ancova() tests whether group means differ after controlling for one or
more continuous covariates. It performs a factorial ANCOVA using Type III Sum of
Squares, matching SPSS UNIANOVA output with the WITH keyword.
Think of it as:
An ANOVA that removes the effect of confounding variables first
Testing group differences on "adjusted" means
Combining regression (covariates) and ANOVA (factors) in one model
The test tells you:
Whether each factor has a significant effect AFTER controlling for covariates
The relationship between each covariate and the outcome
Effect sizes for factors and covariates (partial eta squared)
Estimated marginal means (group means adjusted for covariates)
Arguments
- data
Your survey data (a data frame or tibble)
- dv
The numeric dependent variable to analyze (unquoted)
- between
Character vector or unquoted variable names specifying the between-subjects factors (1-3 factors). These must be categorical variables (factor, character, or labelled numeric).
- covariate
Unquoted variable names of the continuous covariates to control for. Use
c(age, income)for multiple covariates.- weights
Optional survey weights for population-representative results. Give a column name (unquoted or as a string), an expression such as
sampling_weight * 2, or a numeric vector with one weight per row.- ss_type
Deprecated; only 3 (Type III, the SPSS default) is implemented. Passing 2 issues a warning and computes Type III.
Value
An object of class "ancova" containing:
- anova_table
Tibble with Source, SS, df, MS, F, p, Partial Eta Squared
- parameter_estimates
Tibble with regression coefficients (B, SE, t, p, CI, partial eta squared) in SPSS coding: one row per category, the last category of each factor is the reference and reported as a redundant 0 (
redundant = TRUE)- descriptives
Tibble with unadjusted cell means, SDs, and Ns
- estimated_marginal_means
Tibble with adjusted cell means (covariates at grand mean)
- emm_main_effects
For 2+ factors: named list of tibbles with the adjusted main-effect means (unweighted average of the cell means, as SPSS
/EMMEANS=TABLES(factor)); NULL for one factor- levene_test
Tibble with Levene's test of equality of error variances (f, df1, df2, p): as in SPSS UNIANOVA, a one-way ANOVA of the absolute residuals of the ANCOVA model across the cells of the design. Also available as
levene_test(result).- r_squared
R-squared and Adjusted R-squared
- model
The underlying lm model object
- call_info
List with metadata (dv, factors, covariates, weighted, etc.)
Use summary() for the full SPSS-style output with toggleable sections.
For data grouped with group_by(), one ANCOVA is computed per group:
the tables carry the group keys as leading columns and
group_results holds the complete result of each group
(NULL, with a warning, for a group that cannot be analysed).
Details
Understanding the Results
Adjusted Means (Estimated Marginal Means): These are the group means after statistically removing the effect of the covariate(s). They answer: "What would the group means be if all groups had the same covariate values?"
Covariate Effects: The covariate row in the ANOVA table shows whether the covariate significantly predicts the DV after adjusting for the factors.
Factor Effects: These show whether the factor affects the DV after controlling for the covariate. This is the primary test of interest.
Partial Eta Squared (Effect Size):
Less than 0.01: Negligible
0.01 to 0.06: Small
0.06 to 0.14: Medium
0.14 or greater: Large
When to Use This
Use ANCOVA when:
You have group comparisons (ANOVA) but want to control for a confound
Your covariate is continuous and linearly related to the DV
You want to increase statistical power by removing known variance sources
The covariate's relationship with the DV is the same across groups (homogeneity of regression slopes assumption)
References
Huitema, B. E. (2011). The Analysis of Covariance and Alternatives (2nd ed.). Wiley.
IBM Corp. (2023). IBM SPSS Statistics 29 Algorithms. IBM Corporation.
See also
factorial_anova for ANOVA without covariates.
linear_regression for regression analysis.
oneway_anova for single-factor ANOVA.
summary.ancova for detailed output with toggleable sections.
Other hypothesis_tests:
binomial_test(),
chi_square(),
chisq_gof(),
factorial_anova(),
fisher_test(),
friedman_test(),
kruskal_wallis(),
mann_whitney(),
mcnemar_test(),
oneway_anova(),
t_test(),
wilcoxon_test()
Examples
# Load required packages and data
library(dplyr)
#>
#> Attaching package: ‘dplyr’
#> The following objects are masked from ‘package:stats’:
#>
#> filter, lag
#> The following objects are masked from ‘package:base’:
#>
#> intersect, setdiff, setequal, union
data(survey_data)
# One-way ANCOVA: income by education, controlling for age
survey_data %>%
ancova(dv = income, between = c(education), covariate = c(age))
#> ANCOVA: income by education, covariate: age, N = 2186
#> age (covariate): F(1, 2181) = 0.030, p = 0.862, eta2p = 0.000
#> education: F(3, 2181) = 466.246, p < 0.001 ***, eta2p = 0.391
#> Use summary() for detailed output.
# Two-way ANCOVA with weights
survey_data %>%
ancova(dv = income, between = c(gender, education),
covariate = c(age), weights = sampling_weight)
#> ANCOVA: income by gender, education, covariate: age [Weighted], N = 2186
#> age (covariate): F(1, 2177) = 0.013, p = 0.911, eta2p = 0.000
#> gender: F(1, 2177) = 0.115, p = 0.735, eta2p = 0.000
#> education: F(3, 2177) = 455.614, p < 0.001 ***, eta2p = 0.386
#> gender:education: F(3, 2177) = 0.298, p = 0.827, eta2p = 0.000
#> Use summary() for detailed output.
# Multiple covariates
survey_data %>%
ancova(dv = income, between = c(education),
covariate = c(age, political_orientation))
#> ANCOVA: income by education, covariate: age, political_orientation, N = 2008
#> age (covariate): F(1, 2002) = 0.018, p = 0.895, eta2p = 0.000
#> political_orientation (covariate): F(1, 2002) = 1.366, p = 0.243, eta2p = 0.001
#> education: F(3, 2002) = 419.350, p < 0.001 ***, eta2p = 0.386
#> Use summary() for detailed output.
# --- Three-layer output ---
result <- ancova(survey_data, dv = income, between = c(education),
covariate = c(age))
result # compact overview
#> ANCOVA: income by education, covariate: age, N = 2186
#> age (covariate): F(1, 2181) = 0.030, p = 0.862, eta2p = 0.000
#> education: F(3, 2181) = 466.246, p < 0.001 ***, eta2p = 0.391
#> Use summary() for detailed output.
summary(result) # full detailed output with all sections
#> ANCOVA (One-Way ANCOVA) Results
#> -------------------------------
#>
#> - Dependent variable: income
#> - Factor(s): education
#> - Covariate(s): age
#> - Sum of squares: Type III
#> - N (complete cases): 2186
#> - Missing: 314
#>
#> Tests of Between-Subjects Effects
#> ---------------------------------------------------------------------------------------------------------
#> Source Type III Sum of Squares df Mean Square F Sig Partial Eta Squared
#> ---------------------------------------------------------------------------------------------------------
#> Corrected Model 1752821002.875 4 438205250.719 349.723 <.001 0.391 ***
#> Intercept 3472400779.183 1 3472400779.183 2771.252 <.001 0.560 ***
#> age 37721.406 1 37721.406 0.030 .862 0.000
#> education 1752630664.965 3 584210221.655 466.246 <.001 0.391 ***
#> Error 2732810163.639 2181 1253007.870
#> Total 35290790000.000 2186
#> Corrected Total 4485631166.514 2185
#> ---------------------------------------------------------------------------------------------------------
#> R Squared = 0.391 (Adjusted R Squared = 0.390)
#>
#> Parameter Estimates
#> --------------------------------------------------------------------------------------------------------------------------
#> Parameter B Std. Error t Sig 95% CI Lower 95% CI Upper Partial Eta Squared
#> --------------------------------------------------------------------------------------------------------------------------
#> Intercept 5349.320 91.774 58.288 <.001 5169.346 5529.293 0.609
#> age -0.245 1.412 -0.174 .862 -3.014 2.524 0.000
#> [education=Basic Secondary] -2577.931 72.359 -35.627 <.001 -2719.830 -2436.032 0.368
#> [education=Intermediate Secondary] -1744.236 76.304 -22.859 <.001 -1893.871 -1594.600 0.193
#> [education=Academic Secondary] -1112.550 76.328 -14.576 <.001 -1262.233 -962.866 0.089
#> [education=University] 0 (a)
#> --------------------------------------------------------------------------------------------------------------------------
#> (a) This parameter is set to zero because it is redundant (SPSS coding: the
#> last category of each factor is the reference).
#>
#> Estimated Marginal Means
#> (Evaluated at covariate means)
#> ------------------------------------------------------------------------
#> education Mean Std. Error 95% CI Lower 95% CI Upper
#> ------------------------------------------------------------------------
#> Basic Secondary 2758.939 41.294 2677.960 2839.918
#> Intermediate Secondary 3592.634 47.822 3498.852 3686.416
#> Academic Secondary 4224.320 47.836 4130.511 4318.130
#> University 5336.870 59.438 5220.309 5453.431
#> ------------------------------------------------------------------------
#>
#> Levene's Test of Equality of Error Variances
#> F(3, 2182) = 103.953, p < 0.001 ***
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05
summary(result, marginal_means = FALSE) # hide estimated marginal means
#> ANCOVA (One-Way ANCOVA) Results
#> -------------------------------
#>
#> - Dependent variable: income
#> - Factor(s): education
#> - Covariate(s): age
#> - Sum of squares: Type III
#> - N (complete cases): 2186
#> - Missing: 314
#>
#> Tests of Between-Subjects Effects
#> ---------------------------------------------------------------------------------------------------------
#> Source Type III Sum of Squares df Mean Square F Sig Partial Eta Squared
#> ---------------------------------------------------------------------------------------------------------
#> Corrected Model 1752821002.875 4 438205250.719 349.723 <.001 0.391 ***
#> Intercept 3472400779.183 1 3472400779.183 2771.252 <.001 0.560 ***
#> age 37721.406 1 37721.406 0.030 .862 0.000
#> education 1752630664.965 3 584210221.655 466.246 <.001 0.391 ***
#> Error 2732810163.639 2181 1253007.870
#> Total 35290790000.000 2186
#> Corrected Total 4485631166.514 2185
#> ---------------------------------------------------------------------------------------------------------
#> R Squared = 0.391 (Adjusted R Squared = 0.390)
#>
#> Parameter Estimates
#> --------------------------------------------------------------------------------------------------------------------------
#> Parameter B Std. Error t Sig 95% CI Lower 95% CI Upper Partial Eta Squared
#> --------------------------------------------------------------------------------------------------------------------------
#> Intercept 5349.320 91.774 58.288 <.001 5169.346 5529.293 0.609
#> age -0.245 1.412 -0.174 .862 -3.014 2.524 0.000
#> [education=Basic Secondary] -2577.931 72.359 -35.627 <.001 -2719.830 -2436.032 0.368
#> [education=Intermediate Secondary] -1744.236 76.304 -22.859 <.001 -1893.871 -1594.600 0.193
#> [education=Academic Secondary] -1112.550 76.328 -14.576 <.001 -1262.233 -962.866 0.089
#> [education=University] 0 (a)
#> --------------------------------------------------------------------------------------------------------------------------
#> (a) This parameter is set to zero because it is redundant (SPSS coding: the
#> last category of each factor is the reference).
#>
#> Levene's Test of Equality of Error Variances
#> F(3, 2182) = 103.953, p < 0.001 ***
#>
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05