Two-Factor Factorial CRD
A two-factor factorial CRD tests two factors and their interaction on homogeneous experimental units. 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 at the same time and, crucially, tells you not only how each factor acts on its own but whether the two factors interact. Laid out in a Completely Randomized Design (CRD), it is the workhorse of controlled trials where experimental units are homogeneous, such as pot-culture, laboratory and growth-chamber studies. RAISINS runs the entire analysis, the factorial ANOVA, post-hoc mean comparisons, effect sizes, diagnostics and multivariate follow-ups, without you writing a single line of code. This tutorial will guide you step-by-step.
1 What is a two-factor factorial design?
Imagine an agronomist who wants to raise the yield of a crop. The yield in each plot depends on two things at once, the variety sown and the fertilizer applied. Suppose there are two varieties, a and b, and three fertilizers, A, B and C. A tempting shortcut is to study one thing at a time: fix the variety, vary the fertilizer in a first experiment; then fix the fertilizer, vary the variety in a second. The difficulty is that the two experiments never meet. In the first, the difference between varieties is never measured; in the second, the difference between fertilizers is never measured. Worse, if one variety happens to respond well to a particular fertilizer and another responds poorly, that dependence, the very thing a breeder cares about, is invisible to both experiments.
A factorial experiment removes this blind spot by combining each level of one factor with each level of the other. With two varieties and three fertilizers there are \(2 \times 3 = 6\) treatment combinations,
\[ aA,\; aB,\; aC,\; bA,\; bB,\; bC \]
and every combination is grown and measured. From a single experiment you can now estimate the effect of variety, the effect of fertilizer, and whether the varieties respond differently to different fertilizers.
Does the effect of one factor depend on the level of the other?
That last question, the interaction, is the reason factorial designs exist. When the experimental units are essentially uniform (pots filled from the same soil, Petri dishes in one incubator, seedlings in a single growth chamber), the six treatment combinations are simply allotted completely at random to the units. That layout, factorial treatments assigned by complete randomisation, is the two-factor factorial in a Completely Randomized Design (CRD) that this module analyses.
A two-factor factorial CRD tests two factors together on homogeneous units so you can read each factor’s main effect and, above all, their interaction, from one experiment.
2 How many observations will you need?
Sizing a two-factor factorial is straightforward. Count the levels of each factor and the number of replications (how many times each combination is repeated):
- 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 each combination is replicated \(r\) times, 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 \(r = 5\) replications, giving \(6 \times 3 = 18\) combinations and \(N = 6 \times 3 \times 5 = 90\) observations.
The replications are what give the analysis something to compare the treatment effects against, the error. In a two-factor factorial CRD the error degrees of freedom are
\[ df_{error} = a \, b \, (r - 1) \]
so that same \(6 \times 3 \times 5\) experiment carries \(18 \times (5 - 1) = 72\) error degrees of freedom, the figure you will see in the ANOVA table.
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. Because \(df_{error} = ab(r-1)\), the lever you control is the number of replications \(r\), add replications rather than levels if the design feels thin. RAISINS’ template builder warns you when the error df drops below 12.
3 Creating your design template
Before you collect a single reading, it helps to lay out the skeleton of the experiment, one row for every treatment combination × replication, ready for you to type the measured values into. RAISINS builds this skeleton for you in two ways: the Create Data tab and the RA-One chat assistant. Both ask only for the sizes you settled on in the previous section.
3.1 Route 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 Replications - how many times each combination is repeated,
- 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, each repeated for the requested replications, with an empty response column (y1, y2, …) waiting for your data. Enter the values by hand, or paste a column straight from Excel with Ctrl + V. When the table is complete, click Download CSV file and upload it under the Analysis tab.
The tick-box above the entry panel controls the row order. Sorted by treatments, the rows are grouped so that all replications 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 columns, not the row sequence.
3.2 Route 2 — the RA-One (AI) assistant
If you would rather just describe your experiment in words, ask RA-One. Tell it, for instance, “make a 2FCRD template with Factor A at 2 levels, Factor B at 2 levels, 2 replications and 1 response”, and it returns a ready-to-fill template (Figure 2). RA-One does something the plain template cannot: it checks the size for you. In Figure 2 it reports Error df = 2 × 2 × (2 − 1) = 4 and warns that, because 4 is below 12, the F-tests will have low power, advising you to add replications. You can edit the table in place, add rows or response columns, and click Download CSV when done.
Notice that both routes ask for exactly the three numbers from Section 2, levels of A, levels of B, and replications, and produce \(a \times b \times r\) rows. The template is the observation count made concrete: one row per observation, and the error-df formula printed back to you.
4 Calculation of Main effect and Interaction effect
The simplest factorial, two factors each at two levels (a \(2^2\) design), is the clearest place to see what “main effect” and “interaction” actually mean. Consider an experiment in which the current and voltage supplied to a fan affect its speed in revolutions per minute. Factor A is current, at 5 A (low) and 10 A (high); factor B is voltage, at 200 V (low) and 220 V (high). The four treatment combinations, and a compact notation for the response at each, are:
| Combination | Current (A) | Voltage (B) | Response symbol |
|---|---|---|---|
| both low | 5 A | 200 V | \((1)\) |
| A high only | 10 A | 200 V | \((a)\) |
| B high only | 5 A | 220 V | \((b)\) |
| both high | 10 A | 220 V | \((ab)\) |
Here \((1)\) is the response with both factors at their low level (the control), \((a)\) raises only current, \((b)\) raises only voltage, and \((ab)\) raises both.
4.1 The main effect of a factor
Read Figure 3 along its edges. The bottom edge holds voltage low, moving current from 5 A to 10 A goes from \((1)\) to \((a)\); the top edge holds voltage high, the same move goes from \((b)\) to \((ab)\).
The main effect of a factor is how much the response changes when that factor moves from low to high, averaged over the levels of the other factor. Raising current from 5 A to 10 A changes the response by \((a)-(1)\) when voltage is low, and by \((ab)-(b)\) when voltage is high. Averaging these two gives the main effect of current (factor A):
\[ A = \tfrac{1}{2}\big[(ab) + (a) - (b) - (1)\big] = \tfrac{1}{2}(a-1)(b+1) \]
By the same reasoning, the main effect of voltage (factor B) averages the change from 200 V to 220 V across both current levels:
\[ B = \tfrac{1}{2}\big[(ab) + (b) - (a) - (1)\big] = \tfrac{1}{2}(a+1)(b-1) \]
4.2 The interaction effect
The interaction asks whether the effect of one factor depends on the level of the other. The effect of voltage when current is high is \((ab)-(a)\); when current is low it is \((b)-(1)\). If these two are equal, the factors act independently and there is no interaction. Their difference, halved, is the interaction effect:
\[ AB = \tfrac{1}{2}\big[(ab) + (1) - (a) - (b)\big] = \tfrac{1}{2}(a-1)(b-1) \]
Reassuringly, you get the same quantity if you instead ask how the effect of current changes with voltage, the interaction of A with B is identical to the interaction of B with A.
Notice the tidy pattern in the algebra: to isolate a factor’s effect you write that factor’s symbol with a minus sign and the other with a plus sign, \(A \to (a-1)(b+1)\), \(B \to (a+1)(b-1)\); the interaction carries a minus sign for both, \((a-1)(b-1)\). Each effect is a difference of two averages, so it is divided by 2, while the overall mean, based on all four combinations, is divided by 4.
A large interaction does not mean the factors matter more, it means the story cannot be told one factor at a time. When \(AB\) is effectively zero, the main effects fully summarise the experiment. When \(AB\) is large, a “best level” of A exists only for a given level of B, which is precisely why the results section below reads the interaction table before trusting the factor averages.
From a 2 × 2 to any number of levels
-
The \((1),(a),(b),(ab)\) algebra is the exact form only for two factors each at two levels, but the ideas carry straight over to the \(6 \times 3\) experiment analysed later, and to any \(a \times b\) factorial. In the general case RAISINS does not compute these contrasts by hand; it fits the linear model
response ~ A + B + A:B and partitions the variation into a Factor A term, a Factor B term and an A × B interaction term in the ANOVA table. Each term’s mean square is compared with the error mean square by an F-test, the significance you read in the results is the many-level generalisation of the simple differences above.
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 open the Data Analysis section. Among the design and analysis tools, choose 2FCRD — Two factor factorial in CRD. No programming is required: you upload your data, point RAISINS at 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 CRD module is at www.raisins.live/module_record/2FCRD.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 4. It is a two-factor factorial in CRD 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 replicated 5 times for \(N = 90\) observations. Several traits were measured on every unit, of which char1–char5 are analysed here. The first two columns hold the factor labels; 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.
Because the full file runs to 90 rows, Figure 4 shows only the first replication (the 18 treatment combinations once each); in the complete dataset every combination appears on five rows.
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 4: 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), and each further column is one response variable (char1, char2, …). Every row is one experimental unit, so a combination replicated five times appears on five rows. Keep each level label spelled consistently (always S1, never s1 or S 1), 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 and response columns
- The first two columns must be the two factors (factorA, factorB); 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.
- Replication
- Repeat each factorA × factorB combination on as many rows as you have replications; do not average replicates into a single row.
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 replications 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, 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 4. To use it:
- Open the Datasets tab
- Click the Download CSV link for the two-factor factorial 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 replications and how many responses you measured, and it returns a ready-to-fill template (Figure 2), 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 tab
With your CSV ready, open the Analysis tab. The first step is to upload the file: click Browse… under Upload data file Excel or CSV here (Figure 5) and select your CSV. A blue Upload complete bar confirms the file has loaded.
Once the file is in, RAISINS reads its column names and asks you to point it at the right columns (Figure 6):
- 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–char5), 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!. RAISINS fits the factorial model to every selected response and fills the results sub-tabs, Analysis Results, Basic Plots, Advanced Plots, Interpretation, Multivariate, FAQs 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.
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. RAISINS offers the three standard choices and applies safe adjustments for awkward values:
| 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.
10 Analysis results
The Analysis Results sub-tab is where the factorial ANOVA lands. A control strip at the top lets you set the multiple-comparison test (LSD by default, with Tukey’s HSD and Duncan’s DMRT as alternatives), 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, a two-factor factorial in CRD with Chemical1 (Factor A) at 6 levels (S1–S6) and Chemcial2 (Factor B) at 3 levels (F1, F2, F3), 18 treatment combinations each replicated 5 times, analysed with the LSD post-hoc test, and then presents four tables: the ANOVA summary, a Factor A table, a Factor B table, and the interaction (A × B) table. Every table carries one column per response (char1–char5).
Table 1: ANOVA summary — mean squares and significance
Each row is a source of variation, Factor A (5 df), Factor B (2 df), the interaction A × B (10 df) and Error (72 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. Scanning the flags tells the whole story at a glance: char5 responds strongly to Factor A (\(342.47^{**}\)); char1 (\(323.72^{*}\)) and char2 (\(420.60^{*}\)) respond to Factor B; and char1 additionally carries a significant A × B interaction (\(157.99^{*}\)). By contrast char3 and char4 are NS for every source, unaffected by either chemical. Each significant cell is unpacked in the tables that follow.
How this table is built
RAISINS fits the linear model response ~ A + B + A:B for each trait and forms the analysis-of-variance table with Type II sums of squares. Each factor’s mean square is its sum of squares divided by its degrees of freedom, and the F-test divides it by the error mean square; the p-value behind each F decides the * / ** / NS flag against the chosen \(\alpha\). This is the many-level generalisation of the \(2 \times 2\) contrasts in Section 4.
Table 2: Factor A — means and comparison statistics
This table averages over Factor B to show the main effect of Factor A (the six levels S1–S6). The one response Factor A affects is char5: \(F = 8.39^{**}\) (\(p = 0.00\)), with a critical difference CD (A) of 4.65 and a large Cohen’s F of 0.76. The letter groupings on char5 rank the chemicals: S4 (\(74.52 \pm 4.76\)) and S1 (\(73.18 \pm 8.35\)) share letter a as the highest, while S5 (\(62.90 \pm 6.47\), letter c) is the lowest. For char1–char4 the F-statistics are non-significant, so their CD (A) rows read “–”, RAISINS withholds the post-hoc yardstick when there is nothing to separate. 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 to show the main effect of Factor B across its three levels (F1, F2, F3). Factor B matters for two responses. For char1, \(F = 4.21^{*}\) (\(p = 0.02\), CD (B) \(= 4.51\)): 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: \(F = 3.68^{*}\) (\(p = 0.03\), CD (B) \(= 5.5\)), with F1 and F2 on par (letter a) and F3 (\(89.83 \pm 11.34\), letter b) lowest. So the third level of Chemcial2 depresses both char1 and char2; for char3–char5 no level of Factor B stands out.
Table 4: Interaction A × B — cell means and comparison statistics
| Combination | char1 | char2 | char3 | char4 | char5 |
|---|---|---|---|---|---|
| S1×F1 | 123.52 ± 5.72abc | 97.04 ± 6.30 | 46.94 ± 4.94 | 16.00 ± 4.74 | 76.39 ± 8.16 |
| S1×F2 | 130.24 ± 10.06a | 100.24 ± 10.06 | 46.63 ± 5.96 | 19.20 ± 3.83 | 70.54 ± 6.80 |
| S1×F3 | 127.00 ± 5.66ab | 91.10 ± 9.31 | 50.60 ± 3.86 | 15.40 ± 2.79 | 72.63 ± 10.45 |
| S2×F1 | 125.68 ± 8.00abc | 92.72 ± 8.70 | 48.21 ± 6.84 | 15.80 ± 4.27 | 63.64 ± 6.21 |
| S2×F2 | 125.48 ± 9.49abc | 95.52 ± 9.55 | 45.93 ± 10.75 | 20.80 ± 6.87 | 65.60 ± 3.60 |
| S2×F3 | 128.48 ± 4.58a | 94.92 ± 8.56 | 53.33 ± 4.86 | 15.80 ± 3.70 | 61.66 ± 4.67 |
| S3×F1 | 124.20 ± 6.45abc | 95.72 ± 8.63 | 48.10 ± 9.42 | 14.80 ± 4.44 | 67.84 ± 6.10 |
| S3×F2 | 128.92 ± 9.50a | 98.36 ± 8.54 | 49.27 ± 7.57 | 19.40 ± 3.36 | 71.20 ± 2.19 |
| S3×F3 | 131.56 ± 5.27a | 95.88 ± 6.84 | 55.61 ± 6.37 | 15.80 ± 1.79 | 65.66 ± 2.68 |
| S4×F1 | 130.64 ± 7.23a | 95.32 ± 12.21 | 48.15 ± 6.85 | 17.80 ± 2.05 | 74.14 ± 3.27 |
| S4×F2 | 127.56 ± 10.32ab | 98.36 ± 11.96 | 52.66 ± 4.67 | 15.80 ± 2.68 | 74.99 ± 7.01 |
| S4×F3 | 115.88 ± 13.19c | 84.04 ± 13.80 | 48.49 ± 6.11 | 16.00 ± 5.48 | 74.44 ± 4.37 |
| S5×F1 | 129.76 ± 8.78a | 94.24 ± 9.39 | 47.47 ± 6.22 | 19.40 ± 2.97 | 61.80 ± 4.24 |
| S5×F2 | 128.68 ± 7.96a | 91.64 ± 11.55 | 52.02 ± 7.08 | 16.20 ± 5.50 | 65.26 ± 4.91 |
| S5×F3 | 116.92 ± 12.38bc | 86.16 ± 13.45 | 49.24 ± 7.83 | 15.80 ± 5.12 | 61.65 ± 9.69 |
| S6×F1 | 130.52 ± 3.24a | 96.96 ± 8.78 | 48.87 ± 8.13 | 19.40 ± 1.95 | 67.40 ± 3.82 |
| S6×F2 | 132.52 ± 7.91a | 97.76 ± 15.04 | 56.15 ± 6.81 | 14.20 ± 3.35 | 69.26 ± 6.41 |
| S6×F3 | 115.80 ± 13.55c | 86.88 ± 14.32 | 55.55 ± 10.60 | 15.60 ± 5.13 | 65.85 ± 10.92 |
| F stat | 2.05^*^ | 0.46NS | 0.73NS | 1.46NS | 0.45NS |
| p value | 0.04 | 0.91 | 0.70 | 0.17 | 0.92 |
| CD (A×B) | 11.05 | – | – | – | – |
| MSE | 76.88 | 114.17 | 50.94 | 17.07 | 40.80 |
| SE(m) | 3.92 | 4.78 | 3.19 | 1.85 | 2.86 |
| SE(d) | 5.55 | 6.76 | 4.51 | 2.61 | 4.04 |
| CV(%) | 6.94 | 11.36 | 14.22 | 24.53 | 9.35 |
| Cohen’s F | 0.53 | 0.25 | 0.32 | 0.45 | 0.25 |
Cell values are mean ± SD. Where the interaction is significant (char1), superscript letters give the post-hoc grouping — combinations sharing a letter are not significantly different. * marks significance at 5%, ** at 1%, NS non-significant. MSE, SE and CV(%) are common to each response.
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: \(F = 2.05^{*}\) (\(p = 0.04\)) with a critical difference of 11.05 and a Cohen’s F of 0.53. 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–char5 the interaction is non-significant, so those columns show no letters and CD (A×B) reads “–”.
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 or the design is unbalanced.
- 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 is governed by Chemical1 (Factor A) alone, S4 and S1 give the highest values, S5 the lowest, and Chemcial2 has no say. char2 is governed by Chemcial2 (Factor B) alone, F3 sits below F1 and F2, regardless of Chemical1. 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 11.05) to judge which combinations are genuinely on par. This is the payoff of the factorial design: it separates a factor that acts alone (char5), a factor that acts alone (char2), and a pair that must be chosen together (char1), from one experiment that one-factor-at-a-time trials could never untangle.
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 0.76), whereas the char1 interaction, though significant, is more modest (0.53). 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 10) offers the everyday comparisons, Boxplot, Violin Plot, Mean Value Plot, Connected Line Plot and Bar Plot. Choose the factor to place on the x-axis (Chemical1, Chemcial2, or the interaction) and the response to plot. Figure 11 draws the same char1-by-Chemical1 comparison four ways, the six levels S1–S6 overlap heavily, the visual echo of the non-significant Factor A effect on char1 from Figure 8.
The Advanced Plots tab (Figure 12) adds the specialised graphics: two Interaction Plots, a Summary Plot, Raincloud and Advanced Raincloud, a Circular Plot, a QQ Plot, a Distribution Plot, a Pair Plot, a Correlation Plot, and 3-D Scatter views.
Two Advanced plots earn special attention in a factorial study:
- The Interaction Plot draws the mean response for each Chemcial2 level as a line across Chemical1 (or vice versa). 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.
- The QQ Plot checks the normality assumption behind the ANOVA: it plots the model residuals against the values expected under a normal distribution. Points hugging the straight line support normality; a strong curve or S-shape suggests a transformation (Section 9) may be needed.
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 14) 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 five traits are treated as one bundle. RAISINS fits the two-way model with interaction and reports Pillai’s trace, the most robust multivariate test statistic, converted to an F-value (Figure 15). Here Factor A is significant across the combined traits (Pillai \(= 0.54\), \(F = 1.75\), \(p = 0.02\)), echoing its strong univariate effect on char5, while Factor B (Pillai \(= 0.21\), \(F = 1.60\), \(p = 0.11\)) and the interaction A × B (Pillai \(= 0.64\), \(F = 1.05\), \(p = 0.39\)) are non-significant when all traits are pooled. The partial \(\eta^2\) column (0.11, 0.10, 0.13) reports effect sizes; a moderate \(\eta^2\) can accompany a non-significant test, so the p-values remain the deciding evidence.
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 (Figure 16) shows how much variance each component captures: PC1 explains 41.75% and PC2 28.42%, together 70.17% of the total variation, more than enough for a meaningful two-component index. The accompanying scree plot shows the same information graphically, the variance explained falling away after the first three components. To decide which index to use, read the loadings of each trait on the components (Figure 17): char3 loads positively on PC1 while char1, char2, char4 and char5 load negatively, so a PC1 score separates char3 from the rest. The biplot places each treatment combination in the direction of the traits it scores highly on.
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. Here the interaction is not significant in the MANOVA, and RAISINS itself 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 (Figure 18). 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 effect on char5 (S4 highest at \(74.52 \pm 4.76\), S5 lowest at \(62.90 \pm 6.47\)), the Factor B effects on char1 and char2 (F3 the lowest level in both), and the significant interaction on char1 (\(p = 0.04\)), 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 Table 4 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 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 char2?”.
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 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. Typical questions include “my design is unbalanced, why is the critical difference shown as a dash?” and “the interaction is significant but neither main effect is, how do I report that?”, both of which arise naturally with factorial data.
16 View data
View Data is the primary diagnostic tool for ensuring data integrity before analysis. When you upload your dataset, RAISINS runs an automated Health Check to validate the file: it confirms that the 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 design it also checks that every factorA × factorB combination is present and reports whether the replication 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 CRD exists to answer one honest question: do two factors act independently, or does the effect of one depend on the other? Everything in this module, the ANOVA summary, 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 are homogeneous, so the treatments can be assigned completely at random. If your units are not uniform and you had to group them to control a gradient, use the factorial RBD module instead; if you are studying a single factor, use the one-way CRD module. And if you get stuck at any point, RA-One is available 24 × 7, or write to us at [email protected].