Inferential Statistics
- Inferential Statistics – Definition, Types, Examples, Formulas
- Observational Studies and Experiments
- Sample and Population
- Sampling Bias
- Sampling Methods
- Research Study Design
- Population Distribution, Sample Distribution and Sampling Distribution
- Central Limit Theorem
- Point Estimates
- Confidence Intervals
- Introduction to Bootstrapping
- Bootstrap Confidence Interval
- Paired Samples
- Impact of Sample Size on Confidence Intervals
- Introduction to Hypothesis Testing
- Writing Hypotheses
- Hypotheses Test Examples
- Randomization Procedures
- p-values
- Type I and Type II Errors
- P-value Significance Level
- Issues with Multiple Testing
- Confidence Intervals and Hypothesis Testing
- Inference for One Sample
- Inference for Two Samples
- One-Way ANOVA
- Two-Way ANOVA
- Chi-Square Tests
Two-Way ANOVA
Two-Way ANOVA with Examples & When To Use It
ANOVA (Analysis of Variance) is a statistical test used to analyze the difference between the means of more than two groups.
A two-way ANOVA is a statistical method used to analyze the effects of two independent variables (or factors) on a continuous dependent variable. It is a type of ANOVA (Analysis of Variance) that allows us to examine how the interaction between two independent variables affects the dependent variable.
In a two-way ANOVA, each independent variable can have two or more levels, and the dependent variable must be continuous. The two independent variables are called factors. For example, if we were interested in how age and gender affect a test score, age and gender would be the two factors, and the test score would be the dependent variable.
The two-way ANOVA allows us to test for the main effects of each independent variable and their interaction effect. The main effects test whether each independent variable significantly affects the dependent variable, while the interaction effect tests whether the combination of the two independent variables has a significant effect on the dependent variable.
When to use a two-way ANOVA
A two-way ANOVA (Analysis of Variance) is used when you have two independent variables or factors and you want to examine how each of these variables affects the dependent variable, as well as the interaction between the two factors.
For example, let’s say you are interested in investigating the effect of a new drug on blood pressure, and you want to know if the effect differs between men and women. In this case, you have two independent variables: the drug treatment (with two or more levels, e.g., placebo vs. active drug) and gender (male vs. female), and your dependent variable is blood pressure. A two-way ANOVA would allow you to test for main effects of the drug and gender, as well as the interaction between them.
In general, a two-way ANOVA is appropriate when you want to test for the effects of two factors on a continuous outcome variable. This type of analysis can help you identify any significant interactions between the factors and determine the unique contributions of each factor to the outcome variable. It is commonly used in experimental research and can provide important insights into the factors that influence a particular outcome.
How does the ANOVA test generally work?
The ANOVA test assumes that the data is normally distributed and that the variances of the groups are approximately equal. To perform the test, the following steps are typically taken:
- Calculate the mean and variance of each group.
- Calculate the grand mean, which is the mean of all the data points from all the groups combined.
- Calculate the sum of squares between groups (SSB), which measures the variation between the group means and the grand mean.
- Calculate the sum of squares within groups (SSW), which measures the variation within each group.
- Calculate the F-statistic by dividing the SSB by the SSW and adjusting for the degrees of freedom for each term.
- Determine the p-value associated with the F-statistic and compare it to a significance level (e.g., 0.05).
- If the p-value is less than the significance level, then the null hypothesis (i.e., that there is no significant difference between the group means) is rejected in favor of the alternative hypothesis (i.e., that there is a significant difference between the group means).
If the ANOVA test indicates that there is a significant difference between the group means, further post-hoc tests (such as Tukey’s HSD or Bonferroni correction) may be performed to determine which groups differ significantly from each other. ANOVA is commonly used in experimental research and can help researchers draw meaningful conclusions from their data and make informed decisions about their experimental designs.
Assumptions of the two-way ANOVA
The assumptions of the two-way ANOVA include:
- Normality: The data should be normally distributed within each group and for each combination of the two independent variables.
- Homogeneity of variances: The variances of the dependent variable should be equal across all groups and for each combination of the two independent variables.
- Independence: The observations should be independent of each other.
- Random sampling: The sample should be selected randomly from the population of interest.
- Additivity: The effect of one independent variable should be independent of the effect of the other independent variable.
- No multicollinearity: The two independent variables should not be highly correlated with each other.
- Sphericity: The variances of the differences between all pairs of conditions should be equal.
These assumptions are important to ensure the validity and reliability of the results obtained from the two-way ANOVA. Violation of these assumptions may lead to inaccurate or unreliable results. It is important to check these assumptions before conducting the two-way ANOVA and to use appropriate statistical tests to address violations of these assumptions if they are present.
Example – Two-Way ANOVA
Suppose a researcher wants to study how two factors, fertilizer type and temperature, affect the growth of a particular crop. The researcher has three types of fertilizers (A, B, and C) and two temperature levels (high and low) to test. The dependent variable is the height of the crop.
To conduct a Two-Way ANOVA, the researcher would randomly assign the crops to the six possible treatment combinations (3 types of fertilizers x 2 temperature levels), with several crops in each treatment group. After a certain period, the researcher would measure the height of the crops in each treatment group.
The results of the Two-Way ANOVA will show whether there is a significant main effect of fertilizer type, temperature, or an interaction between the two. For example, the results might show that there is a significant main effect of fertilizer type, indicating that certain fertilizers resulted in significantly taller crops than others. The results might also show a significant main effect of temperature, indicating that crops grown at a higher temperature were significantly taller than those grown at a lower temperature. Finally, the results might show a significant interaction effect, indicating that the effects of the two factors (fertilizer type and temperature) interacted in a way that significantly affected crop height.
The output of a Two-Way ANOVA typically includes F-values, degrees of freedom, p-values, and effect sizes for each factor and the interaction. Researchers use these statistical measures to determine whether the results are significant and to interpret the direction and magnitude of the effects.
Suppose the researcher has measured the height of 5 crops in each of the six treatment groups, resulting in the following data:
| Fertilizer Type | Temperature | Crop Height |
|---|---|---|
| A | High | 8 |
| A | High | 9 |
| A | High | 10 |
| A | High | 11 |
| A | High | 12 |
| A | Low | 6 |
| A | Low | 7 |
| A | Low | 8 |
| A | Low | 9 |
| A | Low | 10 |
| B | High | 9 |
| B | High | 10 |
| B | High | 11 |
| B | High | 12 |
| B | High | 13 |
| B | Low | 7 |
| B | Low | 8 |
| B | Low | 9 |
| B | Low | 10 |
| B | Low | 11 |
| C | High | 10 |
| C | High | 11 |
| C | High | 12 |
| C | High | 13 |
| C | High | 14 |
| C | Low | 8 |
| C | Low | 9 |
| C | Low | 10 |
| C | Low | 11 |
| C | Low | 12 |
Step 1: Calculate the means
Calculate the mean crop height for each of the six treatment groups.
| Fertilizer Type | Temperature | Crop Height | Mean Crop Height |
|---|---|---|---|
| A | High | 8 | 10 |
| A | Low | 6 | 8 |
| B | High | 9 | 11 |
| B | Low | 7 | 9 |
| C | High | 10 | 12 |
| C | Low | 8 | 10 |
Step 2: Calculate the grand mean
Calculate the grand mean, which is the mean of all the crop height measurements.
Grand mean = (8+9+10+11+12+6+7+8+9+10+9+10+11+12+13+7+8+9+10+11+10+11+12+13+14+8+9+10+11+12)/30 = 9.8
Step 3: Calculate the Sums of Squares (SS)
To calculate the SS, we need to first calculate the deviations of each observation from the grand mean, and then square these deviations. We can then add up the squared deviations for each factor and the interaction, as well as for the error term.
First, calculate the deviation of each observation from the grand mean:
| Fertilizer Type | Temperature | Crop Height | Mean Crop Height | Deviation from Grand Mean |
|---|---|---|---|---|
| A | High | 8 | 10 | -2.8 |
| A | High | 9 | 10 | -1.8 |
| A | High | 10 | 10 | -0.8 |
| A | High | 11 | 10 | 0.2 |
| A | High | 12 | 10 | 1.2 |
| A | Low | 6 | 8 | -1.8 |
| A | Low | 7 | 8 | -0.8 |
| A | Low | 8 | 8 | 0.2 |
| A | Low | 9 | 8 | 1.2 |
| A | Low | 10 | 8 | 2.2 |
| B | High | 9 | 11 | -2.8 |
| B | High | 10 | 11 | -1.8 |
| B | High | 11 | 11 | -0.8 |
| B | High | 12 | 11 | 0.2 |
| B | High | 13 | 11 | 1.2 |
| B | Low | 7 | 9 | -2.8 |
| B | Low | 8 | 9 | -1.8 |
| B | Low | 9 | 9 | -0.8 |
| B | Low | 10 | 9 | 0.2 |
| B | Low | 11 | 9 | 1.2 |
| C | High | 10 | 12 | -2.8 |
| C | High | 11 | 12 | -1.8 |
| C | High | 12 | 12 | -0.8 |
| C | High | 13 | 12 | 0.2 |
| C | High | 14 | 12 | 1.2 |
Next, we square these deviations:
| Fertilizer Type | Temperature | Crop Height | Mean Crop Height | Deviation from Grand Mean | Squared Deviation from Grand Mean |
|---|---|---|---|---|---|
| A | High | 8 | 10 | -2.8 | 7.84 |
| A | High | 9 | 10 | -1.8 | 3.24 |
| A | High | 10 | 10 | -0.8 | 0.64 |
| A | High | 11 | 10 | 0.2 | 0.04 |
| A | High | 12 | 10 | 1.2 | 1.44 |
| A | Low | 6 | 8 | -1.8 | 3.24 |
| A | Low | 7 | 8 | -0.8 | 0.64 |
| A | Low | 8 | 8 | 0.2 | 0.04 |
| A | Low | 9 | 8 | 1.2 | 1.44 |
| A | Low | 10 | 8 | 2.2 | 4.84 |
| B | High | 9 | 11 | -2.8 | 7.84 |
| B | High | 10 | 11 | -1.8 | 3.24 |
| B | High | 11 | 11 | -0.8 | 0.64 |
| B | High | 12 | 11 | 0.2 | 0.04 |
| B | High | 13 | 11 | 1.2 | 1.44 |
| B | Low | 7 | 9 | -2.8 | 7.84 |
| B | Low | 8 | 9 | -1.8 | 3.24 |
| B | Low | 9 | 9 | -0.8 | 0.64 |
| B | Low | 10 | 9 | 0.2 | 0.04 |
| B | Low | 11 | 9 | 1.2 | 1.44 |
| C | High | 10 | 12 | -2.8 | 7.84 |
| C | High | 11 | 12 | -1.8 | 3.24 |
| C | High | 12 | 12 | -0.8 | 0.64 |
| C | High | 13 | 12 | 0.2 | 0.04 |
| C | High | 14 | 12 | 1.2 | 1.44 |
Finally, we calculate the sum of squares (SS) for each factor and the interaction:
- SS between fertilizers = 5.2 + 1.6 + 0.4 = 7.2
- SS between temperatures = 16.8 + 1.6 + 0.4 = 18.8
- SS interaction = 2.4 + 1.6 + 0.4 + 0.2 + 1.6 + 0.4 + 0.2 + 1.6 + 0.4 = 9.2
Next, we calculate the degrees of freedom (DF) for each factor and the interaction:
- DF between fertilizers = 2 – 1 = 1
- DF between temperatures = 2 – 1 = 1
- DF interaction = 1
Then, we calculate the mean squares (MS) for each factor and the interaction:
- MS between fertilizers = 7.2 / 1 = 7.2
- MS between temperatures = 18.8 / 1 = 18.8
- MS interaction = 9.2 / 1 = 9.2
Finally, we calculate the F-value for each factor and the interaction:
- F-value between fertilizers = 7.2 / 2.93 = 2.45
- F-value between temperatures = 18.8 / 2.93 = 6.42
- F-value interaction = 9.2 / 2.93 = 3.15
To determine if the results are statistically significant, we compare the F-values to the critical F-value for the corresponding degrees of freedom and level of significance. Assuming a significance level of 0.05 and the DFs calculated above, the critical F-value for the between-fertilizers factor