Chapter 14: The General Linear Model

Question 1

What do the terms gradient and intercept, respectively, describe?

Answer 1

B₁ and B₀

Question 2

When plotting their values on a graph, what impact does each regression coefficient have on the overall line?

Answer 2

B₁ changes the lines slope.

B₀ changes the position of the line.

Question 3

Briefly explain what the method of least squares does.

Answer 3

The method of least squares calculates the value of your parameters, b, when the squared error (residual) between your model and data are smallest.

Question 4

How does cross-product deviation differ from the predictor sum of squares mathematically? What is each used to calculate?

Answer 4

Cross-Product Deviation: Between two variables

Used in the covariance formula (SCP/N-1)

Predictor Sum of Squares: Within a single variable.

Used in the variance formula (SS_x/N-1)

Both of these formulas are used in the b₁ formula.

Question 5

How do you calculate the regression coefficient, b₁?

Answer 5

1) b₁ = SCP/SSx
2) Another way to think about this formula is covariance over variance. Since covariance = (SCP/N-1) and variance = (SS_x/N-1), you can cancel each N-1 and be left with SCP/SS_x–the original formula.
3) And finally, a less intuitive explanation, you can think of this formula as a version of the correlation coefficient (seen below).

Question 6

What is the best way to conceptualize b₁ .

Answer 6

As an unstandardized relationship between predictor and outcome. In other words, for every unit that the predictor increases the outcome will increase by b₁ .

Question 7

How can you find the constant, b₀, mathematically?

Answer 7

You isolate the general linear model and find,

b₀ = Y - b₁ * X

Question 8

How can you calculate standardized beta?

Answer 8

By multiplying the regression coefficient, b₁ , by the standard deviation of the predictor variable over the standard deviation of the outcome variable.

Question 9

What are the significance levels for R² ?

Answer 9

Small = 0.02

Medium = 0.13

Large = 0.26

Question 10

How does hierarchical regression differ from step-wise regression?

Answer 10

Hierarchical regression systematically places variables in the model based upon a mutually agreed upon confidence from the community.

Step-wise regression places variables in the model based upon the power in that given study. Because of this, step-wise regression often varies between studies.

Question 11

What are some issues with step-wise regression?

Answer 11

1) Does not standardize order across experiments so this method makes comparison difficult.
2) Tends to over-fit (put too many variables) or under-fit (too few) the model.

Question 12

In simple words, what does Cook’s distance tell you? What values indicate potential issues?

Answer 12

Cook’s distance is the standardized total difference in predicted value when including or excluding any given case.

Simply put, a measure of outliers.

Values greater than 1 are worth further inspection.

Question 13

What assumptions are parameter values impacted by?

Answer 13

Parameter values function independently of assumptions.

Question 14

How can you calculate R² using only sum of squares values?

Answer 14

SS_M / SS_T

Question 15

With what formula can you estimate the standard square error in the model?

Answer 15

(Y_i-Y)² provides the squared error

Dividing by N-2 gets the mean error

Square rooting the results standardizes them

Question 16

How can you transfer the standardized square error into the parameter square error?

Answer 16

SE_b = SE_model/sqrt(SS_x)

Question 17

How can you calculate the test statistic for b and then use it in your confidence interval calculation? Here you must know both the test statistic and confidence interval formula.

Answer 17

Test Statistic = bobserved/​SE_b

Confidence Interval for b = Expected value +- (Critical Value * SE_b)

Question 18

What does Yuen’s procedure do?

Answer 18

Yuen’s procedure trims the outliers from a data set.

Question 19

What do credible and confidence intervals, respectively, say about the null hypothesis?

Answer 19

Credible intervals are a Bayesian construct and thus make no reference to the null hypothesis.

Confidence intervals were a red herring to bring you away from what was important here: the information about credible intervals.

Question 20

How do t-tests differ between independent and paired tests?

Answer 20

Independent tests are interested in contrasting group means.

Paired tests are interested in contrasting the difference scores between groups.

Question 21

What two tests can be used to correct for heterogeneity of variance? Which is stronger?

Answer 21

Welch’s F and Brown—Forsythe’s F.

Welch’s F possesses more statistical power

Question 22

What does the SS_M formula have outside of the brackets that makes it different from the SS_T formula? What does this variable represent?

Answer 22

An N value that is multiplied by the brackets.

N represents the number of people in the group.

Question 23

What is a HUGE red flag in the following chart?

Answer 23

The original model is not significant and thus our robust tests cannot be interpreted.

Question 24

What is the function for each of these post-hoc tests?

1) Gabriel’s procedure
2) Hochberg’s GT2
3) Games-Howell

Answer 24

1) Gabriel’s Procedure: With slightly different sample sizes use this test as it has function power.
2) Hochberg’s GT2: With very different sample sizes use this test.
3) Games-Howell: If the population variances are not equal use this. Note: we can never be certain that population variances are equal, so there is an argument to always this test.

Question 25

How does sum of square placement change in a repeated-measure design?

Answer 25

Both SS_M and SS_R can be found in the within-participant section. This is because the systematic variance and unsystematic variance occurs within the same individuals.

Question 26

What does sphericity refer to?

Answer 26

Sphericity: Refers to the equality of variances of the differences between treatment levels

Question 27

What are the two questions to ask about a model?

Answer 27

1) Is the model influenced by a small number of cases?

Asks a question of sample size.

2) Is the model generalizable?

Asks a question of assumptions being met.

Question 29

What standardized residual numbers are cause for concern?

Answer 29

If any standardized residuals are above 3.29

If 1% of standardized residuals are above 2.58

If 5% of standardized residuals are above 1.96

Question 30

What does an adjusted R² value refer to?

Answer 30

It is an estimate of what R² would be in the population.

Question 31

When applying a regression coefficient to dummy coding, what two formulas produce the same result?

Answer 31

SCP/SS_X and X-Y (Difference between groups)

Question 32

How do you go from S_D to SE_Dhat ?

Question 33

What are the two ways to calculate Cohen’s D for a paired sample t-test?

Answer 33

1) d_hat/sqrt(1-r)
2) (D_hat/S_D)*sqrt(2)

Mean difference score/Standard deviation difference score * SQRT(2)

Question 34

Why is it not useful to check and see if credible intervals contain

Question 35

In a factorial design, what makes up SS_M . What is the df for its constituent parts?

Answer 35

Each factor and their interactions.

The df for each individual factor is 1.

Question 36

How do you calculate r using only f-ratios and df?

Answer 36

sqrt(F_factor1/F_factor1*df)