We now consider in detail the case where there are two within-subjects factors: Training and Skill in the following example.

**Example 1**: A company has created a new training program for their customer service staff. To test the effectiveness of the program they took a sample of 10 employees and assessed their performance in three areas: Product (knowledge of the company’s products and services), Client (their ability to relate to the customer with politeness and empathy) and Action (their ability to take action to help the customer). They then had the same 10 employees take the training course and rated their performance after the program in the same three areas. Based on the data in Figure 1, determine whether the program was effective.

We use two factor ANOVA with repeated measures to perform this analysis. We interrupt the analysis of this example to give some background, after which we will resume the analysis.

**Definition 1**: We modify the **structural model** of Definition 1 of Two Factor ANOVA with Replication as follows. Note that we will use *a* to indicate the number of levels for factor A (instead of *r*) and *b* to indicate the number of levels for factor B (instead of *c*). Also *m* = the number of subjects/participants.

We use terms such as *x̄ _{i}* (or

*x̄*) as an abbreviation for the mean of {

_{i.}*x*: 1 ≤

_{ijk}*j ≤ b*, 1 ≤

*k ≤ m*}. We also use terms such as

*x̄*(or

_{j}*x̄*) as an abbreviation for the mean of {

_{.j}*x*: 1 ≤

_{ijk}*i ≤ a*, 1 ≤

*k ≤ m*}.

We define the effects *α _{i} *and

*β*where

_{j}Similarly, we define *a _{i} *and

*b*where

_{j}We use *δ _{ij}* for the effect of level

*i*of factor A with level j of factor B, i.e. the interaction of level

*i*of factor A and level

*j*of factor B. Thus,

*δ*. Similarly, we have

_{ij}= μ_{ij}– μ_{i}– μ_{j}+ μIt is easy to show that

Finally, we can represent each element in the sample as

where *ε _{ijk} *denotes the error (or unexplained) amount, where

where *γ _{k}* is a random effect corresponding to the subjects/participants. All the interaction terms are also random effects; together they make up the error components of our model.

As before we have the sample version

where *e _{ijk}* is the counterpart to

*ε*in the sample. The null hypotheses for main effects are:

_{ijk}H_{0}: *µ _{1.} = µ_{2.} = ··· µ_{a.} *(Factor A)

H_{0}:* µ _{.1} = µ_{.2} = ··· µ_{.b} *(Factor B)

These are equivalent to:

H_{0}: *α _{i}* for all

*i*(Factor A)

H_{0}: *β _{j}* for all

*j*(Factor B)

In addition there is a null hypothesis for the effects due to interaction between factors A and B.

H_{0}: *γ _{ij}* for all

*i, j*

**Definition 2**: Using the terminology from Definition 1 of Two Factor ANOVA with Replication, except that we use *a* for the number of levels in factor A (instead of *r*) and *b* for the number of levels in factor B (instead of *c*), and also adding C = the participant factor and *m* = number of participants, define:

We can also define four types of between group terms.

And similarly for *BetAC* and *BetBC*. There is also the following *BetABC* version:

**Property 1**:

Proof: It is clear that

If we square both sides of the equation, sum over *i, j* and *k* and then simplify (with various terms equal to zero as in the proof of Property 2 of Basic Concepts of ANOVA), we get the first result. The second result is trivial.

**Property 2**: If a sample is made as described in Definition 1 of Basic Concepts of ANOVA, with the *x _{ijk}* independently and normally distributed and with all (or ) equal, then

Proof: The proof is similar to that of Property 1 of Basic Concepts of ANOVA.

**Theorem 1**: Suppose a sample is made as described in Definitions 1 and 2 of Two Factor ANOVA with Replication, with the *x _{ijk }*independently and normally distributed.

If all *μ _{i} *are equal and all are equal then

If all *μ _{j}* are equal and all are equal then

Also, under certain circumstances,

Proof: The result follows from Property 2 and Theorem 1 of F Distribution.

**Property 3**:

**Observation**: We use the following tests:

If the null hypothesis for factor A is true, then = 0, and so

whereas if the null hypothesis is not true then > 1. The results are similar for the other null hypotheses.

**Example 1** (continued): We now show how to apply the analysis to Example 1. There are two treatment factors: Training (with levels pre and post) and Skill (with levels Product, Client and Action) plus the Subject factor. We begin by creating the worksheet in Figure 2.

In addition to the raw data (range A3:G14), Figure 2 shows the means for the interaction between the various factors: Training × Skill (B15:G15), Training × Subject (J5:K14), Skill × Subject (M5:O14), as well as the means for each factor: Training (J15:K15), Skill (M15:O15) and Subject (H5:H14). Finally the grand mean is shown in cell H15.

From the information in Figure 2 we create the tables in Figure 3.

Row 4 contains the values of *a, b, m* and *n* for Example 1. E.g. *a* = the number of Training levels = COUNTA(J3:K3) and *n* = PRODUCT(R4:T4) or COUNT(B5:G14). The various degrees of freedom are calculated from these values as described in Definition 2. Some examples are given in Figure 4.

**Figure 4 – Representative df**

**from Figure 3**

The count values in column R are used to calculate the *SS* values in column S. In all cases the count value is equal to . The values for *SS* are calculated as in Definition 2. Some examples are given Figure 5.

**Figure 5 – Representative SS from Figure 3**

From the table in Figure 3, we arrive directly at the ANOVA analysis in Figure 6.

Note that the p-values for both main effects and the interaction effect are significant, but before we conclude this definitively we need to take sphericity into account. For the Training factor there are only two levels and so sphericity is automatic. Since the Skill factor has more than two levels we need to calculate the GG and HF epsilon correction factors (see Figure 7) as we did in Sphericity for the one within subjects factor analysis.

Here the covariance matrix is calculated using the supplemental array formula =COV(M5:O14). We see that GG epsilon is close to 1 and HF epsilon is 1, and so it appears that there is no problem with sphericity.

We also need to worry about the sphericity of the interaction effect. For this we will simply look at the lower bound which for the interaction effect is equal to = = .5. Figure 8 provides a revised version of Figure 6 taking sphericity into account.

We see from Figure 8 that the main effects and interaction effect are all significant. We now use planned and/or unplanned tests to better understand what is going on.

**Observation**: As described in ANOVA with Repeated Measures with One Within Subjects Factor where the sphericity assumption holds we can use the same types of tests that we used for ANOVA with independent variables. For these tests the standard error is based on *MS _{E} *where the appropriate value of

*MS*is employed:

_{E}*MS*with

_{A×C}*df*= (

_{A×C}*a*–1)(

*m*–1) for the main effect A,

*MS*with

_{B×C}*df*= (

_{B×C}*b*–1)(

*m*–1) for the main effect B and

*MS*with

_{A×B×C}*df*= (

_{A×B×C}*a*–1)(

*b*–1)(

*m*–1) for the interaction effect A × B.

Where sphericity doesn’t hold it is best to use the contrast-specific standard error and not use an *MS _{E} *type value. In fact, it is probably better to use the contrast/specific standard error in all cases. Unplanned tests can also be used as described in ANOVA with Repeated Measures with One Within Subjects Factor.

**Example 2**: For the data in Example 1 determine whether there is a significant difference in the training program for each of the three skills.

As we can see from Figure 9, it appears that training improves performance for each skill, although only slightly in the case of the Product skill. We need to determine whether any of these differences are significant. We do this by performing pairwise comparisons.

The test for the Product skill in Figure 10 shows there is no significant difference between the means for Product pre-test and post-test (*p* > .05). If we perform the same analysis for Client we see there is a significant difference: *t*(9) = -4.63, *p* = .001 < .05, *d* = 1.47. Similarly for Action: *t*(9) = -11.21, *p* = 3E-05 < .05, *d* = 2.43. Note that the contrast analysis is equivalent to the t-test on paired samples. Even if we use a Bonferroni or Dunn-Sidák correction, the results will be similar.

If we had chosen to make the 6 unplanned tests using Tukey’s HSD, including a test of Product pre-training vs. post-training, then from the above analysis we know that *t _{obs} *= -1.34. We need to compare the absolute value of this observed value of

*t*with the critical value of

*t*(as described in Unplanned Comparisons for ANOVA). First we look up the

*q*critical value

*q(k, df, α)*where

*α*= .05,

*k = ab*= 6 and

*df = m*– 1 = 9. From the Studentized Range q Table, we have

*q*(6, 9, .05) = 5.024, and so

*t*= = 3.55. Since |

_{crit}*t*| = 1.34 <

_{obs}*t*= 3.55 we conclude that there is no significant difference between the Product performance before and after the training.

_{crit}**Example 3**: For the data in Example 1 determine whether the training has had the same effect on the Action skill as it has had on the Client skill.

We test the following null hypothesis:

H_{0}: *μ _{Action×Post} – μ_{Action×Pre} = μ_{Client×Post} – μ_{Client×Pre}*

We now compare the differences between Post and Pre training for Action with the differences Post and Pre training for Client.

Here we see that *p* = .64 > .05 = *α*, and so we conclude there is no significant difference between training effect on Client skills vs. Action skills.

**Real Statistics Data Analysis Tool**: The contrasts used in Examples 2 and 3 can also be produced using the **Contrasts** option of the **Repeated Measures Anova and Contrasts** data analysis tool found in the Real Statistics Resource Pack.

First enter** Ctrl-m** and select **Repeated Measures Anova and Contrasts **from the menu.** **A dialog box will appear as in Figure 5 of** **Sphericity.** **Next enter the appropriate range in the **Input Range** field, select the **Contrasts** option, choose the appropriate **Alpha correction for contrasts** option (see Figure 5 of Sphericity) and click on **OK**.

For Example 2 you should choose the range J5:K14 in the worksheet from Figure 2 as the input. For Example 3 you should choose the range M5:O14 as the input. In both cases you will need to deselect the Headings included in the range option and add the headings in manually.

Thanks a lot for your nice stats explanation and useful tool.

I have the following data and I need to find the difference between (old and young), (young and kid) and (kid and young).

Would it be possible that you could please help me?

here is my data:

subjects measures conds values

1 weight old 71

2 weight old 64

3 weight old 79

1 height old 172

2 height old 151

3 height old 179

1 heartbeats old 72

2 heartbeats old 99

3 heartbeats old 90

1 weight young 84

2 weight young 83

3 weight young 59

1 height young 161

2 height young 176

3 height young 198

1 heartbeats young 80

2 heartbeats young 79

3 heartbeats young 62

1 weight kid 23

2 weight kid 25

3 weight kid 30

1 height kid 146

2 height kid 98

3 height kid 58

1 heartbeats kid 68

2 heartbeats kid 86

3 heartbeats kid 84

Based on the type of data you have presented it seems like you have two dependent variables, namely weight and heartbeat and one independent variable age group (kid, young, old). This is compatible with MANOVA. See MANOVA for more details.

Whether this is really the test to use depends on the hypotheses you want to test.

Charles

I’m having some trouble filling out the dialog box, and I’m not sure how arrange the rows and columns in excel. How is a two factor x two level RM ANOVA different from a one-factor four level RM ANOVA in the column and row layout?

My ANOVA table always has missing cells that do not calculate. I’m running iOS 10.11.1 (El Capitan) and Excel 2011 for Mac.

Michael,

The Real Statistics Resource Pack doesn’t currently support a two within subjects factor RM ANOVA. It does support a mixed factor RM ANOVA. The layout of the data for this model is as described in Figure 19 of the following webpage:

http://www.real-statistics.com/anova-repeated-measures/one-between-subjects-factor-and-one-within-subjects-factor/

The data layout for a one factor RM ANOVA model is described in Figure 5 of the following webpage:

http://www.real-statistics.com/anova-repeated-measures/sphericity/

The versions of the algorithms used require balanced models with no missing data. I expect to allow for unbalanced models with missing data in one of the next two releases of the software.

Charles

Hi. Your work is excellent, but I’d appreciate some help. I can’t find the menu entry for ‘two-way anova with repeated measures on both factors’. It’s there in the multivariate workbook , but not on my computer. When I look at the RealStatistics menu in Excel I see ‘Analysis of Variance’; when I click on that I’m offered Anova:one factor, Anova:two factors, Anova:three fixed factors, Manova single factor, Nested anova:two factors, Repeated measures:one factor, Repeated measures: mixed, and Ancova. None of those seem to be what I want.

I’ve had the same problem on both the Windows version and the Mac version. I’m sure I’m missing something! If you could tell me where to find the right button I’d be very grateful.

Best wishes and thank you in advance

Thomas Green

Sorry Thomas, but I don’t yet provide a two-way anova with repeated measures on both factors option.

Charles

Aieee! There’s one in the Workbook, or am I mistaken? It’s exactly my design – each participant did the same 3 programming problems (factor 1) with or without syntax highlighting (factor 2). I could pool the scores over problems to get a simple test of the effect of highlighting, but the graphs suggest an interaction effect – syntax highlighting seems to be more useful on the harder problems, as you’d expect. What’s my best route forward – planned comparisons? It’s a bit cheeky of me to ask for stats advice, when you’ve already put in so much unpaid work just for the benefit of total strangers. Please feel free to ignore me if you like.

Many thanks,

Thomas

Thomas,

Yes, there is an example of your design described on the website, and you can indeed find the example in the Examples Workbook. I just haven’t implemented it yet in the software. You can use planned comparisons to pinpoint the effect. I give quite a few examples (plus support in the Real Statistics Resource Pack) of how to do this for two factor ANOVA (but not for repeated measures) and one factor ANOVA with repeated measures.

Charles

hai charles,

how about if my study is just two-way within subjects?? what do you think are its assumptions???

thank you 🙂

What do you mean by “how about if my study is just two-way within subjects??” How does your study differ from the type of analysis described on the referenced webpage?

The assumptions for Anova with repeated measures (in general) are:

– Dependent variables have intevral measure

– Independent variables should be repeated measure

– No significant outliers

– Distribution of each group should be normal (although the test is pretty robust to violations of this assumption)

– Sphericity (although the correction factors can be used when this assumption is violated).

Charles

Also, I see from your site you can adapt a two-way ANOVA with repeated measures on only one factor for an “Unbalanced model”. Can you do the same here for a two-way ANOVA with repeated measures on both factors?

David,

I have not yet worked on using regression approaches (or general linear models) for this situation. I plan to turn my attention to this situation shortly.

Charles

I’ve installed the Resource Pack for Excel 2007. Is the “Repeated Measures ANOVA” function in the 2007 Resource Pack suitable for Repeated Measures on 2 factors, or only for 1 factor?

After I format my data to be in a table arranged similar to what you show in Figure 1, can I use to Resource Pack to generate the rest of the readouts?

Using the “Repeated Measures ANOVA” option with the “Contrasts” box ticked gives a readout, but it is not what I observe in your Figure 2.

Is Figure 2 generated using some other options in your Resource Pack, or do you have to actually create the treatment means tables yourself every time?

Thank you for this tool pack and useful website!

David,

The Real Statistics software currently only handles the one within subjects factors case for repeated measures. I generated the various outputs on the referenced webpage manually, but if you download the Workbook Examples you can see how it is done (see http://www.real-statistics.com/free-download/real-statistics-examples-workbook/).

I will eventually add more support for repeated measures ANOVA in future releases.

Charles

Hi,

I just would like to thank you for creating something straight forward, with practical examples and yet enough theory that one can remain focus on what we are trying to achieve without getting lost in theory. I am talking, obviously, for those that have a pre- understanding and just want to double check the test and methods chosen with regards to the data and specific parameters are right. Well done indeed and please, keep this going. THANK YOU!

Marianna,

Thanks you for your kind words.

Charles