Psych 300A: Midterm 2 Review (Regression)

Regression

The straight line that fits the data best, can be expressed mathematically

What does the regression line let us do with respect to X and Y

We can predict unknown Y scores from given X scores (assuming scores fall in data range)

How is regression calculated

Y’ = a_y + b_yx (sometimes Ŷ is also used)

Best fit line

Line that makes predictions about people’s beliefs that are as close to the true scores as possible

Error

Difference between a person’s predicted score and the person’s actual score on the criterion variable

What is another name for the regression line

Least squares regression, because it mathematically minimizes errors associated with trying to predict info

How do we know if a line minimizes errors

We take the sum of squared errors ( ∑(Y - Y’)² ), which is a minimum on the best fitting line

b_y

Slope, constant value that dictates the proportional change in Y given a value of X

How is b_y calculated

b_y = r(SDy/SDx)

a_y

Y intercept, when predicting Y it is the point where the regression line intercepts with the Y-axis

How is a_y calculated

a_y = Ӯ - b_y(X̄) 

T or F: If X̄ = 0, then Y’ = a_y

T

What leads to a flatter slope

Higher variability or heteroscedasticity

What 2 values are used to plot a regression line

1. Y intercept (0, ay)

2. The mean of X and mean of Y (X̄, Ӯ)

(Values need to be in range of data)

T or F: The regression line represents the mean for bivariate data

T, the regression lines for Y’ and X’ intersect at the mean for X and Y

What three datapoints are needed to plot Y’ and X’

Y’ - (0, a_y) and (X̄, Ӯ)

X’ - (a_x, 0) and (X̄, Ӯ)

What happens to the angle of the two regression lines if the correlation is very high

The angle will be very small, if r = ± 1.00 then they overlap, if r = 0 they are at a 90 degree angle (angle increases approaching zero)

What happens to Y’ and X’ respectively if r = 0

Y’ → b_y = 0, a_y = Ӯ and the line is flat

X’ → b_x = 0, a_x = X̄ and the line is vertical

What are the equations if the relationship is curvilinear (quadratic, cubic or quartic)

Quadratic: Y′ = a + bX + cX²

Cubic: Y′ = a + bX + cX² + dX³

Quartic: Y′ = a + bX + cX² + dX³ + eX⁴

What is Y - Y’

Measure of variability, it is the residual/error or difference between an observed Y and a predicted Y on a regression line. Measures error or prediction around the regression line

Standard error of estimate

A measure of the average deviation of the errors, the difference between the -values predicted by the multiple regression model and the -values in the sample

How is deviation score calculated for SD and SDy-y’

SD = (x - x̄)

SDy’ = (Y - Y’)

How are squared deviations calculated for SD and SDy-y’

SD = (x - x̄)²

SDy’ = (Y - Y’)²

How is sum of squared deviations calculated for SD and SDy-y’

SS = ∑(x −x̄)²

SSy-y’ = ∑(Y − Y’)²

How are SD calculated for SD and SDy-y’

SD = √(∑(x −x̄)²/N)

SDy-y’ = √(∑(Y − Y’)²/N)

T or F: there is an alternate way to calculate SDy’ 

T, SDy’ = SDy√(1-r²)

T or F: when r is not equal to 0, SDy’ will be smaller than SDy

T, because SDy’ reduces the potential error by using information from 2 sources instead of 1

What would happen to SDy’ if r = 0

SDy’ = SDy (because SDy’ = SDy√1-r2)

What would happen to SDy’ if there is a perfect positive or negative correlation (r = +- 1) 

SDy’ = 0 (because SDy’ = SDy√1-r2)

What do we need to know in order to understand explained variability

Total and unexplained variability

Total variability

Denoted with X_O, Y - ȳ

Unexplained variability

Denoted with Xt, Y - Y’

Explained variability

Denoted with Xe, Y’ - ȳ

How can one conceptually explain explained variability

Total variability = prediction + residuals

In what 2 ways can total variability be expressed

SS_T

∑(Y - ȳ)²

In what 2 ways can explained variability be expressed

SS_R

∑(Y’ - ȳ)²

In what 2 ways can unexplained variability be expressed

SS_E

∑(Y −Y’)²

Why must the values be squared to calculate total variability (e.g why can we not just do ∑(Y - ȳ) = ∑(ȳ – Y’) + ∑(Y-Y’)

Because the sum of deviations is 0, this is why we must us SS in our regression equation

What is the problem regarding SS_R and SS_E

We have trouble interpreting SS_R and SS_E because they are squares

What is the solution to the problem regarding SS_R and SS_E

We calculate the proportion of variability with regard to explained and unexplained variability

How is proportion of variability used to calculate total variability

Conceptual: Total variability = proportion of explained variability + proportion of unexplained variability

Equation: SS_T/SS_T = SS_R/SS_T + SS_E/SS_T

What is the important implication regarding proportion of explained variability and proportion of unexplained variability and their relationship with r

Explained variability = r² and unexplained variability = 1 - r²

What are the 9 attributes of the regression line

1. Regression line represented bivariate data in a linear relationship and predicts scores based on observed data

2. Defined by linear equation for a straight line

3. Two regression lines can represent bivariate data (Y’ and X’)

4. Does not predict values outside of the range of data

5. Is the best descriptor of bivariate data

6. Reflects the method of least squares (e.g. Σ(Y - Y’)2)

7. Always has some error of prediction present and is measured as standard error of estimate SDy - y’ (unless r = ± 1) 

8. Is a traveling normal distribution with a moving mean

9. Allows separate measures of SST, SSR and SSE

What are the two linear equations for Y’ and X’

Y’ = ay + byX

X’ = ax + bxY