4. F-Tests and Standardisation

Question 1

How do we test significance of overall model?

Answer 1

F-test

Question 2

What is an F-Test?

Answer 2

Process of testing the statistical significance of the test stat (F-ratio)

Question 3

What is not sufficient to test the significance of the overall model?

Answer 3

Testing individual predictors and R2

Question 4

What is the F-ratio?

Answer 4

Ratio of explained variance to unexplained variance

f = (SS model/ Df model) / (SS residual/ Df residual)

so

f = MS model / MS Residual

Tests the null hypothesis that all the regression slopes in a model are all zero so that our predictors tell us nothing about our outcome/don’t explain variance

Question 5

What are mean squares?

Answer 5

Mean squares are sums of squares calculations divided by the associated degrees of freedom.

Question 6

How is the null hypothesis explained in terms of F-test?

Answer 6

The null hypothesis for the model says that the best guess of any individuals y value is the mean of y plus error.

Or, that the x variables carry no information collectively about y.

i.e. the slopes all = 0

Question 7

What would the different results from a F-Test demonstrate?

Answer 7

Big F-ratio = ^ Model significance as more model variance than residual variance involved

F > 1 = ^ More model than residual

F = close to one if null is true

Question 8

How do you test if your F-test is significant?

Answer 8

1. Select alpha level
2. Calculate critical value of F
3. Compare value to critical value
4. F-ratio is evaluated against an F-distribution with df model and df residual & pre-defined alpha

If it is more extreme than critical value then we reject the null

Question 9

What is an f-distribution?

Answer 9

Test for equality of variances from two normal populations

Question 10

What are degrees of freedom?

Answer 10

Number of independent values associated with the different calculations

Df are typically the combination of sample size and the number of things you need to calculate/estimate.

Question 11

What are the three different types of degrees of freedom? (Name only)

Answer 11

Residual DF

Total DF

Model DF

Question 12

What are residual degrees of freedom?

Answer 12

Remaining dimensions that you could use to generate a new data set, that looks like current data set

n-k-1

SS residual calculation is based on our model, in which we estimate k β terms (-k) and an intercept (-1)

Question 13

What are total DF?

Answer 13

SS total calculation based on observed yi & mean of y

n-1

Question 14

What is model DF?

Answer 14

Number of parameters in model that are estimated from data = K

SS model is dependent on the slope (beta)

Question 15

What are unstandardized coefficients?

Answer 15

When the coefficients are in the same units they are as when the data was collected

It is useful when the units are meaningful

Question 16

What are standardized coefficients?

Answer 16

Coefficients that have been z-scored (dividing individual deviations by mean deviations)

The interpretation of the coefficients becomes the increase in y in standard deviation units for every standard deviation increase in x

Question 17

Why is standardization useful?

Answer 17

Useful for comparison if variables are on different scales

Useful if scales are arbitrary

Question 18

What happens to R2, F-test, T-test and B0 when coefficients are standardized?

Answer 18

R2, F test and T test stay the same

B0 = Zero when standardised

Question 19

Why should we be cautious in using standardization?

Answer 19

Just because you can put regression coefficients on a common metric doesn’t mean they can be meaningfully compared.

The SD is a poor measure of spread for skewed distributions, therefore, be cautious of their use with skewed variables

Question 20

What does standardization do to the correlation?

Answer 20

Standardized slope ( ^β∗1) = correlation coefficient (r) for a linear model with a single continuous predictor.

They are the same:

r is a standardized measure of linear association
^β∗1 is a standardized measure of the linear slope.
Something similar is true for linear models with multiple predictors.

Slopes are equivalent to the part correlation coefficient

Question 21

What is the value of the intercept when continuous variables are standardised?

Answer 21

0