Regression

Question 1

Regression is…

Answer 1

used to understand relationship between variables

Question 2

Independent variable (X)

Answer 2

Predictor or regressor

Question 3

Dependent variable (y)

Answer 3

Outcome or reponse

Question 4

Goal of regression

Answer 4

Predict changes in Y based on X

Question 5

Correlation vs regression

Answer 5

Correlation - measures the strength and direction of a linear relationship

Regression Predicts Y based on X

Shared variance (r^2); Proportion of Ys variance explained by X

Question 6

Simple linear Regression Equation

Answer 6

Y = B0 + B1 X

Question 7

B0

Answer 7

Intercept: value of y when x is 0

Question 8

B1

Answer 8

Slope: Change in Y per unit X

Question 9

e

Answer 9

Error: Difference between observed and predicted Y

Question 10

Regression using a SAMPLE of the populations

Answer 10

Sample estimates intercept and slope and predicted values of Y

Question 11

Predicted values of Y are…

Answer 11

points on the Regression line that corresponds to the given value of X

Question 12

Residuals (e’hat’) are…

Answer 12

distances between observed and predicted values of Y for corresponding X

Question 13

Equations for the Slope (B1)

Answer 13

Question 14

Equation for the Intercept (B0)

Answer 14

Question 15

Correlation equation

Answer 15

Question 16

What do you do with the r to find the proportion of shared variance?

Answer 16

rxy^2

Square it

Question 17

1-r^2xy is the…

Answer 17

Variance of Y independent of X

Question 18

Suppose we observe a high correlation between a child’s weight and their reading ability. This correlation is likely due to age, how can we combat the confounds?

Answer 18

We can control for the hypothesized influence of age on reading ability by removing the shared variance between age and weight

Question 19

Squared Multiple Correlation (R-squared) formula:

Answer 19

The SMC represents the proportion of variance in Y shared with (or “explained by”) the set of all X variables

Numerator: proportion of non-redundant variance in Y shared with X1 and X2

Question 20

Shared variance in Prediction

Answer 20

In two predictor regression, we are interested in imposing statistical control over X2 to test the unique effects of X1

Question 21

Goal of Multiple Regression

Answer 21

1. Evaluate the unique effect of X predictors on Y outcomes (holding constant other X)
2. Determine the incremental contribution of new X predictors to estimating variance in Y (in addition to X already in the model)
3. Determine the amount of variance explained in Y from a set of X predictors

Question 22

To determine Incremental contribution to the model we use…

Answer 22

squared semi partial correlation

Question 23

To determine variance explained in Y we use…

Answer 23

Squared multiple correlation

Question 24

Regression is a method of finding an equation to describe…

Answer 24

The line of best for a set of data

Question 25

How to define "best fitting" line when there are so many possibilities?

Answer 25

A line that is best fit for the actual data **minimizes prediction errors**

Question 26

Error of prediction is...

Answer 26

the distance each point is from the regression line (Y- Ŷ)

Question 27

Least-squared-error solution

Answer 27

Procedure that produces a line that minimizes the squared error of prediction

Question 28

Linear model with several predictors

Answer 28

The linear model can be expanded to include as many predictors as you like Expanded formula: 𝑌𝑖= (𝑏0+ 𝑏1 𝑋1𝑖+ 𝑏2 𝑋2𝑖 )+𝑒𝑖

Question 29

r can be thought of a standardized version of...

Answer 29

b (slope)

Question 30

Model Estimation

Answer 30

Total

Question 31

Residual Sums of Squares (SSr)

Answer 31

Gauge of how well a particular line fits the data

Question 32

Sums of Squares Regression (SSR)

Answer 32

Tells us how much error there is in the model but not wheter it is a better fit than nothing Need to compare our model against a baseline model Mean is a model of no relationship

Question 33

Sums of Squares Total (SST)

Answer 33

The differences between observed values and the values predicted by the mean

Question 34

Sums of Squares Model (SSM)

Answer 34

The difference between SST and SSR

Question 35

SSY and SST notation is

Answer 35

Sums of Squares total dfy = n-1 Consists of adding Sums of squares Regression df regression = 1 and Sums of Squares residual df residual = n-1

Question 36

if SSM is large the regression model...

Answer 36

is very different from the mean to predict the outcome variable This implies that the regression model has made a big improvement to how well the outcome variable can be predicted. If its small then using the regression model is better than using the mean

Question 37

Variance explained by the regression model (R^2) formula

Answer 37

Question 38

Mean Squares Regression Formula

Answer 38

Question 39

Mean Squares Residual Formula

Answer 39

Question 40

F Statistic Formula

Answer 40

Question 41

Assessing individual predictors

Answer 41

Question 42

Bivariate observations variable measurement scale

Answer 42

Interval

Question 43

Different notation for sample and population regression statistics

Answer 43

Question 44

A test of (rho=0)

Answer 44

If Rho=0 then the sampling distribution of r is almost normal with an expected value of rho and an estimated standard error of (Sr), given below (where n is the number of bivariate 'pairs' of observations...

Question 45

To test the hypothesis of p Formula

Answer 45

Df = n-2

Question 46

Example of

Answer 46

Question 47

Confidence Interval on r (Formula)

Answer 47

Question 48

Confidence Interval on r (example)

Answer 48

Question 50

Residual sums of squares (SSR)

Answer 50

Gauge how well the particular line fits the data

Question 51

Sums of Squares Total (SST)

Answer 51

The differences between the observed values and the values predicted by the mean

Question 52

Sums of Squares Model (SSM)

Answer 52

The difference between SST and SSR