ANCOVA means Analysis of covariance and Covariance is a measure of how much two variables change together and how strong the relationship is between them. The aim of the study will decide the choice of test to infer a conclusion.

Thanks so much for your advice !!!!!

Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether population means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV), while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV). Therefore, when performing ANCOVA, we are adjusting the DV means to what they would be if all groups were equal on the CV. ANCOVA can be used to increase statistical power (the ability to find a significant difference between groups when one exists) by reducing the within-group error variance. While the inclusion of a covariate into an ANOVA generally increases statistical power by accounting for some of the variance in the dependent variable and thus increasing the ratio of variance explained by the independent variables, adding a covariate into ANOVA also reduces the degrees of freedom. Accordingly, adding a covariate which accounts for very little variance in the dependent variable might actually reduce power.

A number of analytic methods are possible when testing hypotheses about treatment effects within the context of a randomized pretest, posttest, follow-up (PPF) design such as analysis of variance (ANOVA), analysisof covariance (ANCOVA), and hierarchical linearmodeling (HLM). Depending on the question(s) of interest, any of these and other analytic techniques maybe possible, oftentimes leaving researchers perplexedwhen attempting to choose the most appropriate ana-lytic technique for a specific question.

KIndly refer to:

Analytic Methods for Questions Pertaining to a Randomized Pretest,Posttest, Follow-Up Design. Joseph R. Rausch, Scott E. Maxwell, and Ken Kelley. Journal of Clinical Child and Adolescent Psychology2003, Vol. 32, No. 3, 467–486.

http://en.wikipedia.org/wiki/Analysis_of_covariance

An ANCOVA that uses the pretest as acovariate will virtually always be more powerful than an ANOVA that utilizes the same dependent variable but ignores the pretest or an ANOVA that incorporates the pretest as a linear component of the dependent variable. Thus, whenever an ANOVA is performed when participants have been randomly assigned to groups and a pretest has been collected prior to treatment, it is virtually always a suboptimal analysis that will result in a loss of statistical power and precision when compared to the corresponding ANCOVA using the pretestas the covariate.

The ANCOVA model typically utilized in practice only accounts for the linear relationship between the dependent variable and the covariate. Thus, not accounting for quadratic (cubic, quartic, andso on) relationships tends to decrease power and precision when compared to an analysis that does account for these relationshipswhen they truly exist in the population.

Useful reference:

Maxwell, S. E., O’Callaghan, M. F., & Delaney, H. D. (1993). Analysis of covariance. In L. K. Edwards (Ed.), Applied analysis of variance in behavioral science (pp. 63–104). New York: Marcel Dekker.

Comment 2 by Tewari appropriately defines the way i would suggest the measure model

It is not easy to understand the differences without all of these methods without giving some elementary regression equations. Some methods are mathematically equivalent to each other. For technical details on the various methods, including their assumptions, and when to use ANCOVA and when not, plus empirical demonstrations, see e.g. my paper on ANCOVA vs Change from baseline in Multivariate Behavioral Research, 2013, vol 48 (6), 895-922. You do not need high tech maths to read the paper, just an understanding of regression models.

"Unless the correlation is perfect (+/- 1.00), the adjusted DV (posttest) values will always contain some error. For this reason, ANCOVA should not be used unless there is no other way to design the study. Random assignment to groups or groups matched on the covariate are the preferred solutions. ANCOVA should only be used as a last resort when random assignment or matching is not possible."

William J. Vincent – Statistics in Kinesiology (3rd Edition)

I believe that ANCOVA will not help you determine what you want, post test.

Please read the following article about this topic: Article Different ways to estimate treatment effects in randomised c...

Dear Iker,

I recommend you to apply Mixed model methods, where the relevant it is not the statistical difference at point (or time) that you assessing the variable...in this statistical analysis, the aim is about the tendency of the phenomenon. (In this study we used this method...10.1007/s00421-018-3958-0).

Best regards!!

Dear all,

Looking at these various responses to dr Amano's 2012 question, I am sorry to see several misconceptions, all of these due to attempts to understand math models without equations. Let me try to clarify things with one simple equation and two references to non- or mildly technical papers by myself.

First of all, in a randomized experiment/RCT, all methods mentioned in the responses give unbiased treatment effect estimation provided there is no selective drop-out/missingness or noncompliance (the effects of the latter two phenomena are way too complex to deal with here, and require study of causal inference and missing data literature).

Ancova, change and posttest-only analysis all estimate the treatment effect as follows: posttest group difference minus beta*(pretest group difference).

In posttest only, beta=0, in change beta=1, and in ancova beta is estimated such that the residual variance is minimized, thereby maximizing power and precision.

Due to the randomized assignment, the pretest group difference is zero in expectation, deviating from zero in a single trial only due chance/sampling error.

This means that all three methods give the same treatment effect in expectation (i.e. across an infinite nr of replications). The fact that methods give different results in a single trial does not mean that one or the other method is biased.

Bias is defined as a nonzero difference between the parameter to be estimated (treatment effect) on the one hand, and the expected value of the treatment effect estimator (e.g. of the group difference in change, or of the covariate adjusted posttest difference) on the other hand. That expected value is the value to be obtained on the average if we can replicate the trial an infinite nr of times and then average the values from all replications. So a difference between the true (simulated) treatment effect and its estimate in a single trial is no proof for bias (except if the difference is stat significant, and even then only if we are willing to ignore the risk of a type I error).

Next to matching:

Ancova is nothing else than a math sophisticated way to match on the covariate. It is more powerful than matching (where you either have to leave out persons without match in the other group, or have to categorize the pretest values). But this gain in power comes at the price of making two assumptions: linearity of the covariate effect, and absence of group by covariate interaction. Both can be tested though. if the assumptions are satisfied, then ancova is no more biased than matching, and so matching is no solution to cases where ancova is biased.

A graphical demo of the relation between matching and ancova, and of their both being biased in a comparison of natural groups (as opposed to a RCT) is found in my paper in J Clin Epid 2006, figure 2 and text around it.

Now to repeated measures anova:

Analysing the data of a pre-post control group design with rep meas anova and looking at the group*time interaction, is 100% equivalent to comparing groups on change from baseline, see test statistic (F = t-square) and p-value. But change gives more than just F or t, and p: it gives an effect estimate (mean group difference in change) with confidence interval, which rep meas anova does not (for the interaction).

Now to mixed models:

Running a mixed model with REML estimation, with unstructured 2*2 residual covariance matrix (so not e.g. compound symmetry or random intercept with homogeneous residual variance !), and with fixed effects of group, time, group*time, the interaction effect is again equivalent to comparing groups re change from baseline.

Further, if you run the mixed model with time coded as 0=pre, 1=post, then the group effect is the simple effect of group (so group difference) at pretest, which in a RCT is zero apart from chance deviation. Leaving out that group effect from the model, while keeping time and group*time in, the group*time effect is almost 100% equivalent to ancova of posttest with pretest as covariate. This also points to a kind of hidden ancova assumption: absence of a true pretest group difference. For this reason, ancova is not to be recommended for comparing natural groups that existed prior to your study.

Now to GEE for those who have heard of that: it is nearly equivalent to the mixed model if the dependent variable is quantitative, and you use the identity link, and you allow the covariance matrix to be unstructured in GEE as well as in mixed.

Almost all of the above theory, and more (a.o. math proofs and empirical and graphical demos) can be found in my papers in J Clin Epid 2006 (non-tech) and Multivariate Behav Res 2013 (mildly tech).

I appreciate the efforts of all to answer the question of which method to use, but please, remember that statistics is a subdomain of mathematics. All attempts to understand, or discuss about, statistical methods without proper training in the math, are doomed to lead to recurrent and persistent misconceptions and misuses of statistics by empirical scientists. Relatedly, in general do not trust statistical tutorials that are published in non-statistical journals without any math in them, as the risk of the authors and reviewers not being knowledgeable is considerable. Or can we write sound tutorials on, say, biochemistry, in a journal of, say, sports medicine or public health, without any biochem formulae ?

kind regards,

Gerard

Multivariate analysis of variance

DOI:10.4324/9781003179757-16

In book: Advanced Statistics for Physical and Occupational Therapy

Along with ANOVA, DUNCAN test may give good results