No basic ANCOVA should NOT be used for repeated measures. ANCOVA assumes that all the error terms independent (which is not true for repeated measures) . You should be using Multilevel Mixed Models ( random-effects models). If there are only two time-points (i.e. pre post test then you can use ANCOVA such that the Co-variate is the Post-test value (Each co-ordinate is the obviously paired pre-post test)

This may help explain: : http://www.oxfordjournals.org/our_journals/tropej/online/ma_chap10.pdf

Instead, you "could" use repeated measure ANOVA (RM-ANOVA) with each time-point as a category - but this is unwise, as there is no "order" implied with a RM-ANOVA just repeated measures under different conditions (which time could be a legitimate condition). Also if there are many time-points the co-variate error estimates get unwieldy. This gives a handy overview of repeated measure ANOVA:  https://statistics.laerd.com/statistical-guides/repeated-measures-anova-statistical-guide.php

And also when not to use ANCOVA: https://statistics.laerd.com/spss-tutorials/ancova-using-spss-statistics.php

BTW I do not work or get paid by LAERD. :-)

Your best bet is time-course Multilevel models. This book explains it all (very good investment): http://www.amazon.com/Applied-Longitudinal-Data-Analysis-Occurrence/dp/0195152964/ref=sr_1_2?s=books&ie=UTF8&qid=1409637000&sr=1-2&keywords=longitudinal+data+analysis

If you have access to STATA this book is awesome: http://www.amazon.com/Multilevel-Longitudinal-Modeling-Using-Stata/dp/159718103X/ref=sr_1_2?s=books&ie=UTF8&qid=1409637059&sr=1-2&keywords=longitudinal+data+analysis+stata

Apologies to Bruce and Jochen, but your answers are very misleading.

Best, D.

Sure one can.

A simple ANCOVA model is

Y ~ TIME +  GROUP + TIME:GROUP

(Y is the response variable [dependent], TIME and GROUP are the predictor variables [independent] and TIME:GROUP is the interaction = the difference in time-dependecy between the groups).

If you have several measurements per individual (all individuals mesured at every time point) the model can adjust for inter-individual differences by

Y ~ INDIVIDUAL + TIME + GROUP + TIME:GROUP

If not all individuals were measured at the same time points, one has to use a mixed model with INDIVIDUAL as random effect.

This model accounts for indvidual offsets. If the time-dependencies between individuals are varying, one can use the interaction INDIVIDUAL:TIME instead of INDIVIDUAL, or specify the random term in a mixed model accordingly.

You can indeed perform repeated measures ANCOVA with SPSS.  But you may find it the output easier to interpret if center the covariates on meaningful in-range values.  See the link below, for example. 

Good luck!

http://journals.cambridge.org/download.php?file=%2FINS%2FINS13_05%2FS1355617707071147a.pdf&code=150e52590cf069c6f111c69b82140018

No basic ANCOVA should NOT be used for repeated measures. ANCOVA assumes that all the error terms independent (which is not true for repeated measures) . You should be using Multilevel Mixed Models ( random-effects models). If there are only two time-points (i.e. pre post test then you can use ANCOVA such that the Co-variate is the Post-test value (Each co-ordinate is the obviously paired pre-post test)

This may help explain: : http://www.oxfordjournals.org/our_journals/tropej/online/ma_chap10.pdf

Instead, you "could" use repeated measure ANOVA (RM-ANOVA) with each time-point as a category - but this is unwise, as there is no "order" implied with a RM-ANOVA just repeated measures under different conditions (which time could be a legitimate condition). Also if there are many time-points the co-variate error estimates get unwieldy. This gives a handy overview of repeated measure ANOVA:  https://statistics.laerd.com/statistical-guides/repeated-measures-anova-statistical-guide.php

And also when not to use ANCOVA: https://statistics.laerd.com/spss-tutorials/ancova-using-spss-statistics.php

BTW I do not work or get paid by LAERD. :-)

Your best bet is time-course Multilevel models. This book explains it all (very good investment): http://www.amazon.com/Applied-Longitudinal-Data-Analysis-Occurrence/dp/0195152964/ref=sr_1_2?s=books&ie=UTF8&qid=1409637000&sr=1-2&keywords=longitudinal+data+analysis

If you have access to STATA this book is awesome: http://www.amazon.com/Multilevel-Longitudinal-Modeling-Using-Stata/dp/159718103X/ref=sr_1_2?s=books&ie=UTF8&qid=1409637059&sr=1-2&keywords=longitudinal+data+analysis+stata

Apologies to Bruce and Jochen, but your answers are very misleading.

Best, D.

Thanks all for your answers,

I have 4 separated groups (4 different interventions), and I measured the dependent variable over 4 time points. I considered my intervention groups as "between subjects factor" and the time points as "within subjects factor". In this case the suitable test would be "mixed ANOVA with repeated measures", right?

I chose this method: mixed ANCOVA: 4 (intervention groups, between subject factor) * 3 (time points, within subject factor), and the pre-test as covariate (in order to control initial differences)

My consulting adviser said it is false, because there are more than 2 time points and we can't use ANCOVA in this case.

Is it wrong or right? If it's wrong, would you suggest me any appropriate method, please?

David, thank you for the clarification and the nice links. However, I do not completely understand what the specific concern is. Maybe our definitions of "ANCOVA" are not identical, and this is the reason for our discepancies. The ANCOVA, in my eyes, is a linear model with an interaction between a categorical and a continuous predictor (the continuous being the time). However I do not restrict this to this special case (that might be termed "ANCOVA" in some software). I would even prefer to avoid using terms like "ANCOVA" completely and precisely state the model which may contain some more covariables, transformed or not, and interactions. Thus, one can model "time" also as categorical, what might be advantagous when the relationship between time and response can not be linearized and when there are only few distinct time points. For two time-points (pre vs. post) there is no difference between a continuous (linear) and a categorical coding of "time".

So this may be the caus of our differences. I should not call it "ANCOVA"?

You further mentioned the assumption of independence between values taken at different time points. This is not clear to me. I mean, surely the response values of adjacent time points are not independent - absolutely similar to a liniear regression, for instance - but this is not point. The residuals should be independent, but this is a general assumption, and this gets violated only when the model is misspecified (the assumption of independent residuals is identical to the assumption of a correctly specified model). For instance when a linear trend is modeled but the real relationship is non-linear. Notably just a model coding "time" as a factor (categorical) automatically reserves sufficient degrees of freedom to the model that the residuals can be independent (provided the model is correctly specified w.r.t. all other predictors).

Lastly, the results of both, a mixed model and a fixed model, give the identical results when the time points are actually fixed (i.e. all subjects were measured at the same time points, and no measurements are missing). So using a mixed model in this case is just a more complicated way to come to the very same conclusions...

I agree that longitudinal experiments with many time-points (and possibly unequal time-points and missing time-points) should (must) be analysed with multilevel ("mixed") models, as you said. Four time points (pre-test plus 3 experimental time points) work well with fixed models (again, given that the same time points are used for all subjects; whether "time" should be coded as a continuous or categorical variable depends on the actual time-course [if is can be reasonably modeled as a function]).

Jochen go read this as you are quite far off the mark. 

http://www.amazon.com/Applied-Longitudinal-Data-Analysis-Occurrence/dp/0195152964/ref=sr_1_2?s=books&ie=UTF8&qid=1409637000&sr=1-2&keywords=longitudinal+data+analysis

Hey Roshanak,

What do you consider the covariate and what do you consider factors/variables of interest?

As a statistics student, I have taken stats classes from several different departments, psychology, education, stats, industrial engineering, Biostats, etc. What one person calls a covariate, another person calls a factor/variable. I was taught in my stats and IE classes that a covariate was a thing you want to test, that is not under your control. In my biostats and psychology classes, a covariate was a thing we wanted to test but was not of interest in our analysis. We included a covariate to remove some of the variability within the analysis.

If I was analyzing your data in a stats class, I would use ANOVA with repeated measures. Depending upon what you call your variables, I would use a Fixed Effects, Mixed Effects of Random Effects model.   

Hi Zehng,

I used "mixed ANOVA with repeated measures"

4 (intervention groups, between subject factor) * 4 (time points, within subject factor)

Dear Zenhng,

Unfortunately no, not yet.

There are two kinds of covariates found in repeated measures analyses; 1) time-invariant covariates or 2) time-varying covariates.  With time-invariant covariates there is just one value that is used for all the repeated observations for a given subject. Time-varying covariates, on the other hand, can take on a different value for each of the repeated observations. Time-invariant covariates can be analyzed with either wide data or long data. However, time-varying covariates require the data to be in the long form............

For more information, please visit the link below.

http://www.ats.ucla.edu/stat/spss/faq/repeat_covar.htm

Hi @David Broadhurst I tried to get the paperfrom Oxfordbut the link doesn't work, could you give me the nameso I can look for it?