The Linear Model (L6)

Question 1

Describe the equation for the linear model.

Answer 1

y = b0 + b1X1 + error

b1; parameter for the predictor, telling us the direction/strength of relationship (gradient)

b0; value of the outcome when predictor(s) = 0 (intercept)

Question 2

Describe what a graph with the same b0 but different b1 would look like.

Answer 2

Same intercept, different gradient.

Question 3

Describe what a graph with a different b0 but same b1 would look like.

Answer 3

Same gradient, different interept (lines run parallel to each other).

Question 4

What does it mean if the line of best fit is close to many data points?

Answer 4

The model is a good fit.

Question 5

What does it mean if the line of best fit is not close to many data points?

Answer 5

The model is a bad fit.

Question 6

What is the regression plane?

Answer 6

A 3-D line (square) that rotates around b0 depending on the value of b1.

Question 7

What is mulitple regression, and what is it’s equation?

Answer 7

y = b0 + b1X1 + b2X2 (…) + error

Same regression, but you add multiple predictors (X), each of which have a parameter (b).

Question 8

How does SPSS estimate parameter values, and where do you find them in the output?

Answer 8

Uses method of least squares; b1 is labelled as the parameter name, b0 is labelled as constant, both found in “B” column.

Question 9

What do significance tests do, in relation to b1 values?

Answer 9

Assess whether the predictor contributes to the model; ie., is the model significantly different from 0 (no effect).

Question 10

What do confidence intervals do, in relation to b1 values?

Answer 10

Assess the range of values b1 is likely to fall in, within the population.

Question 11

What are effect sizes?

Answer 11

Statistic which quantifies the relationship between variables.

Question 12

What are significance tests and CIs dependent on?

Answer 12

1) Sampling distribution normality
2) Homodescadicity.
3) Independe nt observations.

Question 13

When are parameter estimates optimal?

Answer 13

1) Residuals are normal

2) Homodescadicity

Question 14

What is bootstrapping?

Answer 14

The empirical estimation of CIs and variation of sample (SD). ‘An empirical guesstimate’. Used with small samples when you cannot assume normality.

Question 15

How is bootstrapping carried out?

Answer 15

1) A ‘bootstrap sample’ is calculated by randomly picking scores from data, which are then returned to the main sample so they could be chosen again. The bootstrap sample has the same number of data as original sample (eg. N=25 in both sample)
2) Mean estimated for bootstrap sample
3) Second bootstrap sample created, and mean estimated.
4) Steps 1-3 repeated 1000 times.

Question 16

What p-value do you use when bootstrapping?

Answer 16

SPSS produces a new ‘bootstrap output’ with a new p-value, therefore this value should be used when reporting your data.

Question 17

What is the total sum of squares (SST)?

Answer 17

Tells us the total variability of X from the mean. Calculated by adding up squared errors.

Question 18

What is the residual sum of squares (SSR)?

Answer 18

Tells us the residual/error variability and therefore the fit of the model in total and the error in the model.

Question 19

What is the model sum of squares (SSM)?

Answer 19

Tells us about the model variability, therefore if a model is better than no model (mean vs. model). Tells us through assessing the difference between data point and two lines (model and no model).

Question 20

How do you test the fit?

Answer 20

ANOVA or F-Test; compares the model to the error in the model using the average sum of squares;

F = MSm / MSr.

Question 21

What is R2 (R squared)?

Answer 21

Uses the sum of squares directly to express SSm as a portion of total variability (how much variability predictors account for)

R2 = SSm / SSt.

Question 22

What is adjusted R2?

Answer 22

The estimate of R2 in population (how the model generalizes to the world)

Question 23

What is shrinkage?

Answer 23

Difference between R2 and adj. R2;

R2 is smaller = sample isn’t representative.

Question 24

What is the predicted value?

Answer 24

The value from a solved equation.

Question 25

How do you calculate the t-statistic?

Answer 25

t = Bobserved / SEb.