Scale Analysis

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, McDonald’s Omega)
`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), McDonald's Omega = 0.047, N = 2135
#> Use summary() for detailed output.

Detailed Output

rel <- reliability(survey_data, trust_government, trust_media, trust_science)
summary(rel)
#> 
#> Reliability Analysis Results
#> ----------------------------
#> - Items:
#>     trust_government  Trust in government (1=none, 5=complete)
#>     trust_media       Trust in media (1=none, 5=complete)
#>     trust_science     Trust in science (1=none, 5=complete)
#> - N of Items: 3
#> 
#> Reliability Statistics
#> ---------------------------------------- 
#>   Cronbach's Alpha:              0.047
#>   Alpha (standardized):          0.048
#>   McDonald's Omega:              0.047
#>   Omega (standardized):          0.048
#>   N of Items:                    3
#>   N (listwise):                  2135
#> 
#> Item Statistics
#>   ---------------------------------------------------------------------------------------
#>   Item                                                         Mean  Std. Deviation     N
#>   ---------------------------------------------------------------------------------------
#>   trust_government  Trust in government (1=none, 5=complete)  2.621           1.162  2135
#>   trust_media       Trust in media (1=none, 5=complete)       2.430           1.156  2135
#>   trust_science     Trust in science (1=none, 5=complete)     3.624           1.034  2135
#>   ---------------------------------------------------------------------------------------
#> 
#> Inter-Item Correlation Matrix
#>   -----------------------------------------
#>                           (1)    (2)    (3)
#>   -----------------------------------------
#>   (1) trust_government  1.000  0.014  0.020
#>   (2) trust_media       0.014  1.000  0.015
#>   (3) trust_science     0.020  0.015  1.000
#>   -----------------------------------------
#> 
#> Item-Total Statistics
#>   ------------------------------------------------------------------------
#>                     Scale Mean  Scale Var.   Corrected  Alpha if  Omega if
#>   Item              if Deleted  if Deleted  Item-Total   Deleted   Deleted
#>   ------------------------------------------------------------------------
#>   trust_government       6.054       2.440       0.024     0.029          
#>   trust_media            6.245       2.467       0.020     0.040          
#>   trust_science          5.051       2.723       0.025     0.027          
#>   ------------------------------------------------------------------------
#> Note: Omega if item deleted requires at least 4 items
#> (a one-factor model on the remaining 2 items is not identified).

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.

McDonald’s Omega

Alongside alpha, reliability() reports McDonald’s Omega, a factor-model-based reliability coefficient. Alpha assumes all items measure the construct equally well; omega fits a one-factor model and lets each item carry its own loading, which usually makes it the more accurate estimate when item loadings differ. The same thresholds as for alpha are commonly applied. Omega needs at least 3 items (with 2 items it is reported as NA), and Omega if Item Deleted appears in the item-total table for scales of 4 or more items.

Note: omega is currently an R-only statistic (no SPSS reference run yet); see vignette("spss-compatibility") for its validation status.

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), McDonald's Omega = 0.053, N = 2150
#> Use summary() for detailed output.

Using tidyselect

reliability(survey_data, starts_with("trust"))
#> Reliability Analysis: 3 items
#>   Cronbach's Alpha = 0.047 (Poor), McDonald's Omega = 0.047, N = 2135
#> Use summary() for detailed output.

Grouped Analysis

Check whether reliability holds across subgroups:

survey_data %>%
  group_by(region) %>%
  reliability(trust_government, trust_media, trust_science)
#> [region = East]
#> Reliability Analysis: 3 items
#>   Cronbach's Alpha = 0.037 (Poor), McDonald's Omega = not computed, N = 422
#> [region = West]
#> Reliability Analysis: 3 items
#>   Cronbach's Alpha = 0.050 (Poor), McDonald's Omega = 0.071, N = 1713
#> Use summary() for detailed output.

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%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.

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  Political orientation (1=left, 5=right)
#>     environmental_concern  Environmental concern (1=low, 5=high)
#>     life_satisfaction      Life satisfaction (1=dissatisfied, 5=satisfied)
#>     trust_government       Trust in government (1=none, 5=complete)
#>     trust_media            Trust in media (1=none, 5=complete)
#>     trust_science          Trust in science (1=none, 5=complete)
#> - Extraction: Principal Component Analysis
#> - Rotation: Varimax with Kaiser Normalization
#> - N of Components: 3
#> - N (smallest pairwise): 2168
#> 
#> Descriptive Statistics
#>   --------------------------------------------------------------------------------------------------------------------
#>   Variable                                                                 Mean  Std. Deviation  Analysis N  Missing N
#>   --------------------------------------------------------------------------------------------------------------------
#>   political_orientation  Political orientation (1=left, 5=right)          2.722           1.086        2299        201
#>   environmental_concern  Environmental concern (1=low, 5=high)            3.573           1.194        2400        100
#>   life_satisfaction      Life satisfaction (1=dissatisfied, 5=satisfied)  3.628           1.153        2421         79
#>   trust_government       Trust in government (1=none, 5=complete)         2.621           1.163        2354        146
#>   trust_media            Trust in media (1=none, 5=complete)              2.452           1.163        2367        133
#>   trust_science          Trust in science (1=none, 5=complete)            3.641           1.028        2398        102
#>   --------------------------------------------------------------------------------------------------------------------
#> 
#> KMO and Bartlett's Test
#> ---------------------------------------- 
#>   Kaiser-Meyer-Olkin Measure:     0.505
#>   Bartlett's Chi-Square:          932.068
#>   df:                             15
#>   Sig.:                           <.001
#> 
#> Communalities
#>   -------------------------------------------------------------------------------------------
#>   Variable                                                                Initial  Extraction
#>   -------------------------------------------------------------------------------------------
#>   political_orientation  Political orientation (1=left, 5=right)            1.000       0.786
#>   environmental_concern  Environmental concern (1=low, 5=high)              1.000       0.783
#>   life_satisfaction      Life satisfaction (1=dissatisfied, 5=satisfied)    1.000       0.668
#>   trust_government       Trust in government (1=none, 5=complete)           1.000       0.347
#>   trust_media            Trust in media (1=none, 5=complete)                1.000       0.475
#>   trust_science          Trust in science (1=none, 5=complete)              1.000       0.598
#>   -------------------------------------------------------------------------------------------
#> Extraction Method: Principal Component Analysis.
#> 
#> Total Variance Explained
#>   -------------------------------------------------------------------------
#>              Initial Eigenvalues   Extraction Sums      Rotation Sums
#>   Component  Total % Var.  Cum. %  Total % Var. Cum. %  Total % Var. Cum. %
#>   -------------------------------------------------------------------------
#>           1  1.600 26.666  26.666  1.600 26.666 26.666  1.598 26.635 26.635
#>           2  1.041 17.358  44.024  1.041 17.358 44.024  1.039 17.324 43.959
#>           3  1.017 16.955  60.979  1.017 16.955 60.979  1.021 17.020 60.979
#>           4  0.980 16.334  77.313
#>           5  0.949 15.814  93.127
#>           6  0.412  6.873 100.000
#>   -------------------------------------------------------------------------
#> Sums = sums of squared loadings.
#> Extraction Method: Principal Component Analysis.
#> 
#> Component Matrix (unrotated)
#> ----------------------------------------
#>                                                                            PC1
#> political_orientation  Political orientation (1=left, 5=right)          -0.885
#> environmental_concern  Environmental concern (1=low, 5=high)             0.885
#> trust_science          Trust in science (1=none, 5=complete)                  
#> trust_government       Trust in government (1=none, 5=complete)               
#> trust_media            Trust in media (1=none, 5=complete)                    
#> life_satisfaction      Life satisfaction (1=dissatisfied, 5=satisfied)        
#>                                                                            PC2
#> political_orientation  Political orientation (1=left, 5=right)                
#> environmental_concern  Environmental concern (1=low, 5=high)                  
#> trust_science          Trust in science (1=none, 5=complete)             0.672
#> trust_government       Trust in government (1=none, 5=complete)          0.547
#> trust_media            Trust in media (1=none, 5=complete)               0.524
#> life_satisfaction      Life satisfaction (1=dissatisfied, 5=satisfied)        
#>                                                                            PC3
#> political_orientation  Political orientation (1=left, 5=right)                
#> environmental_concern  Environmental concern (1=low, 5=high)                  
#> trust_science          Trust in science (1=none, 5=complete)                  
#> trust_government       Trust in government (1=none, 5=complete)               
#> trust_media            Trust in media (1=none, 5=complete)               0.448
#> life_satisfaction      Life satisfaction (1=dissatisfied, 5=satisfied)   0.809
#> Extraction Method: Principal Component Analysis.
#> 
#> Rotated Component Matrix
#> ----------------------------------------
#>                                                                            PC1
#> political_orientation  Political orientation (1=left, 5=right)          -0.887
#> environmental_concern  Environmental concern (1=low, 5=high)             0.884
#> trust_science          Trust in science (1=none, 5=complete)                  
#> trust_government       Trust in government (1=none, 5=complete)               
#> life_satisfaction      Life satisfaction (1=dissatisfied, 5=satisfied)        
#> trust_media            Trust in media (1=none, 5=complete)                    
#>                                                                            PC2
#> political_orientation  Political orientation (1=left, 5=right)                
#> environmental_concern  Environmental concern (1=low, 5=high)                  
#> trust_science          Trust in science (1=none, 5=complete)             0.762
#> trust_government       Trust in government (1=none, 5=complete)          0.566
#> life_satisfaction      Life satisfaction (1=dissatisfied, 5=satisfied)        
#> trust_media            Trust in media (1=none, 5=complete)                    
#>                                                                            PC3
#> political_orientation  Political orientation (1=left, 5=right)                
#> environmental_concern  Environmental concern (1=low, 5=high)                  
#> trust_science          Trust in science (1=none, 5=complete)                  
#> trust_government       Trust in government (1=none, 5=complete)               
#> life_satisfaction      Life satisfaction (1=dissatisfied, 5=satisfied)   0.789
#> trust_media            Trust in media (1=none, 5=complete)               0.620
#> Extraction Method: Principal Component Analysis.
#> Rotation Method: Varimax with Kaiser Normalization.
#> Rotation converged in 4 iterations.

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%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.

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%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.

Oblimin — another oblique rotation, common in psychology:

efa(survey_data,
    political_orientation, environmental_concern, life_satisfaction,
    trust_government, trust_media, trust_science,
    rotation = "oblimin")
#> Exploratory Factor Analysis: 6 items, 3 components (PCA/Oblimin)
#>   KMO = 0.505 (Miserable), Variance explained: 61.0%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.

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 factors (ML/Varimax)
#>   KMO = 0.505 (Miserable), Variance explained: 24.1%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.

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 factors (ML/Promax)
#>   KMO = 0.505 (Miserable), Variance explained: 24.1%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.

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%, N = 2168 (smallest pairwise)
#> Use summary() for detailed output.

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%, N = 2182 (smallest pairwise)
#> Use summary() for detailed output.

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   3.000   0.691    4.000   1.000     0.020  2484       16
#>   trust_pomps  47.892  50.000  17.283  100.000  25.000     0.020  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(2457.6) = -2.362, p = 0.018 *, g = -0.095 (negligible), N = 2499
#> Use summary() for detailed output.

# 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) = 177.02, p < 0.001 ***, N = 2113
#> Use summary() for detailed output.

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), McDonald's Omega = 0.047, N = 2135
#> Use summary() for detailed output.
summary(rel)
#> 
#> Reliability Analysis Results
#> ----------------------------
#> - Items:
#>     trust_government  Trust in government (1=none, 5=complete)
#>     trust_media       Trust in media (1=none, 5=complete)
#>     trust_science     Trust in science (1=none, 5=complete)
#> - N of Items: 3
#> 
#> Reliability Statistics
#> ---------------------------------------- 
#>   Cronbach's Alpha:              0.047
#>   Alpha (standardized):          0.048
#>   McDonald's Omega:              0.047
#>   Omega (standardized):          0.048
#>   N of Items:                    3
#>   N (listwise):                  2135
#> 
#> Item Statistics
#>   ---------------------------------------------------------------------------------------
#>   Item                                                         Mean  Std. Deviation     N
#>   ---------------------------------------------------------------------------------------
#>   trust_government  Trust in government (1=none, 5=complete)  2.621           1.162  2135
#>   trust_media       Trust in media (1=none, 5=complete)       2.430           1.156  2135
#>   trust_science     Trust in science (1=none, 5=complete)     3.624           1.034  2135
#>   ---------------------------------------------------------------------------------------
#> 
#> Inter-Item Correlation Matrix
#>   -----------------------------------------
#>                           (1)    (2)    (3)
#>   -----------------------------------------
#>   (1) trust_government  1.000  0.014  0.020
#>   (2) trust_media       0.014  1.000  0.015
#>   (3) trust_science     0.020  0.015  1.000
#>   -----------------------------------------
#> 
#> Item-Total Statistics
#>   ------------------------------------------------------------------------
#>                     Scale Mean  Scale Var.   Corrected  Alpha if  Omega if
#>   Item              if Deleted  if Deleted  Item-Total   Deleted   Deleted
#>   ------------------------------------------------------------------------
#>   trust_government       6.054       2.440       0.024     0.029          
#>   trust_media            6.245       2.467       0.020     0.040          
#>   trust_science          5.051       2.723       0.025     0.027          
#>   ------------------------------------------------------------------------
#> Note: Omega if item deleted requires at least 4 items
#> (a one-factor model on the remaining 2 items is not identified).

# 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%, N = 2227 (smallest pairwise)
#>   Only one component was extracted. The solution cannot be rotated.
#> Use summary() for detailed output.

# 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    N  Missing
#>   ----------------------------------------------------------------------------
#>   m_trust       2.924   3.000   0.696    4.000   1.000     0.024  845        3
#>   trust_pomps  48.093  50.000  17.390  100.000  25.000     0.024  845        3
#>   ----------------------------------------------------------------------------
#> 
#> Group: education = Intermediate Secondary
#> -----------------------------------------
#> 
#>   ----------------------------------------------------------------------------
#>   Variable       Mean  Median      SD    Range     IQR  Skewness    N  Missing
#>   ----------------------------------------------------------------------------
#>   m_trust       2.920   3.000   0.697    4.000   1.000     0.035  636        5
#>   trust_pomps  48.002  50.000  17.421  100.000  25.000     0.035  636        5
#>   ----------------------------------------------------------------------------
#> 
#> Group: education = Academic Secondary
#> -------------------------------------
#> 
#>   ----------------------------------------------------------------------------
#>   Variable       Mean  Median      SD    Range     IQR  Skewness    N  Missing
#>   ----------------------------------------------------------------------------
#>   m_trust       2.922   3.000   0.691    4.000   0.833    -0.006  634        8
#>   trust_pomps  48.053  50.000  17.267  100.000  20.833    -0.006  634        8
#>   ----------------------------------------------------------------------------
#> 
#> Group: education = University
#> -----------------------------
#> 
#>   ----------------------------------------------------------------------------
#>   Variable       Mean  Median      SD    Range     IQR  Skewness    N  Missing
#>   ----------------------------------------------------------------------------
#>   m_trust       2.885   3.000   0.686    4.000   1.000     0.063  385        0
#>   trust_pomps  47.132  50.000  17.157  100.000  25.000     0.063  385        0
#>   ----------------------------------------------------------------------------

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")