You do not state the number of subjects included, but the warning you get suggest that you try to estiamte too many parameters in the model than the number of subjects at your hand. This problem is not solved by changing estimation method. If it is impossible to increase the number of subjects, you must reduce the number of free parameters in the model. To make such a decision further information is required. Is it possible to assume that the tretament effect is identical for the subject at all 4 occassions for instance?

Dear Hui Wei 

I think you can use the generalized estimating equation (GEE) method.

You do not state the number of subjects included, but the warning you get suggest that you try to estiamte too many parameters in the model than the number of subjects at your hand. This problem is not solved by changing estimation method. If it is impossible to increase the number of subjects, you must reduce the number of free parameters in the model. To make such a decision further information is required. Is it possible to assume that the tretament effect is identical for the subject at all 4 occassions for instance?

@ Kenneth: I agree with your statements about the parameter, but I understood the design as follows: replicants seem to be the "subjects". Therefore, he seems to have a 3 (condition/group) x 12(time points/measurements) design with repeated measures on the second factor, but only 4 "subjects" per group.

On the other hand, this would lead to different df for the Mauchly test. So, it seems to be important to have more information about the design.

Dear Hui,

According to your explanation you have split unit design structure. The experiment contains three treatment factors. Into each treatment factor is four subjects, in each subject 12 samples were measured vs. similar time interval were measured under each treatment factor, and the four treatment factors also occurred versus time. If you have this design of experiment then you have hierarchical structure where cluster from four subjects should be defined as experimental units, so, there are three experimental units, each unit contains four subunits (subjects), and each subunit includes 12 individual subunits (samples). In this context you can use the hierarchical repeated measurement ANOVA.

Best regards,

Oleg

Thank you all for the replies. @Kenneth and Rainer, The description on the experimental design by Rainer is right; my experiment contains 3 (condition/group) x 12(time points/measurements) x 4 subjects per group. I want to know whether my treatments led to significant differences in CO2 efflux rate of treated samples from that of controlled samples over my investigation period. The point I want to know is significance of the treatment effect but not of the time effect. Somebody suggested that I could average the 12 measure values at different sampling times for each subject and then use one-way ANOVA to compare the treatment effect. I think it's feasible. But I want to know if there is a better statistical method for such a repeated measures dataset when taking into account of time effect at the same time. If yes, then which information are still needed before suggestion can be made? Thanks again.

In SAS proc Glm might do it

Dear Hui Wei,

your concern being the values of CO2 it might be appropriate to simplify a bit the situation in the line of the proposition you received to use the mean of the 12 values instead of the 12 distinct values thesmselves. I felt you were reluctant to treat the mean by a mere anova and I agree that sounds like a (too) strong simplification and that you may miss some interesting features in the time variations. But you may develop this idea by going further than the simple mean, using orthogonal polynomial decomposition of your 12 individual observations. With 12 points yous may go as high as 11 degree; 11 would be silly, but up to 2 is advisable. The linear trend (d°1) is usually of some value, as is the quadratic trend (d°2) also. Then you will have some information about the mean level, the tendancy to in/decrease (d°1), linearly or not (d°2). It is generally admitted that up d°2 or 3 is sensible with experimental data.  My opinion is that your subject matter relies in the interactions with the treatments. Googling "orthogonal polynomial anova" wil give you nice references. Best wishes.

Don't bother trying to estimate parameters, at least initially. Treat each factor in the design as a synthetic feature, now part of a long vector of predictors for each subject. Apply any good model-free learning machine to the data, where the machine is working in regression mode. Any interactions over the features can be studied using additional features composed of the product over all possible outcomes for those features. Evaluate the machine predictions using mean square error, where this must be measured in terms of the scientific consequences of the error made for each subject. For this, the machine could be asked to generate an error bar estimate for each prediction, on each subject. Next, the effect sizes of each feature in the design can be examined using the same machine. Finally, any truly predictive features of the framework can be interrogated using some, smaller parametric model.

You may use regular ANOVA where you estimate  treatment effects and a slope on time for each treatment (i.e, you treat  time in the model as regression variable):

Y = treatment + time * treatment

which could be done in PROC GLM of SAS as Jay indicated

Dear Jay, Jean, James and Jihad, thank you all for the kind suggestion. They are helpful to me. I will try to do the analyses as you suggested.

i also suggest you SAS for the analysis

If I understand correctly you have 3 x 4 = 12 cases, and each case is measured 12 times, so df total is (3 x 4 x 12) - 1 = 143 and the df for the residual term should be 99 . With only 2 terms in your model (NForm and Time), it should run. I'm guessing this kind of error is due to a data structure problem.

How are your data structured in the dataset? I believe the SPSS procedure you are using expects the data to be in "wide" format, with all repeated observations for the i-th case in a single row, one row per case. If you have a different structure, e.g., "long" format, in which case you have one row per repeated observations (and therefore 12 rows per case), then that may cause some problems.

Dear Oluwafemi, thanks for your suggestion. I most often use SPSS and sometimes R software to analyze my data. I have no SAS in my computer TT... I believe the inner algorithm for RM-ANOVA in different softwares should be similar and perfer to figure out this problem in SPSS :)

Dear Scot, yes your understanding is completely right on my dataset. The data structure was in wide format. But I find that at each sampling point there are only 12 subjects. When the number of subject is 16 in each sampling point (one treatment more with 4 replicates), RM-ANOVA can run with all the results available. Is this the source of error because there are not enough cases for comparisons? Thanks a lot.

Missing data could cause problems. For example, If all cases within one group are missing the same repeated measure, that might result in that kind of error.

Missing data treatments are imputation or aggregation (restructuring). With a smallish sample, imputation might be prohibitive. Aggregating might  be more straightforward, e.g., combining successive observations so that you take the mean of the first 3, or first 4 observations for each person. Of course your theory and design should take precedence in deciding how best to aggregate the data, but those are some examples.

Hui,

Just trying to affirm that I understand your design:

One (1) between-subjects factor, treatment, with three (3) levels

One (1) within-subjects factor, measure, with twelve (12) levelsFour (4) cases per treatment.

As Scot indicated, that should yield 3 x 4 x 12 = 144 data points.

First, you should have gotten legitimate univariate (e.g., repeated measures) tests for all three of the testable effects: treatment (between-subjects, df = 2,9); measure (within-subjects, df=11,99); and treatment x measure (also within subjects, df = 22,99) , so all of your tests are there. 

Second, the reason the multivariate test fails is that you'll need more total cases than you have measures (at present, you have 12 cases, 12 measures).  Ideally, you'd like more cases than measures for each of the three treatment groups, implying far more replicates than you used.  For the special case of a one-way manova with three groups and no covariates, the error df can be computed as: 2(N - p - 2), where N is total cases (12 for your example), p  is number of DVs (12 for you), and the "2"s are constants.  So, you'd need at least 15 cases overall (e.g., replicates per group) to have a non-zero error df value.  Hence, the problem is insufficient sample size.  Bigger N would also solve the Mauchly test's non-report of a significance level.

Finally, the multivariate test evaluates a different hypothesis, that the three treatment groups are equivalent on the set of all 12 measures, so it's not clear why you'd want to look at that test.  If the 12 measurements were 1 each of 12 different variables, then the multivariate test would be more suitable than the univariate repeated measures.  However, your design appears to be the same variable measured on 12 occasions, for which (univariate) RM makes more sense.

Oops!  next to last paragraph: You'd need 15 cases overall...5 replicates per group...

Dear David, thank you for the detailed explanation. Because time effect is not the important point I want to test, it seems multivariate test can be performed to test the significance of treatment effect at each occassion.