Do a search on repeated measures (or within-subjects) ANOVA. Add the name of your software to the search for info about how to carry it out using your stats package. HTH.

...or use a mixed model with "subject" as random factor.

You may try one way repeated measures ANOVA. Let me share the assumptions:

1) You have one dependent variable that is measured at the continuous level.

2) You have one within-subjects factor that consists of three or more categorical levels.

3)There should be no significant outliers in the three or more levels of the within-subjects factor.

4) The distribution of the dependent variable in the three or more levels of the within-subjects factor should be approximately normally distributed.

5) The variances of the differences between all combinations of levels of the within-subjects factor must be equal (known as the assumption of sphericity).

Dear Bruce Weaver , Mosharop Hossian , Jochen Wilhelm ,

Thank you for your time and the valuable information you shared.

I believe that we can’t use repeated measures ANOVA either, because we have multiple measurements of each subject in each group (and not the same number in each group).

Vasileios Skaramagkas in my reading Jochen Wilhelm did not suggest to use a reapeated measures ANOVA, but a mixed model, which also is known as hierarchical linear model oder multilevel model. It is possible to model exactly what you need: it catches the dependencies of the measures (what Jochen meant by "subject as random factor"), i.e. repeated measures, but also that you have several measures within each subject within each condition.

The main options are probably a repeated measures ANOVA or a multilevel (aka linear mixed) model. The issue of multiple measurements of each individual could be addressed by analysing the mean in the RM ANOVA. In a perfectly balanced design this would be equivalent to a three level nested* multilevel model with measurements nested in time points nested in individuals (assuming a normal response and estimating the multilevel model with restricted maximum likelihood). If the design isn't completely balanced the multilevel model is likely a better approach.

* Its possible it isn't a nested design but a cross-classified one but there isn't enough info in your question to tell. That would definitely require a multilevel model.

Some information in the link

https://www.ibm.com/docs/en/search/Linear%20Mixed%20Models%20to%20Model%20Random%20Effects?scope=SSLVMB_subs

You have to use repeated measures ANOVA with between-subjects factors (aka mixed model). In your example, you have to use X as the dependent variable, multiple measurements as the within-subject factor, and 'conditions' as the between-subject factor.

If you are using SPSS you can do it in the following order-

1. Go to analyze - general linear model - repeated measures (box opens)

2. Add a name for the repeated measures (ie. multiple measures). (suppose you give it a name t) - put the number of times multiple measurements of the sample variable X was taken - click add - then click define

3. put the variables of X (ie. variables containing each measurement of X) into withing subject variable box

4. put the variable 'condition' in the between-subject box

5. In the contrasts tab - select 'repeated' from the drop-down menu - click change - click continue

6. In the EM Means tab - select variables and interactions you want to check the estimated means for

7. In the options tab select descriptive statistics, homogeneity tests (and others as required)

8. Then click ok.

Now you can test both the within-subject effects (ie. changes within multiple measures) and the between-subject effect (ie. impact of your 'condition' variable on the repeated measurements. If there is an impact you may want to do multiple comparisons of estimated means between different conditions. By some tweaking of the SPSS Syntax, you can also do that. (You can find necessary resources on the internet for that).

Please note that you want to look for two important assumption tests here. One is the Box's test of homogeneity of Covariance, another is sphericity assumption. However, you can use the p-values from the 'multivariate test' box, instead of looking for Mauchly's test for sphericity assumptions. So if Box's test of homogeneity of covariance is violated you use the Pilai's Trace row for p-value, if not violated you need to use the Wilks' Lamda row from the 'multivariate test'.

I hope this helps. Thanks.

Dear all,

Thank you for your answers.

I believe that we can’t use RM ANOVA because there are some subjects that do not have observations in some conditions. That is, a subject may have 3 observations in “low” condition, 5 in “medium” condition and none in “high” condition, whereas another subject may have 4 in “low”, 5 in “medium” and 1 on “high”.

We tried linear mixed model analysis with no repeated variable, with “subject” as a random factor and “condition” as a fixed factor. I hope this is what some of you meant with the multilevel model.

PS. We are using SPSS