To install and load all the packages used in this chapter, run the following code:

for (pkg in c("car", "broom", "emmeans", "glue", "tidyverse")) {
  if (!require(pkg, character.only = TRUE)) install.packages(pkg)
}

library(car)
library(broom)
library(emmeans)
library(glue)
library(tidyverse)

An analysis of variance on a linear model can be computed in three different ways, traditionally labelled Type I, Type II and Type III. This is not a quirk of R: the same three types appear in SAS, SPSS and other statistics packages, because they reflect a general question about how to attribute variation when the predictors overlap. For perfectly balanced data they all return the same numbers, which is why the distinction is often glossed over in introductory courses. The moment the data are unbalanced (and real experiments often end up unbalanced, even if only because a plot was lost or a plant died), the three types can give noticeably different p-values for the same effect. This chapter explains what the three types actually compute, shows a small example where the differences become visible, and gives practical recommendations for which one to use.

A Quick Overview

The three types differ in how the sum of squares for each term is calculated, not in the underlying model. The model is fit once with lm(). The “type” only decides which comparisons of nested models are used to attribute variation to each term.

Type	R call	How SS are computed	Typical use
I (sequential)	`stats::anova(mod)`	Each term added on top of the previous ones, in the order they appear in the formula.	Balanced data, or genuinely hierarchical / nested models where term order reflects a causal sequence.
II (hierarchical)	`car::Anova(mod, type = "II")`	Each main effect adjusted for all other main effects, but not for interactions that contain it.	Unbalanced data with only main effects, or when interactions are present but not of primary interest. Recommended default for most applied analyses (Langsrud 2003).
III (marginal)	`car::Anova(mod, type = "III")`	Each term adjusted for all other terms, including higher-order interactions containing it.	When interactions are present and main effects should be tested “at the margin”. Requires sum-to-zero contrasts to be meaningful.

Two points are worth emphasising before we look at any data. First, stats::anova() always computes Type I sequential sums of squares — it has no type argument, and changing the order of predictors in the formula will change the result. Second, car::Anova() (note the capital A) is the standard tool for Type II and Type III and is essentially required for the latter, because correct Type III tests need a particular kind of contrast coding (more on this below).

Why Balance Matters

When a dataset is balanced — every factor combination has the same number of observations — the main effects are orthogonal to each other. Orthogonality means that the variation explained by factor A does not overlap with the variation explained by factor B, so it does not matter in which order we attribute it. All three ANOVA types give the same sums of squares and the same p-values.

Unbalanced data destroy this orthogonality. The variation explained by A and by B now shares some common ground, and the three types differ in how they handle this shared portion:

Type I gives all shared variation to whichever term comes first in the formula.
Type II tests each main effect after removing the contribution of the other main effects, ignoring interactions.
Type III tests each main effect after removing the contribution of all other terms, including interactions.

A Concrete 2 by 2 Example

To make this visible, we construct a small unbalanced two-factor dataset. The two factors diet and supp each have two levels, and the cell counts are deliberately uneven:

set.seed(42)

dat <- tibble(
  diet = rep(c("low", "high"), times = c(14, 10)),
  supp = c(rep(c("A", "B"), times = c(10, 4)),  # low: 10 A, 4 B
           rep(c("A", "B"), times = c(3,  7))), # high: 3 A, 7 B
  response = c(
    rnorm(10, mean = 10, sd = 1.5),  # low + A
    rnorm(4,  mean = 12, sd = 1.5),  # low + B
    rnorm(3,  mean = 14, sd = 1.5),  # high + A
    rnorm(7,  mean = 17, sd = 1.5)   # high + B
  )
) %>%
  mutate(across(c(diet, supp), as.factor))

xtabs(~ diet + supp, data = dat)

      supp
diet    A  B
  high  3  7
  low  10  4

The design matrix is clearly unbalanced: the combination low diet + supplement A has 10 observations, whereas high diet + supplement A has only 3.

We now fit the two-way model with interaction:

mod_ds <- lm(response ~ diet * supp, data = dat)

Type I Depends on Term Order

stats::anova() always computes Type I (sequential) sums of squares. To make the order dependence visible, we fit the same model with the two factors swapped and compare the tables side by side:

mod_sd <- lm(response ~ supp * diet, data = dat)

anova(mod_ds)

Analysis of Variance Table

Response: response
          Df Sum Sq Mean Sq F value    Pr(>F)    
diet       1 95.469  95.469 27.3858 4.031e-05 ***
supp       1 17.563  17.563  5.0381   0.03627 *  
diet:supp  1  0.002   0.002  0.0006   0.98005    
Residuals 20 69.722   3.486                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

anova(mod_sd)

Analysis of Variance Table

Response: response
          Df Sum Sq Mean Sq F value    Pr(>F)    
supp       1 61.275  61.275 17.5770 0.0004485 ***
diet       1 51.758  51.758 14.8470 0.0009915 ***
supp:diet  1  0.002   0.002  0.0006 0.9800473    
Residuals 20 69.722   3.486                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The sums of squares and p-values for diet and supp differ between the two tables, simply because of term order. Whichever factor is listed first “absorbs” the shared variation. This is exactly the behaviour that makes Type I unattractive as a default: a scientifically meaningless decision (which factor to write first) changes the reported test statistic.

Type II Ignores Both Order and Coding

car::Anova() (note the capital A) produces order-independent tables. Type II adjusts each main effect for the other main effect, but not for the interaction:

Anova(mod_ds, type = "II")

Anova Table (Type II tests)

Response: response
          Sum Sq Df F value    Pr(>F)    
diet      51.758  1 14.8470 0.0009915 ***
supp      17.563  1  5.0381 0.0362658 *  
diet:supp  0.002  1  0.0006 0.9800473    
Residuals 69.722 20                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

This table is identical whether the formula reads diet * supp or supp * diet, and, just as importantly, it does not depend on how the factors are coded. That double robustness is the main reason Type II is the recommended default for unbalanced data whose interaction is not of central interest (Langsrud 2003).

Type III Needs Sum-to-Zero Contrasts

Type III adjusts each main effect for all other terms, including the interaction that contains it. This is the point that trips up almost everyone who meets Type III for the first time: a correct Type III test requires sum-to-zero contrasts for the factors, and this is the one place in the chapter where the contrast coding actually matters.

Our model mod_ds was fit with R’s default contr.treatment (reference coding). That was harmless for Type I and Type II, which do not depend on the coding, but it is not harmless for Type III. Calling it naively gives:

Anova(mod_ds, type = "III")

Anova Table (Type III tests)

Response: response
            Sum Sq Df  F value    Pr(>F)    
(Intercept) 597.21  1 171.3119 2.888e-11 ***
diet         24.95  1   7.1576   0.01454 *  
supp          7.25  1   2.0785   0.16486    
diet:supp     0.00  1   0.0006   0.98005    
Residuals    69.72 20                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Under reference coding these main-effect rows do not test the average effect one usually wants; they test the effect at the reference level of the other factor (a simple effect). To obtain a correct Type III test we refit the same model with sum-to-zero contrasts:

mod_sum <- lm(response ~ diet * supp, data = dat,
              contrasts = list(diet = contr.sum, supp = contr.sum))
Anova(mod_sum, type = "III")

Anova Table (Type III tests)

Response: response
            Sum Sq Df  F value    Pr(>F)    
(Intercept) 3479.7  1 998.1784 < 2.2e-16 ***
diet          51.7  1  14.8209 0.0009994 ***
supp          17.1  1   4.9035 0.0385802 *  
diet:supp      0.0  1   0.0006 0.9800473    
Residuals     69.7 20                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The main-effect rows are very different between the two tables: in this example the supp effect even flips from clearly non-significant under the default coding to significant under sum contrasts. Only the sum-contrast version answers the intended “average main effect” question. The interaction row, by contrast, is identical in both tables, because the highest-order term is always coding-invariant.

Type III without sum contrasts is almost always wrong

If you need Type III sums of squares, set the contrasts before or at the time of fitting: either options(contrasts = c("contr.sum", "contr.poly")) before the lm() call, or pass contrasts = list(factor1 = contr.sum, factor2 = contr.sum) to lm() as we did above. The contrasts are baked into the model matrix at fit time, so changing the option after fitting has no effect: you must refit. Note also that making a factor ordered (which switches it to contr.poly) is not a safe substitute once the factor has three or more levels: contr.poly is orthogonal but does not reproduce the equally-weighted Type III hypothesis for such factors.

A shortcut that avoids the contrast question entirely

If you would rather not think about contrast coding at all, emmeans::joint_tests() returns coding-invariant Type III tests directly. It gives the correct answer regardless of how the model was fitted, even from the default-contrast fit mod_ds:

joint_tests(mod_ds)

 model term df1 df2 F.ratio p.value
 diet         1  20  14.821  0.0010
 supp         1  20   4.903  0.0386
 diet:supp    1  20   0.001  0.9800

The F-ratios match Anova(mod_sum, type = "III") above, because joint_tests() works from the estimated marginal means (each level averaged equally over the other factor) rather than from the raw model coefficients. In practice this is often the most robust way to obtain a Type III table.

Side-by-side Comparison

To make the differences plain, we collect the p-values for all three types into a single tidy table. Everything is computed on the sum-contrast model mod_sum, so the Type III column is the correct one:

get_p <- function(model, term, type) {
  tbl <- if (type == "I") {
    broom::tidy(anova(model))
  } else {
    broom::tidy(Anova(model, type = type))
  }
  tbl %>% filter(term == !!term) %>% pull(p.value)
}

terms <- c("diet", "supp", "diet:supp")
types <- c("I", "II", "III")

crossing(term = terms, type = types) %>%
  rowwise() %>%
  mutate(p = get_p(mod_sum, term, type)) %>%
  ungroup() %>%
  pivot_wider(names_from = type, values_from = p,
              names_prefix = "Type ") %>%
  mutate(across(starts_with("Type"), \(x) round(x, 4)))

# A tibble: 3 × 4
  term      `Type I` `Type II` `Type III`
  <chr>        <dbl>     <dbl>      <dbl>
1 diet        0         0.001      0.001 
2 diet:supp   0.98      0.98       0.98  
3 supp        0.0363    0.0363     0.0386

Note that the interaction row (diet:supp) is identical across all three types — the highest-order interaction is always tested the same way. The differences are in the main effects.

Which Type Should One Use?

There is no single answer that satisfies every situation, but the following guidelines cover the overwhelming majority of applied analyses:

For balanced data with no covariate, the choice does not matter numerically. Type I is the simplest report and is perfectly adequate. (A numeric covariate breaks the orthogonality even under perfect balance, so the types can then differ despite equal cell counts.)
For unbalanced data with main effects only, Type II is the statistically most powerful option and avoids the arbitrary order dependence of Type I. Langsrud (2003) gives a careful argument for preferring Type II over Type III in this setting; the core point is that Type II exploits the assumption of no interaction (which one is making anyway, if no interaction is in the model) to recover more degrees of freedom.
For unbalanced data with interactions that are of scientific interest, Type III is the conventional choice in many fields (and the SAS default, which is why it is so widespread). It tests each main effect adjusted for the interaction, which matches the “what is the average effect of A, averaging over levels of B” interpretation — but only with sum-to-zero contrasts.
For mixed models (e.g. lmerTest::lmer), the question is more subtle and is usually handled via Satterthwaite or Kenward-Roger approximations rather than by choosing an SS type. Those are covered in the mixed-models material of this course.

In plain words: which average are you comparing?

The choice between Type II and Type III is, at heart, a choice of which marginal mean the main-effect test refers to. Type III compares the equally weighted marginal means — each level of a factor averaged over the other factor with equal weight, as if the design had been balanced. Type II compares the n-weighted marginal means — each level averaged as the data actually fell. So the honest question is rarely “which sum of squares” but “do I want each treatment averaged fairly (equal weight) or as observed (n-weighted)?”. When the unbalance is accidental (a lost plot, a dead plant), equal weighting usually recovers the question one meant to ask; when the unbalance is deliberate (proportional allocation, checks in an augmented design), n-weighting can be the more honest summary.

A practical workflow is to fit the model, check assumptions (see A1. Model Diagnostics), then report Type II by default, and switch to Type III only when the interaction is both present and scientifically meaningful. Whatever is reported, the chosen type and the contrast coding should be stated explicitly in the methods section, because the same model can produce three different ANOVA tables.

One caveat outranks the type choice altogether: when the interaction is itself significant, no type produces a main-effect p-value that is straightforward to interpret, because a main effect has no clean meaning while the interaction is real. In that case the analysis moves on to simple effects or emmeans contrasts rather than a main-effect ANOVA row.

Additional Resources

Langsrud (2003) — clear statistical argument for preferring Type II over Type III for unbalanced data.
Fox, J. and Weisberg, S. (2019), An R Companion to Applied Regression, 3rd ed. — discussion of Type II and Type III in the car package context. See also ?car::Anova.
Anova - Type I/II/III SS explained
How to interpret Type I, II, and III ANOVA? (CrossValidated)

Key Takeaways

Balanced data: all three types agree (unless a covariate is in the model, which breaks the orthogonality even under perfect balance). Worrying about the type only becomes necessary once the design is unbalanced.
stats::anova() is Type I and depends on the order of terms in the formula. Two scientifically equivalent models can produce different p-values.
car::Anova() provides Type II and Type III and is order-independent. Type II is additionally coding-invariant; Type III is not.
Type III needs sum-to-zero contrasts. Use options(contrasts = c("contr.sum", "contr.poly")) before fitting, otherwise the main-effect tests are not what they appear to be. As a robust alternative, emmeans::joint_tests() returns the correct Type III tests regardless of the coding.
Default recommendation: Type II for most unbalanced designs without a scientifically central interaction, Type III when interactions are central. Always report which type and which contrasts were used.

References

Langsrud, Øyvind. 2003. “ANOVA for Unbalanced Data: Use Type II Instead of Type III Sums of Squares.” Statistics and Computing 13 (2): 163–67. https://doi.org/10.1023/A:1023260610025.

Citation

BibTeX citation:

@online{schmidt2026,
  author = {{Dr. Paul Schmidt}},
  title = {A3. {ANOVA} {Types} {(I,} {II,} {III)}},
  date = {2026-08-13},
  url = {https://biomathcontent.netlify.app/content/lin_mod_exp/a3_anovatypes.html},
  langid = {en}
}

For attribution, please cite this work as:

Dr. Paul Schmidt. 2026. “A3. ANOVA Types (I, II, III).” August 13, 2026. https://biomathcontent.netlify.app/content/lin_mod_exp/a3_anovatypes.html.