L8 - Research Questions for Differences

Question 1

how is df of SSbetween calculated?

Answer 1

k-1

Question 2

How is df of SSwithin calculated?

Answer 2

n-k

Question 3

Whats the difference between running ANOVA through regression and on its own?

Answer 3

Different labels

In ANOVA, SSreg is labelled as SSbetween
and SS res is SS error

Question 4

What are planned comparisons?

Answer 4

This is a linear combination of means for 2 or more levels of a factor that is formulated and specified on A PRIORI grounds, in 1 way anova

Question 5

What is a linear combination of means?

Answer 5

This involves a weighted sum of the means of all levels of the between-subjects factor.

Question 6

What is a linear contrast?

Answer 6

When a planned comparison equals zero, then it is a linear contrast.

Question 7

What is each planned comparison analogous to?

Answer 7

It is analogous to an independent samples t test.

Linear contrast effectively reduces the k levels in the factor down to a comparison between 2 sets of linear combination of means.

Question 8

What is the main important statistical assumption in planned comparisons?

Answer 8

homogeneity of variances.

Question 9

What are the two ways we can assess the null hypothesis of a linear contrast?

Answer 9

using a t test (student’s t dist) or f test

Question 10

For the F test, what is it’s df?

Answer 10

it always has 2 df, (x,y)

The numerator df is always 1 in planned comparison

Question 11

What is the relationship between f and t values?

Answer 11

t contrast = square root f contrast

f contrast = MS contrast / MS within

Question 12

What are orthogonal contrast weights?

Answer 12

When the waits of two planned comparisons are independent and NON-REDUNDANT/NO OVERLAP.

This is checked by multiplying their comparison weights up, and adding them up. This should equal zero.

Question 13

What are the advantages of using orthogonal comparisons?

Answer 13

• k-1 orthogonal comparisons can bee defined and constructed for k levels of a factor in ANOVA
• orthogonal comparisons enable decomposition of SSbetween into additive sum of k-1 contrasts sum of squares…

implying that each SScontrast accounts for its own unique part of SSbetween…
and that the set of k-1 orthogonal comparisons fully and independently account of the omnibus result, but in a focussed and immediately interpretable way.

Question 14

So what do we ideally want in a planned comparison?

Answer 14

The weights of the linear contrasts to be orthogonal

The sample size of each diagnostic group to be equal

The sum total of the sum of squares for the two contrasts equaled the between sum of squares (SSBetween).

Question 15

What is a balanced design?

Answer 15

If all levels of a factor in a one-way design have the SAME SAMPLE SIZE, then the design is said to be balanced.

An unbalanced design is one in which sample size differs across levels (i.e. in the cells of the one-way design). Additive decomposition of between-subjects sums of squares requires (i) orthogonal linear contrasts and (ii) a balanced design.

When a design is unbalanced, the between-groups sum of squares will not be equal to the sum of SSContrast.

Question 16

What is an appropriate standardiser for the standardised mean comparison?

Answer 16

s* = squareroot MS within

standardised mean comparison = 
M Contrast (which is contrast weight x Mean) /  s*

FOR THIS TO WORK ALL +VE WEIGHTS MUST SUM TO +1 AND -VE WEIGHTS TO -1

this is analogous to hedge’s g

Question 17

What is the standardiser used to calculate a standardised mean contrast that is analogous to Hedges’ g in a two group design?

Answer 17

The square root of the MSwithin from the omnibus ANOVA Table is typically used as a
standardiser to transform the raw contrast mean different into a standardised metric.

Question 18

What assumption does the standardised mean contrasts based on the pooled within groups variance rest on for it to be robust? Under what conditions will it definitely not be robust to violation of assumptions?

Answer 18

The use of MSwithin in calculating a standardiser for the standardised contrast mean difference rests on the assumption of HOMOGENEITY OF GROUP VARIANCES.

Its use will definitely not be robust when this assumption is violated and the design is unbalanced (i.e., when the k level of the factor do not all have the same group sample size). Even when the design is balanced, there has not been detailed investigation into the robustness of the pooled standardised mean contrast using MSwithin when heterogeneity of group variances is definitely present.

Question 19

What standardiser should we use when there is a violation of homogeneity of variances? (in relation to standardised mean contrast)

Answer 19

Bonnet’s delta

in both BALANCED AND UNBALANCED DESIGNS

It gives slightly larger CI’s than by s*