Factorial ANOVA

We now extend the One-way ANOVA methodology to more than one factor. We will initially restrict our description to two factors where all the samples have the same size.


43 Responses to Factorial ANOVA

  1. DASYAM RAMESH says:

    I need how to check normality procedure for RBD (8 Treatments & 3 Replications)

    • Charles says:

      Hi Dasyam,

      You can check for normality using the techniques described on the webpage http://www.real-statistics.com/tests-normality-and-symmetry/, in particular using QQ plots, box plots or the Shapiro Wilk test. These tests are implemented in the Real Statistics Resource Pack, which you can download for free from the website.

      In your example, I understand that each treatment will only contain a sample of three items. If this is so you won’t be able to test normality since a sample of three is too small.


  2. Gasper says:

    Dear Charles,

    this is very useful site for my career. I am just beginning to play with statistical modeling for research


  3. saif says:

    dear Charles,
    I’m student of Data Analysis, i am working on an assignment and I’m suppose to apply 2-way ANOVA, but when I apply Levene’s test for homogenity, my p-value is .02. i am afraid that i can’t run the 2-way anova because it seems to be violating the normality assumption.
    what to do now?

    • Charles says:

      Levene’s test checks the homogeneity of variance assumption, not the normality assumption.
      If Levene’s test shows that your data violates the homogeneity of variance assumption, then you have the following choices:
      – make sure that the homogeneity of variance assumption isn’t being being violated because of outliers; if so you need to deal with the outliers first
      – perform a one-way analysis instead using Brown-Forsythe or Welch’s test, which do not require homogeneity of variances
      – transform the data to try to eliminate the problem. Typical transformations are log(x), x^2, 1/x

  4. Jaronie says:

    Hi Charles,

    Do you have a sample solutions for ANOVA with more than 2 factors, such as 3 or 4 factors. If possible with 3 levels and at least 2 replications.

    Thanks in advance.

  5. abiy says:

    dear Charles, my name is Abiy Birhanie, i am from Ethiopia, Dire Dawa. I am conducting a thesis with a title of assessing training effectiveness…,the questionnaire follows scale and there is no continuity b/n qns for instance; qn 1 training increase productivity 1 very low 2 low 3 moderate 4 high 5 very high NA not applicable the total qn is 22, distributed to two industries, my qn for u is; when i use two factor anova with out replacement, the result wasn’t satisfactory but when i used ANOVA single factor with excel it is getting better. what shall i do?

    • Charles says:

      Dear Abiy,

      I am not sure what you mean by “the result wasn’t satisfactory”. If not satisfactory means that it is not what you want, then so be it: the result is the result.

      You used the term “two factor anova with out replacement”. I assume that you mean “two factor anova without replication”. To use this test it is important that there is only one sample element for each intersection between levels (groups, treatments) in factor A and factor B. Otherwise the results won’t be meaningful.

      It is entirely possible for ANOVA single factor and ANOVA two factor tests to differ in their results. Both could be valid since they measure different things. If the problem you are investiating lends itself to two factor ANOVA I would start with that test and draw conclusions. I would then look at the single factor ANOVA as a follow up test. If you really get contradictory results then I would double check to make sure that the assumptions for ANOVA are not drastically violated (normality and equal variance).


  6. abiy says:

    my data were needs ANOVA single factor, i have done the steeps u suggest, now i fill full confident to draw my conclusion. i thank you!! please don’t stop helping people because it is more than medicine! such academic support cures mind and gives u special rest. thank u again. if i have qn i will be back.

  7. Hana says:

    Hi, I just have a question about a 2×2 ANOVA. When looking at significant effects, do we refer to Pairwise comparisons adjusted to minimize false positives with a Bonferroni Correction? Or is Bonferroni only used if we have more than two levels?
    Thank you!

    • Charles says:

      Hi Hana,
      You are correct that a Bonferroni correction is only used if there are more than two levels.
      If there are only two levels for a factor, there is no need to do any comparisons for that factor since the omnibus ANOVA test has already has determined whether or not there is a significant difference between the two levels. Since there is no need to do any comparisons, a Bonferroni correction is not needed.

      • Hana says:

        Thank you so much for a quick and clear reply!

        However, if I carry out an ANOVA in SPSS and choose to compare main effects with the Bonferroni confidence interval adjustment, it gives me different results in the Pairwise comparison table than if I choose the option LSD(none), which claims to be the same as if no adjustment had been made. Does that not mean that Bonferroni has corrected something, even though there are only 2 levels?

        I hope that makes sense 🙂

  8. Dhruvita says:

    In my study I have used 2 independent variables having 2 levels each(1). Psychiatric Conditon: a. Autism b. Intellectually Disabled & 2). Gender a. Mothers b. Fathers). Also I have 2 dependent variables (Parental Stress and Marital Satisfaction). So I am comparing Parental Stress and Marital Satisfaction between parents of Autistic children with Parents of Intellectually Disabled children. Also I am comparing the dependent variables within the groups i.e. mothers of autistic vs fathers of fathers and similarly in Intellectually Disabled. Which statistical test is best suitable for this design?

    • Charles says:

      Hi Dhruvita,
      It sounds like you are likely to require MANOVA.

    • Charles says:

      Sorry, but I don’t completely understand the design. E.g. In “mothers of autistic vs fathers of fathers” do you really mean fathers of fathers? Please explain a bit more clearly.

  9. Dhruvita says:

    Hi Charles
    Thanks for the advice. How do I do it in excel?

  10. Jim says:

    Hello Charles,

    I just want to thank you for such a wonderful resource, it’s making my life a lot easier!

    I also have a quick question about the resourcepack. I’ve performed a two way ANOVA and I wanted to know if it’s possible to run a post-hoc test (a Tukey or Dunnett’s test) using your toolset?

    Would the post-hoc Tukey test under the single factor ANOVA options to give meaningful results?


    • Charles says:

      You can run the Tukey HSD test for the main effects since this is the same as Tukey HSD for single factor ANOVA. I have not included Tukey HSD for the interactions between factors. In fact, I had thought that I had done this already, but I see that this is not the case.
      I now plan to implement this in the next release of the software

  11. Rachael says:

    Hi Charles,

    I am trying to test the effectiveness of education programming on age. Therefore, I passed out the same knowledge test before and after an education program for three age levels–elementary, middle/high school, and adult. I wanted to get an idea of what they knew about monarch butterflies before I taught them to then see how much they learned (and how well they listened) after the program. I paired each individual’s before test with their after test based on the birthdate and grade level they provided.

    However, I have unequal sample sizes of the before and after tests collected (due to the fact that the programs were for the public). I have 45 elementary students’ tests, 21 middle/high school students’ tests, and 68 adult tests. If I want to compare the scores before the program between the age groups with the scores after the program between the age groups, which type of statistical test do I use and how do I account for the unequal sample size? (P.S. I do not have access to SPSS, so Excel will have to work.)

    Thank you for your help!


    • Charles says:


      Except for the fact that you have unequal sample sizes, you could use the Real Statistics Repeated Measures Mixed ANOVA data analysis tool, which works in Excel.

      Paired samples, by definition, requires that the paired samples be equal in size. You state that the reason that the samples are unequal in size is “due to the fact that the programs were for the public.” Please explain why this means that the samples are unequal in size.


      • Rachael says:

        Hi Charles,

        Thanks for your quick reply! Considering this was open to the public, I had no way of controlling how many people showed up. Therefore, for those that did show up, I also could not control their ages.

        This event drew in more elementary aged children and their parents. We had less middle and high school aged students participate. Therefore, I received surveys from 45 elementary students, 68 adults, and only 21 middle/high school students. The reason I clarified this is because I read online that if sample size differences are due to “unwillingness to take a survey” then that could be a problem. However, all the participants were willing and comfortable taking it, I just could not have an equal representation of each age group.

        Knowing this, do you still recommend the Repeated Measures Mixed ANOVA?

        Thanks for your help!


        • Charles says:


          The Repeated Measures Mixed ANOVA seems to be the correct approach. I believe that the version of the test that I describe on the webpage Repeated Measures Mixed ANOVA requires a balanced model (i.e. the number of elementary students, adults and middle/high school students are the same).

          I can think of two choices for addressing this problem: (1) randomly select subjects from the larger groups for elimination, so that all the groups have the same number of subjects (21 in this case). This will make the model balanced, but at the cost of eliminating more than half your sample. (2) using a regression approach for repeated measures ANOVA. I described how to do this for fixed factor ANOVA on the webpage ANOVA using Regression. I plan to show how to do this for repeated measures ANOVA in the next few weeks. Stay tuned.


  12. Nnacho Emmanuel says:

    Hello Charles, I am experimenting the effectiveness of 3 concentrations of three avian albumen in the management of a maize weevil. I got data on mortality, F1 progeny emergence and grain damage assessment. I really don’t know how to input the data collected in Microsoft Excel. i have an undergraduate defense hot on my heels, any help please?

    • Charles says:

      It sounds like you are trying to use ANOVA. Microsoft Excel supports three kinds of ANOVA: (1) one-way ANOVA, which could be used to compare the 3 concentrations of avian albumen and (2) two types of two factor ANOVA.

      The data format for one-way ANOVA is shown in Figure 5 of ANOVA Basic Concepts

      The data format for two factor ANOVA is shown in Figure 1 of Two Factor ANOVA with Replication.

      The Real Statistics software extends these three types of ANOVA to many more types. The formatting of the data depends on the type of ANOVA you want to use.


  13. jason says:

    hello, dr. charles:

    would you show me how to use excel to calculate
    1) the reliability based on cumulative binormial distribution given ss=50, defect=1, p=0.98, confidence=0.95, what is reliability ?
    2) what is the sample size if I want to demonstrate R=0.95, CL=0.95, with defect =0,1,2,3 ?

    • Charles says:

      1. Sorry, but I don’t understand what “reliability” means in the context of the binomial distribution.
      2. What test are you referring to in item 2. What do you mean by “defect”?

  14. CB says:

    I am in a Stat class and honestly am struggling with all of it, the class is online and therefore not as hands on teaching and I just do not get it. The lab that I am working on now is Factorial Analysis of variance. I know how to open up in excel and compute the row and column means, I think I got which means to compare to test for main effect of each factor. Now, it says to draw a graph of the cell means, as in your slides and it says to place factor A on the horizontal line. Well I have tried to insert a line graph with the sparklines and cannot insert it and it will not allow me to highlight the information that I am trying to graph, so I have no idea what to do. The last question says to suppose the sample size is huge and from a visual inspection of the figure created, does there seem to be an interaction and explain why? Honestly, I am lost trying to more or less learn this independently. Any help would be welcomed.

    • Charles says:

      Sorry, but since I don’t have access to the online course that you are using, I don’t know how I can help you.

  15. Gami Nasir says:

    Hello Dr. Charles,

    I need your help on this problem. How can I solve this problem?

    Given the data below, using ANOVA and ignoring all interactions, what is the value of the sum of squares for factor A?

    I provided the data in this link


    Answer is = 31437.78, but how



    • Charles says:

      I don’t understand the data format that you are using. It is not one of the formats that the Real Statistics software supports.

  16. SHIN says:

    Hello Dr Charles
    I need help asap!!!
    I am trying to do the results for my dissertation but I dont know what the p value is for my anova factorial test…how do I know?
    A quick response would be appreciated

  17. hl sangappa says:

    Hi Dr Charles

    I need small favor from your side

    I need analyze factorial RBD for my experiment
    No of factors: 5
    No of replications
    Factor levels
    Factor 1: 2
    Factor 2: 2
    Factor 3: 2
    Factor 4: 3
    Factor 5: 3

  18. L.A. says:

    Good evening sir! May I know if your program supports ANOVA for split plot designs? If so, which function is it?

  19. Ahmed Hikal says:

    first of all, I finally have access to the program, it was blocked and I unblocked it as per your instructions. Thanks
    Now I have a question about ANOVA.
    My experimental design has 3 factors:
    Factor 1 (formulation): 2 levels
    Factor 2 (Sequence): 2 levels
    Factor 3 (Period): 4 levels
    So I did 3 factor ANOVA
    1. In the output, how does the program assign A, B, C to the factors?
    2. There is no designation of which factor is between and which is within
    3. It did not make a difference if the factors are numerical or categorical

    • Charles says:

      1. The first column in the input range represents the A factor, the second the B factor and the third the C factor. See example on the webpage.
      2. The factors are neither between or within. I guess you can think of all the output lines as between and the error line as within
      3. All factors are considered to be numeric. If you have a categorical factor then you need to use a tag code. How to do this is described on the website.

  20. SK says:

    I am trying to figure out how to do statistical analysis and which tools to use fro 188 participants undergoing leadership development programs. I have 9 demographic factors including sector, age, working level and 24 variables collected from questionnaire survey on questions like feedback on program effectiveness, tools, HR support etc. I am lost on Anova, factor analysis etc. Can u pl advise how to interpret the data

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