One-Way ANOVA – When and How to Use It with Examples

22.03.23 Types of ANOVAs Time to read: 5min

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ANOVA, which stands for Analysis of Variance, is a statistical test used in statistics to discern and evaluate the variations in the average values among more than two groups. One-way ANOVA, a specific type of ANOVA, focuses on a single independent variance.

This blog post delves into the intricacies of One-way ANOVA.

One-Way ANOVA – In a Nutshell

  • One-way ANOVA is a statistical test that compares the means of more than two independent groups to determine if a significant statistical difference exists between the means of the corresponding populations.
  • One-way ANOVA applies one independent variable.
  • Performing One-way ANOVA tests is typically cumbersome. Therefore, you can use statistical software like Stata and SPSS.

Definition: One-way ANOVA

ANOVA, which is short for, Analysis of Variance is a statistical test that analyses the differences between the means of three or more groups. Therefore, one-way ANOVA is a statistical method that uses one independent variable to test the differences in the means of more than two groups.

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One-way ANOVA: Function

The key role of One-way ANOVA is to analyse data from field studies, experiments, and quasi-experiments. This statistical test helps determine if there is notable statistical evidence that proves a significant difference between the means of the associated populations.

Using One-way ANOVA

You can use One-way ANOVA when your collected data features one categorical independent variable and a single quantitative dependent variable. Furthermore, your independent variables should contain at least three levels. ANOVA will help you determine if the dependent variable changes depending on the independent variable level.

Examples

  • Internet use is your independent variable. So, you assign three groups (i.e., low, medium, and high levels of internet use).
  • Soda flavour is your independent variable. So, you can collect information on three flavours (i.e., strawberry, pineapple, and passion) to determine if there is a price difference.
  • The type of soil is your independent variable. So, you can plant crops in pots with three types of soil to determine the difference in the yield.

How does One-way ANOVA work?

A One-way ANOVA applies the following hypotheses:

  • Null hypothesis: μ1 = μ2 = μ3 = … = μk, this implies that all the population means are equal
  • Alternative hypothesis: at least one of the population means is not equal to the rest.

ANOVA typically works hand in hand with:

F-tests – they are great for statistical significance. These tests allow you to compare several means simultaneously. With f-tests, the error is calculated for an entyre group of comparisons instead of individually.

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One-way ANOVA: Assumptions

For the outcomes of One-way ANOVA to be valid, the assumptions below apply:

  • Normal distribution
    Each sample is obtained from a normally-distribution population.
  • Observation independence
    The data was obtained using statistically valid sampling techniques. Therefore, there are no hidden correlations between the observations.
  • Equality of variances
    The variance within all the groups in the comparison are similar in each group. Barlett’s test will help verify this assumption.

One-way ANOVA: Beginning

You can perform One-way ANOVA by hand. However, this approach is challenging and cumbersome with several observations. Therefore, the R statistical program may come in handy. This program is free, powerful, and readily available.

Example

Sample dataset from a crop yield experiment:

  • Type of soil (Type A, B, and C)
  • Planting density (X- high, Y- low density)
  • Planting pots (Pot, 1,2,3, and 4)
  • Final crop yield in the quantity of plants

You can use a one-way ANOVA to analyse only the effect of the soil type on the final crop yield. Use the program to load the data set in your R environment, then run an ANOVA test. With the program, you do not need to perform any calculations. Instead use, the command aov() to run an ANOVA.

One-way ANOVA: Interpretation of the results

You can cheque out a summary of a statistical model in R by utilizing the summary () command. It should provide the following summary R code:

Summary (one.way)

The output of an ANOVA usually gives you an estimate of the dependent variable variation that the independent variable can explain.

mean error The initial column lists the independent variable together with the model residuals also known as the model error.
Df The column labelled Sq showcases the total variation or sum of squares between the group and the overall mean expressed by the variable.
Sq The column labelled Sq showcases the total variation or sum of squares between the group and the overall mean expressed by the variable.
Mean Sq The column labelled Mean Sq showcases the sum squares’ means, which are found by dividing the summation of the squares by the freedom degrees.
F value The F value column represents the test statistic from the F test.
Pr(>F) The Pr(>F) column represents the F statistic’s p-value, which indicates the likelibonnet of the calculated F value occurring in the null hypothesis.

One-way ANOVA: Post-hoc testing

One-way ANOVA will show you if there is a difference in the independence variable levels. However, it does not tell you how significant the difference is. This is where the Post-hoc (Tukey’s Honestly-Significant Difference) test comes in. This test runs comparisons among each group in pairs using a conservative error estimate.

The Post-Hoc test code is: TukeyHSD(one.way)

  • The table reports the tested model and lists the differences among the groups in pairs.
  • The soil section shows the mean differences for each level.
  • The pairwise comparisons display the level with the highest yield.

One-way ANOVA: Reporting the test results

In one-way ANOVA, the reported results should feature a small description of the following: tested variables, f value, freedom degrees, and each independent variable’s p-value. The report should also have a detailed explanation of the mean of the results.

Example

We discovered a statistically significant difference in the crop yield depending on the soil type. The posthoc test showed a difference between soil type A and C.

FAQs

One-way ANOVA is a statistical test that compares the means of more than three independent groups to determine if a significant statistical difference exists between the means of the corresponding populations.

ANOVA (Analysis of Variance) is a statistical test that analyses the differences between the means of three or more groups.

You can use computer programs to run one-way ANOVA tests because doing so by hand can be cumbersome.

The difference between One-way ANOVA and two-way ANOVA is that the latter contains two independent variables.