Repeated Measures Two-way ANOVA
A repeated-measures two-way ANOVA tests one between-subject factor (treatment) and one within-subject factor measured repeatedly on the same units (time/day), together with their interaction, correctly handling the correlation between repeated observations. Read more …
Repeated-measures two-way ANOVA is used when the same experimental units are measured more than once over time under different treatments - a design common in agricultural, biological, and clinical research. It separates the effect of the treatment, the effect of time (the repeated factor), and their interaction, while accounting for the fact that measurements taken on the same unit are correlated. In RAISINS, this analysis can be performed easily without writing any code. This tutorial will guide you step-by-step.
1 What is a repeated-measures two-way ANOVA?
Suppose you are testing three fertiliser treatments on a crop, and instead of measuring each plot only once, you return on several days and measure the same plots again - on day 1, day 2, and so on. Under treatment T1 a plot averages about 48 units on day 1 and rises to about 53 on day 2; under T2 it starts near 39 and climbs more slowly. Two things are clearly going on at once: the plots respond differently to the treatments, and every plot also changes over time. And there is a third, subtler possibility - the treatments might change over time at different rates.
A one-way test cannot untangle these. A repeated-measures two-way ANOVA is built exactly for this situation. It asks three separate questions from the same data:
- Does the treatment matter? - the between-subject effect (T1 vs T2 vs T3, averaged over all days).
- Does time matter? - the within-subject effect (day 1 vs day 2 vs …, averaged over all treatments).
- Do the treatments behave differently over time? - the interaction (Treatment × Day).
The word repeated is the key. Because the same plot is measured again and again, those measurements are not independent - a plot that is high on day 1 tends to be high on day 2. Ordinary ANOVA assumes independence and would be misled by this. Repeated-measures ANOVA models the plot itself as a source of variation (an error stratum), so the treatment and time effects are judged against the right yardstick.
A repeated-measures two-way ANOVA separates the effect of the treatment, the effect of time, and their interaction, while correctly accounting for the fact that the same units are measured over and over.
2 Between-subject and within-subject factors
Every repeated-measures design has two kinds of factor, and telling them apart is the whole game.
A between-subject factor splits your units into groups, where each unit belongs to only one group. In the working example, Treatment (T1, T2, T3) is between-subject: a given plot receives one treatment and keeps it. Differences here are between different plots.
A within-subject (repeated) factor is measured within each unit, more than once. In the working example, Day is within-subject: the same plot is measured on day 1, day 2, and so on. Differences here are within the same plot, across time.
The third piece, the subject (here the Replication, the individual experimental unit), is what ties the repeated measurements together. RAISINS treats it as a nested error term so that the within-subject comparisons are made plot-by-plot rather than by pooling unrelated plots.
| Factor in the example | Type | What it varies | Compared… |
|---|---|---|---|
| Treatment (T1–T3) | Between-subject | Across different plots | Between units |
| Day (day1, day2, …) | Within-subject (repeated) | Within the same plot, over time | Within units |
| Treatment × Day | Interaction | Whether the time pattern depends on treatment | - |
| Replication | Subject / error stratum | Identifies the repeatedly measured unit | - |
If the Treatment × Day interaction is significant, the treatments are changing over time at different rates, and you should be cautious about reading the treatment main effect on its own - the “best” treatment may depend on which day you look at. Always read the interaction first, then the main effects.
3 Sphericity and the repeated-measures correction
Repeated-measures ANOVA adds one assumption that ordinary ANOVA does not have: sphericity. In plain terms, sphericity means that the variances of the differences between every pair of time points are roughly equal - the spread of (day1 − day2) is similar to the spread of (day1 − day3), and so on. When measurements are taken close together in time, nearby days are often more alike than distant days, and sphericity breaks down.
When sphericity is violated, the within-subject F-test becomes too liberal - it reports significance too easily. The standard fix is to shrink the degrees of freedom of the within-subject effect, which makes the test more conservative. RAISINS offers the two usual corrections:
- Greenhouse–Geisser (GG) - a widely used, safe correction.
- Huynh–Feldt (HF) - a slightly less conservative alternative.
By default RAISINS applies no correction; you can switch on GG or HF from the Sphericity Correction selector in the Analysis tab. The correction changes the degrees of freedom and hence the p-value of the within-subject (Day) and interaction terms; it does not change the between-subject (Treatment) term.
If your time points are closely spaced or you have many of them, sphericity is unlikely to hold - choosing Greenhouse–Geisser is the cautious default. With only two time points sphericity is automatically satisfied and no correction is needed.
4 Assumptions of repeated-measures ANOVA
The analysis is trustworthy only when its assumptions hold. The main ones are listed below.
| Assumption | What it means | How RAISINS helps |
|---|---|---|
| Normality | The response is approximately normally distributed within each treatment–time combination | Normality tests and Q–Q plots (?@sec-8) |
| Sphericity | Variances of the differences between time points are equal | Greenhouse–Geisser / Huynh–Feldt corrections (Section 3) |
| Homogeneity of variance | Spread is similar across treatment groups | Transformations (Section 10) when it fails |
| Independence of subjects | Different experimental units are independent of one another | Ensured by your experimental design |
| Complete, balanced data | Every subject is measured at every time point | Health Check in View Data (Section 20) flags gaps |
Repeated-measures ANOVA expects every subject to be measured at every time point. If a plot is missing a day, that subject cannot contribute a complete profile. Check the View Data Health Check (Section 20) before running the analysis.
5 Getting to the module
Now that the theory is clear, let us run the analysis. Visit the RAISINS home page at www.raisins.live and go to Data Analysis. In this tutorial we use the Repeated Measures Two-way ANOVA module, shown in Figure 1.
5.1 Computational Provenance & Reproducibility Record
CPRR (Computational Provenance & Reproducibility Record) provides a transparent and comprehensive record. Click on the icon shown in Figure 1 to access CPRR and know about the computational workflow performed during the analysis. The record for this module states the R version and the exact version of every package used, names the specific function behind each reported result. CPRR lists every default parameter and decision rule applied by the module and provides fully runnable R code that reproduces each analytical step. Users can execute the code in R to independently reproduce and verify the results. It carries its own DOI.
To cite the platform itself in a paper, thesis, or report, use the RAISINS citation, available in APA, Harvard, and BibTeX formats at www.raisins.live/citation.html. That is the primary reference, and for most manuscripts it is all you need.
The CPRR for the repeated-measures two-way ANOVA is at www.raisins.live/module_record/rm-twoway-anova.html.
Cite the RAISINS paper as your primary reference for the platform. Add the CPRR as supporting documentation when a journal asks for details of the computing environment, or when you want your methods section to be precise about versions and functions rather than saying “analysis was carried out using an online tool.” The CPRR supports the citation and ensures computational reproducibility.
6 Preview mode and Quick Tour
Before subscribing, you can explore the entire module using Preview mode, accessible from the Welcome page. Preview mode loads built-in datasets so you can try every feature (analysis, sphericity corrections, multiple comparisons, normality tests, plots, the multivariate tab, and the RA-One assistant) without uploading your own data. First-time users are also offered a Quick Tour, an interactive, step-by-step walkthrough that highlights each control and explains what it does. You can retake the tour at any time from the Quick Tour tab.
7 A working example
This example uses three treatments (T1, T2, T3), each applied to replicate experimental units that were measured repeatedly across several days (day1, day2, …). For every unit and day, four response characters (char1, char2, char3, char4) were recorded, illustrating how RAISINS can analyse several variables in a single run. Here Treatment is the between-subject factor, Day is the within-subject (repeated) factor, and Replication identifies the subject that is measured over time. The task is to decide whether the treatments differ, whether the response changes over time, and whether the treatments change over time at different rates.
8 How to prepare your data
Your analysis is only as good as your data. Feed RAISINS high-quality data and it will deliver powerful insights; feed it messy data and the results will not be trustworthy. You have four routes:
- Create your dataset in MS Excel
- Build your dataset directly within the RAISINS app
- Using the Model datasets in RAISINS as a reference
- Create your dataset using the RA-One chat assistant
8.1 Preparing data in MS Excel
Open a new blank sheet in MS Excel containing only one sheet, and avoid adding any unnecessary content. The dataset should follow a column-based format. The first column holds the Treatment label (the between-subject factor, e.g. T1, T2, T3). The second column holds the Day/Time label (the within-subject repeated factor, e.g. day1, day2). A third column identifies the Replication (the subject - the unit that is measured on every day). All response characters under study (char1, char2, …) then occupy separate columns. Each Treatment–Day combination repeats once per replication. The file can be saved as CSV, XLS, or XLSX, but CSV is recommended as it is lighter and loads faster. Ensure there are no unwanted spaces in column names or labels. For reference, see the structure in Figure 3.
Dataset creation rules
- Column naming convention
- No spaces allowed in column names.
- Use underscores (
_) or full stops (.) for separation. - Avoid symbols and special characters such as %, #.
- Data arrangement
- Start the data towards the upper-left corner.
- Ensure the row above the data is not blank.
- Cell management
- Avoid typing or deleting in cells without data.
- If needed, select the affected cells, right-click, and choose Clear Contents.
- Column relevance
- Name all columns meaningfully.
- Exclude unnecessary columns not required for the analysis.
- Factor and subject labels
- The Treatment column holds the between-subject factor labels; the Day/Time column holds the repeated factor labels.
- Every Replication (subject) should appear once at each Day within its Treatment, so each subject has a complete profile across time.
- Keep spelling and capitalisation of all labels consistent throughout.
How to save as CSV in MS Excel
Open your workbook. Ensure your data is arranged properly with only one sheet.
Click the ‘File’ menu. Go to the top-left corner and click File.
Choose ‘Save As’ or ‘Save a Copy’. Select the location where you want to save your file.
Set file type to CSV. In the ‘Save as type’ dropdown, choose CSV (Comma delimited) (*.csv).
Name your file. Enter a relevant file name without spaces (use underscores if needed).
Click ‘Save’. Click Save to export the file.
💡 Tip: Before saving, double-check that your data is on the first sheet and follows the required format: no empty rows above the data, meaningful column names, and a complete Treatment–Day–Replication layout.
8.2 Prepare using Create Data in RAISINS
If you are unsure about the correct format, do not worry, RAISINS can create the data layout for you using the prescribed template. Here is how:
- Navigate to the Create Data tab
- Select the number of Treatments
- Select the number of Day/Time points (the repeated factor)
- Select the number of Variables
- Click the Create button
The model layout appears as shown in Figure 4, with the Treatment and Day/Time columns pre-filled and locked. You may enter the observations manually into the CSV once downloaded, or paste them straight into the table provided. Once the observations are entered, download the CSV and upload it under Analysis.
8.3 Download Model Datasets
If you are unsure about the required data format or would like to explore the module before using your own data, RAISINS provides model datasets for reference. To download them:
- Navigate to the Datasets tab
- Click the Download CSV link corresponding to the required dataset
- Save the file to your computer
- Use the model dataset as a reference for preparing your own data or upload it directly to explore the analysis
8.4 Creating a dataset using RA-One chat
RA-One, the built-in chat assistant, can help you create a properly formatted dataset through a simple conversation. To get started, open the RA-One chat by clicking the chat icon available within the app or by heading over to the RA-One tab. Now you can enter the number of treatments, day/time points, and response variables, and RA-One will generate a dataset in the required repeated-measures format. You can review the generated dataset in the chat, download it as a CSV file, and upload it directly under the Analysis tab. The full workflow is illustrated in Figure 6 below.
9 The Analysis tab
Figure 7 shows the Analysis tab in detail, with each option explained. Upload your prepared file by clicking Browse in the sidebar. Once uploaded, selectors appear for the three structural columns of a repeated-measures design: the Treatment column (the between-subject factor), the Day/Time column (the within-subject repeated factor), and the Subject / Replication column (the unit measured over time). You then select the response Variables you wish to analyse - one or many.
Below these, three analysis controls shape the test:
- Sphericity Correction - None (default), Greenhouse–Geisser (GG), or Huynh–Feldt (HF), as discussed in Section 3.
- Multiple comparison test - LSD (default) or TUKEY, used to group treatment and day means once an effect is significant.
- Level of significance (α) - 0.05 (default) or 0.01.
You can also set the number of decimal digits and the font for the tables. Click Run Analysis and all outputs appear across the sub-tabs: Analysis Results, Multivariate, Normality Test, Plots & Graphs, Interpretation, FAQs, and View Data. If your variables turn out not to be normally distributed, RAISINS provides a built-in transformation option (Section 10).
10 Transformation
Log, square root, and arcsine transformations are used to make data more normal and to even out uneven variation. You can apply them directly in RAISINS as shown in Figure 8.
Logarithmic transformation converts a skewed distribution into a more symmetrical one by replacing each data point (x) with its logarithm. It is applied to positive, continuous data where the variance grows in proportion to the mean, a pattern common in phenomena that grow multiplicatively or exponentially.
Square root transformation stabilises variance and reduces right-skewness by replacing each data point (x) with its square root. It is primarily used for non-negative count data, such as those following a Poisson distribution, where variance increases with the mean. By compressing the upper end of the scale more than the lower end, it brings the data closer to normality.
Arcsine transformation (the angular transformation) is designed for proportions or percentages bounded between 0 and 1. By taking the inverse sine of the square root of the proportion, it stretches the ends of the distribution near 0 and 1, where variance is naturally small. It is chiefly used to achieve homoscedasticity in binomial data.
After choosing the appropriate transformation, proceed to Section 11 for the analysis.
11 Analysis results
Once your dataset is uploaded and you click Run Analysis, the repeated-measures two-way ANOVA is fitted for every selected variable. The model RAISINS fits is
\[\text{response} \sim \text{Treatment} \times \text{Day} + \text{Error}(\text{Subject}/\text{Day})\]
which reads: the response depends on Treatment, on Day, and on their interaction, with the repeated measurements nested inside each Subject. The Analysis Results sub-tab presents the ANOVA summary first, then a detailed table for each significance source - Treatment, Day, and the Treatment × Day interaction. Read them in that order.
Table 1: ANOVA summary table
The ANOVA summary reports, for each source of variation (SoV - Treatment, Day, and Treatment × Day), the degrees of freedom (DF) and the mean squares for every selected character, with a significance mark. This is where you answer the three questions from Section 1: a significant Treatment row means the treatments differ overall; a significant Day row means the response changes over time; a significant Treatment × Day row means the treatments change over time at different rates. If you enabled a Greenhouse–Geisser or Huynh–Feldt correction (Section 3), the degrees of freedom and p-values of the Day and interaction rows reflect that correction. Significance is flagged with asterisks (* at the 5% level, ** at the 1% level) and NS for non-significant.
Table 2: Detailed results for Treatment
This table shows, for each treatment and each character, the mean ± SD together with a letter grouping. Treatments that share a letter are not significantly different from one another (at your chosen α), while treatments with no letter in common differ significantly. The grouping comes from the multiple-comparison test you selected - LSD or TUKEY. The table also reports the MSE and CV (%), which are common to the experiment. Use the Show / Remove letter grouping toggle above the table to display the means with or without the letters.
Table 3: Detailed results for Day
This table is read the same way as Table 2, but across the Day (repeated) factor: it shows the mean ± SD for each day and character, with a letter grouping that tells you which time points differ. Because Day is the within-subject factor, this is where any sphericity correction has its effect.
Table 4: Detailed results for the Treatment × Day interaction
When the interaction is significant, this is the most important table. It shows the mean for every Treatment × Day combination, so you can see exactly how the treatments diverge over time - for example, whether one treatment overtakes another only on later days. The footnotes report two standard errors of the difference: SE(d) DF (comparing day means of different treatments) and SE(d) SF (comparing day means of the same treatment), because these two kinds of comparison have different precision in a repeated-measures design.
11.1 Interpretation from Figure 9
Read the three tables together, following the order Treatment → Day → interaction, but let the interaction lead your conclusion. If the Treatment × Day term is significant, report that the treatment effect depends on time, and use Table 4 to describe the pattern day by day rather than quoting a single “best” treatment. If the interaction is not significant, you can interpret the Treatment and Day main effects on their own: the letter groupings in Tables 2 and 3 then give a clean ranking of treatments and of time points. In every case, pair the significance mark with the size of the difference in the means - a difference can be statistically significant yet too small to matter agronomically.
A tiny difference can be statistically significant if your experiment is large enough, and a large, agronomically valuable difference can miss significance in a small trial. Always read the mean difference and the size of the effect alongside the significance mark, and ask whether the difference matters in the field, not just on the page.
A quick glossary of the result columns
Overview of the ANOVA and detailed tables
- Sources of variation
Treatment: the between-subject factor - differences between the treatment groups, averaged over time.
Day: the within-subject (repeated) factor - differences between time points, averaged over treatments.
Treatment × Day: the interaction - whether the pattern over time depends on the treatment.
- Test statistics
DF (Degrees of Freedom): determines the reference distribution for each F-test. For the within-subject terms these are reduced when a Greenhouse–Geisser or Huynh–Feldt correction is applied.
Mean squares: the variance attributable to each source; the F-ratio is a mean square divided by the appropriate error mean square.
Generalized eta-squared (ges): the effect-size measure reported by the model, giving the proportion of variance explained by each effect.
- Descriptive columns
Mean ± SD: the cell mean and its standard deviation. A transformed mean, when a transformation is applied, is shown in parentheses.
Letter grouping: levels sharing a letter are not significantly different; the grouping is derived from the LSD or TUKEY comparison at the chosen α.
MSE and CV (%): the mean square error and coefficient of variation, common to the experiment.
SE(d) DF / SE(d) SF: standard errors of the difference for comparing means of different treatments versus the same treatment in the interaction table.
12 Summary statistics
The Summary stats sub-tab gives you the full descriptive picture of your data before you lean on the inference. For every Treatment × Day cell and each character it reports the sample size (N), Mean, Standard Deviation (SD), Standard Error (SE), Minimum, Maximum, Coefficient of Variation (CV %), Skewness and Kurtosis. Use the character selector to switch variables, and the Copy / Excel / CSV / PDF buttons to export the table (Figure 13).
In the working example (char1), the T1 group on day1 has a mean of 48.37 with an SD of only 0.21 (CV 0.43 %), while T2 on day1 averages 39.33 - the descriptive gap between treatments is already visible here, and the very small CVs tell you the replicates are tight. The Skewness and Kurtosis columns are an early warning for non-normal cells: values far from 0 (skewness) or 3 (kurtosis) flag a cell worth checking on the Normality Test tab before trusting the parametric result.
The ANOVA answers “are the differences bigger than the noise?” - the Summary stats show you what the noise and the means actually are. Reading them first makes the ANOVA and letter groupings easier to interpret, and quickly reveals data-entry problems (a wildly large SD, a stray minimum) that would otherwise distort the analysis.
13 Basic plots
The Basic Plots sub-tab turns your results into quick, publication-ready charts. Five plot types are available as clickable icons - Boxplot, Violin Plot, Mean Value Plot, Connected Line Plot and Bar Plot - and each one is generated the moment you click it (Figure 14). A Plot Settings panel on the left lets you customise everything: Plot Display Mode (single character or all characters at once), Title & Labels, Colors & Patterns, Show/Hide Options, and Line & Theme Settings. When you are happy, choose a Format (PNG, JPEG, TIFF, PDF or SVG) and download.
The a / b / c / d labels above each box are the same letter groupings from the multiple-comparison test in Section 11 - groups that share a letter are not significantly different. The plot and the detailed tables always tell the same story.
14 Advanced plots
The Advanced Plots sub-tab is where the repeated-measures design really comes to life. Alongside diagnostic and multivariate graphics - Summary Plot, Raincloud and Advanced Raincloud, Circular Plot, QQ Plot, Distribution Plot, Pair Plot, 3D Scatter and 3D Scatter + Line - it offers two dedicated Interaction Plots (I and II) that draw the Treatment means across the days. Pick the response variable, click a plot icon to generate it, and refine it through the Plot Settings panel (Title & Axes, Error Bars, Line & Point, Colours & Theme, Axis Text, Legend, Plot Styling), then download (Figure 15).
This is the visual counterpart of Section 2. If the lines for the treatments run roughly parallel across the days, there is little interaction - the treatments keep their ranking over time. If the lines cross or fan apart, the Treatment × Day interaction is real, and you should read the interaction table (Table 4) rather than the treatment main effect alone.
15 Individual ANOVA
The Individual ANOVA sub-tab shows the complete, classical ANOVA table for one selected character at a time - the fuller counterpart of the multi-character summary in Section 11. It lays out every Source of Variation with its DF, Sum of Squares (SS), Mean Sum of Squares (MSS), F value and p value, split into the two error strata that make a repeated-measures design work: the between-subject stratum (which tests Treatment) and the within-subject stratum (which tests Day and the Treatment × Day interaction). Beneath it, a pairwise comparison table lists every pair with its estimate, SE(d), DF, t ratio, p value and CD (Figure 16).
Reading the working example (char1): Treatment is highly significant, F(2, 6) = 1123.26, p < 0.01, tested against the between-subject error (MSS 0.51 on 6 df); Day is F(4, 24) = 2712.37, p < 0.01; and the Treatment × Day interaction is F(8, 24) = 61.56, p < 0.01 - both tested against the within-subject error (MSS 0.58 on 24 df). Because the interaction is significant, the treatments change over time at different rates (exactly the case Section 2 warns about). The pairwise table confirms the treatment gaps: T1 − T2 = 12.30 against a critical difference of just 0.64, so the difference is far larger than the CD and highly significant (p < 0.01); T1 − T3 (5.34) and T2 − T3 (−6.96) are likewise significant.
Between-subject vs within-subject error strata
A repeated-measures ANOVA does not have one error term - it has two. The between-subject stratum captures how whole subjects differ from each other, and it is the correct yardstick for the Treatment effect (differences between subjects that received different treatments). The within-subject stratum captures how measurements vary inside each subject across time, and it is the correct yardstick for Day and the Treatment × Day interaction. Splitting the error this way is exactly what Error(Subject/Day) does in the model (Section 11); an ordinary ANOVA, using a single pooled error, would test the within-subject effects against the wrong, inflated denominator.
16 Multivariate analysis
When you record several characters at once, you may want to rank or select treatments using all the characters together rather than one at a time. The Multivariate sub-tab does this with Principal Component Analysis (PCA). Select the characters you want to include, and RAISINS computes a PCA-based index score that combines them into a single ranking, along with the eigenvalues (how much variance each principal component explains), the variable loadings (how each character contributes), and a biplot. The index score is rescaled to a 0–1 range so treatments are easy to compare.
The PCA index is a convenient way to combine several characters into one score, but it assumes the characters are reasonably correlated and is best treated as an exploratory ranking tool rather than a formal hypothesis test. The ANOVA tables remain the basis for inference.
17 Interpretation
RAISINS provides a clear and concise interpretation of your repeated-measures ANOVA results to help you understand the findings with ease. The interpretation summarises the key results, states whether the Treatment, Day, and Treatment × Day effects are significant, explains what the letter groupings mean, and presents the findings in a publication-ready format. You can access this from the Interpretation sub-tab of the Analysis tab (Figure 18).
18 Chat with your data using RA-One
RA-One is the built-in conversational assistant for the repeated-measures two-way ANOVA module, available from the RA-One tab. You ask questions in plain language and it answers using your own analysis rather than generic statistical advice. Every result it discusses is drawn from what the module actually computed - it never invents numbers, and if a value isn’t available it says so instead of guessing. All answers are in plain English, with no code or software commands.
RA-One works directly with your Analysis Results, Multivariate, Normality Test, and Interpretation outputs. It can explain what the Treatment, Day, and interaction effects mean for your specific data, walk you through why a sphericity correction was or wasn’t applied, interpret the letter groupings from the LSD or TUKEY comparison, and answer general concept questions - between- versus within-subject factors, effect sizes, and how to read the interaction - so you can build understanding alongside your results.
The same chat window can also prepare your data. It can build a correctly formatted dataset template (Section 8.4) for you to fill in, or fetch a model dataset (Section 8.3) so you can try the module straight away - so you never need to leave the tab to get a file ready.
RA-One can also generate plots on request. Ask for a line plot of the treatments over time, a bar plot of treatment means with letter groupings, or a Q–Q plot for a variable, and the app renders the graphic directly in the chat, where you can view it and refine it by asking for changes.
Within a single conversation, RA-One can interpret your results, build a data template, fetch a model dataset, and produce plots - so most of a routine analysis session can be conducted without ever leaving the chat window.
19 FAQs
The module includes a dedicated FAQs section to clarify common doubts and guide you through the features. It offers detailed answers, additional information, and helpful tips for a smooth experience. If you are ever unsure how something works - say when to apply a sphericity correction, how to read the interaction table, or how to choose between LSD and TUKEY - the FAQs are a good place to start.
20 View data
View Data is the primary diagnostic tool for ensuring data integrity before analysis. When you upload your dataset, the system performs an automated Health Check to validate column types and formatting. For the repeated-measures two-way ANOVA this step is especially important: it confirms that the Treatment and Day columns are read as factors, that the Subject/Replication column identifies each unit, that every subject has a complete profile across all time points, and that all response variable columns are numeric with no missing or badly formatted entries that could distort the results.
21 Wrapping up
The repeated-measures two-way ANOVA rests on three honest questions: does the treatment matter, does time matter, and do the treatments behave differently over time? Everything else - the Error(Subject/Day) stratum, the sphericity correction, the normality checks - exists only to make sure those questions are answered fairly, given that the same units are measured again and again. RAISINS automates the machinery so you can concentrate on what the answers mean for your research.
If your design is simpler or different, the companion modules are there: the one-way repeated-measures ANOVA when time is your only factor, and the two-way ANOVA (CRD/RBD) when each unit is measured only once. And if you get stuck at any point, RA-One is available 24 × 7, or write to us at [email protected].