Scale Analysis • mariposa

library(mariposa)
library(dplyr)
data(survey_data)

Overview

Scale analysis combines multiple survey items into a single score that reliably measures a concept. The typical workflow is:

Check reliability with reliability() — do the items measure the same thing?
Explore structure with efa() — how do items group into dimensions?
Create scores with row_means() — compute mean indices
Standardize with pomps() — transform to a comparable 0–100 scale

This guide uses the trust items from survey_data (trust in government, media, and science).

Function	Purpose
`reliability()`	Internal consistency (Cronbach’s Alpha)
`efa()`	Discover underlying dimensions (factor analysis)
`row_means()`	Row-wise mean indices
`row_sums()`	Row-wise sums
`row_count()`	Count specific values per row
`pomps()`	Percent of Maximum Possible Scores (0–100)

Reliability Analysis

Basic Usage

reliability(survey_data, trust_government, trust_media, trust_science)
#> Reliability Analysis: 3 items
#>   Cronbach's Alpha = 0.047 (Poor), N = 2135

Detailed Output

rel <- reliability(survey_data, trust_government, trust_media, trust_science)
summary(rel)
#> 
#> Reliability Analysis Results
#> ----------------------------
#> - Items: trust_government, trust_media, trust_science
#> - N of Items: 3
#> 
#> Reliability Statistics
#> ---------------------------------------- 
#>   Cronbach's Alpha:              0.047
#>   Alpha (standardized):          0.048
#>   N of Items:                    3
#>   N (listwise):                  2135
#> 
#> Item Statistics
#> ---------------------------------------- 
#>              item  mean    sd    n
#>  trust_government 2.621 1.162 2135
#>       trust_media 2.430 1.156 2135
#>     trust_science 3.624 1.034 2135
#> 
#> Inter-Item Correlation Matrix:
#> ------------------------------ 
#>                  trust_government trust_media trust_science
#> trust_government            1.000       0.014         0.020
#> trust_media                 0.014       1.000         0.015
#> trust_science               0.020       0.015         1.000
#> ------------------------------ 
#> 
#> Item-Total Statistics
#> ---------------------------------------- 
#>              item scale_mean_deleted scale_var_deleted corrected_r
#>  trust_government               6.05             2.440       0.024
#>       trust_media               6.25             2.467       0.020
#>     trust_science               5.05             2.723       0.025
#>  alpha_deleted
#>          0.029
#>          0.040
#>          0.027

The detailed output shows item-total correlations, alpha-if-item-deleted, and inter-item correlations — matching SPSS RELIABILITY.

Interpreting Cronbach’s Alpha

Alpha > 0.90: Excellent
0.80 – 0.90: Good
0.70 – 0.80: Acceptable
0.60 – 0.70: Questionable
Below 0.60: Reconsider your items

Item-Total Correlation shows how well each item fits the scale. Values above 0.40 indicate good fit; below 0.20 suggests the item does not belong.

Alpha if Item Deleted shows what happens without each item. If alpha increases when you remove an item, that item weakens the scale.

With Survey Weights

reliability(survey_data, trust_government, trust_media, trust_science,
            weights = sampling_weight)
#> Reliability Analysis: 3 items [Weighted]
#>   Cronbach's Alpha = 0.052 (Poor), N = 2150

Using tidyselect

reliability(survey_data, starts_with("trust"))
#> Reliability Analysis: 3 items
#>   Cronbach's Alpha = 0.047 (Poor), N = 2135

Grouped Analysis

Check whether reliability holds across subgroups:

survey_data %>%
  group_by(region) %>%
  reliability(trust_government, trust_media, trust_science)
#> [region = 1]
#> Reliability Analysis: 3 items
#>   Cronbach's Alpha = 0.037 (Poor), N = 422
#> [region = 2]
#> Reliability Analysis: 3 items
#>   Cronbach's Alpha = 0.050 (Poor), N = 1713

A scale that works well overall might be unreliable in specific subgroups. Always check when your sample spans diverse populations.

Exploratory Factor Analysis

Basic Usage

When you have many items, efa() reveals how they group into underlying dimensions:

efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science)
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Varimax)
#>   KMO = 0.505 (Miserable), Variance explained: 61.0%

Detailed Output

efa_result <- efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science)

summary(efa_result)
#> 
#> Exploratory Factor Analysis (PCA, Varimax) Results
#> --------------------------------------------------
#> - Variables: political_orientation, environmental_concern, life_satisfaction, trust_government, trust_media, trust_science
#> - Extraction: Principal Component Analysis
#> - Rotation: Varimax with Kaiser Normalization
#> - N of Factors: 3
#> 
#> KMO and Bartlett's Test
#> ---------------------------------------- 
#>   Kaiser-Meyer-Olkin Measure:     0.505
#>   Bartlett's Chi-Square:          932.068
#>   df:                             15
#>   Sig.:                           0.000
#> 
#> Communalities
#> ---------------------------------------- 
#>               variable initial extraction
#>  political_orientation       1      0.786
#>  environmental_concern       1      0.783
#>      life_satisfaction       1      0.668
#>       trust_government       1      0.347
#>            trust_media       1      0.475
#>          trust_science       1      0.598
#> Extraction Method: Principal Component Analysis.
#> 
#> Total Variance Explained
#> ---------------------------------------- 
#>   PC1  Eigenvalue: 1.600  Variance: 26.666%  Cumulative: 26.666%
#>   PC2  Eigenvalue: 1.041  Variance: 17.358%  Cumulative: 44.024%
#>   PC3  Eigenvalue: 1.017  Variance: 16.955%  Cumulative: 60.979%
#>   PC4  Eigenvalue: 0.980  Variance: 16.334%  Cumulative: 77.313%
#>   PC5  Eigenvalue: 0.949  Variance: 15.814%  Cumulative: 93.127%
#>   PC6  Eigenvalue: 0.412  Variance: 6.873%  Cumulative: 100.000%
#> 
#> Rotation Sums of Squared Loadings
#> ---------------------------------------- 
#>   PC1  SS Loading: 1.598  Variance: 26.634%  Cumulative: 26.634%
#>   PC2  SS Loading: 1.039  Variance: 17.325%  Cumulative: 43.959%
#>   PC3  SS Loading: 1.021  Variance: 17.020%  Cumulative: 60.979%
#> 
#> Component Matrix (unrotated)
#> ---------------------------------------- 
#>                           PC1     PC2     PC3
#> political_orientation   0.885                
#> environmental_concern  -0.885                
#> trust_science                  -0.672        
#> trust_government               -0.547        
#> trust_media                    -0.524  -0.448
#> life_satisfaction                      -0.809
#> Extraction Method: Principal Component Analysis.
#> 
#> Rotated Component Matrix
#> ---------------------------------------- 
#>                           PC1     PC2     PC3
#> political_orientation   0.887                
#> environmental_concern  -0.884                
#> trust_science                  -0.762        
#> trust_government               -0.566        
#> life_satisfaction                      -0.789
#> trust_media                            -0.620
#> Extraction Method: Principal Component Analysis.
#> Rotation Method: Varimax with Kaiser Normalization.

Understanding the Output

KMO (Kaiser-Meyer-Olkin) measures sampling adequacy for factor analysis:

Above 0.80: Good to excellent
0.60 – 0.80: Acceptable
Below 0.60: Factor analysis may not be appropriate

Bartlett’s Test should be significant ( $p < .05$ ), confirming that meaningful correlations exist.

Eigenvalues show variance explained per component. By default, components with eigenvalue > 1 are retained (Kaiser criterion).

Factor Loadings show item-component associations:

Above 0.70: Strong
0.40 – 0.70: Moderate
Below 0.40: Suppressed by default

Rotation Methods

Varimax (default) — assumes uncorrelated factors:

efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science,
    rotation = "varimax")
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Varimax)
#>   KMO = 0.505 (Miserable), Variance explained: 61.0%

Promax — allows correlated factors, produces Pattern and Structure matrices:

efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science,
    rotation = "promax")
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Promax)
#>   KMO = 0.505 (Miserable), Variance explained: 61.0%

Oblimin — another oblique rotation, common in psychology:

# Requires GPArotation package
efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science,
    rotation = "oblimin")

Extraction Methods

By default, efa() uses PCA (Principal Component Analysis). For a true factor analysis model, use Maximum Likelihood:

efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science,
    extraction = "ml")
#> Exploratory Factor Analysis: 6 items, 3 components (ML/Varimax)
#>   KMO = 0.505 (Miserable), Variance explained: 61.0%

ML extraction provides a goodness-of-fit test and uses SMC (squared multiple correlations) as initial communalities. Combine any extraction with any rotation:

efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science,
    extraction = "ml", rotation = "promax")
#> Exploratory Factor Analysis: 6 items, 3 components (ML/Promax)
#>   KMO = 0.505 (Miserable), Variance explained: 61.0%

Fixing the Number of Factors

efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science,
    n_factors = 2)
#> Exploratory Factor Analysis: 6 items, 2 components (PCA/Varimax)
#>   KMO = 0.505 (Miserable), Variance explained: 44.0%

With Survey Weights

efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science,
    weights = sampling_weight)
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Varimax) [Weighted]
#>   KMO = 0.505 (Miserable), Variance explained: 61.0%

Creating Scale Scores

After confirming reliability, create scores using the row operation functions. For details on row_means(), row_sums(), row_count(), and pomps(), see vignette("data-transformation").

Quick Scale Construction

# Create mean index
survey_data <- survey_data %>%
  mutate(m_trust = row_means(., trust_government, trust_media, trust_science,
                             min_valid = 2))

# Transform to 0-100 scale
survey_data <- survey_data %>%
  mutate(trust_pomps = pomps(m_trust, scale_min = 1, scale_max = 5))

# Check the result
survey_data %>%
  describe(m_trust, trust_pomps)
#> 
#> Descriptive Statistics
#> ----------------------
#>     Variable   Mean Median     SD Range IQR Skewness    N Missing
#>      m_trust  2.916      3  0.691     4   1     0.02 2484      16
#>  trust_pomps 47.892     50 17.283   100  25     0.02 2484      16
#> ----------------------------------------

Using the Scale in Analysis

# Group comparison
survey_data %>%
  t_test(m_trust, group = gender, weights = sampling_weight)
#> t-Test: m_trust by gender [Weighted]
#>   t(2457.2) = -2.362, p = 0.018 *, g = -0.095 (negligible), N = 2499

# As a predictor in regression
survey_data %>%
  linear_regression(life_satisfaction ~ m_trust + age + income,
                    weights = sampling_weight)
#> Linear Regression: life_satisfaction ~ m_trust + age + income [Weighted]
#>   R2 = 0.201, adj.R2 = 0.200, F(3, 2109) = 177.00, p < 0.001 ***, N = 2113

Complete Example

# 1. Check reliability
rel <- reliability(survey_data, trust_government, trust_media, trust_science)
rel
#> Reliability Analysis: 3 items
#>   Cronbach's Alpha = 0.047 (Poor), N = 2135
summary(rel)
#> 
#> Reliability Analysis Results
#> ----------------------------
#> - Items: trust_government, trust_media, trust_science
#> - N of Items: 3
#> 
#> Reliability Statistics
#> ---------------------------------------- 
#>   Cronbach's Alpha:              0.047
#>   Alpha (standardized):          0.048
#>   N of Items:                    3
#>   N (listwise):                  2135
#> 
#> Item Statistics
#> ---------------------------------------- 
#>              item  mean    sd    n
#>  trust_government 2.621 1.162 2135
#>       trust_media 2.430 1.156 2135
#>     trust_science 3.624 1.034 2135
#> 
#> Inter-Item Correlation Matrix:
#> ------------------------------ 
#>                  trust_government trust_media trust_science
#> trust_government            1.000       0.014         0.020
#> trust_media                 0.014       1.000         0.015
#> trust_science               0.020       0.015         1.000
#> ------------------------------ 
#> 
#> Item-Total Statistics
#> ---------------------------------------- 
#>              item scale_mean_deleted scale_var_deleted corrected_r
#>  trust_government               6.05             2.440       0.024
#>       trust_media               6.25             2.467       0.020
#>     trust_science               5.05             2.723       0.025
#>  alpha_deleted
#>          0.029
#>          0.040
#>          0.027

# 2. Explore factor structure
efa_result <- efa(survey_data, trust_government, trust_media, trust_science)
efa_result
#> Exploratory Factor Analysis: 3 items, 1 component (PCA/Unrotated)
#>   KMO = 0.506 (Miserable), Variance explained: 34.7%

# 3. Create mean index (Alpha was acceptable)
survey_data <- survey_data %>%
  mutate(m_trust = row_means(., trust_government, trust_media, trust_science,
                             min_valid = 2))

# 4. Transform to POMPS
survey_data <- survey_data %>%
  mutate(trust_pomps = pomps(m_trust, scale_min = 1, scale_max = 5))

# 5. Use in further analysis
survey_data %>%
  group_by(education) %>%
  describe(m_trust, trust_pomps, weights = sampling_weight)
#> 
#> Weighted Descriptive Statistics
#> -------------------------------
#> 
#> Group: education = Basic Secondary
#> ----------------------------------
#> ----------------------------------------
#>     Variable   Mean Median     SD Range IQR Skewness Effective_N
#>      m_trust  2.924      3  0.696     4   1    0.023       829.7
#>  trust_pomps 48.093     50 17.390   100  25    0.023       829.7
#> ----------------------------------------
#> 
#> Group: education = Intermediate Secondary
#> -----------------------------------------
#> ----------------------------------------
#>     Variable   Mean Median     SD Range IQR Skewness Effective_N
#>      m_trust  2.920      3  0.697     4   1    0.035       617.7
#>  trust_pomps 48.002     50 17.421   100  25    0.035       617.7
#> ----------------------------------------
#> 
#> Group: education = Academic Secondary
#> -------------------------------------
#> ----------------------------------------
#>     Variable   Mean Median     SD Range    IQR Skewness Effective_N
#>      m_trust  2.922      3  0.691     4  0.833   -0.006       616.4
#>  trust_pomps 48.053     50 17.267   100 20.833   -0.006       616.4
#> ----------------------------------------
#> 
#> Group: education = University
#> -----------------------------
#> ----------------------------------------
#>     Variable   Mean Median     SD Range IQR Skewness Effective_N
#>      m_trust  2.885      3  0.686     4   1    0.062       389.6
#>  trust_pomps 47.132     50 17.157   100  25    0.062       389.6
#> ----------------------------------------

Practical Tips

Always check reliability first. A mean index from unreliable items produces meaningless results. Aim for Alpha > .70.
Use min_valid wisely. For 3–5 item scales, min_valid = 2 is a reasonable compromise. For longer scales, require at least half the items.
Specify theoretical scale_min / scale_max in pomps(). Using observed values makes scores sample-dependent and non-comparable.
Run EFA when you have 6+ items. Factor analysis can reveal unexpected dimensions before you create indices.
Check grouped reliability. A scale that works well overall may be unreliable in specific subgroups (e.g., different regions or education levels).

Summary

reliability() checks whether items form a consistent scale (Cronbach’s Alpha)
efa() discovers underlying dimensions (PCA or ML extraction, Varimax/Oblimin/Promax rotation)
row_means() creates mean indices; row_sums() and row_count() provide alternatives
pomps() transforms scores to a comparable 0–100 scale
Always validate reliability before creating scale scores

Next Steps

Learn about row operations and transformations — see vignette("data-transformation")
Use scales in regression models — see vignette("regression-analysis")
Compare scale scores across groups — see vignette("hypothesis-testing")
Apply survey weights — see vignette("survey-weights")