Appendix U — SPSS Tutorial: Analysis of Covariance

Conducting ANCOVA in SPSS: assumption testing, adjusted means, and effect sizes

NoteLearning objectives

By the end of this tutorial, you will be able to:

  • Test the homogeneity of regression slopes assumption before running ANCOVA.
  • Run a one-way ANCOVA using SPSS General Linear Model → Univariate.
  • Locate and interpret SPSS output for adjusted means, the covariate slope, and the Group F-ratio.
  • Obtain partial eta-squared from SPSS output and partial omega-squared via the Statistical Calculators appendix or SPSS 31+.
  • Write a complete APA-style ANCOVA results paragraph.

U.1 Overview

This tutorial walks through a complete one-way ANCOVA using the core_session.csv dataset. The research question is: after statistically controlling for pre-test strength, does the training program produce greater post-test strength than the control condition?

Factor: Group (control, training) — between-subjects Covariate: Pre-test strength_kg Outcome: Post-test strength_kg Sample: N = 60 (30 control, 30 training)

The tutorial is structured as a strict two-stage workflow:

  • Stage 1 — Test the homogeneity of regression slopes (a prerequisite; must pass before Stage 2)
  • Stage 2 — Run the ANCOVA and interpret output

U.2 Dataset preparation

The core_session.csv file is in long format (one row per observation). ANCOVA requires a wide layout with pre-test and post-test strength in separate columns, and only one row per participant.

U.2.1 Step 1: Restructure to wide format

If you completed Chapter 16’s SPSS tutorial, your dataset may already be in wide format. If not:

  1. Click Data → Restructure
  2. Select Restructure selected cases into variables → Next
  3. Move id to Identifier Variable(s)
  4. Move time to Index Variable
  5. Click Next through remaining dialogs → Finish

After restructuring, rename the new columns in Variable View: - strength_kg.1 → strength_pre (verify this corresponds to pre-test) - strength_kg.2 → strength_mid - strength_kg.3 → strength_post

You should also have a group variable in the dataset (one value per participant, since Group is a between-subjects factor).

TipVerify the column-to-time-point mapping before proceeding

After restructuring, inspect the first few rows and cross-check participant IDs against the original long-format file to confirm that strength_pre, strength_mid, and strength_post contain the correct time-point values. SPSS sorts restructured columns alphabetically by the index variable value — for string values “mid,” “post,” “pre” the alphabetical order would be mid → post → pre, not pre → mid → post.

U.3 Stage 1: Testing homogeneity of regression slopes

Before running the ANCOVA, you must verify that the regression slope of the covariate (pre-test strength) on the outcome (post-test strength) is approximately equal across groups. This is done by adding a Group × Covariate interaction to the model. A significant interaction here means the assumption is violated and you should not proceed with standard ANCOVA.

U.3.1 Step 2: Run the slopes test

  1. Click Analyze → General Linear Model → Univariate
  2. Move strength_post to Dependent Variable
  3. Move group to Fixed Factor(s)
  4. Move strength_pre to Covariate(s)
  5. Click Model:
    • Select Custom
    • Move group to the Model box (Main effects)
    • Move strength_pre to the Model box (Main effects)
    • Now create the interaction: hold Ctrl and click both group and strength_pre in the Factors & Covariates list, then click the arrow — this adds the group * strength_pre interaction term
    • Click Continue
  6. Click OK

U.3.2 Step 3: Interpret the slopes test output

Look at the Tests of Between-Subjects Effects table. Find the row labelled group * strength_pre (the interaction):

Source SS df MS F p
group 274.55 1 274.55 116.72 < .001
strength_pre 26,810.02 1 26,810.02 11,398.52 < .001
group * strength_pre 0.22 1 0.22 0.09 .769
Error 131.72 56 2.35

The interaction term is F(1, 56) = 0.09, p = .769 — not significant. The slopes are parallel across groups. The homogeneity of regression slopes assumption is satisfied. Proceed to Stage 2.

WarningIf the slopes test IS significant

A significant Group × Covariate interaction (p < .05) means the two groups’ regression lines are not parallel — the covariate predicts the outcome differently in each group. In this case, do not proceed with standard ANCOVA. Instead, consider: (1) the Johnson–Neyman technique, which identifies the specific covariate range where groups do and do not differ; (2) reporting within-group regression slopes separately; or (3) consulting a statistician about alternative modeling approaches. See[1] for detailed guidance.

U.4 Stage 2: Running the ANCOVA

U.4.1 Step 4: Run the ANCOVA (without the interaction term)

Return to the Univariate dialog. This time, use the Full factorial model (the default), which includes only main effects — not the Group × Covariate interaction you added in Stage 1.

  1. Click Analyze → General Linear Model → Univariate
  2. Move strength_post to Dependent Variable
  3. Move group to Fixed Factor(s)
  4. Move strength_pre to Covariate(s)
  5. Click Model → Select Full factorial (or verify that only main effects are listed) → Continue
  6. Click Options:
    • Under Display Means for, select group
    • Check Compare main effects → set to Bonferroni (relevant if Group has 3+ levels)
    • Check Descriptive statistics, Estimates of effect size, and Homogeneity tests
    • Click Continue
  7. Click Plots:
    • Move group to the Horizontal Axis box
    • Click Add → Continue
  8. Click OK

U.4.2 Step 5: Interpret the ANCOVA output

Levene’s Test of Equality of Error Variances

Check that Levene’s test is non-significant, confirming equal error variances across groups.

Levene Statistic df1 df2 p
0.12 1 58 .731

The test is not significant (p = .731). The assumption of equal error variances is met.

Tests of Between-Subjects Effects

Source SS df MS F p η²_p
Corrected Model 27,528.53 2 13,764.27 5,908.09 < .001
Intercept 372,645.01 1 372,645.01 159,930.21 < .001
strength_pre 27,225.35 1 27,225.35 11,686.00 < .001 .995
group 303.18 1 303.18 130.18 < .001 .695
Error 132.82 57 2.33
Total 414,278.03 60

Key values to record:

  • Group F-ratio: F(1, 57) = 130.18, p < .001, η²_p = .695 — the training effect after adjusting for pre-test strength
  • Covariate F-ratio: F(1, 57) = 11,686.00, p < .001 — confirms the covariate is strongly related to the outcome (do not interpret this as a group comparison)
  • MS_error: 2.33 — needed for ω²_p calculation
  • Error df: 57 (= N − k − c = 60 − 2 − 1)

Estimated Marginal Means

This table gives the ANCOVA-adjusted means for each group, expressed at the grand mean of the covariate (M_pre = 78.01 kg):

Group Adjusted M (kg) SE 95% CI
Control 78.83 0.28 [78.27, 79.39]
Training 83.37 0.28 [82.81, 83.93]

The adjusted mean difference is 83.37 − 78.83 = 4.53 kg in favour of the training group.

TipCompare adjusted and unadjusted means side by side

The unadjusted post-test means (from Descriptive Statistics in the output) were Control = 77.14 kg and Training = 85.06 kg — a raw difference of 7.92 kg. The adjusted difference is 4.53 kg. This reduction reflects the removal of the 3.33 kg pre-test baseline advantage that the training group held at the start of the study. Always report both sets of means so readers can see the magnitude of the adjustment.

U.5 Part 3: Effect sizes

U.5.1 Partial eta-squared

SPSS reports η²_p = .695 for the Group effect directly in the Tests of Between-Subjects Effects table. No additional steps are needed.

U.5.2 Partial omega-squared

SPSS does not compute ω²_p. Use the Statistical Calculators appendix or SPSS 31+ (Analyze → Power Analysis → General Linear Model) to obtain ω²_p without hand calculation.

For reference, the formula for ANCOVA with one covariate is the same as for between-subjects ANOVA:

\[\omega^2_p = \frac{SS_{\text{Group}} - df_{\text{Group}} \times MS_{\text{error}}}{SS_{\text{Group}} + (N - df_{\text{Group}}) \times MS_{\text{error}}}\]

The values \(SS_{\text{Group}}\) (303.18), \(df_{\text{Group}}\) (1), \(MS_{\text{error}}\) (2.33), and \(N\) (60) all come from the Tests of Between-Subjects Effects table. Entering these into the calculator yields ω²_p = .683 for the Group effect.

U.6 Part 4: Visualizing ANCOVA results

U.6.1 Scatterplot with regression lines (homogeneity check)

A scatterplot of pre-test vs. post-test strength with separate regression lines per group is the most informative visual for ANCOVA. In SPSS:

  1. Click Graphs → Legacy Dialogs → Scatter/Dot → Simple Scatter → Define
  2. Move strength_post to Y Axis and strength_pre to X Axis
  3. Move group to Set Markers by
  4. Click OK
  5. Double-click the chart to open the Chart Editor
  6. Click Elements → Fit Line at Subgroups — this adds separate regression lines for each group
  7. Verify the lines appear approximately parallel (confirming the homogeneity assumption)

U.6.2 Adjusted means bar chart

SPSS generates a basic estimated marginal means plot when you specify it in the Plots dialog during Step 4. To produce a cleaner version:

  1. Click Graphs → Legacy Dialogs → Error Bars → Summaries of separate variables → Define
  2. Move strength_post to Error Bars (for unadjusted means)
  3. Click OK

For the adjusted means specifically, export the Estimated Marginal Means table and create the figure using the R code provided in Chapter 17 (fig-ancova-adjusted).

U.7 Part 5: APA-style write-up

A one-way ANCOVA was conducted to examine the effect of training group (control, training) on post-test muscular strength (kg), with pre-test strength serving as the covariate. Levene’s test confirmed homogeneity of error variance, F(1, 58) = 0.12, p = .731. The homogeneity of regression slopes assumption was verified; the Group × Pre-test interaction was non-significant, F(1, 56) = 0.09, p = .769, indicating that regression slopes were parallel across groups. After adjusting for pre-test strength (b = 1.01, p < .001), there was a significant main effect of Group, F(1, 57) = 130.18, p < .001, η²_p = .695, ω²_p = .683. The training group (M_adj = 83.37 kg, SE = 0.28) demonstrated significantly greater post-test strength than the control group (M_adj = 78.83 kg, SE = 0.28), with an adjusted mean difference of 4.53 kg.

U.8 Troubleshooting

“I cannot find the group * strength_pre interaction in the Model dialog”: In the Custom model builder, you must select both group and strength_pre simultaneously (hold Ctrl while clicking) before clicking the arrow to move them to the model box. Clicking them separately adds two main effects; clicking them together adds the interaction term.

“My Estimated Marginal Means table is missing”: Return to the Univariate dialog → Options → and move group to the Display Means for list. The adjusted means only appear when the factor is explicitly requested in this panel.

“The adjusted means are identical to the unadjusted means”: This occurs when the covariate was not successfully entered in the Covariate(s) box, or when the Model dialog was set to a Custom model that omitted the covariate main effect. Verify that strength_pre appears in the Covariates field and re-run.

“My η²_p for the covariate is greater than .99 — is that possible?”: Yes. In a pretest–posttest design where the same measure is taken before and after the intervention, the covariate–outcome correlation is often r ≥ .95. When virtually all within-group variance in post-test scores is explained by pre-test scores, the covariate’s η²_p approaches 1.00. This is expected behaviour, not an error — it reflects the precision of the measurement instrument.

“The Group F-ratio from the ANCOVA is much larger than the F from a simple post-test ANOVA. Is that right?”: Yes — and it is the point of ANCOVA. By removing the covariate’s variance from the error term, MS_error drops dramatically. In this example it fell from approximately 180 kg² to 2.33 kg², making the same-sized group effect appear far more statistically distinguishable against the smaller residual noise.

U.9 Practice exercises

Exercise 1: Run the same ANCOVA described in this tutorial but use vo2_mlkgmin as both the covariate (pre-test) and the outcome (post-test). First verify the homogeneity of slopes assumption. Report the adjusted means, the Group F-ratio, and η²_p. How does the adjusted group difference compare to the unadjusted difference?

Exercise 2: Repeat the ANCOVA for post-test strength_kg but now include both strength_pre and pre-test vo2_mlkgmin as covariates. Does adding the second covariate meaningfully change the Group F-ratio or the adjusted means? What does this tell you about the additional contribution of aerobic capacity as a predictor of post-test strength?

Exercise 3: To understand the power advantage of ANCOVA, run two analyses on the same data: (1) a simple independent-samples t-test comparing post-test strength_kg between groups (ignoring the pre-test), and (2) the ANCOVA from this tutorial. Record the t and F statistics, their p-values, and the effect sizes. Write 3–4 sentences explaining why the F-ratio from ANCOVA is so much larger, connecting your explanation to the concept of error variance reduction.

Exercise 4: Open GPower and use the F-test: ANOVA: Fixed effects, special, main effects and interactions* option to compute the required sample size to detect the observed ANCOVA Group effect (η²_p = .695, converting to Cohen’s f) with power = .80, α = .05, and 1 numerator df. Remember to add 1 to the GPower estimate to account for the covariate’s degree of freedom. How does this compare to the N* = 60 in the study?