New Exam - Comparisons Flashcards
When making inferences between the differences between group means (as is with ANOVA etc), what is used to make the comparison?
error variance
What is the pooled error term?
The pooled error term is the error term based on all the cells of the between-subjects design. It is the error term used to test the main and interaction effects in the overall (omnibus) ANOVA, which showed a significant interaction or A pooled error term is usually the error term from an omnibus ANOVA. Imagine you had a 1-way layout with 3 or more cells. The error term from this anova (MSerror) would be used to test any pairwise comparisons.
What in contrast does the t-test use?
The t-test on specific pairs of cells just uses the data from the cells being compared, and so the error term is based just on those 2 cells involved in the comparison or The pairwise t-tests of specific cells, by contrast, uses only the data from the cells being compared to generate its error term.
what is the advantage of using pooled error terms?
1) The main advantages of using a pooled error term are that it uses data from more participants and so generates a larger error df than the t-test method (roughly twice as big for a 2x2 design). This of course reduces the critical value of the statistic and raises power for the contrast cf the t-test. 2) In addition, given that the pooled error term is based on more data it will generally be a more reliable estimate for the error and so may generate a smaller error term. Use of the pooled error term is recommended in designs like this subject to certain assumptions (see next).
What are the key assumptions for using pooled error terms ?
From youtube video: 1) Independent samples 2) Normally distributed populations 3) Equal population variances From exam paper: (3 from above) (i) The homogeneity of variance assumption. This is so because the pooled error term, as the name implies, pools error variance across cells of the design and so… these variances must be homogeneous in order for the pooling to be justified and to lead to an unbiased measure of error variance. Note that the homogeneity test was included in part 1 of the printout which showed that there was homogeneity across all 4 cells.
If the homogeneity of variance assumption is violated, which unpooled procedure can we use?
Welch unpooled t proceedure - which does not assume equal variance
Outline the reasons why the planned contrasts will generally be more powerful than the planned t-tests
Planned contrasts using the pooled error term are generally more powerful because the pooled error term is based on more data (ie from the whole study not just 2 cells). This increases the df (approx doubles df in the current example). Furthermore, the larger amount of data is likely to lead to a more reliable estimate of error in the error term and so should make it easier to detect real effects. This increase in power based on the pooled error term critically depends on whether the pooled sources of error are homogeneous and so the homogeneity assumption is specifically important for this issue (the homogeneity output results were included in the printout as a cue to the students’ memories on this point).
The psychologist used the following syntax commands to compute the contrasts. Explain how the LMATRIX subcommands are constructed, bearing in mind that the extraversion group variable (extgp) is coded 1=extravert, 2=introvert. The noise condition variable is coded 1=low, 2=high. (10 marks)
- the LMATRIX command specifies the contrast. The simple main effect (SME) of noise at each level of extraversion requires that the noise effect be specified first by the variable (noise) followed by contrast coefficients and then the interaction term within which the SME lies (in this case extgp*noise) followed by appropriate coefficients.
- Coefficients of contrasts need to sum to zero
- noise has two levels and so we can use 1 -1 to see if the mean for noise level 1 (low) minus noise level 2 (high) is different from zero
- Given the coding of the factors extgp and noise the 4 cells of this interaction are laid out in SPSS as follows for the effect extgp*noise:
Thus the coefficients given 0 0 1 -1 contrast the low and high noise groups among introverted subjects and 1 -1 0 0 contrast the noise groups among extraverts.
The contrast for the simple main effect of noise in extraverts yielded t[25]=2.75, p=0.011 (two-tailed); the contrast for the simple main effect of noise in introverts yielded t[25]=-0.27, p>0.5 (two-tailed).
- What did the researcher conclude based on these contrast findings? Explain your reasoning carefully. (5 marks)
- (5 marks) Should conclude that performance under high noise and under low noise differs significantly in extraverts but not introverts. Answer should stress that the p-value (0.011) is judged against a Bonferroni-corrected per comparison alpha rate of 0.05/2 (since two contrasts were used). From inspecting the means, the performance of extraverts was significantly faster (lower RT) under high noise cf low noise.
How should the researcher interpret the findings from study 2? Explain your arguments carefully.
None of the simple main effects in study 2 are significant. They should be judged against a Bonferroni-corrected alpha level per comparison of 0.05/4 (for 4 contrasts).
- this sometimes happens when there is a significant interaction, none of the components of that interaction (ie the SMEs) are individually significant
- might note that the contrast t-values are opposite in sign within each pair of SME contrasts (ie SMEs are in opposite direction, albeit not significantly so)
- thus the interaction tells us, for example, that the effect of reward vs punishment conditions for extraverts is significantly different from the reward vs. punishment effect for introverts, although neither effect is significantly different from zero
Describe what is meant by a set of orthogonal contrasts. Illustrate your answer with reference to a one-way design with 3 levels of the factor. Describe one set of contrasts for the factor which are orthogonal, and one set which are not.
A set of orthogonal contrasts is one in which each pair of contrasts within the set is independent of one another, thus the effect measured by one contrast has no effect on the effect measured by any other contrast. The coefficients specifying each contrast should be uncorrelated with the coefficients specifying every other contrast.
For the specific example which is supposed to be used:-
A factor with 3 levels – A, B and C
contrast 1 – is A vs. average of B and C
contrast 2 – is B vs C
1 and 2 are an orthogonal pair of contrasts
contrast 3 – A vs C
contrast 4 – B vs C
contrasts 3 and 4, though informative (eg if C is a control condition) are not orthogonal
To demonstrate this we can explore the coefficients used and their correlation
A B C
contrast 1 2 -1 -1
contrast 2 0 1 -1
cross multiply the coefficients and add up the result
(2 x 0) + (-1 x 1) + (-1 x -1) =0 hence orthogonal. Do the same for contrasts 3 and 4 and does not add up to zero.
What is the rule for how many contrasts can be conducted?
If a group main effects has N levels (N groups) then there are N-1 df for the GROUP factor, which can be decomposed into N-1 single df contrasts.
In a design one has a between-subjects factor called group consisting of 4 different groups of participants (groups A, B, C and D), with one of the groups (D) being a control group.
Into how many contrasts can one decompose the group main effect? Write a set of contrast coefficients to compare each of the three groups individually to the control group. (5 marks)
Is this an orthogonal set of contrasts? Explain how you arrived at your answer
How to check the orthogonality of a set of contrasts?
Sum the cross-products of the coefficients for every pair of contrasts; if sums=0, this shows the contrasts are
uncorrelated
orthogonal = uncorrelated
non-orthogonal=correlated
No it is not an orthogonal contrast ( 2 marks)
(3 marks) To test a set of contrast for orthogonality, the cross products of the coefficients of each pair contrasts are computed and these cross products are summed. The result should be zero for each pair of contrasts.
Contrast 1 * contrast 2 = 0 + 0 + 0 + 1
Ditto 2 * 3 and 1 * 3.
This guarantees that the vectors formed by each set of contrast coefficients is a right angles to every other vector (ie the coeffs are mutually uncorrelated)
What is a simple main effect? When are analyses of such effects particularly useful? (5 marks).
- (3 marks)
A simple main effect (SME) is an effect of one factor at a particular level of another factor in a factorial design. The four cells of a 2 x 2 design (factors A and B) might be Ab AB ab aB, where upper and lower case letter denote different levels of the two factors. A simple main effect of factor A would compare Ab with ab, (or AB with aB) whereas a SME of B would compare aB with ab (or AB with Ab).
- (2 marks)
SMEs are particularly useful in trying to establish the nature/location of a more complex effect in a factorial design, in particular interactions between factors.
