📜 The Scholar’s Path: A Quest for Statistical Truth

As you step onto the Scholar’s Path, the air hums with the weight of ancient scrolls and modern datasets. Before you stands a towering Library of Comparisons, where the means of independent groups whisper their secrets to those who know how to listen.

Your mission, should you choose to accept it:

🔍 Decipher the language of t-tests and ANOVA
⚖️ Judge whether differences are real or mere chance
🎯 Wield effect sizes to reveal practical truths

"In this chapter, you’ll unlock the statistical tools that separate coincidence from causation—just like the great scholars of old."

Your Creative Mission

🚀 Task: Remix This Template with AI

Review the page design and then copy the HTML from the accordion dropdown near the bottom of the page.
Paste it into a generative AI tool (e.g., ChatGPT, DeepSeek).
Prompt: "Adapt this statistical methods page for [your discipline]. Keep the interactive elements but replace examples with relevant ones from my field (e.g., clinical trials, education research). Maintain the same visual style."
Test your AI-generated code in Canvas’s HTML editor!

💡

Try topics like "Comparing therapy outcomes" or "Athlete performance analysis"

Comparing Means of Independent Groups

Broadly, Comparing means is essential in determining if interventions or group characteristics significantly affect outcomes. For example, a study might compare the mean test scores of students taught by different methods to assess each method's effectiveness. Are the control group and intervention group broadly the same at baseline? Did the control or intervention group's data change from pre-testing at baseline to post-testing?

More specifically, sometimes we need to compare the means of independent groups to determine if there are significant differences between them. The example above of a control group and an intervention group having their means compared can give us the confidence to say that both groups were broadly similar in a specific respect before any interventions were conducted. Other examples include:

Evaluating the effectiveness of interventions or treatments: For example, comparing the anxiety levels of a group receiving a new therapy to a control group.
Assessing differences between distinct populations: Such as comparing graduate salaries based on gender.
Analysing the impact of different methods or approaches: For instance, comparing test scores of students who learned using two different teaching methods.
Medical research: Evaluating the efficacy of a new drug by comparing weight loss in treated and untreated groups.

The figure below shows 3 groups and their respective data points presented in a box-and-whisker plot. The statistical tests on this page are designed to show whether the differences between the groups are significant, or not.

📖 Introduction to Mean Comparison

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Mean comparison is a fundamental statistical technique used to assess differences between groups or conditions in research.

It allows researchers to determine if observed differences in sample means are statistically significant, providing valuable insights for decision-making and hypothesis testing.

There are different methods of comparing means and the choice of method depends on context such as:

how many groups or conditions there are
whether the data is the same group in different conditions/times or each of the groups is independent of each other
how "normal" the data is
the type of data it is
the sample size

Key points:

Purpose: To evaluate whether the average values of a variable differ significantly between two or more groups.
Common scenarios:
- Comparing means between two independent groups (e.g. Are the baseline means of the control group and intervention group similar enough to say that they are comparable?)
- Comparing means from paired or repeated measures (e.g. are the means of the data for the same group after the intervention different enough to say that a change has occurred as a result of the intervention?)
- Comparing a sample mean to a known population value
Statistical tests: The choice depends on the research design and data characteristics:
- t-tests (e.g. one-sample, independent samples, paired samples)
- Analysis of Variance (ANOVA) for multiple group comparisons
Considerations:
- Sample size and normality of data distribution
- Homogeneity of variances between groups
- Type I and Type II errors
Interpretation: Results typically include:
- p-values to assess statistical significance
- Effect sizes to determine the magnitude of differences

Understanding mean comparison techniques is crucial for researchers to draw valid conclusions from their data and contribute to evidence-based practices in their respective fields.

Image from: https://www.sthda.com/english/wiki/comparing-means-in-r

📖 Independent Samples t-test

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The independent samples t-test is a statistical method used to determine whether there is a significant difference between the means of two unrelated groups.

It is commonly used in research to compare outcomes between different populations or experimental conditions (e.g. control and intervention).

Key points:

Purpose: To compare means between two independent groups on a continuous dependent variable.
Assumptions:
- The data are numeric and continuous. This can be either interval or ratio data.
- Observations are independent of one another
- The sample mean is normally distributed
- Equal variances between groups (homogeneity of variance)
Hypotheses:
- Null hypothesis (H0): The population means from the two unrelated groups are equal
- Alternative hypothesis (HA): The population means are not equal
Interpretation:
- If the p-value is less than the significance level (typically 0.05), reject the null hypothesis
- This indicates a statistically significant difference between the two group means
Reporting results:
- Include the t-statistic, degrees of freedom, and p-value
- Example: t(19) = 4.37, p < .01
Considerations:
- Check for outliers and normality of data distribution
- Use Welch's t-test if the assumption of equal variances is violated

The independent samples t-test is a valuable tool for researchers comparing two groups, such as treatment and control groups in experimental studies or demographic differences in observational research.

Checking Distribution of Data

Parametric statistical tests (which most commonly used tests are) require data that meets certain assumptions. Parametric statistics use parameters such as the mean, median, standard deviation etc. This form of statistics uses the data that can be observed to estimate the parameters of the data distribution. If the data doesn’t meet the assumptions the parametric tests don’t work.

Normal distribution

The first assumption is that the data is normally distributed. This is described as a Gaussian distribution or more commonly a ‘bell curve’. With a normally distributed sample if we drew a line through the middle of the curve it would look the same on both sides.

The easiest way to examine the distribution of a data set is to plot a graph of how many times a value occurs within the data set. This is known as a frequency distribution or histogram. This is a graph which plots the values of observations on the horizontal axis with bars showing how many times each value occurred in the data set.

As you can see below, in a normal distribution the tallest bars (values with the highest frequency) occur in the middle of the distribution and the shortest bars (values with the lowest frequency) appear around the edges of the distribution.

Skewness and Kurtosis

There are two main ways that the distribution of a data set might deviate from a ‘normal distribution’. These are called skew (defined as lack of symmetry) and kurtosis (described as pointiness).

Skewed distributions are not symmetrical and the most frequent values (the tallest bars) are clustered at one end of the scale (one side of the histogram) and the values tail off (bars get smaller) on the opposite end of the scale (the other side of the histogram). A skewed distribution can be positively skewed (the tallest bars at the lower end of the scale) or negatively skewed (the tallest bars at the upper end of the scale).

As well as varying in their level of skew distributions also vary in their level of kurtosis. Kurtosis refers to the degree to which the values are clustered at the ends of the distribution (referred to as the ‘tails’ of the distribution. This is usually reflected in how ‘pointy’ a distribution is. A distribution that has a lot of values in the tails (called a heavy tailed distribution) is usually pointy. This is called a leptokurtic distribution and is said to have positive kurtosis. A distribution that is thin in the tails (has light tails) is usually flatter than normal. This is called a platykurtic distribution and is said to have negative kurtosis. If a data set has a normal distribution of data we call it a mesokurtic distribution.

Checking normality in JASP

When you conduct an independent t-test in JASP you can select data for the Shapiro-Wilks test and Levene’s test which are tests for normality. If both of these tests return a p value of > 0.05 then you can assume a normal distribution.

If a distribution has values of skew or kurtosis above zero then this distribution has positive skew or kurtosis. If a distribution has values of skew or kurtosis below zero then this distribution has negative skew or kurtosis. A value of zero indicates a normal distribution

Have A Go!

Independent Samples t-test in MS Excel

If you are starting with the default version of MS Excel, you can download an additional Analysis Toolpak and run additional statistical analysis quite easily in this software. Find out more here: https://statisticsbyjim.com/hypothesis-testing/t-tests-excel/

Independent Samples t-test in JASP

The gif below is an example of how you can undertake this analysis in JASP. Have a go with your dataset and think about the best type of figure/plot for your data.

Find out more here: https://jasp-stats.org/2018/04/18/how-to-conduct-a-classical-independent-sample-t-test-in-jasp-and-interpret-the-results

To find out more about undertaking this analysis in JASP, read the section on Comparing Two Independent Groups in the following textbook

Goss-Sampson, M. A. (2024). Statistical Analysis in JASP 0.18.3: A Guide for Students. March 2024. (Free PDF) (Relevant Section)

📖 One-way ANOVA (Analysis of Variance)

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A One-way ANOVA is a statistical method used to compare the means of three or more independent groups to determine if there are any statistically significant differences between them

An Example from Sports Science:
A researcher wants to compare the effects of different training programmes on sprint performance in athletes. The study involves three groups: one using traditional sprint training, one using plyometric exercises, and one using resistance training. The researcher measures 100m sprint times after 8 weeks of training. One-way ANOVA could be used to determine if there are significant differences in sprint performance between the three training groups

An example from Sports Therapy:
A sports therapist aims to investigate the effectiveness of different rehabilitation protocols on ankle stability following an injury. The study includes three groups of athletes recovering from ankle sprains: one group using traditional physiotherapy, one using proprioceptive training, and one using a combination of both. The therapist measures ankle stability using the Star Excursion Balance Test after 6 weeks of rehabilitation. One-way ANOVA could be used to analyse if there are significant differences in ankle stability scores between the three rehabilitation groups

Key points:

Purpose: To test whether the means of multiple groups are equal
Null hypothesis (H0): All population means are equal (μ1 = μ2 = μ3 = ... = μk)
Alternative hypothesis (H1): At least one population mean is different from the others
Assumptions
- Independent observations
- Normality of dependent variable distribution
- Homogeneity of variance
Interpretation: If the test returns a statistically significant result, it indicates that at least two group means are significantly different from each other
Post-hoc tests: Required to determine which specific groups differ if ANOVA results are significant
Robustness: ANOVA is relatively robust to slight violations of normality, especially with larger sample sizes

One-way ANOVA in MS Excel

If you are starting with the default version of MS Excel, you can download an additional Analysis Toolpak and run additional statistical analysis quite easily in this software. Find out more here: https://www.datacamp.com/tutorial/excel-anova-guide

One-way ANOVA in JASP

To find out more about undertaking this analysis in JASP, read the section on Comparing More Than Two Independent Groups in the following textbook

- Goss-Sampson, M. A. (2024). Statistical Analysis in JASP 0.18.3: A Guide for Students. March 2024. (Free PDF) (Relevant Section)

See this gif and others related to ANOVA here: https://jasp-stats.org/how-to-use-jasp/

📖 Non-Parametric Alternatives

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The most common statistical methods work when the data being analysed is parametric.

Data is considered parametric when it meets specific assumptions and characteristics that allow for the use of parametric statistical methods.

Data is considered parametric when there is:

Normal Distribution: The data follows or closely approximates a normal (Gaussian) distribution. This "bell-shaped" curve is essential for many parametric tests.
Known Parameters: The data can be described by a fixed set of parameters, such as mean and standard deviation for normal distributions.
Homoscedasticity: The data exhibits similar variances across different groups being compared[3]. This is also known as homogeneity of variance.
Continuous (Interval or Ratio Data): Parametric methods typically work with continuous variables (e.g. height) rather than categorical (red, or blue) or ordinal (rank) data.
Large Sample Size: While not strictly necessary, parametric methods are generally more reliable with larger sample sizes (typically n ≥ 30).

When these conditions are met, parametric tests such as t-tests, ANOVA, and linear regression can be applied. These tests are often preferred due to their greater statistical power and ability to provide more detailed insights about the population parameters.

When the assumptions of parametric tests are violated, non-parametric alternatives can be used for comparing means. These tests are more robust to violations of normality and homogeneity of variance.

How do I know if my data is parametric/non-parametric?

Check for normal distribution: Parametric data typically follows a normal or "bell-shaped" distribution. You can visually check this by creating a histogram of your data. If it looks roughly symmetrical, it's likely parametric.
Look at your measurement scale: Parametric data is usually continuous (like height or weight) and measured on an interval or ratio scale. If your data is categorical or ordinal (like rankings), it's likely non-parametric.
Sample size: If you have a large sample size (generally more than 30), your data is more likely to be suitable for parametric tests, even if it's not perfectly normal.
Use statistical tests: There are formal tests like the Anderson-Darling or Shapiro-Wilk test that can help you determine if your data is normally distributed.

Key non-parametric alternatives include:

Mann-Whitney U Test
- Alternative to independent samples t-test
- Compares two independent groups
- See p62 in Goss-Sampson (2024) for this method in JASP https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf#page=66
Kruskal-Wallis H Test
- Alternative to one-way between-groups ANOVA
- Compares three or more independent groups
- See p103 in Goss-Sampson (2024) for this method in JASP https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf#page=107

Considerations: Non-parametric tests analyse ranks of scores rather than raw scores. They are less powerful than their parametric counterparts but more robust to assumption violations For complex designs (e.g., mixed ANOVA), there may not be direct non-parametric equivalents

When using non-parametric alternatives: Clearly state the reasons for choosing a non-parametric approach Report median and interquartile range instead of mean and standard deviation Interpret results in terms of distribution differences rather than mean differences

Find out More:

Parametric and Nonparametric: Demystifying the Terms - https://www.mayo.edu/research/documents/parametric-and-nonparametric-demystifying-the-terms/doc-20408960

📖 Effect size and Power Analysis

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Effect size and power analysis are crucial components in statistical research, helping researchers design studies and interpret results meaningfully.

Effect Size:

Effect size is a quantitative measure of the magnitude of a relationship between variables or the difference between groups. It provides information about the practical significance of findings, beyond mere statistical significance.

Key points about effect size:

Quantifies the strength of relationships or differences
Helps determine practical importance of findings
Can be used to compare results across different studies

Common effect size measures include:

Cohen's d: For comparing two means
Pearson's r: For correlations between variables
Odds ratio: For binary outcomes

Interpreting effect sizes (Cohen's guidelines):

Small: d = 0.2 or r = 0.1
Medium: d = 0.5 or r = 0.3
Large: d = 0.8 or r > 0.5

Find out more:

Effect Size on Wikipedia - https://en.wikipedia.org/wiki/Effect_size
To find out more about Effect Size in JASP, read the section on Effect Size in the following textbook:
- Goss-Sampson, M. A. (2024). Statistical Analysis in JASP 0.18.3: A Guide for Students. March 2024. (Free PDF) (Relevant Section)

Power Analysis:

Statistical power is the likelihood of a significance test detecting an effect when one truly exists.
Power analysis helps determine the sample size needed to detect an effect of a given size with a specified level of confidence.

Components of power analysis:

Statistical power (typically set at 80% or higher)
Sample size
Significance level (alpha, usually 5%)
Expected effect size

Benefits of power analysis:

Ensures statistical significance in results
Reduces the risk of Type II errors
Helps determine appropriate sample sizes for studies

Relationship between effect size and power:

Larger effect sizes are easier to detect and require smaller sample sizes
Smaller effect sizes require larger sample sizes to maintain adequate statistical power

Calculating Statistical Power:

GPower is free software with a graphical user interface that makes it easy to calculate sample size and power for various statistical methods, including F-tests, t-tests, χ2 tests, Z tests, and exact tests. It can be downloaded from www.psycho.uni-duesseldorf.de/abteilungen/aap/gpower3. GPower also provides effect size calculators and graphics options.

Here's an example of calculating statistical power for a two-independent groups t-test:

Open G*Power
Click on the "Test family" dropdown menu and select "t-tests".
In the "Statistical test" dropdown menu, choose "Means: Difference between two independent means (two groups)".
In the main window, you'll need to input the following parameters:
- Effect size: Let's say we expect a medium effect size of 0.5
- α (significance level): Typically set to 0.05
- Power (1-β): Let's aim for 80% power, so input 0.80
- Allocation ratio N2/N1: If the groups are of equal size, leave this as 1

After inputting these values, G*Power will calculate the required sample size for each group. In this case, it would show that you need approximately 64 participants in each group (128 total) to achieve 80% power. If your sample size is fixed due to constraints, you can instead use G*Power to calculate the power you can achieve with your given sample size, or determine the minimum detectable effect size for your study. It's important to note that to calculate observed power after conducting your study, you would input the actual observed effect size from your test, the sample size of each group, and the alpha level you used. The calculation often involves determining the non-centrality parameter based on the observed effect size and then using this to find the power

Understanding effect size and power analysis allows researchers to design more robust studies and interpret results in a way that goes beyond mere statistical significance, focusing on the practical importance of findings.

💾 Step-by-Steps & Sample Data

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Each of the analysis methods above has links to step-by-step guidance and signposting to sample datasets
Step-by-Step Guides
- The online textbook by Goss-Sampson has many clear and concise guides to undertaking all of the most common statistical analyses in JASP.
- Goss-Sampson, M. A. (2024). Statistical Analysis in JASP 0.18.3: A Guide for Students. March 2024. (Free PDF)
Sample Data
- JASP has many example datasets built into its user interface. Download the software and have a look around for them.
- The same datasets are also available at this link where they can also be downloaded and saved as csv files and also used in MS Excel if preferred. https://johnnydoorn.github.io/DataLibraryBookdown

📚 Further Reading

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Statistics Textbooks in the Marjon Library

Dancey, C. P., & Reidy, J. (2020). Statistics without maths for psychology (8th ed.). Pearson education.
Dancey, C. P., Reidy, J. & Rowe, R. (2012). Statistics for the Health Sciences: A Non-Mathematical Approach. Sage.
Field, A. (2018) Discovering statistics using IBM SPSS statistics. SAGE.
Geher, G., & Hall, S. (2014). Straightforward statistics: Understanding the tools of research. Oxford University Press.
Spiegel, M. R., & Stephens, L. J. (2018). Schaum's outline of statistics. McGraw Hill.

Online

Goss-Sampson, M. A. (2024). Statistical Analysis in JASP 0.18.3: A Guide for Students. March 2024. (Free PDF)
- JASP Resources - https://jasp-stats.org/jasp-materials/
Descriptive Statistics - https://sites.nicholas.duke.edu/statsreview/descriptive
Comparison of Means - https://sites.nicholas.duke.edu/statsreview/means
Comparing Means - https://minusthemath.com/comparing-means.html
Online textbook: Statistics for Research Students https://usq.pressbooks.pub/statisticsforresearchstudents

❓ Assess your Knowledge

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Task: The questions and tasks below are to help you engage with the subject matter of this block of learning. Please choose to complete whatever tasks you feel able to but at some point, if you have identified this block as one of the choices for your Learning Pathway, please complete this Canvas Quiz LINK TO QUIZ GOES HERE

Multiple Choice Questions

10 multiple choice questions on this "statistical methods for comparing means of independent groups" learning block.

1. Which statistical test is most appropriate for comparing the means of two independent groups?
a) Paired t-test
b) Independent samples t-test
c) ANOVA
d) Chi-square test

2. What assumption is NOT required for an independent samples t-test?
a) Normal distribution of data
b) Homogeneity of variances
c) Independence of observations
d) Equal sample sizes

3. When comparing means of more than two independent groups, which test should be used?
a) Independent samples t-test
b) Paired t-test
c) One-way ANOVA
d) Correlation analysis

4. What does the null hypothesis typically state in an independent samples t-test?
a) The means of the two groups are different
b) The means of the two groups are the same
c) One group's mean is larger than the other's
d) The variances of the two groups are equal

5. Which of the following is a non-parametric alternative to the independent samples t-test?
a) Wilcoxon signed-rank test
b) Mann-Whitney U test
c) Kruskal-Wallis test
d) Friedman test

6. What does Levene's test assess in the context of comparing means?
a) Normality of distribution
b) Equality of variances
c) Independence of observations
d) Effect size

7. When the assumption of equal variances is violated, which correction should be applied to the t-test?
a) Bonferroni correction
b) Welch's correction
c) Tukey's correction
d) Holm's correction

8. What does the effect size measure in the context of comparing means?
a) The statistical significance of the difference
b) The practical significance or magnitude of the difference
c) The probability of Type I error
d) The sample size required for the test

9. Which of the following is NOT a post-hoc test used after ANOVA?
a) Tukey's HSD
b) Scheffe's test
c) Bonferroni test
d) Pearson's correlation

10. What is the purpose of conducting a power analysis before comparing means?
a) To determine the minimum sample size needed
b) To calculate the p-value
c) To test for normality
d) To adjust for multiple comparisons

Questions for Reflection and Tasks for Learning

Reflective Questions:
1. How might your choice of statistical test affect the conclusions you draw from your data?

2. In what ways could violating the assumptions of a parametric test impact your research findings?

3. Why is it important to consider both statistical significance and effect size when interpreting results?

4. How might your understanding of power analysis influence your approach to planning a research study?

5. What ethical considerations might arise when deciding on sample size based on power analysis?

Tasks:
1. Find a published research article in your field that uses one of the statistical tests we've discussed. Critically evaluate their choice of test and how they've reported the results.

2. Design a hypothetical study in your area of interest. Outline which statistical test you would use and why, considering the nature of your variables and research question.

3. Using a dataset from a previous project or a public dataset, calculate and interpret the effect size. Reflect on how this adds to your understanding beyond the p-value.

4. Conduct a power analysis for a study you're planning or interested in. Use online tools or statistical software to determine the required sample size for different effect sizes.

5. Create a decision tree to help guide the choice between parametric and non-parametric tests based on different data scenarios.

6. Write a brief explanation of the concept of statistical power for a non-scientific audience, such as a newspaper article or blog post.

</> HTML code for the content above

<h2><span>Comparing Means of Independent Groups</span></h2>
<p><span>Broadly, <strong>Comparing means</strong> is essential in determining if interventions or group characteristics significantly affect outcomes. For example, a study might compare the mean test scores of students taught by different methods to assess each method's effectiveness. Are the control group and intervention group broadly the same at baseline? Did the control or intervention group's data change from pre-testing at baseline to post-testing? </span></p>
<p>More specifically, sometimes we need to <strong>compare the means of independent groups</strong> to determine if there are significant differences between them. The example above of a control group and an intervention group having their means compared can give us the confidence to say that both groups were broadly similar in a specific respect before any interventions were conducted. Other examples include: </p>
<ol style="list-style-type: decimal;">
<li>Evaluating the effectiveness of interventions or treatments: For example, comparing the anxiety levels of a group receiving a new therapy to a control group.</li>
<li>Assessing differences between distinct populations: Such as comparing graduate salaries based on gender.</li>
<li>Analysing the impact of different methods or approaches: For instance, comparing test scores of students who learned using two different teaching methods.</li>
<li>Medical research: Evaluating the efficacy of a new drug by comparing weight loss in treated and untreated groups.</li>
</ol>
<p><span>The figure below shows 3 groups and their respective data points presented in a <a href="https://en.wikipedia.org/wiki/Box_plot">box-and-whisker plot</a>. The statistical tests on this page are designed to show whether the differences between the groups are <a href="https://en.wikipedia.org/wiki/Statistical_significance">significant</a>, or not.</span></p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" role="presentation" src="https://www.ncss.com/wp-content/uploads/2013/08/One-Way-ANOVA-Box-Plot.png" alt="" width="522" height="515" /></p>
<p style="text-align: center;"> </p>
<p style="text-align: center;"> </p>
<h3 class="mb-2 mt-6 text-lg first:mt-3">📖 Introduction to Mean Comparison</h3>
<details>
<summary style="background-color: whitesmoke; cursor: pointer;"><small>(click to reveal)</small></summary>
<p> </p>
<p><span>Mean comparison is a fundamental statistical technique used to assess differences between groups or conditions in research. </span></p>
<p><span>It allows researchers to determine <strong>if observed differences in sample means are statistically significant</strong>, providing valuable insights for decision-making and hypothesis testing.</span></p>
<p><span>There are different methods of comparing means and the choice of method depends on context such as: </span></p>
<ul>
<li><span>how many groups or conditions there are</span></li>
<li><span>whether the data is the same group in different conditions/times or each of the groups is independent of each other</span></li>
<li><span>how "normal" the data is</span></li>
<li><span>the type of data it is</span></li>
<li><span>the sample size</span></li>
</ul>
<h4><strong>Key points:</strong></h4>
<ol class="marker:text-textOff list-decimal pl-8">
<li><span><strong>Purpose</strong>: To evaluate whether the average values of a variable differ significantly between two or more groups.</span></li>
<li><span><strong>Common scenarios</strong>:</span>
<ul class="marker:text-textOff list-disc">
<li><span>Comparing means between two independent groups (e.g. Are the baseline means of the control group and intervention group similar enough to say that they are comparable?)</span></li>
<li><span>Comparing means from paired or repeated measures (e.g. are the means of the data for the same group after the intervention different enough to say that a change has occurred as a result of the intervention?)</span></li>
<li><span>Comparing a sample mean to a known population value</span></li>
</ul>
</li>
<li><span><strong>Statistical tests</strong>: The choice depends on the research design and data characteristics:</span>
<ul class="marker:text-textOff list-disc">
<li><span>t-tests (e.g. one-sample, independent samples, paired samples)</span></li>
<li><span>Analysis of Variance (ANOVA) for multiple group comparisons</span></li>
</ul>
</li>
<li><strong>Considerations:</strong>
<ul class="marker:text-textOff list-disc">
<li><span>Sample size and normality of data distribution</span></li>
<li><span><a class="inline_disabled" href="https://en.wikipedia.org/wiki/Homogeneity_and_heterogeneity_(statistics)" target="_blank" rel="noopener">Homogeneity</a> of variances between groups</span></li>
<li><a class="inline_disabled" href="https://en.wikipedia.org/wiki/Type_I_and_type_II_errors" target="_blank" rel="noopener"><span>Type I and Type II errors</span></a></li>
</ul>
</li>
<li><span><strong>Interpretation</strong>: Results typically include:</span>
<ul class="marker:text-textOff list-disc">
<li><span>p-values to assess statistical significance</span></li>
<li><span>Effect sizes to determine the magnitude of differences</span></li>
</ul>
</li>
</ol>
<p><span>Understanding mean comparison techniques is crucial for researchers to draw valid conclusions from their data and contribute to evidence-based practices in their respective fields.</span></p>
<p><span><img style="display: block; margin-left: auto; margin-right: auto;" role="presentation" src="https://www.sthda.com/sthda/RDoc/images/comparing-means.png" alt="" /></span></p>
<p style="text-align: center;">Image from: <a href="https://www.sthda.com/english/wiki/comparing-means-in-r" target="_blank" rel="noopener">https://www.sthda.com/english/wiki/comparing-means-in-r</a> </p>
</details>
<hr />
<h3>📖 Independent Samples t-test</h3>
<details>
<summary style="background-color: whitesmoke; cursor: pointer;"><small>(click to reveal)</small></summary>
<p><span>The independent samples t-test is a statistical method used to determine whether there is a significant difference between the means of <strong>two unrelated groups</strong>. </span></p>
<p><span>It is commonly used in research <strong>to compare outcomes between different populations or experimental conditions (e.g. control and intervention)</strong>.</span></p>
<h4><strong>Key points:</strong></h4>
<ol class="marker:text-textOff list-decimal pl-8">
<li><span><strong>Purpose</strong>: To compare means between two independent groups on a continuous dependent variable.</span></li>
<li><span><strong>Assumptions</strong>:</span>
<ul class="marker:text-textOff list-disc">
<li><span>The data are numeric and continuous. This can be either <a href="https://www.brookes.ac.uk/students/academic-development/maths-and-stats/statistics/types-of-data">interval or ratio data</a>.</span></li>
<li><span>Observations are independent of one another</span></li>
<li><span>The sample mean is normally distributed</span></li>
<li><span>Equal variances between groups (homogeneity of variance)</span></li>
</ul>
</li>
<li><strong>Hypotheses:</strong>
<ul class="marker:text-textOff list-disc">
<li><span>Null hypothesis (H0): The population means from the two unrelated groups are equal</span></li>
<li><span>Alternative hypothesis (HA): The population means are not equal</span></li>
</ul>
</li>
<li><strong>Interpretation:</strong>
<ul class="marker:text-textOff list-disc">
<li><span>If the p-value is less than the significance level (typically 0.05), reject the null hypothesis</span></li>
<li><span>This indicates a statistically significant difference between the two group means</span></li>
</ul>
</li>
<li><strong>Reporting results:</strong>
<ul class="marker:text-textOff list-disc">
<li><span>Include the t-statistic, degrees of freedom, and p-value</span></li>
<li><span>Example: t(19) = 4.37, p < .01</span></li>
</ul>
</li>
<li><strong>Considerations:</strong>
<ul class="marker:text-textOff list-disc">
<li><span>Check for outliers and normality of data distribution</span></li>
<li><span>Use Welch's t-test if the assumption of equal variances is violated</span></li>
</ul>
</li>
</ol>
<p><span>The independent samples t-test is a valuable tool for researchers comparing two groups, such as treatment and control groups in experimental studies or demographic differences in observational research.</span></p>
<h4><strong>Checking Distribution of Data</strong></h4>
<p><em>Parametric statistical tests</em> (which most commonly used tests are) require data that meets certain <em>assumptions</em>. Parametric statistics use <em>parameters</em> such as the mean, median, standard deviation etc. This form of statistics uses the data that can be observed to estimate the parameters of the data distribution. If the data doesn’t meet the <em>assumptions</em> the parametric tests don’t work.</p>
<strong>Normal distribution</strong>
<p><span>The first assumption is that the data is <em>normally distributed</em>. This is described as a <em>Gaussian distribution</em><strong> </strong>or more commonly a <em>‘bell curve’</em>. With a normally distributed sample if we drew a line through the middle of the curve it would look the same on both sides.</span></p>
<p><span>The easiest way to examine the distribution of a data set is to plot a graph of how many times a value occurs within the data set. This is known as a <em>frequency distribution</em> or <em>histogram</em>. This is a graph which plots the values of observations on the horizontal axis with bars showing how many times each value occurred in the data set.</span></p>
<p><span>As you can see below, in a normal distribution the tallest bars (values with the highest frequency) occur in the middle of the distribution and the shortest bars (values with the lowest frequency) appear around the edges of the distribution.</span></p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" role="presentation" src="https://mathbitsnotebook.com/Algebra2/Statistics/normalturqa.jpg" alt="" /></p>
<h4><strong>Skewness and Kurtosis</strong></h4>
<p>There are two main ways that the distribution of a data set might deviate from a ‘normal distribution’. These are called <em>skew</em> (defined as <em>lack of symmetry</em>) and <em>kurtosis </em>(described as <em>pointiness</em>).</p>
<p><em>Skewed distributions </em>are <em>not symmetrical</em> and the most frequent values (the tallest bars) are clustered at one end of the scale (one side of the histogram) and the values tail off (bars get smaller) on the opposite end of the scale (the other side of the histogram). A skewed distribution can be <em>positively skewed</em><strong> </strong>(the tallest bars at the lower end of the scale) or <em>negatively skewed</em><strong> </strong>(the tallest bars at the upper end of the scale).</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="https://miro.medium.com/v2/resize:fit:1350/1*eqSYm-EeXdMOTz6QrrTWew.jpeg" alt="" /></p>
<p>As well as varying in their level of skew distributions also vary in their level of <em>kurtosis</em>. Kurtosis refers to the degree to which the values are clustered at the ends of the distribution (referred to as the <em>‘tails’</em> of the distribution. This is usually reflected in how <em>‘pointy’</em> a distribution is. A distribution that has a lot of values in the tails (called a <em>heavy tailed</em> distribution) is usually pointy. This is called a <em>leptokurtic </em>distribution and is said to have <em>positive kurtosis</em>. A distribution that is thin in the tails (has <em>light</em> tails) is usually flatter than normal. This is called a <em>platykurtic</em><strong> </strong>distribution and is said to have <em>negative kurtosis</em><strong>. </strong>If a data set has a normal distribution of data we call it a <em>mesokurtic</em><strong> </strong>distribution.</p>
<p><strong>Checking normality in JASP</strong></p>
<p>When you conduct an independent t-test in JASP you can select data for the Shapiro-Wilks test and Levene’s test which are tests for normality. If both of these tests return a p value of > 0.05 then you can assume a normal distribution. </p>
<p>If a distribution has values of skew or kurtosis <em>above zero</em> then this distribution has positive skew or kurtosis. If a distribution has values of skew or kurtosis <em>below zero</em> then this distribution has negative skew or kurtosis. A value of <em>zero</em><strong> </strong>indicates a normal distribution</p>
<h4><strong>Have A Go!</strong></h4>
<h4><strong>Independent Samples t-test in MS Excel</strong></h4>
<p>If you are starting with the default version of MS Excel, you can download an additional Analysis Toolpak and run additional statistical analysis quite easily in this software. Find out more here: <a href="https://statisticsbyjim.com/hypothesis-testing/t-tests-excel/" target="_blank" rel="noopener">https://statisticsbyjim.com/hypothesis-testing/t-tests-excel/</a> </p>
<h4><strong>Independent Samples t-test</strong><strong> in JASP</strong></h4>
<p>The gif below is an example of how you can undertake this analysis in JASP. Have a go with your dataset and think about the best type of figure/plot for your data.</p>
<p>Find out more here: <a href="https://jasp-stats.org/2018/04/18/how-to-conduct-a-classical-independent-sample-t-test-in-jasp-and-interpret-the-results" target="_blank" rel="noopener">https://jasp-stats.org/2018/04/18/how-to-conduct-a-classical-independent-sample-t-test-in-jasp-and-interpret-the-results/</a> </p>
<p>To find out more about undertaking this analysis in JASP, read the section on<span> </span><strong>Comparing Two Independent Groups</strong> in the following textbook</p>
<ul>
<li><span>Goss-Sampson, M. A. (2024). </span><em>Statistical Analysis in JASP 0.18.3: A Guide for Students</em><span>. March 2024. (</span><a class="external" href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf" target="_blank" rel="noopener"><span>Free PDF</span></a><span>) (<a href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf#page=61">Relevant Section</a>)</span></li>
</ul>
<p style="text-align: center;"><img role="presentation" src="https://jasp-stats.org/wp-content/uploads/2020/02/Independent_Samples_T_Test.gif" alt="" width="781" height="488" /></p>
<p style="text-align: center;"> </p>
<p style="text-align: center;"> </p>
<p> </p>
</details>
<hr />
<h3 style="text-align: left;">📖 One-way ANOVA (Analysis of Variance)</h3>
<details>
<summary style="background-color: whitesmoke; cursor: pointer;"><small>(click to reveal)</small></summary>
<p> </p>
<p><span>A One-way ANOVA is a statistical method used to <strong>compare the means of three or more independent groups</strong> to determine if there are any statistically significant differences between them</span></p>
<p><span><span class="whitespace-nowrap"><strong>An Example from Sports Science</strong>:<br />A researcher wants to compare the effects of different training programmes on sprint performance in athletes. The study involves three groups: one using traditional sprint training, one using plyometric exercises, and one using resistance training. The researcher measures 100m sprint times after 8 weeks of training. One-way ANOVA could be used to determine if there are significant differences in sprint performance between the three training groups</span></span></p>
<p><span><span class="whitespace-nowrap"><strong>An example from Sports Therapy:</strong><br />A sports therapist aims to investigate the effectiveness of different rehabilitation protocols on ankle stability following an injury. The study includes three groups of athletes recovering from ankle sprains: one group using traditional physiotherapy, one using proprioceptive training, and one using a combination of both. The therapist measures ankle stability using the Star Excursion Balance Test after 6 weeks of rehabilitation. One-way ANOVA could be used to analyse if there are significant differences in ankle stability scores between the three rehabilitation groups</span></span></p>
<h4><strong>Key points:</strong></h4>
<ol class="marker:text-textOff list-decimal pl-8">
<li><span><strong>Purpose</strong>: To test whether the means of multiple groups are equal</span></li>
<li><span><strong>Null hypothesis</strong> (H0): All population means are equal (μ1 = μ2 = μ3 = ... = μk)</span></li>
<li><span><strong>Alternative hypothesis</strong> (H1): At least one population mean is different from the others</span></li>
<li><span><strong>Assumptions</strong></span>
<ul class="marker:text-textOff list-disc">
<li><span>Independent observations</span></li>
<li><span>Normality of dependent variable distribution</span></li>
<li><span>Homogeneity of variance</span></li>
</ul>
</li>
<li><span><strong>Interpretation</strong>: If the test returns a statistically significant result, it indicates that at least two group means are significantly different from each other</span></li>
<li><span><strong>Post-hoc tests</strong>: Required to determine which specific groups differ if ANOVA results are significant</span></li>
<li><span><strong>Robustness</strong>: ANOVA is relatively robust to slight violations of normality, especially with larger sample sizes</span></li>
</ol>
<h4><strong>One-way ANOVA in MS Excel</strong></h4>
<p>If you are starting with the default version of MS Excel, you can download an additional Analysis Toolpak and run additional statistical analysis quite easily in this software. Find out more here: <a href="https://www.datacamp.com/tutorial/excel-anova-guide" target="_blank" rel="noopener">https://www.datacamp.com/tutorial/excel-anova-guide</a> </p>
<h4><strong>One-way ANOVA</strong><strong> in JASP</strong></h4>
<p>To find out more about undertaking this analysis in JASP, read the section on<span> </span><strong>Comparing More Than Two Independent Groups</strong> in the following textbook</p>
<ul>
<li style="list-style-type: none;">
<ul>
<li><span>Goss-Sampson, M. A. (2024). </span><em>Statistical Analysis in JASP 0.18.3: A Guide for Students</em><span>. March 2024. (</span><a class="external" href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf" target="_blank" rel="noopener"><span>Free PDF</span></a><span>) (<a href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf#page=99">Relevant Section</a>)</span></li>
</ul>
</li>
</ul>
<p><span><img style="display: block; margin-left: auto; margin-right: auto;" role="presentation" src="https://jasp-stats.org/wp-content/uploads/2020/05/Simple_Main_Effects.gif" alt="" width="930" height="581" /></span></p>
<p style="text-align: center;">See this gif and others related to ANOVA here: <a href="https://jasp-stats.org/how-to-use-jasp/" target="_blank" rel="noopener">https://jasp-stats.org/how-to-use-jasp/</a> </p>
</details>
<hr />
<h3>📖 Non-Parametric Alternatives</h3>
<details>
<summary style="background-color: whitesmoke; cursor: pointer;"><small>(click to reveal)</small></summary>
<p> </p>
<p>The most common statistical methods work when the data being analysed is parametric. </p>
<p>Data is considered parametric when it meets specific assumptions and characteristics that allow for the use of <a href="https://en.wikipedia.org/wiki/Parametric_statistics">parametric statistical methods</a>.</p>
<p>Data is considered parametric when there is:</p>
<ol style="list-style-type: decimal;">
<li><strong>Normal Distribution</strong>: The data follows or closely approximates a normal (Gaussian) distribution. This "bell-shaped" curve is essential for many parametric tests.</li>
<li><strong>Known Parameters</strong>: The data can be described by a fixed set of parameters, such as mean and standard deviation for normal distributions.</li>
<li><strong>Homoscedasticity</strong>: The data exhibits similar variances across different groups being compared[3]. This is also known as homogeneity of variance.</li>
<li><strong>Continuous (Interval or Ratio Data)</strong>: Parametric methods typically work with continuous variables (e.g. height) rather than categorical (red, or blue) or ordinal (rank) data.</li>
<li><strong>Large Sample Size</strong>: While not strictly necessary, parametric methods are generally more reliable with larger sample sizes (typically n ≥ 30).</li>
</ol>
<p><img style="display: block; margin-left: auto; margin-right: auto;" role="presentation" src="/courses/5569/files/759186/preview?verifier=4DzWFCCxHiYIKntDZ9MX9l6uQV9iye47tyuQB0zM" alt="" width="479" height="445" data-api-endpoint="https://marjon.instructure.com/api/v1/courses/5569/files/759186" data-api-returntype="File" /></p>
<p><strong>When these conditions are met</strong>, parametric tests such as t-tests, ANOVA, and linear regression can be applied. These tests are often preferred due to their greater statistical power and ability to provide more detailed insights about the population parameters.</p>
<p><strong>When the assumptions of parametric tests are violated</strong>, non-parametric alternatives can be used for comparing means. These tests are more robust to violations of normality and homogeneity of variance.</p>
<h4><strong>How do I know if my data is parametric/non-parametric?</strong></h4>
<ol style="list-style-type: decimal;">
<li><strong>Check for normal distribution</strong>: Parametric data typically follows a normal or "bell-shaped" distribution. You can visually check this by creating a histogram of your data. If it looks roughly symmetrical, it's likely parametric.</li>
<li><strong>Look at your measurement scale</strong>: Parametric data is usually continuous (like height or weight) and measured on an interval or ratio scale. If your data is categorical or ordinal (like rankings), it's likely non-parametric.</li>
<li><strong style="color: var(--ic-brand-font-color-dark); font-family: inherit; font-size: 1rem;">Sample size</strong><span style="color: var(--ic-brand-font-color-dark); font-family: inherit; font-size: 1rem;">: If you have a large sample size (generally more than 30), your data is more likely to be suitable for parametric tests, even if it's not perfectly normal.</span></li>
<li><strong style="color: var(--ic-brand-font-color-dark); font-family: inherit; font-size: 1rem;">Use statistical tests</strong><span style="color: var(--ic-brand-font-color-dark); font-family: inherit; font-size: 1rem;">: There are formal tests like the Anderson-Darling or Shapiro-Wilk test that can help you determine if your data is normally distributed.</span></li>
</ol>
<h4><strong>Key non-parametric alternatives include: </strong></h4>
<ol style="list-style-type: decimal;">
<li><strong>Mann-Whitney U Test</strong>
<ul style="list-style-type: disc;">
<li>Alternative to independent samples t-test</li>
<li>Compares two independent groups</li>
<li>See p62 in <span>Goss-Sampson (2024) for this method in JASP </span><a href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf#page=66" target="_blank" rel="noopener">https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf#page=66</a> </li>
</ul>
</li>
<li><strong>Kruskal-Wallis H Test</strong>
<ul style="list-style-type: disc;">
<li>Alternative to one-way between-groups ANOVA</li>
<li>Compares three or more independent groups</li>
<li>See p103 in Goss-Sampson (2024) for this method in JASP <a href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf#page=107" target="_blank" rel="noopener">https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf#page=107</a> </li>
</ul>
</li>
</ol>
<p>Considerations: Non-parametric tests analyse ranks of scores rather than raw scores. They are less powerful than their parametric counterparts but more robust to assumption violations For complex designs (e.g., mixed ANOVA), there may not be direct non-parametric equivalents</p>
<p>When using non-parametric alternatives: Clearly state the reasons for choosing a non-parametric approach Report median and interquartile range instead of mean and standard deviation Interpret results in terms of distribution differences rather than mean differences</p>
<p><strong>Find out More: </strong></p>
<p>Parametric and Nonparametric: Demystifying the Terms  - <a href="https://www.mayo.edu/research/documents/parametric-and-nonparametric-demystifying-the-terms/doc-20408960" target="_blank" rel="noopener">https://www.mayo.edu/research/documents/parametric-and-nonparametric-demystifying-the-terms/doc-20408960</a> </p>
</details>
<hr />
<h3>📖 Effect size and Power Analysis</h3>
<details>
<summary style="background-color: whitesmoke; cursor: pointer;"><small>(click to reveal)</small></summary>
<p>Effect size and power analysis are crucial components in statistical research, helping researchers design studies and interpret results meaningfully.</p>
<h4><strong>Effect Size: </strong></h4>
<p>Effect size is a quantitative measure of the magnitude of a relationship between variables or the difference between groups. It provides information about the practical significance of findings, beyond mere statistical significance.</p>
<p><strong>Key points about effect size:</strong></p>
<ul>
<li>Quantifies the strength of relationships or differences</li>
<li>Helps determine practical importance of findings</li>
<li>Can be used to compare results across different studies</li>
</ul>
<p><strong>Common effect size measures include</strong>:</p>
<ol style="list-style-type: decimal;">
<li>Cohen's d: For comparing two means</li>
<li>Pearson's r: For correlations between variables</li>
<li>Odds ratio: For binary outcomes</li>
</ol>
<p><strong>Interpreting effect sizes (Cohen's guidelines):</strong></p>
<ul>
<li>Small: d = 0.2 or r = 0.1</li>
<li>Medium: d = 0.5 or r = 0.3</li>
<li>Large: d = 0.8 or r > 0.5</li>
</ul>
<p><strong>Find out more: </strong></p>
<ul>
<li>Effect Size on Wikipedia - <a href="https://en.wikipedia.org/wiki/Effect_size" target="_blank" rel="noopener">https://en.wikipedia.org/wiki/Effect_size</a> </li>
<li>To find out more about Effect Size in JASP, read the section on <strong>Effect Size </strong>in the following textbook:
<ul>
<li><span>Goss-Sampson, M. A. (2024). </span><em>Statistical Analysis in JASP 0.18.3: A Guide for Students</em><span>. March 2024. (</span><a class="external" href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf" target="_blank" rel="noopener"><span>Free PDF</span></a><span>) (<a href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf#page=47">Relevant Section</a>)</span></li>
</ul>
</li>
</ul>
<p> </p>
<hr />
<h4><strong>Power Analysis</strong>:</h4>
<ul>
<li>Statistical power is the likelihood of a significance test detecting an effect when one truly exists.</li>
<li>Power analysis helps determine the sample size needed to detect an effect of a given size with a specified level of confidence.</li>
</ul>
<p><strong>Components of power analysis:</strong></p>
<ol style="list-style-type: decimal;">
<li>Statistical power (typically set at 80% or higher)</li>
<li>Sample size</li>
<li>Significance level (alpha, usually 5%)</li>
<li>Expected effect size</li>
</ol>
<p><strong>Benefits of power analysis:</strong></p>
<ul>
<li>Ensures statistical significance in results</li>
<li>Reduces the risk of Type II errors</li>
<li>Helps determine appropriate sample sizes for studies</li>
</ul>
<p><strong>Relationship between effect size and power:</strong></p>
<ul>
<li>Larger effect sizes are easier to detect and require smaller sample sizes</li>
<li>Smaller effect sizes require larger sample sizes to maintain adequate statistical power</li>
</ul>
<p><strong>Calculating Statistical Power:</strong></p>
<p><span class="">G</span><span class="">Power is free software with a graphical user interface that makes it easy to calculate sample size and power for various statistical methods, including F-tests, t-tests, χ2 tests, Z tests, and exact tests. It can be downloaded from <a class="break-word hover:text-super hover:decoration-super dark:hover:text-superDark dark:hover:decoration-superDark underline decoration-from-font underline-offset-1 transition-all duration-300" href="http://www.psycho.uni-duesseldorf.de/abteilungen/aap/gpower3" target="_blank" rel="noopener">www.psycho.uni-duesseldorf.de/abteilungen/aap/gpower3</a>. G</span><span class="">Power also provides effect size calculators and graphics options.</span></p>
<p><span class="">Here's an example of calculating statistical power for a two-independent groups t-test:</span></p>
<ol class="marker:text-textOff list-decimal pl-8">
<li><span class="">Open G*Power</span></li>
<li><span class="">Click on the "Test family" dropdown menu and select "t-tests".</span></li>
<li><span class="">In the "Statistical test" dropdown menu, choose "Means: Difference between two independent means (two groups)".</span></li>
<li><span class="">In the main window, you'll need to input the following parameters:</span>
<ul class="marker:text-textOff list-disc">
<li><span class="">Effect size: Let's say we expect a medium effect size of 0.5</span></li>
<li><span class="">α (significance level): Typically set to 0.05</span></li>
<li><span class="">Power (1-β): Let's aim for 80% power, so input 0.80</span></li>
<li><span class="">Allocation ratio N2/N1: If the groups are of equal size, leave this as 1</span></li>
</ul>
</li>
</ol>
<p><span class="">After inputting these values, G*Power will calculate the required sample size for each group. In this case, it would show that you need approximately 64 participants in each group (128 total) to achieve 80% power. </span><span class="">If your sample size is fixed due to constraints, you can instead use G*Power to calculate the power you can achieve with your given sample size, or determine the minimum detectable effect size for your study. </span><span class="">It's important to note that to calculate observed power after conducting your study, you would input the actual observed effect size from your test, the sample size of each group, and the alpha level you used. The calculation often involves determining the non-centrality parameter based on the observed effect size and then using this to find the power</span></p>
<p> </p>
<p>Understanding effect size and power analysis allows researchers to design more robust studies and interpret results in a way that goes beyond mere statistical significance, focusing on the practical importance of findings.</p>
<p> </p>
</details>
<hr />
<h3>💾 Step-by-Steps & Sample Data</h3>
<details>
<summary style="background-color: whitesmoke; cursor: pointer;"><small>(click to reveal)</small></summary>
<p> </p>
<ul>
<li>Each of the analysis methods above has links to step-by-step guidance and signposting to sample datasets</li>
<li><strong>Step-by-Step Guides</strong>
<ul>
<li>The online textbook by Goss-Sampson has many clear and concise guides to undertaking all of the most common statistical analyses in JASP. </li>
<li><span>Goss-Sampson, M. A. (2024). </span><em>Statistical Analysis in JASP 0.18.3: A Guide for Students</em><span>. March 2024. (</span><a class="external" href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf" target="_blank" rel="noopener"><span>Free PDF</span></a><span>)</span></li>
</ul>
</li>
<li><strong>Sample Data</strong>
<ul>
<li>JASP has many example datasets built into its user interface. Download the software and have a look around for them. </li>
<li>The same datasets are also available at this link where they can also be downloaded and saved as csv files and also used in MS Excel if preferred. <a href="https://johnnydoorn.github.io/DataLibraryBookdown" target="_blank" rel="noopener">https://johnnydoorn.github.io/DataLibraryBookdown/</a> </li>
</ul>
</li>
</ul>
<p> </p>
</details>
<hr />
<h3>📚 Further Reading</h3>
<details>
<summary style="background-color: whitesmoke; cursor: pointer;"><small>(click to reveal)</small></summary>
<p> </p>
<h3><span>Statistics Textbooks in the Marjon Library</span></h3>
<ul>
<li><span>Dancey, C. P., & Reidy, J. (2020). </span><i>Statistics without maths for psychology (8th ed.)</i><span>. Pearson education. </span></li>
<li><span>Dancey, C. P., Reidy, J. & Rowe, R. (2012). </span><i>Statistics for the Health Sciences: A Non-Mathematical Approach. </i>Sage<span>. </span></li>
<li><span>Field, A. (2018) Discovering statistics using IBM SPSS statistics. SAGE. </span></li>
<li><span>Geher, G., & Hall, S. (2014). <i>Straightforward statistics: Understanding the tools of research</i>. Oxford University Press. </span></li>
<li><span>Spiegel, M. R., & Stephens, L. J. (2018). <i>Schaum's outline of statistics</i>. McGraw Hill. </span></li>
</ul>
<p> </p>
<h4><strong>Online</strong></h4>
<ul>
<li><span>Goss-Sampson, M. A. (2024). </span><em>Statistical Analysis in JASP 0.18.3: A Guide for Students</em><span>. March 2024. (</span><a class="external" href="https://jasp-stats.org/wp-content/uploads/2024/03/Statistical-Analysis-in-JASP-2024.pdf" target="_blank" rel="noopener"><span>Free PDF</span></a><span>)</span>
<ul>
<li>JASP Resources - <a href="https://jasp-stats.org/jasp-materials/" target="_blank" rel="noopener">https://jasp-stats.org/jasp-materials/</a> </li>
</ul>
</li>
<li>Descriptive Statistics - <a href="https://sites.nicholas.duke.edu/statsreview/descriptive" target="_blank" rel="noopener">https://sites.nicholas.duke.edu/statsreview/descriptive/</a> </li>
<li>Comparison of Means - <a href="https://sites.nicholas.duke.edu/statsreview/means" target="_blank" rel="noopener">https://sites.nicholas.duke.edu/statsreview/means/</a> </li>
<li>Comparing Means - <a href="https://minusthemath.com/comparing-means.html" target="_blank" rel="noopener">https://minusthemath.com/comparing-means.html</a> </li>
<li>Online textbook: Statistics for Research Students <a href="https://usq.pressbooks.pub/statisticsforresearchstudents" target="_blank" rel="noopener">https://usq.pressbooks.pub/statisticsforresearchstudents/</a> </li>
</ul>
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<h3>❓ Assess your Knowledge</h3>
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<summary style="background-color: whitesmoke; cursor: pointer;"><small>(click to reveal)</small></summary>
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<div style="background-color: #e6f7ff; border-left: 5px solid #003d4f; padding: 15px; margin: 10px 0; font-family: Arial, sans-serif; color: #003d4f;"><strong>Task:</strong> The questions and tasks below are to help you engage with the subject matter of this block of learning. Please choose to complete whatever tasks you feel able to but at some point, if you have identified this block as one of the choices for your Learning Pathway, please complete this Canvas Quiz <strong>LINK TO QUIZ GOES HERE</strong></div>
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<h3>Multiple Choice Questions</h3>
<p>10 multiple choice questions on this "<span>statistical methods for comparing means of independent groups</span>" learning block. </p>
<p>1. Which statistical test is most appropriate for comparing the means of two independent groups?<br />   a) Paired t-test<br />   b) Independent samples t-test<br />   c) ANOVA<br />   d) Chi-square test</p>
<p>2. What assumption is NOT required for an independent samples t-test?<br />   a) Normal distribution of data<br />   b) Homogeneity of variances<br />   c) Independence of observations<br />   d) Equal sample sizes</p>
<p>3. When comparing means of more than two independent groups, which test should be used?<br />   a) Independent samples t-test<br />   b) Paired t-test<br />   c) One-way ANOVA<br />   d) Correlation analysis</p>
<p>4. What does the null hypothesis typically state in an independent samples t-test?<br />   a) The means of the two groups are different<br />   b) The means of the two groups are the same<br />   c) One group's mean is larger than the other's<br />   d) The variances of the two groups are equal</p>
<p>5. Which of the following is a non-parametric alternative to the independent samples t-test?<br />   a) Wilcoxon signed-rank test<br />   b) Mann-Whitney U test<br />   c) Kruskal-Wallis test<br />   d) Friedman test</p>
<p>6. What does Levene's test assess in the context of comparing means?<br />   a) Normality of distribution<br />   b) Equality of variances<br />   c) Independence of observations<br />   d) Effect size</p>
<p>7. When the assumption of equal variances is violated, which correction should be applied to the t-test?<br />   a) Bonferroni correction<br />   b) Welch's correction<br />   c) Tukey's correction<br />   d) Holm's correction</p>
<p>8. What does the effect size measure in the context of comparing means?<br />   a) The statistical significance of the difference<br />   b) The practical significance or magnitude of the difference<br />   c) The probability of Type I error<br />   d) The sample size required for the test</p>
<p>9. Which of the following is NOT a post-hoc test used after ANOVA?<br />   a) Tukey's HSD<br />   b) Scheffe's test<br />   c) Bonferroni test<br />   d) Pearson's correlation</p>
<p>10. What is the purpose of conducting a power analysis before comparing means?<br />    a) To determine the minimum sample size needed<br />    b) To calculate the p-value<br />    c) To test for normality<br />    d) To adjust for multiple comparisons</p>
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<h4><strong>Questions for Reflection and Tasks for Learning</strong></h4>
<p><strong>Reflective Questions:</strong><br />1. How might your choice of statistical test affect the conclusions you draw from your data?</p>
<p>2. In what ways could violating the assumptions of a parametric test impact your research findings?</p>
<p>3. Why is it important to consider both statistical significance and effect size when interpreting results?</p>
<p>4. How might your understanding of power analysis influence your approach to planning a research study?</p>
<p>5. What ethical considerations might arise when deciding on sample size based on power analysis?</p>
<p><strong>Tasks:</strong><br />1. Find a published research article in your field that uses one of the statistical tests we've discussed. Critically evaluate their choice of test and how they've reported the results.</p>
<p>2. Design a hypothetical study in your area of interest. Outline which statistical test you would use and why, considering the nature of your variables and research question.</p>
<p>3. Using a dataset from a previous project or a public dataset, calculate and interpret the effect size. Reflect on how this adds to your understanding beyond the p-value.</p>
<p>4. Conduct a power analysis for a study you're planning or interested in. Use online tools or statistical software to determine the required sample size for different effect sizes.</p>
<p>5. Create a decision tree to help guide the choice between parametric and non-parametric tests based on different data scenarios.</p>
<p>6. Write a brief explanation of the concept of statistical power for a non-scientific audience, such as a newspaper article or blog post.</p>
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