Lecture 04: ANOVA Assumptions Checking
Experimental Design in Education
Jihong Zhang*, Ph.D
Educational Statistics and Research Methods (ESRM) Program*
University of Arkansas
2025-02-10
Class Outline
- Go through three assumptions of ANOVA and their checking statistics
- Post-hoc test for more group comparisons.
- Example: Intervention and Verbal Acquisition
- After-class Exercise: Effect of Sleep Duration on Cognitive Performance
ANOVA Procedure
- Set hypotheses:
- Null hypothesis (\(H_0\)): All group means are equal.
- Alternative hypothesis (\(H_A\)): At least one group mean differs.
- Determine statistical parameters:
- Significance level \(\alpha\)
- Degrees of freedom for between-group (\(df_b\)) and within-group (\(df_w\))
- Find the critical F-value.
- Compute test statistic:
- Calculate F-ratio based on between-group and within-group variance.
- Compare results:
- Either compare \(F_{\text{obs}}\) with \(F_{\text{crit}}\) or \(p\)-value with \(\alpha\).
- If \(p < \alpha\), reject \(H_0\).
ANOVA Type I Error
- The way to compare the means of multiple groups (i.e., more than two groups)
- Compared to multiple t-tests, the ANOVA does not inflate Type I error rates.
- For F-test in ANOVA, Type I error rate is not inflated
- Type I error rate is inflated when we do multiple comparison
- Family-wise error rate: probability of at least one Type I error occur = \(1 − (1 − \alpha)^c\) where c =number of tests. Frequently used FWER methods include Bonferroni or FDR.
Family-Wise Error Rate
- The Family-Wise Error Rate (FWER) is the probability of making at least one Type I error (false positive) across a set of multiple comparisons.
- In Analysis of Variance (ANOVA), multiple comparisons are often necessary when analyzing the differences between group means. FWER control is crucial to ensure the validity of statistical conclusions.
1 ANOVA Assumptions
Overview
- Like all statistical tests, ANOVA requires certain assumptions to be met for valid conclusions:
- Independence: Observations are independent of each other.
- Normality: The residuals (errors) follow a normal distribution.
- Homogeneity of variance (HOV): The variance within each group is approximately equal.
Importance of Assumptions
- If assumptions are violated, the results of ANOVA may not be reliable.
- We call it Robustness as to what degree ANOVAs are not influenced by the violation of assumption.
- Typically:
- ANOVA is robust to minor violations of normality, especially for large sample sizes (Central Limit Theorem).
- Not robust to violations of independence—if independence is violated, ANOVA is inappropriate.
- Moderately robust to HOV violations if sample sizes are equal.
Assumption 1: Independence
- Definition: Each observation should be independent of others.
- Violations:
- Clustering of data (e.g., repeated measures).
- Participants influencing each other (e.g., classroom discussions).
- Check (Optional): Use the Durbin-Watson test.
- Solution: Repeated Measured ANOVA, Mixed ANOVA, Multilevel Model
- Consequences: If independence is violated, ANOVA results are not valid.
Durbin-Watson (DW) Test
Note
The Durbin-Watson test is primarily used for detecting autocorrelation in time-series data.
In the context of ANOVA with independent groups, residuals are generally assumed to be independent.
- However, it’s still good practice to check this assumption, especially if there’s a reason to suspect potential autocorrelation.
Important
- Properties of DW Statistic:
- \(H_0\): Independence of residual is satisfied.
- Ranges from 0 to 4.
- A value around 2 suggests no autocorrelation.
- Values approaching 0 indicate positive autocorrelation.
- Values toward 4 suggest negative autocorrelation.
- P-value
- P-Value: A small p-value (typically < 0.05) indicates violation of independency
lmtest::dwtest()
Performs the Durbin-Watson test for autocorrelation of disturbances.
The Durbin-Watson test has the null hypothesis that the autocorrelation of the disturbances is 0.
This results suggest there are no autocorrelation at the alpha level of .05.
Assumption 2: Normality
- The dependent variable (DV) should be normally distributed within each group.
- Assessments:
- Graphical methods: Histograms, Q-Q plots.
- Statistical tests:
- Shapiro-Wilk test (common)
- Kolmogorov-Smirnov (KS) test (for large samples)
- Anderson-Darling test (detects kurtosis issues).
- Robustness:
- ANOVA is robust to normality violations for large samples.
- If normality is violated, consider transformations or non-parametric tests.
Normality within Each Group
Assume that the DV (Y) is distributed normally within each group for ANOVA
ANOVA is robust to minor violations of normality
- So generally start with homogeneity, then assess normality
shapiro.test()
set.seed(123) # For reproducibility
sample_normal_data <- rnorm(200, mean = 50, sd = 10) # Generate normal data
sample_nonnormal_data <- runif(200, min = 1, max = 10) # Generate non-normal data
shapiro.test(sample_normal_data)
Shapiro-Wilk normality test
data: sample_normal_data
W = 0.99076, p-value = 0.2298
Shapiro-Wilk normality test
data: sample_nonnormal_data
W = 0.95114, p-value = 2.435e-06# Perform Kolmogorov-Smirnov test against a normal distribution
ks.test(scale(sample_normal_data), "pnorm")
Asymptotic one-sample Kolmogorov-Smirnov test
data: scale(sample_normal_data)
D = 0.054249, p-value = 0.5983
alternative hypothesis: two-sided
Asymptotic one-sample Kolmogorov-Smirnov test
data: scale(sample_nonnormal_data)
D = 0.077198, p-value = 0.1843
alternative hypothesis: two-sidedAssumption 3: Homogeneity of Variance (HOV)
- Variance across groups should be equal.
- Assessments:
- Levene’s test: Tests equality of variances.
- Brown-Forsythe test: More robust to non-normality.
- Bartlett’s test: For data with normality.
- Boxplots: Visual inspection.
- What if violated?
- Welch’s ANOVA (adjusted for variance differences).
- Transforming the dependent variable.
- Using non-parametric tests (e.g., Kruskal-Wallis).
Practical Considerations
## Computes Levene's test for homogeneity of variance across groups.
car::leveneTest(outcome ~ group, data = data)
## Boxplots to visualize the variance by groups
boxplot(outcome ~ group, data = data)
## Brown-Forsythe test
onewaytests::bf.test(outcome ~ group, data = data)
## Bartlett's Test
bartlett.test(outcome ~ group, data = data)- Levene’s test and Brown-Forsythe test are often preferred when data does not meet the assumption of normality, especially for small sample sizes.
- Bartlett’s test is most powerful when the data is normally distributed and when sample sizes are equal across groups. However, it can be less reliable if these assumptions are violated.
Decision Tree
library(ellmer)
chat = ellmer::chat_ollama(model = "llama3.2")
chat$chat("How large can be considered as large sample size in one-way ANOVA")In one-way ANOVA (Analysis of Variance), the sample size is a critical factor
that affects the analysis's power and accuracy. While there isn't a universally
agreed-upon "large" sample size, here are some general guidelines:
1. **Minimum sample size**: The minimum sample size for one-way ANOVA can vary
depending on the effect size and level of significance desired. A common rule
of thumb is to have at least 10-15 observations per group (i.e., more than 5
groups).
2. **Moderate sample size**: For a moderate sample size, many researchers use
values around:
* 30-50 observations per group for small effect sizes (<0.1) and moderate
significance levels (e.g., α=0.05).
* 60-80 observations per group for medium effect sizes (≈ 0.1-0.3) and
modest significance levels.
* 100 observation s per group for larger effect sizes (> 0.3) or more
stringent significance levels.
3. **Large sample size**: When there are:
* Very small cells (≈ 10 observation s per group): often, the recommended
approach would be to conduct pairwise comparisons using post-hoc tests like
Bonferroni, Newman-Keuls, or Tukey's honestly significant difference (HSD)
test.
For larger samples (>=50 observations s per group), you should consider
doing an ANOVA. To see if there is a "real" difference in mean values between 2
samples, use the F-statistic to check for significance:
4. **Effect size**: Keep in mind that no sample size is "large enough" for a
given effect size. The required sample size depends on the magnitude of the
expected effect.
Keep in mind that these are general guidelines and may not always hold true
depending on your specific research design, data distribution, and assumptions
(e.g., homogeneity of variance).
Always consult with a knowledgeable statistical consultant and consider various
sources if you're unsure about your study's sample size or need additional
guidance.ANOVA Robustness
Robust to:
- Minor normality violations (for large samples).
- Small HOV violations if group sizes are equal.
Not robust to:
- Independence violations—ANOVA is invalid if data points are dependent.
- Severe HOV violations—Type I error rates become unreliable.
The robustness of assumptions is something you should be carefull before/when you perform data collection. They are not something you can do after data collection has been finished.
Robustness to Violations of Normality Assumption
ANOVA assumes that the residuals (errors) are normally distributed within each group.
However, ANOVA is generally robust to violations of normality, particularly when the sample size is large.
Theoretical Justification: This robustness is primarily due to the Central Limit Theorem (CLT), which states that, for sufficiently large sample sizes (typically \(n≥30\) per group), the sampling distribution of the mean approaches normality, even if the underlying population distribution is non-normal.
This means that, unless the data are heavily skewed or have extreme outliers, ANOVA results remain valid and Type I error rates are not severely inflated.
Robustness to Violations of Homogeneity of Variance
The homogeneity of variance (homoscedasticity) assumption states that all groups should have equal variances. ANOVA can tolerate moderate violations of this assumption, particularly when:
Sample sizes are equal (or nearly equal) across groups – When groups have equal sample sizes, the F-test remains robust to variance heterogeneity because the pooled variance estimate remains balanced.
The degree of variance heterogeneity is not extreme – If the largest group variance is no more than about four times the smallest variance, ANOVA results tend to remain accurate.
ANOVA: Lack of Robustness to Violations of Independence of Errors
The assumption of independence of errors means that observations within and between groups must be uncorrelated. Violations of this assumption severely compromise ANOVA’s validity because:
- Inflated Type I error rates – If errors are correlated (e.g., due to clustering or repeated measures), standard errors are underestimated, leading to an increased likelihood of falsely rejecting the null hypothesis.
- Biased parameter estimates – When observations are not independent, the variance estimates do not accurately reflect the true variability in the data, distorting F-statistics and p-values.
- Common sources of dependency – Examples include nested data (e.g., students within schools), repeated measurements on the same subjects, or time-series data. In such cases, alternatives like mixed-effects models or generalized estimating equations (GEE) should be considered.
2 Omnibus ANOVA Test
Overview
- What does it test?
- Whether there is at least one significant difference among means.
- Limitation:
- Does not tell which groups are different.
- Solution:
- Conduct post-hoc tests.
Individual Comparisons of Means
- If ANOVA is significant, follow-up tests identify where differences occur.
- Types:
- Planned comparisons: Defined before data collection.
- Unplanned (post-hoc) comparisons: Conducted after ANOVA.
Planned vs. Unplanned Comparisons
- Planned:
- Based on theory.
- Can be done even if ANOVA is not significant.
- Unplanned (post-hoc):
- Data-driven.
- Only performed if ANOVA is significant.
Types of Unplanned Comparisons
- Common post-hoc tests:
- Fisher’s LSD
- Bonferroni correction or Adjusted p-values
- Sidák correction
- Tukey’s HSD
Fisher’s LSD
- Least Significant Difference test.
- Problem: Does not control for multiple comparisons (inflated Type I error).
Bonferroni Correction
- Adjusts alpha to reduce Type I error.
- New alpha: \(\alpha / c\) (where \(c\) is the number of comparisons).
- Conservative: Less power, avoids false positives.
Family-wise Error Rate (adjusted p-values)
- Adjusts p-values to reduce Type I error
- Report adjusted p-values (typically larger that original p-values)
Tukey’s HSD
- Controls for Type I error across multiple comparisons.
- Uses a q-statistic from a Tukey table.
- Preferred when all pairs need comparison.
ANOVA Example: Intervention and Verbal Acquisition
Background
- Research Question: Does an intensive intervention improve students’ verbal acquisition scores?
- Study Design:
- 4 groups: Control, G1, G2, G3 (treatment levels).
- Outcome variable: Verbal acquisition score (average of three assessments).
- Hypotheses:
- \(H_0\): No difference in verbal acquisition scores across groups.
- \(H_A\): At least one group has a significantly different mean.
Step 1: Generate Simulated Data in R
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.1.4 ✔ readr 2.1.5
✔ forcats 1.0.0 ✔ stringr 1.5.1
✔ ggplot2 3.5.1 ✔ tibble 3.2.1
✔ lubridate 1.9.4 ✔ tidyr 1.3.1
✔ purrr 1.0.4
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors# Set seed for reproducibility
set.seed(123)
# Generate synthetic data for 4 groups
data <- tibble(
group = rep(c("Control", "G1", "G2", "G3"), each = 30),
verbal_score = c(
rnorm(30, mean = 70, sd = 10), # Control group
rnorm(30, mean = 75, sd = 12), # G1
rnorm(30, mean = 80, sd = 10), # G2
rnorm(30, mean = 85, sd = 8) # G3
)
)
# View first few rows
head(data)# A tibble: 6 × 2
group verbal_score
<chr> <dbl>
1 Control 64.4
2 Control 67.7
3 Control 85.6
4 Control 70.7
5 Control 71.3
6 Control 87.2Step 2: Summary Statistics
Step 3: Check ANOVA Assumptions
Assumption Check 1: Independence of residuals Check
# Fit the ANOVA model
anova_model <- lm(verbal_score ~ group, data = data)
# Install lmtest package if not already installed
# install.packages("lmtest")
# Load the lmtest package
library(lmtest)Loading required package: zoo
Attaching package: 'zoo'The following objects are masked from 'package:base':
as.Date, as.Date.numeric# Perform the Durbin-Watson test
dw_test_result <- dwtest(anova_model)
# View the test results
print(dw_test_result)
Durbin-Watson test
data: anova_model
DW = 2.0519, p-value = 0.5042
alternative hypothesis: true autocorrelation is greater than 0- Interpretation:
- In this example, the
DWvalue is close to 2, and the p-value is greater than 0.05, indicating no significant autocorrelation in the residuals.
- In this example, the
Assumption Check 2: Normality Check
# Shapiro-Wilk normality test for each group
data %>%
group_by(group) %>%
summarise(
shapiro_p = shapiro.test(verbal_score)$p.value
)# A tibble: 4 × 2
group shapiro_p
<chr> <dbl>
1 Control 0.797
2 G1 0.961
3 G2 0.848
4 G3 0.568- Interpretation:
- If \(p>0.05\), normality assumption is not violated.
- If \(p<0.05\), data deviates from normal distribution.
- Since the data meets the normality requirement and no outliers, we can use Bartlett’s or Levene’s tests for HOV checking.
- Alternative Check: Q-Q Plot
Assumption Check 3: Homogeneity of Variance (HOV) Check
Warning in leveneTest.default(y = y, group = group, ...): group coerced to
factor.Levene's Test for Homogeneity of Variance (center = median)
Df F value Pr(>F)
group 3 1.1718 0.3236
116
Bartlett test of homogeneity of variances
data: verbal_score by group
Bartlett's K-squared = 3.5385, df = 3, p-value = 0.3158
Brown-Forsythe Test (alpha = 0.05)
-------------------------------------------------------------
data : verbal_score and group
statistic : 14.32875
num df : 3
denom df : 109.9888
p.value : 6.030497e-08
Result : Difference is statistically significant.
------------------------------------------------------------- - Interpretation:
- If \(p>0.05\), variance is homogeneous (ANOVA assumption met).
- If \(p<0.05\), variance differs across groups (consider Welch’s ANOVA).
- It turns out our data does not violate the homogeneity of variance assumption.
Step 4: Perform One-Way ANOVA
Df Sum Sq Mean Sq F value Pr(>F)
group 3 3492 1164.1 14.33 5.28e-08 ***
Residuals 116 9424 81.2
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- Interpretation:
- If \(p<0.05\), at least one group mean is significantly different.
- If \(p>0.05\), fail to reject \(H0\) (no significant differences).
Step 5: Post-Hoc Tests (Tukey’s HSD)
diff lwr upr p adj
G1-Control 7.611 1.545 13.677 0.008
G2-Control 10.715 4.649 16.782 0.000
G3-Control 14.720 8.654 20.786 0.000
G2-G1 3.104 -2.962 9.170 0.543
G3-G1 7.109 1.042 13.175 0.015
G3-G2 4.005 -2.062 10.071 0.318- Interpretation:
- Identifies which groups differ.
- If \(p<0.05\), the groups significantly differ.
- G1-Control
- G2-Control
- G3-Control
- G3-G1
multcomp: Family-wise Error Rate Control
Other multicomparison method allow you to choose which method for adjust p-values.
Loading required package: mvtnormLoading required package: survivalLoading required package: TH.dataLoading required package: MASS
Attaching package: 'MASS'The following object is masked from 'package:dplyr':
select
Attaching package: 'TH.data'The following object is masked from 'package:MASS':
geyser# install.packages("multcomp")
### set up multiple comparisons object for all-pair comparisons
# head(model.matrix(anova_model))
comprs <- rbind(
"G1 - Ctrl" = c(0, 1, 0, 0),
"G2 - Ctrl" = c(0, 0, 1, 0),
"G3 - Ctrl" = c(0, 0, 0, 1),
"G2 - G1" = c(0, -1, 1, 0),
"G3 - G1" = c(0, -1, 0, 1),
"G3 - G2" = c(0, 0, -1, 1)
)
cht <- glht(anova_model, linfct = comprs)
summary(cht, test = adjusted("fdr"))
Simultaneous Tests for General Linear Hypotheses
Fit: aov(formula = verbal_score ~ group, data = data)
Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|)
G1 - Ctrl == 0 7.611 2.327 3.270 0.00283 **
G2 - Ctrl == 0 10.715 2.327 4.604 3.20e-05 ***
G3 - Ctrl == 0 14.720 2.327 6.325 2.94e-08 ***
G2 - G1 == 0 3.104 2.327 1.334 0.18487
G3 - G1 == 0 7.109 2.327 3.055 0.00419 **
G3 - G2 == 0 4.005 2.327 1.721 0.10555
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- fdr method)Side-by-Side comparison of two methods
TukeyHSD method
FDR method
- The differences in p-value adjustment between the
TukeyHSDmethod and themultcomppackage stem from how each approach calculates and applies the multiple comparisons correction. Below is a detailed explanation of these differences
Comparison of Differences
| Feature | TukeyHSD() (Base R) |
multcomp::glht() |
|---|---|---|
| Distribution Used | Studentized Range (q-distribution) | t-distribution |
| Error Rate Control | Strong FWER control | Flexible error control |
| Simultaneous Confidence Intervals | Yes | Typically not (depends on method used) |
| Adjustment Method | Tukey-Kramer adjustment | Single-step, Westfall, Holm, Bonferroni, etc. |
| P-value Differences | More conservative (larger p-values) | Slightly different due to t-distribution |
Step 6: Reporting ANOVA Results
Result
We first examined three assumptions of ANOVA for our data as the preliminary analysis. According to the Durbin-Watson test, the Shapiro-Wilk normality test, and the Bartletts’ test, the sample data meets all assumptions of the one-way ANOVA modeling.
A one-way ANOVA was then conducted to examine the effect of three intensive intervention methods (Control, G1, G2, G3) on verbal acquisition scores. There was a statistically significant difference between groups, \(F(3,116)=14.33\), \(p<.001\).
To further examine which intervention method is most effective, we performed Tukey’s post-hoc comparisons. The results revealed that all three intervention methods have significantly higher scores than the control group (G1-Ctrl: p = .003; G2-Ctrl: p < .001; G3-Ctrl: p < .001). Among three intervention methods, G3 seems to be the most effective. Specifically, G3 showed significantly higher scores than G1 (p = .004). However, no significant difference was found between G2 and G3 (p = .105).
Discussion
These findings suggest that higher intervention intensity improves verbal acquisition performance, which is consistent with prior literatures [xxxx/references]
3 Aftre-Class Exercise: Effect of Sleep Duration on Cognitive Performance
Background
Research Question:
- Does the amount of sleep affect cognitive performance on a standardized test?
Study Design
- Independent variable: Sleep duration (3 groups: Short (≤5 hrs), Moderate (6-7 hrs), Long (≥8 hrs)).
- Dependent variable: Cognitive performance scores (measured as test scores out of 100).
Data
# Set seed for reproducibility
set.seed(42)
# Generate synthetic data for sleep study
sleep_data <- tibble(
sleep_group = rep(c("Short", "Moderate", "Long"), each = 30),
cognitive_score = c(
rnorm(30, mean = 65, sd = 10), # Short sleep group (≤5 hrs)
rnorm(30, mean = 75, sd = 12), # Moderate sleep group (6-7 hrs)
rnorm(30, mean = 80, sd = 8) # Long sleep group (≥8 hrs)
)
)
# View first few rows
head(sleep_data)# A tibble: 6 × 2
sleep_group cognitive_score
<chr> <dbl>
1 Short 78.7
2 Short 59.4
3 Short 68.6
4 Short 71.3
5 Short 69.0
6 Short 63.9Go through all six steps.
Answer:
# Step 5: Perform One-Way ANOVA
anova_sleep <- aov(cognitive_score ~ sleep_group, data = sleep_data)
summary(anova_sleep) Df Sum Sq Mean Sq F value Pr(>F)
sleep_group 2 3764 1881.9 15.88 1.32e-06 ***
Residuals 87 10311 118.5
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1A one-way ANOVA was conducted to examine the effect of sleep duration on cognitive performance.
There was a statistically significant difference in cognitive test scores across sleep groups, \(F(2,87)=15.88\),\(p<.001\).
Tukey’s post-hoc test revealed that participants in the Long sleep group (M=81.52,SD=6.27) performed significantly better than those in the Short sleep group (M=65.68,SD=12.55), p<.01.
These results suggest that inadequate sleep is associated with lower cognitive performance.
ESRM 64503