Lecture 4_MLR, sr^2, pr, Flashcards
Multiple Linear Regression with 2 predictors, Squared semipartial correlation, partial correlation, and residuals
What are the building blocks of
multiple linear regression (MLR)?
Correlation coefficients (r)
The Regression Line (2 predictors)
Y’ = a + b1X1 + b2X2
• Y′ is the predicted value of the DV, Y.
• a is the intercept of the regression line.
• b1 and b2 are the partial regression coefficients for the predictor variables X1 and X2, respectively
What is Y’ in the regression equation for 2 predictors?
the predicted value of the DV, Y.
What are b1 and b2 in the regression equation for 2 predictors?
the Partial regression coefficients for the predictor variables X1 and X2, respectively
In a path diagram, what does the e path represent?
(1 - R²) the coefficient of non-determination
• error not explained by the model
In a path diagram, what does the curved, double-headed arrow between two predictors represent?
the bivariate correlation between the 2 variables.
• indicates that the researcher is not going to explain the correlation between X1 & X2, but acknowledges that it is not zero.
In a path diagram, what do squares represent?
measured variables
In a path diagram, what do circles represent?
Factors (latent variables hypothesized by “fun” math)
In a path diagram, what do the straight single-headed arrows represent?
the standardized coefficient (β) between the particular predictor variable (IV) and the outcome variable (DV)
A path model must have …
a coefficient for every arrow
What is the interpretation for the partial regression coefficient?
the expected change in the outcome variable (DV) when the predictor variable (IV) changes by 1 unit, controlling for all other predictor variables (holding all other IVs constant)
Why can we not compare the partial regression coefficients (b) to determine the relative importance of each predictor variable (IV)?
the size of b is influenced by the scale/metric in which the IV is measured, and may be different for each IV
What do the standardized regression coefficients (β) indicate?
the relative influence of the IVs in the equation.
• the expected change in the DV (in st. dev. units) when the IV changes by one st. dev. unit, holding all other IVs constant
Which is better for comparing across groups, b or β? Why?
comparisons across groups should be based on bs and not βs.
• βs are population specific; they are sensitive to fluctuations in variances and covariances across populations
What does the Squared multiple correlation (R²) represent?
the proportion of variance in the DV accounted for by all the IVs
What does the Standard Error of the Estimate express?
the amount, on average, that the predicted value of Y will deviate from the observed value (Obtained from R² and the standard deviation of the DV)
When will R² be equal to the sum of the squared bivariate correlations?
When the 2 predictors are not correlated with one another (r12 = 0)
What does the squared semipartial (part) correlation relate?
- how much R² will increase when a predictor is added last to the model
- the unique contribution of a predictor to the variance explained by the model
When predictors are correlated with each other, what will give us the unique proportion of variance in Y that is attributable to each predictor in the model?
Squaring the semipartial correlations
How can we calculate an estimate of the explained variance that is ambiguous?
Take the difference between R² and the sum of all semipartial correlations
Given the correlation matrix, means, and standard deviations of the variables, what can we calculate?
the regression coefficients (standardized and unstandardized), R² and the St. Error of the Estimate
What is the notation for a zero-oreder (Pearson) correlation?
r12, rY1, rY2, etc
What is the notation for a partial correlation?
prY1 = rY1.23 prY2 = rY2.1 pr12 = r12.Y34
What is the notation for a Semipartial correlation?
srY1 = rY(1.23) srY2 = rY(2.1) sr12 = r1(2.Y34)
What does the order (zero-order, first-order, second order, etc.) of a correlation coefficient (r, sr, pr) indicate?
the number of variables that have been “partialed” out
• rY1 = zero-order
• rY2.1 = first-order [also rY(2.1)]
• rY1.23 = second-order
What is a partial correlation (pr)?
the correlation between two variables with the influence of one (or more) other variables removed from Both.
What does the squared partial correlation between the DV and an IV express?
unique variance of the IV as a proportion of previously unexplained variance in the DV
For a given set of variables in a MR, which will be larger (in absolute value): the partial correlation or the semipartial correlation?
the partial correlation (think about the difference in the denominators)
sr = a/(a + b + c + d)
pr = a/d
What are 2 interpretations for e?
- the error in prediction.
* Or a transformed version of Y with the influence of the predictor(s) removed, or partialed out.
What is an alternative method for a partial correlation using SPSS?
Run 2 regressions:
- regress one variable, IV(1), onto other IVs whose effects want removed and save residuals as a new variable (new residual variable is a transformation of the original variable with the influences of the other two removed
- repeat for 2nd variable, DV (dido)
- correlate new IV(1) and DV residual variables
What does the correlation between the two residual variables tell us?
the partial correlation between IV(1) & DV when the influences of other IV have been removed from both of them.
What are 2 measures of how well regression model fits our data?
- R²
* size of the SE of the estimate
What are the units of the SE of the estimate?
the units in which the DV is measured
