Quick Answer: Fishers LSD Test Result

The worked example compares G3 final grade across four studytime groups. The sample contains 649 students. The omnibus one-way ANOVA was significant, F(3, 645) = 15.876, p < .001, so follow-up pairwise comparisons were justified. Because the ANOVA was significant first, this example is interpreted as a Fisher’s protected LSD test.

The group means increased from studytime group 1 to group 3. Group 1 had the lowest mean G3 score, M = 10.84. Group 2 had M = 12.09. Group 3 had the highest mean, M = 13.23. Group 4 had a similar high mean, M = 13.06. Fishers LSD pairwise comparisons found 4 significant pairs out of 6 total pairwise comparisons.

What Is Fishers LSD Test?

Fishers LSD Test is a post hoc pairwise comparison method used after ANOVA. It compares each pair of group means and asks whether the mean difference is large enough to be considered statistically significant. The method uses the pooled within-group error term from the ANOVA table, so it is directly connected to the one-way ANOVA model.

The word LSD means Least Significant Difference. It is the smallest mean difference that must be exceeded before a pair is declared statistically significant. If the absolute mean difference is larger than the calculated LSD critical difference, the pair is significant. If the absolute mean difference is smaller than the LSD value, the pair is not significant.

Fishers LSD Test is often called Fisher’s protected LSD test when it is used only after a significant omnibus ANOVA. The “protected” part means the researcher first checks whether the overall ANOVA suggests that at least one group mean differs. Only after that overall test is significant are the pairwise comparisons interpreted.

Before using Fishers LSD Test, it is useful to understand one-way ANOVA, ANOVA assumptions, p-values, confidence intervals, and effect size.

When to Use Fishers LSD Test

Use Fishers LSD Test when you have one categorical factor with three or more groups, a continuous dependent variable, and a significant one-way ANOVA result. In this example, studytime is the grouping factor and G3 final grade is the dependent variable. The ANOVA found that not all group means were equal, so the LSD post hoc test was used to identify which groups differed.

Fishers LSD Test Formula

The basic Fishers LSD pairwise test compares two group means using the pooled ANOVA mean square error. For groups i and j, the mean difference is:

The standard error for the mean difference is:

The t statistic for the pairwise comparison is:

The least significant difference, or LSD critical difference, is:

Null and Alternative Hypotheses for Fishers LSD Test

Fishers LSD Test is interpreted pair by pair. Each comparison has its own null and alternative hypothesis. The null hypothesis says the two group means are equal. The alternative hypothesis says the two group means are different.

Dataset and Variables Used

The worked example uses a student performance dataset. The dependent variable is G3 final grade. The grouping variable is studytime, coded into four weekly study-time groups. The analysis asks whether average final grade differs across the four studytime categories and which pairs of studytime groups are significantly different.

Before interpreting post hoc results, review the descriptive statistics, group distributions and ANOVA assumptions. Helpful related guides include descriptive statistics, box plot interpretation, Levene test, and ANOVA in SPSS.

SPSS Output Interpretation for Fishers LSD Test

The SPSS output begins with group descriptives, checks variance homogeneity, runs the one-way ANOVA, and then reports the LSD multiple comparisons table. The key point is that the omnibus ANOVA is significant and the Levene test is not significant, so the pooled-error LSD comparison is reasonable for this teaching example.

SPSS Group Descriptives

SPSS Homogeneity of Variances

The Levene result supports the use of a pooled error term. That matters because Fishers LSD Test uses the ANOVA within-group mean square error as its pooled variance estimate. If variances were strongly unequal, a different post hoc method would be safer.

SPSS One-Way ANOVA Table

SPSS Fishers LSD Multiple Comparisons

Python Chart-by-Chart Interpretation

The Python charts show the complete Fishers LSD Test workflow visually. They begin with group distributions, then show group means, pairwise p-values, mean difference confidence intervals and LSD critical differences.

Python Chart 1: Group Distributions Before Fishers LSD Test

The group distribution chart shows that studytime group 1 has the lowest central position, while groups 3 and 4 are centered higher. Group 2 sits between the lowest and highest studytime categories. This visual pattern explains why the omnibus ANOVA was significant and why Fisher’s LSD post hoc comparisons are useful after the ANOVA.

The chart also shows spread and outlying values. Groups 1 and 2 include some very low G3 scores, while group 3 has a tighter distribution and a higher center. Fishers LSD Test compares means, so the boxplot should be read as context rather than as the final decision table.

Python Chart 2: Group Means with Approximate 95% Confidence Intervals

The group means chart shows the same mean pattern as the SPSS descriptive table. Group 1 has the lowest mean G3 score, about 10.84. Group 2 rises to about 12.09. Group 3 has the highest mean, about 13.23, while group 4 is very close at about 13.06.

This visual result supports the Fisher LSD conclusion. The largest differences involve group 1 compared with groups 3 and 4. The smallest difference is between groups 3 and 4, so it is not surprising that the 3 vs 4 LSD comparison is not significant.

Python Chart 3: Pairwise Fishers LSD p-values

The p-value chart shows which pairwise comparisons fall below α = .05. The significant comparisons are 1 vs 3, 1 vs 2, 1 vs 4, and 2 vs 3. These p-values are to the left of the α = .05 reference line.

The comparisons 2 vs 4 and 3 vs 4 are not significant. The 3 vs 4 comparison has a much larger p-value, showing that groups 3 and 4 are very similar in mean G3 score. The chart makes the post hoc decision easier to read than a long SPSS table.

Python Chart 4: Fishers LSD Mean Differences with Confidence Intervals

The mean difference chart uses zero as the no-difference line. Intervals that do not cross zero are significant by Fishers LSD Test. The intervals for 1 vs 2, 1 vs 3, 1 vs 4, and 2 vs 3 stay away from zero, so those comparisons are significant.

The intervals for 2 vs 4 and 3 vs 4 cross zero. That means the observed mean differences for these two pairs could reasonably be explained by sampling variation at α = .05. This is the clearest visual explanation of the LSD confidence-interval decision rule.

Python Chart 5: LSD Critical Difference by Pair

The LSD critical difference chart shows how large a mean difference must be before each pair is significant. The threshold changes by pair because the studytime groups have different sample sizes. For example, group 4 has a smaller sample size, so comparisons involving group 4 have larger critical differences.

The smallest LSD critical difference is for 1 vs 2, about 0.549, because those groups have larger sample sizes. The largest is for 3 vs 4, about 1.210. This explains why a small difference between groups 3 and 4 is not significant.

R Chart-by-Chart Validation

The R validation charts confirm the Python and SPSS conclusions. The same mean pattern appears: group 1 is lowest, group 3 is highest, group 4 is close to group 3, and only four of the six pairwise comparisons are significant by Fishers LSD Test.

R Chart 1: Colorful Group Means

The R group means chart validates the same ranking shown in Python and SPSS. Group 1 is clearly the lowest, group 2 is higher, and groups 3 and 4 are the highest. This agreement across software strengthens confidence that the pattern is not a charting artifact.

The figure also shows why Fisher’s LSD identifies several group 1 comparisons as significant. Group 1 is separated from the higher studytime groups by a meaningful mean gap.

R Chart 2: Colorful Fishers LSD p-values

The R p-value chart confirms the same pairwise decision pattern. The significant comparisons fall below the α = .05 reference line, while the non-significant comparisons remain above it. The visual separation between significant and non-significant comparisons is especially useful for explaining the post hoc result to students.

The chart confirms that 2 vs 4 and 3 vs 4 should not be reported as significant differences. Their p-values do not support rejecting equality of means at the .05 level.

R Chart 3: Colorful Mean Differences

The R mean difference chart validates the confidence interval interpretation. Comparisons whose intervals do not cross zero are significant. Comparisons whose intervals cross zero are not significant. This is a direct visual version of the SPSS LSD multiple comparisons table.

The strongest mean difference is group 1 compared with group 3. The weakest is group 3 compared with group 4. This matches the descriptive means, where groups 3 and 4 are almost identical.

R Chart 4: Colorful LSD Critical Difference by Pair

The R critical difference chart confirms that each pair has its own threshold. Pairs with larger sample sizes have smaller LSD thresholds. Pairs involving the smaller group 4 have larger thresholds because there is more uncertainty in those comparisons.

This chart is important for Excel users because it shows that the LSD value is not always one fixed number when group sizes are unequal. In unequal sample-size designs, each pair can have a different LSD critical difference.

SPSS, R, Python and Excel Workflows for Fishers LSD Test

The same Fishers LSD Test can be reproduced in SPSS, R, Python and Excel. SPSS gives the easiest menu-based workflow. R and Python are useful for reproducible analysis and custom charts. Excel is useful for understanding the manual calculation behind the LSD critical difference.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

In Excel, Fishers LSD Test can be calculated manually after running a one-way ANOVA. The key values needed are the group means, group sample sizes, MSE from the ANOVA table, and error degrees of freedom.

Code Blocks for Fishers LSD Test

SPSS Syntax

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

R Code

data <- read.csv("dataset.csv")

data$studytime <- factor(data$studytime)

model <- aov(G3 ~ studytime, data = data)
summary(model)

# Fisher LSD using agricolae

# install.packages("agricolae")

library(agricolae)
lsd_result <- LSD.test(model, "studytime", p.adj = "none")
print(lsd_result)

Python Code

import pandas as pd
import itertools
from scipy import stats
import statsmodels.api as sm
import statsmodels.formula.api as smf

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

model = smf.ols("G3 ~ C(studytime)", data=df).fit()
anova = sm.stats.anova_lm(model, typ=2)

mse = anova.loc["Residual", "sum_sq"] / anova.loc["Residual", "df"]
df_error = anova.loc["Residual", "df"]
alpha = 0.05

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

rows = []
for g1, g2 in itertools.combinations(summary.index, 2):
n1, n2 = summary.loc[g1, "count"], summary.loc[g2, "count"]
m1, m2 = summary.loc[g1, "mean"], summary.loc[g2, "mean"]
diff = m1 - m2
se = (mse * (1/n1 + 1/n2)) ** 0.5
t_value = diff / se
p_value = 2 * stats.t.sf(abs(t_value), df_error)
lsd = stats.t.ppf(1 - alpha/2, df_error) * se
rows.append([g1, g2, m1, m2, diff, se, t_value, p_value, lsd])

lsd_table = pd.DataFrame(rows, columns=[
"group_1", "group_2", "mean_1", "mean_2",
"mean_difference", "standard_error", "t_value",
"p_value", "lsd_critical_difference"
])

lsd_table["decision"] = lsd_table["p_value"].apply(
lambda p: "Significant" if p < alpha else "Not significant"
)

print(anova)
print(summary)
print(lsd_table)

Excel Formula Pattern

Mean Difference:
=Mean_Group_i - Mean_Group_j

Standard Error:
=SQRT(MSE*(1/n_i + 1/n_j))

t Statistic:
=Mean_Difference / Standard_Error

Two-tailed p-value:
=T.DIST.2T(ABS(t_statistic), df_error)

LSD Critical Difference:
=T.INV.2T(0.05, df_error) * Standard_Error

Confidence Interval:
Lower = Mean_Difference - LSD_Critical_Difference
Upper = Mean_Difference + LSD_Critical_Difference

APA Reporting Wording for Fishers LSD Test

A one-way ANOVA was conducted to compare G3 final grade across four studytime groups. The ANOVA was statistically significant, F(3, 645) = 15.876, p < .001, indicating that mean final grade differed across studytime groups. Levene's test was not significant, p = .400, supporting the homogeneity of variance assumption.

Fisher's LSD 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 lower studytime is associated with lower final grade performance, while the highest studytime groups do not differ clearly from each other.

Common Mistakes in Fishers LSD Test

Downloads and Resources

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

Related Guides

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