
plt.tight_layout()
plt.savefig("irt_analysis.png" , dpi=150 , bbox_inches="tight" )
plt.close()
print ("\nFigure saved: irt_analysis.png" )
Step 2: Learning Curves and Forgetting Models import numpy as np
import pandas as pd
import scipy.optimize as opt
import matplotlib.pyplot as plt
np.random.seed(42 )
n_students = 30
n_trials = 20
trials = np.arange(1 , n_trials + 1 , dtype=float )
T1_true = np.random.uniform(20 , 40 , n_students)
alpha_true = np.random.uniform(0.2 , 0.6 , n_students)
times = np.zeros((n_students, n_trials))
for s in range (n_students):
times[s] = T1_true[s] * trials**(-alpha_true[s]) * np.exp(
np.random.normal(0 , 0.05 , n_trials))
def power_law (n, T1, alpha ):
"""Power law of practice."""
return T1 * n**(-alpha)
mean_times = times.mean(axis=0 )
popt, pcov = opt.curve_fit(power_law, trials, mean_times,
p0=[30 , 0.3 ], bounds=([1 , 0.01 ], [200 , 2.0 ]))
T1_fit, alpha_fit = popt
print (f"Power law fit: T1 = {T1_fit:.2 f} s, α = {alpha_fit:.3 f} " )
print (f"95% CI for α: [{alpha_fit - 1.96 *np.sqrt(pcov[1 ,1 ]):.3 f} , "
f"{alpha_fit + 1.96 *np.sqrt(pcov[1 ,1 ]):.3 f} ]" )
def exponential_learning (n, T_inf, T1, k ):
"""Exponential learning model."""
return T_inf + (T1 - T_inf) * np.exp(-k * (n - 1 ))
popt_exp, _ = opt.curve_fit(exponential_learning, trials, mean_times,
p0=[5 , 35 , 0.1 ], bounds=([0 , 1 , 0 ], [50 , 200 , 5 ]))
print (f"Exponential fit: T_inf = {popt_exp[0 ]:.2 f} s, T1 = {popt_exp[1 ]:.2 f} s, "
f"k = {popt_exp[2 ]:.4 f} " )
def aic (n_data, k_params, sse ):
"""Compute AIC."""
sigma2 = sse / n_data
return n_data * np.log(sigma2) + 2 * k_params
sse_power = ((mean_times - power_law(trials, *popt))**2 ).sum ()
sse_exp = ((mean_times - exponential_learning(trials, *popt_exp))**2 ).sum ()
print (f"\nAIC (power law): {aic(n_trials, 2 , sse_power):.2 f} " )
print (f"AIC (exponential): {aic(n_trials, 3 , sse_exp):.2 f} " )
def ebbinghaus_retention (t, S ):
"""Ebbinghaus forgetting curve: R(t) = e^{-t/S}."""
return np.exp(-t / S)
retention_days = np.array([1 , 2 , 7 , 14 , 30 ], dtype=float )
true_S = 12.0
retention_obs = ebbinghaus_retention(retention_days, true_S) + np.random.normal(0 , 0.03 , 5 )
retention_obs = np.clip(retention_obs, 0 , 1 )
S_fit, _ = opt.curve_fit(ebbinghaus_retention, retention_days, retention_obs,
p0=[10 ], bounds=([1 ], [100 ]))
print (f"\nEbbinghaus S (stability) = {S_fit[0 ]:.1 f} days" )
print (f"Predicted 50% retention at: {S_fit[0 ] * np.log(2 ):.1 f} days" )
fig, axes = plt.subplots(1 , 3 , figsize=(15 , 5 ))
for s in range (min (10 , n_students)):
axes[0 ].plot(trials, times[s], color="lightblue" , alpha=0.5 , lw=0.8 )
axes[0 ].plot(trials, mean_times, "k-" , lw=2 , label="Mean" )
axes[0 ].plot(trials, power_law(trials, *popt), "r--" , lw=2 ,
label=f"Power Law (α={alpha_fit:.2 f} )" )
axes[0 ].set_xlabel("Trial" ); axes[0 ].set_ylabel("Response Time (s)" )
axes[0 ].set_title("Learning Curves" )
axes[0 ].legend()
axes[1 ].plot(np.log(trials), np.log(mean_times), "ko" , label="Data" )
axes[1 ].plot(np.log(trials), np.log(power_law(trials, *popt)), "r-" ,
label=f"Fit (slope={-alpha_fit:.2 f} )" )
axes[1 ].set_xlabel("ln(Trial)" ); axes[1 ].set_ylabel("ln(Time)" )
axes[1 ].set_title("Power Law Verification (Log-Log)" )
axes[1 ].legend()
t_cont = np.linspace(0 , 35 , 300 )
axes[2 ].scatter(retention_days, retention_obs, color="red" , s=80 , zorder=5 ,
label="Observed" )
axes[2 ].plot(t_cont, ebbinghaus_retention(t_cont, S_fit[0 ]), "steelblue" , lw=2 ,
label=f"Ebbinghaus (S={S_fit[0 ]:.1 f} d)" )
axes[2 ].set_xlabel("Days Since Learning" ); axes[2 ].set_ylabel("Retention (fraction)" )
axes[2 ].set_title("Forgetting Curve" )
axes[2 ].set_ylim(0 , 1.1 )
axes[2 ].legend()
plt.tight_layout()
plt.savefig("learning_curves.png" , dpi=150 , bbox_inches="tight" )
plt.close()
print ("\nFigure saved: learning_curves.png" )
Step 3: A/B Testing Educational Interventions import numpy as np
import pandas as pd
import scipy.stats as stats
import statsmodels.api as sm
import matplotlib.pyplot as plt
np.random.seed(42 )
n = 200
n_t = n // 2
pre_test = np.random.normal(55 , 15 , n)
pre_test = np.clip(pre_test, 0 , 100 )
treat = np.concatenate([np.ones(n_t), np.zeros(n_t)])
np.random.shuffle(treat)
te_individual = 8 + np.random.normal(0 , 3 , n)
post_test = (pre_test * 0.7
+ treat * te_individual
+ np.random.normal(0 , 10 , n))
post_test = np.clip(post_test, 0 , 100 )
df = pd.DataFrame({"student_id" : range (n), "treat" : treat,
"pre_test" : pre_test, "post_test" : post_test})
df["gain" ] = df["post_test" ] - df["pre_test" ]
t_gains = df[df["treat" ] == 1 ]["gain" ]
c_gains = df[df["treat" ] == 0 ]["gain" ]
t_stat, p_val = stats.ttest_ind(t_gains, c_gains)
print ("=== A/B Test: Treatment vs. Control ===" )
print (f"Mean gain (treatment): {t_gains.mean():.2 f} " )
print (f"Mean gain (control): {c_gains.mean():.2 f} " )
print (f"Difference: {t_gains.mean() - c_gains.mean():.2 f} " )
print (f"t-statistic: {t_stat:.3 f} , p-value: {p_val:.4 f} " )
pooled_sd = np.sqrt((t_gains.var() + c_gains.var()) / 2 )
cohens_d = (t_gains.mean() - c_gains.mean()) / pooled_sd
j = 1 - 3 / (4 * (n - 2 ) - 1 )
hedges_g = cohens_d * j
print (f"Cohen's d: {cohens_d:.3 f} " )
print (f"Hedges' g: {hedges_g:.3 f} " )
X_ancova = sm.add_constant(df[["treat" , "pre_test" ]])
ancova = sm.OLS(df["post_test" ], X_ancova).fit(cov_type="HC3" )
print ("\n=== ANCOVA (post ~ treat + pre) ===" )
print (ancova.summary().tables[1 ])
print (f"\nANCOVA treatment effect: {ancova.params['treat' ]:.3 f} "
f"(SE={ancova.bse['treat' ]:.3 f} )" )
from scipy.stats import norm
alpha_level = 0.05
observed_d = cohens_d
se_d = np.sqrt(2 /n + observed_d**2 / (2 *n))
power_obs = 1 - stats.norm.cdf(stats.norm.ppf(1 - alpha_level/2 ) - observed_d / se_d)
print (f"\nPost-hoc power: {power_obs:.3 f} " )
df["baseline_group" ] = pd.cut(df["pre_test" ], bins=[0 , 50 , 100 ],
labels=["Low" , "High" ])
subgroup = df.groupby(["baseline_group" , "treat" ])["gain" ].agg(["mean" , "std" , "count" ])
print ("\n=== Subgroup Analysis (by baseline) ===" )
print (subgroup.round (2 ))
fig, axes = plt.subplots(1 , 3 , figsize=(15 , 5 ))
colors = {1 : "steelblue" , 0 : "orange" }
labels = {1 : "Treatment (n=100)" , 0 : "Control (n=100)" }
for t_val in [0 , 1 ]:
mask = df["treat" ] == t_val
axes[0 ].scatter(df.loc[mask, "pre_test" ], df.loc[mask, "post_test" ],
color=colors[t_val], alpha=0.5 , s=20 , label=labels[t_val])
axes[0 ].plot([0 , 100 ], [0 , 100 ], "k--" , lw=0.8 , label="No change" )
axes[0 ].set_xlabel("Pre-test Score" ); axes[0 ].set_ylabel("Post-test Score" )
axes[0 ].set_title("Pre-Post Scores by Treatment" )
axes[0 ].legend(fontsize=8 )
axes[1 ].hist(c_gains, bins=20 , alpha=0.6 , color="orange" , label="Control" , density=True )
axes[1 ].hist(t_gains, bins=20 , alpha=0.6 , color="steelblue" , label="Treatment" , density=True )
axes[1 ].axvline(c_gains.mean(), color="orange" , ls="--" , lw=2 )
axes[1 ].axvline(t_gains.mean(), color="steelblue" , ls="--" , lw=2 )
axes[1 ].set_xlabel("Score Gain" ); axes[1 ].set_ylabel("Density" )
axes[1 ].set_title(f"Distribution of Gains\n(d={cohens_d:.2 f} )" )
axes[1 ].legend()
axes[2 ].scatter(ancova.fittedvalues, df["post_test" ], alpha=0.4 , s=15 ,
c=df["treat" ].map ({0 : "orange" , 1 : "steelblue" }))
axes[2 ].plot([20 , 100 ], [20 , 100 ], "k--" )
axes[2 ].set_xlabel("ANCOVA Fitted" ); axes[2 ].set_ylabel("Observed" )
axes[2 ].set_title(f"ANCOVA Fit (R²={ancova.rsquared:.3 f} )" )
plt.tight_layout()
plt.savefig("educational_ab_test.png" , dpi=150 , bbox_inches="tight" )
plt.close()
print ("\nFigure saved: educational_ab_test.png" )
Advanced Usage
Bayesian Knowledge Tracing (BKT) import numpy as np
from scipy.optimize import minimize
def bkt_forward (responses, p_L0, p_T, p_S, p_G ):
"""Forward pass of BKT (single skill, single student).
Args:
responses: binary list of responses (1=correct, 0=incorrect)
p_L0: prior probability of knowing skill at trial 1
p_T: probability of transitioning to learned state
p_S: slip probability (knows but answers wrong)
p_G: guess probability (doesn't know but answers correctly)
Returns:
p_Ln: posterior P(learned) at each trial
log_likelihood
"""
p_Ln = np.zeros(len (responses) + 1 )
p_Ln[0 ] = p_L0
log_lik = 0.0
for n, r in enumerate (responses):
p_L = p_Ln[n]
p_correct = p_L * (1 - p_S) + (1 - p_L) * p_G
log_lik += np.log(p_correct + 1e-10 ) if r == 1 else np.log(1 - p_correct + 1e-10 )
if r == 1 :
p_L_given_r = p_L * (1 - p_S) / (p_correct + 1e-10 )
else :
p_L_given_r = p_L * p_S / (1 - p_correct + 1e-10 )
p_Ln[n + 1 ] = p_L_given_r + (1 - p_L_given_r) * p_T
return p_Ln[1 :], log_lik
np.random.seed(42 )
true_params = {"p_L0" : 0.2 , "p_T" : 0.15 , "p_S" : 0.08 , "p_G" : 0.20 }
n_trials = 25
p_L_true = np.zeros(n_trials + 1 )
p_L_true[0 ] = true_params["p_L0" ]
for i in range (n_trials):
if np.random.random() > p_L_true[i]:
p_L_true[i+1 ] = p_L_true[i] + (1 - p_L_true[i]) * true_params["p_T" ]
else :
p_L_true[i+1 ] = p_L_true[i]
learned = p_L_true[1 :] > 0.5
sim_responses = []
for is_learned in learned:
if is_learned:
sim_responses.append(1 if np.random.random() > true_params["p_S" ] else 0 )
else :
sim_responses.append(1 if np.random.random() < true_params["p_G" ] else 0 )
def neg_ll (params ):
p_L0, p_T, p_S, p_G = params
if not all (0.01 < p < 0.99 for p in params):
return 1e10
_, ll = bkt_forward(sim_responses, p_L0, p_T, p_S, p_G)
return -ll
result = minimize(neg_ll, x0=[0.3 , 0.2 , 0.1 , 0.25 ], method="Nelder-Mead" )
p_L0_f, p_T_f, p_S_f, p_G_f = result.x
print (f"BKT MLE: L0={p_L0_f:.3 f} , T={p_T_f:.3 f} , S={p_S_f:.3 f} , G={p_G_f:.3 f} " )
p_mastery, _ = bkt_forward(sim_responses, p_L0_f, p_T_f, p_S_f, p_G_f)
print (f"Final mastery probability: {p_mastery[-1 ]:.3 f} " )
print (f"Trial when mastery > 0.95: "
f"{np.argmax(p_mastery > 0.95 ) + 1 if (p_mastery > 0.95 ).any () else 'Not reached' } " )
Differential Item Functioning (DIF) Detection import numpy as np
import pandas as pd
from scipy.stats import chi2_contingency
def mantel_haenszel_dif (responses, group, ability_strata=5 ):
"""Mantel-Haenszel DIF statistic for each item.
Args:
responses: (n_students, n_items) binary matrix
group: (n_students,) binary array — 0=reference, 1=focal
ability_strata: number of ability level strata
Returns:
DataFrame with item DIF statistics
"""
n_items = responses.shape[1 ]
total_score = responses.sum (axis=1 )
strata = pd.qcut(total_score, ability_strata, labels=False , duplicates="drop" )
results = []
for item in range (n_items):
A = B = C = D = 0
for s in range (ability_strata):
mask_s = strata == s
if mask_s.sum () < 4 :
continue
ref_mask = mask_s & (group == 0 )
foc_mask = mask_s & (group == 1 )
n_s = mask_s.sum ()
a = (responses[foc_mask, item]).sum ()
b = (responses[ref_mask, item]).sum ()
c = foc_mask.sum () - a
d = ref_mask.sum () - b
A += a * d / n_s
B += b * c / n_s
C += (a + d) * (a + c) * (b + c) * (b + d) / (n_s**2 * (n_s - 1 ) + 1e-10 )
alpha_mh = A / (B + 1e-10 )
delta_mh = -2.35 * np.log(alpha_mh + 1e-10 )
results.append({"item" : item, "MH_alpha" : alpha_mh, "delta_MH" : delta_mh})
return pd.DataFrame(results)
np.random.seed(42 )
n_s = 400
n_i = 15
theta = np.random.normal(0 , 1 , n_s)
group = np.random.binomial(1 , 0.4 , n_s)
theta[group == 1 ] -= 0.5
a_dif = np.ones(n_i)
b_dif = np.linspace(-2 , 2 , n_i)
b_dif[5 ] += 0.8
P_dif = 1 / (1 + np.exp(-a_dif[None , :] * (theta[:, None ] - b_dif[None , :])))
resp_dif = (np.random.uniform(size=P_dif.shape) < P_dif).astype(int )
dif_results = mantel_haenszel_dif(resp_dif, group)
print ("=== DIF Analysis (Mantel-Haenszel) ===" )
print (dif_results.round (3 ).to_string())
flagged = dif_results[dif_results["delta_MH" ].abs () > 1.0 ]
print (f"\nFlagged items (|ΔMH| > 1.0): {flagged['item' ].tolist()} " )
Troubleshooting Problem Cause Fix IRT estimation diverges Extreme item p-values (< 0.05 or > 0.95) Remove extreme items; increase sample Power law fit R² < 0.80 Plateau or ceiling effects Check for ceiling; use exponential model BKT mastery oscillates Slip > 0.5 or guess > 0.5 Constrain params: p_S < 0.4, p_G < 0.4 Negative Cohen's d Score direction Verify higher score = better; check treatment coding ANCOVA violation Non-parallel slopes Test treatment × pre interaction; use moderated regression Negative DIF delta Focal group advantaged Expected; classify DIF direction (A favors reference, B favors focal)
External Resources
Lord, F. M. (1980). Applications of Item Response Theory . Educational Testing Service.
Embretson, S. E., & Reise, S. P. (2000). Item Response Theory for Psychologists . Erlbaum.
Anderson, J. R., et al. (1995). Cognitive tutors: Lessons learned. JLS , 4(2).
mirt R package documentation EdPsych Measurement MOOC materials
Examples
Example 1: Test Reliability Analysis import numpy as np
import pandas as pd
def cronbach_alpha (responses ):
"""Compute Cronbach's alpha reliability coefficient.
Args:
responses: (n_students, n_items) matrix of item scores
Returns:
alpha coefficient
"""
n_items = responses.shape[1 ]
item_variances = responses.var(axis=0 , ddof=1 )
total_variance = responses.sum (axis=1 ).var(ddof=1 )
alpha = (n_items / (n_items - 1 )) * (1 - item_variances.sum () / total_variance)
return alpha
def item_statistics (responses ):
"""Compute classical test theory item statistics."""
total = responses.sum (axis=1 )
stats = []
for j in range (responses.shape[1 ]):
p_val = responses[:, j].mean()
corrected_total = total - responses[:, j]
r_it = np.corrcoef(responses[:, j], corrected_total)[0 , 1 ]
stats.append({"item" : j+1 , "p_value" : p_val, "r_item_total" : r_it})
return pd.DataFrame(stats)
np.random.seed(42 )
n_s, n_i = 300 , 25
theta_r = np.random.normal(0 , 1 , n_s)
b_r = np.random.normal(0 , 1 , n_i)
P_r = 1 / (1 + np.exp(-(theta_r[:, None ] - b_r[None , :])))
resp_r = (np.random.uniform(size=P_r.shape) < P_r).astype(int )
alpha = cronbach_alpha(resp_r)
print (f"Cronbach's α = {alpha:.3 f} " )
if alpha >= 0.90 : print ("Excellent reliability" )
elif alpha >= 0.80 : print ("Good reliability" )
elif alpha >= 0.70 : print ("Acceptable reliability" )
else : print ("Poor reliability — revise items" )
item_stats = item_statistics(resp_r)
print ("\nItem Statistics (first 5):" )
print (item_stats.head().round (3 ))
poor_items = item_stats[item_stats["r_item_total" ] < 0.20 ]
print (f"\nPoor discriminating items (r < 0.20): {poor_items['item' ].tolist()} " )
Example 2: Spaced Repetition Optimization import numpy as np
import matplotlib.pyplot as plt
def sm2_schedule (initial_easiness=2.5 , n_reviews=10 , min_ease=1.3 ):
"""Simulate SM-2 spaced repetition algorithm schedule.
Args:
initial_easiness: starting E-factor (2.5 by default)
n_reviews: number of review sessions
min_ease: minimum E-factor
Returns:
DataFrame with intervals and retention estimates
"""
import pandas as pd
records = []
interval = 1
easiness = initial_easiness
stability = 1.0
for rev in range (1 , n_reviews + 1 ):
q = 4
easiness = max (min_ease, easiness + 0.1 - (5 - q) * (0.08 + (5 - q) * 0.02 ))
if rev == 1 :
next_interval = 1
elif rev == 2 :
next_interval = 6
else :
next_interval = round (interval * easiness)
stability_at_review = stability * easiness
retention = np.exp(-interval / (stability_at_review * 10 ))
records.append({"review" : rev, "interval_days" : interval,
"easiness" : easiness, "retention" : retention})
stability = stability_at_review
interval = next_interval
return pd.DataFrame(records)
schedule = sm2_schedule()
print ("SM-2 Spaced Repetition Schedule:" )
print (schedule.to_string(index=False ))
print (f"\nTotal study sessions to review 10 times: {schedule['interval_days' ].sum ()} days" )
