PSYC3010 all

• Factorial Design


  • Has at least two factors (IVs), each with at least two levels
  • Two IVs can be examined simultaneously
• Advantages of Factorial Design

  • More economical in terms of participants
  • Allows us to examine the interaction of independent variables (assess generalisability)
• Interactions in Factorial Designs


  • One IV interacts with another when the effects of one are different depending on the level of the other
  • And when it changes (moderates or qualifies) the impact of a second IV on the DV
• Variance


  • Dispersion or spread of scores around a point of central tendency, e.g. mean
  • Error Variance: cannot be explained; should go up with more observations
  • Treatment Variance: systematic differences due to our IV
• Three Questions of Two-Way ANOVA


  • Variance due to factor A? (df a-1)
  • Variance due to factor B? (df b-1)
  • Variance due to AxB interaction? (df(a-1)(b-1))
• Structural Model of 2-way ANOVA
• Variance and Significance


  The more variability attributable to the effects, the more significant they are
• Assumptions of ANOVA


  • Population: normally distributed (normality) and have the same variance (homogeneity of variance)
  • Samples: Independent; obtained by random sampling; at least two observations and equal n
  • Data (DV Scores): measured on continuous scale for mathematical operations (mean, SD, variance)
• Effect Sizes


  • Been proposed as an accompaniment, if not replacement, for significance testing, as it relays implications of findings (ANOVA is binary)
  • Offers another way of assessing reliability of results in terms of variance
  • Can compare size of effects within a factorial design: Cohen's d (0.2, 0.5, 0.8)
• Eta-Squared (n)


  • Describes the proportion of variance in the SAMPLE'S DV scores that is accounted for by the effect
  • Considered biased
• Omega Squared (w)


  • Describes the proportion of variance in the POPULATION'S DV scores that is accounted for by the effect
  • Less biased
  • Larger difference between n and w with smaller sample
• Partial Eta-Squared


  • Proportion of residual variance accounted for by the effect (variance left over to be explained)
  • Usually more inflated
  • Can add up to >100%
  • Hard to make meaningful comparisons
• Following-Up Main Effects


  Use linear contrasts (protected t test) to determine if a set of groups is different from another set using weights (aj)
• Following-Up Interactions


  Test of simple effects: simple effects test the effects of one factor at each level of the other factor
• Variance Partitioning of Omnibus Tests

  Variance partitioned into four parts: Effect due to first factor, Effect due to second factor, Effect due to interaction, Error/Residual/Within-group variance
• Partitioning of Simple Effects

  • Simple effects re-partition the main effect and interaction variance
  • The simple effects of factor 2 should be equal to the combination of the main effect and the interaction
• Simple Comparisons


  1. Follow up simple effects of interactions, comparing cell means rather than marginal

  2. Somewhat redundant, explaining the same thing more than once

  3. Increases family-wise error rate (use Bonferroni or conduct test a priori to avoid)
• Higher-Order Factorial Designs


  • More than two independent factors
  • Allow for designs with higher external validity
• Effects in HO Designs

  • Main Effects: Differences between marginal means of one factor averaging over levels of another
  • Two-Way Interactions: The effect of one factor changes depending on the level of another
  • Three-Way Interactions: The two-way interaction between two factors changes depending on the level of a third
• Partitioning the Variance in Three-Way ANOVA (2x2x2)

  • 7 Omnibus tests: Main effects, 2-Way interactions, Error/residual, 3-way interaction
  • Larger partitions represent that the marginal means for that factor are very different from each other
• Structural Models in Factorial ANOVA
• Following-Up Significant Omnibus Effects in a 3-Way ANOVA

  Interaction followed by simple effects, if significant and more than 2 levels, simple simple effects (effect of factor A at each level of factor B, at each level of factor C ), and simple simple comparisons if still significant
• Type 1 and 2 Errors

  • Type 1: Finding a significant difference in the sample that actually doesn't exist (a)
  • Type 2: Finding no significant difference in the sample when one does actually exist (B)
• Power


  The probability of correctly rejecting the null hypothesis (=1-B)
• Factors that affect power

  • MESS or SALE: Mean differences, Error Variance, Significance Level, Sample Size
• Reducing Power


  • Improve operationalisation of variables (increases validity)
  • Improve measurement of variables (increases internal validity)
  • Improve design of your study (account for variance from other sources, e.g. blocking designs)
  • Improve methods of analysis (control for variance from other sources)
• Blocking Designs


  • Introducing a variable into the design to reflect additional sources of variation or pre-existing differences on DV score
  • Variable is control or concomitant, associated with the DV, but the relationship is neither novel nor interesting
  • Generally match participants to blocking variable through stratified random assignment
  • Shouldn't interact with focal IV (often confound if it does)
• Experimental vs Correlational Research

  • Experimental: Determine causation through manipulation of IVs in controlling setting, assessing effect on DV
  • Correlational: Measures IVs (predictors) and assesses level of association with outcome/DV (criterion)
• Covariance


  • Average cross-product of the deviation scores
  • Positive value = positive relationship
  • When one variable is above/below the mean, the other is likely to be the same
  • Limitations: Covariance is scale-dependent
• Correlation


  • Standardised covariance
  • Expresses the relationship between two variables in terms of standard deviations
  • Pearson's r, always -1 to +1
  • ZERO ORDER CORRELATION
• Interpreting Pearson's r in terms of variance

  • r2 = the coefficient of determination, the proportion of explained variance
  • 1 - r2 = error or residual variance in data
• Testing r for Significance

  1. Is r large enough to conclude that there is a non-zero correlation in the population?

  2. t = systematic variance divided by error variance

  3. df = N-2
• r as a population estimate = radj

  • r is a sample statistic and is biased to the sample (like eta-squared)
  • Can calculate rho (p), the population correlation coefficient - rho is estimated by the "adjusted r" (like omega squared)
  • radj is always smaller than r (more conservative)
  • The difference between r and radj becomes smaller as sample size increases
• Correlation and Predictions


  • = REGRESSION, estimating a score on one variable (Y, criterion) on the basis of scores on another variable (X, predictor)
  • Correlational designs infer causality based on theory
  • Objective is to find the best fitting line
• Bivariate Regression Equation


  • Yhat = bX + a
  • Yhat = predicted value of Y (DV)
  • b = slope of regression line (change in Y with 1-unit change in X)
  • X = value of predictor (IV)
  • a = intercept (value of Y when X = 0)
• Standardising the Regression Slope

  • b would become a standardised regression coefficient (beta, B)
  • B = Z-score change in Y predicted from a 1 SD increase in X
• Error in Regression


  • If a score is different from the average, it is error (lenient)
  • If a score is different from the predicted value (regression line) it is error (conservative)
  • Average deviation of scores from the regression line
  • Called STANDARD ERROR OF THE ESTIMATE
• Standard Error of the Estimate

  • Sy.x reflects the amount of variability around the regression slope (Yhat, variable conditional on X)
  • Can be calculated as SSerror over df, or SD of DV times sqroot of 1 - error variance in data
  • If r2 is zero, there is no error variance, and there is no association between the IV and the DV (standard error estimate is equal to SD of DV)
  • But should be much smaller than SD of DV
  • The regression line is fitted according to the least squares criteria: Such that E(Y - Yhat)2 is a minimum, i.e. such that errors of prediction are a minimum
• What SEE tells us

  • Bigger rxy leads to smaller Sy.x
  • A high correlation between X and Y reduces the SEE and enhances the accuracy of the prediction
  • R2 is overly liberal with small samples, and so Sy.x is underestimated for small samples
• Significance of the Regression Slope

  1. b and B, like r, can be tested for significance using a t-test

  2. t = (b)(sx)(Sqrt N-1)/Sy.x

  3. H0 is that b = 0 (no change in Y when X increases 1 unit)