3 Activity 3 – Between Groups Factorial ANOVA: Pairwise Follow-Up Tests in jamovi

Recall the last 4 x 2 experiment in which we found a significant main effect of the drug dosage factor (look back for a reminder of the experimental design). As dosage has four levels (i.e., more than two), the F test for the main effect of dosage does not tell us exactly where the significant difference(s) in performance actually occur between the different drug dosage levels. At this stage, we can only state that the main effect is significant, but we cannot fully address our hypothesis because we cannot yet describe the direction of this effect.

However, you learnt in second-year statistics (in simpler, one-way between groups ANOVA) that when you get a significant F for a factor with more than two levels, the effect can be followed up with linear comparisons/ contrasts and tested for significance with a Bonferroni t’. The same is true in factorial ANOVA, though due to current technology we can now rely on statistical programs to perform most of these follow-up tests for us, rather than calculating them all by hand. This being said, there are three important aspects to recognise when we perform ANOVA follow-up tests in jamovi: (1) we can only easily perform pairwise t-tests in the program (rather than linear contrast t-tests), (2) we do not tend to perform a Bonferroni adjustment to our critical α-level to correct for the number of comparisons made unless it is deemed necessary (see Conservative Follow-Up Tests for Main Effect Comparisons for an overview of when this is recommended), and (3) if a Bonferroni correction is warranted, the jamovi option for performing this does not actually address the Bonferroni adjustment in an appropriate and theoretically meaningful manner. Therefore, the jamovi option should never be used. Instead, a Bonferroni adjustment to the α_critical should be performed by hand (see Conservative Follow-Up Tests for Main Effect Comparisons later where this method is discussed).

In relation to point 1 above, we can follow up significant main effects in a two-way ANOVA using a series of (a) pairwise comparisons, or (b) linear contrasts.

Q: So which do you choose?

A: It depends on your hypotheses.

• If a series of pairwise comparisons will answer your hypotheses, you can conduct these follow-up tests in jamovi.
• If your hypotheses require the comparison of multiple groups in the same contrast, you have to calculate the necessary linear contrasts by hand (download Extra Materials (PDF, 1.14 MB) for how these linear contrast calculations are performed).

Regardless of which technique you choose, be aware that you are performing a two-way between groups ANOVA and so your calculations for the effects of drug dosage will average over the levels of the other factor, in this case, sex. Further, although the statistical technique used may be ‘pairwise t-tests’ or ‘linear contrasts’, because we perform them to follow up a significant main effect, the tests themselves are called ‘main effect comparisons’ or ‘main comparisons’.

Our second hypothesis regarding the main effect of drug dosage was: “a) Overall, rats will perform better when they receive any amount of the drug than when they do not receive the drug. b) However, a small dose will tend to lead to the best performance compared to a large dose.” Hence our second hypothesis does lend itself to investigation via pairwise comparisons through jamovi!

Exercise 5 Task 1

The answer is that there are four pairwise comparisons we really want to perform to address Hypothesis 2. These are:

1. zero vs. small drug dosage,
2. zero vs. moderate drug dosage,
3. zero vs. large drug dosage, and
4. small vs. large drug dosage.

The first three comparisons address Hypothesis 2a) regarding the benefits of no drug vs. drug, while the last comparison addresses Hypothesis 2b) with reference to the advantage of a small drug dose over a large dose. Together, these four forms the set of pairwise comparisons necessary to assess our explicit predictions for Hypothesis 2.

***NOTE: If we were to conduct five or more follow-up t–tests (rather than four), we would need to carry out an adjustment to the α_critical level (e.g., employ a Bonferroni correction). This would be because despite being based on a priori hypotheses, there would be five or more comparisons being performed (use Conservative Follow-Up Tests for Main Effect Comparisons, as a guide to when adjustment to the critical α-level is appropriate). Prior to conducting follow-up pairwise t-tests, we always need to check whether an adjustment to the α_critical level is necessary. This will guide our analysis decision regarding whether to employ a Bonferroni correction or not.

Since we entered the data last time, you just need to open up the data file in jamovi. Here is the data file again in case you need it – Factorial BG ANOVA data file.omv (click on the three dots and select download). If you saved the data file on your computer at the end of the previous exercise you will have your analysis and menu options also saved with it. If you need to recreate the analysis the steps are outlined in full below. If the analysis has been saved with your data, click in the ANOVA output in the results view and you will see the ANOVA menu appear on the left side with your previously selected options and you can just select the extra options.

1. Select the Analyses menu.
2. Click ANOVA, then ANOVA … to open ANOVA analysis menu that allows you to specify a factorial ANOVA.
3. Select the dependent variable (i.e., Performance) and then click on the arrow button to move the DV into the Dependent Variable box.
4. Select the independent variables in turn (i.e., Sex and Dosage) and click on the arrow button to move each variable into the Fixed Factors box.
5. Select Overall model test under Model Fit and select η² and ϖ² in the Effect size check boxes
6. Click on the Model drop-down tab. The Sum of squares (bottom right) should be Type 3, which is the default method for partitioning the variance (decomposing sums of squares). With a balanced or orthogonal design, it doesn’t really matter which of the methods you use because the tests of the main effects and interaction are independent – all methods will yield equivalent results.
7. NEW STEP: Under the Post Hoc Tests drop-down tab select Dosage in the listing to the left and click the arrow to add it to the box on the right. Under Correction tick the “No Correction” checkbox and ensure that other options are not selected. Under Effect Size tick the Cohen’s d checkbox along with Confidence interval which will be set at 95% as a default.

Task 1

Take a look at the output, paying special attention to the main effect comparisons. Note what pieces of information are present and which are missing.

Figure 3.1

Omnibus ANOVA and Main Effect Comparisons for Dosage

Note.  In the settings menu for jamovi (which you can access by clicking the three vertical dots on the far right of the jamovi menu ribbon) you can change the number of decimal places in your output for p values as well as numbers. APA 7 format suggests that p values be reported to 3 decimal places, and inferential statistics (e.g., t) are reported to 2.  You also can request the means within ANOVA using the emMeans function – but it is more straightforward using the commands below.

Getting Our Marginal Means and Standard Deviations in jamovi

1. Select the Analyses menu.
2. Click Exploration, then Descriptives.
3. Select the dependent variable (i.e., Performance) and then click on the arrow button to move the DV into the Variables box.
4. Select the independent variables for our main effect (i.e., Dosage) and click on the arrow button to move each variable into the Split by box.
5. In the Statistics drop-down menu ensure that you have the Mean checkbox under the Central Tendency section ticked, and the Std. deviation checkbox under the Dispersion section ticked. Finally, check the Confidence interval for Mean checkbox under Mean Dispersion. You can unselect all other default selections to get a table that has only the means and standard deviations you need.

Here is the table you will get when you follow this process.

Figure 3.2

Descriptive Statistics for the Main Effect of Dosage

Task 2

• Determine which of our four focal pairwise comparisons were significant, and
• For those comparisons that were identified as significant, revisit the marginal means to determine their direction of effect (e.g., identify the larger mean for each).

Task 3

Write in words what the findings mean as you would in a Results section.

That is, report:

• The overall type of analysis that was performed, stating what the IV(s) and DV were. This includes specifying the levels of the IV(s). Again, recall that by convention, we tend to use specific notation to present this information in a condensed manner.
• The main effect of drug dosage, stating its significance.
  • Then, report the main effect comparisons (stating the significance and direction for each).

Don’t forget to include relevant:

• Statistical notation after the omnibus test (i.e., F, df, p, η² or ω²).
• Statistical notation after each main effect comparison (i.e., t, df, p, d).
• Marginal means and standard deviations when describing the direction of effect. NB: These statistics are only reported once each and not repeated i.e., they are given the first time the relevant group is mentioned in the text.
• Information about the type of main comparisons that were performed, including:
• Whether they were pairwise t-tests or linear contrasts,
• The number of comparisons conducted,
• The critical α-level employed for each comparison, and
• What type of correction was used (NB: if no adjustment to the critical α-level was performed, this last piece of information is typically omitted).

As a reminder:

• Statistical notation in English/ Latin letters needs to be italicised (e.g., F, t, p, d), while that in Greek (e.g., η², ω²) appears as standard text. Likewise, notation for subscripts and superscripts are presented in standard text (i.e., these do not appear in italics).
• Inferential statistics (e.g., F, t), effect sizes (e.g., η², ω², d) and confidence intervals should be presented to two decimal places, while p-values go to three.
• Descriptive statistics (such as means and standard deviations) are given to the number of decimals suited to their precision of measurement. In our case, maze running performance was scored as whole numbers (i.e., scaled 1-15). Therefore, descriptive statistics regarding the means and standard deviations for this should be taken to one decimal place for reporting.
• Any value that happens to be very small but can potentially exceed a threshold of ±1 (e.g., M, SD, F, t, d), should have a 0 preceding the decimal point. However, any value that cannot potentially exceed ±1 (e.g., p, η², ω²), should not have a 0 before the decimal.
• The direction of effect for main comparisons should also be reflected in the positive or negative valence of the reported t-value. That is, a positive t-value indicates that the first reported group involved in the comparison had a higher mean than the second group. In contrast, a negative t-value indicates that the first reported group involved in the comparison had a lower mean than the second group.
• The jamovi output above gives the CIs for the means, and when the CIs for two means do not overlap, we interpret the difference as significant.  The jamovi output for the t-tests also gives CIs, but for the effect size measure, d.  If you are not troubled by limitations regarding your word count, you can comprehensively report CIs for all of the various parameters reported (and in some other software you would also get CIs for the t and report that too).  However, if you are trying to write tersely, the CIs for the effect sizes are the most conventional to prioritise including.

*** You have not yet finished all the follow-up tests. More are needed in order to decompose the significant interaction (i.e., simple effects and simple comparisons). These will be covered in Exercises 7 and 8. ***

Conservative Follow-Up Tests for Main Effect Comparisons

In the jamovi exercise above, we followed up our significant main effect of drug dosage (i.e., a factor with more than two levels) using a series of pairwise t-tests where no correction to α_critical was made for the number of comparisons performed. There are some circumstances when it is more appropriate to use a Bonferroni correction for this. The Bonferroni procedure adjusts the critical value to which we compare our t-test statistic, maintaining the overall familywise error rate at α = .05 (or α = .01). It does this by taking the critical α-level of .05 and dividing it by the number of comparisons being made.  For example, ten follow-up comparisons would make use of a critical significance level of α = .005 each (i.e., α = .05 / 10). This, in turn, makes the t-critical value more conservative (i.e., larger) and thus more difficult for our t-obtained value to exceed in order to be considered significant.

The advantage of applying a Bonferroni correction to follow-up tests is that it ensures no inflation of our familywise error rate. To illustrate, hypothetically let’s say we had five (rather than four) comparisons, where we also wanted to look at the effect of small versus moderate dosage. Consider what would happen if we performed these five comparisons without correction, each at α = .05. Our familywise error rate for these would be a whacking great α = .250 (i.e., α = .05 x 5)! Yet if we applied a Bonferroni adjustment to α_critical based on these same five follow-up tests, familywise error would remain at α = .05 (where each comparison would be evaluated against a new α_critical of .05 / 5 = .01). To identify the use of a Bonferroni correction in a write-up, the author(s) should explicitly state that a Bonferroni correction was used for [number of] comparisons, where familywise error rate was maintained at α = .05 or where each comparison was evaluated against α = [.05 / number of comparisons], and change the statistical notation from t to t’ (note the addition of the apostrophe/ prime sign).

When to Apply a Bonferroni Correction to the Critical α-level

The decision of whether or not to employ a Bonferroni correction to α_critical for follow-up comparison t-tests rests on the answers to three major questions. These are:

1. Were my predictions for these comparisons made post hoc?

A priori predictions are those made before data collection and analysis, and are based upon theory. In contrast, post hoc comparisons are more exploratory and are performed after initial data analysis, when an unexpected result emerges (e.g., significant main or simple effect involving a factor with more than two levels) and we wish to determine where the difference(s) between groups lies. Hence, post hoc comparisons are not planned and generally not grounded in appropriate theory. In such cases, performing a correction to the critical α-level for our t-tests is a prudent option.

2. Am I performing five or more comparisons?

Follow-up comparisons numbering five or more drastically boost our familywise error rate because – as stated above – each comparison carries with it an α-level of .05. Therefore, our total error rate for a series of follow-up comparisons is .05 multiplied by the number of comparisons being made. This means that our chances of making a Type I error (i.e., claiming a result to be significant when in reality there is no genuine relationship) are also dramatically increased. So whenever we are performing five or more comparisons to follow-up a single significant main (or simple) effect, the familywise error rate is considered too great and an adjustment to our α_critical is needed.

3. Do I want to be conservative?

Times when we want to adopt a conservative approach are cases where (a) the data set we are working with is overly large (e.g., observations numbering in the hundreds to thousands), and/ or (b) the research area is novel and therefore, our theory driving predictions is weak or exploratory. If either of these conditions is not present, however, there is no real need for us to be conservative and adjust α_critical for the follow-up t-tests.

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In a nutshell, if you answer “yes” to any of the above three questions, then a Bonferroni adjustment to the critical α-level for your follow-up t-tests should be employed! However, if you answer “no” to all three, then a Bonferroni correction is not warranted.

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Having said this, some researchers may still err on the side of caution and adjust the α_critical regardless of their answers to the previous questions. Yet although there may be the desire to always be conservative, we must consider the potential disadvantages of doing so. In psychology, a Bonferroni adjustment is often seen to be a bit too conservative to apply as standard practice (i.e., it should only be applied as required, following the above guidelines). Specifically, although a Bonferroni correction to α_critical is acknowledged to protect against the danger of overclaiming differences between groups as being significant, it does so at the cost of potentially underclaiming these results. As a consequence, we may miss out on detecting important and genuine relationships. That is, by overcompensating for a Type I error, we risk a problem of Type II error.

In the previous activity, you were asked to identify the series of pairwise comparisons that would be required to address Hypothesis 2a and 2b of the Dosage x Sex study. These were:

1. zero vs. small drug dosage,
2. zero vs. moderate drug dosage,
3. zero vs. large drug dosage, and
4. small vs. large drug dosage.

Therefore, a Bonferroni correction to our critical α-level is NOT recommended because we are only conducting four follow-up main effect comparisons for drug dosage in line with our a priori study predictions, on a small data set in a research area that is not novel (thus answering “no” to all three questions regarding when to employ a Bonferroni correction). However, when a Bonferroni adjustment to the critical α for follow-up comparisons is warranted, we can conduct the vast majority of this procedure through jamovi, following the steps we completed earlier in Exercise 5. However, an additional two steps are required at the end. These involve (1) a simple hand calculation to determine the new Bonferroni adjusted critical α-level, and (2) a series of visual comparisons between this new critical α-level cut-off and the p-values obtained in jamovi for each comparison, in order to determine significance/ non-significance. [Note again that all this only applies when pairwise comparisons are appropriate to follow up the research hypotheses – linear contrasts would need to be calculated fully by hand].

Based on the above steps, look at the main effect comparisons in the Post hoc Comparisons table in Figure 3.1 (this is from your earlier analysis involving no correction to α_critical via the LSD adjustment method). Apply a Bonferroni correction to our α_critical and use this to assess each of the new five hypothetical key comparisons.

In the space below (a) state whether or not each of these five comparisons was significant, (b) for the significant comparisons, determine their direction of effect (hint: revisit the marginal means to help with this), and (c) provide the t value, and Cohen’s d effect size for each.

Simple Effects in jamovi

Below is the data from the Drug Dosage x Sex experiment again:

Table 3.3

The Effect of Steroidal Dosage and Sex on Maze Running Performance in Rats

Go to a version of Table 3.3 formatted for accessibility.

Step 1: Re-Open the Data File

Since we were working with this data set in the previous exercise, just re-open/ maximise this jamovi again.

Step 2: jamovi for Simple Effects in Between Groups Factorial ANOVA

Simple effects are not available to us within the ANOVA menu we have used so far. We will need to use an additional jamovi module called gamlj (General analysis for linear models in jamovi).

1. Click on + Modules on the far right of the menu ribbon then select jamovi library.
2. Within the jamovi library search for gamlj and click to install it. You should now have a “Linear Models” menu appearing in your Analyses ribbon.  (There’s an updated module, gamlj3, that looks groovy and has some new features, but we haven’t checked it ourselves yet so we recommend you DL the original gamlj version for the steps below.)
3. Select Linear Models then General Linear Model.
4. Move the dependent variable (i.e., Performance) into the Dependent Variable box.
5. Ensure that the independent variables (i.e., Sex and Dosage) are in the Factors box.
6. Under Effect Size ensure the η2 checkbox is ticked.
7. Go to the Simple Effects drop-down menu. Move Dosage to the Simple effects variable spot and Sex to the Moderator spot.
8. As an additional task, you could also test the simple effects of sex at the four levels of drug dosage. To do this, you would repeat step 6 but put Sex in the Simple effects variable spot and Dosage in the Moderator spot. Note though, that we never conduct both sets of simple effects! Only one set should ever be performed (i.e., that dictated by our theory/ hypotheses).

Step 4: Examine the Output

The output for the simple effects is displayed in the Simple Effects section in the Simple effects of Dosage – Omnibus Tests table as shown below.

Figure 3.6

Jamovi output Simple Effects of Dosage for each Sex Level

(This output does not have the partial eta squared or beta, but if you left those boxes ticked in Step 2.6, you will see them too.)

1. Figure out where the output displays the cell and marginal means (in the ‘Descriptive Statistics’ section). Note that the marginal means will be the ‘total’ means for each variable level.
2. Determine in which output table the simple effects test results are displayed.

Task 1: Take a look at the output above and write the key bits of information into the summary table below (as we did for the omnibus tests in Activity 2: Exercise 4).

Download Table 3.4 (PDF, 133 KB) to write the information.

Table 3.4

Summary Table for Simple Effects

Go to a version of Table 3.4 formatted for accessibility.

If exact p-values were not provided, you could compare your F obtained for each simple effect to the F critical value in F tables (with the appropriate df). This critical value would be:

Fα = .05(3, 32) = 2.86

Double-check that your decision regarding significance/ non-significance for each of the above simple effects remains the same using this other criterion.

Task 2:

*** You have not yet completely finished with the simple effects of drug dosage. More follow-up is needed, because we don’t yet know where the differences in dosage are for males or females (i.e., we will need to do simple comparisons). This will be covered next in Exercise 8. ***

Given that the simple effects of drug dosage were significant (note that they were significant both for males and females), we need to go one step further to locate where the differences in maze running performance at the different dosage levels lie. Therefore, this time we will be comparing CELL means. As we are performing the comparisons to follow up significant simple effects (which in themselves were conducted to follow up a significant interaction), we call these ‘simple comparisons’.

Remember, our third hypothesis was:

“The effect of drug dosage will differ for males compared to females. While both sexes will exhibit increased performance when they receive any amount of the drug than when they do not, the particular benefits of the small drug dosage (compared to the large dose) will be more noticeable in female rats than in male rats.”

So again, our hypothesis lends itself to investigation via pairwise comparisons in jamovi!

Task 1

The answer is that there are eight pairwise comparisons needed to address Hypothesis 3, which mirror that for the main effect follow-up tests. Namely, for male and for female rats we want to examine:

1. zero vs. small drug dosage,
2. zero vs. moderate drug dosage,
3. zero vs. large drug dosage, and
4. small vs. large drug dosage.

These four will be performed once for males and again for females, providing eight simple comparisons in total. However, since we will be conducting four comparisons to follow up each significant simple effect (i.e., the simple effect of dosage for males, and the simple effect of dosage for females), this means that each set contains only four follow-up tests. Hence when we are assessing whether or not to employ a Bonferroni correction to our α_critical level, we will use this number of comparisons in each set to help form our decision (i.e., four). In our case, since we plan to conduct four pairwise t-tests to follow up each of our significant simple effects, which are specifically designed to test our Hypothesis 3 predictions within a small data set in an established research area, then an adjustment should NOT be applied to α_critical for the number of tests performed (i.e., no Bonferroni correction; see Conservative Follow-Up Tests for Main Effect Comparisons as a guide to when an α_critical adjustment is appropriate).

As stated above, since Hypothesis 3 is worded in such a way that pairwise simple comparisons can be performed (i.e., comparing only two groups at a time rather than multiple groups), we can run simple comparison results in the jamovi via the ANOVA menu results we have run before. Remember however, at the end of the day, you have a choice to adopt either a linear contrast or pairwise approach to your main and/ or simple comparisons. This choice should be driven by your research questions and underlying theory!

Recall that we will ask for pairwise simple comparisons with no correction to α_critical (i.e., it employs the LSD adjustment method). Therefore, if we had want to apply a Bonferroni correction we will have to do so by hand, just as we would for main comparisons. Consequently, the procedure for this is exactly the same as that taught earlier in Exercise 6 for the hypothetical example, except this time the method would be applied to simple comparisons. Hence, we would find the relevant p-values to compare to our new Bonferroni adjusted α_critical in a different output table i.e., the Pairwise Comparisons table.

Step 1: Obtain the Simple Comparisons That Accompany These Simple Effects.

The best way to obtain these is to scroll up in your jamovi and go back to the original ANOVA in your results output (the one produced in Exercise 5 Step 2). Go to the Post Hoc Tests drop down list and this time select the interaction term (Sex*Dosage) and move it across to the right hand side box. Ensure that No correction is selected under the Correction heading.

The relevant output you will obtain from this is shown below.

Figure 3.7

Simple Comparisons for the Dosage Simple Effects for Male and Female Rats

Step 2: Getting Our Cell Means and Standard Deviations in jamovi

1. Select the Analyses menu.
2. Click Exploration, then Descriptives.
3. Select the dependent variable (i.e., Performance) and then click on the arrow button to move the DV into the Variables box.
4. Select the independent variables in turn (i.e., Sex and Dosage) and click on the arrow button to move each variable into the Split by box.
5. In the Statistics drop down menu ensure that you have the Mean checkbox under the Central Tendency section ticked, and the Std. deviation checkbox under the Dispersion section ticked. Finally, check the Confidence interval for Mean checkbox under Mean Dispersion. You can unselect all other default selections to get a table that has only the means and standard deviations you need.

Here is the table you will get when you follow this process.

Figure 3.8

Descriptive Statistics for the Simple Comparisons

Task 2

Referring to the jamovi output containing the pairwise simple comparisons, determine which of our key simple comparisons are significant. Write your answers in the space provided.

State the significance/ non-significance and direction of each of the eight relevant simple comparisons.

Task 3

Write up the results for the simple comparisons of drug dosage for both males and females.

State in words what the results mean. That is, explain first that simple comparisons were performed via a series of pairwise t-tests in order to follow up the significant simple effect of drug dosage for males/ females. Report the number of comparisons conducted, and the α-level used for each comparison. If a correction to α _critical was employed, state what type (NB: if no adjustment was performed, this last piece of information is typically omitted). Then, for each comparison, state the significance and report the direction of effect (complete with the relevant statistical notation, and cell means and standard deviations). Be sure to group all the follow-up tests for males together, and then report all the follow-up tests for the females, as this makes for a more logical and easy-to-follow presentation of results.

Task 4: The Full Write Up

Now you can piece together the write up from Exercise 4 in Activity 2 with the write ups you have just done. That is, you can report (in the following order):

• The overall type of analysis that was performed, stating what the IV(s) and DV were. This includes specifying the levels of the IV(s). Again, recall that by convention, we tend to use specific notation to present this information in a condensed manner.
• The main effect of sex – remember to state significance and the direction of effect.
• The main effect of drug dosage, stating its significance.
  • Then, report the main effect comparisons (stating the significance and direction of effect for each).
• The interaction – state its significance.
• The simple effects of drug dosage – state the significance of each (i.e., for both males and females).
  • Report the simple comparisons of drug dosage, for males and then for females. Be sure to group these following their relevant simple effect that they are following up. I.e., the simple comparisons of dosage for males should be reported immediately after the significant simple effect of dosage for males, while the simple comparisons of dosage for females should be reported immediately following the significant simple effect of dosage for females.

Don’t forget to include relevant:

• Statistical notation after each of the omnibus and simple effects tests (i.e., F, df, p, η² or ω²). Remember that all the omnibus and simple effects tests will need an effect size!
• Statistical notation after each main/ simple comparison (i.e., t, df, p, d, and 95% CI).
• Marginal/ cell means and standard deviations when describing the direction of effect. NB: These statistics are only reported once each and not repeated i.e., they are given the first time the relevant group is mentioned in the text.
• Information about the type of main/ simple comparisons that were performed, including:

a) Whether they were pairwise t-tests or linear contrasts,
b) The number of comparisons conducted,
c) The critical α-level employed for each comparison, and
d) What type of correction to α_critical was used (NB: if no adjustment was performed, this last piece of information is typically omitted).

As a reminder:

• Statistical notation in English/ Latin letters needs to be italicised (e.g., F, t, p, d), while that in Greek (e.g., η², ω²) appears as standard text. Likewise, notation for subscripts and superscripts are presented in standard text (i.e., these do not appear in italics).
• Inferential statistics (e.g., F, t), effect sizes (e.g., η², ω², d) should be presented to two decimal places, while p-values go to three.
• Descriptive statistics are given to the number of decimals commensurate with their precision of measurement. In our case, means and standard deviations should be taken to one decimal place for reporting.
• Values that can potentially exceed a threshold of ±1 (e.g., M, SD, F, t, d), should have a 0 preceding the decimal point. However, values that cannot potentially exceed ±1 (e.g., p, η², ω²), should not have a 0 before the decimal.
• The direction of effect for main/ simple comparisons should also be reflected in the positive or negative valence of the reported t-value. That is, a positive t-value indicates the first reported group involved in the comparison had a higher mean than the second group. In contrast, a negative t-value indicates that the first reported group involved in the comparison had a lower mean than the second group.

With all this information, you have a complete write-up for a 4 x 2 between groups factorial ANOVA!

Note. It is common to provide a figure that accompanies a write up such as this, in order to illustrate the significant interaction. Recall that this should have the DV on the y-axis [i.e., Maze Running Performance (1-15)], the key factor on the x-axis (i.e., drug dosage) and the proposed moderator as separate lines on the graph (i.e., sex). The points presented on the graph will be the dosage x sex cell means. Don’t forget that this will also require error bars.