17: Basic Variance
17.1 Understanding Variance
Understanding variance and comparing group differences are fundamental aspects of statistical analysis in applied research. Variance tests help to determine whether the means or distributions of different groups or samples differ significantly. These tests are essential for comparing experimental groups, treatment effects, or any situation where you want to assess if the data shows significant differences. This chapter will explore several basic variance tests, including the One-Sample T-Test, the Independent Samples T-Test, the Mann-Whitney U Test, the One-Way ANOVA, and the Kruskal-Wallis Test. We will also delve into the assumptions for each test, which will help you understand when and how to apply them effectively in your research.
17.2 One-Sample T-Test
The One-Sample T-Test is used to determine whether the mean of a single sample is significantly different from a known or hypothesized population mean. This test is often used in cases where researchers want to compare the sample’s mean against a theoretical value or standard, such as testing whether the average weight of a population matches a known standard weight. The assumptions for this test include independence (the data points should be independent of each other), normality (the data should be approximately normally distributed), and scale of measurement (the data should be continuous, such as interval or ratio scales).
Running the One-Sample T-Test in Jamovi is straightforward. After opening your dataset, navigate to Analyses > T-Tests > One-Sample T-Test, select the variable you want to test, and input the hypothesized population mean. Upon running the test, Jamovi will generate results that include the t-statistic, degrees of freedom, and the p-value, which indicate whether the observed mean is significantly different from the hypothesized value.
How To: One-Sample T-Test
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Phrasing Results: One-Sample T-Test
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17.3 Independent Samples T-Test
The Independent Samples T-Test is used to compare the means of two independent groups to determine if there is a statistically significant difference between them. This test is beneficial when comparing two groups, such as treatment and control groups in a clinical study. The assumptions for the Independent Samples T-Test include independence of observations (the groups must be independent with no overlap of subjects), normality (the data in each group should be normally distributed), and homogeneity of variance (the variance within the two groups should be approximately equal, which can be tested using Levene’s test). The dependent variable should also be continuous.
To run the Independent Samples T-Test in Jamovi, open your dataset, go to Analyses > T-Tests > Independent Samples T-Test, select the grouping variable (e.g., control vs. treatment group) and the dependent variable (e.g., test scores). Jamovi will then calculate the t-statistic, degrees of freedom, and p-value to assess the difference between the two groups.
How To: Independent Samples T-Test
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Phrasing Results: Independent Samples T-Test
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17.4 Mann-Whitney U Test
The Mann-Whitney U Test is a non-parametric test used to determine whether there is a difference between two independent groups when the assumptions for the Independent Samples T-Test (such as normality) are not met. This test is ideal when the data are ordinal or the normality assumption is violated. The assumptions for the Mann-Whitney U Test include independence of groups (the groups being compared must be independent) and the use of ordinal or continuous data. The data does not need to be normally distributed.
To perform the Mann-Whitney U Test in Jamovi, open your dataset and go to Analyses > Non-Parametric Tests > Mann-Whitney U. After selecting the grouping variable and dependent variable, Jamovi will output the U statistic, p-value, and other relevant results to help you assess the significance of the difference between the groups.
How To: Mann-Whitney U Test
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Phrasing Results: Mann-Whitney U Test
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17.5 One-Way ANOVA
The One-Way ANOVA (Analysis of Variance) compares the means of three or more independent groups to determine if at least one of the group means differs significantly from the others. It tests the null hypothesis that all group means are equal. The assumptions for One-Way ANOVA include independence of observations (the groups being compared must be independent), normality (the data within each group should be approximately normally distributed), and homogeneity of variance (the variance within each group should be approximately equal).
Running One-Way ANOVA in Jamovi is simple: you navigate to Analyses > ANOVA > One-Way ANOVA, select the grouping variable (e.g., different treatment types) and the dependent variable (e.g., test scores). Jamovi will generate the ANOVA table, which includes the F-statistic, degrees of freedom, and p-value, helping you assess whether there is a statistically significant difference between the groups.
How To: One-Way ANOVA
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Phrasing Results: One-Way ANOVA
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17.6 Kruskal-Wallis Test
The Kruskal-Wallis Test is a non-parametric alternative to the One-Way ANOVA. It is used to compare the medians of three or more independent groups when the assumptions of One-Way ANOVA (normality and homogeneity of variance) are not met. The Kruskal-Wallis test is ideal for ordinal data or data that do not follow a normal distribution. The assumptions for the Kruskal-Wallis test include the independence of groups (the groups being compared must be independent) and the use of ordinal or continuous data.
To perform the Kruskal-Wallis Test in Jamovi, go to Analyses > Non-Parametric Tests > Kruskal-Wallis Test and select the grouping and dependent variables. Jamovi will calculate the H statistic and p-value and provide other relevant results to determine if at least one group differs significantly from the others.
How To: Kruskal-Wallis Test
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Below is an example of the results generated when the steps are correctly followed.
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Phrasing Results: Kruskal-Wallis Test
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17.7 Choosing the Right Test
Selecting the appropriate test for comparing group means depends on the nature of your data and the research question. If you are comparing the means of two independent groups, use the Independent Samples T-Test to see if the data meet the assumptions of normality and homogeneity of variance. If the data is not normally distributed or assumptions are not met, use the Mann-Whitney U Test as a non-parametric alternative.
The one-way ANOVA is suitable for comparing the means of three or more independent groups if the assumptions of normality and homogeneity of variance are met. If these assumptions are not met, the Kruskal-Wallis Test can be a non-parametric alternative.
The One-Sample T-Test is appropriate for comparing a sample mean to a known population mean.
Chapter 17 Summary and Key Takeaways
In this chapter, we explored several basic variance tests commonly used in applied research, including the One-Sample T-Test, which compares the mean of a sample to a known population mean; the Independent Samples T-Test, used to compare the means of two independent groups; the Mann-Whitney U Test, a non-parametric alternative to the independent samples t-test; the One-Way ANOVA, which compares the means of three or more independent groups; and the Kruskal-Wallis Test, a non-parametric alternative to One-Way ANOVA. By understanding the assumptions and appropriate contexts for each test, researchers are better equipped to select the most suitable method for comparing groups and analyzing differences in their data.
- T-tests and ANOVA are used for comparing group means, but they require certain assumptions, such as normality and homogeneity of variance.
- Non-parametric tests like the Mann-Whitney U Test and Kruskal-Wallis Test are used when the assumptions for parametric tests are not met.
- Correctly choosing the right test based on your data and research question is essential for valid and meaningful analysis.