Factorial ANOVA

In this part of the website, we extend the One-way ANOVA methodology to more than one factor. The focus is on support for ANOVA with two or three factors where all the samples have the same size. See Multiple Regression for topics related to unbalanced ANOVA models.

Topics

Reference

Howell, D. C. (2010) Statistical methods for psychology (7th ed.). Wadsworth, Cengage Learning.
https://labs.la.utexas.edu/gilden/files/2016/05/Statistics-Text.pdf

69 thoughts on “Factorial ANOVA”

  1. Sir, After getting Anova results how can find out SE.M. Value and C.D., C.V. Values in two and Three factors Anova.

    Reply

  2. Hello,

    I am working on an assignment, and we have the following data. However, we only measured the U-value once (we have a very small sample size for each set of parameters). We tried using three-factor ANOVA, since there are three independent variables but we received errors. In this simply due to the fact that our sample sizes are too small (1 measurement per set of parameters).

    Thanks in advance!

    Number of Tubes Flow Rate Orientation Sample: U-Value
    One High Counter 2113.302919
    One Low Counter 2099.177058
    One High Co 2074.911618
    One Low Co 1734.849517
    Four High Co 1918.758744
    Four Low Co 1637.986514
    Four High Counter 1657.227586
    Four Low Counter 1423.046233

    Reply

    • The Real Statistics Three Fixed Factor ANOVA data analysis tool works even when there is no replication.
      Make sure that you choose the correct option for the Column headings included with data checkbox.
      If you are still having problems, you can email me an Excel file with your data and resulsts from the analysis.
      Charles

      Reply

  4. Doctor buenas noches quería ver si me podría ayudar con la interpretación de un análisis factorial que hice en excel, no tengo la menor idea de como interpretar los datos que me arrojó.

    Reply

  5. Hello dear sir,
    as i asked before but i did not got the answer, sir i wan to conduct a pot experiment, i have 3 factors, among 2 factors having 4 levels each and the 3rd on having 3 levels, i.e.4x4x3=48 ,so can i do this in CR design ????? as i mentioned that i will conduct a pot experiment????? please sir suggest me that CR design will be ok or not ?????

    Reply

  6. Hi sir,
    My problem has 27 experiments (4 factors within 3 levels).
    I have graded the experiment using an MCDM technique and shown the values in column against each experiment
    I am creating a factor level table taking the average based on factors under each level
    I need the values of ANOVA table. I tried and getting errors. Can you Help me if i send the data…
    Thanks

    Reply

  7. Dr good afternoon, I would like to know if there is any error in executing ANOVA ii ways, because when executing them it shows, for example, a significant difference in the interaction, and verify it with the Excel formulas, however, when applying the posthoc test, it does not detect it, will not it be necessary to make an adjustment to the p-value of the interaction?
    Thank you

    Dr buenas tardes, quisiera saber si es que existe algun error al ejecutar ANOVA ii ways, pues al jecutarlos muestra por ejemplo diferencia significativa en la interacción, y la verifique con las formulas de Excel, sin embargo al aplicar ls prueba poshoc no la detecta, no será que hay que hacer una ajuste al valor p de la interacción?.
    Muchas gracias

    Reply

  8. Hi,

    My survey is on the perception of BMI in one specific culture. The controlled variable is the ethnicity of the respondents (all Thai) and the independent variables is the BMI of the respondent. I have 10 dependent variables, as participants were asked to estimate whether a model in a given picture (10 pictures total) is Underweight, Normal Weight, Overweight, Class I Obese, Class II Obese, or Class III Obese.

    How would I go about performing statistics to measure the correlation between one’s own BMI and their estimates? I would also like to measure whether there is a correlation between gender and their estimates? My research is based on a study I duplicated that says this: “Differences in perceptions between genders and individuals in the various weight classifications were examined using 2×3 analyses of variance (ANOVAs) with gender (female andmale) and weight class (normal, overweight and obese) as the factors. Tukey’s HSD tests were used for post hoc analyses.

    I do not know anything about statistics, please help me!

    Reply

    • Hello Amy,
      It sounds like if you have two factors, Gender and Weight Class as you have described, you can perform Two Factor Anova as described on the Real Statistics website. I have a few questions, however, about the objectives of your study before I can say whether this is the correct approach. In fact, the way you present the scenario I have doubts as to whether ANOVA is the correct approach.
      1. I am confused about what you are considering the dependent and independent variables. Does each of the 76 participants evaluate one model 10 times based on 1 different pictures or 10 different models?
      2. When you say that the BMI is that of the respondent, I assume this is the same as the participant
      3. What hyportheses are you trying to test?
      Charles

      Reply

  9. How do you conduct a factorial ANOVA when all you have are the percentage means from a National census? I am looking at how gender and the percentage of screen time affects the percentage of getting daily physical activity.

    Reply

    • Ryan,
      If you only have the average in each group, you won’t be able to use ANOVA. If you have multiple averages for each group (e.g. the averages for each of the 50 states), then perhaps you can use ANOVA.
      Charles

      Reply

  10. Hi professor Charles,

    I’m a Doctorate degree student, and I’m doing research by asphalt mixture materials. I had tested two type of mixtures(A) by two conditions of tests(B). I have a three replicates to generate my statistical analysis for each condition.

    A fatorial ANOVA is the best way to see significant effect of factor A, B and A*B? I just want to recognize if the type of mixtures have influence, or the conditions of test specimens is the most important factor that influences on my stiffness average measurements.

    I’v also tried a single-factor ANOVA with Tuckey test, if p-value is less then 0.05, to see the diferences between two mixtures of asphalt concrete and two type of test conditions. This is better?

    Thanks a lot.

    Reply

    • Hello Igor,
      It sounds like a reasonable approach. Tukey HSD actually yields all the pairwise comparisons, and so you pay a little price in terms of the p-value if you only need one of these comparisons. You could use contrasts in this case.
      Charles

      Reply

  11. Hi.

    I just need a piece of advice from you Dr. Charles.
    I am currently doing my analysis for my experiment.

    I have 2 factors.

    Age and portion of tree (butt, middle and top)

    And i have 3 replications. (Tree 1, tree 2 and tree 3).

    I would like to ask what type of anova can be used to see if significant effect of factor A, B and AB.

    Thank you.

    Reply

    • Hi Dr. Charles, I’m Shara. I just want to consult you about my action research. My independent variable was the students learning styles and my dependent variable was the classroom settings. The title of my action research is all about how classroom settings impacts the students learning style.
      My survey was like this. For the dependent variable, I asked students to rate 5-1, 5 is the highest and 1 is the lowest, about their perception to the different learning styles and then for the dependent variable i list all the activities that can be held inside the classroom and they will choose on the different classroom settings that they think they are comfortable. I just want to know what kind of correlation will i use for my action research

      Reply

          • Hello Shara,
            Does the following describe the situation that you have? If so ANOVA is a reasonable approach for addressing this situation. If this is not the situation, please clarify.
            Situation: There are a few distinct learning styles (say 3 styles) and we select a certain number of students (say 10 students for each of the 3 styles) at random who use one of these learning styles (or we assign the students at random to one of these learning styles. Each of the students rate their classroom setting from 1 to 5. We are interested in testing whether there are significant differences between the average rating of the classroom setting among the groups (3 groups in the situation described). You can use ANOVA to do this.
            Charles

  12. 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
    Reply

    Reply

  13. Charles;
    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

    Reply

    • Ahmed,
      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.
      Charles

      Reply

  15. 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

    Reply

  16. 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
    Thanks
    S

    Reply

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

      Reply

  18. 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.

    Reply

    • CB,
      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.
      Charles

      Reply

  19. 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 ?
    thanks
    jason

    Reply

    • Jason,
      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”?
      Charles

      Reply

  20. 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?

    Reply

    • 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.

      Charles

      Reply

  21. 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!

    Rachael

    Reply

    • Rachael,

      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.

      Charles

      Reply

      • 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!

        Rachael

        Reply

        • Rachael,

          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.

          Charles

          Reply

  22. 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?

    Thanks
    Jim

    Reply

    • Jim,
      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
      Charles

      Reply

  24. Hi,
    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?

    Reply

    • 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.
      Charles

      Reply

  25. 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!

    Reply

    • 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.
      Charles

      Reply

      • 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 🙂

        Reply

  26. 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.

    Reply

  27. 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?

    Reply

    • 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).

      Charles

      Reply

  28. 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.

    Reply

  29. 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?
    thanks

    Reply

    • Saif,
      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
      Charles

      Reply

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