ANCOVA Flashcards
what dos ANCOVA stand for?
analysis of covariance
what is the broad idea?
to analyse between group differences invariance (such as with ANOVA) - but after the effect from the variance from a covariate has been estimated and removed from the DV.
In this way, the DV variance gets smaller, and the comparison between the (theoretically untouched) IVs is comparatively larger, creating more power/effect.
what was the take home message from the lecture?
These techniques are often wrongly used in research with nonrandom assignment to groups.
What happens with non-random assignment of groups?
Variance is not random and overlapping variance between the covariant factor and factor of interest compromises the stats and becomes meaningless.
what are the three main purposes of ANCOVA?
1) To increase power by reducing error term in experimental work
2) To adjust for mismatch on some nuisance variable in non-experimental
3) follow up to MANOVA (not covered step-down analysis)
what is the issue with adjusting for mismatch on some nuisance variable?
Comparing across groups by shifting group means in line with another ‘nuisance’ variable mean held constant
Removal of variance from a covariate that is inextricably bound with the independent variable of interest would render the results of ANCOVA meaningless.
E.g LORDS PARADOX = comparing boys and girls for weight change over the course of a year. The boys weigh more than the girls at the start and end, and neither groups average weight changes over time. Did diet affect boys n girls differently throughout the year? by estimating and removing the effect of weight from the beginning of the year (covariate), one is adjusting all samples to the same mean weight (heavy boys regress toward the mean + light girls move toward to heavier mean) - so the ANCOVA analysis is meaningless, as OF COURSE boys get heavier as a function of weight, as they, in reality, were heavier at beginning and end.
ANCOVA is equivalent to …
multiple regression (with categorical predictors)
A potential covariate is any variable that is significantly
correlated with the outcome variable, DV (if not it’s just noise = no relation – so you start to put things in that you know etc, bit of swindling)
We assume a xxx relationship between the covariate (x) and the DV (y)
linear r (it’s a general linear model GLM
ANCOVA removes the xxxx variation, explained by the xxxxx
ANCOVA removes the portion of the DV variation, explained by the covariate
ANCOVA thus increases xxxx xxxx to assess the effects of the group factor(s).
and thus increases statistical power to assess the effects of the group factor(s).
The MEAN is adjusted by using …
regression slopes
Ð We make a slope toward the DV for every IV.
Ð We then slide the mean up-or-down for each IV group toward the set mean of the covariate
Ð In this way, all are compared at a set point
technical
ANCOVA uses the regression line between IVs on each subject and calculating each predicted score on the DV based on just the covariate, as if they had scored at the mean of ALL participants on the covariate. Everyone’s score on the DV is adjusted AS IF they all scored at the mean on the covariate.
ANCOVA addresses the same questions about IVs that ANOVA does ……
main and interaction effects, specific comparisons and contrasts
The effects of IVs are assessed holding covariates …
constant (i.e., treating each subject as if they scored at the overall mean for the covariate)
what are the usual ANOVA ASSUMPTIONS
Absence of outliers
Homogeneity of variance
what are the usual MR ASSUMPTIONS
USUAL MR ASSUMPTIONS
Multicollinearity and singularity
Linear relationships
what are the 3 ANCOVA SPECIFIC ASSUMPTIONS?
1) The covariate needs to be independent of treatment (not correlated with any IV in a meaningful way)
2) Homogeneity of regression slopes (HOR)
3) The covariate is measured with as little error as possible (reliable covariates e.g. if you measured IQ really badly – like with one question 2x2)
what are the 4 Minimum Preliminary Data Checks?
check histogram for Y and each of the Xc
and homogeneity of variance
Ð CV was measured before the onset of treatment
Ð CV was measured reliably
why check histogram?
¬ All distributions should be approximately normal
¬ No extreme outliers
how check homogeneity of variance ?
scatter plots Should be approximately linear
What is Homogeneity of regression slopes?
Homogeneity of regression slopes (HOR) for each group
– relationship between the DV and the covariate has to be same for each group. This is because the ANCOVA takes the overall slope (e.g. slopes in the pics provided are all parallel) and applies to all IVs.
how do we test Homogeneity of regression slopes?
Ð Include covariate-by-IV interaction term(s) in the model, as well as main effects
Ð If these interactions are significant then there is heterogeneity of regression and ANCOVA is inappropriate
the more covariates you add in…. the
more you approach diminishing return
what does diminishing return mean?
Ð Ideal is small number of orthogonal covariates, each correlated with the DV
Ð This gives maximum adjustment of the DV for minimum reduction in df for the error term (each covariate reduces error df by 1) – every covariate put in takes 1df away from error term – so harder to become significant, diminishing return as critical F ratio gets larger. Rule of thumb never more than 3.
what is the non-random assignment issue again?
With nonrandom assignment (common in psychology) covariate differences across groups may reflect meaningful substantive differences related to group membership
what is the TAKE HOME MESSAGE for MAKing A STATISTIC ‘CORRECTION’ FOR NUISANCE DIFFERENCES BETWEEN YOUR NON-RANDOMLY ASSIGNED GROUPS ?
IT DOESN’T WORK, AND CONCEPTUALLY IT DOES THE WRONG THING.
COVARIATION BETWEEN GROUPS MAY WELL REFLECT INTRINISC DIFFERENCES BETWEEN THE GROUPS = WEAKENING EFFECT, BUT EVEN WHEN NOT IT INTRINCIALLY CHANGES THE NATURE OF THE GROUP
