Explains advanced use of multiple linear regression, including residuals, interactions and analysis of change, then introduces the principles of ANOVA starting with explanation of t-tests.
20. If a partial correlation is the same as the 0-order correlation, then the IV operates independently on the DV.
21.
22. An IV may have a sig. 0-order correlation with the DV, but a non-sig. partial correlation. This would indicate that there is non-sig. unique variance explained by the IV.
23.
24. In PASW, the sr s are labelled “part”. You need to square these to get sr 2 .
38. Residual analysis “ The Normal P-P (Probability) Plot of Regression Standardized Residuals can be used to assess the assumption of normally distributed residuals. If the points cluster reasonably tightly along the diagonal line (as they do here), the residuals are normally distributed. Substantial deviations from the diagonal may be cause for concern.” Allen & Bennett, 2008, p. 183
40. Residual analysis “ The Scatterplot of standardised residuals against standardised predicted values can be used to assess the assumptions of normality, linearity and homoscedasticity of residuals. The absence of any clear patterns in the spread of points indicates that these assumptions are met.” Allen & Bennett, 2008, p. 183
41. Why the big fuss about residuals? assumption violation Type I error rate (i.e., more false positives)
42.
43. If the residuals don’t meet assumptions these formulae tend to underestimate coefficient standard errors giving overly optimistic p -values and too narrow CIs .
48. Interactions Some drugs interact with each other to reduce or enhance other's effects e.g., Pseudoephedrine Arousal Caffeine Arousal Pseudoeph. X Caffeine Arousal X
49. Interactions Physical exercise in natural environments may provide multiplicative benefits in reducing stress e.g., Natural environment Stress Physical exercise Stress Natural env. X Phys. ex. Stress
68. Analysis of change Example research question: In group-based mental health interventions, does the quality of social support from group members (IV1) and group leaders (IV2) explain changes in participants’ mental health between the beginning and end of the intervention (DV)?
71. Analysis of change Strategy : Use hierarchical MLR to “partial out” pre-intervention individual differences from the DV, leaving only the variances of the changes in the DV b/w pre- and post-intervention for analysis in Step 2.
94. Which difference test? (2 groups) How many groups? (i.e. categories of IV) More than 2 groups = ANOVA models 2 groups: Are the groups independent or dependent? Independent groups Dependent groups 1 group = one-sample t -test Para DV = Independent samples t-test Para DV = Paired samples t -test Non-para DV = Mann-Whitney U Non-para DV = Wilcoxon
95. Parametric vs. non-parametric statistics Parametric statistics – inferential test that assumes certain characteristics are true of an underlying population, especially the shape of its distribution. Non-parametric statistics – inferential test that makes few or no assumptions about the population from which observations were drawn (distribution-free tests).
96.
97. Non-parametric tests are generally used when assumptions about the underlying population are questionable (e.g., non-normality).
104. Some Commonly Used Parametric & Nonparametric Tests Compares groups classified by two different factors Friedman; 2 test of independence 2-way ANOVA Compares three or more groups Kruskal-Wallis 1-way ANOVA Compares two related samples Wilcoxon matched pairs signed-rank t test (paired) Compares two independent samples Mann-Whitney U; Wilcoxon rank-sum t test (independent) Purpose Non-parametric Parametric Some commonly used parametric & non-parametric tests
116. Homogeneity of variance In general, t -tests and ANOVAs are robust to violation of assumptions, particularly with large cell sizes, but don't be complacent .
117.
118. Is the t large enough that it is unlikely that the two samples have come from the same population?
119. Decision: Is t larger than the critical value for t ? (see t tables – depends on critical and N )
120. Ye good ol’ normal distribution 68% 95% 99.7%
121.
122. One-tailed test rejects null hypothesis if obtained t -value is extreme is one direction (you choose – too high or too low)
123. One-tailed tests are twice as powerful as two-tailed, but they are only focused on identifying differences in one direction.
124.
125. Do uni students sleep less than the recommended amount? e.g., Given a sample of N = 190 uni students who sleep M = 7.5 hrs/day ( SD = 1.5), does this differ significantly from 8 hours hrs/day ( = .05)?
139. Comparison b/w means of 3+ independent sample variables = 1-way ANOVA (e.g., what is the difference in Educational Satisfaction between students enrolled in four different faculties?)
7126/6667 Survey Research & Design in Psychology Semester 1, 2009, University of Canberra, ACT, Australia James T. Neill Home page: http://ucspace.canberra.edu.au/display/7126 Lecture page: http://en.wikiversity.org/wiki/Survey_research_methods_and_design_in_psychology http://ucspace.canberra.edu.au/pages/viewpage.action?pageId=48398957 Image sources http://commons.wikimedia.org/wiki/File:Vidrarias_de_Laboratorio.jpg License: Public domain http://commons.wikimedia.org/wiki/File:3D_Bar_Graph_Meeting.jpg ' License: CC-by-A 2.0 Author: lumaxart ( http://www.flickr.com/photos/lumaxart/ ) Description: Explains advanced use of multiple linear regression, including residuals, interactions and analysis of change, then introduces the principles of ANOVA starting with explanation of t-tests. This lecture is accompanied by two computer-lab based tutorial, the notes for which are available here: http://ucspace.canberra.edu.au/display/7126/Tutorial+-+Multiple+linear+regression http://ucspace.canberra.edu.au/display/7126/Tutorial+-+ANOVA
Image source: http://commons.wikimedia.org/wiki/File:Information_icon4.svg License: Public domain
Image source: http://commons.wikimedia.org/wiki/File:Information_icon4.svg License: Public domain
These residual slides are based on Francis (2007) – MLR (Section 5.1.4) – Practical Issues & Assumptions, pp. 126-127 and Allen and Bennett (2008) Note that according to Francis, residual analysis can test: Additivity (i.e., no interactions b/w Ivs) (but this has been left out for the sake of simplicity)
Image source: http://www.gseis.ucla.edu/courses/ed230bc1/notes1/con1.html If IVs are correlated then you should also examine the difference between the zero-order and partial correlations.
Image source: http://www.gseis.ucla.edu/courses/ed230bc1/notes1/con1.html If IVs are correlated then you should also examine the difference between the zero-order and partial correlations.
Image source: Unknown Residuals are the distance between the predicted and actual scores. Standardised residuals (subtract mean, divide by standard deviation).
These residual slides are based on Francis (2007) – MLR (Section 5.1.4) – Practical Issues & Assumptions, pp. 126-127 and Allen and Bennett (2008) Note that according to Francis, residual analysis can test: Additivity (i.e., no interactions b/w Ivs) (but this has been left out for the sake of simplicity)
Image source: James Neill, Creative Commons Attribution-Share Alike 2.5 Australia, http://creativecommons.org/licenses/by-sa/2.5/au/ Histogram of the residuals – they should be approximately normally distributed. This plot is very slightly positively skewed – we are only concerned about gross violations.
Image source: James Neill, Creative Commons Attribution-Share Alike 2.5 Australia, http://creativecommons.org/licenses/by-sa/2.5/au/ Normal probability plot – of regression standardised residuals. Shows the same information as previous slide – reasonable normality of residuals.
Image source: James Neill, Creative Commons Attribution-Share Alike 2.5 Australia, http://creativecommons.org/licenses/by-sa/2.5/au/ Histogram compared to normal probability plot. Variations from normality can be seen on both plots.
Image source: James Neill, Creative Commons Attribution-Share Alike 2.5 Australia, http://creativecommons.org/licenses/by-sa/2.5/au/ A plot of Predicted values (ZPRED) by Residuals (ZRESID). This should show a broad, horizontal band of points (it does). Any ‘fanning out’ of the residuals indicates a violation of the homoscedasticity assumption, and any pattern in the plot indicates a violation of linearity.
Interactions occur potentially in situations involving univariate analysis of variance and covariance (ANOVA and ANCOVA), multivariate analysis of variance and covariance (MANOVA and MANCOVA), multiple linear regression (MLR), logistic regression, path analysis, and covariance structure modeling. ANOVA and ANCOVA models are special cases of MLR in which one or more predictors are nominal or ordinal "factors.“ Interaction effects are sometimes called moderator effects because the interacting third variable which changes the relation between two original variables is a moderator variable which moderates the original relationship. For instance, the relation between income and conservatism may be moderated depending on the level of education.
Example hypotheses: Income has a direct positive influence on Conservatism Education has a direct negative influence on Conservatism Income combined with Education may have a very -ve effect on Conservatism above and beyond that predicted by the direct effects – i.e., there is an interaction b/w Income and Education
Likewise, power terms (e.g., x squared) can be added as independent variables to explore curvilinear effects.
For example, conduct a separate regression for males and females. Advanced notes: It may be desirable to use centered variables (where one has subtracted the mean from each datum) -- a transformation which often reduces multicollinearity. Note also that there are alternatives to the crossproduct approach to analysing interactions: http://www2.chass.ncsu.edu/garson/PA765/regress.htm#interact
Image source: http://commons.wikimedia.org/wiki/File:PMinspirert.jpg Image license: CC-by-SA, http://creativecommons.org/licenses/by-sa/3.0/ and GFDL, http://commons.wikimedia.org/wiki/Commons:GNU_Free_Documentation_License Image author: Bjørn som tegner , http://commons.wikimedia.org/wiki/User:Bj%C3%B8rn_som_tegner
Step 1 MH1 should be a highly significant predictor. The left over variance represents the change in MH b/w Time 1 and 2 (plus error). Step 2 If IV2 and IV3 are significant predictors, then they help to predict changes in MH.
Step 1 MH1 should be a highly significant predictor. The left over variance represents the change in MH b/w Time 1 and 2 (plus error). Step 2 If IV2 and IV3 are significant predictors, then they help to predict changes in MH.
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Correlation/regression techniques reflect the strength of association between continuous variables Tests of group differences ( t -tests, ANOVA) indicate whether significant differences exist between group means
In MLR we see world is made of covariation. In ANOVA we see the world as made of differences. In MLR, everywhere we look, we see patterns and relationships. In ANOVA we view everything as having the same, more or less than other things.
Image soruce: Unknown
Image source: James Neill, Creative Commons Attribution-Share Alike 2.5 Australia, http://creativecommons.org/licenses/by-sa/2.5/au/ “ A t-test is used to determine whether a set or sets of scores are from the same population.” - Coakes & Steed (1999), p.61
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“ A t-test is used to determine whether a set or sets of scores are from the same population.” - Coakes & Steed (1999), p.61
Image soruce: Unknown
Image source: Unknown
Also called: One sample t -test
Adapted from slide 23 of Howell Ch12 Powerpoint: IV is ordinal / categorical e.g., gender DV is interval / ratio e.g., self-esteem Homogeneity of Variance If variances unequal (Levene’s test), adjustment made Normality t-tests robust to modest departures from normality (often violated without consequences) look at histograms, skewness, & kurtosis; consider use of Mann-Whitney U test if departure from normality is severe, particularly if sample size is small (e.g., < 50) Independence of observations (one participant’s score is not dependent on any other participant’s score)
Also called related samples t-test or repeated measures t-test
Also known as: Related samples t -test and Repeated measures t -test How much do you like the colour red? How much do you like the colour blue? Is there a difference between people’s liking of red and blue?