Week 7 - SMR/HMR Flashcards
• List the two types of tests in multiple regression (MR)
Standard
Hierarchical
• Explain the different questions that are addressed in bivariate regression (x1)
o Does the predictor account for significant variance in the DV?
• Explain the different questions that are addressed in multiple regression (x5)
Extending on bivariate regression
Do predictors jointly account for significant variance in the DV?
• F test of Model R2
For each IV: does it uniquely account for variance in the DV?
• t-test of β for each IV
• Explain (in words) what R2 represents in each of the following cases:
o Bivariate regression
o Multiple regression with uncorrelated predictors
o Multiple regression with correlated predictors
(x2)
• Strength of overall relationship between criterion and predictor/s
Variance accounted for by the IV/set of IVs
What is r2 in bivariate regression? (x3)
coefficient of determination -
Proportion of variance in one variable that is explained by variance in another
(like eta2 in ANOVA - SSeffect/SStotal)
What is error variance in bivariate regression? (x2)
1 -r2
SSresidual/SSy
• Conceptually speaking (i.e. in words), how is shared variance determined in:
o Multiple regression with uncorrelated predictors
(x1)
There isn’t any,
So R2 is just r2 for each IV/predictor added together
• Conceptually speaking (i.e. in words), how is shared variance determined in:
o Multiple regression with correlated predictors (x3)
It’s the overlap of the predictors with each other
So you have to account for it - so it doesn’t get used twice in calculating R2 (which measures non-redundant variance)
(can’t just add, as you do for uncorrelated)
• Explain the difference between a zero-order (Pearson’s) correlation, a partial correlation and a semi-partial correlation (x1, x1, x1)
Zero-order - only 1 source of variance in DV, so none shared
Partial: relationship between predictor 1 and the criterion, with the variance shared with predictor 2 partialled out of BOTH dv and iv
Semi-partial: relationship between predictor 1 and criterion, after partialling out of predictor 1 variance shared between predictor 1 and 2
• In a Venn diagram representing one criterion and three predictors, indicate how shared variance between predictors 1 and 2 would be represented, and how the unique variance of predictor 3 would be represented. (x2)
1 and 2 would overlap with each other, and the DV
While 3 would only overlap the DV
• List the 4 key differences between the structure of ANOVA tests and MR tests
No test of overall model in ANOVA - auto in MR
Main effect of IV in ANOVA disregards effect of other variables - MR tests unique variance in each (controls for other variables)
Interactions auto in ANOVA - hard in MR (need MMR)
ANOVA = Fs, effect sizes for IVs/interaction, plus follow-ups
*MR = Rs F-test, beta t-tests, plus follow-ups
What is semi-partial correlation squared (sr2)? (x3)
proportion of variance in DV uniquely accounted for by IV1, out of total
- A/ A + B + C + D
- (where C and D are shared IV2/DV variance)
What is partial correlations squared (pr2)? (x3)
Proportion of residual variability in DV accounted for by IV1, after IV2 variance removed
- A/A + B
- (where a is unique, b is unaccounted for DV variance)
• Explain the difference between the semi-partial correlation squared (spr2/sr2) and the partial correlation squared (pr2)
Partial is like partial eta2 - leftover variability in DV that IV accounts for
Semi-Partial is like eta2 - bit of total DV variability that uniquely due to IV
*the go to effect size for regression
• Identify the linear model for a multiple regression analysis with 2 predictors (x2),
And explain what b1, b2, and a represent
Ŷ = b1X1 + b2X2 + a
Ŷ - predicted score is still a function of slopes, X scores and constant
b1 - slope of plane relative to x1-axis
b2 - slope of plane relative to x2-axis
a - the constant (y-intercept, when x1 and x2 = 0)
• Explain why means and standard deviations are not as critical for interpreting MR results as they are for t-tests and ANOVAs
Because you’re not interpreting/comparing means, but the direction relationship between variables
Although SD still tells us change in DV for every SD change in it’s IV
• Explain what Cronbach’s is (x2), and what values of this index we would ideally like to have (x2)
index of internal consistency (reliability) for a continuous scale
*how well items “hang together”
Use scales with high reliability ( > .70) if available – less error variance
• Define validities (x1), and identify their ideal levels (i.e. high or low) (x1)
Relationship between IV and DV
High - show a strong association, yay!
• Define collinearities (x1), and identify their ideal levels (i.e. high or low) (x1)
Relationships between IVs
Low, because the higher they are, more chance of redundant IV
• Explain the principle of parsimony (x2)
The simplest explanation for the data is good
ie predictors explain different variance in the model
How does parsimony relate to validities and collinearities (x2)
More parsimonious when high validity (predictor/criterion relationships)
And low collinearity (predictor relationships)
• Define R (x1) R2, (x1) Radjusted, and R2adjusted (x1) in the context of multiple regression
Multiple correlation coefficient (R) is bivariate correlation between criterion (Y) and best linear combination of the predictors (Ŷ)
R2 - square R to find variance in Y accounted for by composite (Ŷ)
The adjusted versions attempt to correct positive sample bias by correcting for sample size (so no point for 30+)
• Identify the test used to assess the overall variance explained (x1, plus explain x1), and explain what a significant result means (x1)
F-test: MSregression/MSerror
*variance accounted for (R2/p) divided by that not accounted for (error/N - p - 1)
The relationship between predictors (as a group) and criterion is different from zero
• If the sr values for the predictors are known, explain how to work out the unique variance (x1) explained by each predictor and the variance shared between all predictors (x1)
Square the semi-partial correlation to get the coefficient
Shared variance = R2 - sum of all sr2
• Explain the difference between a zero-order coefficient, a first-order coefficient, and a second-order coefficient (x3)
Zero-order - doesn’t take other predictors into account
First-order - 1 other taken into account
Second-order - 2 accounted for
• Identify the type of coefficient that SPSS provides in its output (x2)
Highest order possible -
Auto control of all other variables
• Identify the test used to assess predictor importance, and explain what a significant result means
Use a t-test - beta divided by its standard error
That the variable contributes significantly to prediction of DV, independently of other predictors
• Define a partial regression coefficient and it’s symbol/notation (x1, x1)
But… (x2)
Importance of the predictor, after correlation with other predictors has been adjusted for
b
Can’t use relative magnitude of b - scale-bound
*Importance of given b depends on unit/variability of measure
• Define a standardised regression coefficient and it’s symbol (x2, x1)
Rough estimate of relative contribution of predictors
*b divided by SD of the DV
beta
• Explain why standardised coefficients can be more useful than unstandardised coefficients (x1, but remember… x1)
Can compare samples within the study
but not across studies - different sample Sds
• Explain the key difference between standard multiple regression (SMR) and hierarchical multiple regression (HMR) (x2)
SMR - simultaneous entry of predictors, shared variance never ‘owned’
HMR - sequential, shared variance goes to previous block
• Explain how model R2 and b’s are assessed in SMR (x2), and at each step of HMR (x2)
SMR - b’s are evaluated for contribution beyond that of all others, R2 assessed in 1 step
HMR - b evaluated for unique contribution, controlling for other IVs in current and earlier steps, but not later ones
*R2 takes more than 1 step
• In HMR, explain the difference (if any) between R2 and R2change:
o at Step 1 of the analysis (i.e. Model 1)
o at Step 2 of the analysis (i.e. Model 2)
(x2)
R2 and R2change are same in step 1 - block and total accounting for same variance
At step 2, R2 becomes total of previous step’s R2, plus the R2 change between step 1 and 2
• Identify the two rationales for the order in which predictors may be entered in HMR (plus explain/eg)
.Partial out effect of control variable
• Like ANCOVA: predictor(s) at step 1 like covariate
Build sequential model according to theory
• e.g., personality measure entered at step 1, attitudinal measure entered at step 2
• Explain the link between results at Step 2 of HMR (e.g. where predictor A was entered at Step 1, and predictors B and C were entered at Step 2), and the results of an SMR (e.g. which includes predictors A, B, and C in the model) (x1)
R and R2 are the same as our full SMR
• Explain the similarities and differences in the structure of SMR and HMR (with 2 steps) tests
R2:
*At step 1, model fit of IVs added so far
*At step 2, same F/p-value as SMR
Coefficients:
*Step 1, beta/p-value of unique contribution at that step
*Step 2: same as you’d get in SMR (so step 1 beta/p-value drops)
• Explain what statistics need to be reported in SMR and HMR tests (x4, x5)
SMR: o Table of M, SD, r o Model R2 with F test o Each IV’s β with p values o Any relevant follow-ups
HMR:
o Table of M, SD, r
o Each block R2 change with F change test
o IVs’ βs with p-values from each block as entered
o Final model R2 with F test
o Any relevant follow-ups
• Explain what multicollinearity and singularity are, and why they are a problem (x1, x1, x2)
Multicollinearity when predictors are highly correlated
Singularity when their explaining the same thing
Instability of regression coefficients:
*More type 1 AND 2, and changeable by moving single scores around
• List the main assumptions re distribution of residuals in multiple regression analyses (x1, x2, x2, x2)
Conditional Y (y-hat) values normally distributed around regression line
Homoscedasticity: variance of Y values are constant across different values of Ŷ (homogeneity of variance)
• Similar DV variance at all levels of continuous IVs
No linear relationship between Ŷ and errors of prediction
• Error the same across different y-hat values
Independence of errors
• Can’t tell anything about error at one point from error on another – different people, different scores
• List the main assumptions re predictor/criterion scales/scores in multiple regression analyses (x1, x1, x2, x1)
Variables are normally distributed
Linear relationship between predictors and criterion
Predictors are not singular (extremely highly correlated)
• If predictors are highly related, form a singularity – Ivs collapsing in on each other
Measured using a continuous scale (interval or ratio)
What is the effect of correlated IVs on betas in MR?
bs and ßs (coefficients) measure unique contribution of each IV to variance left in DV after controlling for other IVs (i.e., they test the partial IV : DV relationship)
When IVs correlated, effects of each IV depend heavily on what other predictors are in model.
Variables can pop in/out of significance, even change directions (+ vs – ßs) when you add other variables.
*betas can be a lot stronger than the r, weaker, or even in opposite direction
Why do we have to plan carefully (based on theory) the order of entry in MR? (think effect of correlated IVs… ) (x2)
If you just mess around, the instability associated with correlated IVs can produce both Type 1 and Type 2 error
And you will not know which.
What is homoscedasticity? (x2)
What is the effect of violating this assumption of regression? (x1)
Homogeneity of variance of residuals
*residuals at each level of predictors should have same variance
Invalidates significance tests
