ANCOVAPowerPoint PresentationSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23ANCOVA•ANCOVAphiles & GLMers & Regressionists•Workings of ANOVA & ANCOVA•ANCOVA, Semi-Partial correlations, statistical control•Using model plotting to think about ANCOVA & Statistical control•Homogeneity & Heterogeneity of Regression slopeIt’s all OLS GLM – “But be careful where you say that, friend!”Most of the statistical models (excluding some of the “nonparametric” ones) you know have been applied, advanced, improved, integrated, separated, named and renamed by a plethora of research areas, resulting in, well… a real mess…There are 3 (main) parts to this:•General Linear Model•Expressing all models as linear combinations of linearly weighted variables (including coded categorical, nonlinear & interactions terms, etc & their combinations)•Distributional assumptions & robustnesses•Multivariate normal distributions•Various “homogeneities” (variance, covariance, slope)•Defining “best fitting model” •OLS or “ordinary least squares” min ∑(y-y’)2There have been multiple attempts to reintegrate the “various named things” under a single central model…The nice folks over in “math stats” have long recognized that these are varieties and variations of a single math model.But, different research areas have “acquired” the models in different orders, for different reasons, from different sources, allowing different “acceptable variations” and, perhaps most importantly…. … calling them different things ... developing software to perform them that accepts different inputs, produces different outputs and labels things differently!Combine this with “market considerations” & a tendency not to change software but rather to add new things with new names whenever a competitor adds new things & you get the current mess….In psychology (& friends) there have been 2 “paths to GLM”Path #1  ANOVA & Enhanced ANOVAExperimentalists used ANOVA •Categorical IVs (mostly – but rem “trend analyses” for “parametric designs with quant IVs)•Always included main effect & interactions among IVsWith the increase in non-Experimental designs, there was an increased use of ANCOVA to provide statistical control•Categorical IVs & (usually) quantitative “Covariates” (confounds, controls, etc)•Always included main effects & interactions among IVs•Assumed (hoped for) homogeneity of regression slope – just the main effects of the covariates•Grudgingly tolerated interactions with and among covariates (“failure of regression homogeneity”)Path #2  Regression & Enhanced RegressionNonexperimentalists used Regression •Quantitative predictors (mostly – but figured out that binary predictors have the same interpretation)•Linear main effects models (steadfastly!!!!)•Unique contribution of each variable “controlling for others” Need Enhancements to allow inclusion other variable types & comparisons…•Multiple-category variables (coding)•Nonlinear terms ( X2 )•Interactions•Comparison of nested models (linear vs “embelishments”)Either “path” gets you to GLM•All the variable types•Interactions•Unique contributions controlling for other variables in modelSo, today we’re gonna talk about ANCOVA as… … ANOVA with “enhancements”Instead of asking…“What is the mean DV difference between the groups assuming the only difference between the groups is the IV?” We’re gonna ask…“What is the mean DV difference between the groups, holding the value of one or more covariates constant at specified values?”You know how ANOVA works• the total variation among a set of scores on a quantitative variable is separated into between groups and within groups variation• between groups variation reflects the extent of the bivariate relationship between the grouping variable and the quant variable -- systematic variance• within groups variation reflects the extent that variability in the quant scores is attributable to something other than the bivariate relationship -- unsystematic variance• F-ratio compares these two sources of variation, after taking into account the number of sources of variability • dfbg-- # groups - 1• dfwg -- # groups * (number in each group -1) • the larger the F, the greater the systematic bivariate relationshipANCOVA allows the inclusion of a 3rd source of variation into the F-formula (called the covariate) and changes the F-formulaLoosely speaking… BG variation attributed to IVANOVA Model F = ----------------------------------------------------- WG variation attributed to individual differences BG variation BG variation attributed to IV + attributed to COVANCOVA F = ----------------------------------------------------------------- WG variation attributed + WG variation attributed to individual differences to COVImagine an educational study that compares two types of spelling instruction. Students from 3rd, 4th and 5th graders are involved, leading to the following data. Control Grp Exper. Grp S1 3rd 75 S2 4th 81 S3 3rd 74 S4 4th 84 S5 4th 78 S6 5th 88 S7 4th 79 S8 5th 89Individual differences (compare those with same grade & grp)• compare Ss 1-3, 5-7, 2-4, 6-8Treatment (compare those with same grade & different grp) compare 5,7 to 2,4Notice that Grade is:• acting as a confound – will bias estimate of the treatment effect• acting to increase within-group variability – will increase error Grade (compare those with same group & different grade) compare 1,3 to 5,7 or 2,4 to 6,8ANOVA • ignores the covariate • attributes BG variation exclusively to the treatment • but BG variation actually combines Tx & covariate• attributes WG variation exclusively to individual differences• but WG variation actually combines ind difs & covariate• F-test of Tx effect “ain’t what it is supposed to be”ANCOVA• considers the covariate (a multivariate analysis)• separates BG variation into Tx and Cov• separates WG variation into individual differences and Cov• F-test of the TX effect while controlling for the Cov, using ind difs as the error term• F-test of the Cov effect while