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Multiple regression, factor analysis and ANCOVA
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- 1What are the assumptions of a multiple regression? How do we test them?DV measured continuously
Normality of raw data and residuals
Linear relationship between IV & DV
No multicollinearity - Tolerance (<.1) and VIF (>10)
Homoscedastic residuals
Linear relationship between residuals and Y
Orthogonal residuals - 2When should you perform a multiple regression?Principle of parsimony
N = 104 + k - 3What is a suppressor variable?Suppressor variables suppress irrelevant variance by their relationship to another IV, inflating R^2
If b and r have opposite signs, or b is significant but r is low, a suppressor variable might be at play - 4What is hierarchical MR?Predictors are added sequentially in a pre-specified order, evaluated as what they add to R^2 at the point of entry
There is no shared variance, as initial predictor(s) get their own portion; shared variance between predictors may reflect causality, so X2 variance 'belongs' to X1
Used to partial out nuisance variables
The contribution of X to R^2 is critically affected by order of entry - 5What is stepwise, mediated and moderated MR?Stepwise MR is also sequential, but based on computer-generated statistical criteria
Attempts to reach the highest R^2 with the fewest predictors, capitalising on chance and ignoring causal chains
Mediated MR tests the plausibility of a 'chain of prediction' where the shared variance between X + Y is explained by Z
The X -> Y relationship shrinks in the presence of Z
Use a Sobel test/95% CI to test for significant shrinkage
Moderated MR tests whether the X -> Y relationship varies as a function of X2
An interaction may contribute significantly to the prediction of Y (hierarchical MR)
Use standardised interaction terms to reduce collinearity
Analyse simple slopes at critical values of Z to interpret interaction
Aside from a median split, SEM is the only continuous alternative to interactions - 6Regression coefficients tell us the importance of individual predictors. What do different regression coefficients tell us?b coefficients show the slope of the regression line, and are weighted to satisfy the least squares criterion to minimise prediction errors
Partial ('b') regression coefficients reflect the change in Y per unit X *Scale-dependent so incomparable across predictors*
Standardised partial ('β') coefficients reflect SD change in Y per SD change in X *scale-independent* - 7What do different correlation coefficients tell us?Simple ('zero-order') correlation coefficients indicate the direct relationship between X -> Y *Useless in MR as it doesn't account for inter-correlations*
Partial correlation coefficients show the correlation between X1 and Y, controlling for X2; indicates proportion of criterion left unexplained by X1
Semi-partial ('part') correlation coefficients show the correlation between X1 and Y after removing X2 variance from X1, thereby indicating unique contribution of X1 as a portion of R^2 - 8What is the difference between FA & PCA?PCA measures all variance - including error - whereas FA only measures shared variance
PCA initial communalities = 1 whereas in FA = R^2
Underlying factors cause variation in FA; PCA data is reduced to components
FA appropriate to test hypotheses
Generally provide similar solutions - 9What is the difference between confirmatory and exploratory FA?Exploratory is outcome-driven, used to explore present factors and their importance
Confirmatory is theory-driven, used to estimate model fit by placing constraints on relationships between observed variables and latent factors - 10What are practical issues & assumptions associated with FA?Requires a large sample, but this depends on factorisability of the correlation matrix
300 is good, 500 is very good; N:variable ratio should be 10:1 or 5:1
Missing data could harm factorisability in small Ns
Normally distributed variables help derivation; multivariate normality is required
Outliers may distort correlations between variables
Multicollinearity doesn't apply to PCA, but could present a problem to initial communalities in FA (>.9) - 11How do we check for factorisability of a correlation matrix?At least some correlations must be >.3
Bartlett's test of sphericity can check but is notoriously sensitive to N
Anti-image correlation matrix shows negative partial correlations - off-diagonal values should be small
KMO measure of sampling adequacy should be >.6 (>.8 better) - 12What are initial and extracted communalities?Initial communalities are estimates of variance accounted for by all components/factors
Extracted communalities are estimates of variance accounted for by the factors extracted
High & low values problematic; low values could mean a variable is unexplained by factors - 13How do we extract factors?Eigenvalues = variance explained by each factor
Extract factors where eigenvalue > 1 or those above the elbow of a scree plot
Could use parallel analysis (larger eigenvalues than factor solution = random noise in the real data) - 14How do we interpret factor solutions?Factor rotation rotates factor axes so variables load maximally onto one factor
Orthogonal rotation keeps factors uncorrelated
Oblique rotation allows correlation of factors
Component loading matrix shows correlation of factor & variables - significant factor loading >.32
Best to use pattern matrix for oblique rotation as this partials out overlap - 15How do we define factors?Pattern of factor loadings define factors
Marker variables have very high factor loadings used to define factor - 16What are factor scores?Estimate score if factor had been measured directly
Better to sum scores on high-loading variables - 17What are the types of comparisons and error rates?A priori comparisons more liberal
Post hoc comparisons more conservative
Per-comparison error rate (PC)
Familywise error rate (FW)
*Most comparisons don't require significant omnibus effects*
A priori comparisons test a subset of comparisons, reducing Type I error, so more powerful - 18What are potential A priori comparisons?Multiple t-tests - use Welch test if heterogeneity of variance, although equal sample sizes will yield same results in homogenous samples
Linear contrasts - compare sets of means using weighted sum of means
Bonferroni t-tests - 19What is the difference between orthogonal and non-orthogonal contrasts?Orthogonal contrasts are independent of each other; they analyse non-overlapping variance
Non-orthogonal contrasts are not independent; their p-values may correlate so a more conservative α level should be used - 20What is the false discovery rate (FDR) correction?FDR controls the expected proportion of Type I errors among rejected H0
If R > 0 then FDR = V/R where V = Type 1 error and R = # of rejected H0
With many significant effects, concern becomes reduction of proportion of Type I error, rather than errors themselves - 21When should we use post hoc tests?Unplanned comparisons more powerful than Bonferroni t-tests when many hypotheses are tested
Trade off between loss of power vs. FW error rate - 22When should we use ANCOVA?Increase power by reducing error variance
Stepdown analyses for MANOVA
Adjust for mismatch on nuisance variables *don't do this!* - 23What is ANCOVA?The effect of a covariate on the DV is removed, decreasing error variance and increasing power
DV scores adjusted as if all Ss score equally on CVs
Cannot be used to adjust if CV & IV correlate, as some variance will be removed - 24What are ANCOVA assumptions?CV should have a linear relationship with DV
CV should be measured at baseline, or it could correlate with the change in DV, thereby removing part of the IV effect
No outliers (check histograms)
Homogeneity of variance (check Levene's)
No multicollinearity
No curvilinearity and homogeneity of regression slopes (check via scatterplot)
Small number (2-3) of orthogonal CVs; C < .1 x (N-J+1) where C = CVs and J = # of groups
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- Fichas sobre Sin título, creado por Peter Panayi el 20/01/2019.
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