University:
California State University, Northridge
Course:
PSY 524/L | Multivariate Analysis Computer and Lab
Academic year:

2004
Views:

495

Pages:

20
Author:

Beeled9z

ANCOVA Lecture 9 Part 2 Normality of Sampling Distribution ► it is assumed that the sampling distribution of means is normal. ► This cannot be shown unless you take multiple samples and form sampling distribution. ► It is assumed normal when the error has degrees of freedom of 20 or more (central limit theorem) Homogeneity of Variance ► Equal variances on the DV across all of the levels of the IV(s) and the CV(s). σ ≅ σ ≅ Kσ 2 1 ► This 2 2 2 p is most important after adjustments have been made, but if you have it before adjustment you are likely to have it afterwards. Homogeneity of Variance ► If CV or IV fail this test a more stringent alpha can be used (.01) or drop the variable from the analysis. ► Tested by Levene’s test of equality of error variances, but this is a very conservative test so evaluate at probability greater than .001. If it fails at this level test with Fmax test (largest variance/smallest variance <= 10) Linearity ► is assumed that each CV has a linear relationship with the DV and other CVs Homogeneity of Regression ► the slope of the line predicting the DV from the CV should be the same for each level of the IV. ► In other words the regression coefficient (B) relating a CV to the DV should be the same for each group. ► In still other words, this means no IV by DV interaction Homogeneity of Regression Reliability of Covariates ► it is assumed that each CV is measured without error (this is unrealistic). ► So it is recommended that CVs only be used when they meet a reliability of .8 or more (not very realistic either). Equations ► Just like in ANOVA the total variance can be separated into within and between groups variance: 2 ( Y − GM ) = n Y − GM + ( Y − Y ) ∑∑ ij ∑( j ∑∑ ij j ( y) ( y) ) 2 2 i j j i j SSTotal ( y ) = SSbg ( y ) + SS wg ( y ) Equations ► But in ANCOVA you also have a partitioning of the variance in each CV: 2 ( Z − GM ) = n Z − GM + ( Z − Z ) ∑∑ ij ∑( j ∑∑ ij j (z) (z) ) 2 2 i j j i j SSTotal ( z ) = SSbg ( z ) + SS wg ( z ) Equations ► And also a partitioning of the covariation between them: SPTotal = SPbg + SPwg Equations ► This covariation is used to adjust the between and within groups sums of squares: SS ' bg ( y )  ( SP + SP )2 ( SP )2  bg wg wg   = SS bg ( y ) − −  SS bg ( z ) + SS wg ( z ) SS wg ( z )    SS ' wg ( y ) = SS wg ( y ) SP ) ( − 2 wg SS wg ( z ) Equations ► The adjustment made to the between group scores above can also be conceptualized as the adjustment is obtained by taking each individuals deviation around Y grand mean (before adjustment) and subtracting from it each persons deviation around the Z grand mean weighted by the relationship between the two: (Y − Y ') = (Y − GM ( y ) ) − β y⋅ z ( Z − GM ( z ) ) Equations ► Each regression coefficient we need to estimate (one for every CV) eats up a degree of freedom. This accounts for the smaller error degrees of freedom when compared to regular ANOVA. Equations ► Hand computation formulas are available on page 286 in the T and F book. ► An analysis of variance is then performed on the adjusted scores ► Any significant findings need to be followed up with either planned or post hoc comparisons. Equations ► Percent of variance accounted for can be found for cases with one IV: η = SS 2 ► where SS + SS SS + SS ' bg ' bg ' bg ' wg ' wg is the adjusted total Equations ► Can be found in factorial designs by: ' SSeffect 2 η = ' , where SStotal is the sum SSTotal of all main effect and interactions but not the SS for the CVs. ► or partial η = SS 2 SS ' effect ' effect + SS ' error Evaluating Covariates ► Each ANCOVA test also gives you a test of whether the covariate is doing an adequate job of adjusting the DV scores ► Usually in the form of an ANOVA table where significant F values for the CV indicate significant adjustment Alternatives to ANCOVA ► Test of difference scores If the CV and DV are the same but just collected at different times (e.g. pre/post design) the you are interested in change Find the difference between the two and use that as the DV in an ANOVA Alternatives to ANCOVA ► Block designs You use what could be a CV to match people Then run an with a matched blocks design.

ANCOVA. Normality of Sampling Distribution