Two-Factor Factorial RBD
A two-factor factorial RBD tests two factors and their interaction while blocking out environmental heterogeneity. This tutorial explains where the design is used, how to size it, how main effects and interaction are defined, and how to run the whole analysis in RAISINS… Read more …
A two-factor factorial experiment studies two factors simultaneously, allowing you to understand not only the individual effect of each factor but also whether the two factors interact. When arranged in a Randomized Block Design (RBD), the experiment provides additional control over field or environmental variability by grouping treatment combinations within relatively homogeneous blocks. This helps separate block-to-block variation from experimental error, leading to more precise comparisons among treatments. RAISINS performs the complete analysis, including factorial ANOVA, post-hoc mean comparisons, effect size estimation, diagnostic checks, and relevant multivariate follow-up analyses, without requiring you to write a single line of code. This tutorial will guide you through the entire analysis step by step.
1 What is a two-factor factorial RBD?
Consider an agronomist who wishes to improve crop yield, where yield depends on two factors: the variety sown and the fertilizer applied. Suppose two varieties, a and b, and three fertilizers, A, B and C, are under study. One approach is to vary a single factor at a time, holding the other fixed. This is inefficient and incomplete: neither experiment estimates both effects, and neither can reveal whether one variety responds better to a particular fertilizer than another does.
A factorial experiment avoids this limitation by combining every level of one factor with every level of the other. Here this gives \(2 \times 3 = 6\) treatment combinations,
\[ aA,\; aB,\; aC,\; bA,\; bB,\; bC \]
each of which is grown and measured. A single experiment therefore yields estimates of the variety effect, the fertilizer effect, and the extent to which the two depend on each other.
Does the effect of one factor depend on the level of the other?
This dependence is the interaction effect. But a field is rarely uniform: soil fertility, moisture and slope vary across it, and if the six treatment combinations were simply scattered at random over the whole field, it can effect the precision of the experiment. The Blocking controls for this. The field is divided into blocks, groups of plots that are as alike as possible within a block. Every treatment combination appears once in every block, allotted at random within the block. Because each block has a full set of treatments, the block-to-block differences can be estimated and removed from the error. Together, factorial treatment combinations and blocking form a two-factor factorial experiment arranged in a Randomized Block Design (RBD).
A two-factor factorial RBD tests two factors together while blocking out environmental heterogeneity, so you can read each factor’s main effect and, above all, their interaction, from one experiment measured with greater precision.
2 How many observations will you need?
Sizing a two-factor factorial RBD is straightforward. Count the levels of each factor and the number of blocks (in an RBD the number of blocks is the number of replications, because every combination appears once per block):
- Let factor A have \(a\) levels and factor B have \(b\) levels.
- The number of distinct treatment combinations is the product
\[ a \times b \]
- If the experiment is laid out in \(r\) blocks, the total number of observations is
\[ N = a \times b \times r \]
For example, the working dataset used later in this tutorial has \(a = 6\) levels of Factor A, \(b = 3\) levels of Factor B and \(r = 5\) blocks, giving \(6 \times 3 = 18\) treatment combinations and \(N = 6 \times 3 \times 5 = 90\) observations.
The blocks are what give the analysis something to compare the treatment effects against, the error, but first the design spends a few degrees of freedom on the blocks themselves. In a two-factor factorial RBD the error degrees of freedom are
\[ df_{error} = (r - 1)(ab - 1) \]
so that same \(6 \times 3 \times 5\) experiment carries \((5 - 1)(18 - 1) = 4 \times 17 = 68\) error degrees of freedom, the figure you will see in the ANOVA table, while the blocks themselves take \(r - 1 = 4\) degrees of freedom.
The F-tests in a factorial ANOVA are only powerful when the error has enough degrees of freedom. A common rule of thumb is to aim for at least 12 error degrees of freedom.
3 Creating your design template
Entering collected data systematically in an excel spreadsheet helps to reduce any chance for miscalculation, RAISINS builds this template for you in two ways: the Create Data tab and the RA-One chat assistant. Based on the levels of factors under study.
3.1 Method 1 - the Create Data tab
Open the Create Data tab and fill in four boxes (Figure 1):
- Levels of Factor A - how many levels the first factor has,
- Levels of Factor B - how many levels the second factor has,
- Number of Blocks - how many blocks (replications) the experiment has,
- Number of characters to analyze - how many response variables (traits) you plan to measure.
Click Create, and the Data entry Panel on the right fills with every combination of Factor A and Factor B, repeated once in each block, with response columns (y1, y2, …) (Figure 2). Enter the values by hand, or paste a column straight from Excel. When the table is complete, click Download CSV file and upload it under the Analysis tab.
y1 column ready for your readings
The tick-box above the entry panel controls the row order. Sorted by treatments, the rows are grouped so that all blocks of a1 b1 sit together, then a1 b2, and so on, which is the easiest order to type into from a field book. The order of rows never affects the analysis; RAISINS reads the factor and block columns, not the row sequence.
3.2 Method 2 - the RA-One (AI) assistant
If you would like to chat and create template, ask RA-One, “make a 2F-RBD template with Factor A at 2 levels, Factor B at 2 levels, 3 blocks and 1 response”, and it returns a ready-to-fill template. Ra-one also calculate the df using the levels of the factor and the number of replication
4 Calculation of Main effect and Interaction effect
To learn about the main effect and interaction effect, let us use a simple agricultural experiment with 2 factors, each at 2 levels. Consider an experiment in which the effect of fertilizer and irrigation on yield of a crop is studied. Factor A is fertilizer, at 50 kg/ha (low) and 100 kg/ha (high); factor B is irrigation, at 3 irrigations (low) and 6 irrigations (high). Suppose the crop yield observed under each treatment combination is given below:
| Combination | Fertilizer (A) | Irrigation (B) | Yield (t/ha) | Response symbol |
|---|---|---|---|---|
| both low | 50 kg/ha | 3 irrigations | 3.0 | \((1)\) |
| A high only | 100 kg/ha | 3 irrigations | 3.5 | \((a)\) |
| B high only | 50 kg/ha | 6 irrigations | 4.0 | \((b)\) |
| both high | 100 kg/ha | 6 irrigations | 6.0 | \((ab)\) |
Here \((1)\) is the yield when both factors are at their low levels, \((a)\) is the yield when fertilizer is high and irrigation is low, \((b)\) is the yield when fertilizer is low and irrigation is high, and \((ab)\) is the yield when both fertilizer and irrigation are at their high levels. These four responses will be used to calculate the main effects and interaction effect.
4.1 The main effect of a factor
Read Figure 3 along its edges. The bottom edge holds irrigation at 3 irrigations, so increasing fertilizer from 50 kg/ha to 100 kg/ha changes the yield from \((1)\) to \((a)\). The top edge holds irrigation at 6 irrigations, so the same increase in fertilizer changes the yield from \((b)\) to \((ab)\).
The main effect of a factor is the average change in the response when that factor moves from low to high, averaged over the levels of the other factor. Increasing fertilizer from 50 kg/ha to 100 kg/ha changes the yield by \((a)-(1)\) when irrigation is low, and by \((ab)-(b)\) when irrigation is high. Averaging these two changes gives the main effect of fertilizer (factor A):
\[ A = \tfrac{1}{2}\big[(a)-(1) + (ab)-(b)\big] \]
or equivalently,
\[ A = \tfrac{1}{2}\big[(ab) + (a) - (b) - (1)\big] \]
By the same reasoning, the main effect of irrigation (factor B) averages the change in yield from 3 irrigations to 6 irrigations across both fertilizer levels:
\[ B = \tfrac{1}{2}\big[(b)-(1) + (ab)-(a)\big] \]
or equivalently,
\[ B = \tfrac{1}{2}\big[(ab) + (b) - (a) - (1)\big] \]
4.2 The interaction effect
The interaction asks whether the effect of one factor depends on the level of the other factor. The effect of fertilizer when irrigation is high is \((ab)-(b)\), whereas the effect of fertilizer when irrigation is low is \((a)-(1)\). If these two changes are equal, fertilizer produces the same change in yield at both irrigation levels, indicating no interaction. If they differ, the effect of fertilizer depends on the irrigation level, indicating an interaction between fertilizer and irrigation.
The interaction effect is calculated as half the difference between these two fertilizer effects:
\[ AB = \tfrac{1}{2}\big[(ab)-(b)-(a)+(1)\big] \]
or equivalently,
\[ AB = \tfrac{1}{2}\big[(ab) + (1) - (a) - (b)\big] \]
Please note that the interaction of A with B is identical to the interaction of B with A.
A large interaction does not mean that the factors are more important. Instead, it means that the effect of one factor depends strongly on the level of the other factor, so the main effect of a single factor should not be interpreted in isolation. When the interaction effect (\(AB\)) is close to zero, the factors act more independently, and the main effects can be interpreted with greater confidence.
5 Getting to the module
Now let us run the analysis. Visit the RAISINS home page at www.raisins.live and oopen the Two factor experiment section under the Analysis of experiment section. Among the design and analysis tools, choose 2F RBD - Two factor factorial in RBD Where you can upload your data, select the the two factor columns and the response columns, and it produces every table, plot and interpretation for you.
5.1 Computational Provenance & Reproducibility Record
CPRR (Computational Provenance & Reproducibility Record) provides a transparent and comprehensive record. Click the CPRR icon in the module header to see the exact computational workflow behind the analysis. The record for this module states the R version and the exact version of every package used, and 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 two-factor factorial RBD module is at www.raisins.live/module_record/2frbd.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 a built-in factorial dataset so you can try every feature, the factorial ANOVA, post-hoc mean comparisons, the interaction and diagnostic plots, the MANOVA and PCA index, 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
The rest of this tutorial follows one dataset, shown in Figure 5. It is a two-factor factorial in RBD with Chemical1 as Factor A at 6 levels (S1–S6) and Chemcial2 as Factor B at 3 levels (F1, F2, F3), giving \(6 \times 3 = 18\) treatment combinations, each grown once in every one of 5 blocks for \(N = 90\) observations. The block (replication) column here is labelled Replication with levels R1–R5. Eight traits were measured on every unit, char1–char8, all analysed here. The first three columns hold the two factor labels and the block; every remaining column is a response. The question the module will settle is whether each character depends on Chemical1, on Chemcial2, and, crucially, on their interaction, once the differences between blocks have been removed.
Because the full file runs to 90 rows, Figure 5 shows only the first block (R1), the 18 treatment combinations once each; in the complete dataset every combination appears on five rows, one per block R1–R5.
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 (Section 3.1)
- Use the Model dataset in RAISINS as a reference
- Create your dataset using the RA-One chat assistant (Section 3.2)
8.1 Preparing data in MS Excel
Lay the file out exactly as in Figure 5: the first column holds the levels of Factor A (here Chemical1, with levels S1–S6), the second column the levels of Factor B (here Chemcial2, with levels F1–F3), the third column the block or replication (here Replication, with levels R1–R5), and each further column is one response variable (char1, char2, …). Every row is one experimental unit, so a combination grown in five blocks appears on five rows. Keep each level label spelled consistently (always S1, never s1 or S 1), make sure every treatment combination appears once in each block, leave no blank rows above or inside the data, and save the finished sheet as CSV before uploading.
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.
- Factor, block and response columns
- The first three columns must be the two factors (factorA, factorB) and the block/replication column; every remaining column is a numeric response.
- Use the same label for a level everywhere it appears, so RAISINS reads the correct number of levels and blocks.
- Replication within blocks
- Repeat each factorA × factorB combination once in every block; do not average replicates into a single row, and do not leave a combination out of a block (the design must stay balanced).
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.
8.2 Prepare using Create Data in RAISINS
If you are unsure about the correct format, do not worry, RAISINS can create the layout for you from the prescribed template. Enter the levels of Factor A, the levels of Factor B, the number of blocks and the number of response columns in the Create Data tab, then click Create, as described in Section 3.1 and shown in Figure 1. Type or paste your observations into the generated table (Figure 2), download the CSV, and upload it under Analysis.
8.3 Download the Model dataset
If you would like to explore the module before using your own data, RAISINS provides a model dataset, the very dataset in Figure 5. To use it:
- Open the Datasets tab
- Click the Download CSV link for the two-factor factorial RBD dataset
- Save the file, then either study its layout as a reference for your own file or upload it directly under Analysis to see the full analysis at once
8.4 Creating a dataset using RA-One chat
RA-One, the built-in chat assistant, can build a correctly formatted template through a simple conversation, as shown in Section 3.2. Tell it the levels of each factor, the number of blocks and how many responses you measured, and it returns a ready-to-fill template, even warning you when the design is under-powered. Fill in the responses, download the CSV, and upload it directly under the Analysis tab, all without leaving the chat.
9 The Analysis Results tab
In the option provided, upload your csv or excel file
Once the file is uploaded, column names are automatically identified. from the columns in the file:
- Select Block - the column holding the block or replication (here,
Replication, with levels R1–R5). - Select Factor A - the column holding the levels of the first factor (here,
Chemical1). - Select Factor B - the column holding the levels of the second factor (here,
Chemcial2). - Select variables - one or more response columns to analyse; you may select all of them at once (here
char1–char8), and RAISINS analyses each in turn. - Click for Transformation - an optional toggle to transform skewed responses before analysis (see below).
When the columns are chosen, click Run Analysis!. Then the results, table, visualisation and interpretation from your data will be available in the following tabs- Analysis Results, Basic Plots, Advanced Plots, Interpretation, Multivariate, and In case of any confusions use these tabs; FAQs (Frequently asked question) and View Data.
You do not need to run the module once per response. Select all the response columns together and RAISINS produces a complete ANOVA, mean-comparison and effect-size table for each, plus a single MANOVA across all of them. This is why the results tables in the next section carry one column per trait (char1–char8).
9.1 Optional: transforming a response
The factorial ANOVA assumes the response is roughly normal with similar variability across treatments. When a trait is strongly skewed or its spread grows with its mean, tick Click for Transformation and choose a transformation before running the analysis (Figure 6).
| Transformation | Use it when | What RAISINS applies |
|---|---|---|
| Logarithmic | The spread grows in proportion to the mean (multiplicative data) | \(\log_{10}(x)\); if any value is \(\le 0\), it shifts first as \(\log_{10}(x - \min + 1)\) |
| Square-root | Counts, or data following a Poisson-like pattern | \(\sqrt{x}\); if any value is \(0\), it uses \(\sqrt{x + 0.5}\); negative values are not allowed |
| Arcsine | Proportions or percentages between 0 and 1 | \(\arcsin(\sqrt{x})\), with \(0\) and \(1\) nudged by \(1/(4n)\) so the transform stays finite |
A transformation is a temporary change of scale to make the test valid; it does not change your experiment. Report that a transformation was used, but discuss the means and recommendations in the original units. If no assumption is violated, leave the toggle off, an unnecessary transformation only makes the results harder to read.
9.2 Optional: adjusting for multiple comparisons (P-adjustment)
When you compare many treatment means, each pairwise comparison carries its own chance of a false positive, and those chances accumulate. The P-adjustment control, in the panel above the results tables (Figure 7), corrects for this. It is offered only for the LSD post-hoc method, and applies to the pairwise comparisons of Factor A, Factor B and the A × B interaction, recomputing the grouping letters accordingly. The choices are:
- None - plain LSD (the default); most powerful, but the highest false-positive rate.
- Bonferroni - the strictest control of the family-wise error rate; use when a single wrong “significant” call is costly.
- Holm - controls the same error rate as Bonferroni but is uniformly more powerful; a good default when you want family-wise control.
- Benjamini-Hochberg (BH) - controls the false-discovery rate; suited to many treatments where a few false positives are tolerable (e.g. screening).
Tukey’s HSD and Duncan’s DMRT already build in their own error control, so the option does not appear for them.
Once an adjustment is applied, a single critical difference (CD) is no longer well defined, the effective threshold differs from one comparison to the next, so RAISINS hides the CD when p-value adjustment is enabled.
10 Analysis results
The Analysis Results sub-tab provides all the formated tabels required for reporting. A control panel at the top lets you set the multiple-comparison test (LSD by default, with Tukey’s HSD and Duncan’s DMRT as alternatives), the P-adjustment (Section 9), the level of significance (\(\alpha\), 0.05 by default), the digits after decimal, and the font. A Quick Summary of the results and the dataset used is given below it.
Table 1: ANOVA summary - mean squares and significance
Each row is a source of variation, Block (4 df), Factor A (5 df), Factor B (2 df), the interaction A × B (10 df) and Error (68 df), matching the degrees of freedom predicted in Section 2 for a \(6 \times 3 \times 5\) experiment. Each cell is that source’s mean square for the response, with a superscript flag: * for significance at 5%, ** at 1%, and NS for non-significant. Super script help to quickly identify significant factors, for example: char5 (\(342.47^{**}\)) and char8 (\(402.58^{**}\)) respond strongly to Factor A; char1 (\(323.72^{*}\)), char2 (\(420.60^{*}\)) and char3 (\(132.60^{*}\)) respond to Factor B; and char1 additionally carries a significant A × B interaction (\(157.99^{*}\)). The Block row is significant for several traits (char2, char3, char5, char6, char8 at 1%), indicating that blocking helped to convert heterogenous field plots into homogenous blocks.
How this table is built
RAISINS fits the linear model response ~ Block + A + B + A:B for each trait and forms the analysis-of-variance table with Type II sums of squares. Each source’s mean square is its sum of squares divided by its degrees of freedom, and the F-test divides the treatment mean squares by the error mean square; the p-value behind each F decides the * / ** / NS flag against the chosen \(\alpha\). The block term is fitted first so its variation is removed before the factor effects are tested. This is the many-level, block-adjusted generalisation of the \(2 \times 2\) contrasts in Section 4.
Table 2: Factor A - means and comparison statistics
This table averages over Factor B and the blocks to show the main effect of Factor A (the six levels S1–S6). Factor A affects two responses. For char5, \(F = 14.22^{**}\) (\(p = 0.00\)) with a critical difference CD (A) of 3.58 and a large Cohen’s F of 1.02: S4 (\(74.52 \pm 4.76\)) and S1 (\(73.18 \pm 8.35\)) share letter a as the highest, while S2 (\(63.63 \pm 4.87\)) and S5 (\(62.90 \pm 6.47\)) share letter c as the lowest. For char8, \(F = 4.33^{**}\) (\(p = 0.00\), CD (A) \(= 7.02\), Cohen’s F \(= 0.56\)): S4 (\(107.76 \pm 8.74\)), S6 (\(104.63 \pm 11.08\)), S1 (\(104.37 \pm 10.49\)) and S3 (\(102.56 \pm 10.50\)) form the top group (letter a), while S2 (\(95.33 \pm 12.60\)) and S5 (\(95.34 \pm 10.87\)) fall to letter b. For char1–char4, char6 and char7 the F-statistics are non-significant, so their CD (A) rows read “–”, The lower rows report the pooled MSE, the standard error of a mean SE(m) and of a difference SE(d), the coefficient of variation CV(%), and the Cohen’s F effect size.
Table 3: Factor B - means and comparison statistics
The Factor B table has the same layout, now averaging over Factor A and the blocks to show the main effect of Factor B across its three levels (F1, F2, F3). Factor B matters for three responses. For char1, \(F = 4.41^{*}\) (\(p = 0.02\), CD (B) \(= 4.42\)): F1 (\(127.39 \pm 6.91\)) and F2 (\(128.90 \pm 8.71\)) share letter a, while F3 (\(122.61 \pm 11.23\), letter b) is significantly lower. For char2 the pattern repeats: \(F = 4.40^{*}\) (\(p = 0.02\), CD (B) \(= 5.04\)), with F1 and F2 on par (letter a) and F3 (\(89.83 \pm 11.34\), letter b) lowest. For char3 the order flips: \(F = 3.07^{*}\) (\(p = 0.05\), CD (B) \(= 3.39\)), with F3 (\(52.14 \pm 6.85\)) the highest (letter a) and F1 (\(47.96 \pm 6.59\)) the lowest (letter b). So the third level of Chemcial2 depresses char1 and char2 but lifts char3; for char4–char8 no level of Factor B stands out.
Table 4: Interaction A × B - cell means and comparison statistics
This is the Interaction effect table. Its rows are the eighteen treatment combinations, each level of Chemical1 crossed with each level of Chemcial2 (S1×F1, S1×F2, S1×F3, …, S6×F3), shown as mean ± SD. When the interaction is significant for a response, a letter grouping appears as a superscript: combinations sharing a letter are not significantly different. The one response with a significant interaction is char1: \(F = 2.15^{*}\) (\(p = 0.03\)) with a critical difference CD (A×B) of 10.82. There the letters come alive, the combination S6×F2 is highest at \(132.52 \pm 7.91\) (letter a) while S6×F3 is lowest at \(115.80 \pm 13.55\) (letter c). For char2–char8 the interaction is non-significant, so those columns show no letters and CD (A×B) reads “–”.
Toggling Two-way table in the strip above the table redraws char1 as a grid, Chemical1 down the rows, Chemcial2 across the columns, with the Mean(A) and Mean(B) margins and all three critical differences (CD (A), CD (B) and CD (A×B)) printed beneath (Figure 11). This is the most natural way to read a significant interaction: you can see at a glance that the F3 column falls away sharply for S4, S5 and S6 while staying level for S1–S3, which is precisely what “the effect of Chemcial2 depends on Chemical1” looks like.
Reading every row of the factor tables
- Mean ± SD - the treatment (or cell) mean and its standard deviation; a transformed mean, if a transformation was applied, is shown in parentheses.
- F stat / p value - the F-test for that source and its probability; the superscript (
*,**,NS) marks significance at the chosen \(\alpha\). - CD - the critical difference (from the selected post-hoc test); two means differing by more than the CD are significantly different. Shown as “–” when the source is non-significant, and hidden entirely when a P-adjustment is applied.
- MSE - the pooled error mean square, common to all comparisons of that response.
- SE(m) / SE(d) - standard error of a single mean and of a difference between two means.
- CV(%) - coefficient of variation, the error SD as a percentage of the grand mean; it gauges experimental precision and is common to the whole experiment.
- Cohen’s F - a standardised effect size for the source (roughly: below 0.10 very small, 0.25 small, 0.40 medium, above 0.40 large).
10.1 Interpretation from Figure 7
Put the four tables together and each character tells its own story. char5 and char8 are governed by Chemical1 (Factor A) alone, S4 and S1 give the highest char5 values and S5/S2 the lowest, with Chemcial2 having no say; the char5 effect is not just significant but large (Cohen’s F 1.02). char2 is governed by Chemcial2 (Factor B) alone, F3 sits below F1 and F2, regardless of Chemical1, and char3 too, though there F3 is the highest level. char1 is the interesting one: it responds to Chemcial2 and carries a significant A × B interaction. This is exactly the situation flagged back in Section 4, the effect of Chemcial2 on char1 depends on which Chemical1 it is paired with, so you cannot name a single best Chemcial2 level for char1. Instead you read the cell means, char1 peaks at S6×F2 (\(132.52\)) and bottoms out at S6×F3 (\(115.80\)), and use the letter grouping (and the CD of 10.82) to judge which combinations are genuinely on par. This is the payoff of the factorial RBD: it separates factors that act alone (char5, char8, char2, char3) from a pair that must be chosen together (char1), from one experiment, and it does so against an error already sharpened by blocking.
A significant effect tells you the pattern is unlikely to be chance; it does not, by itself, tell you the difference is large enough to matter. Read the Cohen’s F effect size alongside the p-value, the char5 Factor A effect is not only significant but large (Cohen’s F 1.02), whereas the char1 interaction, though significant, is more modest (0.36 for Factor B, 0.16 overall on char1). Weigh those sizes against what is meaningful in your own study before acting on them.
11 Visualising the results
Numbers in a table are convincing; a picture is persuasive. RAISINS groups its graphics into two sub-tabs, Basic Plots and Advanced Plots, and every plot can be redrawn for any factor and any response, then exported as PNG, JPEG, TIFF, PDF or SVG.
The Basic Plots tab (Figure 12) offers the, Box Plot, Violin Plot, Mean Value Plot, Connected Line Plot and Bar Plot. Use the Plot Settings panel to choose the factor to place on the x-axis (Chemical1, Chemcial2, or the interaction) and the response to plot, then pick a download format. Figure 13 shows the same family of comparisons drawn four ways.
The letters printed above each box, bar or point are the very letter groupings from the Factor A, Factor B and interaction tables. A plot and its table can never disagree, they are two views of one computation, so use whichever communicates best to your audience.
The Advanced Plots tab (Figure 14) adds the specialised graphics: an Interaction Plot, a Summary Plot, Raincloud views, a Circular Plot, a QQ Plot, a Distribution Plot, a Pair Plot, a Correlation Plot, and 3-D Scatter views. Exactly as on the Basic Plots tab, a Plot Settings panel beside each graphic lets you choose the factor and the response and export the result as PNG, JPEG, TIFF, PDF or SVG. Click any button to draw its plot, then fine-tune it in the settings panel.
Interaction plot
The Interaction Plot is the single most important graphic on this tab. It draws the mean response for each level of one factor as a line across the levels of the other, RAISINS shows both orientations (Chemcial2 profiled across Chemical1 and Chemical1 across Chemcial2) alongside each factor’s own main-effect line. Parallel lines signal no interaction; lines that cross or fan apart signal one, this is the visual form of the interaction contrast from Section 4 and the natural companion to the char1 finding in Table 4 and Figure 11.
The remaining advanced graphics look past a single response to the shape of the data and the relationships between traits. Figure 16 gathers five of them.
3-D scatter views
When you present a significant interaction, show the interaction plot beside the interaction table. The audience sees at a glance that the Chemcial2 effect on char1 changes with Chemical1, whereas the table proves it. Together they make the case the single most important claim a factorial experiment can support.
12 Looking at all traits together: MANOVA and PCA index
The Analysis Results tab tests each trait one at a time. Sometimes you want to know whether the treatments differ when all the traits are considered together, or to combine the traits into a single selection index. The Multivariate tab (Figure 18) provides both tools: a MANOVA and a PCA-based index score. RAISINS lists the response variables entering the multivariate study and offers two buttons, Click here for MANOVA and Click here for PCA Index.
12.1 MANOVA across all responses
MANOVA (Multivariate Analysis of Variance) asks whether the treatment combinations differ when the eight traits are treated as one bundle. RAISINS fits the block-adjusted two-way model with interaction and reports Pillai’s trace, the most robust multivariate test statistic, converted to an F-value, together with a partial \(\eta^2\) effect size for each source (Block, Factor A, Factor B and A × B). Read the p-value of each source to decide whether the treatments separate on the traits collectively.
12.2 PCA-based index score
When you want to rank treatment combinations on all traits at once, the PCA-based index condenses the correlated traits into a few independent components and builds a score from them. The eigenvalue table shows how much variance each component captures, and the accompanying scree plot shows the same information graphically. To decide which index to use, read the loadings of each trait on the components: traits with high positive loadings on PC1 push a combination’s PC1 score up, and the biplot places each treatment combination in the direction of the traits it scores highly on. RAISINS then converts the chosen index to a scaled 0–1 score, and the Select cutoff for Scaled Index Score control lets you highlight the top (or bottom) percentage of treatment combinations.
MANOVA is inferential, does any treatment differ across the traits collectively? PCA is exploratory, how can the traits be combined into an index to rank treatments? They complement each other. RAISINS advises running the PCA index with caution when the interaction is non-significant, since that is when ranking a single “best combination” is least clear-cut.
13 Interpretation
RAISINS provides a clear, plain-language interpretation of your results so you can understand the findings without decoding the tables yourself. Open the Interpretation sub-tab, confirm you are not a robot, and click Click here for interpretation. RAISINS restates the design, then reads every response for you: which main effects and interactions were significant, what the post-hoc letter groupings mean, and how large the effects are. For the working dataset it reports the Factor A effects on char5 (S4 highest at \(74.52 \pm 4.76\), S5 lowest at \(62.90 \pm 6.47\)) and char8, the Factor B effects on char1, char2 and char3 (F3 the lowest level for char1 and char2, the highest for char3), and the significant interaction on char1 (\(p = 0.03\)), with S6×F2 highest (\(132.52 \pm 7.91\)) and S6×F3 lowest (\(115.80 \pm 13.55\)), spelling out which combinations are statistically on par. The text is written to be pasted almost directly into a results section.
The interpretation is generated from the same computation as the tables, so use it as a guide, not a substitute. Read it with Figure 7 and Figure 10 open beside it, the numbers in the prose should match the numbers in the tables exactly, which is your quickest check that you selected the right columns.
14 Chat with your data using RA-One
RA-One is the built-in conversational assistant for the two-factor factorial RBD 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.
Once your results are loaded, RA-One greets you and offers quick-start prompts (Figure 19): Guide me through my results for a step-by-step walkthrough, Interpret results to summarise the significance of each character, Best treatment to ask which combination performs best across all traits, and Report results to help draft the write-up. You can also type your own question, for example “is the interaction significant for any trait?” or “which Chemcial2 level gives the highest char3?”.
Choosing Guide me through my results walks you through the analysis in numbered steps (Figure 20). For the working dataset it opens with the experiment overview, 6 levels of Factor A (Chemical1), 3 levels of Factor B (Chemcial2), 5 blocks, 90 observations and eight responses char1–char8, then explains what “significant” means for each trait (char1: Factor B and A × B significant; char5 and char8: Factor A highly significant; char4, char6, char7: nothing significant), and finally points to the best treatments, reading the interaction means for char1 and the Factor A means for char5 and char8. It is the same reading the Interpretation tab gives, delivered conversationally.
The same chat window can also prepare your data. It can build a correctly formatted design template (Section 3.2) for you to fill in, or fetch the 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.
Within a single conversation, RA-One can interpret your results, build a data template, fetch the model dataset, and guide you through the tables, so most of a routine two-factor factorial RBD session can be conducted without ever leaving the chat window.
15 FAQs
The module includes a dedicated FAQs sub-tab to clear up common doubts and guide you through the features, with detailed answers and practical tips (Figure 21). Typical questions include “what is Cohen’s F in the results?”, “how is the transformation algorithm applied?”, and, matching the P-adjustment control in Section 9, “when should I use P-adjustment?”, which explains that the option is for LSD, when to prefer Bonferroni, Holm or Benjamini–Hochberg, and why the CD and p-value rows disappear once an adjustment is switched on.
16 View data
View Data is the primary diagnostic tool for ensuring data integrity before analysis (Figure 22). When you upload your dataset, RAISINS runs an automated Health Check to validate the file: it confirms that the block and two factor columns are present and read as categorical labels, that the response columns are numeric, and that there are no stray blanks or misaligned rows. For a factorial RBD it also checks that every factorA × factorB combination is present once in every block and reports whether the design is balanced, the condition under which the standard errors and critical differences are exact. Fix any flagged issue here before trusting the results.
17 Wrapping up
A two-factor factorial RBD exists to answer one honest question: do two factors act independently, or does the effect of one depend on the other?, and to answer it against an error sharpened by blocking. Everything in this module, the ANOVA summary with its Block row, the Factor A and Factor B tables, the interaction table, the plots, the MANOVA and the interpretation, exists to answer that question fairly, and to tell you the cell means to act on when the answer is “they interact.”
This module fits the case where your experimental units fall into natural groups, blocks of field, bench or batch, so a block column controls the background variation. If your units are genuinely homogeneous and need no blocking, use the two-factor factorial CRD module instead; if you are studying a single factor, use the one-way RBD module. And if you get stuck at any point, RA-One is available 24 × 7, or write to us at [email protected].