PSYC3010 questions wih appropriate answers

1. Factorial Design - Has at least two factors (IVs), each with at least two levels - Two IVs can be examined simultaneously 1. Advantages of Factorial Design - More economical in terms of participants - Allows us to examine the interaction of independent variables (assess generalisability) 1. 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 2. 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 2. Three Questions of Two-Way ANOVA 1. Variance due to factor A? (df a-1) 2. Variance due to factor B? (df b-1) 3. Variance due to AxB interaction? (df(a-1)(b-1)) 2. Structural Model of 2-way ANOVA Xijk = mew. + aj + Bk + aBjk + eijk - X (Specific DV) e.g. height, age, gender - mew. - the grand mean (e.g. 1.5m) for IV - aj - the effect of the j-th treatment of factor A (e.g. effect of being male or female) - Bk - the effect of the k-th treatment of factor B (e.g. effect of age) - aBjk - effect of differences in factor A treatments at different levels of factor B treatments (interaction between age and gender) - eijk = error for i person in the j-th and k-th treatments (anything left over after main effects are removed which is not error is due to the interaction) 2. Variance and Significance The more variability attributable to the effects, the more significant they are 2. 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) 3. 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) 3. Eta-Squared (n) Describes the proportion of variance in the SAMPLE'S DV scores that is accounted for by the effect - Considered biased 3. 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 3. 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 3. 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) 3. Following-Up Interactions Test of simple effects: - simple effects test the effects of one factor at each level of the other factor 3. 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 3. 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 3. Simple Comparisons Follow up simple effects of interactions, comparing cell means rather than marginal. - somewhat redundant, explaining the same thing more than once - Increases family-wise error rate (use Bonferroni or conduct test a priori to avoid) 4. Higher-Order Factorial Designs - More than two independent factors - Allow for designs with higher external validity 4. Effects in HO Designs Main Effects: - Differences between marginal means of one factor averaging over levels of another (e.g. Driving ability; car size/age/experience/gender) Two-Way Interactions: - The effect of one factor changes depending on the level of another (e.g. age is beneficial but gender is not) Three-Way Interactions: - the two-way interaction between two factors changes depending on the level of a third (when graphed, if lines do not form same pattern, there is interaction) 4. Partitioning the Variance in Three-Way ANOVA (2x2x2) -- 7 Omnibus tests - Main effects • Variance due to a, b, and y - 2-Way interactions • Variance due to ab, by, ay - Error/residual • Variance due to e - 3-way interaction • Variance due to aby Larger partitions represent that the marginal means for that factor are very different from each other 4. Structural Models in Factorial ANOVA 2-way : Xijk = u. + aj + Bk + aBjk + eijk 3-way: Xijkl = u. + aj + Bk + yl + aBjk + Bykl + ayjl + aByjkl + eijkl 4. 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 5. Type 1 and 2 Errors 1. Finding a significant difference in the sample that actually doesn't exist (a) 2. Finding no significant difference in the sample when one does actually exist (B) 5. Power The probability of correctly rejecting the null hypothesis (=1-B) 5. Factors that affect power MESS or SALE - Mean differences (larger differences mean more power) - Error Variance - eta2e or MSerror (less means more power) - Significance Level - Sample Size or Sample Size - increase Alpha level - increase Larger effects - focus on Error variance - decrease 5. 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) 5. 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) 6. Experimental vs Correlational Research Experimental: - Determine causation through manipulation of IVs in controlling setting, assessing effect on DV. - Some factors are impossible or unethical to manipulate (e.g. brain damage) Correlational: - Measures IVs (predictors) and assesses level of association with outcome/DV (criterion) - Uses bivariate regressions (1 predictor) or multiple regression ( 1 predictor) 6. 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 6. Correlation Standardised covariance - Expresses the relationship between two variables in terms of standard deviations - Pearson's r, always -1 to +1 - ZERO ORDER CORRELATION 6. Interpreting Pearson's r in terms of variance Can also be used to determine effect size from correlation - r2 = the coefficient of determination * the proportion of explained variance - 1 - r2 = error or residual variance in data 6. Testing r for Significance Is r large enough to conclude that there is a non-zero correlation in the population? t = systematic variance divided by error variance df = N-2 6. 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. 6. 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 6. 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) 6. 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 6. 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 6. 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 o 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 criterions: o Such that E(Y - Yhat)2 is a minimum o i.e. such that errors of prediction are a minimum * ei = Yi - Yhati = errors of prediction 6. 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 6. Significance of the Regression Slope - b and B, like r, can be tested for significance using a t-test. t = (b)(sx)(Sqrt N-1)/Sy.x - H0 is that b = 0 (no change in Y when X increases 1 unit) 6. Partitioning the Variance (Regression) Where ANOVA = SS between, SS residual, Regression = SS predicted, SS residual o Yi = bXi + c + ei 6. ANCOVA vs Blocking Error term is adjusted statistically rather than at the level of the design - Blocking o Conceptually simpler o Requires fewer assumptions - ANCOVA - Easier to administer - Can use continuous covariate - Removes effect from error term AND DV o Useful in two situations: * Covariate related to IV and DV (confound) * Covariate related to DV only 6. Covariance in ANCOVA - Variance is the tendency for scores to vary around some mean value - Co-variance is the tendency for two scores to vary together o If a participant's score on one variable deviates from the mean, the score on the other (covarying) variable also deviates o Positive covariance = both deviate in the same direction - A covariate is like the control variable used for blocking, with a couple of differences: o The covariate is a continuous variable and treated as such (i.e., participants are not matched at discrete levels) o In ANCOVA, the covariate is used to remove error from both the error term and treatment effect 6. Structural Model of ANCOVA Xij = u + aj + BZij + eij - BZij = score in variable Z multiplied by a fixed weight (beta) because of continuous nature - Score on DV goes up or down depending on score on Z - No specified interaction between covariate and the IV, as the presence of such is a violation of assumptions 6. How ANCOVA reduces error variance - Covariate = another IV or predictor (continuous) - If the covariate is associated with the DV, the relationship accounts for some systematic variance unexplained by the focal IV. This in turn reduces unexplained variance in the design which lends a smaller error term, because partitioning out the covariate's variance means an increase in power. 6. How ANCOVA differs from blocking - Treatment means are adjusted to account for differences on the covariate - Random assignment to IV conditions normally prevent differences in covariate means (confounds should be designed out) - but in case covariate does differ across groups, ANCOVA effectively partials out the effects of the covariate from the IV as well as the error term - If the covariate differs between levels of the IV it won't be clear which variable is explaining differences in DV treatment means = CONFOUND - Effectively don't care about effect of covariate - ANCOVA does this by asking "would the focal IV have an effect on the DV is all participants were equivalent on the covariate?" 6. Statistically Adjusting Means in ANCOVA 1. Calculate overall covariate sample mean and assume to be population mean 2. Assume that in an unconfounded population, all groups of the focal IV have this covariate mean (e.g. if all drivers were the same age, they would all get the same result) 3. If a group's mean is different to the overall covariate mean on the covariate, there is a confound. 4. You can adjust the group's "expected" mean on the DV to be what it would be if the group's covariate mean were the overall covariate mean, by using the regression line 5. Super dodgy lol 6. Logic of ANCOVA - Adjusted treatment means assume that covariate means are the same at each level of the focal IV - Thus any differences in the adjusted treatment means can be attributed to the focal IV only * Refines treatment effect to adjust for any systematic group differences on covariate that existed before experimental treatment 6. Uses of ANCOVA 1. To control unwanted variation that would otherwise inflate the error with which we test our models (classical use) 2. To control for group differences, especially in the analysis of clinical trials or other pre/post designs. - Best use is when you do not expect differences across levels of focal IV in the population on the covariate. If there's a strong association between the focal IV and the covariate in the population, then ANCOVA is deeply problematic because it's testing a hypothetical situation that would never exist 6. Assumptions of ANCOVA - Regular ANOVA assumptions (homogeneity of variance, normal distribution, independence of erros) - Plus: * Relationship between covariate and DV is LINEAR (regression slope) * Relationship between covariate and DV is linear WITHIN EACH GROUP * Relationship between DV and covariate is equal across treatment groups - HOMOGENEITY OF REGRESSION SLOPES (must be relatively parallel) 7. Bivariate Regression vs Multiple Regression - Correlation and bivariate regression = single predictor - Multiple regression: *Variation as a function of multiple predictors usually acting simultaneously * Multiple correlation (R): relation between criterion Y and a set of predictors * Multiple regression: scores on criterion Y are predicted using 1 predictor 7. Tests in Multiple Regression 1. Strength of overall relationship between criterion and set of predictors: R2 (F test) 2. Importance of individual predictors: b, B, sr (t test) 7. MR with Uncorrelated Predictors - Rare - Predictor importance indicated by ry1 squared and ry2 squared respectively - Can unambiguously identify proportion of variance accounted for by each predictor - R2 = ry1sq + ry2sq 7. MR with Correlated Predictors - Common - Predictors share overlapping variance with each other, as well as DV - Therefore R2 is ry1sq + ry2sq - Importance of individual predictors myst be calculated in an extra step 7. The Partial Correlation - Examines the relationship between predictor 1 and the criterion, with the variance shares with predictor 2 partitioned out of both DV and IV - pr2 = the proportion of residual variance in the criterion uniquely accounted for by predictor 1 [A / (A + B)] - A + B = residual variance in the DV that is available to be explained by IV1 after controlling for IV2 (eliminating C and D) 7. The Semi-Partial Correlation - Examines the relationship between predictor 1 and the criterion, after removing the variance shared between 1 and 2, from predictor 1 - spr2 = the proportion of total variance in the criterion uniquely accounted for by predictor 1 [A / (A + B + C + D)] - all variance in DV 7. The Linear MR Model - 2 Predictors - Criterion scores are predicted using the best linear combination of the predictors (like line-of-best-fit, but plane instead) * b1 is the slope of the plane relative to the X1 axis, b2 for X2 * a is the point where the plane intersects the Y acis (when X1 and X2 are equal to zero) - Too messy for 3+ predictors 7. Regression Models - 1 predictor: model Yhat with a line described by 2 parameters (bX + a) - 2 predictors: model Yhat as a plane described by 3 parameters (b1X1 + b2X2 = a) - P predictors: model Yhat as a p-dimensional depiction with p + 1 parameters (for each IV) - 1 slope per IV + constant 7. Standardised Regression Coefficients - Rough estimate of relative contribution of predictors (importance), can be compared - Cannot necessarily be compared across groups (SD may change) - When IVs are not correlated, B = r - When IVs are correlated, Bs (magnitudes, signs) are affected by pattern of correlations among predictors 8. Uses for HMR - CMM - To account for control (nuisance) variables - To test mediation - To test moderated relationships (interactions) 8. Assumptions of HMR Residuals: Never Lick Hairy Inbreds Scale: Never Condone Lazy Sex - Distribution of residuals o Conditional Y values are NORMALLY distributed around the regression line o No LINEAR relationship between Yhat and errors of prediction o HOMOSCEDASTICITY: variance of Y values are constant across different values of Yhat (homogeneity of variance) o INDEPENDENCE of errors - Scales (predictor and criterion scores) o Variables are NORMALLY distributed o Measured using a CONTINUOUS scale (interval or ratio) o LINEAR relationship between predictors and criterion o Predictors are not SINGULAR (extremely highly correlated) 8. Multicollinearity and Singularity - Occurs when predictors are higher correlated ( .80 to .90) - Diagnosed with high intercorrelations of IVs and a statistic called TOLERANCE - Tolerance - (1 - R2) - Where R2x is the overlap between a particular predictor and all the other predictors - Low tolerance = multicollinearity/singularity - High tolerance = relatively independent predictors - Multicollinearity leads to unstable calculation of regression coefficients (b), even though R2 may be significant 8. Moderation Analyses In a regressional interaction, the second predictor is usually called a moderator. The moderator enhances or attentuates the relationship between criterion and predictor 8. Linear Model for Moderation - Additive effects: Yhat = b1X + b2Z + a - Non-additive effects: Yhat = b1X + b2Z + b3XZ + a 8. Mean Centering - Reduces multicollinearity - Easier to interpret coefficients in presence of interaction - b1 represents the relationship between X and U at the mean of Z (i.e. 0), making direct effects more meaningful. Usually b1 would be different at different values of Z, so using Z = 0 doesn't make sense. 9. Mediation Analyses - Mediation is a third variable, or mediator, that may explain or account for the relationship between an IV and a DV 9. Correlations for Mediation in MR MUST HAVE STRONG THEORY 1. IV should predict mediator 2. IV should predict DV in Block 1 3. Mediator should predict DV in Block 2 4. Coefficient for IV should decrease to non-significant (full mediation) or in size but still significant (partial mediation) 5. Sobel test or bootstrapping analyses should be significant 10. AxP Design The extent to which the effect of Factor A changes across participants - no interaction means the factor has the same effect on every participant (effect change = error) - To determine treatment effects, must use Tau j = Muj - Mu dot (i.e., the effect of being in the jth condition is equal to the condition mean minus the overall mean) - In this type of design, error is the inconsistencies in treatment effect for each individual (e.g. Ps cell deviation score from marginal mean - marginal mean deviation from grand mean)2 10. Within-Participants ANOVA Designs - Total variance = between participants + within-participants - F test = Treatment/Treatment*Participant interaction 10. Structural Model of WP ANOVA Xij = u +Pii + Tj + eij u = grand mean Pi.i = variation due to the i-th person (ui - u) Tj = variation due to the j-th treatment (uj - u) eij = error - variation associated with the i-th cases in the j-th treatment - error = piTij (plus error) 10. Partitioning of Variance in WP ANOVA 1. Error (TRxP) 2. Treatment 3. Participants - Treatment F should be greater than residual TRxP 10. Follow-up Tests in WP ANOVA Error terms need to be separated when following up main effects - Inconsistency is expected in TR effect x Participants, so in simple comparisons, use only data for conditions involved in the comparison and calculate SEPARATE ERROR TERMS each time 10. Two-Way WP Designs - Test for main effects of A and B, plus AxB interaction - But there is a separate error term for each effect, rather than the same as in BP designs. - This error term simply corresponds to an interaction between the effect due to participants, and the treatment effect: * Main effect of A - error term is MS(AxP) * Main effect of B - error term is MS (BxP) 10. Following Up Interactions - Simple effects error term is MS(A at B1xP) - The interaction between factor A and participants at B1 - Simple comparisons error term is MS(Acomp at B1xP) - interaction between factor A (only the data contributing to the comparison, Acomp) and participants at B1 10. Mixed Model Approach to WP Designs Treatment is a fixed factor (You chose the levels of the IV, by either sampling all or selecting them based on theory), participants is a random factor (The levels of the IV are chosen at random, different error terms) - Powerful when assumptions are met - Mathematically user-friendly 10. Assumptions of the Mixed Model Approach 1. Sample is randomly drawn from population 2. DV scores are normally distributed in the population 3. Compound symmetry - Homogeneity of variances in levels of RM factor (variances are roughly equal) - Homogeneity of covariances (equal correlations/covariances between pairs of levels) 10. Mauchley's Test of Sphericity - Broader and less restrictive assumption than compound symmetry - Assumes we have ROUGHLY the same variances and covariances, allows for some differences to emerge - Evaluated as chi-square - if sig, sphericity is violated - NOT ROBUST: very commonly fail even with violations 10. When Sphericity Doesn't Matter - In BP designs, because treatments are unrelated (different Ps) - When WP factors have two levels, because only one estimate of covariance can be computed 10. When Sphericity Does Matter - In WP designs with 3+ levels - When the sphericity assumption is violated, F-ratios are positively biased - leads to increase Type I error. Best to assume a problem will come up and proactively adjust df 10. Adjustments to Degrees of Freedom Epsilon Adjustments - Simply a value by which the df for the test of F-ratio is multiplied - Equal to 1 when sphericity assumption is met and 1 when assumption is violated - The lower the epsilon value (further from 1), the more conservative the test becomes 10. Different Types of Epsilon Adjustments Lower-Bound: - Act as if only two treatment levels with maximal heterogeneity - Used for conditions of maximal heterogeneity, or worst-case violation of sphericity - often too conservative Greenhouse-Geisser: - Size of epsilon depends on degree to which sphericity is violated - 1 = E = 1/(k-1): varies between 1 (sphericity intact) and lower-bound epsilon (worst-case violation) Huynh-Feldt: - An adjustment applied to the GG-epsilon - Often results in E exceeding 1, in which case it is set to 1 - Used when 'true value' of epsilon us believed to be greater than or equal to .75. - Increases power, can be too liberal otherwise 10. Multivariate Approach to WP Designs MANOVA - Creates linear composite of multiple DVs - In MANOVA approach to RM designs, our RM variable is treated as muyltiple DVs and combined/weighted to maximise the difference between levels of other variables • i.e. weight levels based on the differences they produce • Multivariate tests: Pillai's Trace, Hotelling's Trace, WILK'S LAMBDA, Roy's Largest Root • Does not require restrictive assumptions that mixed model within-participants design does * More complex underlying maths - NOT USED IN REPEATED MEASURES 10. Advantages and Disadvantages of Multivariate Approach + More efficient and simple (N Ps in j treatments generate nj data points) + More sensitive (estimate individual differences (SSparticipants) and remove from error term - Restrictive statistical assumptions - Sequencing effects (e.g. direct carry over - learning something in previous condition that alters latter) 11. Mixed (Split-Plot) ANOVA - Has BP and WP factor - Combines the best features of both (power and independency of observations - no carry-over) 11. Assumptions of Mixed ANOVA - DV is normally distributed - BP Terms: homogeneity of variance within levels of BP factor - WP Terms: * Homogeneity of variance: assume WPFxP interactions constant at all levels of BP factor * Variance-covariance matrix exhibits compound symmetry (sphericity) * Usual epsilon adjustments apply when WP assumptions are violated. 11. In a two by three mixed ANOVA in which gender (male; female) serves as a between-participants variable and time of test (start of semester, mid-semester, end of semester) serves as a repeated measures variable, participant is crossed with ______ and nested within _______ . Within-participants factor block and between-participants factor group 11. Partitioning the Mixed ANOVA Variance - Between Ps factor (treatment and error term) - Within Ps factor (main effect of block, and interaction of block and group; error term, block * participants within groups) 11. Omnibus tests in a Mixed ANOVA Three Omnibus Tests: 1. Main effect of group (BP) 2. Main effect of block (WP) 3. Group x Block Interaction (WP) 11. Understanding the Mixed Design BP - For BP variability, take vertical marginal means, subtract grand mean and square for the sum of squares - Variability within the group is considered to be error (so participants' simple means (vertical) differing from the marginal mean create error) WP - Variation in marginal means (horizontal) creates variability - Error is observed when there are inconsistencies in results for each participant in each block (e.g. in Block 1, participant 1 differs the marginal mean for each group, etc.) Interaction - The impact of one factor changes based on levels of moderator - Error term is the interaction of the P factor with the WP factor after taking into account group differences 11. Following Up Main Effects in Mixed Design BP Factor - Rule is the same as it would be if this were just a 1-way BP ANOVA - Use original error term from test of BP main effect - MS(Ps within groups) in this case WP Factor - Use a separate error term - MS(Bcomp*P within group) 11. Simple Effects in Mixed Design BP: 1. Use a separate error term for each simple effect - Run four 1-way BP ANOVAs to compare groups at each of the four blocks, then use MSPs within G at B1; MSPs within G at B2; etc. ** Recommended 2. Use a special pooled error term: MSPs within cell - This error term is an estimate of the average error variance within the 12 cells * In both instances, the sums of squares for the simple effects are derived just as we have seen in the case of between participants ANOVA * The separate error term method is a little quicker, but you compromise on df Add or remove terms You can also click the terms or definitions to blur or reveal them Re