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.

1. Lecture 9 Survey Research & Design in Psychology James Neill, 2011 Multiple Linear Regression II & Analysis of Variance I

3. Analysis of Variance I

5. Partial correlations

6. Residual analysis

7. Interactions

8. Analysis of change

12. Choose type – Standard, Hierarchical, Stepwise, Forward, Backward

13. Interpret – Overall ( R 2 , Changes in R 2 (if hierarchical)), Coefficients (Standardised & unstandardised), Partial correlations

14. Equation – If useful (e.g., is the study predictive?)

15. Partial correlations ( r p ) r p between X and Y after controlling for (partialling out) the influence of a 3rd variable from both X and Y.

17. Does time management (IV 1 ) predict university student satisfaction (DV) after general life satisfaction is controlled for (IV 2 )?

19. Partial correlations will be equal to or smaller than the 0-order correlations

20. If a partial correlation is the same as the 0-order correlation, then the IV operates independently on the DV.

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.

24. In PASW, the sr s are labelled “part”. You need to square these to get sr 2 .

25. For more info, see Allen and Bennett (2008) p. 182

26. Part & partial correlations in SPSS In Linear Regression - Statistics dialog box, check “Part and partial correlations”

27. Multiple linear regression - Example

28. .18 .32 .46 .52 .34 Y X 1 X 2

29. Residual analysis

31. Normality of residuals

32. Equal variances (Homoscedasticity)

34. Sometimes positive, sometimes negative but, on average, 0

35. Normally distributed about 0

36. Residual analysis

37. Residual analysis

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

39. Residual analysis

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)

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 .

44. Interactions

46. However, there may also be interaction effects - when the magnitude of the effect of one IV on a DV varies as a function of a second IV.

47. Also known as a moderation effect .

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

51. Caffeine

54. b 12 can be interpreted as the amount of change in the slope of the regression of Y on b 1 when b 2 changes by one unit.

59. Pseudo only (incr./decr.)

60. Caffeine only (incr./decr.)

61. Pseudo + Caffeine (additive inc./dec.)

62. Pseudo x Caffeine (synergistic inc./dec.)

63. Pseudo x Caffeine (antagonistic inc./dec.)

65. An alternative approach is to run separate regressions for each level of the interacting variable .

66. Interactions – SPSS example

67. Analysis of Change

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)?

70. IV3 = Support from group leader

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.

73. Regression coefficients for predictors in step 2

76. Student questions ?

79. Independent

80. Paired

82. Ch4 Sampling Distributions and Hypothesis Testing

83. Ch7 Hypothesis Tests Applied to Means Readings – Analysing differences

85. Which difference test?

86. Parametric vs. non-parametric

88. Tests of differences reflect differences in central tendency of variables between groups and measures.

90. In ANOVA we see the world as made of differences. Everywhere we look we see differences .

92. t -test/ANOVA e.g., What is the difference between the heights of human males and females?

93. Are the differences in a sample generalisable to a population?

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).

97. Non-parametric tests are generally used when assumptions about the underlying population are questionable (e.g., non-normality).

99. Non-parametric statistics can be used to analyse DVs that are non-normal or are nominal or ordinal.

100. Non-parametric statistics are less powerful that parametric tests. Parametric vs. non-parametric statistics

102. Small number of observations ( N ).

103. DVs have nominal or ordinal levels of measurement.

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

106. One-sample t -tests

107. Independent sample t -tests

108. Paired sample t -tests

110. These are inferential tools for examining differences between group means.

111. Is the difference between two sample means ‘real’ or due to chance?

113. Independent Compares mean scores on the same variable across different populations (groups)

114. Paired Same participants, with repeated measures

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 .

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%

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.

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)?

126. One-sample t -test

128. Do Americans vs. Non-Americans differ in their approval of Barack Obama?

129. Do males & females differ in the amount of sleep they get?

132. Normality : t -tests robust to modest departures from normality, otherwise consider use of Mann-Whitney U test

133. Independence of observations (one participant’s score is not dependent on any other participant’s score)

134. Do males and females differ in in amount of sleep per night?

135. Do males and females differ in memory recall?

136. Adolescents' Same Sex Relations in Single Sex vs. Co-Ed Schools

137. Adolescents' Opposite Sex Relations in Single Sex vs. Co-Ed Schools

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?)

141. Data is sampled within subjects

142. Pre- vs. post- treatment ratings

143. Different factors e.g., Voter’s approval ratings of candidates X and Y

146. Independence of observations (one participant’s score is not dependent on another’s score)

147. Does an intervention have an effect? There was no significant difference between pretest and posttest scores ( t (19) = 1.78, p = .09).

148. Adolescents' Opposite Sex vs. Same Sex Relations

150. Comparison b/w means of 3+ within subject variables = 1-way ANOVA (e.g., what is the difference in Campus, Social, and Education Satisfaction?)

152. Develop a conceptualisation of when to each of the parametric tests from one-sample t -test through to MANOVA (e.g. decision chart).

154. Independent-samples

155. Paired samples

157. 1-way repeated measures ANOVA

158. Factorial ANOVA

159. Mixed design ANOVA

160. ANCOVA

161. MANOVA

162. Repeated measures MANOVA

164. Francis, G. (2007). Introduction to SPSS for Windows: v. 15.0 and 14.0 with Notes for Studentware (5th ed.). Sydney: Pearson Education.

165. Howell, D. C. (2010). Statistical methods for psychology (7th ed.). Belmont, CA: Wadsworth.

167. Free and open source software.

168. http://www.openoffice.org/product/impress.html