Lecture 9 Flashcards Preview

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1
Q

What is the Regression equation used to for?

A

Used to describe the linear relationship between X and Y

2
Q

What is the one assumption of Regression?

A
  1. X and Y are linearly related.

The assumptions of Regression are NOT the same as ANOVA.

3
Q

Give the equation for Y’ (Yprime - i.e. Y predict)

A

Y’ = a + bx

4
Q

Which of X or Y is the dependent variable? Which is the independent variable?

A

Y is always the dependent variable.

X is always the independent variable.

5
Q

What if you have two dependent variables?

A

The score that you are trying to predict goes on the Y axis.

6
Q

Explain the “least squared line”

A

The method for determining a and b makes Σe² (i.e. the deviations) as small as possible. Therefore often called the “least squared line”.

7
Q

Give the equation for the regression line, and explain each component part.

A
Y' = a + bx where:
Y' = Ypredict
a = intercept (i.e. the value of Y when X = zero - a = Ybar - bXbar
b = Regression coefficient - the slope of the least squae line. b = σxy/σ²x = Covariance of x and y/variance of x --> = Σ[(x -Xbar)(y - Ybar)]/n-1 / Σ(x - Xbar)²/n-1 (the n-1's cancel each other out)  so we will end up with the slope of the line.
8
Q

Give more detail for b (the slope).

A

The numerator is telling us how much X and Y co vary, or the degree of statistical association.
The denominator adjusts the covariance (i.e. the rate of change information) so that it corresponds to a unit increase in X.
i.e. the slope indicates how many units Y increases for every one unit increase in X.

9
Q

How are ANOVA and Regression similar?

A

They are mathematically identical. Compare and contrast in notes.

10
Q

How do we calculate SS for a regression?

A

Conceptually, as in ANOVA, we sum the squared deviations.
- using computational formulas which don’t actually involve calculating the deviations. Calculating SS in regression is easier to track, if we exploit our knowledge of the Pearson r.

11
Q

How do we find SSreg?

A
r²xy = variability in y attributable to x/total variability in y
r²xy = Σ(Y' - Ybar)²/Σ(Y-Ybar)²
r²xy = SSreg/SSy

with algebra, we get:
SSreg = (r²xy)(SSY)

12
Q

Explain SSresidual i.e. SSres

A

If r²xy = variability in y attributable to x, then 1 - r²xy must = variability in y NOT attributable to x. i.e. residual variability
Therefore SSres = (1 - r²xy)(SSy)
This is the computational table.

13
Q

Explain the F table as it relates to Regression.

A

Once we calculate the degrees of freedom, we can use the F ratio to determine if the statistical association is due to chance.
dfreg = 1
dfres = n - 2

14
Q

Describe the equation for the F ratio.

A

F = (SSreg/dfreg)/(SSres/dfres) = (r²xy)(SSy)/dfreg/(1 - r²xy)(SSy)/dfres –> SSy in numerator and denominator cancel each other out.
important:
F = r²xy/dfreg/(1 - r²xy)/dfres

15
Q

Discuss “Test Significance of Slope”

A
  • Could be that the slope (b) we estimate from our data is just due to chance.
  • The significance test for slope is the F ratio, or the F test.
16
Q

Give the hypotheses for testing the significance of the slope (b)

A

H0: population b = 0 (i.e. slope of regression line = 0)
H1: population b =/= 0 (i.e. slope of regression line =/= 0)

17
Q

Describe testing the significance of r (correlation). Including hypotheses.

A
  • Testing the significance of r (the correlation) is equivalent to testing the significance of b. Therefore do one or the other, not both.
  • H0: population r = 0
  • H1: population r =/= 0
    Compare the obtained r (correlation) to rcrit.
  • rcrit is a tabled value (able B.6, yellow handout).
  • to enter table, need to know degrees of freedom for r (dfr) = n - 2 (where n = pairs of scores, i.e. subjects)
  • If robtained is greater than rcrit, reject H0
18
Q

What is AdjR²? (Adjusted r²)

A
  • Sample r or R fluctuates from sample to sample
  • And we square r or R to get r² or R²
    and R² can only be positive
    therefore all fluctuations of R² are in a positive direction. Therefore, R² is biased.
  • it tends to overestimate R
    so we make an adjustment.