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RMII - Stats - Coggle Diagram
RMII - Stats
Calculations
- SST (basic) => scores - grand mean
- SSM (best) => group means - grand mean x n (group size)
- -> dfM = k - 1 (k = no. groups)
- -> MSM = SSM / dfM
- SSR (residual) => scores - group means
- -> dfR = N - k = k(n - 1)
- -> MSR = SSR/dfR
- SST = SSM + SSR
- F = MSM/MSR
- If F > 1 + significant
Assumptions
- Homogeneity of variance within groups = Levene's
** -> Welch procedure
- Normality of variables
** Moderate skew: OK
** Kurtosis: affects Type I errors
** Unequal group sizes: bigger problem if not normal
** -> Transformation or Kruskal-Wallis non-parametric test
- Independence of variables between & within groups
** Type I errors
** -> Clustering, multilevel modelling
Logic
- F = var. between groups / var. within groups
- F = MSM / MSR
Effect size
- Diff between groups in SD units
- Eta-squared = SSM/SST
- Omega squared (less biased)
- Interpret as squared correl. coefficient
- .02 = small, .13 = medium, .26 = large
- Interpret in relation to previous research
- x% variance in DV attrib. to diff in IV
Power
- Type I = alpha = .05 = no effect, but found one
- Type II = beta / b = effect, but didn't find it
- Power = 1 - b
- If a larger -> b smaller -> higher power -> low Type II -> easier to find sig (even if small effect)
- To increase power
- Increase alpha
- Increase subjects per condition
- Control extraneous variables
- Large effect size
A Priori / Post Hoc
- Multiple comparisons: tell us which groups means differ sig.
- A priori: chosen before, few comparisons -> reduces Type I error rate, groups of groups, match hypotheses
- Post hoc: after data collection, all pairwise comparisons, pairwise only
- Pairwise: 2 groups at a time
Error rates
- Error rate = likelihood of Type I error
- Per comparison (PC): consider error rates separately for each contrast (.05)
- Familywise (FW): consider error rates jointly for series of contrast
* Approx*: c(α)
* Actual*: 1 – (1 – α')^c (overestimates)
A priori / planned contrasts
- Compare 1 condition with set of conditions
- Use coefficients to weight group means (balance)
- Linear contrast: psi = a1X1 + a2X2... (X = group mean)
- Sum of weights/coefficients = 0
- Fractional weights -> meaningful mean diff.
- SScontrast = n x psi^2 / SUM(a^2) (n = sample size / group)
- MScontrast = SScontrast (df = 1)
- F = MScontrast/MSerror (MSerror from ANOVA table)
Orthogonal contrasts
- Independent - no overlapping info
- SS of linear comparisons = SStreat/model
Conditions
- Coefficients for any contrast sum to 0
- Products of all pairs of coefficients sum to 0
- No. comparisons = df for treatment
- -> never compare from one branch to another
FW error rate
- Reduce FW error rate by reducing a (e.g. .01 instead of .05)
- FW = .05
- Bonferroni t/Dunn: PC = FW/c - conservative!
- Modified Bonferroni
- If c =< df, don't adjust PC
- If c > df, use df x FW / c = df x .05 / c
Factorial ANOVA
- Multiple IVs: pair every factor at level A with level B, etc.
- Factors = IVs (e.g. gender)
- Levels = categories within factors (e.g. male/female)
- Two-way -> 2 factors
- 3 x 2 factorial -> 3 levels, 2 levels
Two-way ANOVA
- 2 IVs
- Main effects: effect of 1 IV, ignoring levels of other IV
- Interaction effects: effect of 1 factor modifies effect of other
- Simple effects: analyse part - same as one-way, but use error term from overall ANOVA
- Same as main effect if no interaction
- Can cross factors
- If interaction, interpret main effects with caution
Calculations
- SST => scores - grand mean
- SSR => scores - cell means
- SSA => mean for individual level of each factor A (A1, A2) - grand mean x nb (group size x no. levels other factor)
- SSM => cell means - grand mean x n
- SSAB => SSM - SSA - SSB
- SST = SSA + SSB + SSAB + SSR
- -> dfT = dfA + dfB + dfAB + dfR
- MS = SS / df
- F = MS / MSR
Simple effects
- SSB at A1: A1 cell means - A1 mean x n
- SSB at A2: A2 cell means - A2 mean x n
- SSB at A1 + SSB at A2 = SSB + SSAB
- MS = SS / df
- F = MSB at A1 / MSR
Considerations
- If interaction and/or RQ -> simple effects
- Plot data first
- A priori simple effects (to reduce Type I errors)
- If no a priori, choose set which explains clearly
Advantages
- Greater generalisability
- Can see interactions
- Economy (less participants)
Three-way ANOVA
- 3 IVs
- 3 x main effects
- 3 x 2-way interaction effects
- 1 x 3-way interaction effect
- Simple interaction effects
- Higher order = 3+ IVs
Flowchart
- If 3-way interaction (2nd order)
- -> Simple interaction effects: if sig
- ---> Simple effects
- If NO 3-way interaction
- -> 2-way interactions (1st order): if sig
- ---> Simple effects
- -> if NO 2-way / simple interactions
- ---> Main effects
Repeated-Measures ANOVA
- Every participants takes part in every condition
- -> participants treated as another IV
- -> Less contaminated by individual diff.
Advantages
- Removes individ. diff from error term
- -> Smaller error
- -> More powerful -
detects smaller treatment effects
- More efficient (less participants)
Calculations
- SST => scores - grand mean
- SSM => treatment means - grand mean x n
- SSW => scores - individual mean
- SSR => SSW - SSM
- MS = SS / df
- F = MSM / MSR
- k = groups/levels, n = participants/group
Sphericity assumption
- Equality of variances of the differences between pairs of treatment levels
- Compare variance across all pairs of time points
- Only for 3+ time points
- Problem for repeated-measures - same participants in each treatment -> scores correlated
- Not problem for independent ANOVA - diff. participants -> scores have covariance = 0
- If violated -> F-ratio not valid -> more Type I errors
- If Mauchley's W p < .05
- -> Check epsilon (0 - 1 no violation)
- -> If epsilon < .75 => Greenhouse-Geisser correction
- -> If epsilon > .75 => Huynh-Feldt correction
- => Multiply both dfs by epsilon -> harder to get sig F -> reduces Type I error
Mixed designs
- At least 1 within-subjects variable + 1 between-subjects variable
- Error terms
- Between-subjects: standard error term
- Repeated measures & interaction: interactive error term (participant x treatment)
Error term
- Participant x treatment interaction
- SStotal = SSB + SSW
- SSW = SSM + SSR
- -> Error = SStotal - SSB - SSM
- -> Error = SSW - SSM
Regression = prediction
- Categorical or continuous
Bivariate Regression
Coefficient of determination R2
- Proportion of variance in DV predicted from IV
Residuals
- Diff between predicted & actual
- Residual variance (var) & SE of estimate (SD)
- Ordinary least squares: minimises sum of squared residuals
Regression line / line of best fit
y = bX + a
- b: slope, increase in DV, for 1 unit increase in IV, unstandardised regression coefficient, strength & direction
- a: constant/intercept, DV when IV = 0
- Centre of relationship
- Significance: t-test
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- IV predicts DV
- Stronger correlation -> more reliable prediction
Multiple Regression
Sequential Multiple
- Variable entry controlled by researcher (based on logic / theory)
- Predictve importance of IV: unique contribution
* at point of entry - R2 change
* in final model - sr2
Calculations
Adjusted R2
- Adjusts for no. IVs
- Adjusts for sample size
- As more IVs / small sample -> higher R2
- 1 – [(1 - R2)x(N – 1)]/(N – k – 1)
R2 = SSM / SST = squared multiple correlation
- How well equation fits data
- Variance predicted by equation
- Significance: F test
- 0.30+ is meaningful
sr2 = squared semi-partial correlation
- Unique variance IV accounts for in DV, after controlling for assoc. with other IVs
- Reduction in R2 if IV removed from equation
Variance
- SST = SSM + SSR
- SST (total) => actual scores - mean
- SSM (model) => predicted scores - mean
- SSR (residual) => actual - predicted scores
R2 change
- How much R2 is improved by adding IVs
R
- Strength of relationship (correlation) between actual & predicted scores of DV
- Strength of relationship (correlation) between DV & IV (because predicted scores use IVs)
sr = semi-partial correlation
- Unique correlation b/n IV & DV after controlling for correlation between
- IV & other IVs
- DV & other IVs
Regression coefficients
- b weight = unstandardised coefficient
- => Predict scores in original scale
Beta weight = standardised regression coefficients
- Converted to z-scores
- Equation has no a (intercept), because mean = 0
- Influenced by correlations between IVs + effect of IV on DV
- => Relative importance of IV, after controlling for assoc. with other IVs
Standard Multiple
- Variables entered simultaneously
- Predictive importance of IV: unique association with DV
Significance tests
- R2: F ratio = MSM/MSR; MSM = SSM/dfM, MSR = SSR/dfR
** dfM = k, dfR = N - k - 1
** k = no. IVs, N = no. cases
- sr2 - unique predictveness: t-test
- R2 change: F change-test
- b / beta weight: t-test
R = multiple correlation
Strength of relationship between
- DV & predicted DV scores
- DV & combined IVs
Considerations
-
Practical
Sample size
- N > 50 + 8m: R (overall model - combined IVs)
- N > 104 + m: (individual IVs)
- m = no. IVs
- For medium effect size, alpha .05, beta .20
Multicollinearity / singularity
- Check intercorrelation matrix -> delete 1 / combine variables
- Multicollinearity: correlation of .90+
- Singularity: correlation of 1
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GLM Assumptions
Normality
- Skewness: +ve (tail to R), -ve
- Kurtosis: +ve/lepto/high, -ve/platy/low
- Screening
- Kolmogorov-Smirnov: stat test
- z-test: skew/kurtosis / SE should be < 3 (higher for bigger samples)
- Histograms
Linearity
- Curvilinear/part-curvilinear (U-, J-shaped)
- Screening: scatterplots
- => Transform data -> transform to dichot.
- => Use diff analysis
Homogeneity of variance in arrays (homoscedasticity)
- Variance of Y scores similar for all values of X
=> Transform data
Transformations
- Leaves underlying relationship in tact
- Transform raw data
- Square root -> log -> inverse/reciprocal (more intense)
- Larger numbers reduced more than small numbers
- Use for skewed data (reflect -ve skew values first)
- If doesn't work: run with original & flag OR use non-parametric analysis
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