Quick Answer: Type I Sum of Squares Result

The worked Type I Sum of Squares ANOVA uses sequential model entry. Because school is entered first, its sum of squares is credited before studytime is considered. Then studytime is tested after school. Finally, the school × studytime interaction is tested after both main effects.

The Type I ANOVA table reports that school is significant, SS = 546.629, F = 59.738, p = 4.202e-14, sequential η² = 0.08082, and partial η² = 0.08525. Studytime is also significant after school, SS = 341.212, F = 12.430, p = 6.557e-08, sequential η² = 0.05045, and partial η² = 0.05498. The school × studytime interaction is not significant, SS = 9.982, F = 0.3636, p = 0.7793, and partial η² = 0.001699.

What Is Type I Sum of Squares?

Type I Sum of Squares, also called sequential sum of squares, tests ANOVA model terms one at a time in the exact order they are entered into the model. The first term receives the variation it explains before the second term is considered. The second term receives only the extra variation it explains after the first term. The interaction receives the variation it explains after the main effects already entered the model.

In this example, the model order is school → studytime → school × studytime. Therefore, school is tested first, studytime is tested second, and the interaction is tested third. This model order matters because the dataset is not perfectly balanced across all school and studytime cells. When designs are unbalanced, Type I SS can change when the model order changes.

This makes Type I SS useful when the order is theoretically meaningful. For example, a researcher may want to control school first and then ask whether studytime adds extra explanatory value. However, Type I SS is not ideal when the researcher wants order-independent tests of each main effect after all other effects are included.

This topic connects closely with One Way ANOVA, Factorial ANOVA, Fixed Effects ANOVA, Balanced ANOVA, ANOVA Effect Size, Eta Squared, Omega Squared, Cohen’s F Formula and F Distribution.

Type I Sum of Squares Formula

Type I Sum of Squares is calculated from a sequence of nested models. Each effect receives the reduction in residual sum of squares when that effect is added to the model at its position in the sequence.

For this model order, the sequential effects are:

F Statistic Formula

The effect mean square is the effect sum of squares divided by the effect degrees of freedom. The error mean square is the residual sum of squares divided by the residual degrees of freedom. In this example, the residual mean square is approximately 9.150.

Sequential Eta Squared Formula

Sequential eta squared describes the share of the corrected total variation assigned to an effect at its point in the sequence. In this example, school has sequential η² about 0.08082, studytime has about 0.05045, and the interaction has about 0.001476.

Partial Eta Squared Formula

Partial eta squared compares each effect against that effect plus residual error. In this output, school has partial η² = 0.08525, studytime has partial η² = 0.05498, and the interaction has partial η² = 0.001699.

Type I vs Type III Sum of Squares

Many students search for type i vs type iii sum of squares, type i ii and iii sums of squares, and SPSS type i vs type iii sums of squares because ANOVA software often asks which sum-of-squares type should be used. The most important difference is model-order dependence.

Type I Sum of Squares Hypotheses

Because Type I SS is sequential, each hypothesis is conditional on the model terms already entered before it.

Dataset and Variables Used

The worked example uses student performance data. The dependent variable is G3 final grade. The first factor is school, with GP and MS groups. The second factor is studytime, with four levels. The interaction term is school × studytime. The SPSS output reports 649 cases.

Cell Mean Pattern

The means show why school and studytime both matter. GP has a higher overall mean than MS, and higher studytime groups generally have higher G3 means. However, the interaction pattern is not strong enough to be statistically significant.

For supporting concepts, review Descriptive Statistics, Mean Median and Mode, Standard Deviation, Variance, Confidence Interval, P Value, Null and Alternative Hypothesis and Effect Size.

ANOVA Assumptions for Type I Sum of Squares

Type I Sum of Squares is a way of partitioning ANOVA variation. It does not remove the usual ANOVA assumptions. The dependent variable should be numeric, observations should be independent, residuals should be reasonably normal, and group variances should be reasonably similar.

For assumption support, use ANOVA Assumptions, Levene Test, Bartlett’s Test, Brown-Forsythe Test, Brown Forsythe ANOVA, Q-Q Plot Normality Check, P-P Plot Normality Check and Outlier Detection.

SPSS Output Interpretation for Type I Sum of Squares

The SPSS output uses UNIANOVA G3 BY school_id studytime with /METHOD=SSTYPE(1). This means SPSS reports Type I Sum of Squares, not the default Type III table commonly seen in many GLM outputs.

SPSS Reading Order

SPSS Type I ANOVA Table

Python Chart-by-Chart Interpretation

The Python chart sequence explains Type I Sum of Squares through sequential SS, model-order comparison, F statistics, p-value decisions, factor means, interaction pattern, residual diagnostics and a final ANOVA summary table.

Python Chart 1: Type I Sum of Squares by Effect

The first chart shows the sequential SS values in the chosen model order. School receives the largest sum of squares because it is entered first and explains 546.63 units of corrected G3 variation before studytime is considered.

Studytime still explains 341.21 additional units after school is already in the model. The interaction explains only 9.98 additional units, which is very small compared with the main effects.

Python Chart 2: Type I SS Changes When Model Order Changes

This chart demonstrates the most important feature of Type I Sum of Squares: order dependence. When school is entered first, school receives the larger first share. When studytime is entered first, studytime receives more of the shared explanatory variation.

This is why Type I SS should not be reported without the model order. In unbalanced designs, the same dataset can produce different sequential sums of squares depending on whether school or studytime is entered first.

Python Chart 3: Type I ANOVA F Statistics

The F statistic chart shows school with the strongest test statistic, about 59.74. Studytime has a smaller but still strong F statistic, about 12.43.

The interaction F statistic is almost zero compared with the main effects. This matches the final decision that the interaction is not statistically meaningful in the sequential model.

Python Chart 4: Type I ANOVA p-value Decision

The p-value decision chart shows school and studytime below alpha = .05. The interaction p-value is far above alpha, near 0.7793.

This chart gives the clearest decision summary. The sequential main effects are significant, while the interaction should be reported as non-significant.

Python Chart 5: Mean G3 by School

The school mean chart provides practical context for the school effect. GP has a higher mean G3 than MS, and the confidence intervals are separated enough to support a visible group difference.

This explains why school receives a large sequential SS and a significant F statistic when entered first in the Type I model.

Python Chart 6: Interaction Mean Pattern

The interaction pattern chart shows that GP is higher than MS across studytime levels. The lines are not perfectly identical, but they move in a broadly similar direction.

Because the formal interaction p-value is 0.7793, this chart should be interpreted descriptively. It does not support a significant school × studytime interaction.

Python Chart 7: Residuals vs Fitted Values

The residual chart shows vertical fitted-value bands because the model uses categorical group means. Most residuals are distributed around zero, but several large negative residuals are visible.

This diagnostic means the model explains group mean differences, but individual cases can still be far below the fitted cell mean. The significant Levene result also suggests that variance differences should be reported cautiously.

Python Chart 8: Type I Summary Table

The summary table gives the final Python result in one place. School and studytime are significant. The school × studytime interaction is not significant.

This is the best table for reporting because it includes both sequential SS and effect-size columns. It also makes clear that this result depends on the order in which the model terms entered.

R Chart-by-Chart Validation

The R charts repeat the same Type I SS workflow in a second software environment. R’s basic anova(lm()) table is naturally sequential, so it is a useful tool for teaching Type I Sum of Squares.

R Chart 1: Type I Sum of Squares by Effect

The R chart confirms the same sequential SS pattern as Python. School has the largest Type I SS, studytime is second, and the interaction is very small.

This agreement confirms that the result is not a software artifact. It is a property of the chosen model order and dataset.

R Chart 2: Type I Order Comparison

The R order-comparison chart confirms that Type I SS is order-dependent. If studytime is entered before school, the sequential allocation of explained variation changes.

This chart is central to the teaching purpose of the post. It shows why analysts must justify the sequence before interpreting Type I Sum of Squares.

R Chart 3: Type I ANOVA F Statistics

The R F statistic chart confirms that school has the strongest F statistic and studytime has a significant but smaller F statistic.

The interaction has a very small F statistic, matching the non-significant interaction decision.

R Chart 4: Type I p-value Decision

The R p-value chart validates the same decision: school and studytime are significant, while the interaction is not.

This supports the final reporting conclusion across both Python and R.

R Chart 5: Mean G3 by School

The R mean chart confirms that GP has a higher mean G3 than MS.

This visual difference explains why school is significant when it enters the Type I model first.

R Chart 6: Interaction Mean Pattern

The R interaction chart confirms the same descriptive pattern as Python. GP remains higher than MS across studytime levels.

The formal interaction remains non-significant, so the chart should not be used to claim that the studytime effect differs by school.

R Chart 7: Residuals vs Fitted Values

The R residual plot confirms the same model diagnostic message. Most residuals are centered around zero, but several large negative residuals are present.

This supports a transparent assumption statement: the group mean results are clear, but residual spread and variance assumptions deserve caution.

R Chart 8: Type I Summary Table

The R summary table confirms the same final result as Python and SPSS. School and studytime are significant in the chosen order, while the interaction is not.

This agreement across tools makes the interpretation stable for a teaching article and practical reporting.

SPSS, R, Python and Excel Workflows for Type I Sum of Squares

The same Type I Sum of Squares workflow can be reproduced in SPSS, R, Python and Excel. SPSS requires /METHOD=SSTYPE(1). R’s standard anova(lm()) output is sequential. Python can use statsmodels.stats.anova.anova_lm(..., typ=1). Excel can support the logic with sequential model comparisons, but SPSS, R or Python is better for official reporting.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Type I Sum of Squares

SPSS Syntax for Type I Sum of Squares

* Type I Sum of Squares in SPSS.
* Dependent variable: G3.
* Model order: school first, studytime second, interaction third.

TITLE "Type I Sum of Squares: G3 by School and Studytime".

UNIANOVA G3 BY school_id studytime
  /METHOD=SSTYPE(1)
  /INTERCEPT=INCLUDE
  /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY PARAMETER
  /CRITERIA=ALPHA(.05)
  /DESIGN=school_id studytime school_id*studytime.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="type_i_sum_of_squares_spss_output.pdf".

Python Code for Type I Sum of Squares

import pandas as pd
from statsmodels.formula.api import ols
from statsmodels.stats.anova import anova_lm

df = pd.read_csv("dataset.csv")

df["G3"] = pd.to_numeric(df["G3"], errors="coerce")
df["school"] = df["school"].astype("category")
df["studytime"] = df["studytime"].astype("category")

data = df[["G3", "school", "studytime"]].dropna().copy()

# Model order: school first, studytime second, interaction third
model = ols(
    "G3 ~ C(school) + C(studytime) + C(school):C(studytime)",
    data=data
).fit()

type1_table = anova_lm(model, typ=1)

error_ss = type1_table.loc["Residual", "sum_sq"]
corrected_total = type1_table["sum_sq"].sum()

type1_table["seq_eta_sq"] = type1_table["sum_sq"] / corrected_total
type1_table["partial_eta_sq"] = type1_table["sum_sq"] / (type1_table["sum_sq"] + error_ss)

print(type1_table)

# Compare order dependence
model_reversed = ols(
    "G3 ~ C(studytime) + C(school) + C(studytime):C(school)",
    data=data
).fit()

print(anova_lm(model_reversed, typ=1))

# Cell means
print(data.groupby(["school", "studytime"])["G3"].agg(["count", "mean", "std"]))

# Residual diagnostics
data["fitted"] = model.fittedvalues
data["residual"] = model.resid
print(data[["G3", "school", "studytime", "fitted", "residual"]].head())

R Code for Type I Sum of Squares

# Type I Sum of Squares in R

library(tidyverse)

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

df$G3 <- as.numeric(df$G3)
df$school <- as.factor(df$school)
df$studytime <- as.factor(df$studytime)

data <- df %>%
  select(G3, school, studytime) %>%
  drop_na()

# Model order: school first, studytime second, interaction third
model_type1 <- lm(G3 ~ school + studytime + school:studytime, data = data)

# R's anova() gives sequential Type I sums of squares
anova(model_type1)

# Reversed order to demonstrate order dependence
model_reversed <- lm(G3 ~ studytime + school + studytime:school, data = data)
anova(model_reversed)

# Cell means
data %>%
  group_by(school, studytime) %>%
  summarise(
    n = n(),
    mean_G3 = mean(G3),
    sd_G3 = sd(G3),
    .groups = "drop"
  )

# Diagnostics
par(mfrow = c(1, 2))
plot(fitted(model_type1), residuals(model_type1),
     xlab = "Fitted values", ylab = "Residuals",
     main = "Residuals vs Fitted")
abline(h = 0, lty = 2)
qqnorm(residuals(model_type1))
qqline(residuals(model_type1))

Excel Notes for Type I Sum of Squares

Excel support workflow:

1. Arrange the data:
   G3 | school | studytime

2. Create a PivotTable:
   Rows = school
   Columns = studytime
   Values = average of G3, count of G3, standard deviation of G3

3. Create dummy variables:
   school dummies
   studytime dummies
   school × studytime interaction dummies

4. Fit sequential regression models:
   Model 0: intercept only
   Model 1: school
   Model 2: school + studytime
   Model 3: school + studytime + interaction

5. Calculate sequential SS:
   SS_school = SSE_Model0 - SSE_Model1
   SS_studytime = SSE_Model1 - SSE_Model2
   SS_interaction = SSE_Model2 - SSE_Model3

6. Calculate F values:
   F = MS_effect / MS_error

7. Formal Type I ANOVA:
   Use SPSS, R or Python for the final publishable table.

APA Reporting Wording

When reporting Type I Sum of Squares, always mention the model order. Without the order, the result is incomplete because Type I SS changes when terms are rearranged.

Common Mistakes

When to Use Type I Sum of Squares

Use Type I Sum of Squares when the order of predictors or factors is meaningful. It is especially useful in hierarchical ANOVA, planned model-building, polynomial terms, nested logic, and situations where earlier variables should receive credit before later variables.

Compare this guide with Factorial ANOVA, One Way ANOVA, Balanced ANOVA, Fixed Effects ANOVA, Nested ANOVA, Brown Forsythe ANOVA, ANOVA Effect Size, ANOVA in SPSS, ANOVA in R and ANOVA in Python.

Downloads and Resources for Type I Sum of Squares

Use these resources to reproduce the Type I Sum of Squares workflow. The Python report, R report and SPSS output PDF are included as verification files. Script and workbook placeholders can be replaced after the final downloadable files are uploaded to the WordPress Media Library.

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