Psychometrics

What I do

• Design and analyze psychological measurements
• Build and validate questionnaires
• Measure latent traits
• Assess reliability and validity
• Create scoring systems
• Handle survey data analysis

When to use me

Use me when:

• Building assessment tools
• Measuring user attitudes/personality
• Validating questionnaires
• Creating scoring models

Key Concepts

Psychometric Models

import numpy as np
from scipy import stats

# Classical Test Theory
def cronbach_alpha(items):
    """Calculate Cronbach's alpha for reliability"""
    item_scores = np.array(items)
    n_items = item_scores.shape[1]
    
    # Item variances
    item_vars = item_scores.var(axis=0, ddof=1)
    total_var = item_scores.sum(axis=1).var(ddof=1)
    
    alpha = (n_items / (n_items - 1)) * (1 - item_vars.sum() / total_var)
    return alpha

# Item Response Theory (Simplified)
# 1PL Model: P(correct) = 1 / (1 + exp(-(theta - b)))
def logistic_1pl(theta, difficulty):
    return 1 / (1 + np.exp(-(theta - difficulty)))

Reliability Measures

• Cronbach's Alpha: Internal consistency
• Test-retest: Stability over time
• Inter-rater: Agreement between raters
• Split-half: Correlation between halves

Validity Types

• Content: Measures intended construct
• Criterion: Correlates with outcome
• Construct: Theoretical underpinning
• Face: Appears valid to examinees

Factor Analysis

from sklearn.decomposition import FactorAnalysis

# Exploratory factor analysis
fa = FactorAnalysis(n_components=3)
factor_scores = fa.fit_transform(X)

# Interpretation
loadings = fa.components_.T
# High loadings indicate strong factor relationships

Item Analysis

• Difficulty index (p-value)
• Discrimination index
• Item-total correlation
• Response distribution