Quick Answer: Games Howell Test Result

The worked example compares G3 final grade across four studytime groups. The sample contains 649 students. The standard one-way ANOVA was significant, F(3, 645) = 15.876, p < .001. The Welch ANOVA context was also significant, F(3, 139.101) = 18.183, p < .001. Because Games Howell Test is designed for unequal variances and unequal group sizes, it is a strong post hoc option for this type of comparison.

The group means increased from studytime group 1 to the higher studytime groups. Group 1 had the lowest mean G3 score, M = 10.84. Group 2 had M = 12.09. Group 4 had M = 13.06. Group 3 had the highest mean, M = 13.23. Games Howell pairwise comparisons found 4 significant comparisons out of 6 total comparisons.

What Is Games Howell Test?

Games Howell Test is a post hoc multiple comparison method used to compare group means after ANOVA. It is commonly recommended when sample sizes are unequal, variances are unequal, or both problems may exist. Unlike Tukey HSD, it does not require the assumption that all groups have the same variance.

The test compares every pair of group means and calculates a separate standard error for each pair. This pair-specific logic is important because two groups may have different sample sizes and different standard deviations. Games Howell Test adjusts the comparison using Welch-style degrees of freedom and a studentized range approach.

The purpose of Games Howell Test is to answer the post hoc question: which specific group means are different? A significant omnibus ANOVA or Welch ANOVA tells the researcher that at least one group mean differs, but it does not identify the specific pairs. Games Howell Test identifies those pairs while being robust to unequal variance conditions.

Before using Games Howell Test, it is useful to understand one-way ANOVA, Welch’s t-test, ANOVA assumptions, Levene test, p-values, and confidence intervals.

When to Use Games Howell Test

Use Games Howell Test when you have one categorical factor with three or more groups, a continuous dependent variable, and a need to compare group means pair by pair under unequal variance or unequal sample-size conditions. In this example, studytime is the four-level factor and G3 final grade is the continuous outcome.

Games Howell Test Formula

The pairwise mean difference for groups i and j is:

The pairwise standard error uses each group’s variance and sample size:

The Welch-style degrees of freedom for each pair are calculated as:

The Games Howell test statistic is based on the absolute mean difference divided by the standard error and compared with a studentized range critical value:

Null and Alternative Hypotheses for Games Howell Test

Games Howell Test is interpreted pair by pair. Every group comparison has its own null and alternative hypothesis.

Dataset and Variables Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade. The factor is studytime, coded into four weekly study-time categories. The analysis asks whether average final grade differs across studytime groups and which specific pairs differ using the Games Howell post hoc method.

Before interpreting Games Howell pairwise comparisons, review the group means, standard deviations, variances and distribution shapes. Useful related guides include descriptive statistics, box plot interpretation, Levene test, Brown-Forsythe ANOVA, and ANOVA in SPSS.

SPSS Output Interpretation for Games Howell Test

The SPSS output provides the official Games Howell multiple comparisons table. It includes group descriptives, variance assumption context, standard one-way ANOVA, Welch robust test and the Games Howell post hoc comparisons.

SPSS Group Descriptives

SPSS Homogeneity of Variances

Even though Levene’s test is not significant in this example, Games Howell remains useful because the group sizes are unequal and the procedure does not depend on pooled equal-variance logic. It also provides a robust comparison that fits well with the Welch ANOVA context.

SPSS ANOVA and Welch Robust Test

SPSS Games Howell Multiple Comparisons

Python Chart-by-Chart Interpretation

The Python charts show the Games Howell workflow visually. They include group spread, group means, variance context, adjusted p-values, mean difference intervals, a pairwise p-value heatmap and a group size with standard deviation chart.

Python Chart 1: Group Spread Boxplots

The group spread boxplot shows that group 1 has the lowest central position and includes several very low G3 scores. Groups 3 and 4 are centered higher, while group 2 sits between the lowest and highest categories. This pattern explains why the overall Welch and standard ANOVA results are significant.

The spread view is especially important for Games Howell Test because this method is designed for unequal variance and unequal sample-size contexts. The chart gives a visual check of how much spread exists inside each group before interpreting the pairwise post hoc results.

Python Chart 2: Group Mean Comparison

The group mean comparison chart shows the same mean ordering reported in SPSS. Group 1 has the lowest mean, about 10.84. Group 2 rises to about 12.09. Group 4 is about 13.06, and group 3 is highest at about 13.23.

This chart shows why most significant differences involve group 1. The mean gap from group 1 to groups 3 and 4 is large, while the mean gap between groups 3 and 4 is very small.

Python Chart 3: Group Variance Context

The variance context chart explains why a robust post hoc method is useful. Group variances are not identical: group 3 has the smallest variance, about 6.26, while groups 1 and 2 are above 10. Group 4 has a variance near 9.23.

Games Howell does not force all groups to share one pooled variance. Instead, each pairwise comparison uses the two group variances and sample sizes involved in that pair. This makes the method safer when variance equality is uncertain.

Python Chart 4: Games Howell Adjusted p-values

The adjusted p-value chart shows which comparisons fall below α = .05. The significant pairs are 1 vs 3, 1 vs 2, 1 vs 4, and 2 vs 3. These comparisons have adjusted p-values below .05 after the Games Howell correction.

The non-significant comparisons are 2 vs 4 and 3 vs 4. The p-value for 3 vs 4 is very large, showing that those two high-studytime groups have almost no statistical separation in final grade.

Python Chart 5: Mean Difference Intervals

The mean difference interval chart uses zero as the no-difference line. Intervals that do not cross zero are significant. The intervals for 1 vs 2, 1 vs 3, 1 vs 4, and 2 vs 3 exclude zero, so these pairs are significant.

The intervals for 2 vs 4 and 3 vs 4 cross zero. That means those observed mean differences are not statistically clear after the Games Howell adjustment.

Python Chart 6: Pairwise p-value Heatmap

The pairwise p-value heatmap gives a compact view of all group comparisons. Cells involving group 1 against groups 2, 3 and 4 show significant differences. The group 2 vs group 3 cell is also significant.

The heatmap makes the non-significant areas easy to see. The group 2 vs group 4 and group 3 vs group 4 comparisons do not show enough evidence for a significant difference at the .05 level.

Python Chart 7: Group Size and Standard Deviation

The group size and standard deviation chart shows why Games Howell is appropriate for this dataset. Group sizes are unequal: group 2 has 305 students, group 1 has 212, group 3 has 97, and group 4 has only 35.

The standard deviations also differ across groups. Games Howell accounts for this by using pair-specific standard errors and Welch degrees of freedom. This chart gives practical evidence for choosing a robust post hoc method instead of a strictly equal-variance post hoc test.

R Chart-by-Chart Validation

The R charts validate the same Games Howell interpretation using a separate workflow. The group mean order, p-value decisions and confidence interval conclusions match the Python and SPSS results.

R Chart 1: Group Spread Boxplots

The R boxplot confirms the same distribution pattern seen in Python. Group 1 has the lowest central position, while groups 3 and 4 are higher. The chart also shows that there are low-scoring observations in the lower studytime groups.

This validation matters because Games Howell Test should be interpreted with both means and spread in mind. The R chart supports the conclusion that lower studytime is associated with lower final grade performance.

R Chart 2: Group Mean Comparison

The R group mean comparison chart confirms that group 1 is lowest and group 3 is highest. Group 4 is close to group 3, and group 2 sits between group 1 and the highest groups.

This visual pattern matches the post hoc decisions. The small difference between groups 3 and 4 explains why that pair is not statistically significant.

R Chart 3: Group Variance Context

The R variance chart confirms that group variances are not identical. This does not automatically invalidate ANOVA, but it explains why a robust post hoc method is useful.

Games Howell handles this situation by avoiding a single pooled variance for all comparisons. This is why it is often preferred when variance equality is uncertain.

R Chart 4: Games Howell Adjusted p-values

The R adjusted p-value chart validates the same four significant pairs: 1 vs 2, 1 vs 3, 1 vs 4, and 2 vs 3. These p-values are below the .05 threshold.

The two non-significant pairs remain the same in R: 2 vs 4 and 3 vs 4. This software-to-software agreement strengthens confidence in the final interpretation.

R Chart 5: Mean Difference Intervals

The R interval chart confirms the same confidence interval decisions. Significant comparisons have intervals that stay away from zero, while non-significant comparisons cross zero.

The largest difference is group 1 compared with group 3. The smallest difference is group 3 compared with group 4. This matches the group mean table and the p-value chart.

R Chart 6: Pairwise p-value Heatmap

The R heatmap summarizes the pairwise decisions in a compact matrix. It shows that group 1 differs from every higher studytime group and that group 2 differs from group 3.

The heatmap also shows that groups 3 and 4 are not clearly different from each other. This is important because both groups have high means, but statistical significance depends on the difference relative to uncertainty.

R Chart 7: Group Size and Standard Deviation

The R group size and standard deviation chart confirms the unequal group-size structure. Group 4 is much smaller than groups 1 and 2, so pairwise uncertainty is larger when group 4 is involved.

This chart supports the use of Games Howell because the test adjusts each comparison according to the specific groups being compared rather than forcing all pairs to use the same pooled structure.

SPSS, R, Python and Excel Workflows for Games Howell Test

The same Games Howell Test workflow can be reproduced in SPSS, R, Python and Excel. SPSS gives the easiest official menu-based output. Python and R are best for reproducible scripts and charts. Excel can reproduce the main calculation logic but does not provide a simple built-in Games Howell post hoc menu.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Excel can reproduce the main Games Howell calculation logic by computing group means, group variances, group sample sizes, pairwise standard errors and Welch degrees of freedom. However, Excel does not have a simple built-in Games Howell button, so SPSS, R or Python is better for final post hoc output.

Code Blocks for Games Howell Test

SPSS Syntax

ONEWAY G3 BY studytime
  /STATISTICS DESCRIPTIVES HOMOGENEITY WELCH
  /POSTHOC = GH ALPHA(0.05).

R Code

data <- read.csv("dataset.csv")
data$studytime <- factor(data$studytime)

# Group summary

aggregate(G3 ~ studytime, data = data, function(x) {
c(n = length(x), mean = mean(x), sd = sd(x), variance = var(x))
})

# Welch ANOVA

welch_result <- oneway.test(G3 ~ studytime, data = data, var.equal = FALSE)
print(welch_result)

# Games-Howell can be run with a suitable post hoc package,

# or calculated manually using group means, variances, sample sizes,

# Welch degrees of freedom, and studentized range probabilities.

Python Code

import pandas as pd
import itertools
import numpy as np
from scipy import stats

df = pd.read_csv("dataset.csv")
df["studytime"] = df["studytime"].astype("category")

summary = df.groupby("studytime")["G3"].agg(["count", "mean", "std", "var"])
print(summary)

rows = []
for g1, g2 in itertools.combinations(summary.index, 2):
n1 = summary.loc[g1, "count"]
n2 = summary.loc[g2, "count"]
m1 = summary.loc[g1, "mean"]
m2 = summary.loc[g2, "mean"]
v1 = summary.loc[g1, "var"]
v2 = summary.loc[g2, "var"]

```
diff = m1 - m2
se = np.sqrt(v1/n1 + v2/n2)

numerator = (v1/n1 + v2/n2)**2
denominator = ((v1/n1)**2/(n1-1)) + ((v2/n2)**2/(n2-1))
welch_df = numerator / denominator

t_value = diff / se

rows.append([g1, g2, m1, m2, diff, se, welch_df, t_value])
```

games_howell_table = pd.DataFrame(rows, columns=[
"group_1", "group_2", "mean_1", "mean_2",
"mean_difference", "welch_standard_error",
"welch_degrees_of_freedom", "t_value"
])

print(games_howell_table)

Excel Formula Pattern

Group mean:
=AVERAGEIF(group_range, group_id, value_range)

Group variance:
=VAR.S(group_values)

Group sample size:
=COUNTIF(group_range, group_id)

Pairwise mean difference:
=Mean_Group_i - Mean_Group_j

Pairwise standard error:
=SQRT(Variance_i/n_i + Variance_j/n_j)

Welch degrees of freedom:
=((Variance_i/n_i + Variance_j/n_j)^2) /
(((Variance_i/n_i)^2/(n_i-1)) + ((Variance_j/n_j)^2/(n_j-1)))

Confidence interval:
Lower = Mean_Difference - Critical_Value*Standard_Error
Upper = Mean_Difference + Critical_Value*Standard_Error

Decision:
If the confidence interval excludes 0, the pair is significant.
If the confidence interval includes 0, the pair is not significant.

APA Reporting Wording for Games Howell Test

A one-way ANOVA was conducted to compare G3 final grade across four studytime groups. The standard ANOVA was statistically significant, F(3, 645) = 15.876, p < .001. The Welch robust test was also significant, F(3, 139.101) = 18.183, p < .001, indicating that mean final grade differed across studytime groups under unequal-variance robust testing.

Games Howell post hoc comparisons indicated that group 1 had significantly lower G3 scores than group 2, group 3 and group 4. Group 2 also had significantly lower G3 scores than group 3. The comparisons between group 2 and group 4 and between group 3 and group 4 were not statistically significant. These results suggest that the lowest studytime group had lower final grades than the higher studytime groups, while the highest studytime categories did not differ clearly from each other.

Common Mistakes in Games Howell Test

Downloads and Resources

Use the following downloadable outputs to verify the Games Howell Test result and compare the SPSS, Python and R workflows.

Related Guides

Continue with these related Salar Cafe guides to understand ANOVA, robust post hoc testing, assumptions and reporting.